P R O G R E S S I N LOW TEMPERATURE PHYSICS 111
CONTEh-TS OF VOLU-ME I
c.
J . GORTER,
The two fluid model for superconductors and helium I1 (16 pages)
I<.
P.
J.
R. PELLAM, Rayleigh disks in liquid helium I1 (10 pages)
A.
c. HOLLIS
FEYNMAN,
Application of quantum mechanics to liquid helium (37 pages)
HALLETT,
Oscillating disks and rotating cylinders in liquid helium I1
(14 pages) E. F. HAMMEL,
The low temperature properties of helium three (30 pages)
J . J . M . BEENAKKER
and
K.
w. TACONIS, Liquid mixtures of helium three and four
(30 pages) B . SERIN,
c.
The magnetic threshold curve of superconductors (13 pages)
F. SQUIRE,
T. E .
The effect of pressure and of stress on superconductivity (8 pages)
and A. u. PIPPAKD, Kinetics of the phase transition in superconductors (25 pages)
FAHER
K. MENDELSSOHN, J . G . DAUNT, A . H. COOKE,
N . J . POULIS
Heat conduction in superconductors (18 pages)
The electronic specific heats in metals (22 pages) Paramagnetic crystals in use for low temperature research (21 pages) and c.
D. DE KLERK
and
J . GORTER,
M.
Antiferromagnetic crystals (28 pages)
J . STEENLAND,
Adiabatic demagnetization (63 pages)
L. NEEL,
Theoretical remarks on ferromagnetism at low temperatures (8 pages)
L. WEIL,
Experimental research on ferroinagnetism at very low temperatures
(11 pages) A. VAN ITTERBEEK,
Velocity and attenuation of sound at low temperatures
(26 pages) J . DE BOER,
Transport properties of gaseous helium at low temperatures (26 pages)
CONTENTS O F VOLUME I1
Quantum effects and exchange effects on the thermodynamic properties of liquid helium (55 pages)
J . DE BOER,
H.
c. KIIAMERS, Liquid helium below 1°K (24 pages) and D. H . N. W ANSINK, Transport phenomena of liquid helium I1 in slits and capillaries (22 pages)
P. WINKEL
K . R. B.
ATKINS, Helium films (33 pages)
T. MATTHIAS, Superconductivity in the periodic system (13 pages)
E. H. SONDHEIMER,
v.
A. JOHNSON
Electron transport phenomena in metals (36 pages)
and
K.
LARK-HOROVITZ, Semiconductors a t low temperatures
(39 pages)
D. SHOENBERG, The De Haas-van Alphen effect (40 pages)
c.
J . GORTER,
Paramagnetic relaxation (26 pages)
and H. A. atures (46 pages)
M. J . STEENLAND
c . DOMB and
TOLHOEK,
J. s. DUGDALE,
Orientation of atomic nuclei a t low temper-
Solid helium (30 pages)
F. H. SPEDDING, s. LEGVOLD, A. H. DAANE and L. D. JENNINGS, Some physical
properties of the rare earth metals (27 pages) The representation of specific heat and thermal expansion data of simple solids (36 pages)
D. BIJL,
H . VAN D I J K
and
(34 pages)
M. D U R I E U X ,
The temperature scale in the liquid helium region
F R O M TIIE S E R I E S I N PIIYSICS Geueral Editors: J.
Professor of Physics, 1:niversity of Amsterdam l'rofessor of Physics, University of Groningcn H. G. CASIBIIK, Director of the Philips' 12aboratories, Eindhovcn
nE
BOER,
xi. BRINKMAN, H.
Monogruph c : n.
13.4~.Elcinrntarp
Introduction t o Molecular Spectra c. ~ R I P I ' I C MApplication .~N, of Spinor Invariants in Atomic Physics H. G . V.\N IIUEIIEN, Imperfections in Crystals s. R. DE t i ~ o o r Thermodynamics . of Irrevcrsi1)lc Processes 1:. A . GucC;ENHm,v, Thermodynamics E. A . G U G C E N H E I M , Uoltzmann's D i s t r i h t i o n IAW I and J . E. m u E , Physicocheniical Calculations Cory of Brillouin Zones and Electronic States in Crystals H. A . K R A h I R R S , Qnantum Xlechanics 11. A . KR.WI:RS,Tlic Foundations of Quantuni Tlicory J . G. L I N I I A R T , Plasma l'hysics J . nIccoNmxLL, Quantum Particle 1)ynamics A . M E R C I E R , Analytical and Canonical Formalism in Physics I. PRIGOGINE, The Molecular Theory of Solutions E , G. R I C H A R D S O N , Relaxation Spectrometry P. R O M A N , Theory of Elementary Particles M . E. ROSE, Internal Conversion Coefficients j. L. S Y N C B , Relativity: The Special Theory J . L. S Y X G E , Relativity; The i;eneral Theory J . L . SYNGI:, The Rclativistic (;as H . U M H Z A \ V A , Quantuiii 1;icld 'l'hcory A. V A S T ~ K KOptics , of 'Thin Films A . 11. W A P S T R A , G. J . N I J G H and R. V.\N I.IRSHOUT,Nuclear Spectroscopy Tables 11.
Edited Volunzrs
r. M. E x m and ni. U E ~ I E U K(editors), Nuclear Reactions, Vol. I c. J . GORTICR (editor), Progress in I.ow Tcmpcrature Physics, Volumes 1-111 G . I.. DK HIZAS-I.ORENTZ (editor), H. A . Lorentz, Impressions of his Liie and Work 11. J . L t r I u N (editor), Proceedings of the Iictiovotli Confcrencc on Nuclcar Structure N . R . N I I S S O N (rditor), Proccedings of the Fourth International Conference on Ionization Pticnomcna in Gases (LJppsaln, 1959) K . SIEGB.%HN (etlitor), Beta- and Gamma-Kay Spectroscopy Syinposium on Solid State Diffusion (Colloque sur la diffusion 8.1'8tat solide, Saclay 1058) 3c ~011oqueIdc XIdtaIIurgic, sin la Corrosion (Saclay, 1969) Turning l'oints in Physics. A Series of Lectures given at Oxford University in Trinity Term 1958 J . G . WILSON and s. A . WOUTHUYSEN (editors), Progress in Elementary Particle and Cosmic R a y Physics, Volumes I-V RALTII. V A N DER POL, Selected Scientific Papers r. EHRRNFEST, Collcctcd Papers
P R O G R E S S I N LOW
T E M P E R A T U R E PHYSICS E D I T E D BY
C. J. G O R T E R Professor of Experimental Physics Director of the Kainerlingh Onnes Laboratory, Leiden
V O L U M E 111
1961 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
No purt of this book m a y be reproduced i n a n y f o r m by p r i i z t , microfilm or a n y other m e a n s without written permission f r o m the publisher
PUBLISHERS: NORTH-HOLLAND PURLISI-IING CO.,
-
AMSTERDAM
SO1.E DISTHIRIJTORS 1;OR U.S A , : INTERSCIENCE PUBLISHERS INC,. - NEW YORK
PRINTED I N T H E NETHERLANDS
PREFACE TO THIRD VOLUME
When the second volume of “Progress in low temperature physics” had been completed I did not intend to edit more books of the same character. I had found out that I prefer pondering and reading about questions that fascinate me above editing books about them, and that I have already too many duties of an administrational or managerial character. Moreover I was - in spite of the good reception which, in general, the two volunics found - not quite convinced of their indispensability. But a few near colleagues, and also the publishers, insisted on continuing the series. And so I decided to put the question in personal conversations before a number of prominent low temperature physicists whom I mct in 1959 at a few international conferences. In order not to let amiability determine their advice, I questioned the benefit of Progressbooks in general and asked about their experience on the usefulness of such books. Somewhat to my disappointment the result of the inquiry was remarkably positive. The only objections I encountered were about the considerable time required for writing a review paper, which time might have been spent more profitably by doing research or complementing one’s general education. Curiously enough, the few physicists that adhered to that view all belonged, apart from myself, to the same nationality. And two of them were unhappily looking forward to writing a review paper. In the positive reactions it was stressed that students who have to work themselves into a field and research workers in other areas who got interested in such a field derive great profit from a collection of review papers. The existence of an intermediate stage between the tangle of the original publications and the lucid and concise order of the advanced textbook was generally acclaimed and so was the system collecting reviews of neighbouring fields in one volume. After all this encouragement I could hardly do anything else than surrender, and I must say that some of those whom I could blame for that helped me most generously by writing contributions to the present
volume which I not only appreciate from an editor’s point of view, but which I also enjoyed reading as fascinating accounts of recent advances in fields with which I was losing touch. The set-up of the book is not different from that of the first volumes. Some review papers cover a rather large field and are therefore somewhat longer than most articles in the earlier volumes. I feel this has ccrtain advantages. Again the emphasis lies on liquid helium, superconductivity and some problems in the wide field of magnetism. Again a few other subjects have been put upon the stage while certain topics are badly missed. If the necessary co-operation is obtained, we might try to fill up some of the gaps in a fourth volume.
C. J. GORTER
CONTENTS
Chapter Page 1 lv. F. VINEK, \-OKTICX L I NE S I N L I Q U I D HELIUM 1 1 . . . . . . 1 1. Introduction, 1. - 2. Theoretical background, 2. - 3. Vortex lines in uniformly rotating helium, 10. - 3.1. Theoretical equilibrium state, 10. - 3.2. Mutual friction in the uniformly rotating liquid, 12. - 3.3. Vortex waves in the uniformly rotating liquid, 18. - 4. The detection of single quanta of circulation, 25. - 5 . The energy of a free vortex line, 31. - 6. The hydrodynamic stability of ideal superfluid flow, 36. - 7. Superfluid turbulence, 43. - 8. The va!idity of the concept of a simple vortex line, 54.
.
11 G. CAREEII, HELIUM I O N S I N LIQUID HELIUM I1 . . . . . . . . 58 1. Introduction, 88. - 2. Ionic structures in liquid helium, 59. 2.1. Possible structures, 59. - 2.2. Experimental behaviour of ionic structures, 60. - 2.3. ‘The adopted structures, 61. - 3. The ionic motion in liquid helium 11, 62. - 3.1. The theoretical viewpoint, 62. - 3.2. Experiments in non-turbulent flow, 63. - 3.3. Experiments in turbulent flow, 66. - 4.The ionic mobilities, 69. - 4.1. Experimental results, 69. - 4.2. General discussion, 72. 5. Ionization and recombination processes, 75. - 6. Concluding remarks, 7 8 . I11 M. J . BUCKINGHAM and TV. &I. FAIRBANK, THE NATURE THE 1 - T R A N S I T I O N I N LIQUI11 HELIURl
.
.
. . . . . . . .
OF
. . .
80
1. Introduction, 80. - 2. Thc specific heat near the 2-point, 82. 2.1. Description of tlie experiment, 82. - 2.2. Method of measurement, 84. - 2.4. Results, 84. - 2.4. Relation to data far from the transition, 87. - 3. Thcrmodynamics of 2-transitions, 89. - 3.1. A-transitions, 89. - 3.2. Properties at the transitions, 89. - 3.3. Properties near the transition, 90. - 3.4. Properties of the new variables, 92. - 3.8. Connection with other thermodynamic relations, 93. - 4. Analysis of other measurements, 94. - 4.1. Properties of the A-line, 94. - 4.2.Thernial expansion coefficient, 95. - 4.3. Velocity of sound, 99. - 4.4.0ther measured quantities, 100 - 5. The nature of the transition, 102. - 6. Separation of the singularity, 104. - 7. Analysis of the specific heat of liquid helium, 107. - 8. The superconducting transition, 109.
Iv
E. R. GRILLYandE. F. HAMMEL, LIQUIDANDSOLID W E . . 113 1.. Theories of liquid 3He, 113. - 1.1. Landau’s theory of liquid 3He, 113. - 1.2. The Brueckner and Gammel theory of liquid 3He,
X
CONTENTS 115. - 1.3. Goldstein’s theory of liquid 3He, 116. - 1.4 Pair correlation theories of liquid 3He, 116. - 2. Theories of solid 3He, 118. - 2.1. Qualitative, 118. - 2.2. Theory of Bernardes and Primakoff 118. - 3. Pressure-volume-tempcrature relations, 119. - 3.1. At vapor pressures, 119. - 3.2. At intermediate pressures, 122. - 3.3. At melting pressures, 123. - 4. Thermal properties, 129. 4.1. Specific heat, 129. - 4.2. Entropy, 132. - 5. Transport properties of liquid and solid 3He, 134. - 5.1. Thermal conductivity and viscosity of liquid 3He, 134. - 5.2. Heat transport in solid 3He, 136. - 5.3. Self-diffusion coefficient for liquid 3He, 136. - 5.4. Self-diffusion coefficient for solid 3He, 138.-6. Nuclear spin relaxation in condensed 3He, 138. - 6.1. Liquid 3He, 138. - 6.2. Solid SHe, 142. - 7. Velocity of sound in 3He, 143. - 7.1. Experimental results, 143. - 7.2. Attenuation, 145. - 7.3. Zero sound, 146. 7.4. Sound propagation in liquid 3He below the “phase transition”, 147. - 8. Summary, 147.
-
V
VI
.
..
.
.
H . . .~ . . . , ~ . . 153 1. Introduction, 153.-2. First cryostat by Roberts and Sydoriak, 155. - 3. Calorimeter by Seidel and Keesom, 156. - 4. Calorimeter by Taconis and De Bruyn Ouboter, 158. - 5. Refrigerator with glass Dewar by Zinov’eva and Peshkov, 160. - 6. Refrigerator by Keich and Garwin, 163. - 7. Metal Refrigerator by Peshkov, Zinov’eva and Filimonov, 164. - 8. Large metal cryostat by Laquer, Sydoriak and Roberts, 165. - 9. Cryostat construction by Taconis and Le Pair, 166.
I(. W. TACONIS, S
J. RARDEEN and J . R. SCHRIEFFER, RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
.
. .
. . . .
,
. .
.
. . .. .. .
,
1. Introduction, 170. - 2. Historical Survey, 175. - 3. Outline of the London and Pippard Theories, 178. - 4. London’s quantum picture of the superconducting state, 181. - 5. Elementary excitations in normal metals, 184. - 5.1. Quasi-particle excitations, 184. - 5.2. Screening and backflow, 186. - 5.3. Interactions between elementary excitations, 189. - 6. Electron-phonon interactions, 191. - 7. Elementary excitations in superconductors, 193. - 8. Nature of the wave functions for superconductors, 195. - 8.1. Reasons for pair configurations, 195. - 8.2. Ground state wave function, 198. - 8.3. Excitation spectrum, 200. - 8.4. Relation to Einstein-Bose condensation, 200. - 9. Results for simplified model, 201. -9.1. Integral equation, 201. - 9.2. Coulomb interactions and life-time effects, 203. 9.3. Excitation spectrum, 204. - 10. Thermodynamic properties, 207. - 11. Transition probabilities and coherence effects, 212. - 11.1.Theory, 212. - 11.2. Acoustic attenuation, 218. - 11.3. Nuclear spin relaxation, 222. - 12. Electromagnetic properties, 224. - 12.1. Theory, 224. - 12.2. Infrared transmission through thin films, 236. 13.3. Penetration depths, 240. - 12.4. Surface impedance, 246. - 13. Collective excitations, 252. - 14. Two-fluid model and persistent currents, 263. - 14.1. Two-fluid model, 263. - 14.2. Critical currents in thin films, 267. - 14.3. Ginzburg-Landau theory of boundary energies, 268. - 15.
-
-
170
~
XI
CONTENTS
Thermal conductivity, 270. - 15.1. Lattice component, 270. 15.2. Electronic component, 272. - 16. Superconducting alloys and compounds, 275. - 17. Conclusions, 280. VII
M. Y A . AZBEL’ and I. M. LIFSHITZ, ELECTRON RESONANCES METALS
. . . . . . . . . . . . . . . . . . . . . . . . . .
IN
288
Introduction, 288. - 1. Cyclotron resonance, 289. - 1.1.Physical picture of cyclotron resonance, 289. - 1.2. Theory of cyclotron resonance, 294. - 1.3. Analysis of the surface impedance, 302. 1.4. Further development of the theory of cyclotron resonance, 308. - 2. A special type of damping (non-skin) in a metal, 312. 2.1. Physical basis for non-skin damping, 312. - 2.2. New resonance effects related to field “splashes”, 317. - 3. Paramagnetic resonance, 319. - 3.1. Physical picture of paramagnetic resonance, 319. - 3.2. Theory and experimental observation of paramagnetic resonance, 321. VIII
W. J . HUISKAMP and H. A. TOLHOEK, ORIENTATION OF M I C NUCLEI AT L O W TEMPERATURES I1
ATO-
. . . . . . . . . . . . . 333
Introduction, 333. - 1. Theoretical results on nuclear effects with oriented nuclei; general theory; alpha and gamma radiation, 333. - 2. Theoretical results concerning beta radiation emitted from oriented nuclei, 335. - 3. Experimental results on alpha particle emission, 346. - 4. Beta asymmetry experiments, 352. - 5. Experimental results on gamma radiation, 365. - 6. Methods of nuclear orientation, 373. - 7. Nuclear orientation in ferromagnetic and antiferromagnetic substances, 374. - 8. Dynamic methods of nuclear orientation, 380. - 9. Concluding remarks, 390.
IX N. BLOEMBERGEN, SOLID STATE MASERS . . . . . . . . . . 396 1. Introduction, 396. - 2. Paramagnetic resonance in maser materials, 400. - 3. Paramagnetic relaxation, 405. - 4. Maser circuits, 414. - 5. Noise, 420. - 6. Millimeter and infrared solid state masers, 424.
X
J. J . M. BEENAKKER, THEEQUATION
OF STATE AND THE TRANS-
PORTPROPERTIESOFTHEHYDROGENICMOLECULES
. . . . . . . 430
1. Introduction, 430. - 2. The second virial coefficient, 431. - 2.1. Introduction, 431. - 2.2. Absolute determinations of PV isotherms 432. - 2.3. Relative determinations, 433. - 2.4. Direct excess
determinations, 435. - 3. Thermal conductivity, 436. - 3.1. Experimental data, 436. - 4. Viscosity, 437. - 4.1.Experiments in the liquid hydrogen temperature range, 437. - 4.2. Experiments between 20 and 80”K, 438. 4.3. Experiments on binary mixtures, 439.-5. Thediffusioncoefficient, 439.-6. Thermal diffusion, 440. - 6.1. Introduction, 440. - 6.2. Experimental results, 441. 7. The influence of the total nuclear spin on the properties of H, and D,, 442. - 7.1. Introduction, 442. - 7. 2. The difference in the second virial coefficient, 443. - 7.3. The difference in the
-
CONTENTS
XI1
viscosity, 444. - 8. Theoretical calculations, 445. - 8.1. Introduction, 445. - 8.2. Comparison with experimental data, 448.- 8.3. General conclusions, 452.
XI
Z. norcoum,, S
O ~ ~ SOLID-GAS E EQUILIDIUA AT LOW TEMPERATURES
464
1. Introduction, 464. - 2 . Some properties of the phase equilibrium in a binary system, 456. - 3. Experimental methods for the determination of equilibrium curves, 459. - 4. Survey of Leiden results, 463. - 4.1. H,-N,-mixture and 13,-CO-mixture, 464. 4.2. He-N,-mixture, 466. - 4.3. H,-CH,-mixture, 468. - 4.4. H,0,-mixture, 471. - 4.5. Some remarks on S + L equilibrium, 475. - 5. Theoretical determination of equilibrium curves, 477.
. . . . . . . . . . . . . . . 481
ERRATA OF THE
VOLUMES I A N D I I
AUTHOR INDEX
. . . . . . . . . . . . . . . . . . . . . . .
483
SUBJECT INDEX
. . . . . . . . . . . . . . . . . . . . . . .
493
CHAPTER I
VORTEX LINES IN LIQUID HELIUM I1 BY
?V. F. VINEN THEROYAL SOCIETY Rl OXD
~ A R O K A T O R Y , CAMBRIDGE, E N G L A K D
CONTENTS: 1. Introduction, 1. - 2. Throretical background, 2. - 3. Vortex lines in uniformly rotating helium, 10. - 4.The detection of single quanta of circulation, 25. - 5. The energy of a free vortex line, 31. - 6. The hydrodynamic stability of ideal superfluid flow, 36. - 7. Superfluid turbulence, 43. - 8. The validity of the concept of a simplc xrortcx line, 54.
1. Introduction
The first theory to account satisfactorily for many of the curious hydrodynamic properties of liquid helium I1 was that due to Landau1; it explained in particular the success of the two fluid model and the fact that the superfluid component can undergo frictionless flow without interaction with either the normal fluid or any solid boundaries. However, the theory suffered from two serious defects: first, it predicted that the superfluid flow should always be irrotational, whereas this was observed not to be true (see $ 3) ; and secondly, it predicted values for the critical velocity above which superfluid flow ceases t o be frictionless that were much too large (see $ 6). The first defect was particularly serious, since the irrotation condition was fundamental t o the whole theory so that its apparent violation in practice seemed to undermine the whole theory. These difficulties were removed as soon as it was realised that, although the irrotation condition must hold throughout most of the superfluid, it is perfectly possible to have highly localised regions of vorticity in the superfluid, in the form of what are essentially singularities in the velocity field; i.e. in the form of vortex sheets and vortex lines. Furthermore, it was realised that circulation in the superfluid should be quantized in units of him, so that the strength of a vortex line, for example, should be quantized. The idea of vortex sheets was REfcr.crlcr~sp . .i6
.)
LCII. I , $ 2 suggested independently by Onsager2, H. London3, Nott I , and hy Landau and Lifshitz5, and developed to Some extent by F. London(; and by Ginsburg’. The view that superfluid circulation is quanti7c.d and the idea of vortex lines were both discussed first by Onsagerz, y, although they have been developed considerably by Feynmm9. Of these ideas that of the quantized vortex line seems at present to be the most important, although, as will be explained in S 6, it is possible that the vortex sheet is sometimes important in the initial breakdown of ideal superfluid fjow at high velocities. I t seems possible, as will be shown in the present chapter, that with the introduction of these ideas all the essential elements for a complete theory of the hydrodynamics of helium I1 are known. Many problenis remain to be solved, but there is, as yet, no clear evidence that any radically new ideas will be involvcd. The aim of this chapter is to summarise our present knowledge of the properties of vortex lines and vortex sheets in the superfluid and of the part that these forms of vorticity play in the hydrodynamics of helium 11. We shall begin with a straightforward outline of the basic theory, but later we shall be concerned mainly with the interpretation and explanation of specific experimental results, in the course of which we shall present the experimental evidence that the basic theory is correct. IV. 1:. V I X E N
2. Theoretical Background
In summarising the basic theory, we shall follow closely the approach used by I;eynmang, to which reference should be made for details. In this approach a physical picture is first formed of the state of the liquid at rest at absolute zero, i.e. a picture of the ground state wave function, and then the possible low lying excited states, corresponding either to thermal excitations or to a flow of the liquid, are obtained by making all the various small modifications that are possible to this ground state. Let @(Rl, R,, . . . ., R,) be the ground state wave function for the AV atoms in the liquid. This function is always positive, and it must be symmetrical with respect to interchange of the particle coordinates. I t has its maximum value when the coordinates R irepresent a more or less evenly spaced (but randomly arranged) array of atoms, and it will be small when this spacing is not even. I t is convenient to consider first the low lying states that could R c j l ~ r ~ l l cpl ~. s56
CII. I,
s1'
1-ORTEX L I N E S I N LIQCID HELIUM I1
3
be thermally excited at low temperatures. The detailed argument by which these states may be found, which is based directly on the fact that helium atoms are identical particles obeying Bose statistics, is not relevant to the present discussion, and therefore only the results will be quoted. The wave function for all these states is found to be roughly of the form
and the corrcsponding envrgy to he given by
P(k)
= ?12f<2'~112S(k),
(4
\There S ( k ) IS thv form factor for the diffraction of waves (X-rays or neutrons) Ly the liquid, and iii is the mass of a helium atom. This function ~ ( kis)the n.ell-known cnergy-momentum relation first postulatcd by Landau1; the excitations with low values of k are the (longitudinal) phonons, and those with larger values of k (of the order of the reciprocal of thc i n x n t o m i c syncing) are the rotons. The wave function (1) is good for the phonons, but rather poor for the rotons; better wtve functions for the rotons have been obtainedlO, and they suggest that the roton m:~yhave the form of a microscopic vortex ring. The rotons ha\-e a finite niinirriuin energy (equal to about 9" K), and this is consistent with the important general principle that rotational motions in the liquid cannot be excited with arbitrarily low energy. As was emphasisd strongly both in the early work of Landau and in the later rrork of Feynnian, this principle is of the greatest iniportance, for it is in effect responsible for all the characteristic properties of helium 11. The principle is perhaps most easily visualised in terms of the quantization of angular momentum : for a liquid, unlike a solid, may be imagined very roughly to be composed of a large number of sniall quasi-independent units, each containing only two or three atoms, and the lowest allowed rotational state of each such unit corresponds of courbe to a considerable energy. (This argument needs some elaboration if the importance of the Bose statistics is to be brought out.) When the temperature is low the concentration of phonons and rotons is small, and they form in effect a gas. If this gas of excitations is given any drift velocity, the resulting momentum corresponds to a densitv that is less than the total density of the liquid, and this smaller density is identified with the normal fluid density Q , . The gas of excitaRcfeveiicL
7
p 56
4
1%'. 1:. V I X E N
[CH. I ,
52
tions therefore forms the normal fluid, and what remains is the s u p fluid. The next step is to consider wave functions that describe a flow of the whole liquid. This is most easily done for the liquid at absolute zero, and it may be assumed that the results will apply to the superfluid component a t a finite temperature. One wave function that appears to be satisfactory is y = [exp i C 7 s(R,)] @,
(3)
where s ( R ) is any function that varies slowly with R. For it can be shown easily that it corresponds to a mean excess energy density of ~ Q V , "and a mean momentum density of pvS, where v, = (A/.z)grad s,
(4)
and e is the density of the fluid; furthermore, provided that grad s does not vary appreciably in a distance of the order of an interatomic spacing, the wave function is, to a good approximation, a solution of the Schrodinger wave equation, and the complete set of wave functions (3) satisfy also the condition that they should be orthogonal to one another and to the ground state @. Thus the wave function (3) describes satisfactorily a flow of the whole liquid. However, this flow is subject to two conditions. First, as is clear from equation (4),it must be irrotational (curl v, = 0) ; and secondly the function s must be restricted so that the wave function satisfies the Bose symmetry requirements. To sce the meaning of the latter condition, we consider what happens to the wave function as a set of atomic positions forming a ring is changed in such a way that each atomic position R j in the ring moves a distance AR, into the position of its neighbour in the ring. (We are not considering real movements of atoms here, but merely changes in the function Y(Rl, R,, . . ., R , ) as the parameters R,, R,, . . . R, are changed.) Provided temporary changes are allowed in atomic positions outside the ring, this change can be brought about continuously in .wch a way that the wave function never vanishes. There can therefore have been no sudden and hidden change in phase of the wave function, and therefore the phase must have changed simply by an amount
-
C [grad s(R,)] A R
rmg
.
Thus the condition that the wave function should satisfy the Bose symmetry requirements means that this phase change should be an 1:eIl I c n u s
p. 50
C R . I,
g 21
VORTEX LIKES I N LIQUID
mmux
11
5
integral multiple of 2x, and this clearly leads to the condition on the velocity V, that
$ v,
*
dr
h m
= IZ - ,
(5)
where n is any integer. The flow described by the function (3) is therefore not only irrotational but also subject to the condition that the circulation round any circuit is quantized in units of hjm. These conditions on the flow velocity apply of course only to flow described by the wave function (3). However, it is believed that this wave function is the only one that will describe flow and at the same time satisfy the various conditions outlined above, a t least if the flow velocity does not vary too rapidly with position. If this belief is justified, the conditions (4)and ( 5 ) on the flow are perfectly general. In view of the experimental evidence presented later in this chapter, it seems that this belief is in fact justified to a large extent, but it has never been proved to be so. One difficulty is that the wave function (3) is not really an exact solution of the wave equation if the velocity varies with position, especially if i t varies rapidly with position, and it is not clear what modification is required to make it an exact solution, or how extensive this modification would be. The problems here will be discussed again to some extent in 9 8, but for the present they will be ignored, and the wave function (3) and the conditions (4) and ( 5 ) will be accepted as generally valid. It may be noted that both the irrotation condition (4) and the quantization of circulation (5) can be understood crudely in terms of the idea already mentioned that helium may be regarded as composed of quasi-independent units of atomic dimensions. For the irrotation condition implies the absence of any local rotation, and this is a consequence of the fact that the lowest rotational state of each unit has a large energy and will not be excited in an ordinary flow; while the quantization of circulation is equivalent simply to a quantization of angular momentum of each atom about some axis. If the helium is contained in a volume that is simply-connected, the circulation must of course vanish round any circuit. In order that a finite circulation may exist, it is necessary that the volume be multiply-connected; the finite circulation may then be established round any circuit whose area cannot shrink to zero while a t the same time the perimeter of the circuit remains continuously in the liquid. I n the Rrjsrencrs p. 66
ti
w,r. IXXS
Li.€i. 1,
42
simplest case the volume may bc multiply-connected owing to tllp prcsence of suitable solid boundaries ; a practical situation nlrcrc t h l < is so, and \\here it has been possible directly to. verify the truth of the quantization condition, will be described in 3 4. Howelw, it is :~l:,o yossiblc to achieve niultiple-connecti~itS.by having a small cylindrical hole running through the liquid, and a quantized circulation may be established round such a hole. The resulting flow pattern is the ciii;tnti/ed vortex line with which most of this chapter is concerned. A 5 will be seen later, vortex lines that appear in practice involvt, usually only one quantum of circulation, and all subsequent discussion will refer to such a line. The 1,elocity due to a n isolated straight vortex line is then equal to h / u w , whcrc Y is the distance from the axis of the line, and this leads to a kinetic energy per unit length given by
7' = ( q A 2 / m 2 ) In b / n o ,
( (9
\+liere ii,] is the radius ol the hole (core) a t thc centre of the line, and b is some appropriate outer radius. This hole a t thc centre of a line is expected to be very small. In an elementary way, its size may be considered to be determined by a balance of surface tension forces and Ucrnouilli forccs, in which case its radius will be gi\-cn hy a. =
pfi2//%ni20,
(7)
whcre cr is the surface tension of the liquid. The expression ( 7 ) yields a ialue of about 0.5 -4. Thus there is likely to be hardly any real hole in the liquid, but only an axis, on and very close to which the probabilit y of finding an atom is zero. It should be added of course that this naive picture of the conditions near the centre of a vortex line niay well be wrong; for the velocity of flow in this region varies rapidly with position, and, R S already stressed, it is under these conditions that the wave function (3) and the conditions (4)and (5) may cease to be valid. (We shall return to this point in 5 8.) Thus, when we come to interpret any measured value of the energy per unit length of line, the radius a, appearing in equation (6) should not be interpreted too litcrally as the radius of any real hole, but merely as a parameter that reflects the detailed conditions at the centre of the line. (It may be noted that even in the present naive treatment the total energy per unit length of line contains not only the kinetic energy (6) but also the surface energy associated with the hole; it follows from eq. (7) that this latter contribution is equal to .zeh2/m2.) R c f ~ i e i t c e sp , 56
CII. I,
9 21
VORTEX LINES I N LIQUID HELIUM I1
7
Besides the vortex lines it is possible to imagine also the existence of vortex sheets in helium. These will be surfaces in the liquid separating regions of different irrotational velocity fields. For example, one might ha1.e a cylindrical surface, inside which the velocity is everywhere zero, and around which there is a circulation hjm. Now the presence of a vortex sheet must not lead to a violation of the fundamental condition on which the quantization of circulation is based, namely that thc total wave function shall be unchanged by the permutation of atoms round a ring. By considering rings that cut the vortex sheet, it is easily seen that this condition will in fact be violated unless thc wave function vanishes whenever any atomic position passes through the vortex sheet; and as the wave function vanishes there must be a sudden change in phase. Thus the vortex sheet must form a node in the wave iunction. Such a nodal surface must have associated with it a surface energy, roughly equal in magnitude to the ordinary surface tension of the liquid. (This energy, which is analogous to the surface energy at the centre of a vortex liw,is composed of two terms: first, a potential energy term due to a reduction in density; and secondly a kinetic energy term of the type that is always associated in quantum mechanics \kith a rapid spacial change in wave function. The width of the node is determined by the condition that the sum of these two energies be a minimum, and is easily shown to be of the order of an interatomic spacing.) Thus vortex sheets must have considerable energies ( m 0.1 ergs cm-2), and, as will be shown in detail later, it is for this reason that they are not usually formed. Again, however, our picture may be too naive, so that this energy may be somewhat overestimated. We note that the surface energy of a vortex sheet is another example of the large minimum energy associated with any local vorticity. This completes our description of the possible types of flow in helium at absolute zero, and it seems likely, as indicated already, that the results apply also to the superfluid component of the liquid at a finite temperature (this is not quite true close to the I-point; see below). JYe see therefore that throughout most of the superfluid any flow must be irrotational, but that localised regions of vorticity are possible in the form of either quantized vortex lines or vortex sheets. -4s indicated in the introduction, the realisation that the irrotation condition (4)does not preclude all rotational motion in the superfluid led to the explanation of two observations not accounted for in the original Landau theory: the fact that the superfluid will rotate with Kift)tTlCtS
p . 56
S
1%'. F. VINE?;
[CH. I,
$2
a containing vessel; and the fact that ideal frictionless superfluid flow tends to break down at a comparatively small velocity. Observations on the behaviour of the superfluid in a rotating vessel will be discussed in the next scction, and it will be shown that they can be explained very easily in terms of vortex lines. The existence and breakdown of frictionless superfluid flow will be discussed in 9 6, and it will bc shown that, although the problems here are much more difficult, the observed facts can probably be explained, a t least in principle, by considering the possible creation of vortex lines and sheets in the flow. Three important observations should be made about the behaviour 01 vortex lines and vortex sheets. First, vortex sheets and macroscopic lengths of vortex line will not be present in thermal equilibrium, at least a t low temperatures; the energy is too high and the entropy (or number of possible configurations of line or sheet) is too low. Vortex rings of atomic dimensions might be present in thermal equilibrium, but, as already indicated, such rings may probably be identified with the rotons. Whether macroscopic lengths of vortex line can be present at high temperatures is an interesting problem, which has not been solved, but the suggestion has been made by FeynmanQthat as thr temperature is raised it does suddenly become favourable to have a great length of vortex line, 2nd that this sudden transition is to be identified with the A-point. An observation that may have some bearing on this suggestion is described is 4 4. Secondly, any vortex lines and sheets do not form part of the normal fluid, in spite of the fact that they are in a sense excitations in the liquid, like rotons and phonons, and in spite of the fact also that collisions must be possible between the rotons or phonons and the lines or sheets. This is simply because there is no mechanism by which a system of vortex lines or sheets can acquire an arbitrary steady drift velocity relative to the superfluid, such as is possible for the roton-phonon gas. (This is not true of course for microscopic vortex rings, but we have already guessed that these are identical with the rotons.) Thus vortex lines and sheets must bc considered to belong to the superfluid, and the existence of collisions between the phonons or rotons and the lines or sheets will give rise to a frictional force (mutual friction) between the two fluids. Thirdly, vortex sheets and vortex lines will probably behave hydrodynamically in the manner predicted by the classical theory of inviscid fluids. except in three essential respects : (a) They can probably be created in certain circumstances in an p . 5/i
RPjl~rCrl1:L'A
CH. I,
s a]
VORTEX LINES I N LIQUID HELIUM I1
9
initially irrotational flow by quantum mechanical transitions ; this is discussed in detail in $ 6. (b) Two closely spaced and oppositely directed lines or sheets can probably be annihilated by quantum mechanical transitions with the creation of rotonsg; see 9 7 . (c) As just explained, a force will act on a vortex line or sheet whenever it moves relative to the normal fluid. In the case of a vortex linc the Magnus effect will then come into play, SO that the force will cause the line to move relative to the local superfluid velocity in a direction perpendicular to the force. The importance of this force and the associated Magnus effect will be discussed later, particularly in 9 3 . 2 and 9 7. Finally, it should bc added that the conditions existing near the centre of a vortex line may be modified considerably when the temperature is very close to the A-point, and the energy of the line may be appreciably reduced thereby. The modification consists in an increase in normal fluid concentration (perhaps a complete conversion to normal fluid) close to the centre of the line. This happens for two reasons: first, it pays to create extra thermal excitations in order t o eliminate some or all of the large kinetic energy due to flow of the superfluid close to the centre of the line (this can also be viewed in terms of the increase that is known to occur in en/@as (v, - an) is increased1‘) ; and secondly a gradual conversion to normal fluid as the centre of the line is approached tends to eliminate the surface energy associated with the core. The effect will occur to some extent at all temperatures, but it is of negligible importance except very close to the I-point. lTnfortunately,of course, any theory of helium I1 in which the normal fluid is regarded as a gas of weakly interacting phonons and rotons is not applicable close to the A-point, but a phenomenological theory invented by Ginsburg and Pitayevskiil2Sl3 for dealing with behaviour close to the A-point can be applied and leads t o similar results. (In the Ginsburg-Pitayevskii theory the superfluid is described by an “effective wave function” y , which is related to p, by the equation m I y l2 = p,. Then any gradient in e, has associated with it a certain energy. If this idea is correct, the gradual conversion to normal fluid as the centre of the line is approached leads only to a reduction, and not an elimination, of the surface energy term associated with the core.) Similar cmsiderations apply to the conditions on a vortex sheet close to thc A-point. lieferences
p . 56
10
I \ . E . TISEU
[CH. I ,
$3
3. Vortex Lines in Uniformly Rotating Helium As already mentioned, experimental evidence s h o w that if helium I1 is contained In a uniformly rotating vessel the equilibrium motion of the superfluid on a macroscopic scale niust correspond closely to that found with an ordinary liquid, i.e. motion as a solid body with the vessel. (It is probable, as Mill be seen in 9 4, that this is not true for ;1 sinall vessel rotating very slowly, but we shall not be concerned with this situation in the present section.) The evidence is of two types. I'irst, there have been observations on the shape of the free surface of rotating helium 143 15, which have revealed no apparent difference from an ordinnry liquid ; and secondly there have been nleasurements 1'; of the torque required to accelerate or decelerate the rotation of heliunl in ;L vessel, and the angular momcntuin obtaincd by integrating this torque ~ i t respect h to time has been found to correspond closely to the full classical value (although the detailed form of the torque-tirnc curve is oftcn quite different from that for a n ordinary liquid). It must be admitted that in another torque experiment by Walmsley and Lane17 this result was not confirmcd, in that the total angular momentum collected when a rotating vessel was brought to rest appeared to be significantly less than the full classical value, especially at rather IOU angular velocities and in R vessel of large radius; but, since this re.ult appears in effect to contradict all other evidence, it must for the present be viewed with some reserve. If then, as appears generally to be the case, the superfluid will, when in equilibrium, rotate in a manner that is macroscopically indistinguishable from classical solid body rotation, it must, in view of the theoretical arguments of $ 2 , do so though the presence of some system of vortex sheets or lines. The problem therefore arises of determining what particular system of lines or sheets will be favoured. N'e shall first discuss this problem theoretically, and then describe the relevant experiments. 3.1. THEORETICAL EQVILIRRIUM STATE
A coinparison of various arrangements of vortex line and sheet was carried out by Hall and Vinenl*. The condition for equilibrium in the rotating state is that the quantity
shall be a minimum, where o is the angular velocity of the containing R ~ f e r u i c d sp . 5G
CH. I ,
4
31
V O K T L S LINE5 I S LIQUID HELIUJI I1
11
vcssel, F is the free energy of the liquid, and J4 is its total angular momentum. For an ordinary liquid this condition leads to “solid body rotation” with the containing vcssel, but, as we have seen, this is not possible for the supcrfluid of helium 11. The best that is possible is sollie arrangement of vortex line or sheet, and it is necessary to determine which of the various possible arrangements leads to the smallest value of F‘. As usual the calculations are carried out for helium a t ‘~bsolutczero, and it is assumed that the results apply to the superfluid component at a finite temperature. Three arrangements were explicitly considered by Hall and Vinen : (a) a systcm of concentric sheets (suggested independently by Onsager2 and by Landau and Lifshitz5*); (b) a system of vortex lines running parallel to the axis of rotation (Feynmang); and (c) a system of small tubes of vortex sheet running parallel to the axis of rotation, each tube having zero circulation inside it and one quantum of circulation outside it. Detailed calculations, which will not be reproduced here, showed that by far the best arrangement is the array of vortexlines, each with one quantum of circulation, the lines being uniformly spaced with the number crossing a unit area normal to the axis of rotation equal to approximately 2mwlh. This system of lines, each of which moves with the velocity created in its neighbourhood by the others, then rotates with the vessel. Except for the singularities a t the vortex lines, the velocity distribution in the fluid differs little from that corresponding to solid body rotation (provided the angular velocity is not too low) ; and the total angular momentum is also close to the classical value. Thus the picture is consistent with the angular momentum and free surface observations referred to above (there should apparently be small irregularities in the surface, but it can be shown that these irregularities should probably be too small to be observable, especially when the action of the surface tension of the liquid-vapour interface is taken into account). However, certain vortex sheet arrangements are also consistent with these observations, so that more subtle experiments, such as those described below, are required to provide evidence that is specifically in favour of the vortex line arrangement. The reason for the maiked preference for lines rather than sheets lies in the large surface energy of a vortex sheet, and the calculations * In thcir original papers, London took no account of the surface tension in a vortex sheet, and I .an(lau and Lifshitz apparentl? took no account of the quantization o f circulation I& fri eiwt 9 p 6 h
w. I;. T’ISEN
12
[CH. I,
33
make it clear that an arrangement of lines will always be energetically preferable to any arrangement of shects. Expressed in a general way, the important point is that vorticity will always have associated with it a large energy, and therefore the best arrangement is the one that reduces the total volume of vorticity to a minimum and this must clearly be an arrangement of lincs rather than sheets. :I. 2 .
>IUTV.iL
F R I C T I O X IN THE ~ K I F 0 R M I . Y I
As \\.as mentioned in S 2, vortex sheets and vortex lines will act as scattering centres for the thermal excitations that constitute the normal fluid. If the excitations are drifting relative to the background fluid, i.e. if there is a relative velocity between the two fluids, this scattering will give rise to a momentum transfer between the fluids and hence to a mutual friction. A mutual friction should therefore be observed in the uniformly rotating liquid. The existence of this mutual friction has been confirmed expcrimentally by studying the propagation of second sound in the uniformly rotating liquid lY. Mutual friction should lead to an extra attenuation of the second sound; this attenuation is indeed observed, and it provided the first piece of direct experimcntal evidence in favour of the theory developed in 4 2 . The experiments employed a resonance technique, and the attenuation was measured from the Q-value of the resonator. In order to study any anisotropy in thc attenuation, two resonators were used, and they are shown in Fig. 1 . In the first, the second sound was propagated radially between two concentric cylinders and perpendicular to the axis of rotation; in the second the second sound was propagated along the axis of a pill-box shaped cavity, and this axis could be made either to coincide with the axis of rotation or to be perpendicular to it (in the latter case the second sound was propagated at right angles to the axis of rotation b u t not in a purely radial direction). It should be noted that in both resonators the whole assembly rotated, and the helium inside the rotating assembly was in contact with disturbed helium outside through only the film and the vapour; there was thcrefore no reason to suppose that the hcliuni inside each resonator could not reach complete equilibrium with the rotating vessel. Starting or stopping of the rotation was always accompanied by large transient attcnuations, which, as will be clear from $ 7, were h’t./t.irmes
p. 56
CH. I, 9: 3 1
VORTEX LINES I N LIQUID HELIUhl I1
13
probably due to turbulence in the superfluid*. However, there remained an excess attenuation in the rotating state even after equilibrium had been reached, and it is this attenuation that is of interest here. The experimentally observed characteristics of this steady extra attenuation were as follows (except near the I-point, where the results were complicated and are not yet understood). First, it was proportional to the angular velocity of rotation (angular velocities between about 0.5 and 1.5 rev sec-l were used) and independent of the amplitude of the second sound. Secondly, it was unchanged when different harmonics (covering the frequency range from 1.5 to 4.5 kHz) of the
a
b
Fig. 1. Resonators used to study the attenuation of second sound in uniformly rotating helium18. [a) Radial mode resonator; (b) axial mode resonator. The radial mode resonator was rotated about its axis of symmetry A ; the axial mode resonator could bc rotated about either the axis R or the axis C. T, R : second sound transmitter and receiver.
resonators were used. And thirdly, its magnitude for a given angular velocity was the same in both the radial mode resonator and the axial mode resonator on its side (axis C ) , but much smaller (possibly zero) in the axial mode resonator in its other position (axis B). These last observations show that the attenuation probably has the same (large) value for all directions of propagation that are perpendicular t o the axis of rotation, but is negligibly small for directions parallel to the axis of rotation. It should bc added that the velocity of the second sound suffered no detectable change owing to the rotation, and this
* The existence of an excess attcnuation of second sound in a rotatmg turbulent superfluid was first reported by Wheeler, Blakewood and LaneZO. References p . 56
11
rCH. I,
11'. 1;. \'ISBN
4 :l
confirms that the two fluid model, with unchanged values of en ;mcl ", rcriiajns applicable to the rotating liquid (a similar conclusion XIS reached by Andronikashvili and Kaverkin l5 from n-~easurementsof tlir fountain pressure in rotating helium). These experimental results show that there must exist in the uniformly rotating liquid a volume mutual friction force between the two fluids that is probably of the form
per unit volume, where w is the steady angular velocity of rotation, and B is a dimensionless constant. Values of B , deduced from the observed attenuations, are shown by the points in Fig. 2 . We have
I I I
I I u 20
22
T ~ K )
Fig. '7. The m11toa1friction constant B as a fuiictioii of teniperaturc. 0, radial mode resonator; [J, axial mode resonator rotating about axis C . The broken line shows the tIi(md3cal \%luc of B obtaincd from the scattering cross section (9) ; t h e solid lincs show thc theorctical values of B and H' obtained from thc inodilirtl scattering cross section discusscd in the test.
alreacly seen that thc theory developed earlier will account for the existence of some forin of mutual friction in the rotating liquid, and it is now necessary to investigate whether it will account also for this precise form and magnitude of the force. I t is easily seen that the Jornz of the force (8) is indeed easily csplained in terms of the idea that uniformly rotating helium contains an array of vortex lines. For it is clear that, if the plausible assumption is made that a thermal excitation can exchange momentum only in n 1icfr.vrvicr 5
p . .iG
CH. I,
$ 31
VORTEX L I X E S I X LIQCID HELIUlI I1
16
direction normal to the scattering vortex line, then, as is observed, the friction force will have a constant and large value for all directions of (v,,- us) that are perpendicular to the axis of rotation, but will be zero for directions parallel to the axis of rotation. (Any small force that exists when (vn - 11,) is parallel to the axis of rotation could probbably be explained if the lines wobble a little as a result of zero point motion or thermal agitation.) Furthermore, it is also clear that any model of rotating helium with concentric vortex sheets is not consistent with the form of (8). For in this case momentum exchange could probably take place only at right angles to the sheets, so that themutual friction would have its maximum value only if (wn - vS) is directed purely radially, a conclusion that is directly at variance with the observed equality of the mutual friction in the radial mode resonator and that in the axial mode resonator on its side. Thus our theoretical conclusion that concentric vortex sheets will not be the stable state is supported by experiment. It now remains to discover whether the observed magnitude of thc force (8) is consistent with the vortex line model. Only an outline of the necessary calculations can be given here; and for details reference should be niadc to the original papers1*,2*. The first step in the calculation of the mutual friction to be expected from an array of vortex lines is an investigation of the scattering of thermal excitations by a single vortex line. In the temperature rangc for which measurements have been made the normal fluid density is due almost entirely to rotons, so that the phonons can be ignored, and only the scattering of rotons need he considered. It would appear that the interaction energy between a roton of momentum p and a line should bc composed simply of two terms: the first, equal to p . v,, arising from the interaction of the roton and the velocity field v, of the line; and the second arising from the change in density of the fluid near the centre of the line. Of these two terms the first turns out t o be much the more important, and, as shown by Lifshitz and I'itayevskiiZ1, it gives rise to a scattering that may be described b:y an average cross-section per unit length of line equal to roughly
and an average angidar deflection (to the right for a right hancled line) equal to roughly 2 t a r 1 (puF:T/pi)!,where p and fi0 are the parameters h'tftrexcis
p . .i6
If
;w. I ,
IV. I;. \ . I S E S
03
appex-ing in the expression E = A 4 (fJ - @ o ) 2 / %€or p the energy spcctrurri for the rotons, and x is the circulation Ia/ni, round the vortc.x line *. These results may now be combined with a simple kinetic calculation to obtain the force facting on unit length of a line in the second sound velocity field. The value of the force obtained is
f = D(v,
- vL)
+ D’ K A
(VR --
14
v,) >
~~
where D m 1.2 q,d,u1z7‘/fJ0, 1)’ = %en, v, i s the drift velocity of the rotons in thc ncighbourhood of the line, and v, is the velocity with which the vortex line itself is moving. I t must be realised next that the velocities v, and v , are not equal to the velocities v, and v, rcspectively appearing in eq. (8),which must be regarded a5 involving averages over a volume containing several vortex lines. The difference between v, and v, arises because a vortex line tends to drag the nornial fluid in its vicinity, and it can be shownlS that the relationship between these velocities will be given by
vn
-
(11)
v,,= f / E ,
where (12) qn is the normal fluid viscosity, ii = (p,coo/q,)+, oo is the angular frequency of the second sound, and L, is the roton-roton mean free path. The difference between v, and v, arises from the Magnus effect; whcn a force facts on a vortex line it will move sidcways relative to the background fluid, and the difference (vL - us) will be given by
f = eAvL - v,)
x.
(13)
(Detailed arguments to justify thc application of the usual Magnus formula to this case are given i s ref.IR.) Eqs. (ll),(12) and (13) can now be used to rewrite eq. (10) in terms * I n the paper on this subjcct by IIall arid Vinenls thc scattering was calculatcd crudcly by means of a Born approximation, and was found to be symmetrical about thc forward direction. Ho\vcver. it was later pointed out by Lifshitz and I’itayevskii that a W.K.B. approximation \vould he better, a n d thcy uscd this approximation to ol>tainthe results quoted ahovr. KCfL-rcitc1.s
p . 56
CH. I,
s 31
17
VORTEX LINES I N LIQUID HELIUM I1
of (v, - v , ) . The mutual friction force per unit volume is then obtained by multiplying the resulting value of f by the length, 2 w / x , of vortex line per unit volume (where w is the angular velocity of rotation), and the final result is found to be
where 0 2 f 0'2
D'
-
-)'I
1
, (15)
"Qs
and
The theoretical expression (14) differs in form from the expression (8) obtained from experiment in that it contains an extra term representing a force a t right angles to (v, - v,) ; this extra term arises partly from the asymmetry in the fundamental roton-line scattering and partly from the Magnus effect (13). However, it can be shown that the presence of this term would not have been detected in the experiments described so far, so that the more general form (14) for the mutual friction is not inconsistent with these experiments. Experiments of a different type which show indirectly that the term probably does exist (or, strictly speaking, that the contribution to the term from the asymmetry of the scattering exists) will be described in 9 3.3. A possible experiment for observing the term directly, using a square resonator, was described in ref.l9, but it has not yet been carried out. The theoretical value of the mutual friction constant B derived from eq. (15) is shown as a function of temperature by the broken line in Fig. 2. It can be seen that the order of magnitude is correct (there are no adjustable constants), but that the temperature dependence is wrong. I t seems likely that this discrepancy arises from some defect in the treatment of the roton-vortex line scattering process. The discrepancy can in fact be removed by arbitrarily adding to the interaction already discussed an extra temperature independent scattering Kej'freizces
p . 56
18
W. F. V I N E N
[CH. I ,
s3
cross section, equal to a per unit length, corresponding -__ to absorption of the total roton momentum (this adds a term a@,d 2 k T j n p to D , and reduces D’ by a small factor) ; for if ci is put equal to lOA the solid curve in Fig. 2 is obtained, and this is in satisfactory agreement with experiment. The origin of this extra scattering is not yet clear, although Lifshitz and Pitayevskii suggest that it may be associated with a process in which a roton is absorbed by the line, with the excitation of vortex waves (see $ 3.3), and subsequently re-emitted. But a completely satisfactory theory must presumably await a more complete description than we have at present of both the nature of the roton and the conditions near the centre of a line (see 5 8). We may summarise this section by saying that studies of the absorption of second sound provide fairly good evidence for the existence of quantized vortex lines in uniformly rotating helium, but that the interpretation of the experiments is handicapped owing to the absence of an adequate theory of the fundamental roton-vortex line scattering process. 3.3.
VORTEX
WAVES I N
THE UNIFORMLY ROTATING LIQUID
It should be possible to propagate circularly polariscd transverse waves on a vortex line (Thomson22), and it is clear that further evidence for the existence of quantized vortex lines in uniformly rotating helium would be available if effects due to such wave motion could be observed. As we shall see later, such effects have in fact been observed, but before we describe them it is convenient to consider in some detail the theory of these waves. Consider first the case of an isolated vortex line that is initially straight and running along the z-axis, and suppose that it is given some form of wave-like displacement described by coordinates [E(z), ~ ( z ) in ] the (x,y) plane. This displacement will result in the presence of curved portions of line, and it can be seen that these portions will move with a velocity proportional to the local curvatmc of the line and directed a t right angles to the plane in which the curvature takes place (this is because each part of the line will move with the velocity created in its neighbourhood by the rest of the line). Thus the line as a whole will inove according to the equations a2E v-=-
822
I&/rn 11116 p . 56
Pq at ’
8%
],-z=
822
85 2f ’
-~
(171
CH. I,
9
31
VORTEX L I S E S I N LIQUID HELIUM I1
19
where v is a constant. These equations are easily solved to give displacements of the form
5
=
toeiRt
eikz I
q= fit,
(18)
where K 2 = f Q/v. Thus the displacement (6,q) propagates as a dispersive circularly polarised wave : if the vibration is in the opposite sense t o the rotation in the line, the waves are undamped; if it is in the same sense, the waves are evanescent. Evaluation of the constant v requires a more detailed calculation22;for the undamped waves it is found that x
1.046
472
so that strictly speaking the v is not quite constant but varies slightly with the wave number k. Although an argument of the type just given must be used in a rigorous approach to the theory of vortex waves, the behaviour of the waves is more easily visualised by imagining that the lines are moving relative to a stationary fluid, and that the circular wave motion arises from a balance between the resulting Magnus force on the line (equal to esv A x ) and a tension in the line (equal to an effective energy per unit length (es9/4n) In b/a,, where b is a length which must be equal, according to eq. (19), to 1.046/k). If the vortex line is not an isolated one, but is one belonging to an array of lines such as we believe to exist in the uniformly rotating superfluid, the situation is more complicated owing to the interactions between the lines. If we use the simpler approach just indicated, we see that each vortex line can no longer be imagined to be moving relative to a stationary fluid, but instead it must be imagined to be moving relative t o a fluid which is itself moving with a velocity created by the neighbouring vortex lines. A generalisation of the elementary theory to take this effect into account has been given by Hall23;this generalisation is based on the idea of balancing the Magnus force against the line tension, but there is no reason to doubt that the results obtained from it are valid (Hall in his paper gives a more sophisticated and convincing presentation of the argument than is possible here). Hall points out that the average (superfluid) velocity relative to which each vortex line may be imagined t o move is easily obtained from the condition that the total number of vortex lines must be References p. 56
20
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[CH. I,
2
conservedls. If we write O ( = curl v,) as the vorticity averaged over a region containing several lines, this condition takes the form
D
a"
Dt
Ft
__ Q = -
+ (v,
.grad)& =
-
Q
div v,,
where vL is the velocity with which the lines move. Since (0 . gradjv, must be zero, it follows that
a0
_ _ _ curl (v, A o)= 0, Ft
so that V, is given by
av,
-_
at
-
v, A 8 = grad
4,
where 4 is some scalar, which must represent the effect of temperature and pressure gradients and must therefore equal { - (P/e) ST}. This result may now be combined with the equation representing the balance between the Magnus force and the line tension, i.e., with the equation
+
(e,v/x) ( x . grad) x = ps(vL - V,) A x ,
(23)
to give equations for vL and V,, from which the possible motions of the lines and the possible overall motions of the superfluid can be obtained. Application of these equations to the case of the array of vortex lines existing in the superfluid of helium I1 contained in a uniformly rotating vessel shows that wave motions should be possible, although in general they will be rather complicated in form. However, the motion becomes fairly simple in two extreme cases, to which we shall confine our attention. The first of these cases corresponds to weak interaction between lines, and it holds if the spacing between lines is large compared with the wavelength of the waves (this is equivalent to the condition l2 w , where Q is the angular velocity of rotation) ; the situation then is closely similar to that found for an isolated line, and the interactions give rise only to small corrections such that the dispersion relation is changed from K2 = k sZ/u t o k2 = k (Q/v) -~ 2w/v. The other extreme case corresponds to strong interaction between lines, and it holds if the spacing between lines is small compared with the wavelength (i.e. Q
>
<
RcJt7rcnces p . 56'
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VORTEX LISES IN LIQUID HELIUM I1
ably not that given by equation (19) but instead ()1/47c) In b/a,, where b is a length of the order of the line spacing. The results just quoted can be expected t o apply strictly only to helium at absolute zero. At a finite temperature the effect of the normal fluid must be taken into account. This effect will be to add an extra force (mutual friction) to the lines, given by eqs. (10) and (I 1) of 3 3 . 2 ; i.e. by
>
Again a distinction may be made between the cases 2 ! w and 9 w . In the former case the coupling between the two fluids will be small, so that to a good approximation the overall normal fluid motion will not be perturbed by the vortex waves; the first term in eq. (24) then produces a damping of the waves as exp(- ccz), where K rn B,p,k/4e, while the second term gives rise to a fractional change in the effective value of 11 equal to B;,n,/-Le.Both these effects are small. In the latter case the situation is generally more complicated, but it can be shown that in the practical situation described below the mutual friction can have only a negligible effect. Experiments with which this theory of vortex waves can be conipared have been carried out by Hall23.The essential part of the apparatus is a small aluminium can, containing a pile of discs of the type used in the Androniltashvili experiment, which is suspended by a torsion fibre from a head that may be rotated at any constant angular velocity. The experiment consists in filling the can with helium, rotating the torsion head until the whole system is rotating with it, and then measuring the period of small oscillations of the can superimposed on the uniform rotation. Observations were made with various disc separations between 0.B and 7 mm, with periods of oscillat’ion between 3 and 25 sec, and with periods of rotation between 0.1 and 1.1 rad sec-I.
<
RL?jdYeflces
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[CH. I,
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3
The idea behind this experiment is that oscillations of the disc system will set up waves on the vortex lines that are presumed to be present in the uniformly rotating superfluid, and that the presence of these waves will show up as a change in the period of oscillation. In order that waves can be set up in this way, it is necessary of course that the discs should be able to grip the ends of the vortex lines, and in order t o encourage this gripping the disc surfaces were roughened. (It is reasonable to suppose that owing to the tension in a line the ends of a
I
0
0.05
I
1
0.15
0.20
I
0.10 w i l kni
sed)
Fig. 3. Experiments on vortex waves using a disc-filled can. (a) Results for f2 & to: (b) rcsults for 0 w . Thr hroken lines arc the theoretical curves given by eqs. (26) and and (27), with Y = 8.5 x 10-4 cmP src-1; the solid lines are the theoretical curves obtained \fit11 same valuc of Y when some degree of slipping is taken into account. (21 = disc separation.)
line will attach themselves to protuberances in the roughness and will not easily jump from one protuberance to another.) The experimental results are shown by the points in Fig. 3. They are expressed in terms of the effective density e' of the fluid being dragged by the discs, which may be deduced directly from the observed periods. Rrfcrmces
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VORTEX LINES IS LIQUID HELIUM I1
This effective density will be due almost entirely to the superfluid, since all the experiments were carried out at a temperature of about 1.27" K, where the normal fluid density is only about 4% of the total. w, Results were obtained for the two extreme cases, 9 w and 9 and these are shown separately in the figure. It can be seen immediately that there are resonance effects if Q w, but none if Q w ; this is exactly as expected, since, as we have shown, there is the possibility of undamped wave propagation in the former case but not in the latter. Detailed application of the theory, in the form of eqs. (22) and (23), to the present experimental situation yields the following theoretical results
>
>
<
<
(26)
where 21 is the disc separation. These functions are shown by the broken lines in Fig. 3, with the value of v taken to be 8.5 x cm2 sec-I. It can be seen that the agreement between experiment and theory is extremely good, except that there is a damping effect to be observed in the experimental resonances, which has not been taken into account in the theory. It might be thought that this damping is due simply to the mutual friction (24), but it proves to be much too large to be explained in this way. It must therefore be assumed that the damping is due to imperfect reflection of the waves from the disc surfaces; i.e. to slipping of the ends of the vortex lines (in spite of the roughness). If it is assumed, very reasonably, that the rate of slip is proportional to the angle of inclination of the end portion of line away from the perpendicular position, agreement with experiment can be achieved (the full curves in Fig. 3 ) , but only if the further assumption is made that the constant of proportionality decreases (i.e. the surface becomes effectively rougher) with increasing period of oscillation. The need for this latter assumption is not yet understood. The experiments that have just been described provide impressive evidence for the existence of quantized vortex lines in rotating helium 11. Nevertheless they suffer from two defects, both of which owe their existence to the imperfect reflection. First, the resonances for 9 Q are not very sharp, and this precludes any accurate determination
>
Ref1~ c n r z 5p 56
24
[CH. I,
JV. F. V I X E N
$3
of v (and hence a,) ; and secondly, the effects due to the mutual friction (24) cannot be observed. Hall24has therefore carried out recently a modified version of these experiments, in which these defects are not present. This consists in observing resonances between a single disc and a moveable reflector 3 w ) . The free surface of the liquid may be used conveniently as the reflector. A typical set of results, in the form of a plot of period against disc-reflector separation, is shown in Fig. 4. The value of v can be deduced from the resonance positions (extra-
(a
I
2
I
I
I
4
I
6
1
I
I
8
I
I
IC
(iiirn)
Fig. 4. Vortex wavrs betwcen a sin& disc and a movable reflector. Period (If oscillation ( T ) of tlic: disc plotted against srparation ( I ) brtween disc and reflector. (w = 0.140 rad S C C - ~ ; temperature 1.3” K.)
polated to absolute zero) ; the value of B, from the attenuation of the waves; and the value of &, from the temperature variation of the resonance positions (it is assumed that v is independent of temperature, which is probably true to a good approximation except very close to the ,?-point; see 0 2 and 5 8). The parameter v can be dctermincd in this way with fair accuracy, and Hall gives the value (9.7 & 0.2) x lo-* cm2 sec-l, which corresponds to a value of a, (eq. (19)) of 6.8 l.6x. The constant B , (which is approximately equal to B at the low temperatures used; see eq. ( 2 5 ) ) ” can be determined with an accuracy of about 30%, and within this possible error the values obtained agree with those deduced from the measurements of the attenuation of second sound described in 4 3 . 2 . The more interesting quantity BL ( a 3 at the low temperatures *), which cannot be obtained from these second sound measurements, can however be determined only very roughly, but the results do suggest a value that agrees both in order of magnitude and in sign with that predicted by the theory outlined in $ 3 . 2
-+
* These cqualities imply that the cxistencc of the Magnns effect (13) does not inilurnce appreciably thc>values o f B and B’; the existence o f a finitc B’ is then due entirely to t h t : asymmetry of the roton-line scattcring. Hc jercnccs p, 66
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(based on the Lifshitz-Pitayevskii scattering calculation corrected empirically to agree with measured attenuation of second sound) *. 4. The Detection of Single Quanta of Circulation The experiments that have been described so far provide us with good evidence that quantized vortex lines exist in the rotating superfluid, and hence that the theory outlined in $ 2 is correct. However, the evidence for the theory as a whole is incomplete, owing to the fact that strictly speaking the experiments do not establish for certain the law of quantization of circulation; for it can be arguedls that the appearance of vortex lines with circulations of order h/m is an accidental consequence of the existence of a large energy associated with any local vorticity, and not as a consequence of a general +rinci+le that superfluid circulation is quantized. For this reason it became clear that attempts should be made t o establish by direct measurement that superfluid circulation is quantized even for the case of a multiply connected region formed entirely by solid boundaries. The quantum of circulation is comparatively large, so that there was hope that such a measurement should be possible by a direct mechanical method. A successful experiment of this type has now been carried out2j!26,and it will be described in this section. The experiment, as originally conceived, consisted simply in a measurement of the circulation round a fine wire stretched along the axis of a uniformly rotating vessel of helium. This measurement can be made by studying the modes of transverse vibration of the wire. If there is no circulation, each mode is really doubly degenerate owing to the possibility of vibration in two directions a t right angles. However, if there is a circulation it, this degeneracy is removed through the action of the Magnus effect, the normal modes become two circularly polarised modes with opposite directions of rotation, and these modes are split by a frequency difference e,xl'Lnw, where w is the mass per unit length of the wire. In practice, the vessel consists of a brass tube with an internal diameter (2b) of about 5 mni, and the wire consists of a 5 cm length of cm. A transverse magberyllium-copper of diameter ( 2 a ) 2.5 x netic field can be applied to the system, so that transverse vibrations
* :lfter this revieiv \vas completed, tlic author lcarncd of further interesting esperiments on vortex wa\-es by .lndronikashvili and Tsakadze4%5 0 and of detailed calculations on tlic effect of mutual friction on vortex waves by Mamaladze and Jlatinyan"'. Rdjcvenccs
p. 56
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4
of the wire can be generated by passing a current (pulsed or alternating) through the wire and observed on an oscilloscope as a voltage induced bettieen the ends of the wire. The splitting ~ , x / 2 n wand , hence the circulation x , is obtained from the irequency of beats superimposed on freely decaying oscillations of small amplitude generated by discharging a condenser through the wire. At the 'frequency used ( m 500 Hz) the Q factor of the vibrating wire, which is limited by the viscosity of the normal fluid, proves to be just large enough to allow the measurement of a circulation of order hjm to within 2 - 3 0 / , provided the temperature is less than about 1.4" K. Before the experiments are described any further, it should be pointed out that a situation may well arise in practice where there is a circulation of (say) one quantum along only a part of the wire (a situation corresponding to a partial attachment of a free vortex to the wire), in which case an apparent circulation equal to a non-integral multiple of h / m will be observed. As we shall see, this situation does occur rather often, and this has made it more difficult to establish the quantization condition than might have been hoped. It should also be noted that the normal fluid cannot contribute appreciably to the measured circulation. LVhen the apparatus is rotating steadily at an angular velocity w , there should, if the system is in equilibrium, be a certain number fi of quanta of circulation round the wire, together with an array of q free vortex lines in the surrounding helium, the values of fi and q increasing with increasing w . The precise values of p and q to be expected for any given angular velocity must be determined from a detailed calculation to which reference will be made later; a t this point we merely quote the result that, if the simple picture of a free vortex line is correct, the observable quantity fi should be equal to zero for 0) < w o = (h/mb2)In b/a ( m 0.133 revjmin in the present apparatus), equal to one for w o < w < m 3w,, equal to two for 30,
7w,. Thus as m is increased from zero up t o one rev/min, the observed circulation should increase from zero to 3 hjm in steps of hjm. The experimental results show that these predictions are probably more or less correct in principle, but that in practice the situation is m n d e coniplicated by the fact that it proves difficult to achieve equilibrium. Thus, in the first place, it has been found that, a t the low velocities that must be used, it is extremely difficult, if not impossible, Krjmwcra
p . 56
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VORTEX LINES IN LIQUID HELIUM I I
27
to achieve even approximate equilibrium by simply setting the system into rotation below the I-point; the evidence at present is that under these conditions the superfluid may, at least for long periods, simply remain a t rest. The difficulty is of course easily overcome by rotating the apparatus above the A-point and then cooling it, so that the superfluid is created in a state of rotation. This procedure has been used successfully to the extent that it does lead to apparent circulations round the wire that are of the expected order of magnitude. However, it has been found that it does not lead to apparent circulations that are aIways integral multiples of h/m, so that it must be concluded that even this procedure does not always produce the state of complete equilibrium, but instead a state in which a vortex line is only partly attached to the wire. (Presumably the ends of the free line stick to protuberances in the apparatus, due, for example, to roughness.) It is of course necessary to find experimental evidence in favour of this conclusion if the quantization condition is to be verified, and this has been done by studying the behaviour of the apparent circulation after the rotation has been stopped. This behaviour is as follows. As soon as the rotation is stopped, the apparent circulation changes slightly, but in the absence of disturbances it soon settles down to a value that often appears to be quite constant for an indefinite time. This constant value still corresponds in general to a non-integral multiple of hjni (usually between one and two), and its existence is, incidentally, direct evidence that superfluid currents, and even free vortex lines, can persist indefinitely in helium contained in a stationary vessel. However, the vital observation is that, if the system is now fairly violently disturbed by, for example, exciting a Iarge amplitude of vibration on the wire, then the apparent circulation will generally decrease, but often it will decrease only as far as a value that is equal, within the experimental error, to exactly hjm. This is precisely the behaviour t o be expected, since the removal of a partly attached vortex line involves only the movement and, possibly, the stretching of an existing vortex line and should be comparatively easy, while the removal of a vortex attached to the whole length of the wire involves the creation of a new free line, and this process, as we shall see in 3 6, is probably very difficult. Thus the quantization condition is verified. Examples of the actual observations are shown in Figs. 5, 6 and 7. Fig. 5 shows typical examples of the behaviour of the apparent circulation as a function of time after the rotation is stopped. The vertical Rcforcnccs
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W. F. VIXEN
[CH. I ,
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arrows indicate times at which the wire was subjected to vibrations of large amplitude. The stability associated with a circulation of exactly one quantum is clearly seen in (b). In both cases the apparent circulation corresponded initially to between one and two quanta; in (a) this must have been composed entirely of partly attached lines, while 0
I/
0
051 '
I
I I
Fig. 5 . Tlic vibrating wire experiment. Typical examples of thc beliaviour of the apparent circulation (phjnz) as a function of time ( t ) after the rotation is stopped (Q = 3.34 rad min-1). The vertical broken line shows the timc at which the rotation was stopped, and the vertical arrows indicate timcs at which thc wire \\'as repeatedly vibrated with large amplitude.
in (b) one of the lines must have been completely attached. Fig. 6 summarises all the results that have been obtained of the type illustrated in Fig. 5 ; it shows a histogram of the total number of obserILfL~reltCcsp . 56
C H . I,
9 41
VORTEX LINES I N LIQC'ID HELIUM I1
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vations in which large amplitude vibrations have changed the apparent circulation by less than 4%, plotted against the initial value of the circulation. The peak occurring when the circulation is close to one quantum is very clear. (The evidence here would be even more convincing if quantized circulations equal to two and more quanta had also been observed, but no method has yet been discovered of establishing such large circulations; see $ 5 . ) The stability associated with a circulation of one quantum has been confirmed by another type of observation, typical examples of which are shown in Fig. 7. The system is first set into rotation in one direction
Fig. 6. The vibrating wire experiment. Histogram showing the measured circulations ( p h / m ) that have been observed to be stable against repeated vibration of the wire with large amplitude.
(say anticlockwise) by the technique of rotating above the A-point and cooling, and this will produce a number of free and attached vortices, all presumably with the same (anticlockwise) sense. The rotation is then stopped, and the system allowed t o stand for a short time. Finally the rotation is started again, but in the opposite (clockwise) direction. As might be expected, this reverse rotation causes a reduction in the magnitude of the apparent circulation; this is shown in (a), where the apparent circulation falls steadily to zero in about five minutes. However, in the case shown in (b) the reduction went only as far as one quantum, and this demonstrates again the stability of a circulation of one quantum, even this time against a reverse rotation of the whole apparatus. The vibrating wire apparatus can be used for other studies besides the verification of the quantization condition, and one of these will be mentioned in the next section. There is, however, one small but imZiLfcl
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54
portant observation that has been made with the apparatus which may be mentioned a t this point. I t is that even when the system is cooled from the A-point in a non-rotating state there is often a single vortex partly attached to the wire. The explanation of this phenome2FJ-
I
0
0 0
01
15-
I I
Io
!x IO-
(I: 0
l I I I
0
I
I5-
IO-
o
I 0
0 01
0 1 0
lo
00
oooooo
0
I I
I I O5I
I
Fig. 7. The vibrating mire experiment. Typical examples of the effect of reversing the direction of rotation. The apparent circulation ( p h l m ) is plotted as a function of time ( t ) after stopping an anticlockwise rotation (w = 3.34 rad min-') ; the rotation was started again, in a clockwise direction, a t the time indicated by the vertical broken line.
num is not known, although one interesting possibility is that the vortex is a remnant of the mass of vortex line which might, according to Feynman9, be present in helium I (see 5 2). Whatever the explanation, the observation itself shows that isolated vortices are very likely t o be present in apparently undisturbed helium, and we shall find that Rcjerences
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this may be important when we come to consider mechanisms by which superfluid turbulence can be nucleated (4 6). It should be added finally in this section that an ingenious experiment that might have provided direct evidence for the quantization of superfluid circulation has been carried out by Craig and Pellam2'. This experiment consisted in measuring the lift on an aerofoil placed in a uniform stream of superfluid. The lift (Magnus force) should of course be proportional to the circulation round the aerofoil, but the measurement showed no evidence for quantization, although there was an interesting region a t low velocities where the lift appeared to be accurately zero. It now seems very likely that this apparent absence of quantization was associated with the fact that vortex lines may well be only partly attached to the aerofoil.
5. The Energy of a Free Vortex Line As explained earlier, the conditions existing near the centre of a vortex line may depart considerably from those in the simple picture developed in 9 2 (i.e. constant circulation right up to a hole of diameter equal to about the interatomic spacing), and the true conditions can be obtained only from a much more complete quantum mechanical treatment than that given in 5 2 (see 5 8). Any such complete quantum mechanical treatment will almost certainly prove to be very difficult, and it is therefore important that as much information as possible about the conditions near the centre of a line should be obtained from experiment. Now there are three experimentally measurable quantities that reflect these conditions : (a) the roton-vortex scattering cross section"; (b) the energy per unit length of a stationary line; and (c) the parameter v appearing in the dispersion equation for vortex waves. The first of these is probably of little value at present, since the scattering process is probably very complicated (involving the excitation of vortex waves) and too little is known about the roton. We are left therefore with the energy and the parameter v, which are in fact closely related, and the present section will be devoted to a discussion of the extent of our knowledge of these quantities. For the simplest form of vortex line (constant circulation x outside
* Certain other vortex scattering processes might also be observable, e.g. the scattering of phonons**, He3 atoms, and charged He4 ions. However, none of these has yet been studied experimcntally, and some of them, if not all, are likely to involve formidable experimental difficulties. Rcfercnces p . 56
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a holc of radius a, with a surface energy y p x 2 / 4 n )the total energy per unit length of the stationary line is given by (see 3 2)
and the parameter v by (see $ 3 . 3 ) 1z
Y =
I _
4n
(In b/a,
+ y)
(29)
(the parameter y was ignored in $ 3 . 3 , but probably it should be included), where b is a calculable macroscopic length, the value of which depends on the precise experimental conditions. According to simple arguments given in 3 2 , a, m 0.5 A and y m 1. For a real vortex line it is convenient to define effective radii u o Eand a,,, by means of eqs. (28) and (29) respectively (with y = l),where E and Y now have their real values, obtained either from experiment or from a rigorous theory. We do not assume that aoe and a,, are identical; for although, as is clear from the physical arguments of 9 3 . 3 , the product eilv may be interpreted as an energy per unit length of line, it refers to a moving line in a wave and therefore need not necessarily be equal to E , which refers to a stationary line. Furthermore, we should bear in mind the possibility that uoEand aov may vary with the precise experimental conditions. Experiments from which values of a,, may be deduced were described in $ 3 . 3 ; it was stated that under the conditions of these experiments (vortex lines in helium rotating at a uniform angular velocity of about 0.14 rad sec-l), Y is found to be equal to (9.7 f 0.2) X lod4 cm2 sec-l, so that a,, is equal to 19 f 5 A. Thus a,, is not much larger than the value expected from the simple picture, and this suggests that, although this simple picture is not quite right, it is nevertheless not too far from the truth. The direct experimental measurement of uoe is more difficult, and the only value that exists so far is one of uncertain validity deduced from observations made with the vibrating wire apparatus26.As was stated in $4, when the vibrating wire system is rotating in equilibrium a t an angular velocity w , there should be a number of quanta of circulation round the wire, together with an array of q free vortices in the surrounding helium, the numbers p and q increasing with increasing w and to be determined from a detailed calculation. This
+
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detailed calculation26 consists of course in finding the particular arrangement of vortices that minimizes the function F’ = F - M w (see 9 3 . l),and, as might be expected, the results are found t o depend on the energy per unit length of the free vortices in the helium surrounding the wire. The observable quantity that is of interest is of course p , the number of quanta of circulation in equilibrium round the wire, and the theoretical value of p , obtained from the detailed calculation, is shown as a function of a,, and the dimensionless parameter y = (bz.z/ti)win Fig. 8. Thus it is clear that experimental observations of the value of p should yield some information about uoe.
p- 1
p. 2
I 1 0
20
30
Y Fig. 8. Diagram showing the predicted number p of quanta of circulation round the wire of the vibrating wire apparatus as a function of uo6 and y = (b2m/fi)o.
Unfortunately, any straightforward deduction of this type has not been possible owing to the difficulty mentioned in 4 of achieving equilibrium in the rotating apparatus. The best that can be done is to cool the rotating system from above the I-point ; this certainly creates a rotating superfluid, but the state produced cannot correspond in general to strict equilibrium, since it usually involves a circulation round the wire that is neither an integral multiple of hjm nor strictly reproducible. And no method (e.g. prolonged rotation, shaking, etc.) has been discovered that appears to produce any improvement in the equilibrium. Thus all that has been possible so far is to take measurements of circulation in the rotating state (obtained by cooling the rotating apparatus from above the I-point) on a large number of Kcfuenccs p . 56
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occasions, and then to see whether any information can be deduced from the average behaviour. This average behaviour is shown in Fig. 9, where the average apparent circulation in the rotating state is plotted against the parameter y. In discussing these results, it is convenient to divide them according as they relate to values of y less than or greater than 15; for it is only for values of y that exceed 15 that the predictions displayed in Fig. 8 are a t all dependent on the value of aoE. Now it can be seen that in the range of values of y up to 15 the experimental points for the average circulation do follow the predicted
0
1
I
10
20
Y
I
I
x)
40
Fig. 9. Apparent average circulation (ph/ni) plotted against y for the rotating statc in the vibrating wire apparatus. The theoretical variation for aOE= cm is shown by the broken line.
equilibrium behaviour to some extent, in that there is a steep rise from values close to zero to values near to one quantum when y is roughIy 5. However, the rise takes place to a value that is significantly larger than unity, and it is clear therefore that in this range of low angular velocities there is a tendency for the observed circulation to exceed its equilibrium value. This is a satisfying observation, since it is the one to be expected; for there are two effects that might lead to non-equilibrium values of p , and both may be shown to lead to values that are probably too large. The first is connected with the fact that cooling of the helium round the wire probably takes place from the outside, so that as the liquid passes through the kpoint a boundary separating helium I1 from helium I probably moves inwards towards the wire; and the second is connected with the idea mentioned in 9 2 that a vortex line close to the ].-point may have a markedly lower energy than it does at a lower temperature. h'ifermcts p. 56
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When y lies between 15 and 35, the observed average circulations are still in the range from one to two quanta. Thus, if we assume, as seems reasonable, that these average values are still too large for equilibrium, we must conclude that the equilibrium value of the circulation in this range is still equal to one quantum. I t appears therefore from Fig. 8 that under the conditions of these experiments the value of aoEmust exceed the maximum value for which calculations have been carried out, i.e. ciii. (Calculations have not been carried out for values of a,,, exceeding cm or for values of y exceeding 35, owing to the fact that they then become prohibitively laborious.) It must be emphasised of course that the present argument is not rigorous, since it is quite possible that some unknown factor contributing to the lack of equilibrium could act differently in the two ranges of angular velocity. Thus the present estimate of the value of aoEmust be regarded as extremely tentative. The fact that our estimates of aovand aoFare found to be unequal is, as we have seen, not unexpected, but it is surprising that they should be so different. And it must be admitted that the existence of this large difference casts additional doubt on the validity of the estimate of aoe. However, this estimate is believed to be sufficiently reliable to make worthwhile at least some re-examination of the basic theorygiven in 9 2 , and we shall return to this point in 5 8. If the estimate of aoEproves to be correct, there arises the problem of deciding whether the discrepancy is due to a fundamental difference between ao, and a,, or to a difference in the conditions (e.g. angular velocity) under which the two determinations were made, and to settle this point it may be necessary t o carry out further experiments under a wide variety of conditions. Another and more satisfactory method of measuring a,, exists in principle, and attempts to use it in practice are now being made by D. V. Osborne (private communication). It is t o measure directly the “tension” in a stationary vortex line; this tension, equal to the integrated pressure drop over a cross section cutting the line, may easily be shown to be equal t o the energy per unit length and therefore to be related to aoE.The apparatus being used by Osborne consists essentially of two rough horizontal discs placed one above the other in a vessel of helium, the lower disc being fixed to the vessel and the upper disc being suspended from a fibre; and it is arranged that the whole assembly can be rotated with constant angular velocity. The experiRefcrcnces
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ment consists in measuring the steady torque required to twist the upper disc through a small angle when the system is rotating. Provided the vortex lines in the rotating helium do not slip on the disc surfaces, this torque will be directly related to the tension in these lines.
6. The Hydrodynamic Stability of Ideal Superfluid Flow So far our discussion has been concerned almost exclusively with the vortex lines present in helium that is in a state of essentially uniform rotation. Thus we have still to discuss how vortex lines can be created, how this creation process can lead to an explanation of the circunistances in which ideal potential flow of the superfluid breaks down, and how the detailed form of the dissipative processes that accompany this breakdown may be understood. The number of practical situations where these problems arise is bewilderingly large, and no attempt can be made to discuss them all, especially as some are extremely complicated (for example, the periodic boundary layer experiments revicwcd recently by Donnelly and Hollis-Hnllett 2 9 , and even the process by which helium is accelerated and decelerated in a rotating bucket). We shall therefore confine our attention t o one of the simpler cases: the breakdown of ideal superfluid flow in a straight channel. The principles involved in this case are probably of general applicability, and, with some elaboration, are probably sufficient for an understanding of the more complicated cases. Experiments show that when the superfluid flows steadily through a straight channel it can do so without appreciable frictional dissipation provided that its velocity does not exceed a certain critical value, but that as soon as the velocity does exceed this critical value complicated nonlinear frictional forces begin to appear. As we shall show, it now seems very likely that this breakdown of ideal superfluid flow is due to the creation of quantized vortex lines (or possibly vortex sheets); these lines stretch and spread out into a tangled array, so that the superfluid becomes in effect turbulent, and the frictional forces arise from this turbulence. In this section we shall be concerned largely with the initial breakdown process, and the turbulence will be considered in the next section. The principal observed characteristics of the breakdown process are as follows. First, the critical superfluid velocity decreases steadily with increasing channel width; its value in the film (width about 3 x lopo cni) is about 50 cm sec-l, and this falls steadily to a very small value Ktfcrcncrs p . 56
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(probably less than 3 x cm sec-l) as the channel width is increased to a few millimetres. (For a recent review of critical velocity data see Secondly, the velocity that is relevant seems to be v, and not, ~,~~. for example, (vn - v,), at least in the narrower c h a n n e l ~ ~Thirdly, there can in some circumstances be marked hysteresis effects, in that the velocity at which appreciable frictional forces appear as the flow rate is gradually increased may be considerably greater than that a t which they disappear when the flow rate is then d e ~ r e a s e d ~The ~,~~. extent of the hysteresis depends on the previous history of the helium, on the smoothness of the channel, and on the amount of vibration present ; the hysteresis is small if the superfluid has recently been taken above its critical velocity, and it tends to be large only if the channel is very smooth and free from vibration. And finally, the breakdown tends, a t least in some Circumstances, to take place first either a t the ends of the channel, where roughness or some other form of protuberance is likely to be present, or at some point within the channel where there definitely is some roughness 34. These observations suggest very strongly that the initial breakdown of ideal superfluid flow is due to an instability of a type similar to that leading t o turbulence in an ordinary liquid, and this view is supported by the results of experiments to be described in the next section on the behaviour of the frictional forces that build up after the critical velocity is exceeded. I t is often assumed that the forces opposing the flow of the superfluid below its critical velocity are not merely negligible but are accurately zero. I t is clearly difficult to decidc this point experimentally. It seems probable that the superfluid can flow without friction (perhaps the clearest evidence is provided by the observation of persistent currents in the vibrating wire experiment described in $ 4), but it does not follow that the flow is invariably frictionless below observed critical velocities. Indeed there is good evidence that in some cases it is not frictionless (see refs. 35 and 36, and 9 7 ) , and the possibility cannot be completely excluded that all, or many, of the observed critical velocities are of this “non-ideal” type. However, it will be assumed for the present that “ideal” critical velocities do exist, and that most observed critical velocities arc in fact of this type; and the remaining discussion of the present section udl be concerned only with this type. An example of a non-ideal critical velocity will be discussed in the next section. An explanation of the existence of ideal superfluidity was of course K e f u e n c e s p . 56
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included in the original Landau theory1, but, as indicated in 9: 1, this explanation was in one respect unsatisfactory, in that it yielded far too high a critical velocity. We shall now show that this defect can be removed (at least to some extent) if we allow for the possibility of creating quantized vortices. The essence of the Landau argument is as follows. Suppose that superfluid (or helium a t absolute zero) flows through a tube a t a velocity v,. As a result of interaction with the walls of the tube, two types of process could in principle take place: (i) the liquid might slow down as a whole, the velocity remaining irrotational: or (ii) some form of localised motion, or “excitation”, might be generated. The first process is, however, very improbable, so that we need consider only the second. The second will take place only it the liquid can thereby lose energy, and this condition can easily be shown to be satisfied only if 7),
> .I$,
(30)
where E is the energy of the “excitation” and fi is the component of its momentum parallel to u, (both referred to coordinates moving with the liquid). Thus a sufficient condition for the existence of ideal superfluid flow a t velocities less than a critical value u, is that there should be no possible “excitation” with E l f < ZI,. I n his original theory, Landau believed that the only excitations possible were the phonons and rotons; for these the minimum value of E/P is about 6 x lo4 cm sec-1, which is of course much larger than the observed critical velocities. However, we now know that quantized vortices can exist in helium, and these provide other types of localised “excitation”. Thus we might consider excitations in the form of vortex rings of radius 6. It can be shown3’ that such rings have kinetic energy and momentum given by
and
(provided the ring is not too close to a solid boundary), so that they will have associated with them a critical velocity Rrftretzccs
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In a channel of width d , the radius b will have a maximum value of order d, so that the minimum critical velocity will be given roughly by
And it is clear that a similar result is likely to apply to the generation of vortex line of more general shape. This minimum critical velocity agrees roughly with the experimentally observed critical velocities in both its order of magnitude and its dependence on channel width, and it therefore seems likely that the observed breakdown of ideal superfluid flow is indeed due to the creation of quantized vortices. I n practice these vortices will presumably be farmed in a rather irregular way, and we guess that the final result will probably be a tangled mass of line, or, in other words, a kind of turbulence (as will be seen in the next section, a mechanism exists for stretching any small length of line into a concentrated tangled mass). This is an attractive idea, partly because, as we have seen, it accords with the observed characteristics of the critical velocity, and partly because, as we shall see in the next section, it leads to an explanation of many of the observed properties of the dissipative forces that build up under supercritical conditions. However, there are still difficulties, as is clear as soon as we examine the detailed mechanism for the initial production of the vortex line. Let us consider more carefully the example used already: the creation of a vortex ring of radius b ; similar considerations probably apply t o other configurations of vortex line, including those in which a line is partly attached to a solid boundary (the situation is a little more complicated if the ends of the line are actually s t w k to the boundary, but it can be shown that similar conclusions can still be drawn). It is implicit in the discussion so far that this creation process must involve a direct quantum-mechanical transition, induced by interaction with the wall, between a state of uniform flow and a state of flow that includes the ring. Now we know from experiment 36 that critical velocities can be as low as 3 x 10-2 cm sec-1; and we must therefore assume cm can be created. apparently that rings with radii as large as 3 x But the creation of such a large ring would involve enormous numbers of atoms at considerable distances from the wall, and it is very difficult Refcumces p . 56
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to believe that such a process could take place with any appreciable probability, even with a rough wall. Indeed it seems alniost as unlikely as the process of slowing the liquid down as a whole. I t should be emphasized that the ring cannot grow gradually from a sinall size to a large size; for the creation of a small ring involves a considerable increase in energy of the liquid, not a decrease. Thus, in a sense, the difficulty is that the creation of vortex line is opposed by a large potential barrier. The size of the largest ring (or similar configuration of line) that is at all likely to be created by a quantum mechanical transition through a perturbation applied at the wall is not easy to estimate with certaint y ; but a rough consideration of the transition probabilities involved suggests that the size cannot be very much larger than an interatomic spacing3R.Thus it is probably safe to assume that it does not exceed 1 0 P cm, which corresponds to a critical velocity (34) of about 7.5 x l o 2 cm SCC-1. In practice all experiments on critical velocities have been carried out at a finite temperature, and we must therefore enquire whether the presence of normal fluid helps in the creation of vortex line. The presence of phonons and rotons as such certainly does not; for (to use again the example of a vortex ring) it can be shown easily that a small ring will move rapidly (under its own velocity field) relative to the normal fluid in such a direction that the mutual friction force acting on it will, through the Magnus effect, causc it to contract. Howevcr, there exists thc possibility that large lengths of vortex line arc occasionally produced by the interaction of phonons and rotons; i.e. that lengths of vortex may occasionally be present in thermal equilibrium (these lengths would act as the basis for the growth of more line). This possibility was mentioned in $ 2 , and, as indicated there, no calculations on it have yet been carried out. Although, as we shall see in the next section, therc is some slight evidence that this effect might be important, it does on the whole seem unlikely that it really is important. In spite of these difficulties, there is considerable evidence, as will be shown particularly in the next section, that the breakdown of idtxl supcdluid flow does lead in fact to the creation of vortex line, and some detailed mechanism for the creation of this linc must therefore bc found. There seem to be four possible ways out of the difficulty, ancl wc consider these in turn. (a) The first and no st obvious possibility is that apparently unI21.fcreitr.cs p . hG
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disturbed helium does always contain lengths of vortex line, but not in thermal equilibrium, and that these lengths can grow as soon as the superfluid velocity exceeds a certain critical value (growth mechanisms are discussed in 3 7). As was explained in 5 4, observations made with the vibrating wire apparatus have shown that such lengths of line can be present, and it was suggested that they are held in metastable equilibrium by having their ends tied to protuberances in the apparatus. (b) The second possibility is that a protuberance can itself cause the creation of a length of vortex line. Suppose superfluid is flowing past a protuberance with a sharp edge. Near this sharp edge the superfluid velocity will be very large, so that it might be possible to satisfy the condition (30) for a ring (or other similar configuration) of vortex line that is small enough to be created by direct quantum mechanical transition (or thermal excitation). Thus a small length of line might be created close to the edge of the protuberance and then be pulled away from the protuberance by the main flow. Detailed examination of this idea38for various idealised shapes of protuberance suggests that the process is probably possible, but only at rather large velocities. Thus, if the protuberance has the form of a knife edge of height H placed perpendicular to the flow, and if the smallest ring or other shape of line that can be created by direct transition has linear dimension 6, then the critical velocity is probably of order (?i/m)(2/H8)+In 6/a,; and if, say, H = cm and 6 = cm this equals about 30 cm sec-l. (c) Thirdly there exists the possibility of creating vortex sheet 4, ', which can subsequently break up into vortex lines (classical vortex sheets tend to break up in this way, owing to instability against any undulatory motion, and presumably the same tendency will exist in helium, although it will then be opposed to some extent by the surface tension in the sheet). The idea here is that a region of stationary fluid is formed, separated from the moving fluid by the vortex sheet. The critical velocity for a region with linear dimension of order d is given by
where o is the surface tension of the sheet (S 2 ) . For a low critical velocity the region must therefore be very large, and thus the same difficulties arise as in the case of vortex line formation. However, there exists again the possibility that a protuberance can help, and indeed it RCJ~ICII.CES p . 56
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turns out that the critical velocity for creation of vortex sheet in this way is probably about the same as that for vortex line. Thus for the knife edge protuberance of height H the critical velocity is probably of order ( 2 u / H p , ) +(independent of the size of sheet formed); and if H = cm this is again equal to about 30 cm sec-l (it might, however, be smaller if, as is quite possible, the estimate of G given in 9 2 is too high). (d) Finally the possibility ought to be mentioned that, if the conditions near the centre of a vortex line differ appreciably from those in the simplest picture, then the energy of a very small ring may be appreciably less than that given by eq. (31) ; and vortex line formation would then be correspondingly easier by any of the processes that we have been considering *. Nothing is known about the fourth possibility, and we shall therefore for the present ignore it. Wenote that all the others are based directly on the existence of protuberances in the flow channel, and this is satisfactory since, as we have seen, there is experimental evidence that protuberances might be important. (Unfortunately, however, most of the evidence for this comes from certain heat flow experim e n t ~ ~in ~ which, $ ~ as ~ we . ~shall ~ see in 9 7, the critical velocity may not have been of the ideal type; thus the possibility exists that the effectsobserved are quite unrelated to the mechanisms being considered here.) Process (a) seems to be the most likely one a t low velocities; process (b) or process (c) at the higher velocities. It should perhaps be emphasised that if ideal superflow does break down by one of these mechanisms the critical velocity will depend on different and more complicated considerations than those used t o obtain eq. (34) ; and, as will be seen in the next section, the same is likely to apply to nonideal critical velocities. Thus the fact that eq. (34) does sometimes appear to be obeyed in practice may after all be to extent fortuitous. Furthermore, the critical velocity will depend on the precise experimental conditions, and this may provide an explanation of the fact that the measurements of critical velocities by different authors in channels of the same width have sometimes yielded widely differing results.
* In the special case of t h e film, still another mechanism exists: the formation of quantized surface wavcs or “ripplons”48. References
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7. Superfluid Turbulence In this section we shall describe some of the characteristics of the dissipative processes that are found to follow the breakdown of ideal superfluid flow, and show that they are consistent with the idea that these processes are associated with the presence in the superfluid of a tangled mass of vortex line, i.e. a kind of turbulence. I n the preceding section we confined our attention to flow in a straight channel; in this section we must confine our attention still further, and we shall in fact consider only the case of a channel that is fairly wide ( >10-2 cm) in which the flow is a counterflow of the two fluids due to a heat current. This case has the virtue that it is probably one of the most simple, and also that it is the one that has been studied most extensively; and it will certainly serve to illustrate the general principles. We begin with a survey of the experimental observations and the qualitative interpretation of them in terms of superfluid turbulence ; and then we indicate briefly the extent to which it is possible to build a quantitative theory. It turns out that this theory can be developed to a large extent without knowledge of the processes discussed in the preceding section. The reason for this is that to a considerable extent these processes appear merely to nucleate the turbulence ; once a small length of line has been produced, turbulence can be built up and maintained simply by the stretching of existing line. Thus, in the basic eq. (38), only the second term on the right hand side depends on these processes, and this term is in fact unimportant in many circumstances. The type of apparatus used for heat flow experiments is shown in Fig. 10. A known power is generated in the heater H; the thermometers T I and T, record the temperature gradient along the channel C ; and the manometer M records the pressure gradient. The channel may be of either circular or rectangular cross section, and it may be constructed of either metal or glass. If it is desired to study the propagation of second sound in the heat current (see below), the channel may be made into a second sound resonator; for example40, it may be made with a rectangular cross section so that a resonance may be established across the heat current (Fig. 10a), the second sound being generated by a wire heater h running along the length of one side of the channel and detected with phosphor bronze resistance thermometer wires (t) running along the opposite side of the channel. Consider first the observations that have been made on the regime References
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44 W. I;. V I X E N [CH. I, 9 7 obtaining in a heat current in the steady state. Measurements of thc temperature gradient required t o maintain the heat current 33,30, 40 and of the attenuation of second sound in the heat current40 show that the most important force opposing the flow is an essentially linear isotropic mutual friction Fsn = ('8
- vn)
(36)
per unit volume, where in the steady statc thc factor G' takes the form ~
G = Aesen(l v s - v n I - 210)'. (37) A is of order 50 cm sec g-l, increases with increasing temperature, but is approximately independent of channel width and of the nature of
' I Fig. 10. Type of apparatus used for heat flow experiments in lirlium I1 (schematic only). V: vacuum space. (a) Cross-section of channel C (enlarged) whcn used as second suund resonator.
the channel surface (except in the narrower channels) ; (vs - v,) is the instantaneous relative velocity between the two fluids; (vs - v,,) is the mean steady rclativc velocity duc to the steady heat current (these two velocities will not be equal when second sound is superimposed on the heat current); and TI" is a smaIl quantity (less than the velocity corresponding to the critical heat current), which varies sonicHhat with tcmperature and tends to decrease with increasing channel width. Qualitatively, this mutual friction can obviously be 1icfcvr~nr.e.~ p . 56
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accounted for in terms of vortex lines in a turbulent superfluid, the amount of vortex line being determined, as one might expect, by the steady heat current independently of any second sound, and the force will be of the same fundamental type as that observed in uniformly rotating helium (§ 3 . 2 ) . The observed dependence of G on ( v s - v,) shows that the length of vortex line per unit volume of superfluid must increase approximately as the square of the steady heat current; the fact that A is approximately independent of both the channel width and the nature of the channel surface suggests that the turbulence is approximately homogeneous and is maintained by the steady relative motion of the two fluids and not by motion relative to the walls; and the isotropy of the mutual friction suggests that the turbulence is also isotropic. At one time it was believed that the mutual friction is the only force acting in a steady heat current, apart from forces due to the normal fluid viscosity, but recent pressure gradient measurem e n t ~have ~ ~ shown - ~ ~that this is probably not true. If it were true, then, as is easily shown42,the pressure gradient along the flow would be equal simply to the value required to maintain ordinary Poiseuille flow of the normal fluid through the channel * ; but it is found that the pressure gradient is appreciably greater than this value. (Analogous effects have been observed in other experiments ; see, e.g., refs. 43, 44.45. ) The existence of this excess pressure gradient shows that there must be some mechanism for transferring momentum within the superfluid. Such a mechanism does indeed exist in a turbulent fluid, owing simply to the fact that there is a continuous transfer of actual matter across the main flow. The turbulent flow behaves to some extent like a laminar flow, but with an extra effective viscosity, which is usually termed an “eddy viscosity”. Values of the eddy viscosity calculated from the observed excess pressure gradient vary somewhat with the experimental conditions and with the assumptions made in the calculation (particularly with the assumed boundary condition for the superfluid at the channel walls), but are usually of order 10-100 ,UP. It should be noted of course that since the turbulence is homogeneous the concept of a well-defined eddy viscosity is perfectly valid. As already indicated, the mutual friction (37) does not hold in small heat currents, and it has been found that the results then depend on * I t is assumcd here that the normal fluid does not become turbulent, and this seems a reasonable assumption, except perhaps for very wide channels or very large heat currents. References p . 66
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the channel width and perhaps also on the nature of the channel surface. Two sets of detailed experiments have been carried out, and, since the conditions in the two were rather different, it is convenient to discuss them separately. In the first, the “Oxford” experim e n t 33* ~ 34, ~ 44, ~ rather ~ narrow channels (<2 mm) were used and the channel surface was probably very smooth; in the second, the “Cam-
Flg. 11. Form of mutual Iriction observcd a t low velocities in the Cambridge experiments: curves of G1 plotted against (us-vn). (Part of these curves were obtained, as t-xplainecl in the text, by using the dependence of the delay time t on the presence initially of a small amount of turbulence.) (a) channel cross section (rectangular) 3.40 mm x 6.45 m m ; 1.4 ’I<. (b) same channel; 1.6 ’I<. ( c ) clianiiel cross section 4.00mm x 7.83 mm; 1.4‘I<.
bridge” 46, rather wider channels were used ( > 2 mm), and the channel surface was probably rather rough. I n the Cambridge experiments it was found that as the heat current was reduced to a small value the magnitude of the mutual friction fell niarkedly below that given by eq. (37) ; in the narrower of the channels used and at low temperatures, the fall took place discontinuously at n critical velocity, but in the wider channels and at higher temperatures the fall was more gradual (see Fig. 11). Furthermore, there was inh’@tL’IICCS
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direct evidence (see below) that even when there was a sharp critical velocity the mutual friction did not fall completely to zero, so that the critical velocity could not have been of the ideal type discussed in the preceding section. In the Oxford experiments there was again a critical velocity, which was, however, always sharply defined; this is of course perfectly consistent with the Cambridge observations, since the Oxford channels were narrower. Unfortunately, however, it is not clear whether there was any small sub-critical mutual friction in the Oxford experiments, so that the possibility exists that the Oxford critical velocity was of the ideal type and fundamentally different from the Cambridge one. Both sets of experiments indicate a critical value of (us - v,,) that varies from about 10 cm sec-1 in a channel of cm to about 0.5 cm sec-l in a channel of width 4 mm. width Observations have also been made on the manner in which the mutual friction builds up and decays when the heat current is switched on and off. In the Cambridge experiments both build-up and decay were observed; in the Oxford experiments only the build-up (second sound must be used to observe the decay, and no second sound measurements were made in Oxford). In the Oxford experiments the build-up was dominated by an effect mentioned in $ 6, namely that the frictionless (or nearly frictionless) flow tended to break down initially only at the ends of the channel, so that all that was observed was a gradual spread of mutual friction along the tube. This spread took place at a rate that was not always the same, but for a given heat current and given temperature the rate tended (at least in some circumstances) to be an integral multiple of a certain minimum value, this minimum value increasing with increasing heat current. There was also a marked hysteresis effect of the type mentioned in g 6, and the extent of the hysteresis depended markedly on the previous history of the helium and on the amount of vibration present. In the Cambridge experiments the frictionless flow did not appear to breakdown at any particular place, and the mutual friction appeared as a rule to grow in a homogeneous manner throughout the channel *. A typical example of the way in which G (eq. (36)) increased with time is shown in Fig. l2(a). This growth can be conveniently characterized by a delay time z (defined as the time required for G to rise half way to its final value), and this time was found to be of the * It seems reasonable to assumc that this was t h e case, although the evidciice was not conclusive. IZc,ftrcnccs p. 56
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order of one second, to vary with heat current as W-3/2(except close to the critical heat current where it tended rapidly to infinity), and to be independent of channel width (again except near the critical heat current). Furthermore, t also depended on the previous history of the liclium, in that it was appreciably reduced if the heat currcnt had been taken recently above the critical value ; but no hysteresis effects were observed. The manner in which mutual friction was observed to decay in the Cambridge cxperinients when the heat current was switched off
a __.I
0
2
4
6
I
s! u
1;ig. 12. Examples of the form of the growth (a) and decay (b) of mutual friction observed in the Cambridge experiments. The growth shown is for a heat current of 0.078 W cm-2; the decay is for an initial heat current of 0.139 W cm-* and for channel widths of about 2.4 mm (broken line) and 4.0 mm (full line). Temperature 1.4" I<. (The final stages of the decay were studied, as explained in the tcxt, by using t h e dependence of thc delay time T on the presence initially of a small amount of turbulence.)
is shown in Fig. 12(b); it can be seen that during most of the decay period 1/G varies lineary with time, independently of channel width, but that there are anomalous regions at the beginning and end of the decay which do depend on channel width. It is clear that in these experiments one must probably be watching the growth and decay of turbulence, and they therefore provide good evidence that the Gorter-Mellink force is indeed due to vortex lines in a turbulent superfluid. Perhaps the most striking evidence is the observed dependence of both the hysteresis in the Oxford experiments and the delay time t in the Cambridge experiments on previous history, for its is difficult to see any explanation of this other than in terms of a n accelerated growth of turbulence when some turbulence is already Rcfcvences
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present. (In the Cambridge experiments the sensitiveness of z to small amounts of turbulence present initially was used to detect, and roughly to measure, the turbulence present at subcritical velocities and in the final stage of the decay; see refs. 36, 46, 4 7 for details.) The reason for the difference between the Oxford and Cambridge results is not yet entirely clear. Zf the Oxford critical velocity was of the ideal type, then the difference must arise because the initial creation of vortex line was more difficult in the Oxford channels than in the Cambridge channels, and, as we saw in 9 6, this is likely to be due to a difference in surface conditions; but if it is allowed that the Oxford critical velocity may be of the non-ideal type (like the Cambridge one) then there exists a quite different explanation, as we shall see later. The development of any quantitative theory of superfluid turbulence presents extremely difficult problems, probably as difficult as those encountered in classical turbulence theory. However, it seems that some progress can be made provided one is content to rely on a combination of rough physical arguments, dimensional analysis, analogues with classical turbulence, and some appeal to experiment. The case of a heat current in a wide channel, to which we are confining our attention, is fortunately one of the simpler ones, since, as we have seen, the turbulence is probably homogeneous and isotropic; and for this reason more progress has been possible in this than in other cases. I n summarizing the work that has been done, we shall consider first the magnitude of the eddy viscosity, and then go on to consider the processes of growth and decay of the turbulence, including in particular the level at which turbulence can be maintained by these processes in a steady heat current. The eddy viscosity in a homogeneous turbulent flow will be given to an order of magnitude by qe m Qlu,where 1 is the size of the dominant eddies and u is the root mean square turbulent velocity measured relative to the bulk motion. For a tangled array of vortex line Z cannot be less than the spacing between the lines, and u cannot be less than about nlrnl. Therefore the superfluid eddy viscosity cannot be less than about es?i/m rn 10 ,UP,and it may in fact be somewhat larger if vortex lines combine together to make “eddies” that are in effect larger than the line spacing. These conclusions agree with experiment, since, as we have seen, the observed eddy viscosity probably lies between 10 and 100 ,UP. The growth and decay of turbulence in a heat current has been References p . 56
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considered theoretically in refs. and *’. An attempt was made simply to describe the average rate of change of length of line in a channcl containing a high concentration of line (spacing between lines small compared with channel width), and the analysis was therefore restricted to wide channels and to not too small heat currents. The results m a y be summarized in the equation
where I, is the length of line per unit volume, v = (a, - v n ) , d is the channel width (or more precisely the ratio of area t o perimeter of the cross section of the channel), xl, xz, x3 are constants of order unity, and y is a parameter equal to approximately 1.1 T1’cm-: secf. The meanings of the various terms in this equation are as follows. The first term represents a build-up of turbulence through the action of the relative motion of the two fluids : the normal fluid exerts a force on the lines, which causes them to move owing to the Magnus effect, and this movement, combined with interactions between the lines, leads to a continuous stretching of the lines; the form of the term can be justified by physical argument together with dimensional analysis. This dimensional analysis is based on the reasonable assumption that there is “complete self-preservation” in the turbulence; i.e. that apart from a scale factor the geometrical arrangement of the vortex lines a t any instant is always essentially the same. The second term is purely empirical. It is introduced to take account of the mechanism by which turbulence is nucleated (i.e. it provides a small length of line which can be built up by the first term), and its form is chosen to obtain agreement between the experimental and theoretical values of the delay time t observed in the Cambridge experiments. (It is easily seen that t is given by an integral of the form
and that this integral will diverge at its lower limit unless a nucleation term is present.) We guess that this then represents the effect of one of the mechanisms for the initial breakdown of ideal superfluid flow discussed in 9 G , but it may also include other effects. The third term represents a homogeneous decay process. It can be argued that this RcJevemrs p . 66
CH. I,
5 71
VORTEX LINES I N LIQUID HELIUhf I1
51
decay proceeds through a tendency of lines t o cluster and hence for lines of opposite sense to annihilate one another (see 9 2) ; this process is in some respects analogous t o the decay of homogeneous turbulence in an ordinary liquid, and the form of this third term can be obtained by analogy to an empirical law known t o apply to ordinary liquids. However, it can also be obtained by dimensional analysis from the assumption of complete self-preservation, and this derivation has the advantage that it shows that the only effect that the normal fluid can have on the decay is to make x 2 temperature dependent. The fourth term is also to some extent empirical, in that its form was chosen in the first place to fit the experimental results; but it clearly represents the perturbing effects of the walls of the channel, and it can easily be argued36 that this effect might well take this particular form. For example, it is probable that the generation mechanism represented by the first term will be impeded in the vicinity of a wall; and if, as seems quite possible, this mechanism is effectively absent for a distance from the wall of the order of the average line spacing then a term of the required form will indeed be introduced. Comparison of eq. (38) with experiment is of course perfectly straightforward, since the instantaneous value of the factor G in the mutual friction (36) is directly related to the instantaneous value of L through the relation
(this is obtained from eq. (24) if is assumed that the vortex lines do not on the average move relative to the mean superfluid flow). It is found that the equation does give a very satisfactory description of many of the experimental results. Thus the magnitude and velocity dependence (37) of the steady Gorter-Mellink force a t high velocities (obtained by putting dLjdt = 0 ) is satisfactorily accounted for (this does not depend on the empirical second term) ; the form of the build up observed in the Cambridge experiments is correctly described, although it has t o be assumed that the last term in (38) does not act during the early stages (admittedly the second term was chosen t o give the correct values of z, but the general form of the build-up depends largely on the first and third terms, which are not purely empirical) ;and the decay of mutual friction observed in the Cambridge experiments is also satisfactorily described in both its dependence on References p . 56
52
W. F. T’INEN
[CH. 1,
97
time and in its absolute rate (the anomalies at the beginning and end are not described: that a t the heginning is probably associated with the formation of large eddies immediately after the heat current is switched off (see ref.*’); and that at the end is clearly an extra wall effect, since the point a t which it sets in is found to correspond to an average spacing between lines which is of the order of the channel width) However, it is the form of the steady mutual friction predicted by eq. (38) for fairly small velocities that is perhaps most interesting. This is shown for various temperatures and channel widths in Fig. 11. I t can be seen that for channels with d equal to about 1 mm, as used in the Cambridge experiments, a behaviour is predicted that agrees well with what is in fact observed; namely that there is a critical value of (.uh - .un) of order 1cm sec-‘ below which the mutual friction suddenly falls to a smaller but non-zero value, that this critical velocity decreases with increasing channel width, and that it becomes less distinct and eventually disappears as the channel width is increased or the temperature raised (the agreement with experiment is not quantitatively good, but this is to be expected, since the theory is rather crude, particularly at velocities close to the critical value where the spacing between vortex lines is no longer very small compared with the channel width). Thus we have found a mechanism for producing a “non-ideal” critical velocity with just the right characteristics. It is clear that there might be some hysteresis associated with this critical velocity (indicated by the arrows and broken lines in Fig. 13), but for the channels of the width used in the Cambridge experiments it would be very small and might well have been missed. However, for smaller channels, such as those used in the Oxford experiments, there should be a much larger hysteresis effect (Fig. 13(c)).Thus we have another possible explanation of the difference between the Oxford and Cambridge results, which, if true, would imply that the critical velocity observed in the Oxford experiments is not of the ideal type, and that the onset of an observable mutual friction was due, as in the Cambridge experiments, to a transition from a state of weak turbulence to a state of stronger turbulence (this transition might well take place first a t the ends of the channel and then propagate along the channel, as is observed). This is to be contrasted with the situation cnvisaged in the alternative explanation offered earlier : that the large hysteresis in the Oxford experiments was due to a difficulty in creating any vortex line a t all (i.e. to the absence of any source term like p*), with the consequence that the critical I
Irefrrcllces p. 56
CH. I,
3 71
VORTEX LINES I N LIQUID HELIUM I1
53
velocity would be of the ideal type. Which of the two explanations is correct is not yet known, and further experiments are required. Furthermore, the process by which turbulence propagates down a channel is not yet fully understood, and, in particular, no explanation has been given of the fact, mentioned earlier, that in some circumstances this
..* rwr
20
40
60
0.- u ) (cm sec')
80
Fig. 13. Form of mutual friction predicted by rq. (38) for small velocities: curves of G * plotted against (u,-un). (a) curve A, d = 0.088 cm, 1.4 OK; curve B, d = 0.133 cm, 1.4 O K . (b) curve A , d = 0.088 cm, 1.6 " K ; curve B, d = 0.088 cm, 1.8 "K. (c) curve A, d = 0.01 cm, 1.4 "K; curve E, d = 0.01 cm, 1.8 'I<.
process can take place at various rates that are integral multiples of a certain minimum value. Some comment should be added on the significance of the source term yv'. The fact that this term has to be assumed to be independent of channel width (at least to a first approximation) and to be strongly temperature dependent, suggests that the mechanism for the nucleation of the turbulence is independent of the walls and is essentially thermal in origin, involving, for example, the thermal excitation of vortex rings References
p . 56
54 W.F. VINEN [CH. 1, 3 8 (the possibility of which was mentioned in 9 6). However, this kind of process does seem theoretically unlikely, so that perhaps the nucleation mechanism is, after all, one of those based on protuberances (probably (a)), and the particular form of the source term may then be a consequence of processes, not yet understood, in the early growth of the turbulence rather than in its nucleation. The type of approach used here to account for the properties of the Gurter-Mellink friction has also been used by Hall to disciiss his observations on the angular acceleration of helium contained in a disc-filled can. Again there was clear evidence for superfluid turbulence, and although the flow is of a more complicated type than that occurring in a heat current, Hall was able to give a satisfactory interpretation of many of his observations. I t should perhaps be added that the turbulence accompanying angular acceleration appears to have a somewhat different character from that observed in a heat current, so that eq. (38) cannot be taken over unchanged. The success of the heat flow analysis and of Hall’s analysis of the angular acceleration process gives one extra confidence that the ideas on which these analyses are based are correct ; i.e. that the breakdown of ideal superflow leads to a kind of turbulence in the superfluid, in the €orin of a tangled mass of vortex line. Of course, detailed theories remain to be worked out, particularly in reIation to the nucleation process, as also do theories of more general applicability (to cases, for example, where the concentration of vortex line is not high or where turbulence in the normal fluid is important). But there seems at present to be every reason to hope that, except possibly in the case of the nucleation process, the fundamental physical principles involved are understood, at least in essence, and furthermore that the same principles apply to types of flow other than those considered explicitly in the present discussion. 8. The Validity of the Concept of a Simple Vortex Line IVe conclude this chapter with some comments on the extent to which the idealized picture of a vortex line, in which the velocity of flow varies exactly as I / r right up to the centre of the line, is likely to accord with reality. The experimental evidence on thc energy of a vortex line presented in 9 5 certainly suggested that it might represent an oversimplification, and we shall now show that there are theoretical grounds for supposing that this is indeed the case. XrJumcL
p. 56
CH. I,
S 81
VORTEX LINES I N LIQUID HELIUM I1
55
It is immediately obvious that our idealized picture is defective in one straightforward respect, in that we have ignored the possibility that waves on the line can be thermally excited. Wave motion on a line was considered in 4 3 . 3 ; strictly speaking the energy of this wave (where Q is the motion will be quantized, presumably in units of ?iQ frequency of the wave), and clearly the low energy quanta will be thermally excited. (The roton-line collision process probably leads to the excitation of waves; and it is interesting to notice that for drift motions parallel to the length of the vortex line these thermally excited waves may probably be regarded as belonging to the normal fluid.) This will give rise to an apparent variation in effective core radius a , with temperature (even when the temperature is not close to the I-point). However, as has been shown by Hall, the effect is very small and can for most purposes be ignored (except perhaps in the roton scattering process). However, the question then arises whether these waves have associated with them any zero point energy, equal to, say, &Q per mode. ,4t first sight it appears that they should have, but this cannot in fact be so, because it would imply that the energy of a line is greater than the energy given by the simplest vortex line wave function obtained by putting s = d in eq. (3). It is of course very reasonable that there should be some zero point motion, or uncertainty in position, of a line, but this should, if treated properly, lead to a reduction in energy, not an increase. And such a reduction is perfectly possible, because, as mentioned in Q 2 , the wave function (3) is not necessarily correct, especially in the region near the centre of a line where the velocity of flow varies rapidly with position. One is therefore led to reconsider the difficult problem of finding a better wave function for a vortex line. For this purpose it is convenient to consider the particular case of helium in equilibrium in a rotating vessel at absolute zero, since the problem can then be formulated easily in terms of a variational principle. For it can be shown that the correct wave function for this case is given rigorously by minimizing the integral Jy)*(H- M . w ) y d t , where H is the Hamiltonian of the system and M the angular momentum operator. We now know with fair certainty that this wave function must describe some kind of vortex line system, and we guess that the position of the lines will be more or less uncertain. The problem is therefore reduced t o that of devising a variety of trial wave functions representing smeared out I ? < ~i cnces L p . 56
56
W. F. VIXEN
[CH. I
vortex lines and feeding them into the variational integral. Some work along this direction has been carried out recently by the present author, but it is not yet complete and no definite results have yet been obtained.
I am indebted to Dr S. M. Bhagat, Mr P. R. Critchlow, Dr H. E. Hall, and Dr D.V. Osborne for informing me of their work before publication. REFERENCES L. D. Landau, J. Phys. U.S.S.R. 5, 71 (1941); 11, 91 (1947). L. Onsager, Unpublished remark a t a Low Temperature Physics conference a t Shelter Island, 1948. H. London, Report of Physical Society Cambridge Conference 2, 48 (1946). N. F. Mott, Phil. Mag., 40, 61 (1949). L. I>. Landau and E. M. Lifshitz, Dokl. Akad. Nauk. U.S.S.R. 100, 669 (1955). I?. London, Superfluids 2, 151 (Wiley, New York, 1954). 7 V. L. Ginsburg, J. Exp. Theor. Phys. U.S.S.R. 29, 254 (1955). L. Onsager, Nuovo Chim. 6, Suppl. 2, 249 (1949). R. P. Feynman, Progress in Low Temperature Physics, F d . C. J. Gorter, 1, ch. 11. p. 36 (North-Holland Publishing Co., Amsterdam, 1955). lo R. P. Feynman and M. Cohen, Phys. Rev. 102, 1189 (1956). l1 R. 13. Dingle, Adv. in Physics (Phil. Mag. Suppl.) 1, 111 (1952). l2 V. I .. Ginsburg and L. P. Pitayevskii, J. Exp. Theor. Phys. U.S.S.R. 34, 1240 (1958); Sov. Phys. 7, 858 (1958). l 3 L. P. Pitayevskii, J . Exp. Theor. Phys. U.S.S.R. 35, 408 (1958); Sov. Phys. 8, 282 (19.59). l4 I ). V. Osborne, Proc. Phys. SOC.A 63, 909 (1950). l5 E. L. Andronikashvili and I, P. Kavakin, J. Exp. Theor. Phys., U.S.S.R. 28, 126 (1955). l6 H. E. Hall, Phil. Trans. A 250, 359 (1957). ' 1 R. H . Walmsley and C. T. Lane, Pliys. Rev. 112, 1041 (1958). l6 H. E. Hall and W. F. Vinen, Proc. Roy. SOC.A 238, 215 (1956). lS H. E. Hall and W. F. Vinen, Proc. Roy. SOC.A 238, 204 (1956). zo R. G. Wheeler, C. H. Blakewood and C. T. Lane, Phys. Rev. 99, 1667 (1955). E. 31. Lifshitz and L. P. Pitayevskii, J. Exp. Theor. Phys. U.S.S.R. 33, 535 (1957); Sov. Phys. 6, 418 (1958). 22 Sir W. Thomson, Phil. Mag. ( 5 ) 10, 155 (1880). 23 H. E . Hall, Proc. Roy. SOC.A 245, 546 (1958). H. E. Hall, Adv. in Physics (Phil. h$ag. Suppl.) 9, 89 (1960). 28 U '. F. Vinen, Nature, London 181, 1524 (1958). 26 W. F. Vinen, Proc. Roy. SOC.A, in the press. 27 1'. P. Craig and J. R. Pelham, Phys. Rev. 108, 1109 (1957). 28 L. Y.Pitayevskii, J. Exp. Theor. Phys. U.S.S.R. 35, 1271 (1958); Sov. Phys. 35(8), 888 (1959). 20 R. J , Donnelly and .4. C. Hollis-Hallett, Annals of Physics 3, 320 (1958). 30 I<. R. Atkins, Liquid Helium (Cambridge University Press, 1959). 31 C. S. Hung, R. Hunt and P. Winkel, Physica 18, 629 (1952).
CH. I] 32 33
34 35
36 37 38 39 40
41 42 43
44
45
46
47 48 19
50
51
VORTEX LINES I N LIQUID HELIUM I1
57
D. F. Brewer, D. 0. Edwards and K. Mendelssohn, Phil. Mag. (8) 1, 1130 (1966). D. F. Brewer and U. 0. Edwards, Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin (1957). I<. Mendelssohn and W. A. Stecle, Proc. Phys. Soc. A 73, 144 (1959). P. Winkel, A. Brocsc van Groenou and C. J. Gorter, Physica 21, 345 (1955). W. F. Vinen, Proc. Roy. Soc. iZ 243, 400 (1957). H. Lamb, Hydrodynamics (6th Ed.) p. 239 (Cambridge University Press, 1952). W. F. Vinen, To be published. W. H. Iieesom, B. F. Saris and L. Meyer, Physica 7, 817 (1940). W. F. Vinen, Proc. Roy. Soc. A 240, 114 (1959). P . R. Critchlow, Private communication. C. J . Gorter and J. H. Mellink, Physica 15, 285 (1949). S. M. Bhagat, Proc. Phys. Soc. A 75, 303 (1960) R. J. Dannelly and 0. Penrosc, Phys. Rev. 103, 1137 (1956). R . J. Donnelly, G. V. Chestcr, R. H. Walmsley and C. T. Lane, Phys. Rev. 102, 3 (1956). W. F. Vinen, Proc. Roy. Soc. 4 240, 128 (1957). W. F. Vinen, Proc. Roy. Soc. A 242, 498 (1957). C. G. Kuper, Physica 22, 1291 (1956). E. L. Andronikashvili and L). S.Tsakadzc, J. Exp. Theor. Phys. U.S.S.R. 37, 322 (1959). E. L. Andronikashvili and D. S. Tsakadzr, J. Exp. Theor. Phys. U.S.S.R. 37, 563 (1959). Y u . G. Mamaladzc and S. G . Jlatinyan, J. Exp. Theor. Phys. U.S.S.R. 38, 184 (1960)
C H A P T E R I1
HELIUM I O N S I N LIQUID HELIUM I1
G. CAKEKI"
ISTITUTO DI Frsrc.4, UXIVERSITADi PADOVA, ITALY
6 8 . - 2. Ionic structures in liquid hc~lium,59. - 3. Ionic motion in liquid liclium 11, 62. - 4. The ionic mobilities, 69. - 5. Ionization and rrcombinatiun processcs, i s . - 6. Concluding remarks, $8.
CONTENTS:1. Introdiiction,
1. Introduction
Helium ions in liquid helium are of interest not only as forming a rather unusual ionic solution, but also as a new tool for investigating the properties of liquid helium. Since the size and mass of an ion are comparable to those of the elementary excitations of the liquid, and since they are easy to control and to detect, they can be expected to yield valuable information on a microscopic scale. Helium ions were first produced in liquid helium by Gerritsenl in Leiden during the war, with the aim of testing the Kramersz theory of columnar recombination in LY particle tracks. In 1957, Williams3 measured the ionic mobility at very high fields in liquid helium I and 11, but found little explanation for his results. In the same year the author and coworkers* showed that ions were participating in normal fluid flow as had been suggested in 1948 by Landau and Pomeranchukj on general theoretical grounds. This paper concerned evidence suggesting the possibility of the ion method in the investigation of the problems of liquid helium, and was followed the next year by the careful measurement of the mobility at low fields by Meyer and Reif G, which confirmed that ions were indeed scattered by the normal fluid. During the last three years considerable work has been done in this field, and the review of this work will form the body of this paper. * Present permanent address. Istitiito di Fisica T~niversitid i K ~ J I ~ XItal\. I,
CH. 11,
s '21
HELIUM IONS I N LIQUID HELIUM 11
59
2. Ionic Structures in Liquid Helium 2 . 1 . POSSIBLE STR~JCTURES
We will first review the structures which are likely to exist and that have been suggested on intuitive grounds' and independently proposed by Atkinss. Atkins' calculations rest on a sound thermodynamic argument, but suffer from the fact that the atomic polarizability is assumed to be independent of the electrical field and the density, while actually both fields and density can achieve quite large values near an ion. These calculations should, however, give the right order of magni t ude. Tlze positive ion. It is well known that in He gas the He: ion is a stable entity, and elementary calculations indicate that in a dense state a cluster He; must easily be formed, the n atoms surrounding the positive ion being attached to it by polarization forces. These polarization forces are actually so large that the coordinated atoms will stick together a t a distance much shorter than the average atomic distance, behaving like a highly compressed solid droplet. On these grounds Atltins calculated that the mass of a cluster due to this electrostrictive effect should amount to about 50 4He mass units. We point out here that this mass excess must be considered as a static effect, and has nothing to do with any hydrodynamic effective mass which may exist in specific situations, and which may in those situations be added to the above indicated static mass. However, we are faced with another possibility. The positive hole may migrate, jumping from one atom to another in a random walk, as happens in similar situations in crystalline media. I n this case the electrostrictive effect would soon be obliterated and we would have a quite different entity, namely a charge distributed in a fairly large volume of normal density. Therefore, one has to choose between two limiting models : I" The positive ion is a solid cluster of He atoms polarized around a positive charge, which may change its site while still remaining inside the cluster. 11" The positive ion is a charge distributed over a large region with a density comparable to that of the liquid, the charge rapidly jumping from one atom to another. The negative ion. It is also well known that in He the He- ion is unstable, and He; must be loosely bound if it forms a t all. It seems Reft~rritccsp. 79
60
G. CARER1
[CH. 11,
32
unlikely that He, has any stability. The resulting structure would then be similiar to the I" structure proposed abovc for the positive ion. However, if the electron remains free, it must be expected to stay in the empty space; then the interesting possibility exists that it might remain free in an effective cage, big enough to reduce its zeropoint motion and self-trapping inside this cage by polarization forces. It seems that this possibility was first pointed out by Ferrels to explain the anomalously life of a positronium in liquid helium. In this way a positronium atom is able to avoid contact with the liquid and increase its lifetime. However, here a tunneling effect may exist, which allows the electron to leave the cage of polarized atoms and diffuse in a large region. Therefore, we are still faced with the following possibilities : I" The negative ion is in a solid cluster like the structure I" proposed for the positive ion. 11" The negative ion is a cloud of charge in a cage, self-trapped by the shell of polarized atoms. 111" The negative ion is a free electron moving in a large region, escaping by tunnelling any trapping which might occur. To these possibilities, we must add the following one, suggested by Williams3 and supported by analogies in the gas: IV" The negative ion is actually a charged impurity (probably oxygen) which can form a cluster in the same way as the structure I". 2.2.
EXPERIMENTAL BEH.~VIOUI< O F I O N I C STRUCTURES
To choose between the above outlined possibilities, an experiment of a qualitative nature was performed in PadualO. In this experiment the ions were accelerated by an electric field between the plates of a parallel plate capacitor and the resultant current was measured as a function of the applied voltage. The gap between the horizontal plates could be completely filled with liquid or solid helium, or partially filled with liquid helium, in order to have the liquid-gas boundary parallel to, and between the plates. In other words, the apparatus was essentially a diode in which the ions were produced by a layer of polonium at one plate, their sign selected by an appropriate field and collected on the other plate. Different diodes were used, and their performance was first studied in classical dielectric liquids with quite understandable results. When the apparatus was uscd in liquid helium, RtjCYeircrc
p . 79
CH. 11,
3 21
61
HELIUM IONS I N LIQUID HELIUM I1
some important new features were noticed, which may be summarized briefly as follows: a) In the two phase experiment, above the A point, the behaviour of the positive and negative ions was essentially the same. But a t a temperature slightly lower than T,, the positive ion current was dramatically reduced while the negative ion current was only slightly affected. Further lowering of the temperature gave a decrease of the negative current as well, until the currents of both signs fell to unmeasurably small values at about 1.4" K. (See Fig. 1.) b) In solid helium the total currents were unmeasurably small. Since this experiments dealt with ions of both signs simultaneously, E"-
-s0 -
-
-- ...-. ------
'.I
70
-
solid line
DuIU liquid current
dolled
I,
currant
I
,
.
T
r
2 Og°K and T i 1.40°K
in two phase 7. 209'K ,, 7 ,1 4 0 0 ~
.
I
...._ L.. ...L ___.. 4 ____._ b _____I.....+ 0
50
100
150
200
250
E
300
350
400
450
500
(*)
Fig. 1. The current versus applied field in the two phase experiment a t two different temperatures. (See text.)
this meant that the ionic current of at least one sign vanished, the total current then also vanishing, as a result of polarization. To the above quoted results, we must add the information gained in the mobility e ~ p e r i m e n t ~ pthat 6 , ~ ~the negative ion has a lower mobility than the positive ion, both below and above the 1 temperature. 2 . 3 . THE ADOPTEDSTRUCTURES
The results of the inter-phase experiment may be qualitatively understood in terms of the current picture of liquid helium. The rapid References
p . 71)
62
G. CARER1
[CH. 11,
9
:3
fall of the ionic evaporation below the 1 temperature is due to thc rapid decrease of the number of excitations, since excitations are needed to push the ions across the liquid-vapour surface. Then we deduce that the positive and negative ions must be quite different entities, and that it is easier to push the negative ion out of the surface than the positive one. We conclude that the negative ion cannot have the same or a more massive structure than the positive one, and therefore, the structure I" and IV" must be discarded for it. We stress the practical importance of discarding IV" because that means that in liquid helium the concentration of impurities is really negligible, or at least that they have no effect in these experiments with ions. Next we consider the absence of currents in the low temperature runs of the interphase experiment : this indicates that tunnelling effects are negligible. The same conclusion is gained by considering the absence of currents in the solid helium experiment. On semi-empirical grounds, we can then assign to the positive ion the solid droplet structure I" and to the negative ion the cage configuration 11" where the electron cloud is self-trapping. Furthermore, we believe the negative ion to be more extended than the positive one, in order to account for its lower mobility. I t is hoped that the future theoretical work will clarify these picture which for the present time will be adopted as reliable schematic pictures.
3. The Ionic Motion in Liquid Helium I1 3 . 1 . THE THEORETICAL VIEW-POINT
As long as one deals with subcritical velocities, that is, with nonturbulent flow, there is no doubt that the well known Landau picture of liquid helium I1 is quite satisfactory, and that the motion of impurities may be described in this frame as was first shown by Landau and Pomeranchuk5. The argument is well known since it has been applied to dilute 3He mixtures. Essentially one can say that the impurities in a dilute solution in liquid helium I1 behave like a new type of excitations, equal in number to the impurity atoms, which must be added to the phonon-roton spectrum. If the interaction between the impurities can be neglected, they behave as free particles with some appropriate effective mass nz, and the energy spectrum is simply Rcfmetices
p . 70
CH. 11,
$ 31
HELIUM IONS 11; LIQUID HELIUM I1
&=A+-
63
,tJz r
2WZ
p
being the momentum and A a constant which includes the zeropoint motion of the impurity. This point of view is shared by Feynman11, who has interpreted this effective mass as the inertial effect of the atoms surrounding the impurity, moving aside to make way for it as it moves. He was able to calculate this effective mass as 1.9 m,, m, being the true mass of the 3He impurity. The experimentally found value of m for 3He impurities lies around 24 m,. It must be emphasized that Feynman’s result is quite general and is independent of mo as long as the atomic distribution function remains the same. I n the above outlined situation the impurities will collide with the other existing excitations, and therefore will participate in the motion of the normal fluid. This has been demonstrated in the heat flush experiments in dilute 3He solutions12. It is necessary, of course, that the effective masses of the impurities be not much greater than those of excitations, so that the impurities will quickly come to equilibrium with the stream of excitations. All these conditions are, like the requirement for an extremely dilute mixture, easy t o meet in a n ionic solution, and we therefore expect the heat flush effect to be observed. The foregoing discussion is, however, limited to velocities of the impurities so low as not to produce excitations themselves during their motion. According to Landau and Pomeranchuk the lowest velocity at which this could occur in a solution of 3He is approximately the velocity of sound. Since this velocity is never reached in practice, the effect is not observed. I n ionic solutions, however, high velocities can be easily obtained by the application of strong electric fields when the mean free path is relatively long. We may expect therefore t o find such a treshold and its associated new phenomena. Finally, we stress again that the foregoing theoretical viewpoint is limited to a superfluid velocity lower than the critical one, because the collisions of impurities with the vortex lines are not considered. We will return to this effect in the next section. 3.2, EXPERIME~TS IN NON-TUREULENT FLOW
The first experimental test of the Landau-Pomeranchuk theory for an ionic solution was performed by the author and his coworkers4 and was essentially a heat flush experiment. In spite of its semi-quantitative References 9. 7.9
64
[CH. 11,
G. CAREKI
9
3
nature, that experiment gave evidence that the general ideas were correct and that the ions could indeed be used as tracers to folIow the normal fluid motion. The experiment is not described here because it has been followed by a quantitative one7 which is reported in the following. The apparatus shown in Fig. 2 is essentially an ionization chamber with several detecting electrodes. The ionic currents are measured as a function of the heat current 4 produced by a constantan wire heater at the bottom of the chamber. The total length is 60 mm and the cross sectional area of the channel about 80 mm2. The a particles intensely ,o..-
~
~
T=!.58'H Em-125 Vlcm
A g p1.t.s
Plexiglasr
Fig. 2 Fig. 2. Schematic view of the heat flush apparatus (ref.?). Fig. 3. A typical plot of the change in current in plates 2 and 4,due to the heat flush effect, (versus heat input with subcritical heating for positive ions (ref.?).
ionize the helium within a small distance from the active electrode, and an electric field is applied across the chamber drawing a small fraction of the ions of one sign from the ionized region toward the four collecting electrodes. Thc heat flow channel was formed on two sides by plexiglass walls, the other two sides of the channel being formed by the electrodes themselves. Care was taken to ensure that only a negligible amount of heat escaped through the walls of the apparatus, or along the wires and electrodes. Upon heating i t was observed from the change in current on plates 2 and 4 that the ions have a drift velocity v , in the direction of, and proportional to, the heat current density, as is shown in Fig. 3. This proportionality is quantitative evidence of the heat flush phenomenon. References
p . Y9
CH. 11, §
31
HELIUM IOBS I N LIQUID H E L I U M I1
65
A simplified calculation which assumes that vi is constant over the whole channel section and neglects the effects of space charge proceeds as follows. The angle 8, that the trajectory of the ion makes with the field will be given by
z being the displacement along the heat flow and p the mobility. This displacement will be proportional to the change in current observed on plates 2 and 4
z(4) = COi($i).
(3) The constant c can be determined from the current on plate 3 without heat input. Then from ( 2 ) and (3) we have 6
Fig. 4. Experimental values of the mobility of positive ions versus the inverse of absolute temperature. White circles from Meyer and Rcif data; crosses from Padua by space charge; black circles from Padua by heat flush. The dotted line gives the similar data from the negative ions. The experimental points are not shown for sake of clarity. References p . 79
This value of zii deduced from (-1) with the help of the mobilities measured by direct methods, must now be compared with the value of %I= drduced froin the familiar hydrodynamic equation for the entropy conservation in liquid helium 11, which now simply requires thay
4 = p.SV,T, (5) S being the entropy per gram, e the density. and T the absolute t a n peraturc. For the purpose of presentation we reverse the argument and give in Fig. 4 the mobilities derived from our experimental data, assuming the drift velocity equal to the normal fluid velocity, m d compare them with those already measured. A glance shows that indeed, the Landau-Ponieranchuk point of view has been quantitativcly checked, even better than in the mixtures where the analogoiii confirmation relies on second sound experiments l3 which have to be interpreted on the assumption of ideality of the mixtures. 3 . 3 . EXPERIMEXTS 11; TURBL~LEST FLO\V
I n this section we will report some experimental results obtained in Padua by Scaramuzzi, Thomson and McCormick, on the ionic motion in wide channels for superfluid velocities larger than the critical value. Two different channels have been used : one bcing essentially the same used in the heat flush experiment described 13m.e (see Fig. 2 ) , and a new one which is shown schematicalIy in Fig. 5 . Let us start with the apparatus of Fig. 2 and see what happens \\.hen the heat input is rather large. I t was found that after a del‘iy time T , the distribution of negative ions between the four plates suddenly changed, and that a new situation was established in the channel. The positive ions, on the other hand, were almost unaffected. -4 more careful study of this delay time for the iicgative ion currcnts shows that it approaches a very high value at a critical lieat input i*, and that for more heating the empirical law found by Vinen is satisfdctorily verified. Furthermore, the value of q* itself is in agreement with the treshold values found by Vinen14, and these tresholds appvar better defined the lo\ver the temperature. One can certainly conclude that one is observing the same kind of phenomenon now widely accepted as “turbulence”, and that the negative ions collide with the RtfC/‘JU?\
p. 7 3
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HELIUM IOXS IX LIQUID HELIUM I1
67
vortices, ~7hilethe positive ions do not. A further inspection of the distribution of the currents between the four plates, as shown by Fig. 6, gives evidence of the dramatic change which takes place for heat input larger than g*. Plate 1 should not receive current at all if the drift velocity of the ions upwards is simply the normal fluid velocity, but instead, it gets a current as large as that collected on plate 2 ! Altogetht~r,onc has tlic inipression from the many similar
Fig. 3. Schematic vicii o f tlic apparatus to detc.ct turbulcncc in differcnt points of a channc.1. Total cli;t~iuc:l Ic~ngtliis 24 cm ant1 cross scctional area is 0.15 cmz.
experiments nhich hxve been carried out, that the beam of negative charges is driven upwards much more effectively than would correspond to the simple action of the normal fluid, and that its shape is completely deformed. In other experiments, still with the same apparatus, it was found that, connecting all plates in parallel, the total negative ion current decreased sharply after q* and that this decrease was proportional to 4 - 4*. On the other hand, the total current of positive ions did not show any detectable change. I t was decided, therefore, to study the effect of turbulence simply by measuring the change in the total current of the negative ions, and a new apparatus has been used as shown in Fig. 5. It was iound that with or without heating no current was detected on the guard plates 1, 3 and 5, signifying that the decrease of the current was linked to Some phenomenon occurring in the immediate Rtfuenccs
p 79
68
L C H . 11,
s3
Ai arbitrary units
20
10
0
-10
-2c
Fig. 6. Idlcntical to Fig. 3 n i t h supercritical hcating fur rlrgalivc ions. Dotted lincs indicate extrapolation of subcritical behaviour for plates no. 2 and no. 4. 'The oblique crosscs and triangles are relative to plates 1 and 3 respcctively.
neighbourhoocl of the ion source. Next, the aypearencc of the turbulence on plates 2 and 4 was studied as a function of the hcnt input at different temperatures. This w7ork is still in progress and it may be already said that the threshold g* is the same for both plates (see Fig. 7 ) , but at higher temperatures some other complications occur. For instmco, the delay time of plate 2 is always larger than that of plate 4. This and other experimental details suggest that the turbulence arises at the bottom of the channel and propagatcs upwards. The sharp break of the Rc/lrcnces
p . 79
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41
HELIUM IONS I N LIQUID HELIUM I1
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curves of Fig. 7 should be noticed; this always occurred, and does not seem to be connected with turbulence of the normal fluid. To conclude, we give in Fig. 8 the thresholds of turbulence j * detected with the two above apparatuses, and compare them with the ones observed by Vinen. There is very satisfactory agreement. While there is no doubt that negative charges can be used merely as tracers to detect turbulence in liquid helium, one would like to understand more about what happens to them in this process. Actually it is quite conceivable that they interact with vortex lines, because the free charges tend toward empty space to reduce their zero-point energy, and might then take up a position in the cores of the vortex lines15. This is quite in agreement with our previous discussion on the nature of the ionic structures (see 2 . 3 above). According to the adopted structures, only the free electron, due to its so small mass, can substantially lower its zero-point energy by setting it self in the empty core of the vortical line, while the positive ion, being so massive, will remain essentially unaffected. This interaction term of the negative ion with the vortical line will give rise to an appreciable collision cross section, and finally to a lowering of the mobility of the charges themselves. Thislowering of the mobility can be the reason for the anomalous drag upward in the apparatus of Fig. 2, because the charges take a longer time t o travel between the electrodes, and are in the meantime dragged upward by the normal fluid. Again this lowering of the mobility can explain the decrease in the total current suffered by the beam of the negative charges, as shown in all the apparatuses. While the above picture seems selfconsistent, there is no doubt that a direct measurement of the mobility of the negative ions in rotating helium should substantiate this picture in a quantitative way, due to the possibility to evaluate now the scattering cross section by the known array of the vortical lines. 4. The Ionic Mobilities
4 . 1 . EXPERIMENTAL RESULTS
The first mobility measurements of ions in liquid helium were performed by Williams3 using very high fields. His method consisted of a careful measurement of the rise time of the current pulse through a parallel plate capacitor produced by one single CI particle from a polonium source. From this the time of flight can be deduced and the IZeferences
p. 7.9
70
[CH. 11,
5
0
I
I
I
10
15
20
I 25
I
30
I 35
s
I
9 (mW/cm’)
40
Fig. 7. Tile change i n total currcnt of the plates of thc apparatus of Fig. 5 for super critical heating.
0 0.8
I 0.9
I 1.0
I 1.1
I 1.2
I 1.3
I
I
1.4
1.5
r (*K)
Fig. 8. The critical heat input versus temperature in a cliannel of area 0.16 cm2 by different techniques. The sqnares are the Vinen results, and all other points were obtained by the ion technique.
CH. 11,
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H E L I U N IOSS I N LIQUID HELIUM I1
71
mobility calculated. The measurements were made in both helium I and I1 : in helium I the mobility was found to be roughly independent of electric field and temperature, but below the A temperature the mobility decreased nearly as E-3 with increasing fields. Furthermore, the mobility of the negative ions has always been found to be lower than that of the positive ones. Williams has not tried to interpret his results in the frame of the current picture of liquid helium. Actually it is difficult to analyze the situation at such high fields, when excitations can certainly be created and when the “temperature” of the ions is not the same as that of the bath. At about the same time the author and coworkers4 measured the mobility at low fields using the heat flush effect. The first measurements, however, were made with a strongly non-uniform electric field which had the effect of introducing a spurious voltage dependence. This effect disappeared in the succeeding experiments7 in which a plane geometry was used. The next important contribution to the topic reviewed in this paragraph, was made by Meyer and ReifG,who did a careful and direct measurement of the mobility at low fields for a wide range of temperatures, down to 1.18” K. The method used was an adaptation of the one introduced for gases by Tyndall and Powell. Here one measures the time of transit of the ions between two grids which are operated at convenient a.c. potentials and act as shutters. These results show no field dependence up to 250 volts/cm, and an exponential dependence on the inverse temperature with an “activation energy” of 8.3” K for positive ions and 7.8” K for the negative ones. Very recentlylG Meyer and Reif extended their measurements down to 0.6” K, and found results which are also shown in Fig. 4 . Work is in progress in Padua by Dupre and Modena, on the measurement of the mobility at very low temperatures, where the mean free path is sufficiently large so that the magnetic deflection method applied to gases by Townsend and well known in solid state physics as the Hall effect can be applied. In practice one can detect a deflection only if the mobility is lo2 or larger. The experiment can be carried out by an adiabatic cooling technique, at temperatures as low as 0.1” K, and the first data indeed indicate such a large mobility, but not as large as one would obtain by extrapolating the Meyer and Reif data. There is, however, some doubt concerning the interpretation of this experiment because of the small amount of 3He naturally present, which at such a low temperature may be an effective scattering agent. New References p . $9
72
G. CARER1
[CH. 11,
94
attempts with helium purified of 3He by film flow are a t present in progress. We point out that both in the shutter and magnetic deflection methods, one measures essentially the displacement of the center of gravity of an ion packet. Therefore, one gets rid of the ion-ion collisions by virtue of Newton's third law, and the results can be analyzed in tcrnis of a dilute mixture. A new method of measuring the mobility in high fields were the spreading of a beam due to the space charge is measured together with the total current of the beam is being used in Padua laboratory by Cunsolo. The results so far obtained confirm the data of Meyer and Reif down to 1-03"K and up to a field of 500 V/cm, and show no voltage dependance. To the direct methods described above, we should like t o add the indirect method based on the heat flush effect, which has been reviewed in the previous section 3 . 2 . It is worth noting that there is good agreement between the Meyer and Reif results and those obtained by different methods in Padua laboratory (see Fig. 4). 4 . 2 . GLNERALDISCUSSION
The equation of motion of a particle with charge E and effective mass m,moving under the influence of an electric field E in a medium of friction coefficient nzz is evidently
(::
m-+-v
)
=
eE.
It is well known that the steady statc solution of constant velocity is reached exponentially with a time constant z, and that this steady drift velocity vD is ezE
VD
=-
m
(7)
The mobility is then defined as
This relation may be used to obtain z / m measuring p, but in order to derive the relaxation time z, one has to know m from a different source of information. One may, for example, use cyclotron resonance, as in the analogous problems of semi-conductors. At the present time, for Referenu5 p . 79
CH.11, S 41
HELIUM IONS I N LIQUID HELIUM I1
73
ions, only the mobility has been measured and t o get information on a microscopic scale, one has to use some rather crude models. For instance, if the ionic masses are much larger than those of the excitations, a good procedure would be to treat the friction coefficient of the ion in the excitation bath as a viscous effect, using Stokes’ law with the viscosity of the medium taken as the viscosity of the excitation gas. But the viscosity of liquid helium I1 shows no sign of the striking exponential decrease with increasing temperature shown by the ionic mobility, as was realized by Meyer and Reif6, and therefore, this procedure must be discarded. On the other hand, one can use a mean free path treatment, and reasonably assume
il being a mean free path, and (c) an average velocity which can be evaluated according to Wannier17
(cl) and ( c z ) being the average velocities of carriers and scattering centers respectively. To make further progress one has to make a guess about the scattering centers. They can be the excitations as the previous analysis of the heat flush experiment has shown, or any other impurities present, including the ions themselves if their density is high enough. Let us define a “dilute” ionic mixture as one which has such a low density, n, of ions/cm3that we may disregard the ion-ion collisions in comparison with the ion-excitation ones. A good criterion for the presence of this situation is provided, according to Wannier, by the condition that the work done by closely surrounding ions in one free path is negligible if compared to mean free ion energy. That is, at ‘‘low’’fields when their energy is essentially thermal
and at high fields, when the energy obtained from the field predominates, M being the mass of the excitations
Refwences p . $9
74
[CH. 11,
G. C,ZRERI
44
It is easily seen that the above conditions are severely restrictive on IL, and that their realisation must be checked a posteriori when i,
has been determined. Suppose we now deal with a dilute mixture. \Ye can write
A=--
1
(13)
aN
N being the number density of excitations and
(T
the total cross-
section of the process, and finally
If one is interested in the “low field” condition, when the ions have the same “temperature” as the bath then (c> can be calculated from the equipartition law if the masses m and M , and the number of excitations are known. C‘nreri,Scarnmuzzi and Thoinson‘ used ey. (14) with the assumption that the scattering centers were rotons a t rest. This assumption is suggestcd by the value of the experimental slope of In ,u versus 1/T, which is close to the roton energy gap. The somewhat lower value may be due to the change in density around an ion at long range due to the polarization forces, which would make n roton easier to form there. Postulating different values for the effective masses, say 10 and 100 He masses, and supposing now that the rotons be in motion with their thermal velocity, eq. (14) gives: = 284 x cm2 and 23 x l0-ls cm2 respectively, independent of the temperature. To get an idea of the size of these values, let us compare them with the rotonroton cross section. According to Khalatnikovls this amounts t o 50 x 10-l6 T-1 cm2. While it is comforting to see that we are close t o the right figure, to make further progress we must definitely know the effective mass, and also the effect of the persistence of velocity. At larger fields the analysis is difficult because one has to introduce one more parameter, the mean fraction of kinetic energy dissipated in one collision, which determines the ion temperature, being now larger than the bath temperature. I n the limiting case of very large fields, when (c,) is essentially due to the external field and (c,) can be neglected, one obtains the familiar law p = E-t. All the previous discussion can be complicated by the occurrence Kt.ft.rciicrs
p . 79
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$ 51
HELIUM I O N S I S L I Q U I D HELIU3I I1
F'
19
of inelastic collisions, that is, the production of excitations by the too energetic ions. The situation is, therefore, in many aspects similar to the behaviour of the mobility of the hot carriers in semiconductors 19. There is no need to emphasize the extreme interest in the behaviour of hot ions in liquid helium, as an important source of information on the properties of the excitations themselves. However, so far there are few available indications of the effect of the field on the mobility except the pioneering measurements by \.liillianis, and some very recent accurate data by Meyer and ReiflGat a temperature of about 0.6" I<. Especially from these last data, it appears that a kind of mean free path approach can be useful for moderately hot ions, and that a limiting velocity of about 40 nijsec is reached at high fields. This last result, if substantiated by further experiments, will give evidence for the direct production of rotons by the hot ions, a quite remarkable feature. It is, however, not clear why a threshold has not been noticed so far at 1" K in the Padova experiments, in which velocities as large as 20 mjsec have been reached. Tlie absence of any marlwd effect of the field on the mobility at a drift velocity ;I good deal larger than the velocity threshold for production of vortical lines, is a problem which deserves much attention. As a matter of fact, one could expect the formation of quantized vortex rings behind the ion in motion, much in the same way as in the wake produced by a sphere moving a t intermediate Reynolds numbers in an ordinary fluid. It is easy to see that the critical velocity for this effect is n few meters per second, depending on the size of vortex rings that are postulated. Needless to say, some theoretical work is needed to clarify from first principles the hydrodynamics of fast moving charged particles in superfluid helium. An attempt in this direction, has been made by Arkhipov20, but that is limited to charged particles of effective mass comparable to the electron mass, while our picture requires much more massive entities.
5. Ionization and Recombination Processes h good knowledge of the elementary processes occurring during the production and decay of the ions is important in itself, and has obvious applications for the preparation of convenient ion sources. For practical reasons CL sources of polonium have been employed by all workers, and only a few attempts have been made with X rays by h'ejirencrs p .
76
G . CARER1
[CH. 11,
3
5
Gerritsen and KoolhaasZ1and with a source of promethium in Padua by Gaeta. The ionization process in a irradiated liquids and dense gases has been studied theoretically by Jaff222, and was later extended by I
p . 79
CH. 11,
9 51
77
HELIUM IONS IN LIQUID HELIUM I1
Fig. 9.) I t seems that no theory available can account for the complicated state of affairs at low fields, but one has the impression that at the limit of zero field, most of the ions come from diffusion out of the columns, while at larger fields the ions of the track core are extracted,
10‘ I
1 00
0 10
0 20
0 30
LO9,J
Fig. 9. The current versus temperature in a diode ionized by cc particles; the curves are labellcd according t o the constant value of the field, in V/cm.
and in the limit of very large fields a kind of Kramers theory should be correct. This impression is substantiated by some measurements taken in /3 irradiated liquid helium, where the ionization is certainly diffuse, and which show at all fields the same break in the curves of current versus temperature at constant field, as in the low field limit of the a case. The reason for the break in these curves, however, is not easily understood. To achieve a better understanding of the complex problem, it was decided in Padua to measure the recombination coefficient dir e ~ t l y The ~ ~ .apparatus is essentially a system of two beams of ions References p. 79
7s
G . C.\KISRI
[CH. I J ,
56
of opposite sign, the currents of which may at first be measured separately and then again when the two beams are superimposed. 'Tho changc in the total current is obviously related to the rccombiixttion in the overlapped beams. We want to emphasize that the recombination coefficient measured by us, is the true volume recombination coefficient because the range of the M particles is negligible in comparison \\it11 the distance travelled by the bcam. This experiment has not yet, for espcrimental reasons, been carried out at fields larger than n few hundredths volt/cm. So far, it appcars that the recombination coefficient in liquid helium is about one half of that predicted by the Langevin thctory 7 = 4 m ( p + . ,L). The approximate validity of the Langevin cspression is, however, very comforting, because this means that the prohlcm of clilutc ionic solutions can be approached in the same way as in the throry of dilute electrolytic solutions developed by Debye, Hiickel and Onsager. 'The'absolute value of y at 2' K is only one order of magnitude larger than the corresponding value in gaseous hcliuni at room temperature. This suggests that an assenibIy of positive and negative ions in Iiquid helium does not decay rapidly, and that collective effects could also btr observed in the laboratory. As a matter of fact, for a typical ionic solution of los ions/cnP, thc Debye shielding distance is about 10cm, and the plasma frequency about l o 7 sec-l, depending upon thc actual value of the mass. This cold "plasma" is an entirely iic'w subject and may have a promising future.
+
6. Concluding Remarks
From thc preceding discussion it is evident that ions can be used not only a5 traccrs lo detect macroscopic motions in liquid helium 11, but probably also as microscopic probc particles to get inforination on the nature of the scattering centers, namely the excitations thems e l i u . Theoretical work is, however, needed to clarify the actual semiempirical pictures and to provide a safer base for tlie possible development of this technique. The samc need of a tlieory is felt in tlie other aqpects of this work, more related to the ionic solution itself, which could provide an interrsting kind of cold plasma.
CII. 111
HELIUM IONS I N L I ~ ~ l J IHELIUM Il I1
79
The author is deeply indebted to his coworkers for the continuous discussions during the preparation of this paper, and especially thanks are due to: S. Cunsolo, 1;. Iluprk, I?. S. Gaeta, W. Mc Corniick and I;. Scaramuzzi. Further, he would like to acknowledge the cooperation of L. Meyer and F. Reif in letting him know their last results before public at ion. ’
T o t e added iiz proof (October 1960). Since the preparation of this manuscript further work has been carried out in this field, which should be quoted here and which would add some new and important informations expecially to the subject of ionic mobility. Due to the difficulty to make a good account of it here, we prefer to call the attention of thc reader on the papers which havc been submitted for publications in the Proceedings of the VIIth Int. Conference on Low Temperature Physics, Toronto 1960, section 30.
REFERESCES 1 2
3
* 9 10
l1
Lo
21 LL
23 21
-1.N. (;erritsen, I’liysica 14, 407 (1949). 13. .I. liramrrs, I’liysica 18, 665 (1952). R. L. N’illiams, Can. J . Phys. 35, 134 (1957). . Careri, J. Reuss, 14%. Scaramuzzi and J. Thomson, Proc. 5th Int. Conf. on Lf)w emp. Phys. (Madison, m’isconsin (1957) p. 79. dau and I. Ponicranchuk, Dokl. Akad. Nauk USSR 59, 669 (1948). and R. Rrif, Phys. Rev. 110, 279 (1958). F. Scaramuzzi and J. 0. Thomson, Nuovo Cinicnto 13, 186 (1950). K. R. htkins, I-’hys. Iicv. 116, 133!) (1959). R..I.Ferrel, Phys. Rrv. 108, Iti7 (1957). G , Cnrcri, 1,‘. Fasoli ant1 I;. S. Gacta, S u o v o Cimento 15, 774 (1960). li. 1’. 1:c~yiinian.l’hys. Hvv. 94, 2 6 2 (1954). T. Latlc, H. .I. Fairlxmk, I>. ’r.Aldrich and .1. 0. Xier, Pliys. Rev. 73, -756 (1948). . K. htkins, J i q u i d Hcliuni (Cambridge, 19511) p. 254. \V. 1;. \’inen, I’roc. Roy. SOC. A 240, 128 (1057); ibid. -\ 243, 400 (1937).
K.
(;. J\rkhipov, Sov. Pliys. JETP 6, 307 (1938). .L N . C;errits:en arid J. liovlliaas, Physica 10, 49 (1043). G. Jaff&, . - h i . I’liysik 42, 303 (1!)13), R. L. \Villiams a i d 1;. 1). Staccy, Can. J. P h y s 35, 92s (1937). C;. Careri and F. S . Gacta, t o be published in Nuovo Cimmto.
C H A P T E R I11
THE NATURE OF THE 1-TRANSITION IN LIQUID HELIUM
M. J. BUCKINGHAM SCHOOL OF PHYSICS, UNIVERSITY OF SYDNEY, N.S.W., AUSTRALIA and
W. M. FAIRBANK r k P A I I T M E N T OF PHYSICS, STANFORD U NI V E R S I T T , STANFORD,
CALIFORNIA CONTENTS:1. Introduction, 80. - 2. T h c spccific hcat near the A-point, 82. - 3. Thermodynamics of d-transitions, 88. - 4.Analysis of other measuremmts, 94. - 5 . The nature of the transition, 102. - 6. Separation of the singularity, 104. - 7. Analysis of the specific heat of liquid helium, 107. - 8. The supvrconducting transition, 109.
1. Introduction
One of the most interesting properties of liquid helium throughout its exciting history has been the A-transition itself. Not only is the nature of this transition of great theoretical significance, but it represents, out of all cooperative transitions, the one perhaps most accessible to precise experimental measurement. Below the temperature of the transition, liquid helium possesses a unique momentumordering and all the fascinating superfluid properties which have been the subject of such extensive study for more than twenty years. The intrinsic properties of helium conveniently permit experimental studies very close to the transition without the smearing effects -due to crystal structure defects, impurities, strains, etc. -expected for a comparable transition in a solid. However the extremely rapid change near the 1-point, in such properties as the thermal conductivity, makes it difficult to obtain equally reliable results above and below the transition. In fact one of the historical characteristics of the helium transition has been that, as measurements have been made closer and closer to the ]-point, the apparent nature of the transition has changed. In an experiment described in the next section1, the specific heat along the saturated vapour line has becn measured t o within Rt-jcrevites p. I l l
CH. 111,
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13
THE NATURE O F THE 1-TR.4NSITION
81
degree of the 1-transition. The results of this experiment suggest that, in fact, as the temperature approaches that of the transition, the specific heat at constant pressure, c,, becomes infinite like the logarithm of the temperature interval. While, of course, an infinite value could not be measured, the behavior of the specific heat follows this relationship for many orders of magnitude of the temperature interval. Measurements of the expansion coefficient to within 10-3 degree of the ;(-point indicate a similar singular behavior 2-5. The statistical theory of interacting particles which form a system capable of undergoing a cooperative transition is notoriously difficult. No approximation method has yet been devised which is valid in the neighborhood of the transition and thus capable of yielding the true nature of the singularity. The only exact solution for a problem of this type is that, first found by Onsager6, for the two dimensional case of the Ising model. In this, particles with two states, and interacting with their nearest neighbours, occupy the sites of a rigid lattice. It is a striking fact that the nature of the singularity with this exact solution is of just the same form as that observed in the liquid helium transition. I t may well prove to be the case that this particular form is characteristic of cooperative transitions, in spite of the fact that there are, in principle, many possibilities available. When it was first discovered, the form of the exact solution for the Ising model came as something of a surprise. Approximate methods of solution yield an apparently second order transition. The reason for this is discussed in section 6. It is noteworthy that experimental measurements with insufficient resolution also yield an apparent second order transition -and for very similar reasons. A coarse measurement effectively measures what the correspondingly approximate theory can calculate! The suggestion that the specific heat of helium might become infinite a t the transition was first made by Tisza’. Also it was suggested by Atkins and Edwards2 that a logarithmic term could be used to describe the results of their measurements of the thermal expansion coefficient below the ?,-temperature. One of the original objects of our experiment was to test a conclusion obtained by Blatt, Butler, and Schafroths on the assumption cm) that the range of order in superfluid helium, while large ( m is nevertheless finite. The conclusion was that there is no true transition -only a small but finite region of extremely rapid variation. They Refcremes p . 111
82
M. J . 13I’CKISGHAiU AND 1%’. 31. FAIK13ANK
[CII. 111,
3
2
predicted, for example, that the specific heat maximum would be rounded if measured with a resolution better than degree. The results of this experiment amply demonstrate that this conclusion is untrue. It should be mentioned here that, in spite of being based on an apparently invalid assumption, the finite --correlation -length theory led to results for the fraction of normal and superfluid and for the specific heat in remarkably good agreement with experiment except in the vital region very close to the transition. This agreement is not fortuitous. The calculations should, in fact, give a good account of a system composed of helium confined to regions of linear dimensions of about m i . That is, a system artifically prevented from achieving a range of order greater than this length. As can be seen from our discussion in 6, such a system should have thermodynmiic properties indistinguishable from those of actual helium, except very near the transition. In the language of section 6, such a system would have essentially the same “excitation” properties, but no “longr angel ’ proper ties. It is our purpose in this paper to discuss the specific heat experiment in some detail, and to show that the nature of the singularity is determined with very little ambiguity. In section 3 we shall derive sonic general thermodynamic relations Liseful for systems possessing transitions of any type -including A-transitions. We shall use these relations to compare other properties that have been measured near the transition with the specific heat results. These include the expansion coeficient and the velocity of sound. In the following sections we thcii discuss the nature of cooperative transitions and suggest a method of distinguishing the long-range ordering effects from other contributions to the thermodynaniic propertits. In the filial section we examiiic, irom this point of view, the apparently exceptional case of the superconducting transition. We point out that it may not, in fact, be exceptional -experimental results up to now being incapable of yielding a real decision.
2. The Specific Heat Near the 2-Point 2 . 1 . DESCRIPTIOX OF
THE
EXPERIMENT
An experiment has been performed1 in which the specific heat along the saturation line, C,, was measured to within 10-6 degree of the kpoint, T,, with equal prccision both above and below T,. The H r f w o i c c s p. 111
CH. 111,
S 21
THE SATURE OF THE
A-TRASSITION
83
apparatus used is illustrated in Fig. 1. To obtain such high temperature resolution it is essential that the attainment of equilibrium be unaffected by the drastic change of thermal conductivity of liquid helium at the A-point, or by the onset of the creeping film. Both of these requirements wcre met by permanently sealing the helium (0.0587 g)
I-*-
A
/ /
Fig. 1. Schcniatic diagram of adiabatic chamber for specific heat measurements’. - Stainless steel wire for closing heat switch B - Brass cap on filling capillary C - Wires connecting to heater and resistors on sample 1) - Cotton plus dyed with carbon black as radiation trap E - Indium “0”ring F - Filling capillary G -~ Kovar seal used as thermal short for wires to sample H - Radiation shield and thermal short for heat switch I - Nylon cord J - Three prongs of heat switch (copper) I< - Indium coating and suspension for sample L - Temperature sensitive resistor 31 - Heater N - Sample cavity 0 - Temperature sensitive resistor P - Copper shield over resistor R - Calorimeter wall A \
F
H
in a copper container (200 g), the inside of which was in the form of fins so placed that the helium was everywhere within 0.003 inches of a copper surface. Calculation of the temperature distribution with this arrangement showed that with a heat input of, say, 10 ergs/sec to the outer surface of the container, there would be temperature differences in the sample no greater than degree, even in the He I region, while the inem temperature would be increasing by a t least degree per second. This thermal relaxation time of less than one second was verified by actual measurement. In order to keep uncontrollable heat leaks from the bath small, the sealed container was suspended in vacuum without the introduction References p . 111
84
bl. J . BCCIiISGHXM AKD li'. h.1. I:AIKB.~XK
[CH. 111,
3
9
at any stage of helium exchange gas. A mechanical heat switch provided contact with the bath when required. 2.2.
h l E T H O D O F hfE.4SUREMEhT
Measurements were made with various heating and cooling rates depending on the total temperature range to be covered in order to verify that the results correspond to thermal equilibrium. Rates as low as one erg per second were employed. Heat was added by passing a current through a 600 ohm, constantresistance wire, wound on the sample container and bonded to it with hraldite. Cooling was achieved by including a small heat conductor in the form of a constantin wire connecting the sample to the vacuum container, immersed in the well-regulated helium bath. The rate of cooling could be determined by balancing against a measured heat input, and could be adjusted by varying the balance oi the heat input or by adjusting the bath temperature. The temperature was measured with a resistance thermometer consisting of four parallel Allen Bradley 390 ohm, $ watt resistors and an external bridgc circuit coupled through a lock-in detector t o a recorder. The total resistance at the A-point was 46330 ohms and it was possible to detect changes of 0.02 ohm corresponding to a temperature change of 2 x lo-' degree. A measuring power of 0.05 ergs/sec per resistor was used. Measurements were made by recording the resistance as a function of time during a constant rate of heating or cooling. The specific heat is proportional to the inverse of the rate of change of resistance. An electronic differentiator was built to differentiate the output of the lock-in detector; thus it was possible to record directly the time dcrivate of the resistance and thus the specific heat. 'The resistance thermometer was calibrated using the 1958 temperature scale. It may be noted that the relative shape of the specific heat very near the A-point would in no way be affected by errors in calibration, which would affect only the absolute values. 2 . 3 . RESYLTS
The specific heat of liquid helium under its saturated vapour, C,, was measured over the interval between 1.9 "I< and 2.35 O K . The temperature of the transition, T,, was 2.172&0.002 degrees (1958 scale)g. Values of C, up to 0.1 degree above and below the A-point are RefeniaLes p . I l l
CH. 111,
5 21
85
T H E SATI'IIE O F T H E A-TR-4XSITION
tabulated in Tnble 1 and shown in Figs. S and 3. The values are averages of ten runs taken at different heating and cooling rates. The estiinnted errors in incasurcment are indicated by the flags in Figs. 2 and 3 detcrmined by the scatter in the data from the ten runs. Although not affecting the relative shape of the specific heat very 1
TABLE
T : h ~ I a t k ~ oi r\ ~
2'
T* !
~
~~
trmperaturc
(,-s(T;I Ti.)
~~
~
1.3 -. i .) !?.U 1.0
: 10 'i 2, 10 6 <; IOP"
2.0
?'
:i 1 0 ~ j
:. lo-'
.i 0. 1.00 >.
x-mlis
( ~ ~ ~ ~ ~ ~ ~ ~ , s C/ S~ V, A~ ~ T ~ )~ . ~ , )
( d 4 ~~
C,
imw,iiri~tl
10 10
a
i
2.00 v 10 .' 5.00 :.. LO 1 1.00 >: 1 0 8 1.59 :.: 1 0 - ~ 2.51 >< I O - ~ 3.98 x 10-3 6.30 >' 101.00 x 10 2 1.5!) Y 1 0 - 3 ?..>I <: 10 2 3.98 :i 10 2 6.31 Y 1 0 - 2 1.00 x lo-'
~
-~ ~. ~.
~
~-
~~
~
.
~~
.? > . ,).
16 . 3
20.0
14.9
19.4 1 S.45
14.1 13.65 12.40 11.31 10.57
15.6
17.40 16 . 5 1 15,:jO
14.39 13.63 13.08 12.42
!).xi
11.i7 11.11 10.5.; 9.8!)
6.63 3.9.' n.30
!).-"O 8.41 i.66
6.39
8.36 7.T!) i.lY
4.60 1.14
3.661 3.307 2.853
near the transition, there is an estiniated possible error of 1 3 percent in the absolute magnitudes. This is due to the uncertainty in the mass of the heliuni sample and to the correction necessary to convert the measured specific heat to C,. This latter correction, necessary because of the small amount of helium in the vapour phase in the fixed volunie container, was calculatcd in a manner similar to that described by Hill and Louiiasmaa14. The proportion of the volume occupied by vapour at each temperature was detcrmined by observing at what temperature the expanding liquid completely filled the container. At higher temperatures thcre is only one phase present. This temperature was detcrmined by observation of the specific heat and noting the occurance of the apparent second order transition. This information together with the known densities and latent heat along the saturated Refevencrs p . I 1 1
86
If. J . l
[CH. 111,
$2
vapour line is sufficient to determine the correction. A detailed discussion of the correction, especially near Tz is given in the thesis of Kellers 1. It can be seen from Fig. 2 that as the specific heat is displayed on a more and more expanded scale it maintains tlie same geometric shape. In Fig. 3, C, has been plotted as ordinate against [ 1' - T , 1 in degrees Kelvin on a logarithmic scale. This method of plotting is used since
16
b I124 t
6t
4
6
2 0 -1.5 -1.0-0.5 0 0.5 1.0 1.5 IJ- Jn I degrees
I
1 1 1 1 1 -4-2 0 2 4 6 I T - Jnl millidegrees
-20 -10 0 10 20 30 IT-TAI microdegrees
Fig. 2 . Specific heat of Iiquid helium vs T - TI in O K . Iiellers, Fairbank, and Buckinghaml. c+ Represent, above 1.5' I<, data of Hill and Lounasmaa14arid Lounasmaa and Iiojo*'. c+ Represent, below 1.5' I<, data. of Kramers, Wasscher and Gorterla. Solid line represents cmpirical Eqs. (1) and (2). Width of small vertical line just above origin indicates portion of diagram shown expanded (in width) in thc curve directly to the right.
0 rcTprescnt data of
it displays well the behavior near the lambda point and illustrates vividly how accurately the measurements follow the logarithmic dependence. Sear the lambda point, on each side, there is a factor of lo4 in 1 T - T2.1 over which the data fall on two parallel straight lines which are branches of the expression :
CII. 111,
21
THE KATURE OF THE I
A-TKASSITIOS I
I
I
87 1
1
25
IT-TJ
Fig. 3. The specific heat of liquid heat of liquid helium vs log I T - Tn 1. 0 Represents data of I T N .
2 . 4 . RELATIOR' TO DATA4FAR
TRA~VSITION By combining our data near the 1-point with that published by other workers outside this rangel0-I6 it is possible to obtain an empirical function that fits the specific heat data t o within five percent from the lowest temperature at which measurements have been made up to 3.6 "K. The expression is as follows: FROM THE
+ (130 - 90.9log,, 1 T - T , I) exp (- 7.40/T) T < T , (2) C, = C,(T) + (23.5 - 16.4 log,, 1 T T , I) exp (- 3.70/T) T > T , C,
= C,(T)
-
where C, represent the Debye Function, evaluated at each temperature with the appropriate value of density and velocity of sound. It was found empirically that the best simple fit to the data both above and below the A-point could be obtained by inserting, as a factor in Eq. (l),a simple exponential term after first subtracting out a term representing the theoretical Debye .specific heat. The R ~ f i . w n r r sp . 111
88
12. J . HY(:KISGH.I31 AND If’. $1. F.IIRCrlNK
[CH. 111,
s9
constants in the exponential term that gave the best fit to thc data \\rere found to be 7.10 “ K bclow T , and 3.70 “I< above the ],-point. When these had been dctermjned, the other constants shown in the equation above all followed directly from Eq. (1) setting ‘1‘ = T , in the exponential term. Thus all of thc constants except the “energy gap” in the expoi~entialtcrrn were obtained from the data talien within dcgree of the 31-point. I t is interesting that the presence oi the logarithmic term leads to a rclat ively simple emyiricd expression over the wholc temperatiire rnngc removing t h e difficulty found in tlie past of tinding sucli a n expression. We do not necessarily imply any particular theoretical significance to the actual expression, but it will be discussed again in section 7 . I n summary the specific heat data near the A-point can be representctl by n logarithmic singularity. With this term, which can be deterrnincd from the data within 10 -2 degrec ot the 31-point, it is possible by multiplying by a simple exponcritial term with a teInperatui-e independen t. gap, differcnt on the two sides of the ii line, to fit the data over the entire range of tcmpcrature from the lowest temperature up to 3.8 degrees, due allowance being made for the l k b y c pl-ionon specific heat. Although this does not necessarily reprcscnt the most uscful tlieoretical equation, it points up thc need for the logarithmic term in explaining the data. I t is interesting to note that the constant difference, A , between the two straight lines representing the specific heat above and below the A-point (Fig. 3) is equal to -tR. In this section we have discussed the specific heat under the saturated vapour pressnre, C,. A more interesting quantity for the theory is the specific heat at constant pressure, C,. This is related to C, by,
B being the coefficient of
expansion and (6P/ bT)v,p,v. the slopc of the vapour pressure curve. Below 2.5 “K tlie difference between C, and C, is less than 1 percent so, within experimental error, the two yuantitics can be used interchangeably when discussing the behavior of C , near the A-point, as is done in the discussion which follow\.
Hefewiices
p. 111
CH. 111,
9
31
THE N A T ~ R E01; TIE
,&TRANSITION
3. Thermodynamics of ;”.-Transitions 3.1. )~.-TRAKSITIOXS I n this section we derive rigorously the thermodynamic consequences of A-transitions characterized by the absence of a latent heat, but at which the spec.ificheat at constant pressure becomes infinite. I€ any such transition occurs for a range of pressures, the transition points T,(P) form the %-line,which will in general have a finite slope, ( ZT)?,, on the pressure temperature diagram. Since the integral of the specific heat must be finite, C,, must approach infinity less rapidly than I T - 7’, I--l. The observed properties of liquid helium are consistent with those of a transition of this type, in which C,, becomes infinite like log I1’ - T , i . At such a lL-transition,not only does the specific heat C , become infinite, but so also must the thermal expansion coeficient and the isothermal compressibility. There exist relationships between these quantities and also between C,, ( ?P/ Z).and ( aV/ aP)., which remain finite, if the slope o€ the Mine is finite, as we assume hereafter. These relations, which we derive in the next sections, are useful for comparing different measured quantities and play a role analogous to the Clapeyran and Ehrenfest rclations. 3 . 2 . PROPERTIES AT THE
TIMSSITIOKS
At constant pressure, the entropy and volume are continuous functions of temperature, but achieve an infinite slope at T,. Now from the Maxwell relations we have
iP aT ar (d2, ( d(P 7 d S (w)T (dV (dS iiT
27.
=
61’
= -
‘
(3)
At the transition, (aT[as), and factors on the left in each expression vanish, the other factors remaining finite. This is easily seen by using the relations t
a?’
(4) CP
and 7 \l:e
iisc lierc, and frtiqiicntly in what follows, the identity
where W , S, I’, 2 are functions of state, with two indepcndent variables.
References
fi.
111
Xo\r if C, is infinite along the 1,-line, ( a T /aP),=,
( a T /ZP),, and since the second term on the right in (4) vanishes on the line, ( aT/ aP)s is finite and, by (3), (aT/aV\, must vanish. Then (5) shows that (iT/CP)vreaches the same finite value ( 87'1 %P),as does ( FT/ a$').. I n the same way we see that
as
as
=
(Av 0, W C(A,,; p =
aV
1)
c?P
-
thus at the A-transition the specific heat at constant volume reaches the value
For the velocity of sound we require
aV
2V
FT
as
(ds (%JCP(P), (dc; =
--
Thus at the A-point the velocity of sound, c , is given by
The relations (5)-(7) have been used to calculate the values for liquid helium given in Table 2 of section 4. Each of thcse quantities which remain finite a t the A-transition, achieve the finite value at a sharp cusp, varying near thc transition like Cly1. This variation is examined in the next section. 3 . 3 . PROPEIUIES SEAR
THE
TRWSJTION
In order to study the variation of thermodynamic properties near the A-transition it is convenient to introduce a new variable, the "neighborhood temperature", t, a function of state defined by f = t(T,P ) = 2' - T,(P),
is)
whcre T,(P) is the A-temperature for the preswre P . For convenience we will write 1';. for ( ?I1/aT),, the slope of the A-line, and a function of Refureiirrs p . 111
CH. 111,
$ 31
THE XATURE OF THE
A-TRINSITION
91
pressure. Thus the line t = 0 is the I-line, the line t = t o (constant) js a line parallel, in the P - T plane, to the I-line, but displaced parallel to the temperature axis by an amount to. Thus, by definition,
?I'
aP
2P
( F )=,( F )=,
= -
(T) T'
(9)
14-e also note the following properties of thc new variable:
A simple relationship connecting C , and the expansion coefficient can be obtained from the equation
This equation is exact and we note that if C,/T is plotted as a function of ( aV/ 6 T ) , the resulting curve approaches asymptotically the straight line of slope Pi and intercept (as/Z),t . The displacement from the asymptote is (as/aT), - (as/PT},. It is interesting to note that exactly the same straight line is the asymptote for a corresponding plot of the specific heat at constant volume. Since
8.S
=
(T), - p;
(g)t (%) VI
a curve giving CJ7' as a function of ( 2 P /3T),,, in units of -( aP/ ZV),, will approach a straight line of slopc Pi and intercept (as/aT), as did the plot of C,/T. Unlike the latter, however, thc curve for C, ends at the finite value given by Eq. (6). Another simple relation connects the expansion coefficient and the
t Zn approuniate relatzun of this type has bccn given by Pippaic117. Thl5 is discussed briefl) in Sec. 3.5 below. Referetares
p.
111
'32
31. J. FJYCI
\V. $1. FAXIIIB.ihIi
[CH. 111,
$3
isotliermal comprcssihility. This niny bc obtained from tlic equation
Thus a plot of the expansion cocfiicicnt I/ l( L V / t T ) I ,as a function of the coniprcisibility V-l( i i ' / 2'P)T,,approaches an asymptotic straight line of slope Pl :tnd intcrcept IT(: 2V/ 117.)1. Scar the kpoint, the vclocity of sound, c, may be obtained iroin the rclntion ~
'I'hus
A t the A - p i n t , t 0, tlic last torm in (14) ixnishes and we obtain again Eq. ( 7 ) for the finite value of the velocity of sound at that point. Tlie last term is inversely proportional to the specific heat a t constant prehsure, and is much the most rapidly varying term near the transition. : -
3 . 4 . PROPERTIES OF THE NEW Vl\Hr.iiw.s
All of the relations we have obtained arc exact. For the purpose oi comparing diffcrcnt measured quantities we rcquire the value of, for example, ( a s / LnT)t as a function of temperature at n given pressure. Tlic variation uf this quantity is qiialitativcly similar to that of thc entropy itself, in thc ncigliborliood of tlie transition. This can be seen from tlie fact that, near thc transition linc we could write
s
-
,s(l-l)+ a ( P ) f ( t ) ,
where a(P) is a function of the pressure and the functioiif(2) has an infinite derivatc at t = 0 [for heliumj(t) t(1og t ) (Eq. (l)].Then N
CH. 111,
5
31
THE NATURE OF THE
1-TRANSITION
93
so that (aS/2T),has a t dependence which is similar to the t dependence of S. Thus, even though ( as/ 8T),has an infinite temperature derivative at the transition temperature, its total variation over a small temperature range will be relatively small t. Similar remarks apply to the other quantity required, namely ( o‘V/2T),. This behaves qualitatively like the volume itself and, while possessing an infinite derivative at the A-line, nevertheless has little variation in actual magnitude over a small temperature interval. 3.5.
COiYNECTION \VITH O T H E R THERhIODYNAMIC
RELATIONS
The relations, such as (ll), (12) and (la), that involve the variable =0 corresponds to a transition line. Of course, they are of little use unless it does so, although they could be used to derive well known relations. Thus, for example, if the line t = 0 represents the line of constant volume, Eq. (11) becomes
t are completely general, and are exact whether or not the line t
which gives the usual expression for C, - C,. If the line t = 0 represents a first order transition line, the relations correspond to Clapeyron’s equation. For example if Eq. (11) is integrated at constant pressure from a point just in phase 1 across the transition line to a point just in phase 2 , the integral of ( a S / a T ) , , which is everywhere finite, will vanish. Then, if L is the latent heat,
where Pi is (aP/i3T) along the first order transition line. This is just Clapeyron’s equation. If the line t = 0 represents a second order transition line, along ZT), is continuous across the line. The which 2P/i3T = P;, then (as/ t A t the saturation pressure in helium, for cxample, i t is simple to obtain two empirical values. At the tcrnperature of the dcnsity maximum we see from Eq. ( I l ) , that (aSjaT), = P I C p . h rccent measurement5 gives a temperature of 6-8 millidegrce above the &point for this maximum a t which temperature the specific heat is 5.7 joulcs per gm pcr degree. Thus (a.Y/aT), = 2.64 a t 6 - 8 x 10-3 deg. above the A-point. By plotting the entropy data of T.ounasmaa and Kojo16,we find at the 1-point ( a S / a T ) ,M 2 . 3 & 0.3. ’I‘has ( a S / a T ) ,changes according to these figures by pcrhaps 15 pcr cent in 7 millidegrees abuve thc J.-point. Rejereizces ;b. 7 7 1
94
31. J . U U C R I S G H A M . I S D
rr’. If.
FAIRHANK
[CH. 111,
s4
difference of Eq. (11) for points 1 and 2 , one on each side of the line givcs the result
AC,
=
TPiz4(ijV/ aT),,
which is just Ehrenfest’slR relation for a transition of this typc. Similar simple arguments \vould permit the derivation of equivalent relations for any type of transition. Pippard l 7 has given approximate relations similar to some of those derivcd here. His clerivation is based on the assumption that thc entropy surface is cylindrical near the A-line. Since, in general, the “strength” of the singularity will vary continuously along the A-line, the shape of thc “cylinder” changes along its length. Pippard’s assumption effectively replaces our ( ijSj8T)t, for example, by a constant. As we have seen, this “constant” actually changes infinitely rapidly at the transition, but its net change in magnitude over a small tenipernture intcrval is nevcrtheless relatively small. Thus, in fact, Pippard’s relations are a good approximation very near the transition line. In the comparisons of different measured properties with the thermodynamics, made in the next section, most of the uncertainty arises because of the inadequately known values of the derivatives of pressure, entropy, etc. along the A-line. We will discuss a number of the thermodynamic properties, but a similar analysis could be made of others, such as the coniyrcssibility, latent heat of vaporization, surface tension, variations with 3He concentration, etc. The method we have used for deriving the thermodynamic equations of this section could readily be applied to obtain the necessary relations. 4. Analysis of Other Measurements
In this section we discuss various equilibrium thermodynamic properties that have been measured by others in the neighborhood of the 1-transition in helium. We will compare these results with those expected on the basis of the thermodynamic relations derived in the last section. 4 . 1 . PROPERTIES OF THE ~.-LISE
Akalready stated, this comparison requires the values of several properties of the A-line. Unfortunately these values are hard to obtain \vith precision clue to the difficulty of making precise measurements very close to the transition. For example the position of the A-transition Rpferericr? p . 111
CH. III,
$ 41
THE NATURE OF THE
1-TRANSITION
95
changes by only about seven millidegrees when the pressure is increased by one atmosphere from the saturated vapor pressure 19. Cntil recently the measurement of (.“P/aT),a t the saturated vapor pressure represented an extrapolation from measurements taken at several atmospheres pressure. This method gave values ranging from 8 0 . 8 1 0 ~ 2to0 97.g21 atm/deg for ( a P /ZT), at the saturated vapor pressure. I n fact when our comparison was first attempted it became clear that consistancy required a value of (8PI ‘?TI,at the vapour pressure of about 133 atmospheres per degree. Recent measurements 19, 22 of the 1-temperature at pressures below one atmosphere made in connection with other experiments indicate the gratifying value of 130 & 10 atm/degt. This number is the one included in Table 2 , which lists values of the parameters of the Mine at the vapour pressure, near the solidification pressure and at an intermediate pressure. Except for the value of (aP/aT), mentioned, all the values were obtained by estimating the slopes of curves published by Lounasmaa and Kojo16 and Lounasmaa and KaunistoZ1. Include in the table are calculated values of the specific heat at constant volume and the velocity of sound using Eqs. (6) and (7) and the values in Table 2 . TXBLE 2 Values of some quantities for 4He at thc ,l-line5,1%
0.05 2.173 13 2.00 29.7 1.762
4.2.
0.146 0.167 0.180
‘rHERJIAL
130 -z 10 13 3 73 & 3 “7 :f 0.0 60i-L 1.5
7.3 & 0 . 3 0.9 0.6
21, 22
370 50 42 1 4 17 2
216 5 20 310 f 25 440 + 30
EXPASSION COEFPICIEST
We have seen that the singularity in the specific heat at constant pressure implies also a singularity in the expansion coefficient, = l / V ( SV/ i T ) p . Experiments to measure the latter have been performed by Atkins and Edwards2 along the saturated vapour line below the 2-point to within degree of the 1-transition. Atkins and Edwards point out that the data can be described by an equat’ion implying a logarithmic singularity in 0, at T,; t AX reccnt measurement of 110 atrn/deg has been reported by private communication from C. -1.R e p n ~ l d s ? ~ . Rpfwences
p . I11
9(i
XI. J . BVC;KtX;CIIhRI .\KD FT'. M. FhIRB24ii1<
p
= 0.0008
+ 0.0148 log," I T - 7.a 1
T
[CII. 111,
94
< T,.
This data was the first experimental suggestion of a logarithmic singularity at the A-transition. More recently Edwards has measured the index of refraction near the ;.-point at a wave length of 5462.27 A. Assuming that the polarizability of liquid helium is constant and equal to 0.12464 cm3 m o k l for the wave length used he was able to determine from the variation in the index of refraction the thernial expamion coefficient. Edwards noted that the data near the 2.-point may be represented within experimental error by
+ 14.5 loglo [ T 7-AI 103//3,, = - 1.5 -+ 14.5 loglo 1 '1' - T , I
103/1, = 41.5
and
-
for T for T
T,
< Ta.
The data extend to within 0.007" of T , above the A-point and to within 0.002" below. Maxwell, Chase and Millett and Maxwell and Chasez2have measured the dielectric constant of liquid helium under its saturated vapour pressure close to the A-point. The dielectric constant was determined by measuring as a function of temperature the capacitance of a parallel plate condensor immersed in liquid helium. The measurements were made at a frequency of lo5 cps. Assuming that the polarizability is constant in the vicinity of the I-point, the density and thermal expansion coefficient are calculated using the Claucius Mosotti equation. The data obtained from the above three experiments are plotted in Fig. 4. In each of these experiments temperature equilibrium was obtained by means of a 4He bath in which the sample holder was immersed. 1mmcdiately above the A-transition temperature equilibrium is difficult to obtain due to the great reduction of thermal conductivity in He I and the presence of a negative coefficient of expansion directly above the A-point which inhibits convection. As can be seen from the scatter in the data it is difficult to obtain accurate measurements in this region. Recently Kerr and Taylor5 have measured the density directly and avoided these difficulties by using a liquid 3He bath. They have kindly given us a preliminary curve of their clnta prior to publication. This curve is shown in Fig. 5. I t may be noted how well their curve Rejwences p 111
CH. 111,
$ 41
97
T H E NATURE O F T H E 1-TRANSITION
0.05-
Atkins and Edwards Edwards * Chase and Maxwell Kerr and Taylor
0.06
0
+
++
0.04-
.:. +
++?’
+,
++
++++
+
x
x
* *.x x
0.03
0.04 Calculated
-‘--I
0.06
L
0.07 -$ 10
’ ’
lo-‘
I
I
10+
IT-Til
10-l
I
I
OK
Fig. 4. Cocfficicnt of cxpansion vs 1 T - T , 1. The t w o symptotic straight lines in this figure arc dctermined using t h e asymptotic straight line in Fig. 6 and thc specific heat data of Taljle 1 .
y 27. 3
9 w
2
s
M O L A R VOLUME ( V i c i n i t y o f A-point)
W
27.
TEMPERATURE (*K)58
Fig. 5 . Density of liquid hcliuni in vicinity of 1point as measured by Kerr and Taylor5. Note density maximum 6-8 millidegrecs above T . R t f w e w r s .h 7 1 1
98
M. J . 13L'Cl
[CH. 111,
9
4
demonstrates that the I-transition, at the point of maximum slope, occurs 6-8 inillidegrees below the temperature of the density maximum. We have estimated the expansion coefficient above T , from this curve and the results are included in Figs. 4 and 6. Accurate values of fl determined from the original data will be published by Kerr and Taylor in due course. We use the thermodynamic relation (11) to compare C, and 1, as the A-point is approached. Expression (11) can be written for convenience in the form: (15)
where e represents the density. Fig. 6 is a parametric plot of (T,/T)C,against (e,/e)P. The values of the parameter (t = T - TL) are indicated by the arrows in the diagram. The asymptotic straight line give by Eq. (15) with slope PiTAej' and intercept T,( as/aT), is determined, we would emphasize, purely from the data given in Table 2. I t is seen that below the I-point the data plotted (Atkins and EdwardsI2 approaches asymptotically the expected straight line. Above the I-point the same is true of the data of Kerr and Taylor5. From the specific heat data and assuming the assymptotic straight line shown in Fig. 6 to be correct, the values of /? can be determined as near T A as desired. These values are shown as the two parallel straight lines in Fig. 4. It is seen that except for the scatter in the data and the apparent systematic errors above the A-pqint, the coefficient of expansion data is consistent with the specific heat data and the measured values of ( aP/ aT), and ( as/aT), given in Table 2. An exact check of the logarithmic singularity indicated by the specific heat will require measured values of 16 closer to T , and under better conditions of temperature equilibrium than have so far been attempted and also more exact determination of the parameters of the I-line. It should be noted that in the case of the helium transition there is both a logarithmic singularity and a step function in the specific heat and expansion coefficient. Our thermodynamic equation determines the ratio of the coefficients of both these terms. For the step function the relation is just the same as Ehrenfest'sl* equation, and is displayed by the separation of the parallel asymptotes in Figs. 3 and 6. ReJeuences
p . 111
CH. 111,
41
THE NATURE OF THE
A-TRANSITION
99
Fig. 6. Parametric plot of ( T a / T ) C , vs (ea/p)/l. Solid straight linc represents asymptotic ~ (aV/aT)a valuc of C , vs p calculated from Eq. (11) and the valucs for ( a s / a T )and given in Table 2.
4 . 3 . VELOCITYOF SOUND
The velocity of sound has been measured by several investigators 24-33 using standing wave techniques, pulse techniques and optical techniques. The results of all the investigators agree to within experimental error. Since the measurements were taken over a range of frequency from 0.22 to 15 Mc/s, they show that there is less than 4 percent dispersion over this frequency range. Precise determinations in the immediate vicinity of the I-point are difficult due to the appearance ot a large attenuation in this region. Recently Chase33 has designed a special experiment to extend the sound measurements very close to the References p . 111
100
hf. J . BUCKINGHAM A N D 1%'. M . FAIRBANK
[CH. 111,
54
I-point. Below we compare these results with the theoretical curve calculated from the data in Table 2 and Table 1 using the relationships derived above. It was mentioned in connection with the relations (13) and (14) that the change of the velocity of sound near the I-point is dominated by the last term of (14) which is inversely proportional to the specific heat. An approximate expression for the velocity of sound can be written, using (13) and (14), as
(16)
where the variation of quantities other than C, has been ignored and
Calculnted values of crl, using the relation ( 7 ) , are included in Table 2 and shown on Fig. 7 which also shows measured values obtained b y Atkins and Stasior". In Fig. 8 we show the calculated velocity of sound in the neighborhood of the transition obtained from Eq. (161, the experimental values of C, listed in Table 1,and the values of the parameters listed in Table 2 . For comparison with the measured variation we reproduce a figure from the report by Chase33 illustrating his results near the kpoint (Fig. 9). The rounded minimum instead of a very sharp cusp would be expected, since any experimental arrangement would necessarily average the exact values of the velocity over a finite temperature interval. We feel that both thc absolute values of the velocity of sound and its variation near the ].-point indicate a satisfactory agreement between the observed and the thermodynamically expected behavior. has compared his data with C , using Pippard's relations1' and finds satisfactory agreement at least below the transition tomperaturc.
4.4. OTHER bIEASI'RE1) QI'AXTITIES As discussed above, Lounasmaa and KojoI6 have measured C, near the A-line for various densities of liquid helium and Lounasmaa and Rifevciicc5
p.
111
CH. 111,
5 41
THE NATURE OF THE 1-TRANSITION
101
40
100-
I
0
I
I
10 20 Pressure (atmospheres)
I
30
Fig. 7. Calculated values of velocity of sound a t the A-line using Eq. (7). Solid curve represents experimental data of Atkins and Stasior
005
0
T J , , (mlliidegreas)
Fig. 8
Fig. 9
Fig. 8. Calculated velocity of sound in the neighborhood of the A-transition obtained from Eq. (16) using the experimental values of C, listed in Table 1 and values of t h e parameters listed in Table 2. Fig. 9. Experimental values for the velocity and attenuation of first sound near the A point3s. Rejcrcizces
p.
111
102
M. J . BUCKINGHAM AN11 11‘. M. FAIREAKTK
[CH. 111,
$5
Kaunisto21 have measured (aP/ ? T ), under similar conditions. Both these measurements represent a considerable increase in accuracy and resolution over previous data. C, and ( aP/ aT) can be compared by Eq. (12) using the slopes of the Mine determined from the same experiments. We have attempted such a comparison, but the data does not extend close enough to the A-line and the slopes of the A-line are not known with suficient precision to make such a comparison very significant. Lounasmaa has indicated2’ that experiments are in progress to extend the data to within degree of the ,?-line. When this is done such a comparison will be extremely interesting. Other properties such as viscosity and the attenuation of sound (Fig. 9) which have been measured close to the transition are of great interest, but outside the scope of our present discussion of equilibrium properties.
5. The Nature of the Transition \lie have seen that measurements very near the lamba point suggest that the specific heat and expansion coefficient become infinite a t the transition. Of course, actual measurement could never prove an infinite value, particularly in view of the logarithmic behavior found. To obtain a specific heat value 50 per cent greater than the highest obtained would require a temperature resolution better than degrees. ,411 that can be said is that the observations are consistent with a specific heat that becomes infinite like the logarithm of the temperature interval from the lambda point. Higher order transitions of this type appear to be quite widespread in nature. They are the cooperative transitions, typified by the magnetic and order-disorder transitions. The characteristic property is the emergence of a type of long-range order below the transition temperature and what might be termed the “anticipation” of the change above the temperature of the actual transition. It is interesting to note that the particular logarithmic type of singularity may be characteristic of cooperative transitions, rather than exceptional. As already remarked the same singularity is a property of the two-dimensional Ising model6, the only case which has been solved exactly in statistical mechanics. Recent measurements by Robinson and Friedberg35 suggest a logarithmic singularity in the specific heats of hydrated nickel and References
p.
Ill
CH. 111,
5 51
THE NATURE OF THE &TRANSITION
103
cobalt chlorides. Their results, shown in Fig. 10 do not exhibit a symmetrical singularity, but this may be due to the absence of data nearer than about one tenth of a degree from the transition. In fact the same relative temperature range for liquid helium would not yet lie on the straight lines of Fig. 3 (in which the horizontal arrows indicate the equivalent range of the nickel chloride measurements). The close correspondence for the two substances in their equivalent temperature range is indeed noteworthy. An ordinary first-order transition involves a discontinuous change in the structure of the system. A chemical reaction in which the actual 5
10
5 4
38
E& 3
i2
&!6
0
3
‘1.
e $
-E 4
2
8
up 2
1 0
0
0
2
4
6 8 10 12 14 16 1 8 2 0 2 2 Temperature (‘lo
0
Fig. 10. C, vs I T - TN I and C, vs log 1 T - T N I for NiCl, * 6H,OS6.Extent of data is shown on a relative temperature scale by arrows in Fig. 3.
molecular composition of the system changes could be regarded as an extreme example. I n general, the establishment of the new phase involves nucleation phenomena with the accompanying possibility of supercooling and superheating. These effects are not excepted in the A-type of transition. Indeed, the fact that some of the change occurs before the transition temperature is reached shows that there is no choice of paths. The new properties are established gradually rather than suddenly. In the case of a ferromagnetic or antiferromagnetic transition it is the relative directions of the elementary magnetic moments which achieve a long-range order below the transition temperature -and an “anticipatory” short-range order above. I n a binary alloy it is the relative location on lattice sites of the two types of atom which is similarly ordered. In a rotational transition it is the relative phases of the different rotating elements. The analogous property in liquid helium that becomes ordered in this fashion-over a short range above the transition and over a long References
p . 111
104
M. J . BUCKINGHAM A N D W. M. F A I R B A N K
[CH. 111,
9
6
range below -is of a more subtle character. Certainly there is no great change of spacial arrangement of atoms at the lambda point. Recent experiments36in which the pair distribution function is measured by neutron scattering techniques demonstrate how little different is the average arrangement of atoms in the neighborhood of any given atom. The mere fact that the substance remains liquid demonstrates, in itself, an absence of long-range spacial order. As F. London3’ so vigorously stressed, it is not special order, but momentum order which is characteristic of superfluid helium. The momentum arrangement of atoms is not easily measured directly, but it would seem that just above the A-transition, neighboring atoms develop a tendency to move with the same momentum. At the transition the “neighborhood” spreads to a long range and below that temperature the momentum of atoms is correlated, even when very far apart. The superconducting state is the only other system exhibiting longrange momentum correlations, and it possesses many properties analogous to those of superfluid helium. There is a striking difference in the observed specific heat, however, which will be discussed in a later section. 6. Separation of the Singularity The difficulties of calculating by statistical mechanics the properties of a co-operative system very near the transition are no less than those associated with measuring them. The various exact and approximate expansion methods reach their radius of convergence at the transition, and no finite number of terms can reveal the nature of the singularity, which depends on the asymptotic form of the high terms. For this reason it is perhaps profitable to examine the results for helium in an attempt to separate the contributions arising from different causes. In this way, we can hope to simplify the essential theoretical problem. Firstly, it is to be expected that the contributions arising from the modes of compressional oscillation in the liquid are essentially independent of other contributions. This separation of a “phonon” part is usual and the only interaction with the remainder comes from properly using the empirical density and sound velocity appropriate to each temperature. Except at the lowest and highest temperature, the phonon specific heat is a relatively small proportion of the total. References p . 111
CH. 111,
5 61
THE NATURE O F THE A-TRANSITION
105
We now attempt to justify a further separation into “excitation” and “long-range order” contributions. This separation can probably not be carried out exactly, since the contributions are not really additive. Nevertheless, we shall see that each dominates in a different temperature interval. If one regards the transition from the high temperature side, one expects the statistical and dynamical interactions between particles in larger and larger “clusters” to contribute, and cause an increasing departure from the perfect-gas behavior at the high temperature limit. The effect of large clusters is not great until very close to the transition. We would regard the effect of the smaller clusters as due to the short-range order that is developing towards lower temperature. In a similar fashion, if we start from low temperatures, we expect excitations to develop in the fully ordered zero-temperature state. These excitations (rotons) have a finite energy, E , of excitation and thus develop slowly at first, their number increasing as exp ( - ~ / k l “ ) .As the temperature increases the number of excitations becomes sufficient for interaction between them to be effective. This could be described in terms of “clusters” of excitations, the maximum cluster-size contributing significantly increasing with temperature, because of the rapid increase of the density of excitations. A further expected effect of the increasing excitation would be an increasing reduction of the threshold energy E , thus further speeding up the increase of density of excitations. Just as the description in terms of clusters of particles, on the high temperature side of the transition, would diverge at the transition temperature, so would the description in terms of clusters of excitations diverge as the transition is approached from below. On the high side we have a co-operative interaction of particles, on the low side a cooperative interaction of excitationst. Just as with the particleclusters, it is only very near the transition that large-sized excitationclusters would contribute. If the large clusters could be ignored, we could continue the descriptions in terms of the small ones towards and even beyond the transition temperature. In general we would not expect the entropy or specific heat, for example, to join smoothly. At the transition there would be an apparent first-order transition. In some circumstances we would expect this procedure to give
t
This picturesque description is due to L. Onsager (private discussion). References p . 111
106
M. J. BUCKINGHAM AND W. M. FAIRBANK
[CH. 111,
96
accurate results. An example is the liquid-gas transition at constant pressure, well below the critical pressure. I n this case, the influence of the large clusters would be confined to the neighborhood of the critical point. At much lower pressures we would have a good description of the liquid and gas, respectively, and the continuations beyond the transitions would be valid descriptions of the superheated and supercooled states. A t pressures nearer the critical pressure, accuracy would require the inclusion of larger and larger clusters for both the gas and liquid states. Eventually, at the critical pressure we would have a situation similar to our present case of liquid helium. An example in which the procedure is very accurate, indeed trivial, would be, sav, ice changing to water vapor at a pressure well below the triple-point pressure. The gas would be essentially ideal and only clusters of one particle would be needed. Similarly the description of the solid in terms of non-interacting lattice vibrations would be very accurate. So calculated, the entropy difference at the transition would give accurately the heat of sublimation. Returning to liquid helium, we recall that the extrapolations of the small-cluster descriptions would lead to an entropy and specific heat discontinuity at the transition. We would further expect the whole specific heat curve by itself to resemble that of a second order transition-in fact, because the excitation energy, E , is in the nature of an order parameter, we might expect a result somewhat similar to a Brag-Williams model of an order-disorder transition. The entropy change between the two extrapolations is that associated with the long-range ordering. We expect the loss of this entropy to occur over the very small temperature interval in which large clusters are significant. When the long-range ordering is ignored, the change is discontinuous and the specific heat would contain a 6-function at the transition temperature. The long-range terms would spread the &function over a finite temperature interval, but leave a weaker type of singularity. I n the interval very near the transition, the contributions of the small clusters change only slowly, whereas those of the large clusters are changing very rapidly. Thus in the specific heat very near the transition the small clusters contribute a relatively small and essentially constant part to a total that approaches infinity. Thus very near the transition we could ignore the small clusters and obtain an approximation which becomes the more accurate the nearer the transition. References p . 111
CH. 111,
9 71
T H E NATURE O F T H E A-TRANSITION
107
Our method of separating the long- and short-range contributions should be valid sufficiently near and sufficiently far from the transition. In the intermediate range it is not valid, and in any case, would depend on an arbitrary separation into “large” and “small” cluster contributions. When we analyze the experimental results for liquid helium from this point of view, there is a similar arbitrariness in the extrapolations, but this does not affect the result very near or very far from the A-temperature.
7. Analysis of the Specific Heat of Liquid Helium The discussion in the last section forms the basis for our analysis of the specific heat of liquid helium. We expect the singular part to be essentially symmetric about the transition temperature. Thus, having subtracted the phonon contribution, we have extrapolated the remaining specific heat smoothly as far as the transition from the high temperature side. This extrapolation is subtracted and the remainder forms the high-temperature half of a symmetric singular contribution, C,. Thus the low-temperature part of C, has the same value at a given temperature as the high-temperature part has a t the same temperature interval from the transition. We now have three additive contributions to the specific heat, Cobserved
= CDhonon
+ + c,. c x
The “excitation” contribution Cx, and the singular contribution, C,, are shown in Fig. 11 and have the form expected from the discussion. The C, term represents a simple logarithmic singularity, smoothed out to zero away from the transition. We would expect a similar contribution in other A-type transitions. The term C, should represent, below the transition, the contribution of excitations ignoring their high-order interactions. For a system in which a finite energy gap for excitation exists one would expect this term to have the form
where B(T) is the effective gap at the temperature T . The form of the factor F ( T ) would depend on the particular form of the excitation spectrum. If F ( T )is taken as a constant, our values of C , require a gap decreasing somewhat with increasing temperature, but not vanishing at the transition. References p . 111
108
M. J . BUCKINGHAM AND W. M. FAIRBANK
[CH. 111,
57
The empirical expression given in Eq. (2) is not of the form we have discussed. It has been given as a convenient summary of the observed specific heat values. In this expression the parameter analogous t o B(T)is independent of temperature, but takes a different value below and above the transition. While the precision of this property is perhaps surprising we do not attach any particular theoretical significance to it.
10
-
h
s Q 0
P -
-IC
-a
-
Y
ox 5 -
I
-
A
*O
ID
2.0 3.0 Temperature (OK)
I
00
I.o
3.0 20 Temperature(OK)
Fig. l l a . The “excitation” contribution t o the specific heat of liquid helium, C,, vs temperature in OK. Fig. l l b . The singular term in the specific heat of liquid helium, C,, vs temperature in “K. This curve is obtained by subtracting Cphonon C, from Cobserved.
+
We feel there is more significance in the separation into additive contributions. The energy gap and other properties of the excitation spectrum of liquid helium have been measured using neutron diffraction techniques 3a-41. Henshaw and Woods41 have observed a temperature varying minimum in the excitation spectrum both below and above the &point. I t would be interesting to calculate the specific heat corresponding to this measured spectrum and to compare it with our C,, although this has not been attempted by us. References
p . 111
CH. 111,
4 81
THE NATURE OF THE &TRANSITION
109
8. The Superconducting Transition In superconductors it is correlated-pairs of electrons that achieve a long-range order in the low temperature state. Because of the ordinary condensation of the Fermi gas of electrons, there are very few (a fraction T/T,,,,,) “effective” particles per unit volume in the metal a t low temperature compared with the number in liquid helium. Thus above the transition one-particle clusters only should provide a good approximation. Below the transition the same is true of the excitation clusters. The excitations correspond to removing correlated pairs from the ordered background, or to breaking up a pair. The energy, E , necessary to form such excitations is finite, as in helium, and diminishes as the order decreases with rising temperature. Although increasing exponentially with temperature, the density of excitations is nevertheless small and an approximation ignoring interactions between excitations would be a good one. Of course the effects of dimishing order have to be included just as the changing density must be included for the phonons in helium. Both above and below the transition a one-particle cluster picture is satisfactory, but how close to the transition will it remain good? The particle-pairs are vital below the transition ; are the two-particle clusters negligible just above ? Certainly in the presence of a magnetic field, when the transition becomes first-order, the picture will be quite adequate, just as the equivalent one is for the gas-liquid transition below the critical pressure. In zero field it is difficult to rule out the possibility that pairs would contribute significantly near the transition and larger clusters nearer still. If such contributions exist, there should be a small entropy change associated with the emergency of the ordered state, but as in helium, it would occur over a small temperature interval rather than as a latent heat. It would be expected that, if it exists, such an entropy contribution would be relatively very much smaller than in the case of helium and occur over a much smaller temperature interval. This is because of the relatively low density of the system. There is no experimental evidence for a singular contribution to the specific heat near the superconducting transition in zero magnetic field. However, it would be extremely hard to detect. Not only would its magnitude be relatively much smaller than in liquid helium, where it is about one-fifth of the total entropy just above the transition, but Rejerences p . 111
w. M. FAIRBANK [CH. 111, 5 8 it would tend to be spread out because of the inherently imperfect nature of any real specimen. Unlike the situation in liquid helium, it would be impossible to avoid the presence of some strain and imperfection in any solid specimen, and very little would be needed to obscure the small singular contribution we have suggested. It should be noted that such a contribution would imply an anomalous susceptibility in the normal state in the close neighborhood of the zero-field transition temperature. In an attempt to test this suggestion, m e a ~ u r e m e n t son ~ ~a sample of tin were made with the same apparatus which was employed for 110
M. J . BUCKINGHAM AND
1.5
1 E 0
.c=e 1.0
9 Y. .-
-
. I -
2 c u P In
0.5
B .c
d
'3.72
3.721
3.722 3.723 3.724 Temperature( O K )
3.725
Fig. 12. The specific heat and relative susceptibility of tin as a function of temperature very close to the tran~ition'~.
the high resolution experiments on liquid helium. The temperature resolution was actually more than adequate, the transition being spread over a millidegree. Fig. 12 shows the observed specific heat and change of susceptibility for a few millidegrees each side of the transition. There is no evidence for an anomaly in the behavior of either quantity, but we would in any case only expect it to show up in a rather more sharply defined transition. Attempts to find a specimen satisfying this exacting condition are being made. Refevences
p. I l l
CH. 1111
THE NATURE OF THE
1-TRANSITION
111
REFERENCES W. AI. Fairbank, M. J. Buckingham and C. F. Kellers, Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin (1957), p. 50; W. M. Fairbank, M. J. Buckingham and C. F. Kellers, Bull. Amer. Phys. SOC.,Ser. 11, 2, 183 (1957); C. F. Kellers, Thesis, Duke University (1960). This experiment was performed while the authors were a t Duke University. The improvement in the sensitivity of the experiment by an order of magnitude over the initial published results is due t o C. F. Kellers, and is described in his Ph. D. thesis. 2 K. R. Atkins and M. H. Edwards, Phys. Rev. 97, 1429 (1955). 3 M. H. Edwards, Canad. J. Phys. 36, 884 (1958). 4 E. Maxwell, C. E. Chase, and W. E. Millett, Proc. 5th Int. Conf. Low Temp., Phys., Madison, Wisconsin (1957), p. 53. 6 E. C. Kerr and R. Dean Taylor, private communication. I ) Lars Onsager. Phys. Rev. 65, 117 (1944); G. F. Newel1 and E. W. hlontroll, Revs. Modern Phys. 25, 353 (1953). 7 L. Tisza, Phase Transformations in Solids, edited by Smolychowski, Mayer and Weyl (New York, John Wiley and Sons, 1951). J . M. Blatt, S. T. Butler and M. R. Schafroth, Nuovo cimento 4, 677 (1956). 8 F. G. Brickwedde, H. van Dijk, Durieux, J . R. Clement and J . K. Logan, Physica 24, 5128 (Proc. Kam. Onnes Conf., Leiden, 1958). lo W. H. Keesom and K. C h i n s , Proc. Kon. Acad. Amsterdam 35, 307 (1932). l1 W. H. Keesom and Miss A. P. Keesom, Physica 1, 128 (1933-4). l2 H. C. Kramers, J. D. Wasscher and C. J. Gorter, Physica 18, 329 (1952). 13 G. Hercus and J. Wilks, Phil. Mag. 45, 1163 (1954). l4 R. W. Hill and 0. V. Lounasmaa, Phil. Mag. 2, Ser. 8, 145 (1957). l5 A. H. Markham, D. C. Pearce, R. G. Netzel and J. R. Dillinger, Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin (1957), p. 45. 0. V. Lounasmaa and E. Kojo, Series A, VI. Physica 36, 3 (1959). l7 A. B. Pippard, The Elements of Classical Thermodynamics (Cambridge University Press. 1957), Chapter IX. P. Ehrenfest, Comm. Kam. Onnes Lab., Univ. of Leiden Suppl., 75b (1933); Proc. Acad. Sci. Amst. 36, 153 (1933). W. E. Keller and E. F. Hammel, Jr., Annals of Physics 10, 202 (1960). 2o W. H. Keesom, Helium (Elsevier, Amsterdam, 1942) p. 256. 21 0. V. Lounasmaa and Leila Kaunisto (preprint) to be published in Ann. Acad. Sci. Fennicae A VI (1960). 22 E. Maxwell and C. E. Chase, Proc. of the “Kamerlingh Onnes Conference on Low Temp. Physics,” Physica 24, 5139 (1958) ; and private communication. 23 J. B. Burnham, J. B. Pearson, A. H. Spees and C. A. Reynolds, private communication. 24 J. C. Findlay, A. Pitt, H. Grayson-Smith and J . 0. Wilhelm, Phys. Rev. 54, 506 (1938); ibid 56, 122 (1939). 25 J . R. Pelham and C. F. Squire, Phys. Rev. 72, 1245 (1947). 2g K. R. Atkins and C. E. Chase, Proc. Phys. SOC.A 64, 826 (1951). 27 C. E. Chase, Proc. Roy. SOC.A 220, 116 (1953). 28 I<. R. ?Itkins and R. A. Stasior, Canad. J . Phys. 31, 1156 (1953). 29 A. van Itterbeek and G. Forrez, Physica 20, 133 (1954). 30 A. van Itterbeek, G. J. van den Berg and W. Limburg, Physica 20, 307 (1954). 31 G. J. van den Berg, A . van Ittcrbeek, G. M. V. van Ardenne and J. H. J. Herfkens. Physica 21, 860 (1955). 1
112
ni. J . RUCKIXGHAM AND
w.
ni. FAIRBASK
[CH. 111
C. E. Chase and M. A. Herlin, Phys. Rev. 97, 1447 (1955). C. E. Chase, Phys. of Fluids 1, 193 (1958). f 4 C. E. Chase, Phys. Rcv. Letters 2, 197 (19.59). 3 p \V. I<. Robinson and S. A. Friedberg, Phys. Rev. 117, 403 (1960). 38 I ). G. Henshaw, Phys. Rev. 119, 9 (1960). 37 F. London, Superfluids, Vol. I1 (John Wiley and Sons, New York, 1954). 3 8 H. Palevsky, K. Otnes and K. E. Larsson, Phys. Rev. 112, 11 (1959). 38 J. L. Yarncll, G. P. Arnold, P. J. Brendt and E. C. Icerry, Phys. Rev. 113, 1379 (1959). 40 D. G. Henshaw, Phys. Rev. Letters 1, 127 (1959). 4 1 D. G. Henshaw and A. D. B. Woods, Program VII International Conference on Low Temperature Physics (Univ. of Toronto 1900) p. 64. 42 C. F. ICellers and W. M. Fairbank (unpublished). KI I<. R. Atkins, Liqnid Helium (Cambridge University Press, 1959). This book is a good general reference for background information on liquid helium. 32
38
CHAPTER I V
LIQUID AND SOLID 3He BY
E. R. GRILLY
AND
E. F. HAMMELT
UNIVERSITY OF CALIFORNIA, Los ALAMOS SCIENTIFIC LABORATORY LOSALAMOS,NEW MEXICO CONTENTS:1. Theories of liquid 3He, 113. - 2. Theories of solid SHe, 118.
-
3. Pressure-volume-temperature relations, 119. - 4. Thermal properties. 129. 5. Transport properties of liquid and solid SHe, 134. - 6. Nuclear spin relaxation in condensed SHe, 138. - 7. Velocity of sound in SHe, 143. - 8. Summary, 147.
1. Theories of Liquid 3He
Since the early and unsuccessful efforts to describe the properties of liquid 3He in terms of an ideal Fermi-Dirac gas1, numerous attempts have been made to obtain a more accurate theoretical description of this liquid. Although discussion of these theories in detail is not the purpose of this article, it is important for a full appreciation of the experimental results that they be related to theory whenever possible. To facilitate this comparison and simplify references, a very brief summary of several theories of 3He is presented below.Q1 1.1. LANDAU’S THEORY OF LIQUID3He
The theory of an isotropic Fermi liquid was developed by Landau2p3. and extended by Abrikosov and Khalatnikov5*6, 7. Landau’s theory is based upon two fundamental assumptions: a) There is a one to one correspondence between the energy levels in a Fermi liquid (with interactions) and those in an ideal Fermi gas, i.e. the “switching on” of the interactions is adiabatic; the number of quasi-particles thus equals the number of atoms. b) The energy of a quasi-particle in a selfconsistent field is determined by the state of the surrounding particles ; the energy of the system is therefore not equal to the sum of the t Work performed under the auspices of the U.S. Atomic Energy Commission. References
9 . 150
114
E. R. GRILLY A N D E. F. HAMMEL
[CH. IV.
3
1
energies of the quasi-particles but is instead, a functional of their distribution function. From these assumptions, using a simple generalization of the Hartree-Fock method, Landau has been able to derive the properties of a so-called Fermi liquid in terms of integrals over the interaction potential. It is assumed that 3He is such a liquid on the basis oi its apparent failure (to date) to exhibit superfluidity. For Landau's theory to be valid, the excitation energies, of the order of KT, must be considerably greater than the quantum indeterminancy of the energy, i.e. kT hit, where z is the time between collisions and varies as T-2. In order to satisfy this inequality it is required that
>
T
< 0.3 "K
6*7.
For the specific heat C and entropy S Landau finds, in the limit T = 0, C = S = AT where A is the ideal Fermi gas constant with the particle mass m replaced by the effective mass m* of the quasi-particle. The nuclear magnetic susceptibility expression for a Fermi liquid, although including a large exchange interaction term favoring parallel spin orientation, predicts a net paramagnetic susceptibility which becomes constant in the limit T = 0 if the average exchange interaction is less than a certain critical value. The Fermi liquid viscosity and thermal conductivity were obtained by Abrikosov and Khalatnikov6,'. Their results confirm the predictions of Pomeranchuks made earlier, and give for the viscosity 7 = m/T2 (am poise degz) and for the thermal conductivity x = /?/T (b m lo2 - lo3 erg cm-lsec-l). Landau3 has also suggested that a new type of wave motion called zero sound should be observable in a Fermi liquid. This topic will be discussed in Section 7 . 3 . Although comparison of theory with experiment will be made in more detail in subsequent sections of this review, it may be stated here that the properties of liquid 9He are, with the exception of the thermal conductivity, either in agreement with, or at least consistent with Landau's predicted properties of a Fermi liquid 7 . If this theory of a Fermi liquid is applied to 3He at temperatures above a few tenths ot a degree Kelvin (but still much less than the ideal Fermi degeneracy temperature of 5 OK) it is found that the experimental and theoretical results diverge. Khalatnikov and Abrikosov5 have, however, shown that the Fermi liquid formalism can be used to 7 See footnote p. References p . 150
134.
CH.
IV,
9
11
LIQUID AND SOLID
3He
115
improve the fit to the experimental data for the specific heat and magnetic susceptibility up to 2 OK approximately, if the ideal gas type spectrum ~ ( p= ) p2/2m* is replaced by a shell-like distribution in momentum space of the quasi-particles with E ( $ ) = (fi - pJ2/2m* where p , is the radius of the shell. Usiisghas pointed out that such a spectrum as well as the ideal gas type used by Landau leads to a positive thermal expansion coefficient in the limit T = 0, in apparent contradiction with experiment, and suggests an alternative spectrum E(+) = d + p2/2m* which can yield the correct sign of the expansion coefficient.
+
1 . 2 . THE BRUECKNER AND GAMMELTHEORY OF LIQUID3He
Brueckner and Gammels have developed a theory of liquid 3He valid in the limit T = 0 and based on Brueckner’slO general theory for a many-body system. For the interaction between pairs of atoms, the potential of Yntema and Schneiderll was used. A summary of the results of the B-G theory for liquid 3He at 0 OK are as follows: a) The computed binding energy is about -k of the observed value and extremely sensitive to the potential used. Small changes in either the attractive or repulsive part of the potential used could easily improve the fit to experiment. b) The predicted compressibility is about 5.3 %/atm. c) The limiting specific heat is C = C,m*/nz where C, is the specific heat of an ideal Fermi gas with density equal to that of the liquid and m*/m is the ratio of effective mass to actual mass. m*/mwas evaluated as 1.84, giving C = 3.78 T cal mol-l degl. Brueckner and Gammel also predicted the correct variation, qualitatively, of C with pressure (through the increase of m*/m with pressure). d) For magnetic susceptibility, B-G derived X = X,E,(F)/Eswhere X, is the ideal Fermi gas susceptibility and E,(F)/E, is the ratio of magnetization energies for the gas and liquid, respectively. The energy ratio obtained was 12, which is in good agreement with that derived from susceptibility measurements. Such close agreement is probably fortuitous, however, as E,(F)/E, is very sensitive to parameters. The theory also predicts an increase of X with pressure and at sufficiently high compressions antiparallel spin alignment may become energetically unstable with respect to parallel spin orientation. e) Brueckner and Atkins12 and de Boerso extended the B-G calculations to show that the liquid expansion coefficient at zero pressure should be about -0.1 T degl.
References
p.
150
116
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
3
1
1 . 3 . GOLDSTEIN’STHEORY OF LIQUID3He
In order to correlate nuclear magnetic susceptibility measurements with other properties of condensed 3He, Goldstein13-ls devised a theory of the nuclear-spin system. He set the fraction per unit volume of atoms or spin moments which escape orientation by the internal field equal to the ratio of X/X,, where X is the actual susceptibility and X, is the limiting Langevin susceptibility which the liquid would have at its actual density if it were an ideal paramagnet. Since X I X , gives the fraction of free spin configurations, the entropy of the spin system is defined as S , = ( X / X o ) R In 2. Through classical statistical thermodynamic formulas other properties, such as specific heat, volume, and volume derivatives, can be derived for the spin system. Such a property, of course, is to be considered only as a portion of the total or observed property. However, at low temperature, where other influences die out, the behavior of the total property should approach that of the partial spin property. From the susceptibility data down to 0.23 OK, Goldstein obtained the following: a) in the limit T = 0, the temperature derivatives of the spin entropy and the spin specific heat approach 2.31 R ; this value sets a lower limit for the T = 0 slopes of the corresponding total properties. In this same limit, the contribution of the spin system to the expansion coefficient, a = l / V ( a V /aT),, tends to -0.126 T . From an estimate of the non-spin contribution to a,-0.103 T was obtained for the total expansion coefficient as T + 0. b) As T increases, ( aC/ W), and ( as/W), [= - aV] increase from zero, pass through maxima, and eventually become negative. At vapor pressures, these coefficients become negative at about 0.15 OK and 0.5 O K respectively. Since these and other derived results by Goldstein either agree with or are consistent with those obtained from experiment, the implicit assumption that exchange and other interaction effects in the spin system are automatically accounted for by using the experimental susceptibilities appears justified within the stated limitations of this model. 1 . 4 . PAIRCORRELATION THEORIESOF LIQUID3He
Although London’slgconjecture, that Bose statistics are essential for the formation of the superfluid phase, received support in subsequent theories20 of liquid 4He, it was not until recently that the apparently contradictory superfluid behavior of a-system of fermions, i.e. the electron gas in many metals, was resolved by the theory of Bardeen, Refevences
p . 150
CH.
IV, 3 11
LIQUID AND SOLID
3He
117
Cooper and Schriefferzl. The success of this approach prompted similar investigations of other strongly interacting fermion systems, the first of which was carried out by Cooper, Mills and Sessler22.Besides introducing a pair correlation function in the wave function and using the usual approximation21 that only pairs of given total spin and total momentum are strongly correlated, it was also assumed that the “normal” fluid corresponded to an uncorrelated “phase”, i.e. the normal fluid was treated as a system of independent fermions in a momentum dependent potential. It is in the nature of such a theory to search for bound states of “Cooper pairs”. I n this first work, Cooper et al. tacitly assumed the bound state to be of S wave character in angular momentum. A number of trial S state solutions were examined, but no transition to a correlated superfluid phase was found nor did a transition appear likely. Such a transition was not definitely excluded however. Emery and Sessler23 and simultaneously Brueckner, Soda, Anderson, and Morel24 extended the Cooper et aLZ2 treatment to include pair interactions for states of higher angular momentum. It was found that for states of relative angular momentum I > 0 (more probably 1 > 1) cooperative effects arise leading to a phase transition in the temperature range 0.05-0.1 OK. Immediately above the transition temperature, the specific heat should be proportional to the temperature with C m 2 C,, where C, is the ideal Fermi gas specific heat23. At the transition temperature T,, a discontinuity in the specific heat is predicted. The actual AC will depend upon the relative contributions from the various 1 and m modes. For example, for 1 = 2, m = 0 , AC = 0.71 C , where C, is the “normal” fluid specific heat at T,. Below the transition temperature, the particle pairs will be correlated for arbitrary directions in the medium. Since the correlation range is only about 400 A, the unperturbed liquid should break up into randomly oriented cells of roughly this dimension, in each of which a correlation axis will exist. In order to observe certain correlation dependent properties, it will be necessary to establish macroscopic polarization axes in some fashion, e.g. by viscous interaction of the flowing fluid with the walls. The properties then measured will be angle dependent. Below the transition temperature, an enhanced fluidity rather than perfect superfluidity should be observed in liquid 3He. It is well known that the latter phenomenon requires the existence of an energy gap, whereas the formation of a correlated phase involves only a reduction in level densityg4. References p . 150
118
E. R. GRILLY AND E. F. HAMMEL
[CH.
IV,
2
2. Theories of Solid 3He 2 . 1. QUALITATIVE
Pomeranchuk*, assuming negligible exchange effects from nuclear spin in the solid, concluded that the spin alignment temperature should be about lo-’ OK. Although the correct value proved to be about 0.3 OK (see Sect. 3.3a), Pomeranchuck’s discussion disclosed the important anomaly in 3He at T < w 1 “K that the solid entropy should be greater than the liquid entropy, which is observed as the negative thermal effect of melting and the minimum in the melting P-T curve (see Sect. 3.3a). Primakoff z5 predicted that nuclear spin alignment in the solid should decrease with incresing pressure. The corresponding entropy increase, at sufficiently low temperatures, led to the prediction that solid 3He will have a negative expansion coefficient (see Sect. 3 .3a). Goldstein 15, 16, 1 7 extended .his theory of the partial spin properties from the liquid (Sect. 1.3) to the solid, with the result that parallel behavior would be exhibited among certain thermal and PVT properties. 2 . 2 . THEORY OF BEKNARDES AND PRIMAKOFF
A quantitative analysis of the properties of solid 3He was made by Bernardes and Primakoff26, who began with a gas-phase LennardJones “12-6” potential modified at small interatomic distances. I n contrast with Porneranchuk, they concluded that exchange effects represent the predominant mechanism for spin alignment in the solid. Their calculations for P M 30 atm and T w 1 “K led to the following conclusions: a) The cohesive energy per atom is about R X 2.5 “ K ; b) the root-mean-square deviation of an atom from its lattice site is about 0.36 times the nearest neighbor distance; (c) the nuclear magnetic susceptibility X follows the Curie-Weiss law X = c/(T - 0) with a Weiss constant e of antiferromagnetic sign 0 w - 0.1 OK; d) the decrease of - 19 w T , with increasing pressure corresponds with a possible transition to ferromagnetic behavior at fi w 150 atm, which could be connected with an observed crystallographic transition (see Sect. 3.3b); e) at T,, the specific heat and susceptibility exhibit singularities (cusp-like or otherwise well-defined maxima) associated with the alignment of the nuclear spins; f ) the thermal expansion coefficient becomes negative below about 0.6 OK; and g) the melting Refersnrrs
p.
150
CH. IV,
9
31
LIQUID AND SOLID
3He
119
curve is characterized by a minimum at T w 0.37 OK and a maximum at T M 0.08 OK. The most striking predictions of this theory appear to be: 1) The singularities in specific heat and susceptibility; and (2) the maximum in the melting curve. The apparent absence of 1) in the liquid is ascribed ultimately to the difference in character between the associated quasi-particles (phonons and magnons or spin waves in solid; individual atoms with m* # m in liquid).
3. Pressure-Volume-Temperature Relations A rather large amount of work has gone into PVT studies of condensed 3He, beginning in 1949 with the vapor pressure and the density of saturated liquid27. The reasons for this great effort lie in 1) the inherent importance of determining the behavior of a second quantum liquid, whose properties were expected to be significantly different from thcse of 4He; 2) the technical need of knowing how to handle the substance in the course of many experiments; 3) the rapid development in the entire field of 3He studies, which naturally brought on simultaneous duplicative investigations. The total effort now covers the tempxature range 0.3 to 3.2 OK between vapor pressures and melting pressures and up to 30 OK along the melting curve. The measurements involved from 0.02 to 12 cm3 of liquid, used a variety of techniques, and usually attempted to obtain high accuracy. At present, therefore, the PVT data on liquid 3He are comparable in extent and quality to those on the much more available and “older” 4He, which in turn has received greater attention than most liquids. The studies on solid 3He have been limited for the most part to the region of the melting curve. 3 . 1 . AT VAPORPRESSURES
The vapor pressure of liquid 3He was measured originally by Sydoriak, Grilly and HammelZ7, then more accurately over the range 1.O-3.3 OK by Abraham, Osborne, and Weinstock28. Measurements were extended down to 0.45 OK by Sydoriak and R0berts2~,who cooled the sample in a liquid 3He bath and determined temperatures from the susceptibilities of two different paramagnetic salts. Sydoriak and Roberts also recomputed the data of Abraham et al. to derivea single equation accurate over the entire range of 0.45 to 3.327 O K (the critical point) and fairly reliable down to 0.28 OK. This equation is References
p.
150
120
E. R. GRILLY AND E. F. HAMMEL
[CH. I\’,
$3
+
In P(mmHg) = 2.3214 In T - 2.53853/T 4.8153 - 0.20644 T 0.08640 T2 - 0,00919 T3,
+
where T = T , is based on the “55E” scale of Clement30. The vapor pressure as T -+0 OK can be calculated from another equation of Sydoriak and Roberts provided that the spin entropy integral can be evaluated. The first serious attempt at high accuracy in saturated liquid density was made over 1.3-3.2 OK by Kerr3I, who tried to limit the error to 0.2 yo.P t ~ k h followed a ~ ~ with a technique that, unfortunately, allowed a possible error of 1 %, but her results agreed with Kerr’s within 0.2 yo up to 2.2” K, then jumped to 0.6% greater in molar volumes. Peshkov’s measurement^^^, through refractive index observation, yielded changes in density, particularly with pressure, more accurately than absolute values. Sherman and E d e s k ~ t yundertook ~~ an ambitious program to determine all of the PVT surface between the vaporization and melting curves from 0.96 to 3.32 OK with great accuracy. Their estimated possible error was less than 0.1% in molar volume, but their results are consistently higher than all the others, at both vapor and melting pressures (the latter comparison being made with the Grilly and Mills35 data)?. A t the time of this writing, Taylor and Kerr36 are remeasuring the molar volumes of the saturated liquid, particularly to determine the behavior of the expansion coefficient below 1 OK. It seems highly desirable to present here a consistent and “bestvalue” summary of PVT data. For molar volumes of saturated liquid, there appear to be three attempts to obtain high accuracy. In the region of overlap, 1.2-1.6 OK, the volumes of Taylor and Kerr are higher than those of K e n by 0.2Sy0and lower than those of Sherman and Edeskuty by 0.28%. It seems adequate, therefore, simply to average the data from these three sources. The resulting numbers are shown in Table 1. For the thermal expansion coefficient ct = V-l( aV/ aT),, the values in Table 1 were chosen as follows: for 0.3 to 1.2 OK, those of Taylor and Kerr; for 1.2 to 3.0 OK, those of Sherman and Edeskuty. The dividing temperature of 1.2 OK was selected because here both sets of data yield the same value and above it the conversion of the V-T slopes of Taylor and Kerr to isobaric derivatives becomes too uncertain. The low-temperature anomalous behaviour of ci is illustrated in Fig. 1, i.e., the values become negative at sufficiently low temperatures. t For explanation, References p . 150
see footnote on p. 122.
CH. IV,
9 31
LIQUID AND SOLID
3He
121
TABLE1 P V T relations of liquid 3He at vapor pressures
TE (OK)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.7 2.8 2.9 3.0 3.1 3.2
P (mmHg at 0 ' C)
(cw3mole-I)
0.00001 0.001 50 0.024 05 0.141 8 0.4985 1.291 2.744 5.092 8.564 19.765 38.03 64.91 101.93 150.55 212.28 288.60 381.02 433.73 491.00 553.01 619.92 691.88 769.04
36.785 36.746 36.722 36.713 36.719 36.741 36.777 36.830 36.899 37.089 37.354 37.705 38.18 38.83 39.67 40.75 42.15 43.03 44.05 45.30 46.90 49.00 52.30
V
loa a
102 ,B
(deg-')
(ah-')
- 8.46
3.68 3.69 3.70 3.73 3.77 3.82 3.89 3.96 4.18 4.49 4.94 5.61 6.58 8.03 10.36 14.43 17.9 23.4 33.6 58.8
-4.29 -0.13 4.12 8.50 13.1 17.9 23.1 34.0 48.7 66.5 89.4 118.3 155.4 207.4 287.1 344.0 439.0 593.0 932.0
The values of compressibility = -V-l( aV/ aP), of saturated liquid 3He come from these sources: the conventional PVT measurements over 1.0-3.0 OK of Sherman and Edeskuty; the index of refraction observations over 1.6-3.1 OK of Peshkov; and the velocity of sound measurements of Laquer, Sydoriak, and Roberts3' over 0.343.14 "K and of Atkins and Flicker3* over 1.2-3.2 OK. The last method gives adiabatic compressibility, which must be multiplied by C,/C,; therefore the uncertainty becomes excessive above 2 OK. The measurements were taken within ?n atmosphere of the vapor pressure except those of Sherman and Edeskuty, which were extended to the melting pressure. All the results are in remarkable agreement, i.e., within 1yo, except Peshkov's, which are lower by as much as 10% at T < 2.7 OK and still more at T > 2.7 O K . The results of the others for saturated liquid are combined in Table 1.
References
p . 150
1'1'1
E. R. GRILL\'
AND E. F. HAMMEL
Temperature
[CH. IV.
$3
(OK)
Fig. 1. Thermal expansion coefficient versus temperature for liquid 3He a t various pressures [from Lee and Fairbank*O]. The dotted curve represents additional results of Taylor and Kerra6 a t vapor pressures.
3 . 2 . AT INTERMEDIATE PRESSURES
Although the PVT area bounded by the vaporization curve, the melting curve, T = 1.0 OK, and T = 3.3 OK was measured by Sherman and E d e ~ k u t y ~there ~ , exists the possibility that their molar volumes are too high. Near vapor pressures, this excess might amount to 0.28% below 1.6 O K and somewhat more above 1.6 OK. Near melting pressures, their volumes are greater than those of Grilly and Mil1s3j by 0.3% in the region of 2.0-2.8 O K and by greater amounts above and below this region. Presenting their extensive array of data is only possible in tabIes such as theirs 7 . Below 1 OK, the major interest has been in the behavior of the thermal expansion coefficient a. From A T / d P measurements of adiabatic expansions covering pressures 1.7 to 22 ?tm, Brewer and Daunt39 derived a values over the range 0.15 to 0.6"K, all of whichwere negative. Up to 1.15 OK, they also obtained T where a = 0. Thus they showed that : a was negative below a temperature which monotonically t .4 recalibration of the cell volume shows that the molar volumes of Sherman and Edeskuty should be lowered by 0.30%. The corrected molar volumes are within 0.08% of those a t vapor pressures by Taylor and Kerr and of those a t melting pressures by Griily and Mills up t o 2.8 "K. Therefore, in Table 1 the values of V at T > 1.6 "K are now derived solely from the results of Sherman and Edeskuty. References
p . I50
CH. IV,
5
31
LIQUID AND SOLID
3He
123
increased with pressure; and a had, at 0.2 O K , minimal values which increased with pressure. More direct values of u, through dielectric constant measurements, were obtained by Lee and Fairbank40 between 0.2 and 29.5 atm over a 0.15 to 1.2 OK range. The behavior was similar to that seen by Brewer and Daunt. In Fig. 1 the data of Lee and Fairbank are compared with those along the vapor pressure curve derived from data of Taylor and From all the present results, it appears that CI ( P = 0) could approach T = 0 approximately as cc = -0.1 T , which was computed by de Boereo, by Goldsteinl6P1*, and by Brueckner and Atkins12. 3.3. AT MELTINGPRESSURES a) Low Region The melting curve of 3He was measured by Weinstock, Abraham, and Osborne41from 1.5 down to 0.16 OK by using the blocked capillary method. They found the melting pressure leveling off to 29.3 atm below 0.4 “K, which effect is to be expected when their technique is used below the temperature of a pressure minimum. Lee, Fairbank, and 40 used a dielectric constant measurement to distinguish between liquid and solid. They concluded there is a minimuminmelting pressure, Pmin, at 29.1 atm and 0.32 OK. The first to report a detailed study of the minimum were Baum, Brewer, Daunt and Edwards43, who measured pressures with a strain gauge cemented to the cell containing sealed-off 3He. The gauge was sensitive to f 0.02 atm and had been calibrated to f 0.1 atm when the vessel contained liquid 3He. The cooling and thermometry were accomplished through paramagnetic salts. The measurements showed Pmin to occur at 29.3 f 0.1 atm and 0.32 OK. Sydoriak, Mills, and Grilly4*used a system in which pressure on the solid could always be measured as the sum of a spring pressure plus liquid 3He pressure. Before and after 3He measurements, their bourdon gauges were calibrated, in sitw, to i 0.02 atm. Cooling and thermometry involved separate compartments of liquid 3He. The melting pressures were within 0.1 atm of those of Sherman and E d e ~ k u t yin~ the ~ short region of overlap between 1.0 and 1.2 OK. Sydoriak et al. found the pressure minimum at 0.330 & 0.005 OK and 28.91 f 0.02 atm, and their points below 0.5 OK were consistent to & 0.02 atm with the empirical equation (where T = T , of ref.29) P(atm) = 28.91 Referenres
p . 150
+ 32.2 (T - 0.330)2.
124
E. R . GRILLY AND E. F. HAMMEL
[CH. IV,
3
They also observed at 0.308 "K the heating effect of melting connected with dP/dT < 0. The results of the various investigations are given in smoothed form in Table 2. The question of a minimum in the melting curve of 3He arose in 1950 when Pomeranchuks suggested its possibility (see Sect. 2 . 1 ) . Fairbank and Walters45 were first to observe the reversal in the heat of melting, at T M 0.4 O K , which corresponds to a negative dP/dT. TABLE2 3He melting pressures (atm) below 1.2 "K
T
(OK)
0.12 0.16 0.20 0.25 0.30 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20
Ref." 29.3 29.3 29.3 29.3
30.1 31.5 33.2 35.2 37.4 39.9 42.6 45.7
Ref..4s 30.8 30.2 29.8 29.5 29.3 29.3 29.45 30.15 31.4 33.1
Hef.44
Ref.4o
28.94 28.91 29.07 29.80 31.02 32.02 34.50 36.79 39.30 42.08 45.10
29.1 29.1 29.1 30.0 31.5 33.2 35.6
Their nuclear magnetic susceptibility measurements showed that alignment in the solid occurs at about 0.3 OK in contrast with Pomeranchuk's estimate of lo-' OK. From this behavior, Bernardes and Primakoff 26 concluded that solid 3He is a nuclear antiferromagnetic, in the paramagnetic region its Weiss constant O( m - T,) being about - 0.1 OK. The corresponding entropy value led to a predicted minimum in the melting curve at 0.37 OK where P, = 29.1 atm. Below the minimum, their calculated pressures agreed with the measurements of Baum et a1.43,while those computed from T , = 10V "K were much higher. Furthermore, they predicted a maximum in the melting curve at 0.08 OK and 31.7 atm, which did not arise from Pomeranchuk's theory. Such region has not yet been investigated experimentally. To obtain volume measurements below 1 OK and in the vicinity of the melting curve minimum, Sydoriak, Mills and grill^*^ used the apparatus already described for melting curve measurements. The Refcremes p . 150
$ 31
CH. IV,
3He
LIQUID AND SOLID
125
TABLE3 Molar volume of melting liquid (V,) and volume change of melting ( A V,) for SHe below 1.2 "K ~
T("K) Vl(cm3/mole) d V , (cm3/mole)
0.5 25.84 1.19
0.33 25.99 1.20
0.6 25.65 1.18
0.8 25.17 1.14
1.0 24.63 1.10
1.2 24.07 1.06
molar volume of liquid Vl and volume change on melting AVm, with estimated possible errors of 0.1 yo and 1yo,respectively, are presented in Table 3. They joined smoothly the results of Grilly and Mills35 above 1.2 OK. The expansion coefficient is shown in Fig. 2. The anomalous negative values of m1 found a t lower pressures are seen to persist up to the melting pressure. From the relation
and the observation that dVl/dPm was approximately constant as P + Pmin, it appeared likely that ml + 0 as T -+Tmin.Furthermore at Tminthe term in parentheses of eq. (1)seemed to be zero, which would make dorl/dT = 0 , This behavior of ccl does not readily fit in with that expected at low pressure, i.e., the vanishing of al at T = 0 with finite negative slope, and therefore deserves more study. In contrast, the
-
5
I
m
d o T 2
-5
U
-10 0.2
0.01
0.2
1.0
0.6
1
I
0.6
I
I
I .o
1.4
I
I 1.4
Tmt deg K Fig. 2. Thermal expansion coefficient (upper figure) and compressibility coefficient (lower figure) of 3He a t melting pressures [from Sydoriak, Mills and grill^^^]. A Grilly and Mills35; Sherman and Edeskuty34; @ calculated from a , using eq. (1). Broken curves represent gS calculated from eq. (2) - - - : assuming pS = p , ; - - - - - - : assuming ps = 0.99 p,.
References
p . 150
126
E. R. GRILLY A N D E. F. HAMMEL
normal behavior of the liquid compressibility
[CH. IV,
53
is also shown in Fig. 2 ,
as measured and as calculated from eq. (1).
The thermal expansion of the solid was calculated from g,
1 dV, dV, 1 - __ -- + +-
=
V, d T
V, dT
dPm
( B E
- 81) dT
(2)
with the reasonable assumption that 8, 2 /I,. As shown in Fig. 2, it was concluded that g becomes negative at a T, of 1.0 to 1.1 OK, in qualitative agreement with the theories of Goldstein1a.17 and of Bernardes and Primakoff 25,26. b) High Region The melting curve in the region 1.2-31 OK and 50-3400 atm was measured by Mills and grill^*^, using the blocked capillary method under a procedure that insured obtaining equilibrium values within very narrow limits. Estimated possible errors in meltin: pressure were 0.02 and 0.2 atm below and above 240 atm, respectively; temperatures were significant within O.O0lo up to 5", within 0.1' between 5' and 14", and within 0.01' above 14". The results are reproduced by three equations :
+ 15.5053 T2 - 1.35019 T 3 for 1.2 < T < 3.148 = 3.748 + 29.5713 T + 3.95049 T 2 for 3.148 < T < 4.4
P(atm) = 26.379
= 24.35
-
0.62615 T
+ 19.4362 T1.517'38
tor 1.9 < T
OK; O K ;
< 31 OK.
Sherman and E d e ~ k u t obtained y~~ similar results and give the equation P(atm) = 24.559 16.639 T 2 - 2.0659 T3 0.11212 T4 for 1.07 < T < 3.1 O K . Several volume relations near the melting curve were examined by Grilly and Mills35 over the range of 1.3-31 'K and 50-3500 atm. Measured directly were: the molar volume of fluid Vf to 0.1 yo;the volume change on melting d V m to 0.5%; the thermal expansion and compressibility of fluid, and Bf, respectively, to 5%. The A V , measurements led to the unexpected conclusion that there exist two forms of solid 3He. The d V of transition was found, indirectly, to be about 10% of AV,. The P-T curve of transition was determined from the sudden change in compressibility accompanying the phase change. The phase diagram is shown in Fig. 3, and some properties of the transition are listed in Table 4. Subsequently, X-ray diffraction
+
References
p . 150
+
CH. IV,
3 31
LIQUID A N D SOLID
3He
127
T (deg K )
Fig. 3. Phase diagram for condensed 3He [from Grilly and MillsS5].
studies by Schuch, Grilly, and Mills4' showed that the cr-solid, existing at lower pressures, has the body-centered-cubic structure and the ,&solid has the hexagonal-close-packed structure. Furthermore, the lengths of unit cell axes yielded a density equal to 0.154 f 0.004 g/cm3 a t 1.9 "K and 96.8 atm and density equal to 0.172 f 0.004 g/cm3 at 3.3 OK and 177 atm, which are in good agreement with the values derived by extrapolation from the direct measurement^^^ along the melting curve. TABLE4 Properties of the
T (OK)
1.8 2.2 3.6 3.0 3.148
References
p . 150
+ 01 transition in in solid 8He AV
(aW
dPjdT (atmideg)
(cm3/mole)
AS (cal/deg/mole)
107.9 113.2 120.8 131.3 135.9
10.7 16.0 22.4 39.9 33.0
0.068 0.094 0.116 0.123 0.125
0.017 0.035 0.061 0.087 0.098
P
128
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
33
The results on Vf and AVm(cm3/mol)are reasonably represented as functions of melting pressure (atm) by:
V , = -3.248 + 50.841 ( P + 1.04)-0.1a1532for 50 < P < 3440; AV, = 1.55910 - 0.39023 log,, (P - 29.033) for 50 < P < 135.92; A V , = 1.506 15 - 0.30825 log,, ( P - 41.212) for 135.92 < P < 3440. Since AV,(P > 136 atm) was observed to decrease with pressure in a regular fashion, it was interesting to examine the behavior of the
Fig. 4. The thermal expansion coefficient, q ,and the compressibility coefficient, of fluid SHe along the melting curve [from Grilly and Millsss].
pi,
corresponding entropy change AS,, which could be computed from AV, and dP/dT through the Clapeyron equation. A formula for A S , as a iunction of P gave a maximum in A S , at 4080 atm, which is only slightly higher than the experimental range, and indicated that A S , = 0 at 77 x lo3 atm (T = 235 OK). Therefore, while a critical point in a melting curve has never been seen, the requirements of one, A S , = dV, = 0 , could possibly be met in 3He, both in principle and technically. References
p . 150
CH. IV,
3 41
LIQUID AND SOLID
3He
129
The compressibility of fluid seems to behave normally aU along the melting curve, i.e., it decreases regularly with increasing pressure and temperature and never changes sign (see Figs. 2b and 4). Previous discussions brought out the anomalous negative values of thermal expansion below 1.2 OK (see Fig. 2 4 . At higher temperatures, ccI first rises to a maximum at 3.1 OK and 140 atm and thereafter falls in a regular way, as shown in Figs. 4 and 5.
I 0
2000
3000
P,(kg C m " )
Fig. 5. The thermal expansion coefficient of fluid SHeand *He along the melting curve [from Grilly and MillsSS].
4. Thermal Properties 4.1. SPECIFIC HEAT
The history of specific heat measurements on liquid We has reflected the interest in trying to answer the questions: 1) Is there a lambda or other type anomaly in the specific heat-temperature curve ? 2) How does the specific heat extrapolate to 0 OK ? In seeking answers to these questions, investigators successively lowered the temperature limit of measurements. The early measurements of de Vries and Daunt 48 from 0.57 to 2.3 OK were improved and extended by Roberts and Sydoriak'g to 0.37 OK, by Abraham, Osborne and Weinstock50 to 0.23 OK, and by References
p.
150
130
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
54
Brewer, Sreedhar, Kramers and DauntK1to 0.085 OK. The first three series obtained Csst,a specific heat for a change of state while the liquid remains saturated (and thus involves three variables : P,V,T) while the last measured C,, where P was constant at 6 to 14 ern Hg. Qualitative, as well as quantitative, differences can occur between the various specific heats, as shown by Goldstein16 in Fig. 6.
TVK)
Fig. 6. Various molar heat capacities of liquid SHea t vapor pressures [from G o l d ~ t e i n ~ ~ ] The dotted curve represents additional results of Brewer et al."'.
In the region of 0.5 to 1.7 OK, the data of Roberts and S y d ~ r i a k ~ ~ were assigned probable errors of 1.5 to 2.0% and fit the empirical formula C,,, = 0.577 0.388 T 0.0613 T3 cal mol-1 deg-l
+
+
with a mean deviation of 1.0%. Below 1 OK, the merging of C,,, and
C, permits a direct comparison in Fig. 7. These results, combined with the early warm-up observations up to 3.21 "K by Sydonak and Hamme16*, permit us to conclude there is no btype transition in liquid 3He down to 0.085 OK. Furthermore, no maximum of any kind, except possibly in C , at 2.5 "K, appears. Below 0.7 OK, the behavior of C is interesting in that the very small variation down to 0.2 OK rapidly changes, so that the extrapolation of C at 0.085 OK to C = 0 at 0 OK by Brewer et aL61appeared reasonable and consistent with the linearity predicted by the theories of G ~ l d s t e i nl6, ~ ~Landau29 . 4, and Brueckner and GammeP. References p . 150
131
LIQUID AND SOLID 3He
CH. IV, 10
I
09 08
2 07 go6 r05 J
204 0 03
a? 0 0
01
a2
04
03
06
05
07
TEMPERATURE (K)
Fig. 7. C , for liquid *He versus temperature [from Brewer, Daunt, and Sreedhara']. rn De Vries and Daunt48. Roberts and Sydoriak". x Abraham, Osborne and Weinstock &O.
+
TABLE5 Specific heat of liquid SHe in cal mole-' deg-l as a function of P (atmospheres) and T (OK)
P T
V.P.
< 0.1 0.10 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
References p . 150
4.00 T 0.400 0.472 0.555 0.640 0.684 0.714 0.737 0.757 0.777 0.793 0.807 0.823 0.845 0.867 0.890
15
29
0.510 0.565 0.609 0.630 0.640 0.648 0.654 0.662 0.671 0.683 0.693
0.522 0.571 0.602 0.612 0.617 0.622 0.627 0.637 0.647 0.659 0.673
5 -
~
0.494 0.560 0.626 0.662 0.685 0.698 0.706 0.717 0.732 0.748 0.764
. -
132
E. R . GRILLY AND E. I;. HAMMEL
[CH. IV,
34
Brewer, Daunt and S r e e d h a ~measured -~~ C , at pressures up to the melting pressure between 0.12 and 0.6 OK. The results, partially reproduced in Table 5, show that ( aC,/ aP),is negative above T w 0.16 "K and positive below this temperature. Near 0 OK, a positive value was predicted by Brueckner and Gammels, Hammel et u Z . ~ ~ , and Goldstein16,whereas a negative value arises from an ideal Fermi-gas model. The lowest temperature, 0.12 "K, was not sufficient to allow reliable extrapolation of C, to 0 "K (but see Sec. 4 . 2 on the entropy). 4 . 2 . ENTROPY
The early m e a s ~ r e m e n t s ~50~ ~of specific heat were not at low enough temperatures to allow extrapolation to 0 "K so as to yield absolute entropies directly. Brewer et uLsl linearly extrapolated their specific heat results (equivalent to Csat) from 0.085 OK to 0 OK thereby obtaining Saatto i 0.03 cal d e g l mole-l. The possible error inherent in this procedure is emphasized by the p r e d i ~ t i o n 2 ~of9 ~a~specific heat anomaly below 0.1 OK. Such an anomaly would influence the limiting slope of C and S, and it might change the values of S above the anomaly temperature. However, the data of Roberts and S y d ~ r i a k ~ ~ , who obtained their absolute entropy values from the thermodynamic vapor pressure equation1, agree with those of Brewer et ~ 1 . Another ~ ~ . way of deriving S,,, is through combination of calculated vapor entropy and measured vaporization heat AH,,, which was done by Abraham, Osborne and W e i n ~ t o c kTheir ~ ~ . measured 499
AH,
= 10.39 & 0.02 cal
mole-1 at TL5S=1.5 "K,
from which Seat(1.5OK) = 2.614 f 0.03 cal d e g l mole-l.
Combining this value with their specific heat measurements 5 0 , which are quite consistent with those of other investigators (see Fig. 7), one finds that their entropy values down to 0.23 OK are higher than those of Brewer et al. by 0.10 f 0.06 cal d e g l mole-l. As Table 6 is based on the Ssatvalues of Brewer et al., one should understand from the above discussion that there is a slight uncertainty in the reference zero of the data presented. At higher pressures, extrapolation of C, to 0 OK was more uncertain. Therefore, C, was used only to derive AS,, which was combined with entropy of compression (S, - Sgat).The latter was computed originally References p . 150
CH. IV,
9: 41
LIQUID A N D SOLID
3He
133
by Brewer and Daunt 39 from their thermal expansion results to yield values of S , as a function of pressure and temperature up to 22 atm and 1 OK, respectively. However, their values were slightly altered, using the more direct expansion coefficients of Lee and Fairbank40, to those given in Table 6. TABLE 6
Entropy of liquid SHe in cal mole-' deg-' as a function of P (atmospheres) and T ("K) P T
V.P.
5
10
15
22
4.00 T 0.476 0.594 0.766 0.914 1.042 1.153 1.253 1.344 1.426 1.503 1.573 1.703 1.822 1.933 2.036
4.44 T 0.516 0.635 0.807 0.951 1.073 1.180 1.273 1.357 1.432 1.603 1.568 1.684 1.792 1.890 1.982
4.77 T 0.547 0.666 0.836 0.977 1.097 1.200 1.290 1.370 1.443 1.511 1.574 1.686 1.790 1.881 1.966
5.12 T 0.581 0.701 0.871 1.010 1.125 1.225 1.312 1.389 1.460 1.524 1.585 1.696 1.792 1.881 1.962
5.55 T 0.619 0.740 0.910 1.046 1.161 1.258 1.342 1.417 1.487 1.549 1.607 1.712 I. 805 1.887 1.964
______
T+O 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00
Examination of these -Jta led Brewer and Daunt to t,,e conclusion that as T + 0, SPIT= y = Cp/T,where y is a constant. This observation, then, lends support to the Fermi-liquid model theories of Landau23 4 (see calculations by Khalatnikov and Abrikosov5! ') and of Brueckner and Gammel9 as well as to the nuclear spin theory of Goldstein13r15.From the theory of Brueckner and Gammel, y is expected to be 3.78 cal mole-1 d e g 2 at Psatand to increase with pressure. Using the relation of G01dstein~~. l5 between nuclear magnetic susceptibility and spin entropy, Brewer and Daunt obtained y values of 4, 5, and 6 at 0, 11.2, and 27.6 atm, respectively, which are close to their observed values for total entropy. In Table 6, one can see that the normal variation of entropy with pressure is reversed at low temperatures, starting at T M 1 OK for the higher pressures, and becoming completely reversed at T < 0.6 OK. This behavior is also consistent with the predictions of Goldstein15 and of Brueckner and Gamnielg. References p . 150
13.4
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
$5
5. Transport Properties of Liquid and Solid 3He 5.1. THERMAL CONDUCTIVITY AND
LIQUID Several measurements have been made of both the thermal conductivity and viscosity of liquid 3He with the result that the data now extend from approximately 0.26 OK to the vicinity of the boiling point, as shown in Figs. 8 and 9. Although some slight discrepancies exist between the experimental values from different laboratories, the VISCOSITY OF
0.3
0
A,0,
TPK) Fig. 8. The thermal conductivity of liquid SHe. Lee and Fairbanks*, p = 3 a t m ; Challis and Wilks6'.
+
temperature dependence over the indicated temperature range of each of these quantities now appears to be well established. In the thermal conductivity measurements of Lee and F a i ~ - b a n k ~ ~ , anomalous x values were observed at first below the density maximum for high heat fluxes. This was attributed to a contribution from convective heat transfer in the liquid sample ; consequently below this temperature (0.48 OK at 8 atm) and for the higher heat currents, the direction of the heat flux was inverted with respect to the gravitational field. A t high temperatures although the heat transport through the walls of the containing tube was larger than the heat flow through the sample, accurate corrections were made. At low temperatures, where the corrections were smaller, they were slightly less well known due to the perturbing effect of the Kapitza boundary resistance 62 t . The t A t the VIIth International Conference on Low Temperature Physics, University of Toronto, 29 August-3 September, 1960 (see the Programme, p. 22), J. Jeener and References p . 150
LIQUID AND SOLID 3He
CH. IV, 5 51
135
thermal conductivity values of Challis and WilksS7 have not been corrected for the thermal boundary resistance. These authors estimate
T W )
Fig. 9. The viscosity of liquid aHe from Peshkov and Zinov’eva68. Osborne and Abraham68; Taylor and Dashao;0, x Zinov’evas’.
+
A Weinstock,
F’
>
11#
2
. . ..
I
1.
Fig. 10. The ratio of the thermal conductivity t o the product of viscosity and specific heat a t constant volume as a function of temperature for liquid *He and liquid 4He. The kinetic theory value of this ratio, 2.5, is shown b y the horizontal line. The upper curves are calculated using the viscosity data of Zinov’evaal; the lower using the viscosity values of Taylor and DashGo.See footnote p. 134.
that the application of this correction would increase their x values of the order of 10% and hence bring them into closer agreement with G. Seidel showed that the boundary resistance corrections would raise the conductivities of Lee and Fairbank as T falls below 0.5 “K (by 30% at 0.25 OK). Accordingly, x
increases as T decreases. Riferefices p . 150
136
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
5
those of Lee and Fairbank in the range of overlap. Since tabulations and discussion of the viscosity values can be found in the original articles and also in the review article by Peshkov and Z i n ~ v ’ e v a ~ ~ , these data will not be further reviewed here. Attempts to correlate the properties of liquids usually derive from the assumed similarity of this phase either to a disordered solid or a highly compressed gas. The high zero point energy of liquid 3He and its associated expanded structure suggest that the gas model should be the more applicable. A t “high’’ temperatures, this assumption appears to be justified by the fit of x , 7, and C, to the kinetic equation x = 5 / 1 2 C,q as shown in Fig. 10 taken from the work of Lee and F a i r b a ~ ~ k ~ ~ . The apparent failure of this equation below 1 OK suggests that some new process may be contributing in this temperature region to the transport of energy or momentum. Finally it is of interest to note that below 1 OK the marked change in temperature dependence of the viscosity, which has been interpreted as the beginning of a transition to the expected T-2 dependence of 7 for a Fermi liquid’, is not reproduced by the thermal conductivity (see footnote pag. 134). 5.2. HEATTRANSPORT IN SOLID3He
The heat conductivity in solid 3He has been measured by E. J. Walker and H. A. Fairbank63. For a dielectric solid at low temperatures the thermal conductivity should be given by the expression x = ATne-elbT’,
and the results (shown in Fig. 11) demonstrate that this relationship is obeyed by solid 3He. The changes in slope for the lower density curves have been tentatively identified with the change in the sign of the expansion coefficient of the solid reported by Sydoriak, Mills and Grilly44. The discontinuity in the curve BB’ is attributed to the a+ phase change which occurs at this density as the temperature is reduced (see Fig. 3). 5 . 3 . SELF-DIFFUSION COEFFICIENTFOR LIQUID3He
The coefficient of self-diffusion has been measured in liquid 3He, using spin echo techniques, by Garwin and Reich6*and by Hart and Wheatley‘j5t.The former authors determined the pressure as we1 as the t These measurements have recently been extended to 0.03 “K by Anderson, Hart, and Wheatleye2. The magnetic susceptibility was simultaneously determined. References 9. 150
CH. IV,
4 51
LIQUID AND SOLID
I
I
I
I
3He
I
1
137
I
VT PK-~)
Fig. 11. Thermal conductivity of solid SHe from E. J . Walker and H. A . Fairbankss.
temperature dependence of D.Within the experimental error the coefficient of self-diffusion for pure liquid 3He is given as a function both of T and e, by the empirical equation
D
=
5.9 In
0.16 (T) exp (T/2.8)
applicable between about 1.5 and 4 OK for pressures from 2.4 to 67 atm. A t about 0.55 OK the diffusion coefficient in the saturated liquid passes through a minimum and increases rapidly below 0.2 OK as shown in Fig. 12, taken from the work of Hart and Wheatley. Above approximately 0.6 O K , it is apparent from eq. (3) that the References p. 150
138
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
36
diffusion process in liquid 3He is neither thermally activated [requiring an exp (-T,/T) type temperature dependence] nor gas-like (for which D = IAV cc TIJa). Garwin and Reich suggest that the observed dependence is explicable qualitatively by considering the diffusion of 3He to be a quantum mechanical tunneling through potential barriers. The increase in D at low temperatures is, according to Hart and Wheatley, probably caused by a decrease in the probability of atomic scattering processes, (equivalent to an increase in the excitation mean free path predicted by Pomeranchuk* and Landau2).
*-
VERAGED DATA, 5-6 CM-HG ATURATED VAPOR PRESSUR GARWIN AND REICH. 238ATM 2
5
Fig. 12. Logarithm of the self-diffusion coefficient of liquid W e vs the logarithm of the temperature. The data of Garwin and Reich are taken from ref.u4.
5 . 4 . SELF-DIFFUSION COEFFICIENTFOR SOLID3He
The preliminary data available is discussed in Sect. 6 . 2 . 6. Nuclear Spin Relaxation in Condensed 3He 6 . 1 . LIQUID3He
The spin-lattice or longitudinal relaxation time T , of a spin system is defined as the time necessary for all but l/e of the spins, following an instantaneous change of state, to reach thermal equilibrium with the other degrees of freedom of the medium containing the nuclei in question. T , is therefore a measure of the coupling or interaction between the nuclear spin system and the “lattice”. According to the References p , 150
CH.
IV,
9 61
LIQUID AND SOLID
3He
139
theory by Bloembergen, Purcell and Pound66,the spin relaxation of a given nucleus in a pure liquid is caused by the Fourier components, at the Larmor frequency, of the fluctuating magnetic fields generated at a given nucleus by the thermal or Brownian motions of adjacent nuclei. The associated relaxation time is given by
where y is the gyromagnetic ratio, b the average interspin distance, z, is the "correlation time" of the motion (z, is a measure of the time interval during which molecular orientation persists and the local field at the nucleus is approximately constant) and w o is the precession frequency in the field Ho(wo= yH,). For mot, 1, a condition fulfilled in all of the spin relaxation experiments carried out in liquid 3He to date (t,M 10-12 sec, w o rn lo7), eq. (4)can be simplified to yield Ti1 = 0.9 '/41i2b-'tc.
<
For monatomic liquids with diffusion coefficient D , this equation becomes
Ti'
=
0.3ny4@No/aD
(5)
where a is the atomic radius and N o is the number of magnetic moments per unit volume. The Stokes-Einstein expression relating the diffusion and viscosity coefficients may also be used to give? Ti' = 9n2y4?i2qNo/5kT.
(6)
Equations ( 5 ) and (6) in principle permit a comparison of T , values computed from the BPP theory, using either experimental viscosity599 60, or diffusion coefficient64 data, with experimentally determined values of the same quantity 67-75. Although some uncertainty remains with respect to the appropriate numerical coefficient to be used in the application of these equations to liquid 3He7097679, this problem is at the moment of much less import than that of resolving the widely divergent reported results for T , in liquid 3He. These results initially appeared so confusing that the present authors sought information from the investigators involved, all of whom have respond7 However the gas-kinetic relationship, e D / q = constant, describes the relationship between q and D for liquid 3He over a wider temperature range. References
p . 150
140
E. K. GRILLY AlYD E. F. HAMMEL
[CH. IV,
9
6
ed. Their contribution to this section is gratefully acknowledged. The following is then, a summary of the status of spin-lattice relaxation time results based upon these communications and published reports. a) On the basis of our present understanding of spin relaxation processes, systematic errors in the measurement of T , should produce shorter rather than longer values of T I . Consequently from the available experimental results, those measurements yielding consistently the largest values of T , should most closely approximate the true value. Figure 13 shows one such spin-lattice relaxation time measurement along the saturated vapor line. These results were obtained by R ~ m e r and ~ ~ ,represent the largest T,’values reported to date (see ref. 93 however). b) Wall relaxation processes have been shown, particularly by Careri, Modena and Santini’O. 71, to yield spuriously short relaxation times. These processes were probably absent in R o r n e r ’ ~work ~ ~ since 900
I
I
I
I
I
1
1
‘
1
I
/
u I0 I25
‘075
15
-
175 2 0 225 250 TEMPERATURE ( O K )
2 275 30 325
Fig. 13, Relaxation time, T , V ~ Y S U Stemperature for various fields, [from Rorner7*]
his measured T,’s were found to be independent of container surface to volume ratios. Romer also used Pyrex glass containers, the walls of which are known to be poor spin relaxing surfaces. c) Bulk impurities (of the order of 1 part in l o 7 oxygen or other paramagnetic impurities in the liquid) can also yield spuriously short relaxation times. Hence even longer T , values than those measured by Romer are not excluded, and the results shown in Fig. 13 must therefore still be regarded as tentative. References
p . 150
CH. IV,
5 61
LIQUID AND SOLID
3He
141
d) R ~ m e r ’ sT ~, ~ results are qualitatively consistent with the predictions of the BPP theory, i.e. equations ( 5 ) and (6). e) The BPP theory predicts that T , should be independent of magnetic field. Romer’s results (Fig. 13) demonstrate this independence from about 1-3.2 OK, and for fields from 1560-12 200 gauss. According to recent work by Low and Rorschache3 field dependent T , values probably originate in field dependent wall or bulk liquid impurity relaxation processes. f) The BPP theory also predicts that, provided coot, 1, the spinlattice relaxation time T , should be equal to the spin-spin relaxation time T2. Schwettman, Low and Rorschachso have measured T , and T , in liquid 3He over the temperature interval 1.2-2.5’ K and found that T , w ilaT1. The values of T , so found were about 30 sec, independent of chamber size and the value of T,. This result is obviously not in agreement with the BPP theory. On the other hand, neither has it been shown that T , is unaffected by bulk impurities (see also ref. 93). g) Garwin and Reich72 have reported T , as a function of pressure. Their results at 2.38 atm (the lowest pressure studied), showrrin Fig. 13, are considerably lower than R ~ m e r ’ and s ~ ~show a different temperature dependence. For reasons noted above, it is probable that both wall and bulk impurity processes produced, in the low pressure results, the low T , values. As the pressure was increased the temperature dependence of TI at low temperatures changed sign and T , passed, with increasing temperature, through a maximum which shifted progressively to higher temperatures. At the highest pressure studied, T , increased with temperature over almost the entire liquid range (2.0-4 OK). In the region 1.5-3.2 OK, T , was found to decrease with increasing pressure. h) Schwettman, Low and Rorschachso observed a decrease in T , with time in the course of a single run. They have also, when the same sample chamber is used for several successive runs, observed a monotonic increase in values of T , from one run to the next. Observations such as these and those of Careri et aZ.70v71 emphasize the caution which must be exercised in obtaining reliable data from any given experimental apparatus or procedure. RorschachB1has also suggested that by postulating a wall relaxation time rn 1/D, and a bulk relaxation time M D,most of the published results on T , can be understood (see also ref. 93).
<<
Refercwces p . 150
142
E. li. GRILLY -4SD E. F. HAMMEL
[CH. IV,
96
From the above it is clear that although considerable progress has been made in clarifying this subject, additional work is still required to elucidate the nature of wall relaxation processes and the degree of applicability of the BPP theory to liquid He3. And finally in view of Hart and Wheatley's 65 results on the temperature dependence of the diffusion coefficient below 0.5 "K, it is clear that measurements of T , and T , at progressively lower temperatures will prove extremely interesting. 6 . 2 . SOLID3He
Reichs2 and Goodkind and Fairbanks3 have recently measured T , and T , in solid 3He as a function of temperature and pressure, and Reich, in addition, has measured D. Their results are summarized below. a) In the u-phase T , M T , in the high temperature region (wt 1) as would be expected according to the BPP theory of nuclear spin relaxation modified for relaxation by translational diffusion by Torre^'^. Upon reducing the temperature at constant solid density T , and T , both decrease and then diverge, T , increasing and T , decreasing. After passing through the minimum and entering upon the region where ~t 1, TIincreases initially, as exp (T,/T) and then becomes constant for all densities. A t constant temperature (in the vicinity of 2 OK) a discontinuity in T , and T , (TI increasing, T , decreasing) is observed upon passing from the a to ?j 3He solid. At lower temperatures, Goodkind and Fairbank find this discontinuity replaced by a discontinuity in slope of the In T , and In T , curves vs 1/T. b) In the high temperature region (i.e., COT 1) for which T is greater than about Tm/2and in the a-phase, Reich's results indicate that
<
>
<
D
=
Do exp (--T0/T).
(7)
To was found to increase with increasing density according to To = 5.2Tm.Do was shown to be approximately equal to 3.5 x cm2/sec, independent of density. At the lowest density, D becomes constant as the temperature is further reduced. The results of both investigators are tabulated below.
References p . 150
CH. IV,
9 71
LIQUID AND SOLID
3He
143
TABLE 7 Activation energies for diffusion in solid a
Approx.
e
klcvn3)
____
0.13407 0.13881. 0.148* 0.150* 0.1500t 0.1561t
t
- 8He (see eq. (7))
_. -- -
...
.
7.68 9.79 12.5
14.0 13.70
15.45
ReichBe,* Goodkind and Fairbankas
7. Velocity of Sound in 3He
7.1. EXPERIMENTAL RESULTS
Direct measurements of the velocity of sound in liquid 3He along the saturated vapor pressure curve as well as at higher pressures have been carried out by Laquer, Sydoriak, and Roberts37 and by Atkins and 84. The sound velocity has also been measured by the
TEMPERATURE CK)
Fig. 14. Sound velocities in liquid SHe and 4He [from ref.871. 3He data at 5 MHz from Laquer, Sydoriak and Roberts37;4He data from Van Itterbeek and Forrezs6 and from Atkins and Stasiorsa. References
p. 150
144
[CH. IV, 9
E. R. GRILLY AND E. F. HAMMEL
7
TABLE8 Sound velocity in fluid 3He
T
("K) 0.0 0.5 1.0
1.5 2.0 2.5 3.0 3.2 3.40 3.87 4.22 8
b C
d
U,(P) (in m/sec) (from ref.S4)at indicated Bressures i n atmos.
LT,(saf) (iiz m/sec)
from ref.s7)
1 183.4 182.4 177.8 170.0 158.5 141.8 115.0 99.0 -
1938 191b 178 16Ob
136 1286
-
2
4
-
21Oa 206b 197 184b 167 160 -
2348 232b 228 221b 210 204 200c 185C 172bC
6
2588 255b 251 246b 236 231 -
8
9
-
-
-
2748 272b 269 265b 260 257 -
2788 270 266 -
Extrapolated from smoothed published data. Interpolated from smoothed published data. Gas values at 3.99 atm (see original paper for additional data). Inconsistent with value of 99 m/sec a t saturated vapor pressure from refs.s7 andS'J.
1.5. The velocity of 14 MHz sound in liquid SHe as a function of temperature at various constant pressures [from Atkins and Flicker 841.
CH. IV,
0 71
LIQUID AND SOLID
3He
145
latter authors above the critical temperature in the gass4. Along the saturated vapor pressure curve there is effectively no difference, within the experimental error, between the data from the two laboratories (see note d of Table 8, however). The results are displayed in Figures 14 and 15 and in Table 8. 7 . 2 . ATTENUATION
Sound attenuation measurements in 3He have not yet been reported. Although the attenuation in principle was obtainable from measurement of successive echo amplitudes in the sound transmission cell of fixed path length used in the velocity measurements, both groups of investigators found that these amplitudes were too poorly defined to permit their accurate measurement. In part this was due to the small liquid samples available which accentuated wall effects, and to the fixed geometry of the sound cell which prevented in sit% alignment of the crystal transducers. The theoretical implications of the variation of the attenuation with temperature have been discussed by Goldstein 15, Landau3, and by Abrikosov and Khalatnikov7. Pellam and Squire8' first measured sound attenuation in 4He I, and showed that in this liquid the classical expression
was obeyed from about 3.2" to 4.5 O K . Below 3.2 OK an extra attenuation was observed, increasing to a value many times the classical at the A-point. For 3He there is no a ?riori reason for expecting a deviation from the above expression until temperatures of the order of 0.1 OK are reached. Below 1 OK, the attenuation due to the thermal conductivity should be negligible due t o the smallness of the factor (C,/C, - 1). According to Abrikosov and Khalatnikov7 the first viscosity 7 5, the second viscosity for 3He, and the attenuation below about 1 "K should therefore be given almost entirely by the first viscosity term. At sufficiently low temperatures (i.e., less than 0.05-0.1 OK) as has already been pointed out by Pomeranchuck* and by Abrikosov and Khalatnikov' the viscosity and hence also the attenuation of liquid 3He should vary inversely as T 2 .
>
References p . 150
146
E. R . GRILLY A N D E. F. HAMMEL
[CH. I V ,
fi
7
7 . 3 . ZERO SOUND
A t some temperature below 0.3 OK, the quantum statistical properties of liquid 3He should also manifest themselves in its sound transmission behavior. Landau3 has suggested that ordinary compressional waves of sound will continue to propagate in the liquid provided the wave length is long compared with the mean free path of the quasi-particles, i.e. ~t 1 where t is the liquid relaxation time. In this region the classical attenuation (eq. (8)) should continue to be obeyed. For a Fermi liquid z cc T-2 however, so that for any given frequency there will be some temperature below which the above inequality will no longer be fulfilled; the wave length of the sound will approach that of the mean free path of the quasi-particles, and the sound wave will be strongly attenuated. At higher frequencies or lower temperatures, for which at 1, Landau predicts the existence of a new type of sound termed zero sound. Since the wave length of zero sound is very much less than the mean free path of the quasiparticles, collisions between the quasi particles are neither essential for its propagation nor capable of establishing local thermodynamic equilibrium in the path of the sound wave. Zero sound is thus a nonequilibrium type of wave propagation. It is characterized analytically by a periodic deformation of the Fermi surface (ie., a time variation in the distribution function). An example would consist of an extension of the Fermi surface at maximum amplitude in the direction of the wave motion and a lesser flattening of the surface in the opposite direction. Half a cycle later the deformation is reversed. The velocity of zero sound in liquid 3Hein the limit T -+0 is estimated to be slightly larger than that of first sound, namely 192 m/sec. Although in principlc, zero sound modes which differ from one another in their angular dependence of both velocity and amplitude are possible in a Fermi liquid, Landau considers it improbable that such modes can be propagated in liquid 3He. Experimentally, zero sound in liquid 3He should be equivalent to an ordinary compression-rarefaction wave in the medium and should be demonstrable by suitable ultrasonic techniques. The attenuation of zero sound will be proportional to T2 (h., to the number of collisions of the quasi-particles, which in turn result in absorption of the sound quanta), and independent of the frequency provided the energy qf the sound quanta is small in comparison with that of the quasi-particles, i.e., tiw kT. In addition both these latter
<
>
<<
Referemes p . 160
CH. IV,
9 S]
LIQUID AND SOLID 3He
147
quantities must be large with respect to the quantum uncertainty in the energy of the quasi-particles: kT Kw hz-l, the latter condition being a general one for applicability of the whole theory of a Fermi liquid. As the temperature is lowered further or the frequency increased, fio will become equal to or greater than kT. In this region a quantum calculation shows the attenuation t o be dependent only on the square of the frequency. Because of the high frequencies necessary for the propagation of zero sound, Kapitza’ suggested that this phenomenon might be effectively investigated using the satellite lines (Brillouin doublets) from the Rayleigh scattering of visible light. Investigation of this idea led to the conclusion that the frequency shift of the satellite lines would be related t o the velocity of zero sound by the expression d o = f qzc, where u is the zero sound velocity and q = (2o/c) sin #Iwhere , 8 is the scattering angle. There is some question, however, whether the intensity of the scattered beam will be sufficient to permit an accurate measurement of the effect.
> >
7.4. SOUND PROPAGATION IN LIQUID3He BELOW THE “PHASE TRANSITION”
According to pair correlation theories, at the transition temperature the attenuation will increase strongly. For temperatures below T , and low enough so that the number of quasi-particles is small, it has been predictedss that ordinary sound will again be propagated in the “superfluid” with a velocity
where p , is the momentum at the Fermi surface and p is the mass of a 3He atom. In the correlated phase the attenuation will be small (similar to *He) and will decrease with decreasing temperature due to the decreasing density of excitations. 8. Summary Since the writing of the article on 3He for this series in 1955,not only has much more experimental work appeared, but also theoretical descriptions of the liquid and solid have become much more sophisticated. In 1955, although the difficulties inherent in the simple ideal FermiDirac description of 3Hewere beginning to be recognized, no alternative References p . 150
148
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
08
treatment had yet appeared. Subsequently, several attempts to introduce the effects of interactions between the “particles” were made, with the result that, at the present time, our understanding of liquid and solid 3He has progressed considerably. Formidable mathematical difficulties still stand in the way of a quantitative theoretical description of 3He, however, and consequently assumptions, approximations, and experimental data have been required to derive theoretical predictions of new 3He phenomena. The degree to which the theoretical conclusions depend upon these approximations and assumptions is as yet not well established and for those cases in which experimental data is available to compare with theory, the correspondence, although sometimes impressive, is more often only fair. But it is probably naive at the present time to expect any theory to provide a complete and quantitative description of 3He. Hence if the different theories arc viewed by experimentalists as alternative approaches to an exceedingly difficult problem, and if the comparison of theoretical predictions with experiment is used by the theoreticians to draw conclusions concerning the validity of the various approaches employed, 3Hewill continue to be a rich and rewarding raw material for both experimental investigation and its complementary theoretical interpretation for some time to come. In summary, it appears that : a) The incipient linear temperature dependence of the specific heat as T -+ 0 provides a satisfactory agreement between quasi-particle theory and experiment, at least at low pressures. The prediction of a sharp maximum or discontinuity in specific heat at 0.03 < T < 0.08 by the pair correlation theories still lacks an experimental check. Even if a transition to a correlated phase is subsequently demonstrated, the quasi-particle description may still be valid in the temperature range T , < T T,. Above 0.2 OK, the specific heat lacks a basic explanation in much the same sense as in all other theories of the liquid state. Further specific heat work, both experimental and theoretical, is warranted on the compressed liquid and on the solid, including for the latter an investigation of the predicted singularity in specific heat at about 0.1 OK. b) In general, the experimental PVT relations of the liquid are fairly well established. The locus of the minimum in thermal expansion with respect to temperature and pressure requires further definition, however. For both liquid and solid along the melting curve, it seems that an anomaly in thermal expansion ( a = 0 ) might occur at about
<
Referewes p . 150
CH. IV,
9 81
LIQUID AND SOLID
3He
149
0.3 OK. A possible maximum in melting pressure at about 0.08 OK has been predicted but not yet observed. The experimental observation that the cr-type solid is not close-packed was unexpected theoretically and is not understood. c) Experimental determinations of transport properties in general tend to support Landau’s theory of a Fermi liquid. Although the predicted variation of viscosity as T-2 is not inconsistent with experimental data obtained to date, the thermal conductivity was found to have a positive? variation with T in the corresponding temperature range instead of tending toward the predicted T-l. Measurements to still lower temperatures are obviously required. The most recent data on the variation of the diffusion coefficient with temperature tend to support Landau’s predictions. d) Sound absorption in liquid 3He has not yet been the subject of an experimental investigation, and in view of the predictions of “zero sound” this topic appears to be a rewarding if difficult research problem. e) Because the nuclear magnetic susceptibility of liquid and solid 3He had been adequately discussed in previous reviews5*,this subject was omitted from the present article. Recently however the results of two new investigations by Low and Rorschachs3 and by Adams, Meyer, and Fairbankg5have appeared. The former presents additional susceptibility data in the liquid, and the latter work includes new measurements on nuclear resonance in both the liquid and solid phases, including a discussion of ferromagnetism in the compressed liquid and in the solid. Finally, new measurements of the liquid susceptibility to 0.03 OK have also been reported by Anderson, Hart, and Wheatleysz.
Acknowledgement The authors wish to express their appreciation to their many colleagues who contributed to this article by discussion, letter, or the sending of preprints of recently completed research. In particular the authors wish to thank Professor Robert Brout for many helpful comments on the theoretical section.
t
See footnote p. 134.
References p . 150
150
E. R. GRILLY AND E. F. HAMMEL
[CH. IV
REFERENCES
*
See review article by E. F. Hammel, Progress in Low Temperature Physics, Vol. 1. (North-Holland Publishing Company, Amsterdam, 1955). L. D. Landau, J. Exptl. Theoret. Phys. 30, 1058 (1956) [Trans. Soviet Physics J E T P 3, 920 (1957).] L. D. Landau, J. Exptl. Theoret. Phys. 32, 59 (1957) [Trans. Soviet Physics J E T P 5, 101 (1957).]
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6
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I. M. Khalatnikov and A. A. Abrikosov, J. Exptl. Theoret. Phys. 32, 915 (1957) [Trans. Soviet Physics J E T P 5, 745 (1957).] A. A. Abrikosov and I. M. Khalatnikov, J. Exptl. Theoret. Phys. 32, 1084 (1967). [Trans. Soviet Physics J E T P 5, 887 (1957).] See also A. A. Abrikosov and I. M. Khalatnikov, Keports on Progress in Physics 22. 329 (1959).
I. Ia. Pomeranchuk, J. Exptl. Theoret. Phys. 20, 919 (1960). K. A. Brueckner and J. L. Gammel. Phys. Rev. 109, 1040 (1958). lo K. A. Brueckner, Phys. Rev. 100, 36 (1955). I1 J. L. Yntema and W. G. Schneider, J . Chem. Phys. 18, 641 (1950). la K. A. Brueckner and K. R. Atkins, Phys. Rev. Letters 1, 315 (1958). la L. Goldstein, Phys. Rev. 96, 1455 (1954). 1 4 L. Goldstein, Phys. Rev. 102, 1205 (1956). 1s L. Goldstein, Phys. Rev. 112, 1465 (1958). la L. Goldstein, Phys. Rev. 112, 1483 (1958). 17 L. Goldstein, Annals of Physics 8, 390 (1959). 18 L. Goldstein, Phys. Rev. 117, 375 (1960). 19 F. London, Nature 141, 643 (1938); See also F. London, Superfluids, Vol. 2 (John Wiley and Sons Inc., 1954). *O R. P. Feynman, Phys. Rev. 91, 1291, 1301 (1953); 94, 262 (1954). a1 J. Bardeen. L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). *a L. N. Cooper, R. L. Mills and A. M. Sessler, Phys. Rev. 114, 1377 (1959). Is V. J. Emery and A. M. Sessler, Phys. Rev. 119, 968 (1960). 14 K. A. Brueckner. T. Soda, P. W. Anderson and P. Morel, Phys. Rev. 118, 1442 (1960).
H. Primakoff, Bull. Am. Phys. SOC.2, 63 (1957). N. Bernardes and H. Primakoff, Phys. Rev. Letters 2, 290 (1959): 3, 144 (1959); Phys. Rev. 119, 968 (1960). 87 S. G. Sydoriak, E. R. Grilly and E. F. Hammel, Phys. Rev. 75, 303, 1103 (1949). 18 B. M. Abraham, D. W. Osborne and B. Weinstock, Phys. Rev. 80, 366 (1950). 89 S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). 80 J. R. Clement, private communication; see W. E. Keller, Nature 178, 883 (1956). 81 E. C. Kerr, Phys. Rev. 96, 651 (1954) ; by private communication some small temperature corrections were proposed. ** T. P. Ptukha, J . Exptl. Theoret. Phys. USSR 34, 33 (1958) [Trans. Soviet Physics J E T P 7 , 22.1 8s V. P. Peshkov, ibid. 33, 833 (1957) [Trans. Soviet Physics J E T P 4, 607.1 84 R. H. Sherman and F. J . Edeskuty, Annals of Physics 9, 522-547 (1960). 8s E. R. Grilly and R. L. Mills, Annals of Physics 8. 1 (1959). 86 R. D. Taylor and E. C. Kerr, private communication. 37 H. L. Laquer, S. G. Sydoriak and T. R. Roberts, Phys. Rev. 113, 417 (1959). 88 X. R. Atkins and H. Flicker, Phys. Rev. 113, 959 (1959).
Is
so
CH. IV] 89 4O
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LIQUID AND SOLID 3Ht?
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D. F. Brewer and J. G. Daunt, Phys. Rev. 115, 843 (1959). D. M. Lee, H. A. Fairbank and E. J. Walker private communication. B. Weinstock, B. M. Abraham and D. W. Osborne, Phys. Rev. 82, 263 (1951); 85, 158 (1952).
4a
‘8
D. M. Lee, H. A. Fairbank and E. J. Walker, Bull. Am. Phys. SOC.4, 239 (1959). J. L. Baum, D. F. Brewer, J. G. Daunt and D. 0. Edwards, Phys. Rev. Letters 3, 127 (1959).
S. G. Sydoriak, R. L. Mills and E. R. Grilly, Phys. Rev. Letters 4, 495 (1960). 46 W. M. Fairbank and G. K. Walters, Bull. Am. Phys. SOC.2, 193 (1957); Symposium on Liquid and Solid SHe (Ohio State University Press, Columbus, 1957), p. 220. 46 R. L. Mills and E. R. Grilly, Phys. Rev. 99, 480 (1955). 47 A. F. Schuch, E. R. Grilly and R. L. Mills, Phys. Rev. 110, 775 (1958). 4* G. de Vries and J. G. Daunt, Phys. 92, 1572 (1953); 93, 631 (1954). 49 T. R. Roberts and S. G. Sydoriak, Phys. Rev. 93, 1418 (1954); 98, 1672 (1955). Ka B. M. Abraham, D. W. Osborne and B. Weinstock, Phys. Rev. 98, 551 (1955). 61 D. F. Brewer, A. K. Sreedhar, H. C. Kramers and J. G. Daunt, Phys. Rev. 110. 282 (1958); Bull. Am. Phys. SOC.3, 133 (1958). 3 1. S. G. Sydoriak and E. F. Hammel, Proceedings of the International Conference on the Physics of Very Low Temperatures (Massachusetts Institute of Technology, Cambridge, 1949). p. 42. $8 D. F. Brewer, J. G. Daunt and A. K. Sreedhar, Phys. Rev. 115, 836 (1959); Bull. Am. Phys. SOC.Ser. 11, 3, 399 (1958). 54 E. F. Hammel, R. H. Sherman, J. E. Kilpatrick and F. J. Edeskuty, Physica, Suppl. t o 24, 1 (1958). 58 B. M. Abraham, D. W. Osborne and B. Weinstock, Physica 24, 132 (1958). 66 D. M. Lee and H. A. Fairbank, Phys. Rev. 116, 1359 (1959). 5’ L. J. Challis and J. Wilks, Symposium on Liquid and Solid 8He (Ohio State University Press, Columbus, 1957), p. 38. 68 V. P. Peshkov and K. N. Zinov’eva, Reports on Progress in Physics 22, p. 504 (1959). 69 B. Weinstock, D. W. Osborne and B. M. Abraham, Proceedings of the International Conference on the Physics of Very Low Temperatures (Massachusetts Institute of Technology, Cambridge, 1949). p. 47. R. D. Taylor and J. G. Dash, Phys. Rev. 106, 398 (1957). 61 K. N. Zinov’eva, J. Exptl. Theoret. Phys. 34, 609 (1958) [Trans. Soviet Physics JETP 7, 421 (1958).] H. A. Fairbank, private communication. ** E. J. Walker and H. A. Fairbank, Phys. Rev. Letters, 5, 139 (1960). O4 R. L. Garwin and H. A. Reich, Phys. Rev. 115, 1478 (1959). * b H. R. Hart, Jr. and J. C. Wheatley, Phys. Rev. Letters 4, 3 (1960). 66 N. Bloembergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948). 67 W. M. Fairbank, W. B. Ard, H. G. Dehmelt, W. Gordy and S. R. Williams, Phys. Rev. 92, 208 (1953). 68 W. M. Fairbank, W. B. Ard and G. K. Walters, Phys. Rev. 95, 566 (1954). W. M. Fairbank and G. K. Walters, Proc. Symp. on Liquid and Solid We, Ohio State Univ., Aug. 20-23, (1957). 7o G. Careri, I. Modena and M. Santini, Nuovo Cimento, Series X, Vol. 13, 207 (1959). G. Careri, I. Modena and F. Santini, Nuovo Cimento, t o be published. ‘3 R. L. Garwin and H. A, Reich, Phys. Rev. 115, 1478 (1959). 7s R. H. Romer, Phys. Rev. 115, 1415 (1959). 74 R. H. Romer, Phys. Rev. 117, 1183 (1960). F. J. Low and H. E. Rorschach, Post-deadline paper presented t o the American Physical Society. April 30, 1969. 44
152
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[CH. IV
H. C. Torrey, Phys. Rev. 92, 962 (1953); 96, 690 (1954). H. C. Torrey, Suppl. Nuovo Cimento 9, 95 (1958). R. Kubo and K. Tomita, J. Phys. SOC.(Japan) 9, 888 (1954). 79 See footnotes 19 and 20 in ref.74. no H. A. Schwettman, F. J. Low and H. E. Rorschach, Jr., Bull. Am. Phys. SOC.5, 111 (1960). H. A . Rorschach, private communication. 8z H. A. Reich, personal communication and Proc. Second Symposium on Liquid and Solid 3He, Ohio State Univ., Aug. 23-25, 1960. J . M. Goodkind and W. M. Fairbank, Phys. Rev. Letters, 4, 458 (1960). 84 K. R. Atkins and H. Flicker, Phys. Rev. 116, 1063 (1959). 86 A. van Itterbeek and G. Forrez, Physica 20, 133 (1954). 86 K. R. Atkins and R. A . Stasior, Can. J. Phys. 31, 1156 (1953). 87 J. R. Pellam and C. F. Squire, Phys. Rev. 72, 1245 (1957). 88 P. W. Anderson, Phys. Rev. 112, 1913 (1958); See also Bogoliubov, Tolmachev and Shirkov, “New Method in the Theory of Superconductivity,” (Academy of Sciences of the IJSSR, Moscow, 1958). 80 T. Usui, Phys. Rev. 114, 21 (1959). 00 J. de Boer, private communication. 91 For a discussion of cell model theories of liquid SHe see J . de Boer. Progress in Low Temperature Physics, Ed. C. J . Gorter, Vol. 11,(North Holland Publishing Company, Amsterdam, 1957) Ch. I. O2 A. C. Anderson, H. R. Hart and J. C. Wheatley, Phys. Rev. Letters, 5. 133 (1960). ~33 F. J. Low and H. E. Rorschach, Phys. Rev. 120, 1111 (1960). 64 See also L. P. Pitaevskii, J. Exptl. Theoret. Phys. (U.S.S.R.) 37, 1794 (1959) [Trans. Soviet Physics J E T P 10, 22 (1960)l. 8.5 E. D. Adams, H. Meyer and W. M. Fairbank, Proc. Second Symposium on Liquid and Solid 3He, Ohio Statc Univ., Columbus, Aug. 23-25, 1960. 7g
77
CHAPTER V
3He CRYOSTATS BY
K. W. TACONIS
KAMERLINGH OWES LABORATORY, LEIDEN C O N T E N T S1.: Introduction, 153. - 2. First cryostat by Roberts and Sydoriak, 155. 3. Calorimeter by Seidel and Keesom, 156. - 4. Calorimeter by Taconis and De Bruyn Oubotcr, 158. - 5. Refrigerator with glass dewar by Zinov'eva and Peshkov, 160. - 6. Refrigerator by Reich and Garwin, 163. - 7. Metal refrigerator by Peshkov, Zinov'eva and Filimonov, 164. - 8. Large metal cryostat by Laquer, Sydoriak and Roberts, 165. - 9. Crvostat construction by Taconis and Le Pair, 166.
1. Introduction Since 1954 a new technique has been developed in low temperature physics. A small separate cryostat in which a 3He bath evaporates under its saturated vapour pressure is placed in the conventional *He dewar, thus facilitating experiments well below 1"K. The technique has already earned its place next to the well known demagnetisation procedure for the production of low temperatures and, especially from 1" K down to 0.3" K, it has proved to be a much simpler and most successful way of creating a constant temperature bath. I t covers, moreover, almost the same temperature range as the magnetic refrigerator which cools down to 0.25" K. Circumstances are very favourable here in comparison with the *He cryostat as 3He does not shown superfluidity. Therefore, we do not notice the phenomenon of the creeping helium film which gives rise to a large heat influx into the bath of a usual 4He cryostat below the lambda point, due t o the rapid flow of superfluid helium from the bath along the walls of the vessel to regions of higher temperature. Indeed, this helium transported by film flow evaporates and has to be pumped off or otherwise it will condense again on the surface of the bath and deliver there its heat of vaporization. The film transfer cannot be easily suppressed and is often very high, especially when small quantiRefwences p . 168
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1
ties of impurity such as solid air etc. are deposited on the walls of the cryostat. Even such small traces of contaminant as can hardly be avoided are able to increase the flow by a factor two or more. In the extreme experiment of Keesom in 1932, for instance, a pump of 675 liter per sec capacity was necessary to reach a temperature of 0.7" K in a 4He bath. This temperature corresponds to a vapour pressure of 2pHg, whereas at the same pressure in a 3He bath a temperature of about 0.3" K results. which c-an be attained with a very small
Fig. 1. Vapour pressures of 8He and 'He.
diffusion pump with a capacity less than 20 liter per sec. As a consequence of this it is quite common to work with 1 cm3 of liquid 3He for several hours and sometimes even a day. A second advantage of 3He is that its vapour pressure range gives access to a relatively very large temperature range. This is shown in Fig. 1 where the logarithm of the vapour pressure is plotted against the inverse temperature for 3He and 4He. One notes that due to the Refcremes p . 168
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very much smaller heat of vaporization of 3He the curve is less steep than that for 4He. At 1" K the ratio of the vapour pressures for 3He and 4He is a factor 70 whereas at 0.6" K this factor is already 1700. The purpose of this article is to review the various types of 3He cryostats used at present. 2. First Cryostat by Roberts and Sydoriak The first 3He cryostat was built in 1954 by Roberts and Sydoriakl and had a double purpose. It served as an apparatus for the study of
PUMP
Fig. 2. First aHe dewar by Roberts and Sydoriak.
the vapour pressure curve of 3He down to 0.45" K and it supplied specific heat data on 3He below 1" K. A copper sphere 12 mm in diameter, filled for about 50% with 3He and the other 50% with paramagnetic salt particles is silver soldered to a copper tube 25 mm long and this tube is again soldered to a copper nickel tube 200 mm long, 3 mm in diameter and with a 0.25 mm wall References Q. 168
156
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thickness. This capsule is put in a brass vacuum jacket. The long copper nickel tube is hard soldered in a brass block to assure that the upper end of this tube has a temperature of 1.1"K: that of the surrounding 4He bath. A hole is drilled through the block to evacuate the lower part of the dewar. Several inserts can be screwed inside the brass block depending on the experiments done in the cryostats. For vapour pressure investigations a tube system is inserted to measure the static pressure above the 3He bath in the spherez. The lower part has an inside diameter of 1 mm, and the upper part above the screwpiece a diameter of 2 mm. There is another hole in this screwpiece in order to allow the pumping of the 3He vapour out of the copper sphere through the annular space around the pressure sensing tube and this hole. The vapour pressures are measured by means of a mercury or oil manometer above 1" K and an oil or MacLeod manometer beIow 1" K. Thermomolecular pressure corrections have to be applied at the very low pressures obtained in this cryostat. A separate investigation was made of the thermomolecular pressure corrections and it appeared that they were the same as for 4He 3. The temperature of the 3He is derived from the susceptibility of the paramagnetic salt using ballistic measurements. The primary and secondary coils are both wound directly on the brass vacuum jacket. Two paramagnetic salts were used, ferric ammonium alum and chromium methyl amine alum, both calibrated during each run at various temperatures between 2.5" and 1.1"K. The research resulted in an elaborate vapour pressure equation and a table derived with aid of the formula. The table was used in the preparation of the graph in Fig. 1. With a second insert an electrical heater is brought into the copper sphere and specific heat measurement on the 3He were performed below 1" K1. From a known heat input and resulting temperature change the heat capacity of the calorimeter was determined. All parasitic heat absorbing effects, such as the heat capacity of the salt and the heat absorbed in evaporation of 3He into the large dead space for the vapour, were eliminated by measuring with different quantities of 3He in the calorimeter.
3. Calorimeter by Seidel and Keesom Two 3He cryostats were especially developed to measure specific heats. One of them was constructed for the purpose of calorimetry by licftmvtres p . 168
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Seidel and Keesom in 19584. Inside a vacuum jacket, surrounded by a dewar filled with 4He, a copper vessel is mounted which consists of three different compartments. The part d contains the 3He bath and its pressure can be reduced through the german silver tube c (4mm diameter, 0.1 mm wall thickness) with an oil diffusion pump (25 liter per sec). Inside the compartment h the specimen is suspended by a nylon thread connected to a steel wire 0.12 mm in diameter. A heater and a carbon thermometer are attached to the specimen which can be raised or lowered in the space h by pulling the steel wire, as this goes over two
- Helium bath
- Capillary - Pumping tube - 3He space
- Pulleys - Nylon thread - Copper vessel - Specimen compartment
- Specimen
- Copper rod k - Copper plate
i
W
1 - Salt compartment m. - Secondary coil n - Primary coil
Fig. 3. Calorimeter by Seidel and Keesom.
pulleys and through a 1 mm wide thin-walled capillary to outside the cryostat. The specimen can be cooled down to the temperature of the 3He bath by lowering it until it makes thermal contact with the polished copper plate which is connected through a copper rod to the 3He vessel. This thermal contact is necessary because if some exchange gas was admitted for the purpose of cooling it required at least 1 2 hours of pumping to obtain a sufficient thermal isolation for heat capacity measurements. Thus, the exchange gas was pumped out at 4.2" K and References
p. 168
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in further cooling via the thermal contact with aid of the 3He bath it took 6 hours before the specimen finally reached the lowest temperature of 0.35" K which was 0.05" above the 3He bath temperature (0.3" K). The difference is due to the heat produced in the carbon resistance thermometer (0.1 erglsec). In order to determine the temperature of the whole copper can a glass sphere filled with powdered ferric ammonium sulphate is placed in the compartment 1. The susceptibility of this paramagnetic salt is measured with a Hartshoorn type AC bridge. The coils are wound of niobium wire. The inner wall of the compartment 1 is covered with 50% tin solder. Due to the use of these superconducting materials heating by eddy currents is avoided. Good heat contact was achieved between the salt and the 3He bath by filling the space 1 with "e gas at a pressure of 1 atmosphere at room temperature and also filling the glass sphere with this helium gas at a pressure of 1 atmosphere at 77" K. The carbon resistance thermometer was calibrated each time after the heat capacity measurements were finished and a little exchange gas was admitted into the compartment h. The specific heats below 1" K were measured with the specimen situated in contact with the copper plate because upon raising it, too much heat was produced and moreover, the temperature fluctuated due to mechanical vibrations. Above 1 ° K the specimen had to be raised since there the heat insulation was insufficient. The apparatus used 3 liter 3He at normal temperature and pressure to maintain a bath temperature of 0.3" K constant for a period of 80 hours. 4. Calorimeter by Taconis and De Bruyn Ouboter
A second calorimeter was used in the study of the heat capacity of liquid 3He-4Hemixtures in the region of phase separation by Taconis and De Bruyn Ouboter5. It seems to us that it has a much wider applicability since the method is very convenient. The specimen is cooled down to the temperatures of about 0.35' K in excellent thermal contact with the cooling 3He bath, the heater and the thermometer. Subsequently, the contact with the bath is broken, and the specific heat of the substance is derived from the temperature rise due to the known heat input in the heater, nearly without correction as the heat capacity of the calorimeter itself is almost negligible below 1" K. References p . 168
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The solution chosen here is that the specimen and the 3He bath are both situated in the same copper block; each in its own compartment. The contact with the 3He is broken by simply emptying this 3He bath at the lowest temperature. The heater, H, is wound around the copper block and the carbon thermometer, T, deposited on an insulated copper bar, in the usual Leiden way, is also soldered to the copper block. After the condensation of the 3He and the subsequent condensation of the mixture at about 1.2” K is finished, the 3He bath is reduced in
‘cu
Cu /c2
/He ,M
1 ’
/T
v
- Radiation screen
-vacuum space I - Vacuum vessel C , --Capillary C, - Pumping tube 8He - SHe bath M - Mixture space H --Heater T - Carbon thermometer
Fig. 4. The calorimeter by Taconis and De Bruyn Ouboter.
pressure by a specially sealed rotary pump of 20 liter/min capacity t. t The vacuum pumps for ‘He are usually modified in order to prevent the loss of the rather expensive gas. Sometimes the whole pump is placed in a extra oil bath; often the rotor axis is double sealed with a n oil filled space between the two seals. In the Kamerlingh Onnes Laboratory a double stage Edwards pump 2S20 was revised, In addition to the extra axis seal an oil level indicator was mounted on the little oil bath between the two seals. All flanges could be provided with “0” ring seals. The pump had almost no dead space and the total quantity of oil in the pump was reduced to 5 cmy. The oil which is pushed out of the outlet valve is slowly sucked back into the pump again through a narrow channel connected to the brsnze bearing of the pump and from there it takes care of the tightness of the pump and its lubrication. References
p. 168
160
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The exchange 4He gas in the vacuum space was already evacuated at 4.2" K. During the cooling period at certain intervals the inlet valve of the pump is closed for a few minutes and as soon as pressure equilibrium above the 3He bath and in the connection tubes is attained, the carbon thermometer is calibrated against the vapour pressure of the 3He. The latter is read from a mercury manometer, an oil manometer filled with Octoil S or a small MacLeod gauge depending on the magnitude of the pressure. The essence of the method is now to estimate the quantity of 3He which is condensed so that just after the lowest obtainable temperature is reached the 3He chamber is nearly empty. If not, the heater is applied carefully to empty it completely, which is signalled by the first rise of the carbon thermometer. From this moment on the heat capacity of the specimen can be determined since the heat capacity of the empty calorimeter is practically zero. The heat leak in this calorimeter is between 100 and 30 erg/sec. In one run of measurements, heating the sample about 0.015' for each heat capacity determination, fifty points are measured between 0.35 and about 1.5" K. The time for such a run is 2 hours. Two runs are performed in one day's experiment using the same liquid 3He-4He concentration. Altogether 16 different concentrations were studied. Most essential in this particular research was an overfilled calorimeter, because evaporation inside the calorimeter would greatly confuse the specific heat data obtained. Therefore, the capillary C,, connecting the calorimeter with the filling system at room temperature, has a very small volume of 24 mm3 and is for about one quarter filled with liquid mixture. The total length of the capillary is 80 cm, its inner diameter 0.2 mm. The uncertainty in the effective volume is negligible as the volume in the copper block is 535 mm3. Also, the influence of demixing effects due to thermodiffusion, film creep, heat conduction currents etc. can be neglected with such a small dead space for the vapour.
5. Refrigerator with Glass Dewar by Zinov'eva and Peshkov Cryostats of large and very large cooling capacity make use of a continuous refrigerating cycle. The first were constructed in Russia by Zinov'eva and Peshkov6 and in the U S A . by Reich and Garwin7. In the Russian cryostat the liquid 3He boiled in a glass dewar surrounded by a 4He bath. This glass dewar was again a new phase in the deReferences
p . 168
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51
3He CRYOSTATS
161
velopment of the cryostat. Before, the 3He used in large quantities was too expensive to be exposed to possible loss by the breaking of glass apparatus and everything was designed in metal. The metal dewar did not allow the possibility of visual observation. Zinov'eva and Peshkov gave up this principle, perhaps realizing that the price drops every two years by about a factor ten, and the work they did in their glass dewar was very striking. The stratification in a liquid mixture of 3He and 4He below 0.8" K due to the gradual demixing effect with decreasing temperature was observed8, and the viscosity of liquid 3He was measured5. As in the standard refrigerator, the 3He follows in its cycle the path from compressor to condensor ; subsequently through the expansion valve5 to- the evaporator' and finally back to the compressor. The compressor, or more accurately vacuum pump, pumps the 3He gas from, for instance, a pressure of 0.002 mm mercury corresponding to a temperature of 0.3' K, to a pressure of a few centimeters of mercury necessary to condense the gas in a condensor placed in the 4He bath at a temperature of 1.2" K. The 3He enters the evaporator through the expansion valve which can be accurately regulated in order to allow very sinall quantities of liquid to pass from condensor to evaporator. This verj7 simple but proven scheme works most satisfactorily. Some details are: The dewar has an 8 mm inside diameter. The length is 100 mm. The pump line of thin-walled stainless steel is 15 mm in diameter and connected to the 3He dewar by means of a copper-glass seal. Inside this copper junction two copper screens3 are soldered in contact with the surrounding 4He bath to shield off the radiation from above. The condensor spiral, the expansion valve and the liquid inlet are shown in Fig. 5. I n the viscosimeter experiment a glass beaker is filled with liquid by dipping it into the 3He bath as it can be moved up and down by moving a thin thread on which the beaker is suspended from above through the pump tube. The beaker is protected against radiation from above by means of a paramagnetic salt shield, mounted just above the beaker on the suspension thread. After the beaker is pulled upwards the liquid can flow out through a capillary connected to the bottom. In fact two different capillaries were used, 100.2 and 74.5 ,u in diameter. From the velocity with which the level in the beaker fell the viscosity of 3He was derived. A second experimental arrangement is shown in Fig. 5 . It consists Rejcyenrrs p . 168
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[CH.
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5
5
of n small glass vesselGof 3.5 mm inner diameter and 20 mm length. It is closed by a copper top and connected by means of a 0.5 mm inner diameter capillary7 to a filling system outside the cryostat for preparing various mixtures of 3He and 4He. If a mixture of known concentration is condensed and slowly cooled down below 0.S8" K, Zinov'eva and
1 - 3He dewar 2 - Copper tube 3 - Radiation screen 4 - Condensor 5 - Expansion valve 6 - Mixture vessel 7 - Capillary 8 - Heater 9 - Thermometer
Fig. 5 . Refrigerator with glass evaporator by Zinov'eva and Peshkov.
Peshkov observed and photographed very clearly a boundary between two layers of liquid helium9. Stratification takes place, which means that depending on the concentration the liquid separates into two different layers; the lighter one containing a higher 3He concentration and the other a higher 4He concentration. The boundary usually shifts with the temperature. The lower appendage of the vessel is a special expedient to observe very accurately the forming of the first little bit of the heavier composition when a mixture of high concentration ( >70%) is cooled down into the phase separation region. Refercirces p . 168
CH. V,
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3He
61
CRYOSTATS
163
With the heater8 half way in the small vessel the superfluidity of the surrounding liquid could be detected, since the appearance of very small bubbles around the heater during heating was only observed in the upper 3He-rich layer and never in the lower one, which apparently showed superfluidity. With this trick for detecting the superfluidity the lambda transition of mixtures was also investigateds.
6. Refrigerator by Reich and Garwin An all metal refrigerator was used by Reich and Garwin' in their
C A
8-
D
- Expansion valve E - Capillary C - Condensor E Heater F - Carbon thermometer G - Experimental vessel H - Capillary
A
~
Fig. 6. A11 metal refrigerator by Reicli and Garwin.
research on the self-diffusion coefficient in 3He derived from nuclear spin relaxation times. The condensor is most primitively formed by the expansion valve C itself and is surrounded by the 4He bath at 1.2" K. The liquid 3He flows through capillary B to the evaporator, D, in which for control of the temperature a heater E and a carbon thermometer F are available. The measurements are performed in the space G and the surrounding vessel is filled with 4He through capillary H in order to establish a homogeneous temperature. The authors mention that a gas filling of 1 atmosphere at room temperature is also quite satisfactory. The quantity of 3He needed for this kind of apparatus is References p . 168
164
[CH. V,
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97
largely determined by the dead space of the pump housing (about 1 liter) in which the pressure of the 3He must be 30 mm of mercury so that the gas can again be condensed at 1.2” K. The quantity used here amounts to 80 cm3S.T.P. the cost of which in the U.S.A., since January 1960, is only a little more than $lo.-. 7. Metal Refrigerator by Peshkov, Zinov’eva and Filimonov
A similar but larger metal refrigerator was described by Peshkov, Zinov’eva and Filimonov in 1959lO. The 3He gas is condensed in the coil, 1, (length 5 meters, inside and outside diameters 1.4 and 2.0 mm
bw>
1 - Condensor 2 - Expansion valve 3 - Heat exchangcr 4 - Evauorator 5 - Heat contact surface 6 - Copper container 7 - 4He inlet valve 9 Vacuum vessel 10 -- Vacuum tight seals 1 1 - Charcoal trap ~
Fig. 7. Metal refrigerator by Peshkov, Zinov’eva and Filimonov.
respectively) is subsequently expanded in the valve, 2, and flows through coil 3 (length 20 cm, inside diameter 0.2 mm) in heat exchange with the gas evaporating in vessel 4. The valve is in good heat contact with the 4He bath and screens the radiation from above. The volume of the evaporator is 2.5 mms, the connection tube is 100 mm long, is 1 2 mm in diameter and has a wall thickness of 0.2 mm. The material is stainless steel which is welded at the upper end to the outer dewar vessel wall. The copper container, 6, of 200 c1n3 volume was filled with liquid 4He out of the surrounding bath through valve 7 and the german silver capillary, 8. Rrfrrenres
p. 168
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3He CRYOSTATS
165
The lowest temperature reached was only 0.5' K and this relatively poor performance was due to the heat leak along the capillary, 8, (160 mm long, ins'ide diameter 1.4 mm and wall thickness 0.2 mm) since it contains 4He and therefore helium I1 heat transfer mechanisms are in action in the rather wide tube. The authors remark that the displacing of the valve to somewhere within the vacuum space would improve the insulation and bring the lowest temperature down to 0.35" K as this is the temperature which is reached without filling the 4He container. The vacuum space was filled at room temperature with a little 4He exchange gas to a pressure of 0.5 mm mercury. This gas provides an excellent heat exchange during the precooling of the apparatus. The exchange gas is automatically evacuated by a small charcoal trap, 11, when the temperature has reached the working region. The time needed to cool the work space, 6, filled with *He from 4.2"K to 0.5" K was less than one hour, whereas to reach the lowest range below 1" K took only 5 to 10 minutes. The temperature in the 4He bath, 6, was measured with a phosphor bronze resistance thermometer. Two superconducting wires, one of pure aluminium ( 5 0 , ~ and ) one of pure cadmium (67 p ) are put in series with the thermometer wire and their transition temperatures are used as calibration points for the thermometer. The authors do not indicate what kind of experiments are planned with this set-up.
8. Large Metal Cryostat by Laquer, Sydoriak and Roberts A large all metal cryostat of very different construction was used by Laquer, Sydoriak and Robertsll in their research on the velocity of sound in 3He. Also, the vapour pressure of mixtures of 3He and 4He was studied in this apparatus. The gas under examination is condensed in the central vessel of 2 cm3 volume. This contains the quartz crystals to generate and detect the sound signals. It is surrounded by an annular shaped 3He bath of 15 cm3 volume in which a 3He vapour pressure thermometer is also mounted. The whole 3He assembly is constructed of copper and surrounded by a vacuum, maintained in a copper can in perfect heat contact with an 4He bath of about 5 liter volume boiling at 1.05 OK and lasting for 60 hours. The helium bath is again shielded by a nitrogen bath. The various tubes connecting the 3He bath with the auxiliary equipment at room temperature are, on their way upwards, first joined in good heat contact with the 4He bath at 1.05" K and secondly, References p . I68
K. W. TACONIS
[CH. V,
$9
half way to the nitrogen shield temperature at 75" K they are joined by copper straps to the 1" diameter 4He evaporation line in order to reduce the heat influx, as the effluent cold vapour gives extra cooling. The connection tubes are, moreover, provided with radiation traps. All these measures decrease the evaporation rate by 157". The evap oration of the 3He bath is 0.03 cm3 of liquid per hour at the minimum temperature attained of 0.34" K.
HEATER IFOR
1~I
ENOLIC CONE ARTS CRYSTAL
THERMOMETER
He DEWAR VAC D N, DEWAR VAC
Fig 8. All metal cryostats of Laquer, Sydoriak and Roberts. Fig.9. Detailed drawing of $He dewar.
9. Cryostat Construction by Taconis and Le Pair A compromis between a metal and a glass construction when visual observation is necessary is found in an apparatus by Taconis and Le Pair12 used to investigate the helium film flow in equilibrium with a mixture of 3He and 4He below 1" K. The 3He bath, B, is condensed in a metal vessel, actually a 6 mm hole drilled in a copper block, A. It is connected by means of a german silver tube (100 mm long, 2.4 mm inner diameter and 0.1 mm wall thickness) to a 12 mm wide pumping tube. In the center of the block is placed a superleak of strongly compressed jewellers rouge enclosed, after further compacting of the powder, between two steel needles which serve to fix the rouge very solidly. These precautions are rather essential, otherwise very often the superleak can be used only once and is ruined because it cracks when it warms up. The reason is probaRefcrmces p . 16s
CH.
v, 9 91
3He
CRYOSTATS
1GT
bly that the superfluid helium in the rouge is transformed t o normal liquid helium when passing the lambda point and this helium cannot
p - Capillary q - Capillary S - Superleak B --He bath A - Copper block
Th - Thermometer I - Copper joint F - Copper frame C - Glass capillary W - Windows
Fig. 10. 3He cryostat of Taconis and Le Pair.
escape easily enough out of the pores. Both sides of the leak are connected to a Topler system in which various mixtures of 3He and 4He References 9. 168
168
K. W. TACONIS
[cr-I.v
can be prepared and brought into the apparatus through the capillaries p and q (length 100 mm, inside and outside diameter 0.6 and 1.0 mm resp.). A glass capillary, C, of 1.2 mm inside diameter is soldered at its upper end to the copper junction, J, under the superleak, and at its lower end to a copper pen, both via an intermediate platinum seal. In the lower part of the glass tube, C, a liquid mixture is condensed, the height of the meniscus of which can be observed since the lower part of the vacuum jacket is also made of glass. The function of the copper pen which is in good heat contact with the copper frame, F, is to give the liquid helium mixture the temperature of the 3He bath. The helium inside the glass capillary is, moreover, protected against radiation by means of two pairs of infrared absorbing glass windows ; one pair in the outer 4Hebath and one pair, W, mounted in the frame,F. The film transport is measured as follows. After the vacuum jacket is evacuated a t 4.2" K the pressure in the surrounding cryostat is reduced to a temperature of 1.2" K and 3He is condensed in vessel B. More exactly explained, the condensation takes place in the pump line and liquid flows down through the wide tube into the copper block. Subsequently, the capillary, C, is filled with some mixture, and finally a little bit of helium mixture of high 3He concentration is also brought into the space above the superleak. Now the 3He bath is rapidly cooled down to the desired working temperature, and after a few minutes temperature equilibrium is reached as can be observed from the vapour pressure read on manometers connected to capillaries p and q. Then the level in C is followed and from its fall the transfer rate of the helium film can be derived since the film creeps up the wall of the glass tube and through the superleak into the upper mixture space due to the osmotic pressure exerted on the superfluid in the helium film when a concentration difference exists. The lowest temperature obtained with this appartus is, a t the present, 0.5" K. REFERENCES 2
6
7
T. R. Roberts and S. G . Sydoriak, Phys. Rev. 98, 1672 (1955). S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). T. R. Roberts and S. G. Sydoriak, Phys. Rev. 102, 304 (1956). G. Scidel and P. 13. Keesom, Rev. Sci. Instr. 29, 606 (1958). To be published in Physica; see also Physica 25, 723 (1959). IC. N . Zinov'eva, Soviet Physics, J.E.T.P. 34, 609 (1958); translated 7, 421 (19.58). H. A. Reich and R. L. Garwin, Rev. Sci. Instr. 30, 7 (1959). IC. N. Zinov'eva and N. P. Peshkov, J.E.T.P. 37, 33 (1959).
CH. V]
REFERENCES
169
K. N. Zinov’eva and N. P. Pcshkov, J.E.T.P. 32, 1256 (1957); translated 5 , 1025 (1957). 10 V. P. Peshkov, I<. N. Zitiov’eva and -1. J. Filimonov, J.E.T.P. 36, 1034 (1959); translated 9, 734 (1959). n H. L. Laquer, S. G. Sydoriak and T. R. Robcrts, Phys. Rev. 113, 417 (19.59); Low Temperature Physics and Chemistry, Proc. of the 5th International Conference Madison Wisconsin 1958. l2 To be published in Physica. a
CHAPTER V I
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY t BY
J. BARDEEN AND J. R. SCHRIEFFER DEPARTMEKT OF PHYSICS, UNIVERSITY OF
ILLIKOIS, URBANA, ILLINOIS
COSTENTS:1. Introduction, 170. - 2. Historical survey, 175. - 3. Outline of London and Pipard theories, 178. - 4. London’s quantum picture of the superconducting State, 181. - 5. Elementary excitations in normal metals, 184. - 6. Electron-phonon interactions, 191. - 7. Elementary excitations in superconductors, 193. - 8. Nature of wave functions for superconductors, 195. - 9. Results for simplified model, 201. 10. Thermodynamic properties, 207. - 11. Transition probabilities and coherence factors, 212. - 12. Electromagnetic properties, 224. - 13. Collective excitations, 252. - 14. Two-fluid model and persistent currents, 263. - 15. Thermal conductivity, 270. - 16. Superconducting alloys and compounds, 275. - 17. Conclusions, 280.
1. Introduction
There has been a great deal of interest in the problem cf superconductivity1 since its discovery by Kamerlingh Onnes in 1911. Until recently all attempts to construct an adequate microscopic theory of the phenomenon have failed, in spite of the large amount of experimental and theoretical work devoted to th.e problem. There has been great progress in the past few years, hoc.ever, in both experiment and theory, and it is now possible to explain a large body of the existing experimental facts on the basis of a theory originally proposed by L. N. Cooper and the authors of the present article. I n writing this article, we hope to present as clear a view as possible of the physical basis and content of the recent theoretical advances and t o outline the results of recent experiments which have confirmed a number of predictions of the theory. The remarkable properties of superconductors (as well as superfluid flow in liquid helium) are consequences of quantum effects operating on a truly macroscopic scale. General lines along which an explanation might be found were suggested by F. London2; the present theory is in t Supported in part by the Office of Ordnance Research, U.S. Army. References p. 282
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accord with his ideas. The superconducting state has the characteristics of a single quantum state extending throughout the volume, which is not destroyed by local thermal excitations. The basis for the theory, as first suggested by Frohlich in 1950, is an effective attractive interaction between electrons which results from the electron-phonon interaction ($6). The wave function for the superconducting ground state is described in terms of a linear combination of normal state-like configurations in which the individual electron states are occupied in pairs of opposite spin and momentum. Such a wave function gives correlations between electrons of opposite spin which extend over large distances in real space so as to take advantage of the attractive interaction. In their original brief note3&,Cooper and the authors calculated for a simplified model the energy difference between normal and superconducting phases at T = 0" K and showed that there is an energy gap for excitation of electrons from the superconducting ground state. In their main article3b, the spectrum of elementary excitations for higher temperatures was worked out and applied to a calculation of thermal and electromagnetic properties (for low frequency or static fields). It was shown that the theory accounts for a second order phase transition at the critical temperature, and also for the Meissner effect and persistent currents. Included in the article was an evaluation of matrix elements for transition probabilities appropriate to such phenomena as absorption of ultrasonic or electromagnetic waves and nuclear spin relaxation times. An intensive study of the mathematical structure of the theory has been carried on by a number of authors. Bogoliubov4 and Valatin5 have advanced alternative formulations which are often more convenient for calculational purposes and lead to results which in general are in agreement with the original treatment. Largely through the work of Anderson 6, Bogoliubov and coworkers 4, Nambu', Rickayzens, and Pines and Schrieffers, the role of collective excitations of the electrons has been clarified and questions regarding the gauge invariance of the calculation of the Meissner effect based on the original form of the theory have been resolved. An outstanding problem is the precise role of Coulomb interactions between electrons in restricting superconductivity and the proper criterion to distinguish superconductors from nonsuperconductors. Further applications which have been worked out since the original References
p . 282
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J . BARDEEN AND J . R. SCHRIEPFER
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article include electrodynamics for fields oi arbitrary frequency, thcrma1 conductivity, effects of impurities and alloying (including pnramagnetic impurities), critical currents and fields in small specimens, the Knight shift and electron spin paramagnetism, and the problem of the boundary between normal and superconducting regions. Several authors have contributed to these developments. With few exceptions, the theoretical results are in surprisingly good agreement with esperiment in spite of the simplicity of the model on which most of the calculations are based. Perhaps the most direct evidence for the existence of an energy gap for quasi-particle excitations comes from the far infrared transmission and reflection measurements of Tinkham and collaborators lo. At very low temperatures, as the frequency is increased there is no energy absorption until the quantum energy of the radiation exceeds the gap, after which the absorption rises rapidly to that of the normal metal. Microwave measurements of surface impedance, particularly the recent measurements of Biondi and Garfunkelll, give good evidence for a gap which decreases from a maximum a t ?' = 0" K to zero a t the transition temperature, T,, as predicted by the theory. A striking prediction of the theory, verified by experiment, is a strong influence of the coherence properties of the superconducting wave functions on the transition probabilities of the system induced by an external field. Electrons of opposite spin and momentum absorb coherently, and their contributions may add constructively or destructively depending on the particular phenomenon involved. Destructive interference applies to an ordinary interaction such as is involved in attenuation of ultrasonic waves. There is a very rapid drop in the absorption coefficient as the temperature drops below T,. Recent measurements of Morse and coworkers12 are in fairly good agreement with theory. At the time the original theory was being worked out, Hebel and S l i ~ h t e r 'made ~ the first measurements of nuclear spin relaxation times in superconducting aluminum. They found, surprisingly, that as the temperature drops below T , the relaxation rate increases to values more than double that of the normal state, indicating a larger interaction between electrons and nuclear spins in the superconducting state than in the normal state. In this case, the two contributions t o the matrix element add constructively. The increase above that of the normal state is due to a very high density of states in energy just above the gap. More recently, similar measurements have been made K~Jereitces9. 282
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with greater precision and over a more extended temperature range by Redfield14, with results in good agreement with predictions of the theory. Constructive addition also applies to absorption of electromagnetic radiation. What corresponds to the conductivity of the normal component of a two fluid model may, if the frequency is not too high, be higher than that in the normal state a t temperatures near T,. These coherence effects give strong support to the concept of paired electron states on which the theory is based. There has been renewed interest in measurements of thermal conductivity. Earlier work had shown a marked difference in behavior of lead and mercury, in which the electrons are scattered mainly by lattice vibrations, and other superconductors in which impurity scattering dominates. The present theory is in reasonably good agreement for the case of impurity scattering, but fails to account for the rapid drop in thermal conductivity below T , observed in Pb and Hg. TheEe elements, with low Debye temperatures, behave differently from most superconductors in other respects as well, for exemple, in the temperature dependence of the electronic specific heat. It is important to know whether or not the rapid drop is peculiar to Pb and Hg or would be observed in other superconductors if sufficiently purified so that scattering is dominantly by phonons. Recent measurements of Guenaultlj on extremely pure tin indicate that the drop near T,, while greater than for impurity scattering, is much less than that observed in Pb and Hg. Measurements of Androes and Knight l6 on nuclear spin resonance in thin superconducting films confirm earlier results of Reifl' on colloidal mercury that there is a frequency shift from electron spin paramagnetism which apparently persists down to T = 0" K. The shift near T = 0" K is about 70 percent of that found in the normal state. This problem has been a puzzle for some time; possible explanations which have been given will be discussed in tj 13. In view of the Meissner effect, it has long been thought that superconductivity and ferromagnetism are incompatible. It is therefore quite remarkable that recent experiments of Matthias and coworkers show that these phenomena can coexist in the same specimens. This occurs in alloys of rare-earth elements in which the ferromagnetism is attributed to f-electrons. I n certain ranges of composition, the Curie temperature may be higher than the superconducting transition temperature, T,. When cooled below T,, both remnant magnetization and Refevcncss p . 282
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J. BrlRDEEN AND J. R. SCHRIEFFER
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diamagnetic behavior typical of superconductivity are observed. Some of this work is reviewed in 5 16. Most of the applications of the current theory have been based on a simplified model which involves basically three parameters: (1) N(E,), the normal density of states in energy at the Fermi surface, (2) no = &-l 1 a s / % 1 *"e, the average velocity of the electrons in the normal state at the Fermi surface, and (3) a parameter which depends on the effective interaction between electrons leading to superconductivity and which determines the energy gap at T = 0" K and the critical temperature, T,. This last parameter is the only one which involves the superconducting state. To compare with experimental results for a particular metal, these three parameters are determined empirically. The density of states, N(E,), is obtained from the normal electronic specific heat constant, y , defined by c, = yT. The product N(E,)v, may be obtained from measurements of the skin resistance of the normal metal in the extreme anomalous limit. The third parameter usually is obtained from T,, or is eliminated by plotting the results on a reduced temperature scale, t = TIT,. Matrix elements for normal state transition probabilities generally are not required for calculations of the ratio of absorption in superconducting and normal states. The fact that the agreement between experiment and theory obtained in this way usually is surprisingly good is an indication that there is a rough law of corresponding states for superconductors. There are, of course, significant departures from the law, as is to be expected from the complex band structure of superconducting metals. The simplified model appears to fit reasonably well to the average of actual metals. Metals whose properties depart most significantly from the average behavior are those with very low Debye temperatures, notably lead and mercury. It is probable that an extension of the theory is required for these elements. One would like to be able to derive the effective interaction and other parameters from first principles. This is necessary to establish a reliable criterion to distinguish between superconductors and non-superconductors, and to estimate critical temperatures. Only limited progress has been made on this very important problem. Some of the difficulties come from the complexities of the electronic structure of actual metals. However, even calculations for an idealized metal with Coulomb interactions are difficult and, while a great deal of work has I < E ~ ~ Y E ~ p. ~ C 282 CS
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been done, the problem is not yet solved. Various calculations have been made for a Hamiltonian which includes only the electron-phonon interaction and omits the direct Coulomb interaction between electrons. This model, of considerable mathematical interest, has some features which differ qualitatively from actual metals. Most important of these is a spectrum of low lying collective excitations which correspond to longitudinal density fluctuations of the electron gas. In real metals with Coulomb excitations such density fluctuations are plasma oscillations of very high quantum energy.
2. Historical Survey Some of the most important milestones on the way to our understanding of scperconductivity up t o the time of the development of the present theory are listed below: (1) Pioneer work of Iiamerlingh Onnes’* and collaborators which showed an infinite conductivity, most strikingly by a persistent current flowing in a ring, and a critical magnetic field above which superconductitity is destroyed. (2) Experiments of Keesom, Van den Endc and I
p . 28.2
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J . BARUEEN A S D J. R. SCHRIEFFER
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(9) Landau’s3‘ prediction (1037) of a laminar structure for thc intermediate state with subsequent experimental verification by Shalnikov, Rkshkovsky, and othcrs32. (10) Measurements of surface resistance, R,,a t microwave frequencies ( 109 H z ) by H. London33(1940), which showed that there is no discontinuity in X, a t the critical temperature, but that R, drops rapidly below Tc, approaching zero as T --f 0” K. This mcthod was later developed and used extensively by Pippard and others. Pippard34 (1947) showed that information about penetration depths can be obtained from the reactivc part of the surface impedance. ( 1 I ) Phenomcnological extension of the London thcory by Ginzburg and Landau35 (1950) for application to the calculation of boundary energies betwccn normal and superconducting phases and other problems. (17) The isotope effect, Tc cc M-1I8, discovered independently by Maxwell and by Reynolds et al. (1950) which strongly indicates that superconductivity arises from interactions between electrons and lattice vibrations. or phoiionssd. (13) Frohlich’s independent developments7 (1950) of a thcory based on clectronphonon interactions, which yielded the isotope effect, but failed to predict other supcrconducting properties. A somewhat similar approach to a theory by one of the authors also ran into difficultiess8. (14) Pippard’s introductionas (1953) of a coherence distance and a nonlocal niodification of the London equations t o account for several experiments on penetration phenomena. One of the authors40 (1955) showed that the Pippard nonlocal relation would most likely follow from an energy gap model. (15) Experimental evidence from several sources41 (1953-present) of an energy gap for excitations of electrons from the superconducting ground state. This evidence will be discussed in more detail later. Many suggestions for such a gap had been made earlier on theoretical grounds. (16) Investigations by Matthias42 of the occurrence of superconductivity in a large number of alloys, compounds and solid solutions, and the development of empirical rules for the occurrence of superconductivity based on such factors as atomic volume, mass, and number of valence electrons per atom. (17) Cooper’s proof43(1956)that a Fermi sea with net attractive interactions between the particles is unstable against the formation of bound pairs, no matter how weak the intcraction.
For many years, it was assumed that the superconducting state is basically a state of infinite conductivity and the major theoretical efforts were centered around the problem of developing an explanation for this phenomenon. As a consequence of this assumption, Maxwell’s equations predict that the magnetic field in the interior of a massive superconducting specimen cannot change when the external magnetic field is altered; therefore, the field distribution within the specimen will be identical to that which existed when the transition from the normal to superconducting state occurred. An important step in the understanding of the phenomenon came with Meissner and Ochsenfeld’s surprising discovery that the magnetic induction always vanishes inside a massive superconductor. Only then was it realized that the superconducting state is fundamentally a state of perfect diamagnetism and many of the aspects formerly associated with infinite conductivity Refcvcizcfs fi. 2S2
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are actually a consequence of the magnetic properties of the system. In a simply connected body, the supercurrents are associated with and uniquely determined by the magnetic field; thus the problem reduces to that of establishing the property of perfect diamagnetism for a superconductor. There remains, however, the additional problem of explaining the metastability of a current in a ring. The current and magnetic field are determined self-consistently in this case. The Meissner effect shows that there is a unique state of a simply connected superconducting body for given conditions of temperature, pressure and external applied magnetic field. The transition from the superconducting to the normal state is therefore reversible and thermodynamics may be applied to the system. It follows from thermodynamic arguments that the difference in Gibbs free energy density between the two states of a massive body in zero magnetic field is given by :
where H , is the critical field for destroying superconductivity at the pressure and temperature in question. Thus the critical field is determined entirely by thermodynamic considerations and may be calculated from specific heat measurements made in the absence of a magnetic field. Conversely, one may calculate the electronic specific heat from measurements of critical field. This method is particularly useful for metals, such as lead, with low Debye temperatures, for which it is difficult to separate the contributions of the lattice and the electrons to the directly measured values. As indicated in Tab. 1, the isotope effect has now been observed in a TABLE I
T,
Lead Mercury Tin Thallium
K
M-a
0.478 0.504 0.505 0.49
number of elements. All measurements are consistent with a relation T,Ma = const, with u very close to 0.5. Early measurements for lead gave values of u significantly larger than 0.50, but a series of careful References p . 282
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J . BARDEEN AND J . R. SCHRIEFFER
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experiments by Hake, Mapother and Decker44 yield a value of c( = 0.48 f 0.01, as listed. This law puts stringent requirements on a satisfactory theory.
3. Outline of the London and Pfppard Theories The electrodynamical properties of a metal in which the conduction electrons encounter no resistance were derived by Becker, Heller and Sauter. The Londons modified these equations so as to pick out a particular class of solutions which give a unique result for a simply connected body in an external magnetic field. For the case of infinite conductivity, with the electron freely accelerated, the relation between current density, ja, and magnetic field H, is
- Ac curl
~
dj, dH = -. dt dt
The Londons obtained a unique solution by postulating that the constant of the time integration always vanishes, so that
- Ac curl j ,
=
H.
(3.2)
or, if H is derived from a vector potential A such that div A = 0 and AI = 0 on the surface: - Acj, =
A.
(3 * 3)
For a free electron gas, A = m/ne2, where n is the concentration of electrons. More generally, A-l = $e2N(EF)vg. The penetration of a field into a plane surface can be obtained by combining the equation (3.2) with Maxwell's equation c curl H = 474. If the surface is the plane z = 0, and H,(z) is the magnetic intensity parallel with the surface, the appropriate solution is H y ( z ) = N,(0)e-zlA, --
(3.4)
where 1 = dAc2/4nis the penetration depth. With the free-electron value for A , this becomes 1 = d m c 2 / 4 z d , which is of the order of cm for the usual range of electron densities in superconductors. Observed values of 1 are generally about 5 x cin, several times larger than the London value. Pippard's nonlocal modification39 of the London equations was suggested by analogy with the theory of the anomalous skin effect in normal metals. For the latter, one is interested in the normal current Xcfererrcrs p . 282
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
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density, in, when the electric field intensity, &, varies over a mean free path. An expression given by Chambers45,equivalent to an earlier one of Reuter and Sondheimer, may be written
s
inW= e 2 N2n (EF)vo
R [ R * &(r')] exp (- R/Z) dt' I
R4
(3.5)
where R = r - r f and I is the mean free path (m.f.p.). In the extreme anomalous limit, for which the skin depth is small compared with I, the current density and thus the surface resistance, R,, are independent of 1. This limit is obtained with high frequency microwave fields if the m.f.p. from impurity scattering is not too short. As shown by Faber and Pippard 46, one can derive from measured values of Rsan empirical value for the parameter N(E,)v0. From this value and the value of N(EF) derived from the observed electronic specific heat, one can determine empirically the average Fermi velocity, vo. On the basis of empirical evidence, Pippard suggested that the current density in a superconductor, instead of being proportional to the vector potential, A, at the point in question, is given by an integral of A over a region surrounding the point. The form he suggested is similar to Chambers' :
is(r) = -
a1 3
R [ R A(r')] exp (- R/E,,) exp (- R/I) dt' R4
, (3.6)
where R = r - r f and again div A = 0. For slowly varying A, the expression (3.6) reduces to the London form (3.3). A new parameter is introduced, the coherence length to,of the order of lo4 cm. Pippard4' has given a number of arguments in favor of a coherence length in superconductors of this order of magnitude: (1) The sharpness of the phase transition in zero magnetic field (Doidge4' finds a width of 0.002' C or less from resistance measurements) suggests that large numbers of electrons act coherently to reduce local fluctuations which usually play a role near a lambda-point; (2) If the superconducting phase is characterized by some sort of order parameter, one might expect it to change in large magnetic fields so as to permit a greater field penetration into the surface and thus a lower free energy. However, the observed change in penetration depth with magnetic fields up to the critical field is very small, which leads to the conclusion that the change in order parameter is not confined to the penetration region, cm. (3) A but must extend to a greater depth, of the order of References
p . 252
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number of experiments point to the existence of a large interface energy, ans, for the boundary between normal and superconducting phases in the intermediate state. When expressed in the form A x H,/8n, observed values of a,, lead to a length A M 10-4 cm. It would appear reasonable that such a large value of c, arises from a boundary which is spread out over a region of thickness A . (4)Further evidence in favor of a non-local theory comes from Pippard's observations of a large increase of the penetration depth in tin upon addition of up to 3% of indium as an impurity. Such a small addition has very little effect on the transition temperature, and does not have much effect on the electron density. Thus the London theory would predict only a very small change in A. The effect can be accounted for by the factor exp (- R/Z) in the integrand of the non-local theory. Evidence for a decrease in the boundary energy with added impurity concentration has been found by Doidge4', suggesting that the range of coherence is reduced by impurity scattering. From these arguments Pippard concludes that the superconducting state is characterized by a long range coherence such that substantial changes cannot occur over a distance less than about cm in pure metals t. He therefore interprets the non-local form of the equation as arising from the wave function of the super electrons responding slightly to the magnetic field, the perturbation due to the vector potential at any one point being spread over a distance to. With the non-local form of the theory, the change in field, H ( z ) , with depth from a plane surface is not exponential, as in the London theory, but has a more complicated variation, with a small reversal of direction at large depths. Most experiments measure only the total flux. The penetration depth, 1,is defined by:
1
m
AH(0) =
H ( z ) dz.
(3 * 7)
0
The present theory gives a non-local relation between j , and A similar to that suggested by Pippard. In 3 12 a comparison of the predictions with experimental results is given. Also given there are arguments to show why a non-local version is likely to follow from any reasonable model which gives an energy gap. That the order Characteristic of the superconducting state may extend across a normal conducting barrier has been shown by H. Meissner, Phys. Rev. Lett. 2, 468 (1969). References p . 282
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4. London’s Quantum Picture of the Superconducting State F. London2suggested that the currents given by (3.2) are diamagnetic in origin, and indicated how an equation of this sort might follow from quantum theory. That quantum theory is essential iollows from a famous theorem of Bohr and of van Leeuwen that a classical system can exhibit no d i a m a g n e t i ~ mSuperconductivity ~~. is a case of perfect diamagnetism in which the field vanishes in the interior of a bulk specimen. According to London 4 9 : “In thermal equilibrium there is no permanent current in an isolated superconductor except in the presence of an applied magnetic field, and there is no conservation of these currents; they differ for every variation of the strength or direction of the applied field.” A persistent current in a ring is, of course, metastable rather than stable. In this case, the flux through the ring is maintained at a constant value when the external field is changed. One must go beyond the concept of perfect diamagnetism to account for the metastability of currents in multiply-connected bcdies. What sort of a quantum system may be expected to give a large diamagnetism ? It is well known that for atomic or molecular systems, the susceptibility is given by
where N is the number of atoms per cm3, and ~ 7 . sthe mean square radius of the orbit of the electrons. For electron densities typical of a solid, and for a typical atomic radius of cm, this expression gives 1 x 1 rn 10-6-10-7, as observed €or both normal metals and insulators. In the derivations of (4.1) it is assumed that the wave functions are not changed appreciably by the magnetic field. In a normal metal the orbits are modified, and as Landau has shown, the susceptibility is small and of the order indicated above. A perfect diamagnetism corresponds to x = - (4n)-l giving B = H ( l 4 n ~= ) 0. A value of this order requires large orbits for the electrons, an effective of the order of cm. Of course a model with a large and a large diamagnetism does not necessarily have all of the properties of a superconductor, such as persistent currents50. London’s approach to (3.3) from quantum theory is as follows. The general expression for the current density of an 12-electron system with
+
<
References p . 282
<
1a2
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
54
a wave function Y(r,, r 2 . . . .r,,) in a magnetic field described by the vector potential A(r) is: ?ie
-
e2
(Iv* grad, Y - !P grad, Y*) -
1
-A ( y , ) Y * Y x ?TZC
d(r - r,) dr,.
. . .dr,,).
(4.2)
In the absence of a magnetic field ( A = 0 ) , Y = Yo and the current density vanishes. In a normal metal, the wave function responds to the magnetic field in such a way that there is very close cancellation between the “paramagnetic” contribution involving the gradient terms and the “diamagnetic” contribution proportional to the vector potential, A, leaving only the small Landau diamagnetism. London proposed that a superconductor differs in that there is a rigidity or “stiffness” in the wave function such that it is essentially unmodified by the magnetic field, provided that the field is described in the gauge div A = 0. If the wave function is unchanged, one may replace Iv by !Po,and the paramagnetic contribution vanishes even in the presence of the field. This leaves the diamagnetic term, which gives an equation equivalent to (3.3)
To understand the significance of this result, one must be careful to distinguish between the momentum operator, p, and velocity 1 e v = - pm + - A .C
(4.4)
In a normal metal, the average value of p changes in a magnetic field in such a way that v is practically zero. In a London superconductor, (p),,, remains M 0 when the field is applied, so that the average velocity is proportional to A. It was suggested that the reason that (p) does not change when the field is applied is that there is a long range order which maintains the local average value of the momentum constant over large distances in space. This order would be maintained even in the presence of the magnetic field. The ordered ground state is regarded as a single quantum state extending throughout the metal. As stated by London49, a superconductor is a “quantum structure on n References p. 282
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macroscopic scale” which is a “kind of solidification or condensation of the average momentum distribution”. According to the present theory, the “paramagnetic” contribution vanishes only for fields which vary slowly over a coherence distance. When first order changes in the wave functions produced by the field are taken into account, one is led to a non-local theory similar to, although not identical with, that of Pippard. The above considerations apply to the single quantum state appropriate to T = 0” K. As the temperature is raised, thermal excitations appear. The elementary electronic excitations may be regarded most simply as electrons excited out of the ground state. If the temperature is sufficiently low, the number of these excitations is so small that the long range order in the ground state is not destroyed. It is only when the transition temperature, T,, is reached that the number becomes large enough to destroy this order. The ground state may be regarded as the superfluid component and the excitations as the normal component of a two-fluid model. Up to T,, the single quantum state character of the ground state is maintained. There is a similarity in this regard between superconductors and HeII. The ground state of HeII at rest is a single quantum state in which on the average a finite fraction of the atoms have momentum exactly equal to zero, as in a condensed Bose-Einstein gas51. This strong correlation between the momenta is maintained even in the presence of interactions between the atoms. As the temperature is raised, there exist excitations, described as phonons and rotons, but up to the I-point, these are not sufficient in number to destroy the order in the ground state. The single quantum state character of the ground state of liquid He is perhaps shown most strikingly in an experiment of V i n e r ~ He ~ ~ showed . that the circulation, J v * dl about a wire along the axis of a cylindrical container is quantized to multiples of 2nh/M, which is to be expected only with a ground state wave function extending throughout the volume. There is a corresponding quantization of currents in a superconductor. One consequence is that the total flux through a superconducting ring must be an integral multiple of 2n&c/e (about 4 x 10-7 gauss cm2). It can, in fact, be shown that a Bose-Einstein gas of charged particles would exhibit a Meissner effect and other superconducting properties below the transition temperature. Following up this approach as a possible .explanation of the phenomenon, Schafroth and others 53 have References p . 282
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suggested that there may be localized bound pairs of electrons which obey Bose-Einstein statistics. An attempt was made by Schafroth, Blatt and Butler53c to develop such a theory using what they call a quasi-chemical approach. Because of mathematical difficulties, they were not able to carry out calculations based on their general formulation for any model which exhibited superconducting properties. For a qualitative picture, they suggested a model with localized pairs such that the average size of the pair is smaller than the spacing between pairs. Such pairs of “molecules” would have translational motion and there would be a close analogy between the ground state and low lying excited states with those of a Bose-Einstein gas?. According to the present theory, the ground state of a superconductor does have some of the characteristics of an Einstein-Bose condensate. Electrons in momentum space are associated in pairs such that the total momentum of each pair is identically the same. However, there are no pairs localized in space in the way envisaged by Schafroth, Blatt and Butler, and the spectrum of elementary excitations is completely different from that of an Einstein-Bose gas. Thus, while there are some features in common, including the very important one of a long range order of average momentum, the differences are so great that the analogy between superconductors and an Einstein-Bose condensation is actually not a very close one. This point will be discussed later (8 8) in connection with the structure of the wave functions.
5. Elementary Excitations in Normal Metals 5.1. QUASI-PARTICLE EXCITATIONS
An essential difficulty in constructing a microscopic theory of superconductivity is the very small energy difference between the normal and superconducting phases. This energy is H:/8n per unit volume, which is only of the order of 10-’eV/atom. One obtains a value of this order if it is assumed that electrons which have energies within w k,T, of the Fermi surface have their energies reduced by NN k,T, by the t As stated by Schafroth, (see ref.64) : “If it can be shown that the electron pairs, in contrast t o the pairs t o the BCS theory, form fairly localized entities (‘pseudo-molecules’) whose center-of-gravity motion is essentially undisturbed, then one may be able to show that the complicated features of the Hamiltonian affect the internal wave function of the pseudo-molecule only, whereas the center-of-gravity motion is essentially free. This, together with a proof that these ‘pseudo-molecules’ obey some kind of Bose statistics, would be sufficient to establish the Meissner effect.” Rcferenccs
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transition. It is this minute energy which gives the remarkable change in the electromagnetic and other properties of the system. This energy is to be contrasted with a correlation energy of the order of 1 eV/atom from Coulomb interactions and of the order of eV/atom from self energies associated with the electron-phonon interactions. Since it is not possible to calculate the energy of either phase with anything like an accuracy of w lo-' eV, all one can hope to do is to focus on the terms in the energy which bring about the difference between normal and superconducting phases and to estimate this difference as accurately as possible. One way to do this is to express the superconducting ground state wave function in terms of normal state-like configurations which include the correlation effects common to both phases. We shall first discuss the normal state, an! its spectrum of elementary excitations and then in the following section those of the superconducting state. In both normal and superconducting metals, there are three principal types of elementary excitations ;quasi-particles, phonons and plasmons. These are not exact eigenstates of the Hamiltonian, but interact with one another and with the Fermi sea. They thus have a finite lifetime, '6. For most purposes, an elementary excitation is reasonably well defined if the uncertainty in energy, &/r,is small compared with the excitation energy. All of the elementary excitations are waves and may be designated by an appropriate wave vector, k. While a definite value for k corresponds to a wave running throughout the crystal, one may, as usual, form localized wave packets of extent A x by taking a small spread d k w l/Ax around the specified wave vector. Because of their finite life-time, it is best to think of the elementary excitations as so localized. The various low-lying quasi-particle configurations of a normal metal with interactions can be described in one-to-one correspondence with those of the Bloch individual particle model, i.e., by giving the occupied states in k-space. For the ground configuration, at T = 0" K, states below the Fermi surface are occupied, those above unoccupied. The energy of an excited configuration above that of the ground state can be given by a sum of quasi-particle energies. We shall measure the quasi-particle energy ~ ( k from ) the Fermi energy, E,. Thus ~ ( k ) for a normal metal is a continuous function of the wave-vector k , vanishing at the Fermi surface. When k is below the Fermi surface, E ( k ) is a negative number. I t is often convenient to describe occup a t'ion References p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. \’I,
95
below the Fermi surface in terms of unoccupied states, or holes. The energy of a hole in k is then the positive number - E ( k ) . If q(k , a) = 1, 0 gives the occupation of the state k with spin a, the excitation energy above the ground state is
W - W
C
O - k>Fg,0
q ( k , a)e(k)
+ P < P,,C
a
[l -
where 1 - q( k , u) may be regarded as the hole occupation number for k < k,. The excited particles above and the holes below the Fermi surface are to be regarded as the elementary quasi-particle excitations. In a normal metal, e ( k ) --f 0 as k -+ k,, so that only an infinitesimal energy is required to excite an electron at the Fermi surface. As we shall see, the various excited configurations of a superconductor can be described in terms of occupation numbers in k-space as in a normal metal. The excitation spectrum differs in that a finite energy, the energy gap, is required t o excite a particle from the superconducting ground state. Phonons, the quanta of the lattice vibrations, have a Debye specof the order of eV. These fretrum with average energies, &oph, quencies are changed very little by the transition to superconductivity. Plasmons, quanta of plasma oscillations of the electron gas, with energies of rn 10 - 20 eV, are not normally excited and play no direct role in superconductivity. Introduction of the plasma modes is important, however, for an adequate treatment of screening of the electrons, as will be discussed in more detail in 5 13. When a particle is excited out of the Fermi sea, there is a hole left behind. The excited particle and the hole will in general not be bound together in space and they may be regarded as independent excitations and treated in an equivalent manner. Quasi-particle excitations are thus created in pairs from the ground state. 5 . 3 SCREENIXC AND BACKFLOW
A quasi-particle is not a “bare” particle moving independently of all others, but should be regarded as a particle moving in the electron fluid. In the language of field theory, it is “clothed” by interactions with phonons, plasmons and other particles. Surrounding each electron is a “screening hole” with a net deficit of electronic charge which is just equal t o that of the electron in question. A local depletion of the other electrons resulting from Coulomb repulsion leaves a positive uncomKcjerenccs
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pensated ionic charge which balances that of the electron in the center of the hole. The situation is illustrated in Fig. 1, which also shows schematically the motion of the electron from A to B. Since the screening hole moves with the electron, there is no net transfer of charge. When the electron in question is transferred from A to B there must be a compensating “backflow” in the surrounding electron fluid. The concept of backflow was introduced by Feynman and Cohen55in discussing the nature of the rotons in HeII, which also may be regarded as quasi-particles
A
B
Fig. 1. Screening hole suricxnding each electron. When an electron moves from A to B there is a compensating “backflow” in the surrounding electron fluid.
moving with an associated backflow. At large distances the backflow leads to a current distribution which is dipolar in form. The velocity potential a t large distances is given by: (5.2)
The strength of the dipole, ,u, is proportional to the velocity of the particle. Backflow is a collective motion which may be described in terms of collective variables. In the formalism of Bohm and Pines, in which the long range part of the Coulomb interaction is described in terms of plasmon variables, the backflow at large distances may be viewed as a cloud of virtual plasmons which move with the electron. The problem of backflow for electrons in metals has been discussed by Pines and one of the authorsg. They have pointed out that it is essential to include backflow in order that the quasi-particle excitations satisfy the equation of continuity, a problem closely related to the gauge invariance of the theory. In this connection, there is a marked difference in the description of current flow in longitudinal and transverse waves. In Fig. 2 , we have shown schematically the elementary Rrjerrrtces
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J. BARDEEN AND J . R . SCHRIEFFER
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dipoles associated with quasi-particle motion for the two types of waves. There is an exact analogy with longitudinal and transverse waves of magnetization ; the velocity-field corresponds to the H-field from magnetic dipoles and total current density from both particle motion and backflow to B = H 4nM. For long wave lengths, the longitudinal currents are described almost entirely in terms of the collective motion, and thus in terms of plasmon variables. The quasiparticles with their screening holes do not contribute appreciabIy (corresponding to B = 0). The opposite occurs for transverse waves, the backflow from different parts of the wave cancels out (corresponding to H = 0 ) , so that the current is just that which would be obtained from particle motion alone with backflow neglected. The above considerations show that in calculating the response to longitudinal waves, it is necessary to consider collective excitations
+
- - c c c
-+
- - c c c
4-
- - t c c c
- 4
4 - c - c
- 4
Longitudinal waves, B='O
-
c
-
4
c
-
c
-
c
-
4
c
c
c
CCtCCCC
C
C
C
F
I
-
C
C
Transverse waves, H=0
Fig. 2. Longitudinal and transverse waves of magnetization. The velocity-field of the hackflow from the quasi-particles corresponds to thc H-field from the magnetic dipoles and thc total current density from both particle motion and associated backflow to B = H + 4xM. For long wave length longitudinal waves B = 0, while for trans\wse waves H = 0.
explicitly. On the other hand, the response to transverse waves is almost entirely by quasi-particles, and their contribution to the current can be calculated in the usual way, neglecting backflow. This is the essential reason for the London choice of gauge, div A = 0 , which implies transverse waves. In order to make calculations in a general gauge, it is necessary to introduce collective as well as quasi-particle variables, In 5 13 we shall discuss recent gauge invariant calculations of the Meissner effect. Results are essentially the same as those of earlier calculations made in the gauge div A = 0 and which considered explicitly only the quasi-particle excitations. Backflow may also be neglected in considering steady currents flowing in a wire or other conductor. It should be pointed out that a plasmon excitation is simply a coherent superposition of electron-hole excitations which has the form of a density fluctuation of the electron gas. Since the plasmons Rcfucnces
p. 282
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constitute good elementary excitations for long wavelengths, it is clear that one would be over describing the system by including all possible quasi-particle excitations as well as plasmon excitations. I n many problems, no difficulties arise because of the fact that only the special combination of quasi-particle excitations corresponding to a coherent density fluctuation must be suppressed. Actually, it is possible to carry out a consistent calculation in which no extra variables are introduced by use of generalizations of the Hartree self-consistent field method appropriate to normal and superconducting states (see 5 13). I n addition to the plasmon cloud surrounding each electron, there is a displacement of ions in the vicinity which follows the motion of the electron and which may be described in terms of a cloud of virtual phonons. Further, in describing the phonons, it is important to take into account the screening of the fields of the ions by the electrons. Plasmons, while primarily an oscillation of the electron gas, also involve some ionic motion. 5 . 3 . INTERACTIOKS BETWEEN ELEMENTARY EXCITATIOXS
We shall next discuss the interactions of the quasi-particles with the Fermi sea and with each other. Those important for superconductivity are the electron-phonon interaction and the screened electron-electron interaction. An excited electron can decay or be scattered by emitting or absorbing a phonon, by exciting another electron out of the sea (creating two new quasi-particles) or by interaction with another quasi-particle. The lifetime, z, of quasi-particle excitations is reasonably long a t moderate temperatures because of the restrictions on scattering introduced by the Pauli principle. This accounts for the success of the Bloch individual particle model. A particle with an excitation energy E can knock another particle out of the Fermi sea only if the energy of the latter is within E of the Fermi surface, E,. The energies of both particles after scattering must also be within E of E,. The effect of these restrictions on the available phase space is to increase the free path for electron-scattering by a factor of the order of ( E , J E ) ~This , gives for E , NN lOeV and E M 0.OleV (corresponding to T M 100" K) a free path of the order of (lo6 x 10V) cm, or 10W cm. For values of E of this order or smaller, the free path is restricted by the electron-phonon rather than by electron-electron scattering. Heavy elements with a low Debye temperature have a large electron-phonon interaction, so Refmnces
p . 282
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[CH. VI,
$5
that the electron may be readily scattered by excitation of a phonon, with a relatively small mean free path. It is believed that this may account for the anomalous superconducting properties of Pb and Hg, as will be discussed in more detail in 5 9. There is a large number of low-lying excited normal state configurations which correspond to exciting electrons to small energies above the Fermi sea. These may be described, as in the Bloch model, by giving the occupied quasi-particle states in k-space. A typical configuration is shown in Fig. 3. To complete the description, one would have to give the occupation of phonon and plasnion states. It is yresumed that the correlation energies in the ground state are adequately taken into account and that all that prevent the configuration from t
Fig. 3. A typical excited configuration :ii the normal state. Quasi-particle excitations are specified as occupied states aliove and holes below the Fermi surface.
being an exact eigenstate of the Hamiltonian are interactions between the elementary excitations. Thus the configuration is not to be regarded as given by a Bloch determinantal wave function. This type of phenomenological description of a Fermi gas with interactions has been generalized by LandauSs in his theory of the Fermi liquid to include the dependence of the energy of the quasiparticle on the distribution of the particles in k-space in a manner similar to the Hartree-Fock method. He has given a justification for it from basic theory by use of Green's function methods. Long range Coulomb interactions in an electron gas introduce complications not considered explicitly by Landau. This problem has received a great deal of attention from theorists, and considerable progress has been made, but there is as yet no really satisfactory quantitative treatment for the normal range of electron densities. Rq'en~nces p . 2852
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6. Electron-Phonon Interactions Frohlich's suggestion 37 that superconductivity arises from the electron-phonon interaction pointed the way toward the development of a successful theory. His derivation57"of an attractive interaction between electrons from exchange of virtual phonons later was extended by Bardeen and Pines 57b to take Coulomb interactions into account. The interaction may be described in a qualitative way as A particle near the Fermi surface in a state k , emits a virtual phonon of wave vector q and is scattered to a state kl = ki - q. While the electron does not have enough energy to emit an actual phonon, it may do so momentarily because of the uncertainty relation AEAt M fi. A second electron in k , absorbs the phonon and is scattered to kl = k , + q. The effect is to scatter electrons originally in states k,, k , to k;, k; with conservation of wave vector: k,
+ k , = k; + k;.
(6.1)
This corresponds to a pair interaction between the particles, and it is attractive if the energy difference between the electron states involved is less than the energy of the virtual phonon, noph.The criterion for superconductivity is essentially that this attractive interaction dominates the repulsive screened Coulomb interaction. The physical origin of the phonon interaction arises simply from the fact that an electron making a transition from state k , to k , - q gives rise to a charge density fluctuation, de;, of wave vector q and frequency &o(k,, k , - q ) = ~ ( k , ) - &(kl - 9). As a consequence of the electron-phonon interaction, de: can excite a phonon. This phonon will exhibit an ionic (and an associated electronic) charge density fluctuation, de;, which will be out of phase with the initiating electronic charge fluctuation Se; if o(kl, k, - q) is greater than the natural frequency, oqof the phonon. If the reverse is true, de; will be in phase with de;. This process describes the dynamic screening of the electric field set up by the virtual electron transition k , -+ k , - q. I t follows that the strength with which the second electron recoils, ( k , -+ k , q ) , depends on the effectiveness of this screening. If w(k,, k, - q) < w q , over-screening occurs, crudely speaking, by the positive ionic charge fluctuations building up to a value which more than compensates the Coulomb field set up by dp; and the second particle is attracted to rather than repelled from the first. For ~ ( k , k, , 4) > w q ,anti-screening occurs since dp: and are out of
+
+
Ktfrreitces
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J. BARDEEN AND J. R. SCHPIEFFER
[CH.
VI, $ 6
phase. As one might expect, the screened interaction due to virtual phonon exchange is proportional to 1
characteristic of the response of a driven harmonic oscillator. It should be mentioned that the phonon interaction and the screened Coulomb interaction also contribute t o the self-energy of the quasiparticles. One such contribution would correspond to the electron in k, emitting a virtual phonon and going to k;, then reabsorbing the phonon and returning to the initial state. It is presumed that all such self-energy corrections are included in the description of the normal state configurations and that these corrections are essentially unaltered in the superconducting state. All that need appear in an effective Hamiltonian for the electrons and phonons are the true interaction terms. The matrix element for the phonon interaction is:
where M , is the matrix element of the electron-phonon interaction and &GO,the energy of the phonon involved. The interaction is attractive (negative) if the energy difference between the electron states is less than &w,. The criterion for superconductivity is that this attractive interaction dominate the screened Coulomb interaction, which may be written 4ne2 VCOUl
=
I k; - k , 12 + k:
’
where k, is a screening constant. This condition may be written symbolically in the form
where the average is taken over an interaction region near the Fermi surface where I E; - E~ 1 < &wc. Here w , is an average phonon frequency, perhaps half the Debye frequency. This criterion has been studied by Pines and in more detail by Morrel References p. ?S?
CH. VI,
9
71
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on the basis of a simplified model5*. They find that the elements most favored are those with a large number of valence electrons per atom, and within this limitation, a low electron density, in agreement with Matthias' empirical rules. The criterion very roughly separates superconductors from non-superconductors. The largest contribution to Vp,, comes from the Umklapp region where ki - k, lies outside of the first Brillouin zone. If the states k are described in the expanded zone scheme, this applies to most of the possible virtual transitions. In the Umklapp region, I k; - k , I may be relatively large (reducing the Coulomb contribution) while the reduced wave vector q = k , - k; K and thus fico, is small. Here K is a lattice vector of the reciprocal lattice space. The role of the Coulomb interactions in counteracting superconductivity will be discussed further in § 9.
+
7. Elementary Excitations in Superconductors The most striking difference between the excitations in normal and superconducting states is the existence of an energy gap for quasiparticle excitations in the latter. Quasi-particles in superconductors may be designated by a wave-vector k and spin u in one-to-one correspondence with those of normal metals. The energy, E,, may be written in the form
where E, is the Bloch energy in the normal state relative to the Fermi energy and A , is an energy gap parameter which is obtained from the theory as a solution of an integral equation. The excitations correspond roughly to particles above and holes below the Fermi surface, although in a superconductor there is no discontinuity in the nature of the excitation as the Fermi surface is crossed. If q(K, 0) = 1, 0 gives the occupation number of the excitation, the total excitation energy above that of the ground state, W,,, is
we,, w,,= k,2 q(k, +%> -
a
(7.2)
in exact analogy with the normal state. The value of A , depends on the distribution of excitations, and so varies with the temperature. Having a maximum at T = 0" K, the energy gap gradually decreases with increasing temperature and vanishes at the transition. In general, the energy gap may be anisotropic and depend on the direction of k as well as on the energy. There is increasing experimental Hefercnccs
p . ?82
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[CH.VI, 5 7
evidence for some anisotropic effects which are, of course, peculiar to the particular metal. Since such effects are not large and are not essential to an understanding of superconductivity, we shall for the most part ignore them and assume that A , is a function only of the energy E~ of the Bloch state involved. Theory indicates that A , should be appreciable over the range for which the attractive phonon interaction is significant, that is within the order of an average phonon energy, fiw,, of the Fermi surface. A typical plot is shown in Fig. 4. For the simplified interaction used by BCS, A is a constant up to a cut-off, no,,and zero thereafter. One is usually interested in excitation energies of no more than a few k,7,.
c
Fig. 4. Variation of the energy gap parameter A(&) near the Ferini surface. In the case of weak coupling, ?iw, % A .
In what is called the weak coupling limit, and which applies to most superconductors, K,T, noc.In this case, one may take d = const over the interesting range without appreciable error. The weak coupling limit does not apply to metals with very low Debye temperatures, such as lead and mercury. Some of the large amount of experimental evidence for an energy gap will be discussed in more detail in later sections. As mentioned in the introduction, the most direct evidence comes from experiments on absorption of electromagnetic waves, either in the microwave or far infrared part of the spectrum. Other evidence comes from experiments on specific heats, absorption of ultrasonic waves, nuclear spin relaxation times, and thermal conductivity. All of these depend on the presence of excited electrons. Experiments done at very low temperatures indicate that the number of excitations drops exponentially, as exp(- bT,/T), which is suggestive of an energy gap. Most of this evidence has been accumulated since 1953, when Brown, Zemansky and Boorsebg showed that the electronic specific heat of vanadium follows an exponential law and Goodmanso found an exponential drop in the thermal conductivity of tin which he interpreted in terms of an energy gap. In Table 2 we have listed empirical values of the gap for several metals.
<
References
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There have over the years been many suggestions for an energy gap on theoretical grounds. Welker61suggested a gap to account for the Meissner effect. Daunt and Mendelssohn62showed that there is no Thompson heat associated with current flow in a superconductor, and that therefore a supercurrent represents flow of the ground state with entropy equal to zero. They suggested a gap to excited states which contribute to the electronic specific heat. Ginzburglb proposed a twofluid model based on an energy gap. One of the authors lC developed a theory of the Meissner effect based on an energy gap model, and showed that it would lead to a non-local theory of the Pippard type if the gap were no more than a few k,T,. TABLE 11
Values of the energy gap t Columns I-V: 24(0)/kBT, 1 0 3 ~ ~ __
Superconductor -
Indium Tin Mercury Tantalum Vanadium Lead Niobium
__ . 3.39 3.73 4.15 4.39 5.1 7.15 90
-
31 19 52 18 16
76 35
I11
IV
v
_ _ 3.5 3.6 3.7 3.6
3.9 3.6
3.3
3.6
3.6 3.6
3.5
3.9 3.7
4.0
3.9
I1
I
OD ~
4.1 2 0.2 ,0.2 3.6 j 4.6 & 0.2 5 3.0 3.4 rfr 0.2 4.1 & 0.2 2.8 k 0 3
~-
~~
3.9 & 0.3 3 3 & 0.2
4.0 + 0.5
8. Nature of the Wave Functions for Superconductors 8 . I . KEASONSFOR PAIRCONFIGURATIONS
Cooper43showed that if there is a net attraction, two quasi-particles above the Fermi sea of a normal metal may form a bound state with a net gain in energy over the Fermi ground state no matter how weak the interaction. This very important result showed that for attractive interactions, the Fermi sea is unstable against the formation of such bound pairs. He also pointed out that if the binding energy of a pair 7 Measured values of the gap a t absolute zcro, in units of ~ B T , ,compared to values obtained by I Richards and Tinkham from infrared reflection measurements on bulk specimens; I1 by Ginsberg and Tinkham from transmission measurements on thin films, and the last three columns by Goodman: I11 from the relation 2d(0)/kBT, = 2H0/T,(n/6y)1/2given by the microscopic theory, with experimental values of y, Ho, and T,;IV by fitting an exponential function to the experimental specific heat data; and V by fitting the microscopic theory to the experimental specific heat data. 1ZejeYenrcs
p. 282
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J. BARDEEN AND J . R . SCHRIEFFER
[CH.
VI,
58
is of the order of k,T,, with T , M 1" K, the size of the wave function of the pair is of the order of cm. The wave function is made up mainly from states with energies within a few k,T, of E,. Long before Cooper's calculation, it was recognized that a dimension of this order follows from the uncertainty relation. The wave vector difference d k , corresponding to an energy difference k,T, at the Fermi surface, k = k,, is given by:
d k l k , w kB7',/E, w
(8.1)
Since k, is typically of the order of lo8 cm-1, d k is of the order of lo4 cm-l. The uncertainty relation 4kdx w 1 gives a minimum range A x M 10-4 cm for the wave functions. It follows immediately from these dimensional considerations that a picture of an Einstein-Bose condensation of isolated pairs is not a possible one. The number of electrons within M A k of k , and which presumably take part in the formation of the superconducting condensed state is of the order of lop4 x 1022 = 101R/cm3.If these were formed into pairs, the spacing between the pairs would be of the order of ICP cm. Since the size of the pair wave function is of the order of cm, there must be a very great overlapping and a picture of isolated pairs loses its meaning. There is thus a problem involved in applying the idea of pair condensation to account for superconductivity. Cooper and the authors were led to their formulation of the theory by following a mathematical technique used by Cooper in his theory for a single pair. This method may be described briefly as follows. Suppose that one has a Hamiltonian H = H , U , with unperturbed eigenstates H,yt = E,yI and matrix elements U,, of U . For example, H , might represent noninteracting particles and U the interaction between them. Suppose that one can choose a subset of the y i with phases such that the matrix elements U l j between any two members of the set are predominantly negative. Then a trial function for the ground state of H is taken to be a linear combination of members of this subset with coefficients,a,, of the same sign: Y= Cailyt.The corresponding energy is :
+
W
afEi
= 2
+- T,a,a,U,. 11
If the U,, are all negative, the contributions to the interaction energy will add in phase and give a coherent low energy state. The coefficients, a,, may be chosen by a variational method. References
p . 282
CH. VI,
3 81
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
197
A particularly simple example is one for which there is a set of N degenerate ?pi with energies Ei= E o and such that each is connected to m other states by a nonvanishing matrix element U,, = - V . The a , are then all equal and the ground state energy is W
Eo
==
- WZV.
(8.3)
We form the superconducting ground state from a linear combination of normal state-like configurations in which particles are excited to low energies above the Fermi sea. A given configuration, such as that illustrated in Fig. 3 may be designated by the occupation in k-space of particles above and holes below the Fermi sea. We assume that all of the normal state correlation energy and the self-energies of the quasiparticles are included in the description of these configurations. There remains the effective interaction, U , between the quasi-particles from the phonon and screened Coulomb interactions. These give rise to superconductivity if the phonon interaction dominates to give a net attraction for particles near the Fermi surface. The Bloch individual particle model may be used to estimate the matrix elements between these normal configurations, or a more precise method may be derived from basic theory. I n either case, for an effective pair interaction, there are non-vanishing matrix elements between two configurations which differ in the occupation of two particles, for example k,, k , going to k;, ki with conservation of wave vector: k,
+ k, = k; + kb = Q.
(8.4)
For general configurations of Fermi-Dirac particles, the sign of the matrix element depends on the occupation of the other states, k,, k,. . . . k,, which are unchanged in the transition : ( k 2 , k,,. . . . k,, k ; , k; =
1
I/ 1 k,, k,.
i- ( k i , k; I
v
. . . k , , k , k,)
k,, k,).
=
(8.5)
Spin variables are not indicated explicitly. If a definite ordering of all states is chosen, the rrlative sign of the matrix element is given by (- l)A’+-V’, where A’ and N’ are the total number of occupied states between k,a, and k,o, in the initial state and k;a; and kba; in the final state respectively. With interacting particles, the magnitude of the matrix element will also depend on the states occupied by the other particles, but this dependence is small for configurations with IZefcreirces
p. 282
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J . BARDEEN AND J. R. SCHRIEFFER
[CH. V I ,
98
s m d excitations above the Fermi sea such as those we are considering. Since for general configurations, N N' is equally likely to be even or odd, the matrix elements will alternate in sign and one cannot get a coherent low energy state by the method described above. If the various configurations entered with roughly equal weight, the interaction energy would be small. Since the number of matrix elements far outnumbers the number of configurations, there is no way to get consistent signs by changing the signs of the wave functions for the various configurations. To apply the method, we need to select a subset of configurations between which the matrix elements are dominantly negative for an attractive interaction. It is easily seen that this can be done by associating the states in pairs, (k,a,, k,a,) and requiring that if in any configuration one member of the pair is occupied, the other is also. In this way both N and N' are necessarily even, as is N N'. To get a maximum number of matrix elements, and thus the lowest energy, it is desirable to choose the pairs in such a way that any pair can be scattered into any other pair by the interaction. This means that they should have the same net momentum; that is, k , k , = Q should be the same for all pairs. I t is also probable that in most cases the paired states should have opposite spin, since exchange terms usually tend to reduce the matrix elements for parallel spin pairing. For the ground state, Q = 0, and the paired states have opposite spin and momentum ( k f , - k 4). Parallel spin pairing has been considered, however, and may lead to a lower energy state in special cases for which the angular dependence of the interaction is such that the exchange matrix elements are generally of opposite sign to the direct. In cases where impurity scattering is important, the wave vector is not a good quantum number; the appropriate pairing is discussed in 3 12.
+
+
+
8.2.
G R O U N D STATE
WAVE
FUNCTION
One is thus led by these considerations to a coherent superconducting ground state, Y8, which consists of a linear combination of normal state configurations in which the quasi-particle states are occupied in pairs of opposite spin and momentum. Such a pair may be designated by the wave vector k ( k f , - k 4 ). A configuration may be indicated by the occupied pair states. Thus we may write
References p. 282
CH. VI,
5 81
199
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
where the sum is over all configurations. Formation of such a state is favorable if the interaction is dominantly attractive for particles with energies near the Fermi surface. The various configurations, yS,which give the dominant contribution to'the energy all have approximately the same overall distribution of particles in k-space. As illustrated in Fig. 5, the probability, h ( k ) , that a given state is occupied drops conk , to zero for k $ k,, with no distinuously from unity for k continuity at the Fermi surface. Here 1 - h(k) is to be interpreted as the probability of a hole for k < k,. The normal ground state corresponds to h(k) = 1 for k < k , and h(k) = 0 for k > k,. It should be noted that the entire energy difference between normal and superconducting states in this approximation comes from the
<
-1.0
0
I.o
€/A Fig. 5. The probability h(e) that a given state of energy
E
is occupied in the super-
conducting ground state.
+
interaction terms in the Hamiltonian for which k , + k , = k; ki = = Q = 0. These terms have zero weight in the overall interaction and contribute negligibly to the energy of the normal state. If these terms with Q = 0 were treated by a perturbation expansion, it would be found that each order of the expansion would give a vanishingly small contribution in a large system, with an energy per particle of order O(I/%).But they are all-important for superconductivity. The physical consequence of forming these linear combinations of paired states is to give a long range correlation between particles of opposite spin which extends over distances of order cm in real space and is such as to take advantage of the attractive interaction. Since the common value of the momentum of the pairs is everwhere the same, there is also a long range correlation of the average momentum similar to that suggested by London. The state is of just the sort that had been expected to account for the Meissner effect and other superconducting properties, and detailed calculations show that this is indeed the case.
References p. 282
200
J . BARDEEN A N D J . R . S C H R I E F F E R
[CH. VI,
5
8
8.3. EXCITATION SPECTRUM Particle-like excitations of a superconductor in one-to-one correspondence with those of the normal metals can be obtained by specifying occupation of certain states in k-space and using the rest to form linear combinations of paired configurations. In the “single particle” excitations of BCS, one member of a pair, say k f , is occupied and the other, - k 4, unoccupied in all configurations. Each of these configurations is orthogonal to the configurations which make up the ground state. Such a wave function corresponds to an excited particle in k t if k > k , and to a hole in - k 4 if k < k,. Formation of “pair” excitations in which both k f and - k $ are “occupied” requires more care. The ground state may be decomposed into a part vl,in which the pair k is certainly occupied and a part voin which it is unoccupied : ys
= ‘kT’1
f ukvO’
(8-7)
Here v; = 1 - ui = h(k) is the probability that the state k is occupied. The orthogonal combination
(8.8)
= “ k v l - “k?O
is the wave function for an “excited pair” in k. Bogoliubov and Valatin have shown that “single” and “pair” excitations of the superconductor can be treated on the same footing by means of a transformation of the creation and destruction operators for quasi-particle excitations of the normal state ($11.1). 8.4. RELATION TO EINSTEIN-BOSE CONDENSATIOK
If the normal ground state is taken to be one of noninteracting particles, the superconducting ground state may be expressed as an antisymmetrized product of identical pair functions 63 : Y, =
5 (-
l ) pPq(r, - r2)v(r3 - r4).
. . . . . . *P(‘n-1
- rn),
(8.9)
in which the sum is over all permutations, P, of the n particles. Each pair function has a symmetric space part and an antisymmetric spin part, the latter not indicated explicitly. The space part is of the form T(‘1
-
(8.10)
The effect of antisymmetrizing the product is to eliminate all correlations present in p(rl - r2) except the long range correlations associated References
p . 282
CH.VI,
3 91
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
201
with departures of h(k) from the normal values of unity for k > k , and zero for k < k,. The normal ground state may also be expressed as such a product; in this case antisymmetrizing removes all correlations between particles of opposite spin. The expression (8.9) may be used to point out the analogies and differences between the superconducting ground state and that of an Einstein-Bose condensation of pairs and also to give the connection with a theory of Schafroth, Blatt and Butler53. As pointed out by these authors, an E-B condensate of pairs would also be given as a product of identical pair functions. The process of antisymmetrizing would not change the character of the state very much if the size of the pair wave function were small compared with the average spacing between pairs. In this case, functions in the sum differing by the exchange of single members of two or more pairs (e.g., the interchange of rl and r, but not simultaneously r2 and r4)would not overlap very much, and the system would behave qualitatively like a condensed E-B gas. However, antisymmetrization makes a major difference, if, as in actual superconductors, the size of the pair function is large compared with the spacing between pairs. As mentioned earlier, essentially the entire excitation spectrum of a superconductor is that of quasi-particles obeying Fermi-Dirac statistics. There is a small number of collective excitations corresponding to the motion of a pair of quasi-particles in a bound state; however, they posses an excitation energy only slightly less than the energy gap and are very different from the motion of individual pairs as would be the case for an E-B system. The pair wave function p discussed above is a function of the distance, 1 rl - r z ] only, and is thus an s-function. Other possible pair functions have been considered, and may in some cases lead to states of lower energy than given by the s-function. For example, one might have a p-type space function with a symmetric spin part; this corresponds to parallel spin pairing such as considered by Fisher. A d-type function has been suggested for a possible superfluid state in 3He. These will be discussed in 3 13. 9. Results for Simplified Model 9 . I . INTEGRAL EQUATION
As discussed in the preceding section, nearly all of the energy difference between the normal and superconducting ground states References 9. 282
202
J . BARDEEN AND J. R. SCHRIEFFER
[CH. Vf,
59
comes from terms in the Hamiltonian which scatter pairs of zero net momentum. A reduced Hamiltonian in which only these interaction terms are kept is useful for a discussion of the quasi-particle excitations of a superconductor. In this section, we shall give the results for a simplified model used by BCS in which it is assumed that the matrix element V,,, = VkjEfor scattering a pair from k f , - k J to k ' ? , - k' J. is negative (attractive) and constant in an energy shell extending from E , - &wc to E , goc. Here ho, is an average phonon energy which represents the range over which the phonon-induced interaction is large and attractive. This model is not as restrictive as it might appear at first sight, because in the usual weak coupling limit, in which the energy gap is small compared to ho,,nearly all of the results depend only on the magnitude of the energy gap at the Fermi surface. The very important question of just what determines the cutoff energy, nu,,will be discussed later. To calculate the interaction energy, we may decompose the state Ysof (8.7) a second time to give the occupancy of two pairs k and k'. To terms of O(l/n) in the number of particles, the coefficients of the second decomposition are the same a s the firsts4:
+
= vkvk'9?11
+
"k@k'?lO
+
ukvk'?Ol
f
(9.1)
ukuk'pOO'
The energy difference between superconducting and normal states is
x
W S - w W , = 2 Ik; E k r ) ; - 2k < k @ &,+
x,v x 'x
k,k
,lL'
lL'
'
(9.2)
+
where X, c u,v, and u3 v i = 1. The coefficients zt,, v, and thus the energy can be obtained by a variational method. It is convenient to express them in terms of the energy gap parameter A , and quasiparticle energy E , = d,P+di as follows:
The ground state energy is minimum when A , is a solution of the integral equ at'ion
For the special case that V,, = - I' for I E~ I < hwc, A , is a constant d for 1 E, 1 < no, and zero for I E , 1 > liw,, and (9.4) becomes. References p . 282
CH. VI, tj
91
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
203
Here the density of states has been repIaced by N(O),its value at the Fermi surface, clc = 0. In the weak coupling limit,
A
=
2Kw exp (-
l/N(O)V).
(9 * 6)
Practically, the weak coupling limit may be used without appreciable error for N ( 0 )V < 0.5. The energy reduces in this limit to :
w - w = - W ( O ) dm2,
(9 * 7 )
where il(0) is the energy gap parameter at the Fermi surface. Since the interaction, Vkv,is essentially independent of isotopic mass, the isotope effect follows65because the phonon energy, Kw,, which determines the cut-off varies as M-Y2. The solution is such as to make the best use of the available phase space to get a maximum number of pair interactions. The contribution of a pair state k to the energy of condensation is (for k > K,)
The first term represents the Bloch energy of both particles of the pair state k and the second the interaction energy from matrix elements leading to transitions into or out of the pair state k. The maximum contribution comes from states at the Fermi surface, where W , = - A . 9 .2 . COULOMBINTERACTIONS AND LIFE-TIMEEFFECTS
An open question is the role of matrix elements Vllck, of the screened Coulomb interaction, which extend to high energies, of the order of E,, above the Fermi sea. Bogoliubov4 has suggested that if one does not cut off the Coulomb interaction at no,,but allows the pairing to extend t o energies of the order of E,, one can get a superconducting state of lower energy. The result is similar to that one would obtain with a cut-off at Kw,, but with the Coulomb terms reduced by a factor of the order of log (EF/KwC), typically about 5. If this calculation were valid, there would be two serious difficulties: (1) The exponent tc of the isotope effect would be expected to depart significantly from 0.5, contrary to experiment. (2) The effect of the Coulomb interactions would be reduced so much that nearly all metals would be expected to be superconducting. Although one can make only rough estimates, References p . 282
204
J . BARDEEN A N D J . R. S C H R I E F F E R
[CH. VI,
99
a factor of five is difficult to reconcile with Pines’ and Morrel’s calculation~~~. One of the authors has suggested that the cut-off, fiw,, may be determined by the life-time of the quasi-particle excitations. When I E, I is large, the excitation may decay so rapidly that it is not well defined. Mathematically, this may be described as an imaginary part of the energy, which, for I E, I large, may become greater than the real part. An estimate given in 9 13, Fig. 33, shows that this occurs for decay from phonon scattering near the Debye energy, and thus may give the desired cut-off. However, there appears to be an energy region beyond this, extending to perhaps 10 6wc, where the excitations are again well defined. At still higher energies, the life-time for decay from scattering by exciting a particle from the sea becomes short. There is thus an uncertainty as to just what determines the cut-off. A reasonable value for the cut-off is obtained if one requires a life-time long enough for a particle to go a coherence distance, but as yet there is no good mathematical justification for this. The essential difficulty is that the superconducting transition energy is only a tiny fraction of the Coulomb correlation energy, and also of the electron-phonon selfenergy. Life-time effects are likely to play an important role for lead and mercury, for which the cut-off is probably not much larger than the energy gap. In these elements, the electron-phonon interaction is particularly strong. 9 . 3 . EXCITATION SPECTRUM
The energy of a “single” quasi-particle excitation may be determined in the following way. If one of the pair, say k T , is occupied and its partner, - kj,, unoccupied in all configurations, the state k is not available for transitions of pairs of equal and opposite momentum because of the Pauli exclusion principle. This subtracts an energy W , from the ground state, giving an increase in energy of E , - E,. The energy of the particle in k t is E ~ so , that the net increase is E,. This is, of course, the reason for the notation we have used; E , is just the quasi-particle energy in the superconducting phase. The minimum value of E , is O(O), the energy gap parameter at the Fermi surface. A pair excitation in k is described by the function (S. €9, which is orthogonal to the ground state. This is the anti-bonding combination which adds an energy E, + E , in place of W,. The energy relative to Kcfwmrcs
p. 282
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CH. VI,
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RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
the ground state is therefore 2E,, just that of two single excitations. In calculating the energy with the reduced Hamiltonian, one need not distinguish between pairs of “single” excitations and true “pair” excitations. Just as in a normal metal, one can describe an excited configuration by giving an occupation number, q ( k , u), equal to unity if there is an excitation and zero otherwise. Occupancy of both k f and - k 4 implies an excited pair wave function. The total excitation energy is We,,
-
W , = 2 q ( k , u)E,. k0
(9.9)
One can see from the above considerations why there is an energy gap in a superconductor but not in normal metals. The breaking of a pair, say by transferring an electron from k t to another state (k d k ) t , infinitesimally close to k, gives single excitations in - k 4 and ( k A k ) 1 . This eliminates two pair states, k and k dk , from virtual transitions with a corresponding increase in energy of 2 I W , 1. In a normal metal, interaction energy arises from the possibility of making virtual transitions to states above the Fermi sea and unoccupied states below. A transfer of a particle from k t to (k d k) f means that ( k + d k) ? is no longer available for such transitions, but k t now becomes available. Since there is no preference for ( k f , - k .f ) transitions in the normal state, the difference in energy becomes vanishingly small as d k approaches zero. I n the above discussion we have considered only the coherent contributions to the energy from scattering of pairs of opposite spin and momentum. The effect of other terms in the interaction Hamiltonian have been estimated and have little effect except in bringing about the collective excitations. As the temperature is raised above 1’ = 0” K, the number of excitations increases and the pairing energy and energy gap decrease. Since the quasi-particle states in k-space may be occupied independently, the entropy is given by the usual expression for particles obeying Fermi-Dirac statistics,
+
+
+
+
TS=
---/-lX {f(k,o)Inf(k,a) -[1
P, 0
-f(k,u)]ln[1 --f(k,o)]}, (9.10)
where ‘I/ = k,?’ and f ( k , a) is the average occupancy of states in the neighborhood of ( k , a). The energy gap parameter is now determined in such a way that the free energy liefcrcmc\
p.
‘?A’?
206
[CH. V I , 5 9
J . BARDEEN A X D J . R . SCHRIEFFER
F
=
W
-
TS
=
is a minimum. By minimizing F with respect to h,, a quasi-particle representation is determined which best represents states typically excited a t temperature T. This leads to an integral equation of the form (9.12)
The maximum value of T for which there exists a non-vanishing solution for A is the critical temperature, T,. As pointed out by Cooper66, the form of the integral equation is such that if there is an energy gap over part of the Fermi surface, there will be one everywhere, except perhaps a t isolated points or lines. To see this, suppose the contrary is true, and that A, is zero everywhere except in a region R of k-space. Then for a point k not in R, it would be required that A,. Ek’ d k = - v k k 3 -tanh -= 0. (9.13) t’ 2E,, 2k,T But if V,,, does not vanish for all k’, there is no reason why the sum over k’ should vanish, except perhaps accidentally a t isolated points. For a general interaction, A , can take on positive as well as negative values. The energy gap, however, is 2 I A , I. I n Fig. 6 , the energy gap A
.
\ - - - - -1 - .0 tI
I
o.2
\: \ -1
i
big 6 ’fhr \ariation of t h e energy gap parameter 4 ( T ) vitli tcniperature as piedictid
by theory. h’ejereweA
p . 28%
CH. VI,
5 101
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
207
is plotted as a function of temperature for the simplified model with V , , = - V for I .sic 1
(9.14)
In most applications of the theory, this relation is used to determine empirically the average interaction constant. The energy gap at T = 0 may be expressed in terms of T,, M ( O ) = 3.52 k,T,.
(9.15)
Detailed calculations of the free energy and specific heat as a function of temperature have been made for the simplified model of a constant interaction. These will be discussed and compared with experiment in the following section. The theory leads to a second order transition with a jump in specific heat but no latent heat at T,. The reason that there is not a I-point, with a logarithmic singularity, (as observed in He I1 and many orderdisorder transitions) is that there is nothing corresponding to shortrange order. The large coherence distance precludes formation of small superconducting nuclei of atomic dimensions as T , is approached from above. WentzelB7has argued that differences in self-energy terms in normal and superconducting states, neglected in the above treatment, may lead to a very small latent heat.
10. Thermodynamic Properties Early measurements of the electronic specific heat in the superconducting state, cBB, gave c,, a T3,in agreement with the GorterCasimir two-fluid Deviations from the T3 law were first observed by Brown, Zemansky and B o ~ r s ein~ ~ measurements on niobium. These authors found rough agreement with the Koppelc model, which, as GoodmanBohas pointed out, can be interpreted as an energy gap model. Since the current theory has as one of its main features an energy gap in the single particle-like excitation spectrum, the theory predicts an exponentially decreasing electronic specific heat as T 4 0. Explicit calculation based on the expression (9.11) for the free energy gives (10.1) where j3
1/k,T,, andf, is the Fermi function for the quasi-particles.
References p . 282
208
J . BARDEES AND J . R . SCHRIEFFER
[CH. VI,
10
In the case that Vkv = -V for 1 ek I and I ek. 1 < f L o c , and is zero otherwise, A , is a constant, A , given by
for 1 ck < Kw, and A , in the form
3
ces - - - (--)3
yT,
A
2n2 k,T, w
=
0 otherwise. For PA
> 1,c,,
can be expressed
(+)[3K,(PA)+ K,(BA)l 2
8.5 exp (-
\ 26 exp (-
1.44 T J T ) for 2.5 < T,/T < 6 1.62 T J T ) for 7 < T J T < 11,
( 10. 3)
while at extremely low temperatures the coefficient in the exponent tends to 1.76. The specific heat measurements of Corak et al. on vanadium and Corak and Satterthwaite 6 R on tin exhibit an exponential behavior for T J T > 1.3: ~e' 8 -
YT,
a exp [- bT,/T],
(10.4)
where a = 9.17 and b = 1.50, in good agreement with the predicted curve in this region. The measurements on vanadium and tin have been extended to T,/8 by Go0dman6~,and he finds the vanadium data agree with (10.4) down to the lowest temperature measured, while the tin data show an upward curvature for T J T > 4 on a log c,,/yT, vs. T J T plot. More recent experiments by Chou, White and Johnston70 on niobium can be fitted by (10.4) with the parameters used for vanadium, although their data also fall above this law for lower temperatures. Goodman 69, Zavaritskii'l and Phillips'2 have independently carried out measurements on the specific heat of aluminum. Zavaritskii's and Phillips' data closely follow an exponential law down t o T,/T = 6, with a value of b in (10.4) somewhat smaller than that given by the theory, while Goodman's data definitely show an upward curvature beginning at T J T = 4, as may be seen in Fig. 7. Another example of upward curvature is afforded by measurements on zinc by Phillips, although similar measurements by Zavaritskii fall accurately on an exponential curve. It has been suggested that this upward curvature results from 1) a low density of states located in the energy Krferenccs
p . 28?
CH.
VI, 9 101
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
209
gap, e.g., collective states, or 2) an anisotropic energy gap. Calculations indicate that the density of collective states is too low to account for the effect. The second possibility appears to be not unreasonable since Morse, Olsen and G a ~ e n d a 'have ~ found evidence for an anisotropic energy gap in tin, for which the specific heat curve shows upward curvature for T J T > 4.
Fig. 7. Comparison between the theoretical electronic specific heat with empirical data on several weak coupling superconductors.
Measurements of the critical magnetic field as a function of temperature, when combined with the thermodynamic relation (10.5) 1Ceferencc.s p . 282
210
J. BARDEEN AND J. R. SCHRIEFFER
[CH.VI,
3
10
provide an accurate method of determining c,, at low temperature. The data of Maxwell and Lutes 75 on thallium, indium and tin along with the recent curves for lead by Mapother and c o ~ o r k e r s ~ ~are **b plotted in Fig. 8 as deviations from Tuyn’s 1 - ( T / T J alaw, which agrees with the Gorter-Casimir model. Negative deviations for tin, vanadium, etc., reflect the exponential drop of the electronic specific
Fig. 8. The deviation of the critical magnetic field from the Tuyn’s law H,/H, I - (T/T,)zwhich agrees with the Gorter-Casimir two-fluid model.
=-
heat. The positive deviations for mercury and lead suggest that superconductors with smaller ratios of TJ6, must be considered separately; as we will see later, it is likely that these “bad actors” must be treated by an intermediate or strong coupling theory while the weak coupling approximation appears to hold rather well for the remainder of the superconducting materials. The electronic specific heat derived from the magnetic measurements are plotted in Fig. 9. Thus we conclude from the bulk of the thermal and magnetic data that the exponential variation of c,, for temperatures well below T , is a general property of superconductors and is a consequence of an energy gap in the spectrum of elementary excitations. The smaller References p . 282
CH.
VI, 9 101
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
211
rate of decrease below T J T = 4 may be due to either a low density of states with a smaller energy gap or perhaps more likely an anisotropic energy gap.
Fig. 9. The electronic specific heat in the superconducting state for elements requiring an intermediate to strong coupling theory compared with the prediction of the weak coupling theory.
The jump in specific heat corresponding to the second order phase transition at T,, as given by the current theory3, is (10.6)
while the Gorter-Casimir model gives 2.00 and the Koppe mode177 gives 1.71 for this ratio. Empirical values of the ratio c,,/yT, are given in Table 3. References p . 282
212
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
9 11
TABLE I11
Ratio of the electronic specific heat in the superconducting
and normal states at Tc Element .
Pb I-lg Nb
Sn A1
Ta V Zn TI Theory
__
~ t % ( ~ C ) / Y ~ C
-~
3.65 3.18 3.07 2.60 2.60 2.58 2.57 2.26 2.16 2.44
11. Transition Probabilities and Coherence Effects 1 1 . 1 . THEORY
For many applications of the theory, one needs to calculate matrix elements between excited states of the system of an interaction expressed by asingle particle operator of the form (11.1)
in which Bku,pu'is the matrix element for scattering from ko to k'o' and cto and cka are creation and destruction operators for quasiparticle. excitations in the normal state. In the Bloch approximation, individual particle wave functions, v k a ( x ) , may be defined for a selfconsistent field which is not changed very much by small excitations of the system. In this case, (11.2)
in which x may be defined to include the spin variable. More generally, the matrix element is defined for many-particle normal state functions which include correlation effects and which differ by transfer of a quasi-particle from ko to k'a'. In other words, ko is occupied, k ' d unoccupied in the initial state and k'a' occupied, kcr unoccupied in the final state. Occupation of other quasi-particle states is the same in the initial and in the final state. The matrix element Bka,k'o, may depend weakly on the configuration of the other particles, but such dependence usually is negligible for small excitations of the system. Matrix eleReferences p . 282
CH.
VI, 9 111
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
213
ments of H , between excited states of the superconducting phase may then be calculated in a straightforward manner from the corresponding matrix elements for the normal state. A striking difference between superconductors and normal metals arises from coherence effects associated with the paired wave funct i o n ~ I~n. the normal state scattering from ko to k'a' is entirely independent of scattering from -k', -0' to - k , -a, as well as of all other transitions. The probability of the former is proportional to 1 BkU, k'u, (2, the latter to 1 B.-F,-,,,,-k, 12. Because of the nature of the paired wave functions of a superconductor, these two contributions are coherent, and one must add the matrix elements before squaring. To see that this is true, consider the matrix element of a spin-independent interaction between two excited states of a superconductor. Suppose that in the initial state, k t is singly occupied and the pair k' is either in its ground state or has a pair excitation (see 3 9). In either case, the initial state is a linear combination of normal statelike configurations in each of which k ? is occupied and -k J. unoccupied. As illustrated in Fig. 10, in some configurations (a) the pair k' zi (k' 1' , - k ' $ ) is unoccupied, in others (b) the pair is occupied. We have supposed that in the final state, the pair k is either in its ground state or is excited, and that k ' t is singly occupied. In configurations (c) of Fig. 10 the pair k is unoccupied; in (d) it is occupied. --d
initial
k t
(0)
(bl
-kl
final stole
siate kit
-_o_,-P q --?-*-,m
k l
-kil 0
(c)
(d)
0
-kl 0
kit
-kit
0
o o
Fig. 10. Configurations which enter when a quasi-particle makes a transition from a singly occupied state in k f to one in k' f .
There will be nonvanishing matrix elements of H , between the two configurations (a) and (c), corresponding to scattering from k t to k' t , if the occupancy of all other unspecified states is the same. But there is also a nonvanishing matrix element between the same initial and final states corresponding to scattering of a particle in - k ' $ in configuration (b) to - k $ in (d), again with the same occupancy of all other unspecified states. Since the total number of particles in configurations (a) and (c) must be the same as in (b) and (d), there are two more particles in the unspecified states of the former configurations. References p . 282
214
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
5 11
When the total number of particles is large, this difference has a negligible effect on the weight with which different normal state configurations enter into the sum representing a superconducting state wave function. Depending on the nature of the interaction, the two contributions (a) -+ (c) and (b) -+ (d) may add constructively or destructively. In general, B has the same magnitude for k t k‘f as for -k’J to - k 4, because the wave vector differences are the same, but they may differ in sign. The two cases are: Bk’a’,ku
=
-k
B-k.
--o,
-k’, -d
(case I)
(case 11). B-k, -a, -F, --ol where Ooo. = + 1 for 0 = u1 and Oaul = - 1 for u = - IS’. The first applies to an ordinary potential interaction, such as is involved in calculating the absorption of longitudinal ultrasonic waves, the second to the electromagnetic interaction and to the hyperfine interaction involved in nuclear magnetic resonance relaxation times. The coherence factors may be calculated most simply by use of the quasi-particle operators introduced by Bogoliubov4 and by Valatin6. Both “single” and “pair” excitations of a superconductor may be defined through the operators : &,‘,ko
= - O,,
yf+
Y-fJ
Ilkc,*, ukc-z$
VkC-kS
-k
vkCkt>
(11.3)
(11.4)
,--
where uk = V1 - h, and v, = f i kas in 5 9. A single excitation in ( k , a) is defined by y&Yoand an excited pair in k by yf+y-&!P,,. These correspond to the “single” and “pair” excitations discussed in 5 9. The y operators obey the usual Fermi-Dirac commutation relations. In the normal state ( A = 0 ) , yz+ creates a particle in k.f if k is above the Fermi surface and a hole in - k 4 if k is below. The superconducting ground state may be defined as the vacuum for quasi-particle excitations : (11.5)
With these definitions, \yo must be regarded as an admixture of states with different total numbers of particles, peaked about an average number, n. This simplifies the mathematical formalism, and creates no difficulties for systems with large numbers of particles. This may be seen as follows, Matrix elements of a particle (not quasiKefereitccs p. 282
9 111
CH. VI,
215
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
particle) conserving operator, H,, between states !Pamay be calculated by decomposing !Painto components, !Pan.,each with a fixed number of particles :
!Pa = ;I;An,!Pa,,,
(11.6)
n’
where (Yaw,, !Pan,)= 1. Since H , is particle conserving, we have (YP, H I !Pa) =
X A*,Aw,,(!Pp,,Hl!Pant.) =
n’, n”
(11.7)
As in a grand canonical ensemble, the weights [ A , l2 are sharply peaked about the average n. Since the n-particle matrix element is slowly varying with n, the total matrix element of H , between the !Pas is, to order l/n, equal to the corresponding matrix element between states with n particles. In terms of the y operators, we have for the sum of the two coherent contributions to the matrix element (omitting the common factor Bk‘o’$ ka)
C$&k0
& 6ua!c?k-&-k,-a,
= (f’d,,u, ;f
vg*vk) (y$,yYh
+ (“i% .kVx’)
($dYrk-o
8ua,
*
Y-k-oY--k‘--o’)
(11.8)
f
y--k’--o’yko)*
The first term on the right corresponds to the scattering of quasiparticles and the second to the creation or destruction of two quasiparticles. For example, the matrix element for scattering a quasiparticle in the superconducting state from ko to k ‘ d is &oi‘,ko(t+~l, ? vgnk).The upper signs correspond to case I and the lower to case 11. In a normal metal, creation of a pair of excitations corresponds to exciting an electron from below to above the Fermi surface, with formation of a hole below and an excited particle above. The transition probability from a state of energy E to one of energy E no induced by interaction with a field of angular frequency o is proportional to the square of the matrix element and to the density of final states, N ( E no).To get the net rate of absorption of energy one must take the difference between direct absorption and induced emission and sum over initial states. The ratio CC,/OI, of absorption in the superconducting to that in the normal phase can be simply ex-
+
+
Referdnces
+. 282
216
J . BARDEEN A N D J. R . SCHRIEFFER
[CH. VI,
3
11
pressed if it is assumed that the normal matrix elements, Bk'o,,ko,are independent of the energy difference (although they may depend on the angle) between initial and final states. This should be an excellent approximation because the energy differences involved are generally very small compared with the Fermi energy. The result can be expressed most simply if we abandon our usual convention and define E to have the same sign as E , positive above and negative below the Fermi surface. We then have3
where f ( E ) is the usual Fermi distribution function and N , ( E ) , the density of states in energy in the superconductor, is
We may assume that N, is independent of energy and equal to N ( 0 ) . Note that N , becomes infinite at the Fermi surface, E = 0 or E = f d. The ratio of absorption in superconducting to normal state is then
[E(E L-J tc u,
nw
+ nw) '"1
[ f ( E )-.f(E
[(E' - d'){(E +
+ nw)l dE.
(11.11) Kw)' - ~ l ~ } ] ~ ' ~
The upper sign corresponds to case I, which gives destructive interference for 6w < 2d, the lower to case 11. The difference is particularly marked for very low frequencies, &w A . For case I (ultrasonic attenuation) cc,/cc, drops below T , with an infinite slope. In the limit H!W + 0,
<
(11.12) This function, calculated with use of the values for d ( T ) as given in Table 6 is plotted in Fig. 11. For case XI, there is a rapid rise in absorption at low frequencies as the temperature drops below T,, as illustrated in Fig. 12. There is actually a logarithmic divergence of the integral in the limit fiw + 0 , which arises from the singularity in the density of states, N,. As the frequency is increased, the maximum in the absorption decreases, until at a frequency greater than that Referrrtrrs
p . 282
CH. VI, fj 111
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
217
0.81 o
Tin 33.5 MHz
Tin 54.0 MHz 0
A
0.6-
a' a"
\
'v
Indium 28.5 MHz Indium "B"35.2 MHZ
-
0.4 -
0.2-
O.
012
0.4
0.6
0.8
r/rc Fig. 11. The longitudinal acoustic attenuation coeficient in the superconducting state relative t o that in the normal state compared with the theoretical curve for case I for the limit &I) < kBTc.
f=7/7, Fig. 12. Ratio of absorption in superconducting and normal phascs for caw 11. Observed values are reciprocals of ratios of nuclear spin relaxation times in aluminum; probable errors not shown. Curve for fiw = 0.01 kBTc calculated by L. C. Hebells for level broadcning, others bv P 13 Miller for energy transfer no References Q. 282
3 11 the drop starts at T,.
218
J. BARDEEN A N D J . H. SCHRIEFFER
[CH. VI,
corresponding to an energy of about 0.6 k,T,, A t low frequencies absorption occurs only by particles which are already thermally excited. At higher frequencies, there is an additional absorption from excitation of particles across the gap, which can occur when fiw > 2d(T). The gap decreases with increasing temperature; a knee in the absorption curve occurs at the temperature for which 2d(T) w f i w .
A t very low temperatures, where few particles are thermally excited, the absorption is very small until the frequency exceeds the gap frequency, w g = 2d(O)/jL,corresponding to T = 0" K . Fig. 13 indicates
0.8
0.21
L
Oo
6 --L-
- - - -- I2
18
24
30
36
42
Bw/kT,
Fig. 13. Absorption beyond gap for case I1 a t JI' = 0" I< cxpressed as ratio of conductivity in superconducting and normal phases. Experimental points from early measurements of Glover and Tinkliam lo based on transmission through thin films.
how the predicted absorption increases rapidly to that of the normal metal as w is increased beyond og,as observed in transmission of electromagnetic radiation through thin films lo. We shall next review some of the experimental data bearing on the coherence effects. 11.2. ACOUSTICATTENUATION
A major source of the attenuation of ultrasonic waves in metals at very low temperatures is the interaction with the conduction electrons. We shall first discuss longitudinal waves. From the earliest measurements of the attenuation in superconductors by Bommel and M a ~ k i n n i n it ~ ~was , clear that the rapid drop in attenuation as T h'tfeferencea
p . 282
CH. VI,
5
111
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
219
drops below T , reflects a diminishing number of “normal” electrons. However, the observed drop was so abrupt that it was difficult to reconcile with other estimates of the decrease in the normal component of a two-fluid model; for example, the Gorter-Casimir theory24 predicts p,/@ cc (T/T,)4. In the present theory, the destructive interference for an interaction which follows case I, nullifies the effect of the large density of states near the gap edge, leaving just the Fermi factor (11.12). The rapid drop reflects the sharp increase of the gap below T,. The simple theory leading to (11.12) applies when ql 1, where q is the wave vector of the longitudinal wave and I is the mean free path for impurity scattering. One can then regard the interaction with the electrons as corresponding to emission and absorption of phonons. The opposite limiting case, qi 1, has been considered by Kre~in’~; TsunetoBOhas made a general calculation valid for all 1. The temperature dependence of the attenuation is not very much different than for I -+ 00. Some careful measurements to test the theory have been made by Morse and Bohm12. They have found that (11.12) agrees fairly well with their measurements taken on polycrystalline indium and a very pure single crystal of tin, as indicated in Fig. 11. The temperature dependent energy gap of tin determined by combining (11.12) with the empirical results taken at 33.5 and 54 MHz is shown in Fig. 14. The direction of propagation is along the (001) axis. The best empirical value for the energy gap at T = 0” K is 3.54 k,T,, surprisingly close to the value 3.52 k,T, predicted for all metals by the simplified theory (i.e., constant matrix elements with a cut-off at I E I = no,).More recent mea~urernents~~.of attenuation of waves propagated in different directions have shown appreciable crystalline anisotropy in d(O), so that the agreement is partly fortuitous. Thus, a treatment including anisotropy of the normal state parameters must be carried out before a detailed comparison with experiment can be made. The primary deviation between theory and experiment occurs just below T,, where the experimental results indicate the gap opens more rapidly with decreasing temperature than predicted by theory. The relative attenuation coefficient obtained by Morse and Olsen74 from a very pure tin sample, under conditions ql 1 is shown in Fig. 15 for three crystallographic orientations. For T,/T > 1.5, the curves are well approximated by straight lines from which the values of 2A(O) given in Table 4 were obtained. Because of the requirements of
>
<
>
References p . 282
e 4-
0
r 1
-
1
Tin 335 MHz Tin 54 MHz
BCS theory
051 i
t
I
Ob
02
I
-
I-.---
04
06
-L 08
10
T/ T,
Fig. 14. Comparison of the temperature variation of d, as determined by Morse and Bohmr2 from the attenuation of longitudinal acoustic waves in tin with the prediction of the current theory.
[
I
I
2
3
TJT Fig. 15. The crystalline anisotropy of the relative longitudinal acoustic attenuation coefficient obtained by Norse and Olsen from measurements on a very pure tin sample
ql 9 1.
CH. \'I,
9 113
R E C E N T DEVELOPMENTS I N SUPERCONDUCTIVITY
221
TABLE IV
Crystalline anisotropy of d for tin deduced by Morse et af. from attenuation of longitudinal sound waves
2 4 (0) kBT, ~~
parallel to [001] parallel to [I101 perpendicular to [001] and 18" from [loo]
3.2 f 0.1 4.3 0.2 3.5 0.1
*+
energy and momentum conservation in the absorption process, the projection of the quasi-particle group velocity on the direction of sound propagation must equal the speed of sound, s. The majority of the quasi-particles have velocities of the order of vF. Since vF s, only those particles with wave vectors lying in a disc perpendicular to the direction of the wave contribute to the absorption. Experiments on oriented single crystals inherently measure the energy gap averaged over such a disc. Some experimental datas2 on the attenuation of shear waves in
>
-
-
1
g
4t
-
, 1
Discontinuity
0
---
05
,
1.0
-'
I
1
I
I
I
1.5
2.0
2.5
30
3.5
I I
4.0
7 (OK) Fig. 16. The relative acoustlc attenuation coefficient for transverse waves in tin as mrasured by Bohm and Morse. Referencds p . 282
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J . BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
11
polycrystalline tin with ql> 1, plotted in Fig. 16, show an even more abrupt drop in attenuation at T , followed by a more gradual decrease which appears to follow again the law (11.12). The very sharp drop, which is almost a discontinuity, is most likely due to the strong screening of the transverse fields by the supercurrents, since the Meissner effect garantees that the magnetic field generated by the transverse currents associated with the shear wave will be screened in a distance of the order of the A m 5 x cm. Morse suggests that the more gradual drop of the attenuation below T , is due to the shear strain leading to a change in energy of the electron and therefore to an attenuation. The effect may in part be due to relaxation effects of the type suggested by Kittelss for the normal state. As Morse has pointed out, the shear waves can in principle give more detailed information about the anisotropy of the energy gap since a given transverse polarization will favor certain groups of quasi-particles in the disc perpendicular to q. It is important to develop a better understanding of the attenuation of shear waves in the superconducting state. 11.3. NUCLEAR SPINRELAXATION
An example of coherence effects following the constructive interference of case I1 is given by the relaxation of nuclear spins by the quasi-particles. Simultaneous with the development of the current theory, Hebel and Slichter13, using an ingenious method, were able to measure the zero-field nuclear spin relaxation rate in superconducting aluminum from 0.94” K to 4.2” K. The more recent data of Redfield and Anderson14,are presented in Fig. 17. The relaxation rate exhibits an increase by a factor of two just below T, and a subsequent decrease at lower temperatures. Since the dominant relaxation mechanism is provided by exchange of energy with the conduction electrons, this increased relaxation rate would be impossible to explain on the basis of the conventional two-fluid model because the density of “normal” electrons drops sharply below T,. As Hebel and Slichter have shown, the current theory is in good agreement with their results. The actual energy transfer in the relaxation process, and thus the corresponding no,is extremely small. To get agreement with experiment, Hebel and Slichter assumed that the quasi-particle levels are not perfectly sharp, but are broadened by m 0.01 kBT,, or eV. This avoids the singularity which would otherwise occur in the evaluation of the integral (1 1.ll).The source of this level broadening is uncertain. References p . 282
CH. VI,
§ 111
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
223
Anderson and Redfield l4 have recently extended the measurements on aluminum down to T,/T = 6, and their results, shown in Fig. 17, are in good agreement with the predictions of Hebel and Slichter. The two curves in this figure marked “corrected” represent different choices of level widths, where the energy gap was chosen to be the value obtained experimentally by Biondi and Garfunkelll, 24 = 3.25 k,T,, a value 7.5% smaller than that predicted by the theory, 24 = 3.52 k,T,. Hammonds4 has recently observed an increase of relaxation rate below T , by a factor of about 1.7 in Ga. I t appears that the increased 5.0
T
T I Normal state
t
0
Anderson and Redfield Recent data
i
Fig. 17. The nuclear relaxation time T I of superconducting aluminum as measured by Anderson and Redfield (open circles) and by Redfield (solid dots). The theoretical curves are based on the current theory with the density of states near the gap edge smeared by folding the density of states function (11.7) with a square function of width 2d and height (2d)-I, where A / d = r. The dotted and solid curves were calculated with 2d(O)/kBTc = 3.52 and 3.25 respectively, the latter being the value found by Biondi and Garfunkel from microwave measurements.
nuclear relaxation rate below T , is a general feature of superconductors, although the magnitude of the increase depends upon details of the material. It is important to realize that the observed increase in the nuclear spin relaxation rate and the sharp drop in the acoustic attenuation coefficient as the temperature is lowered from T,, imposes contradictory requirements on the conventional two-fluid model. It is one of References p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
$ 12
the major successes of the recent theory that the temperature variations of these independent effects follow in a completely natural way from the general formulation of the ground state and the excited states of the system, and give strong experimental verification of the pairing concept. Although the experimentally observed coherence effects in themselves do not contain enough information to infer that only k t , -k J. interactions are important in contrasting the superconducting and normal states, this pairing is consistent with the empirical facts. Other possible pairings will be discussed in 3 13 on collective excitations. It is possible that systems with strong interparticle interactions in odd angular momentum states or strong spin dependent forces tending to line up the quasi-particle spins would be better described by the parallel spin pairing, So far, no evidence has been found for this case, and we will assume the antiparallel spin pairing to be that which leads to the ground state for the systems under discussion.
12. Electromagnetic Properties 12.1. THEORY
A theory of the electromagnetic properties of superconductors requires an expression for the current density for fields which vary arbitrarily in space and in time. The total field acting on the system, that is the sum of the applied field and that due to currents in the metal, is determined with the aid of Maxwell’s equations in a self-consistent manner. In their basic paper, Cooper, and the authorsab obtained an expression for the current due to weak quasi-static fields by treating the electromagnetic interaction in perturbation theory and including only the particle-like excitations of the system. The theory was later extended to treat fields of arbitrary frequency by Mattis and one of the authorss5 and independently by Abrikosov, Gorkov and Khalatnikovld’ a6. In this section we shall give the results without derivation and compare theory and experiment for several phenomena. As we shall see, in general there is excellent agreement between theory and experiment over a wide range of temperatures and frequencies. Before describing the results, we shall make some general remarks on the methods used and also indicate how the Meissner cffect with a non-local relation between current and vector potential is related to the energy gap model. The derivation of the Meissner effect of Cooper and the authors3b Kefermccs p. 282
CH.VI, 9 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
225
has been criticized because it is not strictly gauge invariant. There are two reasons for this lack of invariance: (1) The effective interaction with a cut-off for 1 E 1 >&m, is a nonlocal momentum dependent interaction, so that the expression for the current contributed by the quasi-particles should be modified from the usual one. However, estimated errors introduced if the usual expression is used are only of order ( A / & o , ) ~in the weak coupling limit, and so are negligible. To see this, one can start from a strictly gauge invariant theory in which the electron-phonon interaction has not been replaced by an effective interaction between electrons. Such a calculation of the Meissner effect has been given by Rickayzens7, who finds that the correction terms are indeed small if the energies and velocities of the quasi-particles are suitably renormalized to include self-energy corrections. Physically, non-local effects are unimportant because the size of the pair wave functions, of which t ow cm is a good measure, is large compared with the range of the non-local interaction, which is typically of order 10V cm. (2) The second reason is more serious and limits the applicability of the original treatment to transverse electromagnetic waves described in a transverse gauge. In the perturbation expansion, only quasiparticle excitations have been included, and the usual expression for the current contributed by a quasi-particle has been used. In the plane wave approximation, the contribution to the transverse current of a quasi-particle described by a “single” excitation in k is v = ?ik/m*, where m* is the effective mass of the electrons in the normal state. It should be noted that v is not the same as the group velocity of the excitation, 7)
1 aE ?ak i
1 aE ae ?i a& ak
=--=---=V-=V-
aE
a&
E
E ‘
(12.1)
Note that vg vanishes for excitations at the Fermi surface, E = 0. As for rotons in He 11, one may picture the quasi-particle as a vortex ring. The ring as a whole, with the accompanying backflow, moves with a velocity vg, but the expectation value of the velocity of the flow through the center of the ring is v. As discussed in 5 5, the backflow cancels out for transverse waves, so that one gets the correct current by summing v rather than v, over the various excitations. To carry out the calculation in a manifestly gauge invariant way one requires a formalism which is general enough to include backflow and Refertiices
p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
$ 12
collective excitations t. This problem which has been discussed by several authorsg0, most completely by Rickayzen t t, will be treated in 5 13, in which evidence for direct absorption by the collective modes will be reviewed. The theory is developed by considering a metal of infinite extent and calculating the response to a transverse electromagnetic field of arbitrary wave vector q and frequency w , described by a vector potential A = A, exp i(q - r
+ wt),
(12.2)
with div A = 0. The field may be due in part to internal sources and in part to currents induced in the metal by the field. A formulation of this type, first used by Klein to discuss the diamagnetic properties of metals and extended by Lindhardgl to determine the complex dielectric constant of normal metals for transverse and longitudinal fields, has been employed in most of the recent discussions of the electromagnetic properties of superconductors. Let Q0 be the many-particle wave function for the ground state of energy W,, or, at a finite temperature, a quantum state with a quasiparticle distribution appropriate to the temperature, T , and let @, with energy W,, ( j = 1, 2, 3 . . .) represent the spectrum of excited states in the absence of the field. Treating the electromagnetic interaction as a perturbation, one may expand
+ ,Xal(t) exp (-iW,t/fi)Ql,
P ! = exp (- iWot/?i)@,
(12.3)
j=O
where to first order in the field a&) =
( j I HI10) exp i(w - is)t W, - q w - is)
w,
(12.4)
Here s is a small positive constant which indicates that the field was The general structure of the equations for the current density for applied fields of arbitrary wave length and frequency has been discussed by S. Nakajima (see ref.88) and by 0. V. Konstantinov and V. I. Perel’ (see ref.80). They show how the conductivity is related to the current-current correlation function, with use of Kubo’s formalism, and also discuss the sum rules. In an infinite medium, there is a &function singularity in the longitudinal conductivity, corresponding to infinite conductivity, only in the long wave length limit. This limit is discussed in 5 14 in connection with the two-fluid model. t t Rickayzen (ref.8) discussed both the Meissner effect and the complex dielectric constant for longitudinal fields. References p . 282
CH. VI,
5
121
227
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
turned on in the remote past; the limit s -+ 0 is taken in the final result. An expression of the form (4.2) is used to calculate the current density from the perturbed wave function. As noted in 3 4,there are two contributions, a paramagnetic current, jp,associated with the gradient operator and a diamagnetic or “gauge” current, jD, proportional to the vector potential, A. The latter, jD,depends only on the electron density and is the same for the normal and superconducting states. The difference comes from the expression for jp,in particular from the terms involving a, for which W , - W , is of the order of the energy gap, E,, or less. In a superconductor, aside from collective modes, there are no terms for which W , - W , < E,, and the contribution is greatly reduced below that of the normal metal for W , - W , < M 2E,. For W , - W , > M 2E,, the difference between normal and superconducting states is small. Since the total current, j , = j,, jD, induced by a static magnetic field in a normal metal is extremely small (corresponding to the weak Landau diamagnetism), we have jD m - in,.The net superconducting current is then j , = j,, jD M M I,, - j,?, which is roughly the negative of the contribution of the series for I,,, in the normal state for W , - W , < M 2E,. Prior to the development of the microscopic theory, one of the authors4, used an argument of this sort to show that an energy gap model would most likely lead to a non-local theory of the Meissner effect similar to that suggested by Pippard, and this has been borne out by subsequent developments. The Pippard limit applies if the dominant terms in the expansion for jnphave energy denominators larger than the gap, the London theory if the denominators are less than the gap. The matrix elements for a wave vector q correspond to exciting a particle from state k below the q above. The energy difference, W , - W,, is Fermi surface to k of order fiqv,, where v 0 is the velocity a t the Fermi surface. The dominant q in penetration phenomena are of the order of the reciprocal of the penetration depth, or about 2 x 105 cm-1. With ZI, w 108 cmisec, this gives W , - W , M eV, which is an order of magnitude larger than the energy gap, indicating that the Pippard theory applies. The London equations would apply for q < lo4 cm-l. As indicated by Ferrell and coworkerss2, these arguments can be made more precise by use of Kramers-Kronig relations. For w # 0 , one may express the current in terms of a complex frequency and wave number dependent conductivity :
+
+
+
References p . 282
228
J . BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
$ 12
(12.5)
u(q, 0)= u,(q, a) - io,(q, o),
where u1 gives the energy loss and u2 the reactive component. These are related by a Kramers-Kronig relation: (12.6)
where P indicates the principle part. The relation between may be written in the form C
j(q, W)
=
-
-K(q,w ) A ( q , o), 4n
i
and A
(12.7)
where (12.8)
This expression may be applied to static magnetic fields by setting o = 0. To have a Meissner effect, K(q, 0 ) must remain finite ( >0 ) as q --f 0. In a London superconductor, for which j , is assumed to vanish, K(q) = l/j12L, where A: = (mc2/4nne2) is the square of the London penetration depth. The matrix elements which enter the sum for K(q) are the same as those which determine absorption of energy at a finite frequency, which can occur for W , - W , M nw. As pointed out by FerrellgZa, one can determine Kfq) from cr,(q, o)if the latter is known for all o. This is closely related to the information that can be obtained about u2(q,w) for small w from ul by use of the Kramers-Kronig relation. An outline of the arguments follows : At frequencies above w, the maximum frequency at which absorption can take place, the response will be that of a system of free electrons, for which u1 = 0 and ne2 u2=-.
(12.9)
mw
When combined with (12.61, this gives the sum rule
me2
~ ~ ( W) 4 ,dm = -, 2m
(12.10)
which must hold regardless of the detailed structure of the system. In a superconductor, part of the contribution to the integral comes References p . 282
CH. VI,
$ 121
R E C E N T DEVELOPhlENTS I N SUPERCONDUCTIVITY
229
from a d-function at w = 0. If S is the strength of the &function, one may write (12.10) in the form (12.11)
Tinkham and Ferrel192bhave applied this relation to an energy gap model by assuming that ols(q,w) = 0 for w < wg, and olS = oln for w > wg, where &wg M 2Eg. For a free electron model
=o
for w
> voq.
No absorption can take place if w > voq because the velocity of the wave is then greater than the velocity of the electron and one cannot conserve energy and momentum. If wg voq (Pippard limit), the sum rule gives
<
3n ne2 3nne2Gg s = wg -= ~
4 mv,q
(12.13)
4mvoq
The d-function contribution to o2 is, from (12.6) : a2 = 2S/nw.
(12.14)
At low frequencies, o1 = 0 and 2SA(q, 0)
j ( q , w) = - io2&(q, w ) = -
nC
< voq
(12.15)
Thus one might expect that for w g
(12.16) The corresponding Pippard relation, from the Fourier transform of (3.6) in this limit (Eoq 1) is
>
(12.17) Comparising these two expressions, we find an expression for toin terms of the energy gap : (12.18) References p . 282
230
J. BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
9 12
The microscopic theory gives an expression similar to (12.18), with Ko, evaluated explicitly as 7z24(0)/2,which is a little over twice the gap. The &function at w = 0 corresponds to acceleration of the whole group of electrons to give a net current flow. This can occur in a metal but not in an insulator or semiconductor with a gap. The sum rule for the latter is satisfied by absorption at finite frequencies. Andersonea has shown explicitly why a long range order is required for superconductivity. The general expression for the current density j(r, t ) resulting from a field defined by a vector potential in the transverse gauge, div A = 0, may be writtenB5in a form similar to that suggested by Pippard: j(r, t) = I; UJ
e2N(O)v,e*mt J- R ( R * Am(r’))I(w, R, T)e-R’l dr’ 2n%c R4
(12.19)
where R = r - r‘. The kernel I ( w , R,T ) is a rather complicated integral over energies which, except for limiting cases, must be evaluated by numerical methods. The derivation was based on the simplified model of 9 9, with constant matrix elements for the effective electronelectron interaction, but should apply more generally for isotropic Fermi surfaces if the energy gap parameter, d ( T ) ,does not vary much with energy over a range extending within a few k,T of the Fermi surface. This is essentially the weak coupling approximation. One may then regard d ( T ) as a parameter to be determined from experiment. Elastic scattering by impurities described by a mean free path, I, introduces the factor exp (- R/Z) in the integrand. That this factor, suggested by Pippard in his phenomenological theory, occurs in the superconducting case in the same way that it does for normal metals was shown in ref.s5. These authors used, as a basis for the manyparticle superconducting wave functions and the perturbation expansion, wave functions for the individual electrons appropriate to the impure metal with scattering centers present. If yn is one such function, another of the same energy is the complex conjugate, y,*. and in general these can be taken to be orthogonal to one another. The paired states for the ground state configurations of a superconductor are taken to be (y,+,yz+); that is if one of these is occupied in a given configuration, the other is also. It can be shown (9 16) that the pairing interaction energy of an impure metal is not much less than that for the pairing (kt - k 4 ) in a pure metal, even though the mean Referencis
p . 282
CH. VI,
3 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
231
free path is much less than the coherence distance. To evaluate the perturbation expansion for the current density, one needs averages of expressionslike (y,*(r)y,(r’)) over random distribution of impurities and over states of the same energy. The averages required for the superconducting case are the same as those required for the evaluation of the normal conductivity and in both cases lead to the factor exp(- R/Z). Derivations by the use of methods of quantum field theory by Edwardsg3and by Abrikosov and Gorkovg4lead to nearly equivalent results. I t is often convenient to use the Fourier transform of (12.19) which gives the relation between the Fourier components of j and A as in (12.7) with ei q Rue-RI1
(1 - d ) I ( w ,R, T ) du dR. (12.20)
Here we have inserted the expression for the London parameter a t T=O: A(O)-l = +e2N(0)vg. (12.21) Several limiting cases of (12.19) are of interest : (1) If the energy gap goes to zero, or more generally, if the frequency is sufficiently high so that Am A , we have
>
I(w,R, 7’)-+ - ninw exp(- iRo/vo),
(12.22)
and the expression for j(r, t) reduces to that of Chambers for normal metals as given by (3.5). The coefficient in front of the integral, N(0)vo,may be evaluated empirically from the surface impedance of the pure metal in the normal state in the extreme anomalous limit (skin depth much less than the mean free path). (2) The limit w --f 0, or Aw A , corresponds to the quasi-static case evaluated by B.C.S. An expression very similar to Pippard’s equation (3.6) is obtained by introducing a function J(R, T ) through
<
(12.23)
One then has for o = 0: j(r) = References
p . 282
-
3 4ncA ( T )t o
1
R ( R A(r’))J(R, T)e-RI1 dr’. R4 0
(12.24)
232
J. BARDEEN A N D
5. R.
SCHRIEFFER
[CH. VI,
9
12
The kernel exp( - R/to)is replaced by the function J(R, T ) , defined so that it has the same integral as the exponential for all T up to T,:
1,
J(R, T ) dR = to ?‘m,/nA(O).
(12.25)
Here to is a temperature independent parameter corresponding to Pippard’s coherence distance, chosen so that J(0, 0) = 1. It turns out that J(R, T ) does not vary much with temperature and does not differ widely from the exponential in form. The limiting value at R = 0 varies from unity at T = 0 to 1.33 at T = T,. With increasing temperatures, A ( T ) increases, corresponding to a decrease in the superfluid component, of a two-fluid model. The ratio may be expressed in terms of the energy gap:
where 9,
=
l/k,T. For the B.C.S. model (12.26a)
A plot of A(O)/A(T)based on calculations of Muhlschlegel (see Table 6 ) is given in Fig. 18. It should be noted that according to (12.26),
“‘R
r0 0.8 7
Fig. 18. The inverse reduced London parameter (A(T)/A(0))-1as a function of [T/d(T)]/[T,/d(O)]. d(0) = 1.76 kBT,.
A(O)/A(T)is a function of BA(T). If the temperature dependence of A for a particular metal differs from that of the model of 9 9, it would be best to estimate A(O)/A(T)from the empirical value of B d ( T ) . References p . 28%
CH. VI,
5 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
233
(3) If the variation in field occurs slowly over a coherence distance, or in impure metals over a free path, one may replace A(r') by A ( r ) and take it out from under the integral sign. In the quasi-static limit, this gives just the London equation, j(r) = - (l/cA)A(r). (4) If the field is confined to a region small compared with the coherence distance, E0, I ( R , T , o) may be replaced by its value at R = 0 and taken out from under the integral sign. This limit applies when the penetration depth 1 t oand also for thin films or other small systems. The remaining integral is the same as that required for the normal state in the same field. The ratio may be expressed as a ratio of the complex conductivity in the superconducting to that in the norma1 state
<
(12.27)
The expression for oJa, is identical with that for u,/u, as given in eq. (11.11). The corresponding expression for a,/o, is
The lower limit is 4 - no for &co < 2 4 and -A for fiw > 2 4 . The ratios for the limit T = 0" K can be expressed in terms of the complete elliptic integrals E ( k ) and K ( k ) . In particular, o1 = 0 for 6co < 24(0), and for no > 2 4 one has 24
01 _
44 E ( k ) --K(K); no
On
( T = 0°K).
(12.29)
The corresponding expression for u,/u, valid for all frequencies is
In these expressions,
k==iii
and K' = (1 - P)'.
(12.31)
Values calculated from these expressions are plotted in Fig. 19. Limiting values (ref. Id) are References
p . 282
234
J. BARDEEN AND J. R. SCHRIEFFER
[CH. V I ,
5
12
TABLE V
The complex conductivity as a function of frequency and temperature
A@
fio
k,T
k,T
‘Jl -
‘Jn
‘Ja ‘Jn ~
A
0.05674 0.08504 0.113A 0.199A 0.2834 0.4254 0.5674 0.850A 1.134 $3 1.994 2.834 Limit a s A + m 3.33 ,I
55.4 36.9 27.6 15.8 11.0 7.30 5.43 3.52 2.53 1.03 0.419
4.99
2.50
0.326 0.257 0.186 0.143 0.0936 0.673 0.0887 0.478
49.7 33.2 24.9 14.3 10.08 6.70 5.01 3.27 2.35 0.883 0.384
0.130 0.198 0.261 0.457 0.652 0.978 1.30 1.96 2.61 4.57 6.52
1.54 1.38 1.24 0.992 0.835 0.665 0.548 0.398 0.304 0.449 0.673
29.7 20.3 15.4 9.00 6.44 4.40 3.35 2.22 1.59 0.499 0.237
1.13
0.105 0.158 0.210 0.368 0.628 0.788 1.05 1.58 2.10 3.68 5.20
1.60 1.47 1.38 1.20 1.08 0.924 0.812 0.715 0.762 0.868 0.921
7.38 5.11 3.97 2.46 1.83 1.31 1.01 0.568 0.352 0.117 0.057
0.101 0.151 0.202 0.353 0.504 0.756 1.01 1.51 2.02 3.53 5.04
1.25 1.19 1.14 1.04 0.972 0.940 0.944 0.950 0.958 0.979 0.987
0.198 0.297 0.398 0.692 0.990 1.48 1.98 2.97 3.90 6.92 9.90
- ~ _ _~. _ 1.74
References p . 252
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.424 0.596 0.505
0.444
~~
,I
0.285 0.428 0.570 0.998 1.43 2.14 2.85 4.28 5.70 9.98 14.3
0.149 0.122 0.104 0.0714 0.0530 0.0359
0.434
54.5 36.3 27.2 15.5 10.9 7.20 5.36 3.48 2.50 0.999 0.414
0.160 0.239 0.319 0.558 0.798 1.20 1.60 2.39 3.19 5.58 7.98
1.064 0.923 0.822 0.632 0.513 0.389 0.308 0.211 0.155 0.222 0.550
42.6 28.4 21.4 12.5 8.74 5.89 4.43 2.91 2.09 0.733 0.326
0.113 0.170 0.227 0.397 0.567 0.850 1.13 1.70 2.27 3.97 5.67
1.73 1.56 1.44 1.24 1.04 0.866 0.737 0.565 0.449 0.97 0.819
17.6 11.4 8.69 5.21 3.78 2.73 2.05 1.36 0.884 0.273 0.125
0.400 0.153 0.203 0.356 0.509 0.784 1.02 1.53 2.03 3.56 5.09
0.102 0.127 1.21 1.09 1.00 0.893 0.886 0.910 0.917 0.954 0.975
-
,, 0.400
,,
n.02~
0.0160 0.0112
0.00604
2.74 1.96 1.58 0.970 0.734 0.504 0.331 0.190 0.119 0.0406 0.019
CH. VI,
3 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY TABLE VI
The thermodynamic functions G o = CedTr!) = ?JTc
t
-
T
'4T)
TC
A (0)
1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 From B.
1.0000 0.9519 0.9048 0.8587 0.8136 0.7694 0.7263 0.6842 0.6432 0.6032 0.5643 0.5266 0.4900 0.4546 0.4203 0.3873 0.3554 0.3249 0.2956 0.2676 0.2410 0.2157 0.1918 0.1693 0.1482 0.1285 0.1103 0.0937 0.0784 0.0646 0.0524 0.0416 0.0322 0.0243 0.0177 0.0124 0.0082 0.0051 0.0030 0.0016 0.0007 0.0003 0.0001 0.0000 Muhlschlegel, Z. Phys. 0.0000 0.2436 0.3416 0.4148 0.4749 0.5263 0.5715 0.6117 0.6480 0.6810 0.7110 0.7386 0.7640 0.7874 0.8089 0.8288 0.8471 0.8640 0.8796 0.8939 0.9070 0.9190 0.9299 0.9399 0.9488 0.9569 0.9641 0.9704 0.9760 0.9809 0.9850 0.9885 0.9915 0.9938 0.9957 0.997 1 0.9982 0.9989 0.9994 0.9997 0.9999 1.0000 1.0000 1.0000
References p . 282
0.5000 0.0000 0.4805 0.0003 0.4619 0.0011 0.4443 0.0025 0.4276 0.0044 0.4117 0.0067 0.3968 0.0096 0.3827 0.0129 0.3694 0.0166 0.3569 0.0207 0.3453 0.0253 0.3344 0.0302 0.3242 0.0354 0.3148 0.0410 0.3060 0.0468 0.2979 0.0529 0.2905 0.0593 0.2837 0.0659 0.2775 0.0727 0.2719 0.0797 0.0868 0.2668 0.2622 0.0940 0.1014 0.2582 0.1087 0.2545 0.2514 0.1162 0.2486 0.1236 0.2462 0.1310 0.2442 0.1384 0.2425 0.1457 0.2410 0.1528 0.1599 0.2399 0.2389 0.1667 0.2382 0.1734 0.2376 0.1798 0.2372 0.1860 0.1919 0.2369 0.2367 0.1975 0.2366 0.2028 0.2365 0.2077 0.2365 0.2123 0.2364 0.2164 0.2364 0.2202 0.2364 0.2236 0.2364 0.2266 155, 313 (1959).
t
1.0000 0.9601 0.9806 0.8814 0.8425 0.8041 0.7660 0.7283 0.6911 0.6544 0.6182 0.5826 0.5475 0.5131 0.4793 0.4463 0.4140 0.3825 0.3518 0.3221 0.2933 0.2656 0.2389 0.2133 0.1890 0.1660 0.1442 0.1239 0.1055 0.0878 0.0721 0.0580 0.0456 0.0348 0.0257 0.0182 0.0123 0.0078 0.0046 0.0024 0.0011 0.0005 0.0001 0.0000
2.4261 2.3314 2.2378 2.1454 2.0541 1.9639 1.8750 1.7874 1.7010 1.6159 1.5321 1.4498 1.3689 1.2894 1.2115 1.1352 1.0605 0.9874 0.9162 0.8467 0.7792 0.7136 0.6501 0.5888 0.5298 0.4731 0.4190 0.3675 0.3188 0.2731 0.2305 0.1913 0.1555 0.1233 0.0950 0.0706 0.0502 0.0338 0.0212 0.0121 0.0061 0.0027 0.0009 0.0002
235
236
J . B A R U EEN AND J . R. SCHRIEFFER
u1
2no
on
w-
- = --(A)
(1 - - ( A ) )
In
[-,I "Y
4k,T
, (tiw
[CH. VI,
< k,T, < 24)
5 12
(12.32)
where y = ec = 1.78 (c = Euler constant). Values of ul/un and u2/un covering a wide range of temperatures and frequencies as calculated by Millergs are given in Table 5. They apply generally to an isotropic superconductor in the weak-coupling limit; so that d ( T )may be taken to be an empirically determined gap. 1 2 . 2 . INFRARED TRANSMISSION THROUGH THINF m i s
Glover and Tinkham lea, and later Ginsberg and Tinkham lot, have measured the transmission of infrared radiation through films whose thickness is small compared with the penetration depth. Since the field strength is approximately constant through the thickness of the film, the results can easily be compared with theory. Frequencies of greatest interest are those in the neighborhood of the energy gap, eV corresponding to a wave which is typically of the order of length of the order of 0.1 cm. This is a most difficult part of the spectrum in which to work, just beyond the range of microwaves. Tinkham and co-workers have developed optical techniques for working in this far infrared region, with use of large gratings and mirrors. These experiments not only give the most direct experimental evidence for the existence of an energy gap, but also some of the strongest evidence for the Pippard non-local form of the theory. A convenient quantity for interpreting the film data is the complex conductance, u = ul - ;a2, per square of surface area. The transmission data were obtained in the form T,/Tn,the ratio of power transmitted when the film was superconducting to that when the film was normal (T > T,) ;this ratio is related to u by
TE Tn
~-
- { [Tn*
+ (1 - T"') ( u , / ~ n+) ] [(I ~ - T,')2u2/~n12)-'. (12.33)
Since a1 and u2 are connected by the Kramers-Kronig relations, a knowledge of T E / T nover a wide range of frequencies suffices to determine ul and u2 separately. The analysis is simplified by the fact that T,/Tn is dominated by either u1 or u2 over all but a narrow band of Kefevences p . 282
CH.VI, § 121
237
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
frequencies in the neighborhood of the gap. A plot of the transmission based on theoretical values of o1 and o2 is given in Fig. 19. The experiments also give a peak in transmission for w M wg, defined by &cog = 2 4 , where both o1 and o2 are small. If the quantity [l - ( C T ~ / U ~ is ) ] -plotted ~ on a logarithmic scale as a function of reduced frequency, Cu/cog, the theory gives approximately a straight line with a slope of about 1.65. Fig. 20 shows values deduced from Ginsberg and Tinkham's experiments on tin films. The empirical slope is 1.83. The temperatures are sufficiently low so that the T 0" K limit should apply reasonably well. Extrapolation to o1 = 0 indicates a gap of about 3.5 k,Tc. Glover and Tinkham attempted to determine 02/an a t lower frequencies by use of microwave transmission measurements. Although their results were not very accurate, they suggested that 02(w)/a,(w) may be expressed as a universal function of frequency exhibiting the lossless inductive type of frequency dependence 3
(12.34)
The parameter a is the same as the one originally introduced by Faber and Pippard in an empirical expression for the coherence distance, supposedly valid for all metals, (12.35)
were no is the velocity of electrons a t the Fermi surface. They estimated that a = 0.15 by fitting the Pippard nonlocal theory to measurement of the penetration depth of aluminum and tin as measured at microwave frequencies. Glover and Tinkham found a value a = 0.27 from analysis of their microwave data. Using the sum rule (12.10) which follows from the Kramers-Kronig relations, Ferrell and Glover92a were able to calculate a from o1 as determined from infrared transmission and found a = 0.21 f 0.05. They believe that this value is much more accurate than that obtained from the microwave data. The theoretical value from (12.25) with A = 1.76 knTc is 0.18, which is intermediate between these empirical results. More recently, Ginsberg and TinkhamlOC have measured with improved accuracy infrared transmission through thin films of lead, tin and indium. Values of [I - (ol/on)]-l deduced from measurements are Refereicces
p . 282
238
J. BARDEEN AND J. R. SCHRIEFFER
[CH.VI, 9 12
w /wp Fig. 19. The frequency dependence of T,/T,, ul/an and u,/u, as calculated by Tinkham from the theory Mattis and Bardeen (see ref.loc).
h W/kT, Fig. 20. A logarithmic plot of the frequency dependence of (1 - ui/un)-l for tin measured by Ginsberg and Tinkham. h'cferences p . 282
CH. VI,
9 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
239
plotted in Fig. 20. Estimated values of the energy gaps from the extrapolated intercepts are 4.0 f 0.5, 3.3 f 0.2, and 3.9 -J= 0.3 k,T, for lead, tin, and indium, respectively. They calculated from these data values for the parameter a from o1by the method described above, and found a m 0.20 for indium, 0.23 for lead and 0.26 for tin. The latter value is somewhat higher than that deduced from the earlier measurements of Glover and Tinkham described above. It is interesting to note that a magnetic field of 8000 gauss had no observable effect on transmission through a lead film. The local London theory predicts the conductivities
for which the value of a is of the order of a hundred times smaller than the values quoted above, unless the ad hoc assumption is made that the concentration of superconducting electrons is strongly reduced for thin films. Thus the thin film experiments give strong support both for an energy gap of the order of 3.5 k,T, and for the validity of a non-local relation between the current density and vector potential. Of considerable interest is the extra absorption peak observed in lead at a frequency somewhat below that of the main absorption edge.
fiW/kTc Fig. 21. Frequency dependence of ul/onfor lead, measured by Ginsberg and Tinkham. Notice the precursor absorption a t fiw/2A 0.85. Whether or not there is a real peak or a shoulder is somewhat uncertain.
Ginsberg and Tinkham also give evidence for such a peak for mercury. Similar precursor absorption peaks were observed in these same materials in reflection measurements on bulk specimens by Richards and Tinkhamlm, as will be discussed later. References p . 28%
240
J . BAKDEEN AND J . R. SCHRIEFFER
CH. VI,
5 121
The origin of the structure in the absorption edge has not been established ; however, it has been suggested that either an anisotropic energy gap or the existence of a set of transverse collective excitations with energies near the edge of the gap may play a role. It appears implausible that the absorption would exhibit a hump if the effect were due to gap anisotropy since each region of the Fermisurface would increase in effectiveness in absorbing as w increases. However, the experiments are not sufficiently accurate to tell whether or not there is a real peak or merely a shoulder in the absorption curve. Recent calculations by Tsunetog6indicate that the d-like transverse collective excitations lead to a precursor absorption ; however, with reasonable assumptions for the interaction between quasi-particles, the peak is too low by almost an order of magnitude. Since the precursor is observed only in lead and mercury, for which a strong coupling theory must be used, an explanation of the phenomenon is likely t o depend upon further theoretical advances along these lines. 12.3.
PENETRATION
DEPTHS
One of the most important applications of the theory is the calculation of the average depth of penetration, 1,of a magnetic field into a plane surface, as defined by ( 3 . 7 ) . As mentioned there, experiments generally give only the integrated flux in the interior and thus provide little information on the way in which the field changes with distance from the surface. I n this section, we shall compare some of the experimentally determined values of ;2 with those deduced from the microscopic theory. Before doing this, we shall give a brief outline of the experimental methods used and of the way in which the theoretical values are obtained. The earliest estimates of penetration depths were based on measurements of magnetic susceptibilities of small particles or thin films. The most extensive measurements of this sort are those of Lock on thin films of tin, indium, and lead. A method suggested by Casimir can be used to find changes of penetration depth with temperature in bulk materials. What is measured is the mutual inductance between coils closely wound around a cylindrical specimen. As il increases with increasing temperature, the flux linking the coils increases in proportion. Laurmann and Shoenberg applied the method successfully to tin and mercury, using a frequency of 70 Hz. The results were analyzed with use of the empirical law References p . 282
CH. VI,
$ 121
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
241
(12.37)
where t = TIT, is the reduced temperature. This law, suggested by the Gorter-Casimir two-fluid model, was found to apply very well to changes in 1 observed in small systems. If this law is valid, measured values of Ail = A(T) - A(0j plotted as a function of y = (1 - t4)-112 should give a straight line with a slope of 1(0). Many of the quoted values for A(0) have been obtained in this way. Another method for determining penetration depths, based on measurements of the surface impedance at microwave frequencies, was suggested and applied by Pippard. The specimen is placed in a resonant cavity and changes in Q and in resonant frequency are observed. The latter depends on changes of penetration of field into the specimen, so that changes in L with temperature can be obtained. It is also possible to estimate absolute values of 1 by comparing the resonant frequency in normal and superconducting states. One difficulty with the method is that there are changes in penetration depth with frequency, and an extrapolation must be made to get the limiting value of 1 for low frequency. We shall discuss surface impedance measurements in more detail in the following section; here we shall simply quote some results of Pippard and coworkers on penetration depths. Recently, Sarachik, Garwin and Erlbachs7" have measured the penetration of field through a thin lead film placed on the outside of a cylinder. The mutual inductance between a coil around the outside of the cylinder and one inside was measured. Precautions were taken to prevent stray fields from leaking around the ends of the cylinder so that only the field going through the lead film affected the inner coil. This method is based on some earlier work of SchawlowQ7b. Depending on the nature of the scattering of electrons at the surface, the penetration depth R can be expressed in terms of integrals involving K(q) as follows: specular reflection
A= References p . 282
7E
(12.38)
random scattering . (12.39)
242
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
9
12
These expressions, due to Pippard, are generalizations of relations derived by Reuter and Sondheimer for the anomalous skin effect. Calculations based on the microscopic theory have been made by deriving analytic expressions for K(q) appropriate to the limiting cases of large q and small q and then using graphical interpolation for the intermediate range where direct calculation is difficult. Experiments on normal metals indicate that the random scattering hypothesis is to be preferred, and this boundary condition has been used in most of the calculations. I n the isotropic model there are three parameters, A(O), to and O ( 0 ) (which can be related to T J . Faber and Pippard have shown that A(0) can best be determined from the anomalous skin effect in the extreme anomalous limit. From A(0) and N(E,), the density of states of one spin at the Fermi surface, one can determine the Fermi velocity, v o , and thus &jl = d ( 0 ) / f i v o . The results t o be given for Sn and A1 make use of the model of 5 9 to determine O(T)from T,, and also for A(T)/A(O).Thus the only parameter involving the superconducting state is the critical temperature, T,. If observed values of A/k,I', differ significantly from predictions of the simplified model, one should use an empirically determined d ( T ) in calculations of the electromagnetic properties. The London relation, 1(T)= ( 4 n / ~ l ( T ) cis~ valid ) ~ / ~ for the limit to 1. Pippard has given an expression valid for the opposite limiting case, E o > 1
<<
(12.40)
As noted earlier, J ( 0 , T ) varies from unity to 1.33 as T goes from 0 to T,. Plots of l(T)/AL(T) versus E0/AL(T)given in ref.3 may be used to determine A in intermediate cases where neither limit is valid. Tab. 7 lists values of the parameters for several metals for which measurements are available as well as a comparison of calculated and observed penetration depths for T = 0' K . In Fig. 22 calculated values of the penetration depths for Sn and Al, based on random scattering, are plotted as a function of y = (1 - P-1/2.The plots are approximately straight lines for y > 1.5, corresponding to temperatures near T,, but there is some bending below the line with a higher slope for y < 1.5. Prior to the development of the microscopic theory, Lewiss* predicted such a bending on the basis of a Refereitcis p . 282
CH. VI,
3 121
243
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY TABLEVII
Y
CLPEL
(erg deg-acm-3) I I1 I11 IV
Sn A1
Pb Cd
1100' 1370k 171@ 5611
17.P 45.58 17-26* 4.lc 12.Od
Do
x 10-8
1061,(0)
(crnlsec)
(cm)
0.65 1.32 0.50 0.29 0.85
3.55 1.57 3.7 11.1 3.8
240 kTe
T,("I<)
3.6 f 0.2e 3.37"," 4.1 4 0.2e 3.39
3.73 1.18 7.15 0.56
106A cm (observed)
I I1 I11 IV
0.23 1.6 0.083 0.76 2.2
6.5 102. 2.2 6.9 59.
5.6 5.3 4.8 18. 11.
1.57 3.36 1.30 1.60 2.8
5.1" 4.9"; 5.15b 3.9" 13 & 1.4C
REFERENCES
a T. E. Faber and A. El.. Pippard, Proc. Roy. SOC.A 231, 336 (1955). b See ref.". c See ref.lo3. d R. G. Chambers, Proc. Roy. SOC.A 215, 481 (1952). e P. L. Richards and & Tinkham, I. to be published. f See ref.76.
g
C. B. Satterthwaite, private communi-
cation. h See ref.O5. i See ref.13*. j J. R. Clement, Phys. Rev. 92, 1578 (1953); B. N. Samoilov, Dok. Akad. Nauk (U.S.S.R.) 86, 281 (1952). k See ref.'*a.
two-fluid model which included an energy gap. The slope for Sn in the range y w 2 to 6 is about 5.4 x 10W cm, in good agreement with measurements of Laurmann and Shoenberg in the same temperature cm, appreciably range. For y rn 1.1 to 1.6, the slope is about 7.0 x cm obtained by Faber and higher than the value of about 5.0 x Pippard for Sn from microwave data. However, the theoretical slope for Al, about 4.8 x cm, is in good agreement with their measurements for this material. The theory gives only approximate agreement between the slope and the intercept at y = 1, or t = 0. The intercept for Sn is about 5.7 x cm and for Al-about 5.2 x 1W6 cm. In some recent measurements of penetration depths in tin, Schawlow and Devlin99 used a modification of the Casimir method in which the self-inductance of a solenoid closely wrapped around a superconducting cylinder is measured. They used a low frequency, 100 kHz. As shown in Fig. 23, they found an increase in slope, dlldy, at low temperatures, References
p . 282
244
[CH. VI,
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$ 12
lx
Lz
4-
W
a
-
2-
I
I
I
I
I
p - I
47
Fig. 22. Calculated values of the penetration depths for tin and aluminum based on random surface scattering.
1
4001
y:-
I
fi Fig. 23. The temperature dependence of dlldy determined by Schawlow and Devlin compared with theory, both showing a rise near T = 0. References
p . 25.'
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in good agreement with predictions of the microscopic theory. The calculations were based on the theoretical gap, ZO(0) = 3.5 k,T,, so that no adjustable parameters were used in obtaining this very striking agreement. Recently Pippard et al. loohave also observed some increase in slope at low temperature, but the increase is only about half that predicted by theory. It is not certain whether or not the discrepancy may be attributed in part to a frequency dependence of the penetration depth. One of the experiments which led Pippard to predict a non-local
Fig. 24. The penetration depth 1 versus y for various values of the mean free path I , expressed through the dimensionless parameter 2Z/nt, as calculated by Miller.
theory was that which indicated a marked increase of penetration depth with decrease in the mean free path, I, from impurity scattering in tin-indium alloys. This effect occurred in alloys in which the indium concentration was so small (<2y0)that there was very little change in electron concentration and thus in A,. In Fig. 24 results of calculations References 9. 262
246
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
3
12
of MillerlO1 of l ( t ) versus y are plotted for various values of I , introduced through the dimensionless parameter 21/nt,. These curves again show appreciable bending for y < 1.5, contrary to the experimental findings. The limiting values of 3, at t = 0 are in reasonably good agreement with Pippard’s results, which give an increase in il by a factor of about two when I is reduced to about cm. We conclude this discussion of penetration depths with a brief account of the measurements of Sarachik et aLg7based on transmission
1.00
1.01
1.02
1.03
1.04
1.05
1.06
yS(i-j4)-l
Fig. 25. The temperature dependence of the penetration depth in lead measured by Sarachik, Garwin and Erlbach. They find good agreement with current theory if 2 4 / k ~ T= , 4.93.
of 2.2 MHz signals through thin lead films. The films ranged in thickness from 1.5 to 4.0 x cm, which is a little less than the penetration depth. They were able to analyze the data to obtain 3,,(T)/l,(O) if the London limit applies, or [ i l m ( T ) / L ( 0 ) ] 3for / 2 the Pippard limit. The former probably applies best to lead, and in Fig. 25 we give their analysis for this case. They find a reasonably good fit to the data for an energy gap with the same temperature dependence as the simplified model, but with 2d(O) = 4.93 k,T,. 12.4. SURFACE IMPEDANCE
Experiments on surface impedance at microwave frequencies have provided extremely valuable information on the electromagnetic proR e f w e m a p. 282
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
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perties of superconductors. From Maxwell’s equations there must be an electric field associated with a time-varying magnetic field in the penetration regions, and this field is appreciable in the microwave range. When T > 0 the electric field can act on quasi-particle excitations and give rise to a dissipation of energy. The first experiments of H. London in 1940 showed that the surface resistance, R,,which measures the loss, is continuous at T , and decreases continuously from the normal value at T , to zero as T -+ 0. In a two-fluid model this loss is attributed to the normal component. The method was further developed by Pippard, who, with his associates, have made extensive series of measurements. As mentioned in the last section, Pippard was able to measure both the resistive and reactive components of the impedance. Although he found that it was not possible to account for the temperature and frequency dependence of the surface resistance by any simple type of two-fluid model, there is excellent agreement with the microscopic theory. More recently other groups have studied the surface impedance. Particularly noteworthy is a very complete and accurate set of measurements on aluminum by Biondi and Garfunkell’, covering a wide range of frequencies and temperatures. Before discussing this and other recent work, we shall summarize the results of the theory. If the superconductor occupies the half-space x > 0 , the surface impedance Z is defined by: (12.41)
J
i y ( x I0 ) dx 0
It can be simply expressed in terms of the complex K(q, co) which relates the Fourier components of j and A (12.7). For random scattering,
>
The ratio of the surface impedance, Z,,, in the Pippard limit, to 1, to that in the normal state in the extreme anomalous limit, Z,, can be expressed in terms of the complex conductivity ratios (see Table 5) : (12.43) References p . 282
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J . BARDEEN AND J . R. SCHRIEFFER
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5 12
Millerg5has evaluated the integral (12.42) for A1 and for Sn and finds that the detailed calculations give large corrections to the surface resistance ratio as calculated in the Pippard limit even for Al, for which E,, m 30 A. Corrections to the reactance ratio are smaller. Biondi and Garfunkel measured energy absorbed from the microwave field by a calorimetric method, and thus determined directly only the resistive part, R, of 2. Their measurements covered such a wide frequency range (from 20 mm to 3 mm wavelength) that they were able to use Kramers-Kronig relations to determine the reactive part, X . The latter may be expressed in terms of an effective penetration depth, d,, by 6,=-.
ex
(12 * 44)
4mo
Figs. 26 and 27 give a comparison of the experimental results with Miller’s calculations. Contrary to the usual practice, the experimental values are plotted as smooth curves and the theoretical values as discrete points, since the former were more complete and more accurate than the latter. The only modification of the simplified model was to take a slightly smaller value for the gap, 3.37 instead of 3.52 k,T,.
ea 0.2 3
ENERGY (in units of kTc)
’
Fig. 26. Frequency dependence of the surface resistance of aluminum as measurcd by Biondi and Garfunkel (smooth curves). Plottcd points are from calculations of Milier. Rejdrmces
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The knees of the surface resistance vs. photon energy plots occur where the photon energy becomes greater than the gap at the corresponding temperature. The increase in absorption beyond the knee comes from excitation of carriers across the gap (creating two quasiparticles). Biondi and Garfunkel have estimated the gap and its temperature variation empirically from the positions of the knee at different temperatures. There have been a great many measurements of the surface impedance of tin, most recently by Kaplan et a1.Io2. They, as well as
9
/I
2
'0
I
2
3
4
5
6
7
8
hv/kT, Fig 27. Frequency dependence of the surface reactance of aluminum, expressed as a penetration depth, from Biondi and Garfunkel (smooth curves). Plotted points are from calculations of Miller.
earlier workers, have found that the experimental results are sensitive to surface imperfections and strain. Pippard has pointed out that over a considerable range (7'from w 0.4 T , to M 0.8 T,, no from M 0.01 k,T, to w 1.0 ksTc), the surface resistance ratio may he expressed as a product of a temperature and a frequency factor,
(12.45) Ikferences
p . 282
250
J. BARDEEN AKD J . R . SCHRIEPFER 10
I
I
I 1 l l l l
'
I
I
,
I I I I ,
I
[CH. VI,
9
12
I 1 1 1 1 I
I
u(kMHz) Fig. -78. Frequency factor A(v)entering the expression for the surface resistance (12.45).
where y ( t ) as a function of reduced temperature is
Pl(t)= t 4 ( 1
(12.46)
- t y (1 - t4)-2.
This empirical result has since been confirmed by other workers. Miller's calculations indicate that these relations are valid within about 10 percent for the same range of t and o.Miller used the simplified model without modification, and thus used the theoretical gap
'P
6
11.2
0.2
T/Tc
Fig. 29. Ilhaikin's measured values of the surface impedance of cadmium compared with theoretical predictions of Abrikosov et al. based on the Pippard limit of the theory. Refcrciicrs p . 262
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of 3.5 k,T,. Fig. 28 gives a comparison of the theory with various experimental determinations of the frequency factor A ( Y ) , The agreement is excellent, considering that no adjustable parameters are involved. The lower curve, based on the Pippard limit, (12.43) is considerably below both the experimental points and the results of the detailed calculations. Khaikin1o3 has measured the surface impedance of cadmium, which has a transition temperature of 0.56" K. The frequency used corre-
a
Vanadium
5.10
0
Tantalum Niobium
4. 9.
o
FREQUENCY (ern-') Fig. 30. Frequency dependence of the electromagnetic power absorbed by bulk superconducting samples in the far infared measured by Richards and Tinkham. Here, Ps and P, are the powers absorbed i n the superconducting and normal states respectively.
sponds to an energy of 0.9 K,T,. Fig. 29 gives a comparison of the observed values with theoretical calculations of Abrikosov et ~ 1 . 8 6based on the Pippard limit (12.43). The agreement again is excellent. We give finally in Fig. 30 some measurements of Richards and Tinkham on the surface impedance in the far infrared region. What is measured is the absorbtion of energy versus frequency of the radiation beyond the gap frequency. Values they have found for the gap in References
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J . BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
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13
various materials are given in Table 11. They depart considerably from 3.5 k,T,, varying from 2.8 k,T, for niobium to 4.6 k,T, for mercury. The general trend is inverse to the Debye temperatures. They found some structure in the absorption curves for lead and mercury, indicating possible absorption by collective modes with energies in the gap or anisotropic effects. In general, the observed absorption curves approach the normal state value more rapidly than indicated by the theory. We conclude that on the whole the microscopic theory of the electromagnetic properties is in remarkably good agreement with experiment, particularly if empirical values are used for the energy gap and its temperature dependence. What discrepancies there are can probably be accounted for by the complex band structure of actual metals as compared with the theoretical isotropic model. Only limited progress has been made in understanding the interesting experiments of Spiewak104on the magnetic field dependence of surface impedancelo5.
13. Collective Excitations Thus far, our discussion has been concerned with applications of the theory to problems emphasizing the quasi-particle aspects of the excitation spectrum, for which an independent quasi-particle approximation is valid. Basic to a gauge invariant description of the Meissner effect and a complete account of the system’s response to space-time varying external fields, are the collective excitationss0. These excitations have energies split off from the continuous spectrum as a result of residual interactions not accounted for in the single quasi-particle approximation. As in the normal state, the collective modes may be viewed as coherent superpositions of quasi-particle configurations. In the superconducting state, the plasmon modes continue to exist and are essentially identical to those occurring in the normal state. Due to their high energies (w15 eV), real plasmon excitations do not enter the low frequency phenomena we have been discussing (no< 10WeV). Virtual excitation of plasmons is, however, essential in obtaining a gauge invariant form of the kernel K relating the current density and vector potential. In the Coulomb gauge, div A = 0 , neither real nor virtual plasmon excitations enter due to the longitudinal character of the plasmon current density. For this reason, calculations are often simplified by choosing div A = 0 , as in 5 12. A new feature of the superconducting state is the possibility of excitation-like modes occurring with energies lying in the energy gap, Rejcrcwces
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as pointed out by Anderson6 and Bogoliubov4. The normal state precludes their existence since the finite density of single particle states near the Fermi surface would lead to rapid decay of the collective modes. In the superconducting state the exciton modes are likely to have rather long lifetimes due to the absence of single particle states within the gap. As opposed to plasmon excitations, the excitons may enter both real and virtual processes and physically observable effects are associated with these modes. Real transitions involving exciton creation are reflected in a resonance of the absorptive part of the wave vector and frequency dependent kernel, K(q, w) for q t , < 1 and 6w < 2 4 , while virtual processes have a small but finite effect on the real part of K for all frequencies of interest. Since the nature of the exciton spectrum is closely related to the angular dependence of the residual two-body interaction V (k , k') , the BCS parameter N ( 0 )V does not suffice to determine the types of excitons which occur in a particular metal. For an L state exciton (corresponding to p, d.. . . . .excitons) to exist lo6,it is required that V , be negative, where V , is the L wave part of the interaction V ( k , k'). A plasmon corresponds to an S state exciton whose energy is greatly increased by the long range Coulomb interaction. An approach sufficiently general to treat both indivdual quasiparticle excitations and the collective modes in the high density limit is given by the generalized time-dependent self-consistent field (SCF) or random phase approximation introduced by Anderson 6, and independently by Bogoliubov, Shirkov and Tolmachev4". This method is a generalization of an approach introduced by Bohm and Pineslo8for a description of the normal state. The most complete discussion of the method has been given by Rickayzena, who used it to derive a fully gauge invariant form of the kernel K(q, w) for the superconducting state. Except for small terms due to the excitons, his result for the gauge div A = 0 is identical to that given in the basic paper of Cooper and the authors3. To illustrate the time-dependent SCF approximation, we begin with a discussion of the elementary excitation spectrum of the normal state, valid in the high electron density limit Y, < 1 = 4n/3 (r,aO)3, a, Bohr radius). In this limit, the Coulomb interaction energy is small compared to the kinetic energy of the electrons. Thus, a perturbation treatment starting from the Fermi sea would be a good approximation were it not for the singular nature of the Coulomb interReferences p . 28,"
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J. BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
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action for long wavelengths, that is V(q) = 4ne2/q2--f 30 as q -+ 0. It would appear that the effect on a given electron of the long range part of V can be represented by an average self-consistent potential arising from the coherent motion of the distant electrons, plus a small fluctuating potential associated with their residual random motion. That the fluctuating potential may be neglected for many purposes in the density limit was discussed in detail in the pioneering work of Bohm and Pines. To derive the spectrum of elementary excitations within the SCF approximation it is convenient to study the motion of electron-hole pairs with total momentum $4. Due to the Coulomb forces, an electron and a hole in st:.ies k + q and k respectively may recombine and excite an elec troll into the state k' q leaving a hole behind in state k'. The riew pair may again recombine transferring the excitation to k" $- 7 2' k.'. This ,rcccss is illustrated in Fig. 31a. There are other S ~ Jc d h ! exchange processes possible in which the pair in k + q and k
+
Fig. 31. Hole-electron scattering processes included in the random phase approximation.
scatters directly to k' + q and k' without recombination and creation taking place, as illustrated in Fig. 31b. Since the matrix element for the processes shown in (a) involve the matrix element 4ne2/q2while those in (b) involve 4ne2/ I k - k' 12, the polarization processes (a) are on the average far more important in the limit q -+ 0. Restricting ourselves to direct processes, which as Gell-Mann and Brueckner lo9 have shown is sufficient for Y, < 1, it is clear that certain linear combinations of the electron-hole states will be eigenstates of the Hamiltonian. The plasmon mode which splits off from the continuous spectrum is given by a superposition of the pair configurations in which all states enter with the same sign and with approximately equal weight. The situation is somewhat analogous to the coherent superposition of many particle configurations used to form the ground References p . 287
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state of the superconductor. The remaining linear combinations of electron-hole pairs correspond to scattering states in which the electron and hole are not strongly correlated. The formal procedure of obtaining the elementary excitation spectrum is carried out by finding those linear combinations, pa(q) *, of electron-hole pair operators @ku(q)
(13.1)
= c&quckuJ
which create an eigenstate of the approximate Hamiltonian Hpz(q)*yO = ('lza(q)
+ wO)pa(q)*yoJ
(13.2)
where H includes only those interaction terms leading to polarization processes. Here &Q,(q) is the excitation energy of the ath linear combination of the @ k , ( q ) . The equation (13.2) will be satisfied if the operators pa(q) * obey
LHJ
pz(4)*1
= nQa(q)pa(q)
*'
(13.3)
If @ k , ( q ) is decomposed into an average value &)(q), taken with respect to a self-consistent state, plus a fluctuation &(p) about this average value, the processes shown in Fig. 31(a) are taken into account if only first order terms in @(l) are kept in the full commutator [ H , @ k , ( P ) ] . This is just the self-consistent field approximation as a straightforward calculation showsllO. In the high density limit, it is sufficient to linearize the equations about the average values &,)(9) appropriate to the Fermi sea. In this case only the zero wavenumber component @ k , ( o ) = nku has a nonvanishing average value : (13.4)
I t is straightforward to show that the elementary excitation energies are given by the dispersion relation (13.5)
The solutions of (13.5) are plotted in Fig. 32. Both individual-particle like states describing electron-hole pairs in scattering states, and collective plasmon states appear. While the energies of the holeparticle pairs within the SCF approximation are unaltered from their values, & k + q - f k , in the absence of interactions, the wavefunctions References
p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
13
are strongly modified. Each particle (electron or hole) is surrounded by a local depletion of the same type of particle (electron or hole, respectively) and the physical picture of backflow discussed in 4 5 follows in a natural manner. In a more complete treatment, &k+q - &k is shifted by a self-energy term which is complex, the imaginary part corresponding to the finite
%F
d
Fig. 32. Elementary excitation spectrum in the normal state.
lifetimes of the excitationsll'. The imaginary part of the single partic e self-energy, calcuIated by a Green's function approach, is plotted as a function of the real part of the excitation energy in Fig. 33. For energies less than the maximum phonon energy &omax,real phonon emission plays the dominant role, while for E > 10 &wmax,hole-electron pair production takes over and the single particle levels become very broad. The effect of finite lifetime on the energy gap equation is discussed in 5 9. In the superconducting state the pairing effects lead to non-vanishing average values for operators of the form bk
2
C-kJCkf
xk
+ bf) (13.6)
'k
=
bkyO),
as well as for %ko. As Anderson6 and Bogoliubov4 have independently pointed out, an improved description of the elementary excitations in the superconducting state may be given by including residual interactions neglected in the original discussion of Cooper and the authors. They discuss a generalized self-consistent field approximation in which both k k and x k are introduced. The analysis is most simply carried References p. 282
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outs by working with the quasi-particle operators discussed in 3 11. One again seeks those linear combinations p$(g), of the operators ~ : + ~ + y *y-k-qlykt, ~ ~ , and y:+qrryko which create elementary excitations of the system. The normal modes of the linearized equations are somewhat morc complicated to determine in the superconducting state than in normal metals. As pointed out by Rickayzene, the electromagnetic response kernel K and the dielectric constant mey be determined without explicit knowledge of the normal mode operators. His results give a fully gauge invariant description of the Meissner effect with the kernel
production
'
u 0
1
2
3
4
5
6
7
8
9
RebpfiuC
1;ig. 33. The cnergy dependence of the imaginary part of the excitation energy in the normal statc. The cxcitations arc poorly dcfined near the Debye energy and above ten times the Debye encrgy for typical electron densities.
K for zero frequency being identical to that derived by Cooper and the authors if only the s wave part of the two body potential is kept, as in
s 9.
It is explicitly seen that the longitudinal collective excitations contribute a polarization current which, when added to the quasiparticle polarization current, just cancels the diamagnetic gauge current, leading to the gauge invariant form for K . If the s and d wave parts of the potential are chosen to have equal magnetudes, corrections to K are of order and may be safely neglected. , is significantly different from its The dielectric constant ~ ( qu)), \.due in the normal metal only if fico m &v0q < A . Rickayzen's treatR c f r t m c r ~p . ?,(?
258
J . BARDEEN AND J. R. SCHRIEFFER
[CH.VI,
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ment also shows the expression for the acoustic absorption coefficient calculated in 5 11 within the single quasi-particle approximation to be valid to order ( o / ~ ~Mq ) ~and (&v,q/A)aQ 1. Tsunetog6 extended Rickayzen’s analysis to treat the surface impedance at finite frequency. Choosing only the s and d wave parts of the potential to be non-zero, he finds a precursor absorption to exist for frequencies below that of the energy gap, ad/&.His results,’when applied to lead and mercury, predict an absorption due to exciton states in the gap which is an order of magnitude smaller than that observed by Ginsberg, Richards and TinkhamlO in these materials. Since lifetime effects are likely to be important in these strong coupling superconductors, it is desirable to extend the theory to the strong coupling regime before drawing conclusions regarding the role played by collective modes in these experiments. The elementary excitation spectrum of the superconducting state is shown schematically in Fig. 34. As mentioned above, the plasmon
2kF
q-
Fig. 34. Elementary excitation spectrum in the superconducting state.
mode is almost identical to that occurring in the normal state. The broad spectrum of quasi-particle pairs in scattering states is bounded from below by the energy gap 24, The exciton states, having energies lying within the energy gap, may be pictured as a pair of quasi-particles bound together in r e d space moving with center of mass momentum 69. The exciton wavefunction is of the formlosa (13.8)
where
describes the relative motion of the pair having an extent
References p. 282
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t oand S,, is the spin function. In the limit q -+ 0, (13.9)
and one has the usual picture of p, d, . . . . state excitons as in an insulator. For larger q, states of different L mix although the mixing is small for qt0 1, while M is a good quantum number for all q if the potential has no crystalline anistropy. Since the quasi-particles are Fermions, the wavefunction must be antisymmetric on interchange of rl, o1 and r2, c2.Thus, a state with L even must have a singlet spin function, while odd L is associated with a triplet spin function. The spectrum of exciton states isstrongly dependent upon the angular dependence of the residual two body potential V (k, k‘). If the potential is decomposed into spherical harmonics
<
and V,( I k
1, 1 k’ 1)
is approximated by
the condition for an L state exciton to exist is go > g, coupling constant g, is given by
g,
= - N(O)VL
> 0. The (13.12)
and go = - N(0)Vo = N(O)V, is the coupling constant introduced in $ 9 . The exciton energy for q = 0 is plotted as a function of g, in Fig. 35. For g, >go, the excitation energy is imaginary and the system is unstable when described by the ground state based on the s state pairing discussed in 9 8. For example, if g, is the largest coupling constant, the ground state will be formed from pair functions p, having p like symmetry (see (8.10))
- ‘2)
‘PIM(‘l
= P)l(lrl - ‘2 1)Y1M(e12J v12)
(13.13)
and the triplet spin pairing considered by FisherlOsbwill be appropriate. If the d wave potential is dominant, singlet d functions are appropriate for the ground state, as considered by Anderson et for 3He. The exciton energy is plotted as a function of center of mass momentum in Fig. 36 for several values of gL. The long range Coulomb potential plays no role for M # 0 and these excitons may be thought of as h’cfmences p . 282
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J . BARDEEX AND J . R . SCHRIEFFER
0.1 -
Oo
i i i 4 4B
+ b 9 Ib1'1
1'2b
[CH. V I ,
9 13
*
1;ig. 35. Energy of the exciton states within the energy gap for zero center of mass momentum, as. a function of the coupling constant.
0 01
0 2
03 0 4
05
06
07
08
09
10
d o
Pig. 36. Euciton cncrgy as a function of the center of mass momentum hq for t h r magnetic quantum numbrr M f 0.
transverse collective excitations. The M = 0 states may be split from the M # 0 states, with the s state exciton being identified as the plasmon mode if the ground state is described by s state pairing. For non-factorizable potentials, that is V(1 k 1, 1 k' I) can not be expressed as W (I k i)W(1 k' I), more than one bound state for given h'efevencea
p . 28?
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L and M may exist, corresponding to states with different principle quantum number, 12, in the hydrogen atom. We turn now to a brief discussion of the electronic spin susceptibility in superconductors. In the normal state the spin susceptibility is given by X, = 2 , 4 N M ( 0 )where , ,u%is the Bohr magneton and N,(0) is an effective density of states at the Fermi surface, differing from N ( 0 ) entering the electronic specific heat by terms arising from the Coulomb exchange energy. In a nuclear resonance experiment the Knight shift K , defined to be the fractional difference in the resonant frequency between a nucleus in a free ion and that same nucleus in a metal, is directly proportional to X,. Since X, is proportional to N,(O), nuclear resonance experiments lead to information about the density of states at the Fermi surface. Reif l7 has measured the Knight shift K , in superconducting mercury colloids consisting of particles mostly less than 500 A in diameter (< A), and finds K , to drop rapidly for T < T,, reaching a saturation value at T m +Tcof about $K,,. Recent data of Androes and Knight l6 taken on thin superconducting platelets of tin ( M 40 A x 140 A) are plotted along with Reif's data in Fig. 37 and show ;I tendency to
0.8 '.OI
x
O
1
0.2
I
O
o
0
:
O/
i
/
,L.-0
r/r, I ig. 37. Temperature dependence of thc rcduccd Knight shift measured b v Kcif on inercury colloids and by 4ndrocs and Knight on tin platelets.
5aturate at K J K , w 0.73 if SnC1, is used as the reference salt. As Yoshida1*8has shown, the microscopic theory leads to a susceptibility X, which vanishes at T + 0" K for a uniform magnetic field, apparent1~Keferenccs
p . 282
262
.
J BARDEEN AND J . R. SCHRIEFFER
[CH.
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in contradiction with experiment. This result follows because the minimum energy 24 required for creation of two quasi-particles from the ground state is larger than the Zeeman energy p B H ogained by the excitations. Heine and Pippardlla have suggested an alternative form for the matrix elements which enter the theory such that a finite Knight shift is obtained. Thus far it has not proved possible to construct wavefunctions which lead to these matrix elements. Ferrelllls and Anderson1l4 have suggested that as a result of the spin-orbit interaction, a finite value of X , in small specimens near T = 0" K would be obtained because the single particle wavefunctions in the normal state are not eigenstates of the spin. In this case, the magnetic interaction, - pB
S, Ho G H , *
will have non-vanishing matrix elements between the ground state and excited states so that a perturbation calculation of X, starting from the ground state defined in the absence of the magnetic field might be appropriate. A number of authors115have pointed out that even in the absence of spin-orbit effects, a non-zero value of the wave vector dependent susceptibility appropriate to space varying fields is obtained. As a result, a positive Knight shift may be seen within the penetration region of a bulk sample, while a region of reversed spins necessary to satisfy X,(q = 0) = 0 would extend to a distance M to beneath the surface and would not be effective in the observed resonance spectrum. There is no empirical evidence in favor of this picture at present. A wave number dependent susceptibility cannot account for the experimental results on tin mentioned above. Since any state derived by a perturbation series from the singlet ground state not involving spin-orbit effects must have vanishing total spin magnetization, the theory would predict a broadening of the resonance line with little or no Knight shift. The experiments on the contrary exhibit a shift which is at least as large as the line width. As suggested by one of the authors116*, a finite value of X , at T = 0' K applicable to uniform fields might be obtained if the ground state in the presence of the magnetic field H , is formed by a pairing different from that appropriate to the case H , = 0. In analogy to the modified pairing ( k + q t , - k q 4 ) introduced to describe current carrying states, one might begin with the magnetized state appropriate
+
References 9. 282
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RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
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to the normal metal, pairing states on the up spin Fermi surface with those on the Fermi surface for down spin. These single particle states are not related by time reversal; however, this condition is not required in the presence of a magnetic field. Thus far no calculations have been carried out with the modified pairing. It would be interesting to investigate the role played by spin-orbit effects by measuring K , for a light metal such as aluminum where the spin-orbit effects are expected to be less effective than in tin and mercury. 14. Two-Fluid Model and Persistent Currents 14.1. TWO-FLUID MODEL
The two-fluid model of He I1 has been extremely successful in predicting and interpreting many of its remarable superfluid properties, such as second sound, heat flow by convection, and various thermomechanical effects. There have been speculations as to whether corresponding effects might be observed in superconductors. In an earlier volume of this series, Gorterll' has given a review and comparison of two fluid models for superconductors and liquid helium. The superfluid component is the part with frictionless flow; it corresponds to flow in the ground state and carries no entropy. The normal component is the part of the flow associated with thermal excitations; it is subject to the usual friction. While the equations for He I1 are formally those of two interpenetrating, non-interacting fluids, Landau118showed that they can be interpreted in terms of the properties of the ground state and the spectrum of elementary excitations of the fluid. Landau's arguments can be formally extended to give a corresponding two-fluid model for superconductors lI9, but for a number of reasons it is much less useful than for He 11. Before outlining the derivation, we shall point out why many of the two-fluid flow phenomena characteristic of He I1 would be difficult to observe in a superconductor. Following the discussion of the theory, we consider applications to persistent currents, critical currents in thin films and the Ginzburg-Landau theory. Some of the complicating factors are: (1) In a superconductor, current flow produces a magnetic field and this field has a strong influence on the flow. For example, when current flows in a superconducting rod, it is confined to a region within a penetration depth of the surface. One can regard References
+. 282
264
J . R A R D E E N A N D J . R. S C H R I E F F E H
LCH. VI,
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thc currcnt as producing a magnetic field and the magnetic field in turn producing Meissner ciirrents which prevent the field from penetrating. The sum of these Meissner currents gives the net current flowing in thc wirc. I t is only whcn thr dimcnsion are of the order of the penetration depth or less that the current density is reasonably uniform. In practicc, this is most easily achieved in thin films. To avoid these complications, w ( s shall in thr following discussion omit effccts of the magnctic field, and suppose that on,' can have a uniform current flow. The electrons still show snperfliiid behavior, wit11 persistcnt currents possible. (2) The excited electrons (normal component) are scattered by and relax to thc lattice. I t is thus difficult to have a normal component of flow in the absence of an electric field; this is one reason why second sound would not be easy t o observe in a superconductor. For simplicity, we shall a t first neglcct relaxation effects, but will consider the consequences later. (3) Uncertainty relations give a much larger minimum size for the excitations in. a superconductor than in He 11, which is mainly a consequence of the large difference in mass. To be reasonably well defined, a quasi-particle excitation in a superconductor cm). I t is only when should bc larger in extent than the coherence distance ( m changcs in motion occur slowly over distances of this order that the local relations of the two-fluid modcl can be used. This puts, for example, limitations on the minimum wavelength a t which one might hope to observe second sound. (4) As indicated above, thermal conduction in He I1 can take place by a counterflow of normal and super components, thc hcat flowing with thc normal component. A similar effcct can occur in a superconductor, but the magnitude is very small compared with the usual electronic thermal conductionlPO.The elementary excitations created when a superconductor is heated correspond to electrons above and holes below the Fermi surface. They are created in nearly equal numbers and tend to flow in the same direction. There is therefore very little net electrical current associated with the hcat flow. In a normal metal this current is related to the thermoelectric effect and is what makes the difference between thermal conductivity with j = 0 and with E = 0, which is known to be very small. The order of magnitude of the effect in a superconductor is correspondingly small.
For the formal derivation of the equations of the two-fluid model, we disregard magnetic fields and electron-lattice relaxation processes and suppose that changes in motion occur slowly over a coherence distance. We also consider for simplicity a free-electron type model for the normal metal. First consider acceleration by an electric field, $. The common momentum wzv, = i(p, - p,) of the ground state pairs (p, 1. , - pz 4) increases according to Newton's cquation m3, = - eb. ,4t the absolute zero there are no thermal excitations and the entire distribution of electrons is displaced in k-space. Because of the energy gap, it is energetically unfavorable to scatter electrons from one side of the Fermi distribution to the opposite until a critical velocity is reached, above which superconductivity is destroyed. If v 0 is the Fermi velocity, this occurs when
+
~ w ( v , v , ) ~- & w ( v~ v , ) ~> E , = 2 4 , Referetiers
p . 282
(14.1)
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or when u8 > O/(mu,,). At higher temperatures, thermal excitations can decrease the current, so that only a fraction of the electrons appear to be freely accelerated by the field. London's second equation is (14.2)
where A ( T ) is the temperature dependent London parameter. One may write j , = - ee,v,/m, where e, is the density of the superfluid component. At 7' = 0, e, = Q = nm. In general, (14.3)
This function, defined by ( 1 2 . 2 6 ) , is plotted in Fig. 18. London]" has shown how one may construct from the ground state wave function, !Po(rl, r 2 . . . . . rn), a function Y
=
exp [i
v(r,)]Yo
(14.4)
1
for which u, is a slowly varying function of position. In a local region about r , this corresponds to a displacement of the distribution in k-space by Bk = grad cp(r), and a corresponding flow velocity
v,(r)
= 6m-l
grad
pl(r).
(14.5)
This expression implies potential flow, curl vs = 0. As pointed out by London la, current flowing around a superconducting ring can be described in this way. In this case, y may change by a multiple of 2n in going around the ring. An identical expression for en is obtained by following Landau's derivation of the two-fluid model as extended by Dingle121for Fermi systems. Thermal excitations may give rise to a net current relative to the ground state. First consider the ground state at rest (v, = 0 ) and suppose that there is a net momentum (mass flow) Jn =
Pf(P)
(14.6)
from excitations with a distribution function f(p). The latter may be determined so as to make the free energy F a minimum subject to a given J, by introducing a velocity v as a Lagrange multiplier (14.7)
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J . BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
9 14
This leads to 1
= 1
+ exp ([E(p)- v - pJ/rZ,T)
(14.8) *
When v is small, J, is proportional to v and the coefficient is defined as the normal density, en:
This expression is in agreement with that derived from (14.3). When there is flow in the ground state, the normal velocity is defined by v, = v, v and the total mass flow is given by
+
J
= ev,
+
env =
egvs
+
envn-
(14.10)
From (14.4) and the kinetic energy associated with v,, the total increase in free energy is found to be (14.11)
One can verify, as DinglelZ1has done, that the entropy flows with the normal component. One must be careful to distinguish between mass flow and flow of the number of excitations, N,,, = X,f(p). The latter move with the normal component so that the flux is given by v,N,,,, which in general differs from v,en/m. Superfluid flow with a velocity v, can be initiated by an electric field in the form of a pulse. All of the electrons are accelerated by the field with an increase in the common velocity of the pairs to v,. Scattering of thermal excitations tends to reduce the current so as to make the free energy a minimum, but such scattering does not change v,. According to (14. l l ) ,the best one can do is to make v, = 0, leaving a net flow J , = e,v,. This is the part which is determined directly from the London equation (14.2). Only a force which acts on all or a large part of the electrons can change v,. In the absence of such a force, the current persists indefinitely. Although second sound would be difficult to observe in a superconductor, it is of interest to estimate the velocity, c2. Formally one can use the same expression'22 as for He I1 (14.12) References p . 282
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where S is the entropy and C, the electronic specific heat per unit mass. For an order of magnitude estimate, we may use the Gorter-Casimir model for which @,/en = 1 - t4, and S and C, are proportional to T3 in the superconducting phase and S is equal to the value in the normal phase, yT,,at the transition temperature : (14.13)
If the free electron value is used for y , we find
for tin. This velocity is of the same order as for He 11. To observe second sound of frequency o,one should have ot > 1, (where t is the electron-lattice relaxation time) and to insure that the wavelength is greater than the coherence distance, c2 >WE,. To satisfy both requires that z > to/c2, or, for tin, z > 10V sec, which would be very difficult to realize in practice, since it corresponds to a m.f.p. of about 1 cm. 14.2. CRITICALCURRENTSIN THIN FILMS
The critical current in a bulk specimen is determined by the critical field. As the current is increased, the field at the surface of the specimen increases until it reaches the critical value, H,, at which the specimen reverts to normal or goes into an intermediate state. This is not true for flow in films so thin that the magnetic field can penetrate throughout. Effects of the magnetic field can be minimized by use of a “compensated” geometry. N. I. Ginzburg and A. I. S h a l n i k ~ v lhave ~~ measured the critical current in films of tin deposited on the outside of a cylinder. There are then no edges where abnormally large fields can occur progressively destroying superconductivity. These authors were particularly interested in the critical current near T,, where they find it varies as (T,- T)3/2,as predicted both by the GinzburgLandau phenomenological theory and by the microscopic theory. Earlier studies, in which fewer precautions were taken to eliminate extraneous effects, gave a variation as (T, - T)lj2or (T,- T)2/3. If the London theory applied, one could write for the increase in free energy for a current density, is:
AF References
p . 282
= +e,v,” = &l(T)if.
(14.14)
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J . BARDEEN A N D J . K. SCHRIEFFER
[CH. V I ,
§ 14
This assumes that A ( T ) is independent of j,, which may not be valid for very large currents. One expects the critical current to be that for which 5 F becomes equal to the energy difference between normal and superconducting phases, H;/Sar, or when (14.15) Near T,, both H , and A(T)plvary as ( T , - T ) ,so that j,, is expected to vary as ( T , - T)3I2.A critical current about 25% smaller is obtained if one takes into account the change in the distribution of quasiparticles and in the energy gap with increase in current, as shown bjRogers 124. The Pippard rather than the London limit applies to very thin films in which the m.f.p., I, is greatly reduced by scattering from the surface. One may express the acceleration of current in terms of the normal conductivity, cr,, and the kernel of the Pippard integral, J ( R ,T ) ,for the limit R + 0 , as follows: (14.16) so that the increase in free energy is
AF
=
$(A(T)t,/J(O, T)Z)j:.
(14.17)
The critical current density is reduced by scattering by a factor of about ( l / E o ) i; the temperature dependence is not affected very much. The critical current is decreased somewhat when changes in the gap with current are taken into account 124. The predicted magnitudes are of the same order as those found experimentally. 14.3. GINZBURG-LANDAU THEORY OF BOUNDARY EX’EHGIES
Some years ago, Ginzburg and Landau3j extended the London phenomenological theory to allow for a space-variation of the cffective concentration of superconducting electrons, n4. This made it possible to treat a number of problems, perhaps the most important of which is the boundary between normal and superconducting regions in the intermediate state. In this case no varies from zero in thc norma1 side to its equilibrium value appropriate to the given temperature in the superconducting side of the boundary, as illustrated in Fig. 38. At the same time, the magnetic field drops from the critical value, H,, IZLfcrcncps
9 . 282
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RECENT DEVELOPMEKTS I N SUPERCONDUCTIVITY
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in the normal side to zero in the superconducting side. The long range of coherence of the superconducting wave functions prevents ns from dropping abruptly at the boundary12j. Ginzburg and Landau assumed that n , ( r ) is proportional to the square of an effective wave function Y s ( r ) . The free energy density F(Y,, T ) depends on Y, (or n J , and the equilibrium value for constant Ysis that which makes F a minimum. They assumed further that in a magnetic field defined by a vector potential, A ( r ) , there is an extra term in the energy proportional to I - ih grad Us ( e A / c ) Y 8 12. To
+
Fig. 38. Variation of the magnetic field and the effective concentration of superelectrons across in normal-supcrconducting phase boundary.
determine the boundary energy, an&,one finds by a variational procedure the functions A ( r ) and Y s ( r which ) make the total integrated free energy a minimum. The parameters of the theory are determined completely from H,(T) and the penetration depth, 1(T).Fairly good agreement is found with values of a,, deduced from experiment, both in absolute value and temperature dependence. However, the theory suffers from the defect that it is based on the London theory rather than on the non-local theory now known to be valid. It is only for temperatures very close to T , that the non-local theory reduces to the London limit. Gor’liov1Z6has extended the microscopic theory so as to allow for a space-variation of the pairing. In the weak coupling approximation valid for most superconductors, the energy gap parameter, O ( r ) , may be regarded as a function of position. Gor’kov formulated the problem in terms of “thermal” Green’s functions, in which temperature is regarded as an imaginary time. While it is not hard to write down the differential equations for the Green’s functions, they are very difficult References p . 282
270
J . BARDEEN AND J . R . SCHRIEFFER
[CH.VI, $ 15
to solve except for limiting cases. Gor’kov carried through the calculation only for temperatures near T , where one expects the London limit to be valid, and found equations almost identical with those proposed by Ginzburg and Landau. The effective wave function, Fs(r) is found to be proportional to A ( r ) . The only difference is that the charge e is replaced by 2e, evidently representing the charge of a pair. Ginzburg127 has pointed out that this change improves agreement with experiment. In his generalized method of compensation, Bogoliubovu has given a different formulation which is also sufficiently general to allow for a space variation of pairing. A pair wave function, p(rl, r 2 ) , need not depend only on the difference, rl - r2, but may depend on rl and r , separately. This approach has not as yet been used to discuss boundary energies. 15. Thermal Conductivity 15.1. LATTICECOMPONENT
The thermal conductivity of superconductors is generally difficult to interpret theoretically because several mechanisms may be effective simultaneously. In the superconducting state, as in the normal state, there are two contributions to the heat current, one due to the conduction electrons and the other due to phononslZ8.The thermal conductivity, x , is given by the sum of the electronic and lattice thermal conductivities, x = x,
+ xg.
(15. 1)
In each case, there are several scattering mechanisms which limit the heat flux. In the normal state one has
B , -1 -- uT2 + Xen T
(15.2)
(15.3)
where the first and second terms in the expression represent the scattering of the electrons by phonons and by static imperfections respectively while the corresponding terms in K& represent the scattering of phonons by electrons and by the boundaries of the specimen. The same scattering mechanisms are effective in the superconducting HefErencm p . 282
CH. VI,
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
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state ; however, their temperature dependences are distinctly different from those in the normal state. For extremely low temperatures (<0.15 T , in general), the thermal conductivity of superconductors appears to follow the T3 dependence characteristic of boundary scattering of phonons. This effect has been well established. At higher temperatures, phonon-electron scattering limits the lattice thermal conductivity; however, this mechanism is difficult to isolate because in pure metals there is a large thermal conductivity associated with the electrons in the relevant temperature range. Since point defects scatter electrons more effectively than they do phonons, the electron component can be reduced by adding impurities. Scattering of phonons by electrons has been estimated by Hulm12Qfrom measurements on tin with mercury impurity and in tantalum. Laredo130 has studied this mechanism in tin with indium impurity and Sladek131 has made a similar study for indium alloyed with thallium. To correct for the electronic contribution to the thermal conductivity, it was assumed that the electronic term could be described by a function xes/xenrdetermined from data taken on relatively pure material
102
--
I
L.
'0
lheory After H u h After Loredo
.L-
02
Fig. 39. Ratio of the lattice thermal conductivity in t h e superconducting state x,, to that in the normal state xgnwhen electronic scattering is predominant. References p . 282
272
J . BARDEEIi A N D J . li. SCHRIEFFER
[CH. VI,
$ 15
for which the thermal conductivity is essentially purely electronic. Rickayzen, Tewordt, and one of the authors132 have derived an expression for the lattice thermal conductivity limited by electron scattering on the basis of the current theory. Their curve along with the experimental results of H u h , Laredo and Sladek are shown in Fig. 39. The experimental values are subject to some uncertainties, since they result from subtracting the large electronic term using the ratio xeS/xenvalid for much purer material. Also, while the normal state values of xgn determined by Sladek and Laredo agree fairly well with the T 2 law expected theoretically, Hulm’s data is more nearly approximated by a T3 law, indicating an admixture of another scattering mechanism. The theoretical curve may be questioned as well. The electron-phonon interactions which are involved here also enter the treatment of phonon scattering of electrons which is of importance in xe8. I n this case, where there is less uncertainty in the experimental situation, there is considerable disagreement between theory and experiment. 15.2. ELECTRONIC COMPONENT
Several authorslS2a, 133 have derived an expression for X , , / H , ~ on the basis of the microscopic theory, valid when impurity scattering is limiting. They find
where y
=
d ( T ) / k , T and (15.5)
Recent measurementsl34 have been made by Zavaritskii and by Satterthwaite on aluminum and zinc in which this mechanism dominates the thermal conductivity over a large range below T,. This condition was also satisfied by the purer tin and thallium specimens of Hulm, by the tin specimens of Zavaritskii, and to a lesser extent by the indium specimens of Hulm and Sladek, although these data are less realiable than in aluminum and zinc. The experimental data and the theoretical curve calculated with A (0) = 3.50 k,T, are shown in Fig. 40. The excellent agreement between theory and experiment is somewhat fortuitous since there is evidence that 24(0) = 3.30 k,?’, for aluminum, and this difference is sufficient to affect the theoretical curve. I \ ~ ~ C Y C ~ Ifi.C P28.2 S
CH. VI,
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
273
The electronic thermal conductivity limited by phonons is less well understood. Most of the data directly related to this mechanism have been taken on lead and mercury samples, which, as we mentioned earlier, exhibit anomolous behavior in their thermal and electromagnetic properties in the superconducting state. Guenault l35 has recently
Fig. 40. The ratio of the electronic thermal conductivity in the superconducting state, xes, to that in the normal state, x,,, when impurity scattering is predominant. The excellent agreement with the theoretical curve is somewhat fortuitous.
measured the thermal conductivity of two series of tin specimens of different purity, as a function of crystallographic orientation. He estimates the fraction of scattering at T , due t o phonons varies from 70 percent down to 15 percent for the two series. His results, taken above 0.7 T,, are shown in Fig. 41. There is some anisotropy between the two orientations shown. For each orientation a systematic change in xJx, is observed as the impurity concentration is varied. This change is consistent with assigning characteristic functions (i.e., functions of T/T,) to both the impurity and phonon scattering terms, as suggested by HulmlZ9.The characteristic fupction for impurity scattering with the thermal currents along the tetrad axis is below the theoretical curve discussed in the last subseLtcon while that for the current along the binary axis lies just above this curve., These results are not inconsistent with Hulm's earlier work on polycrystalline specimens. The corresponding phonon scattering functions show a positive References p . 28.2
274
J. BARDEEN AND J. R. SCHRIEFFER
[CH.
VI, 5 16
slope to T o ;however, slopes of only unity (binary axis) and two (tetrad axis) were observed, in marked contrast to earlier measurements on lead and mercury which show an initial slope of about five for the conductivity ratio, as shown in Fig. 42 for lead.
Fig. 41. Reduced thermal conductivity of several tin specimens of varying purity measured parallel and normal to the tetrad axis by Guenault. The fraction of scattering at Todue to phonons is estimated t o vary from 70% down to lK% for the two orientations.
In a theoretical of an expression for x,,/x,, where lattice scattering was the limiting mechanism, an approximate solution of the transport equation was obtained by use of the Kohler variational
8
-b-4-O TOK
Fig. 42. Thermal conductivity of lead showing a rapid drop below T, and a peak at w Tc/2 premmably when impurity scattering becomes predominant. (See H. M. Rosenberg, Phil. Trans. A 247,441 (196K).) References
p . 282
principk. ?‘tiis gives a lower hound on the thermal conductivity. If it is assumed that thr. nonequilibriurn p ar t o f thc quasi-particle distribu-
tion function is of the form
-a valuc of -0iB.is obtained. for the slope of x,,jx,, versus Xj.X, at.3’;. ‘The calculation gives ;i valut: ze.~x,, = 0.76 :it ‘f*/Tl,= 0.72. Wliilc the limiting slop(: is of the wrong sign to agrw with Giiennult’s measuremcnts, the point ;it T / T c ==,.0..72, .crilculatc!d on thr basis of an isotropic model, rcpresents a reasonablc avcrage of the data. The origin of tlic tliscrcpancy in t ~ i c~ 1 0 1 ;it 1 ~T~is not well indcrstootl at prcsont. In ortlcr to accoiint for thc 1:irgc: positive slopes for 1c;id and Incrcury intcrint!diittt’ to strong coupling c:~lc&hxis should h; carriccl out. .Kadanofi antl JIartin liavo found that thc data on tin can be fittctl approsim;itcly by a thcory in which it is nssumccl that the relaxition timc for clectrori-i)hoiion intcractions is the’samc in normal antl sul)(?rc(~ii(lucting stxtcs. IIowcvcr, this ;issuinption clocs not cl * grec with t h tlct;iild ~ c & d a t i o n s rlwcribcd abovc b:wd on thr I3olt.xmmn eq 11’ t ion. . ‘I‘hat the difficulty m : y bt! ;tssociatctl . . with a short rc:l;ixation time, z for clii;isi-l)articlrs is int1ic;ited by a c:ilculation of V. Z. Kresin l S i , who dctcirniinr.ri the t1icrm;il conductivity in thi?: limit rut ,l. He usc.ti a ini~t~iotl of T:;iniliu ybicli:.takcs i n L : acibunt t ~ i ciircGirsihidty of tilt. process of tlcforming thr latticc. He finds xl!r!zc,M 0.81 for A / / c F ~M T 0.25 antl xeS:’x,,, w (1.72 for A / / t , J ~O.a.,Tliese,.viduusi. ~ kad to ;in eve11 morct r;il)id tlrop n(t;ir Ti: than: is d.xervt?d.h)r lqicl :iud ,. mercurr. lhc assirniption fur.< 1. is an ‘cstrertie.one eseu;:for.thesr ni:itcri:ih with 1;irgc dcctrr,n-i)hc,riaii interactions.
<
16. Superconducting Alloys, and Compounds A number of inttxrc,sting c:gp:rinicnts have been carr,icciuiit rt:ccntly on both rn;ignetic ; m i non-mitgnytic supc:rconductiiig ;dlvys. Of particular' interest :ire.riiatc:ri;ds which :J[)1)wrto tx. simultancuusiy suJ>erconducting ;ind ferromLgnetic. \Vhile. tlic proscnt thcorctical : uncterstancling of the mqwrimvrits is incomplvtt., .some featurcs.of the data have bcen explaint:tl. IVc first discuss the non-magnctic alloys. Tlie cfffccts of 0.01 t o I .o ;itornic pt:rccnt of various nun-maglietic impurities on the criticcil trnil)cratnre o f tin, indiuni :md aluminum have b e c s n investigatvd b!. 1-ynton, Serin, and . %ucker18H. . a n d bv l<'~fer'~m \.'L p.
:'A?
276
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
16
Ghanip,;Lynton, and Serir113~.For sufficiently small concentrations of all solutes, they find that T , decreases linearly with increasing reciprocal electronic mean free path. For larger concentrations, the curves of T , versus concentration fall into two groups, according to the sign of the valence difference, LIZ, between the solute and solvent. The transition temperature has a sharp upward curvature for higher 0.01 0
- 001 rO o .o022
+
F
u
a
k
t
40.03 -0.04
t
v o X
a
Cd TI Go Sn Pb Bi
- 0.05 Fig.43. Lowering of T,of indium due to non-magnetic impurities plotted as a function of mean free path, from Serin et al.
valence solutes while for those with lower valence, T , tends to saturate for large concentration. Their data is shown in Fig. 43 for the two solvent groups in indium. The similarity of the results for the three quite different solvents, indium, aluminium and tin, suggests that the results are characteristic of non-magnetic superconducting alloys. Anderson140 has suggested that the anisotropy of the energy gap 'present in the pure metal may be washed out by the impurity scattering. This could cause the initial decrease in T,. He argues that ground state pairing should be formed by time reversed states vn,, and v&, defined in the presence of the scattering centers. So long as the electronic mean free path is large compared to to,the impurities will mix plane wave states only weakly as an electron moves through a coherence length. When I < to,pnacan no longer be approximated by a single plane wave and states from all parts of the Fermi surface are mixed with more or less equal weight, thereby removing the source of References
p. 282
CH. VI,
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161
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anisotropy in the energy gap. Once the mixing is complete, the effect on T , should saturate. It was the fact that T , does in fact saturate for solutes with negative A Z which lead Anderson to make this proposal. Abrahams and Weiss 141 have considered the effect of impurities on the density of states a t the Fermi surface and on the phonon-electron matrix elements, including the change in electronic screening. Their expression for the change in T , with impurity concentration involves the difference of large terms and has not been accurately evaluated a t this time. Their approach, however, looks like a promising one. Chanin, Lynton and Serin suggest that the more complicated behavior of T , beyond the linear region may be due to the change in concentration of electrons and hence a change in N ( 0 ) . Since Lynton, Serin and Zucker found that y (and therefore N ( 0 ) ) appeared.to.be increased by all solutes, the situation is not clear. Turning now to magnetic alloys, there has been strong interest recently in the relation between superconductivity and ferromagnetism142,143. Matthias, Compton, Suhl, and Corenzwit surprisingly found that magnetic transition metal impurities raise, rather than lower the superconducting transition temperature, T,, as shown in Fig. 44. They suggest the increase is larger than would correspond solely to the variation of electron concentration. Evidence for this is given by comparing solid solution involving the first row of transition elements with those of the corresponding second row of transition elements.
o a
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PERCENT Fe or Ru in Ti Fig. 44. Superconducting transition temperatures of iron or ruthenium solid solutions in titanium (see ref."Q). References p . 282
278
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3
4
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6
7
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PERCENT Gd Fig, 46. Superconducting and ferromagnetic transition temperatures of solid solutions of gadolinium in lanthanum measured by Matthias et at. (see Ref.1*8g),
The latter contribute the same number of electrons but possess no magnetic moment. When small concentrations of rare earth elements are added, it is found that T , is depressed rather than raised. This lowering is roughly a function of the spin, J , of the rare earth atom of the form J ( J 1) (g - l)a,where g is the spectroscopic splitting factor. For larger concentrations, the alloy becomes a dilute ferromagnet. For lanthanumgadolinium solid solutions, results are shown in Fig. 45. Subsequent
+
PERCENT Gd OIL Fig. 46. Superconducting and ferromagnetic transition temperatures of Yl,Gd,OS, solid solution measured by Matthias et al. Rej6rerlces
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results of Hein et al. indicated the possibility of simultaneous occurrence of superconductivity and ferromagnetism at 0.5" K for 1 percent concentration Gd. By forming mixed crystals of Y and GdOs, and of CeRu, and GdRu,, it was possible to obtain systems which were in fact simultaneously superconducting and ferromagnetic, as shown in Fig. 46. There is a region of composition where the Curie temperature 8 is above T,. It one of these specimens is cooled below T,, both remnant magnetization characteristic of ferromagnetism and diamagnetic screening currents characteristic of a perfect conductor are observed simultaneously. I t is possible that there are alternate superconducting and ferromagnetic domains, so that the two effects do not actually co-exist in the same region. It is very likely that in any case the superconducting domains are narrow, because the critical fields must be high. Similar effects are not observed in alloys with transition elements (incomplete d-shell). On the basis of these results, Matthias suggests that the magnetic moment in itself does not adversely effect superconductivity, but rather gives rise to exchange interactions with the s electrons which enhance superconductivity. The long range exchange interactions between f-shell impurities (rare earth atoms) via the conduction electrons leads to dilute ferromagnetism. As Herring and Suhll" have pointed out, the transition temperature in the alloys involving rare earth atoms is depressed by a weakening of the indirect exchange coupling in the superconducting state, since virtual transitions with small excitation energies are absent in this case. A second order perturbation calculation gives the correct spin dependence of the change in T,. On the other hand, for alloys involving transition metals it would appear that the long range indirect exchange potential is very weak. Instead of depressing T,, these magnetic ions may give another mechanism for attractive interactions between electrons to enhance superconductivity. Thus, the dominant effect of introducing rare earth impurities is to lower the free energy by a greater amount for the normal state than the superconducting state, thereby lowering T,.For transition metal alloys, on the other hand, the free energy of the superconducting state is lowered more than that of the normal state, perhaps because of an effective interaction between electrons via spin exchange forces. Further, with transition elements there appear to be no long range indirect exchange interactions between the impurities which lead to ferromagnetism in dilute systems. While these arguments are References p . 282
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plausible, the theory requires a more complete development before a firm understanding of the electron-impurity-electron versus the impurity-electron-impurity interactions is obtained. 17. Conclusions As we have seen, agreement between experiment and predictions of the microscopic theory based on the simplified model is in general much better than might have been expected. There is good experimental confirmation of (1) a temperature-dependent energy gap, (2) marked effects of coherence properties of the paired wave functions on matrix elements and transition probabilities, (3) a non-local theory of the electromagnetic properties and (4) a rapid transition from superconducting to normal behavior as &a becomes greater than the gap. The theory accounts for a second-order transition at T,, the Meissner effect and the metastability of persistent currents. Questions which have been raised about the gauge invariance of the theory have been fully resolved. Possible explanations have been given of the Knight shift and corresponding electron spin-paramagnetism. We have attempted to give a physical picture of a superconductor in terms of the ground state and its spectrum of elementary excitations, Persistent currents and the Meissner effect are related to the long range correlation of momenta of the paired electrons in the ground state. The explanation is along the lines proposed by F. and H. London, as modified by Pippard to take into account a coherence distance. While there is some similarity with Einstein-Bose condensation, there are marked differences as well so that the analogy is not actually a very close one. The most important excitations are the quasi-particles which obey Ferini statistics. Thermal excitations do not destroy the long range correlation of momenta until the critical temperature is reached. An extension of the theory is required to account in detail for the properties of superconductors with large electron-phonon interactions and low Debye temperatures for which the weak coupling approximation is unsatisfactory, This applies particularly to lead and mercury, whose properties depart significantly from a rough law of corresponding states valid for most other superconductors. It is possible that lack of agreement between theory and experiment when the electronic thermal conductivity is limited by phonon scattering is due in part to departures of actual superconductors from the weak coupling limit. In calculations which have been made to date, effects of band Reference5 p . 282
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structure have not been taken into account except in so far as they can be replaced by an isotropic empirical effective mass. It is known that anisotropic effects are significant in normal metals, and the same is true in superconductors. There is evidence from ultrasonic attenuation in tin single crystals that the energy gap is different in different directions. An anisotropic energy gap probably accounts for departures of observed specific heats from theoretical predictions at very low temperatures. One of the most important questions which is still open is the criterion for superconductivity, and in particular just how the repulsive Coulomb interactions counteract the attractive electron-phonon interactions to prevent superconductivity from occurring. It is very likely that effects of a finite life-time for quasi-particle excitations in the normal state must be considered. While the microscopic theory accounts in a qualitative way for the empirical rules of Matthias, a great deal remains to be done before the theory can be used to estimate even roughly the critical temperatures of actual materials and how they vary with composition. In this connection, the experiments of Matthias and co-workers on ferromagnetism and superconductivity raise many interesting questions. Can ferromagnetism and superconductivity co-exist in the same region of a crystal, or only in the same specimen ? Why is there such a close connection between superconducting compounds and those which become ferromagnetic when a small amount of a rare earth (with f-electrons) is added ? Are there other mechanisms for superconductivity than that based on the electron-phonon interaction ? Suhl (private communication) has suggested that there may be an effective interaction between valence electrons from the interaction between valence electrons and d-electrons. In this review, we have discussed the phenomenon of superconductivity only as it occurs in metals. Following the initial suggestion of Bohr, Mottelson, and Pines145,there has been considerable investigation of a similar pairing interaction in nuclei. There is evidence for a gap in the excitation spectra and other phenomena suggestive of superconductivity. There is a direct analogy between the Meissner effect and the fact that the moment of inertia of a rotating nucleus is considerably less than the rigid body value. The Coriolis force in the rotating frame is to first order equivalent to that produced by a magnetic field. A smaller-than-normal moment of inertia means that there References p. 282
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is a large particle current in this frame in a direction opposite to the rotation. This current corresponds' to the diamagnetic current which flows in the Meissner effect. The shell structure and the relatively small size and particle number in nuclei introduce complications not present in metals. We have mentioned the possibility of superfluid behavior in 3He, with pairing most likely to occur through d-type functions146. Nambu14' has proposed a superconductivity theory of elementary particles, in which the mass of nucleons is determined by the energy gap. Mesons are interpreted as low-lying collective or exciton-like states. Thus, superconductivity may be a common phenomenon in Fermi systems.
Acknowledgements The authors are indebted to a number of their associates for supplying information which has been helpful in writing this article. We should like to mention, particularly, the following group: C. M. Androes R. H. Hammond, W. D. Knight, A. Bardasis, M. A. Biondi, M. P. Garfunkel, M. and G. Dresselhaus, E. Erlbach, R. L. Garwin, M. P. Sarachik, D.Ginsberg, P. L. Richards, M. Tinkham, B. B. Goodman, A. M. Guenault, D. E. Mapother, R. W. Shaw, P. B. Miller, C. B. Satterthwaite, B. Serin, and T. Tsuneto. REFERENCES
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. V l
T. Tsuneto, Phys. Rev. 118, 1029 (1960). (a) W. P. Sarachik. R. S. Garwin and E. Erlbach, Phys. Rev. Lett. 4, 52 (1960; (b) A. L. Schawlow, Phys. Rev. 109, 1856 (L) (1958); (C) M. Peter, Phys. Rev. 109, 1857 (1958). H. WT.Lewis, Phys. Rev. 102, 1608 (1956). A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959). loo A. B. Pippard, Superconductivity Conference, Cambridge, 1969. l01 P. B. Miller, Phys. Rev. 113, 1209 (1959). lo* R. Kaplan, A. H. Nethercot Jr. and A. H. Boorse, Phys. Rev. 116, 270 (1959). The magnetic field dependence of the surface impedance of tin has been measured a t 1 kM,/s by M. Spiewak, Phys. Rev. 113, 1479 (1959). She finds [R,(H) - R,(0)] and [X,(H) - X,(O)] may be of either sign, depending upon the relative orientation of the static and r.f. fields and the temperature. Io3 M. S. Khaikin, J. Exptl. Theoret. Phys. (U.S.S.R.) 34(7), 1389 (961) (1958). loo M. Spiewak, Phys. Rev. Lett. 1, 4, 136 (1958), Phys. Rev. 113, 1479 (1959). 106 G. Dresselhaus and M. S. Dresselhaus, Phys. Rev. 118, 77 (1960). 106 (a) A. Bardasis and J.R. Schrieffer, to be published. In addition to particle-particle excitons existing for gL > 0, it is suggested that particle-hole excitons occur for gL < 0. (b) J. C. Fisher (private communication, 1958). 107 L. P. Gor’kov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34(7), 735(605) (1958). 108 D. Bohm and D. Pines, Phys. Rev. 92, 609 and 626 (1953) ; See also D. Pines, Solid State State Physics, Vol. 1, p. 368, (Academic Press Inc., New York, 1956). loo (a) M. Gell-Mann and K. A. Breuckner, Phys. Rev. 106, 364 (1957); (b) K. Sawada, Phys. Rev. 106, 372 (1957); (c) K. Sawada, K. A. Brueckner, N. Fukuda and R. Brout, Phys. Rev. 108, 507 (1957); (d) R. Brout, Phys. Rev. 108, 515 (1957); (e) G . Wentzel, Phys. Rev. 108, 1593 (1957). 110 (a) H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959): (b) J . Goldstone and K. Gottfried, Nuovo Cirnento 13, 849 (1959). 111 (a) J. J. Quinn and R. A. Ferrell, Phys. Rev. 112, 812 (1958); (b) A. B. Migdal, J. Exptl. Theoret. Phys. (U.S.S.R.) 34(7), 1438(936) (1958); (c) The importance of lifetime effects for superconductors was pointed out by J. Bardeen, Superconductivity Conference, Cambridge, 1959. 118 V. Heine and A. B. Pippard, Phil. Mag. Ser 8, 3, 1046 (1958). 118 R. A. Ferrell, Phys. Rev. Lett. 3, 282 (1959). 114 P. W. Anderson, Phys. Rev. Lett. 3, 325 (1959). 116 (a) See ref.112; (b) See ref. 114; (c) P. C.Martin and L. P. Kadanoff, Phys. Rev. Lett. 3,322 (1959). Added in pvoof: A. A. Abrikosov and S. P. Gor’kov, J. Exptl. Theoret. Phys. 39, 480 (1980) have shown that Yosida’s result is valid in the long wave limit in the presence of impurity scattering which involves no spin flip. 116 (a) J. R, Schrieffer, Phys. Rev. Lett. 3, 323 (1959): (b) A similar suggestion has been advanced by J. M. Blatt, Superconductivity Conference, Cambridge, 1959. 117 C. J . Gorter, ProgTess in Low Temperature Physics, Ed. C. J. Gorter, 1, ch. I, (North-Holland Publishing Co., Amsterdam, 1955). 118 L. D.Landau, J. Phys. (U.S.S.R.) 5, 71 (1951); Ibicl 11, 91 (1947). 110 J. Bardem, Phys. Rev. Lett. 1, 399 (1959). 1110 C. J. Gorter, Can. Journ. Phys. 34, 1334 (1956). 181 R. B. Dingle, Phil. Mag. 42, 1080 (1951). Is* K. R. Atkins, Liquid Helium, (Cambridge, 1969). 1*3 N. I. Ginzburg and A. I. Shal’nikov, J. Exptl. Theoret. Phys. (U.S.S.R.) 37(10), 399(285) (1960). 114 K.T. Rogers, thesis; Univ. of Illinois, 1960. 115 A. B. Pippard, Proc. Roy. SOC. (London) A 203, 210 (1950). 97
CH. VI] 1~
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
287
(a) L. P. Gor'kov, J. Exptl. Theoret. Phys. (U.S.S.R.) 36(9), 1918(1364) (1959); (b) L. P. Gor'kov, J. Exptl. Theoret. Phys. (U.S.S.R.) 37(10), 833(593) (1959) (1960). Gor'kov has recently extended the theory to supercooling by finding the temperature a t which the normal state becomes unstable against formation of regions with infinitesmally small values of A . See J. Exptl. Theoret. Phys. (U.S.S.R.) 37(10), 833(593) (1960).
197 1x1
V. L. Ginzburg, J. Exptl. Theoret. Phys. (U.S.S.R.) 36(9), 1930 (1372) (1959). (a) K. Mendelssohn, Progress in Low Temperature Physics, Ed. C. J. Gorter, 1,
p. 185 (North-Holland Publishing Co., Amsterdam, 1955); Can. J. Phys. 34, 1315 (1956); (b) P. G. Klemens, Handbuch der Physik, Vol. 14, p. 198 (Verlag-Springer, Berlin, 1956). 129 J. K.Hulm, Proc. Roy. SOC. (London) A 204, 98 (1950). 180 S. J. Laredo, Proc. Roy. SOC.(London) A 229, 473 (1955). 181 R. J. Sladek, Phys. Rev. 97, 902 (1955). 189 (a) J . Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113, 982 (1959); (b) see also ref."J; and B. T. Geilikman and V. Z. Kresin, Dokl. Akad. Nauk (U.S.S.R.) 123, 259 (1958); J. Exptl. Theoret. Phys. (U.S.S.R.) 36(9), 959(677) (1959). 188 B. T. Geilikman and V. A. Kresin, J. Exptl. Theoret. Phys. 36(9), 959(677) (1959). 184 (a) N.V. Zavaritskii, J. Exptl. Theoret. Phys. (U.S.S.R.) 33(6), 1085(837) (1958); (b) C. B. Satterthewaite, Superconductivity Conference, Cambridge, 1959. 185 A. M. Guenault, Superconductivity Conference, Cambridge, 1959; (private communication). la@ L. P. Kadanoff and P. C. Martin, to be published. 137 V. Z. Kresin, J. Exptl. Theoret. Phys. (U.S.S.R.) 36(9), 1947(1385) (1959). lSB E. A. Lynton, B. Serin and M. Zucker, J. Phys. Chem. Solids 3, 165 (1957). 189 G. Chanin, E. A. Lynton and B. Serin, Phys. Rev. 114, 719 (1959). 140 P. W. Anderson, J. Phys. Chem. Sol. 11, 26 (1959). Time reversal pairing was also used by Mattis and Bardeen, ref.8'. 1'1 E. Abrahams and P. R. Weiss, Phys. Rev. 111, 722 (1958). Also see K. Nakamura, Prog. Theor. Phys. 21,435 (1959), who treats the scattering potential as a perturbation. 1'9 (a) B. T. Mattias, H. Suhl and E. Corenzwit. Phys. Rev. Lett. 1, 92 and 449 (1958); (b) R. A. Hein, R. L. Falge, B. T. Matthias and C. Corenzwit, Phys. Rev. Lett. 2, 500 (1959); (c) H. Suhl, B. T. Matthias and E. Corenzwit and W. H. Zachariasen, Phys. Rev. 112, 89 (1958); (d) H. Suhl and B. T. Matthias. Phys. Rev. 114, 977 (1969); (e) A. I. Akhiezer and I. Ya. Pomeranchuk, J. Exptl. Theoret. Phys. 36(9), 859(605) (1959); (f) B. T. Matthias and H. Suhl, Phys. Rev. Lett. 4, 51 (1960); (9) B. Matthias, V. B. Compton, H. Suhl and E. Corenzwit, Phys. Rev. 115, 1597 (1959). 14a G. S. Anderson, S. Levgold and F. H. Spedding, Phys. Rev. 109, 243 (1958); Phys. Rev. Lett. 1, 322 (1958). 144 C. Herring and H. Suhl, Physica Suppl. 24, 184 (1958). 145 (a) D. Pines, Proc. Rehovoth Conf. Nucl. Structure, September, 1958; (b) A. Bohr. B. R. Mottelson and D. Pines, Phys. Rev. 110, 936 (1958) ; (c) B. R. Mottelson, The Many Body Problem, p. 283 (John Wiley and Sons Inc., New York, 1959) ; (d) S. T. Beliaev, The Many Body Problem, p. 343 (John Wiley and Sons Inc., New York, 1959) also Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959); (e) R. L. Mills, A. M. Sessler, S. A. Moszkowski and D. G. Shankland, Phys. Rev. Lett. 3, 381 (1959); (f) A. B. Migdal, J. Exptl. Theoret. Phys. (U.S.S.R.) 37(10), 249(176) (1960). 14@ (a) L. N. Cooper, R. L. Mills and A. M. Sessler, Phys. Rev. 114, 1377 (1959); (b) K. A. Brueckner, T. Soda, P. W. Anderson and P. Morel, Phys. Rev. 118,1442 (1960). 14' Y. Nambu, Proc. Midwest Theoret. Phys. Conf., Purdue Univ. April 1960. x48 K. Yosida, Phys. Rev. 110, 769 (1958).
CHAPTER V I I
ELECTRON RESONANCES IN METALS BY
M. YA. AZBEL’ AND I. M. LIFSHITZ INSTITUTE OF TECHNICAL PHYSICS OF THE UKRAINIAN ACADEMY OF SCIENCES, KHARKOV Translated from the Russian by W. P. A. HASS
KAMERLINGH ONNESLABORATORIUM, LEIDEN CONTENTS: Introduction, 288. - 1. Cyclotron resonance, 289. - 2. A special type of damping (non-skin) in metals, 312. - 3. Paramagnetic resonance, 319.
Introduction Electron resonance phenomena in metals are at present of increasing interest. The phenomenon of cyclotron resonance, predicted and observed during the last years, provides considerable possibilities for the study of the structure of, the energy spectrum of electrons in metals and, together with other methods, it can give us exhaustive information about the dynamic properties of the conduction electrons. On the other hand, paramagnetic resonance phenomena, apart from giving useful information concerning the nature of the interaction in elementary collision processes, are already of considerable interest because of their applications in the field of nuclear physics. The first two sections of this chapter contain an explanation of the nature of cyclotron resonance, its properties and the possibilities which this phenomenon offers for the study of the electronic structure of metals. Special attention is paid to the quite unusual picture of non-skin damping of a high-frequency field at resonance. This phenomenon, recently predicted by one of the authorslo, has not yet been studied experimentally. The third section is devoted to paramagnetic resonance in metals, its theory, and the analysis of some results obtained in the study of this phenomenon. References p. 330
CH. VII,
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ELECTRON RESONANCES IN METALS
289
1. Cyclotron Resonance
1 . 1 . PHYSICAL PICTURE OF CYCLOTRON RESONANCE
It is well known that in the presence of a uniform magnetic field H a free electron moves in a helical path the axis of which is in the direction of the field. In a plane perpendicular to the magnetic field its motion consists of a uniform rotation around a circle with frequewy D = eH/mc which is independent of both the magnitude and the direction of the electron velocity (e and m are the charge and the mass of the electron). The non-dependence of the frequency of rotation on the velocity of the electron is conserved in the somewhat more general case of a quadratic dispersion law, when the surface of constant energy in momentum space is an ellipsoid. This is exactly the case in semiconductors where the band is either almost empty or almost filled. It is natural to anticipate that in a semiconductor placed in a constant magnetic field H and in a circularly polarized high-frequency field of frequency w = Q perpendicular to H, resonance will occur. This phenomenon was predicted by Dorfmanl in 1951 and independently by Dingle2 and later on it was observed many times in semiconductors, where it played a considerable role in the establishment of the dispersion law for them (see e.g. 3). When considering the resonance of Dorfman and Dingle it is very essential that for a very low concentration of the carriers the electric field in semiconductors can be considered as being uniform (further on it will be shown that this is not always the case even for semiconductors). The arguments given above are not applicable to metals for two reasons. Firstly, the high concentration of carriers in metals leads to a very inhomogeneous electromagnetic field over a distance Y equal to the radius of an orbit (so-called highly anomalous skin effect). In order to obtain a substantial resonance, it is decidedly necessary that the electron succeed in making at least a few revolutions in its orbit between two collisions (with impurities, lattice deformations, phonons, other electrons), i.e. the inequalities wt > 1, I > Y should be fulfilled where I is the mean free path and t the mean free path time. In this case the skin depth 6, does not depend on t and is of the order c/w,, where the plasma frequency wo of the electrons is determined by the electron concentration n : wocc ( m + / l m ) * . For the usual electron h'efcwrces
p. 330
290
M . y.4. AZBEL’ AND I. M . LIFSHITZ
[CH. V I I ,
9
1
density n M ~ m and - ~ velocity v m lO7-lO* cmjs for metals the skin depth is of the order 10-6-10-8 cm, whereas the radius of the orbit Y M v / w is of the order cm for centimetre waves which correspond to a resonance field of the order lo* Oe. The considerable inhomogeneity of the high frequency field coinpletely changes the mechanism by which the electron acquires energy from the field, and makes this mechanism extremely sensitive to the orientation of the constant magnetic field with respect to the metal surface. When the direction of the magnetic field makes an angle
a
b
Fig. 1. Electron trajectories near the metal surface. a) H makes an angle with thc metal surface; b) H is parallel t o the metal surface.
with the metal surface practically all electrons (with the exception of a small fraction-of the order &/r-which have a velocity m wS, along the magnetic field direction) leave the skin depth during the first revolution, accelerated over a small arc only (see Fig. la) which deviates little from a straight line. It is obvious that, in general, the presence of the magnetic field in this case leaves the surface impedance of the metal practically unaltered (this fact is proved in5$6, 441. When the field is strictly parallel to the metal surface there are electrons (see Fig. lb) which do not collide with the surface and which return to the skin layer after each revolution, the layer playing exactly the same role as the gap in a cyclotron. If synchronism occurs and the frequency w is equal to or a multiple of the frequency 52 (the time for one revolution equals or is a multiple of the period of the high frequency field), the electrons in the skin are accelerated 2/2nr times, producing a specific “cyclotron” resonance. This new form of resonance was predicted in 1956 by Azbel’ and Kaner*. We notice that for a quadratic dispersion law the cyclotron resonance is already inferred in formulae (1-2) ofs, where the current density i for 527 1 is proportional to y/sinh(ny) and y = r(1 iwz)/Z = i(m/Q) + l/(Qr), hence j a l/sinn(w/52 - if&) and for w = qD resonance occurs (q = 1, 2, 3 . . .). ’9
+
References p . 330
>
CH. VII,
9
11
ELECTRON RESONANCES I N METALS
291
It should be noted that10 cyclotron resonance can apparently be observed not only in “good” metals but also in “bad“ metals with a small number of carriers such as bismuth, arsenic, antimony, graphite etc. and in doped semiconductors with carrier concentrations of the order 1017 cm-3. This is also related to the fact that at resonance the conductivity sharply increases by a factor wz, causing the skin depth to decrease by a factor d/Ot so that it can be smaller than the radius of the larmor orbit for a given direction of the magnetic field. I t may be that this is responsible for the divergence between experiments and the ordinary theory mentioned by a number of authors (see e.g.ll). The second cause of complication in cyclotron resonance in metals is also the result of their basic property: the high carrier concentration which leads to a partially filled conduction band (an exception is bismuth which will be considered separately). Since the electrons taking part in the conduction are not concentrated at the boundary of the band there are no reasons to expect that their dispersion law, i.e. the relation between energy E and quasi-momentum p, would be even approximately quadratic. In fact all fermi surfaces ~ ( p = ) which are known at present have a very complicated structure (see e.g.ll) ; an exception is bismuth with an anomalously small ?E m 10ls ~ r n - ~ . For a non-quadratic dispersion law the electron trajectory in momentum space is neither a circle nor an ellipse and is described by equations which correspond to conservation of both the energy of the electron in the constant magnetic field and the z-component of its quasimomentum: ~ ( p = ) E, $, = const. These relations can be easily obtained from the equation of motion for the quasi-particle with E = ~ ( p ) in a constant magnetic field4: fi = (e/c)v A H, v = aslap. From the equation of motion it also follows that
i.e. the projection of the trajectory in coordinate space on the plane z = const., perpendicular to the magnetic field, is similar to the trajectory in momentum space, turned through -90” and multiplied by a
factor c/eH 38. Electrons which move along non-closed orbits inside the metal have an infinitely long period of rotation4*,4b, and do not return to the skin layer. These electrons obviously do not take part in resonance (for more details see 9 1.4). Therefore, we shall only consider closed References p. 330
292
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
31
orbits, which practically always exist for any direction of the magnetic field, even if the fermi surface is not closed (cf. e.g. the open boundary fermi surface for copper, established by PippardlZ). Electrons moving along closed orbits describe a periodic motion, however, with frequencies depending on 9,: Q = eH/m*c. The effective mass is determined by the variation with the energy E of the area S(E,pz) of the cross section of the fermi surface m* = (llZn)(i?S/&), m* depends, of course, also on the energy, but this does not need to be considered since in metals only electrons with energies close to the boundary energy E , are of importance. In the case of a quadratic dispersion law ~ ( p = ) +Epttp6pt the effective mass, as already discussed, depends neither on E nor on p , and equals4 4,4a94b,:
m* =
(PyyPza
- Pyz)-'.
Therefore only electrons of isolated cross-sections, tor which
Q(pz) = w , can be exactly in resonance. This is why even the possibility of the resonance is not obvious for an arbitrary dispersion law and the establishment of the conditions under which resonance can exist demands more rigorous consideration. However, it can be understood even from general considerations which electrons are at a relatively favourable position. For a small variation of p , near Po the frequency Q, in general, vanes linearly with p , - p,. However, if Po corresponds to an extremum of the electron frequency (Q'(p,) = 0 ) , the frequency Q varies considerably more slowly, with the square of p, - p,,. Therefore near such a cross-section a considerably greater number of electrons has a frequency of rotation near Q(p,) and hence at this frequency it is natural to expect resonance. In the following it will be shown that resonance occurs at these frequencies which are extreme with respect to a variation of 9, and also at frequencies corresponding to the supporting points of the fermi surface where the electron velocity is parallel to the magnetic field (see Fig. 7 points A and B). Soon after its theoretical prediction cyclotron resonance was observed experimentally by Fawcett l3 in tin and very weakly in copper. After that cyclotron resonance was observed not only in tin1&" and ~ o p p e r l ~but - ~ ~also , in lead20, indium21, zinczz, aluminiumz3~ 24 and bismuthzsz8. The effective masses in various directions have been determined for these metals. h'eferetrces
p . 330
CH. VII,
11
293
ELECTRON RESONANCES I N METALS
Fig. 2. Cyclotron resonance in tin16. d R / d H (arbitrary units) as a function of the magnetic field. w = 2n 24 kMHz, T = 4 OK.
Characteristic curves of cyclotron resonance are depicted in Fig. 2, 3 and 4. The multiplicity of the resonance frequencies is apparent. The sensitivity of the resonance to the angle between the constant magnetic field and the metal surface has been corroborated by experiment ; the anisotropy of the effective masses1e, lg9etc. has been studied; the data for copperlgand aluminiuma3.24 are in agreement with the shape of the fermi surface established on the basis of other effects. In 34a (see also 34) a comparison of theory with experiment is made. 27p
I
0
I
I
I
I
I
.
4
I
a
I
I
I
I
I
I
1
12
HI KILO-OERSTEDS
Fig. 3. Cyclotrorrresonance in copper lo. d R / d H (arbitrary units) as a function of the magnetic field. w = 2n 24 kMHz,
m $ = 1.24 m,,
T = 4.2 O K mj. = 1.29 m,;
m, is the free-electron mass. Refevences p . 330
294
[CH. VII, $ 1
M. YA. AZBEL’ AND I. M. LIFSHITZ
- 20 -30
I
I
I
I
I
I
0
1
2
3
4
5
I& VJ-
H
d lnX/dH
=
Fig. 4. Cyclotron resonance in tin1?. - dFfdH (arbitrary units) as a function of the magnetic field (lower curve is a repeated record) w = 2n 9.4 kMHz T = 2.4 OK.
1 . 2 . THEORY OF
RESONANCE The fact (which can be proved rigorously) noticed in $ 1 . 1 , that the electron is accelerated by the electric field only during the small part of its orbit in the skin layer, is very important for the compounding of a theory of cyclotron resonance. The electron velocity perpendicular to the metal surface is obviously almost zero in this part of the orbit. Therefore, the energy acquired by the electron is almost independent of the electric field E , perpendicular to the metal surface as the field E, performs almost no work on the electron. [This argument has, of course, a limited validity. At resonance the field E, increases in proportion to the “resonant” parameter dot and the relative contribution of E , is proportional to the parameter of “anomaly” d o / ~ , hence it is always legitimate to neglect E , out of resonance and a t resonance for W Z (v/d0)2. [In sections 1 and 2 this inequality is always assumed to be satisfied, if not otherwise stated. As to the consequences of this inequality with respect to the temperature and frequency see § 1 . 3 ).] This means that both the acceleration of the electrons and the current produced by them (parallel to the metal surface) are, in prinCYCLOTRON
<
Rejereitcrs
p . 330
CH. VII,
$ 13
ELECTRON RESONANCES I N METALS
295
ciple, independent of the field E,, which consequently can be ignored. This enables us to use the Maxwell equations only for the field parallel to the metal surface, Ell exp (iwt): €;{
4niw
= -j
.
C2
The fact that only electrons with v,, M 0 (and near resonance also with M Qe,, for a non-quadratic dispersion law) are accelerated and hence give rise to the unbalance in the distribution function, leads, at collisions, to a much larger transition probability from this characteristic state to all other states, than for the reverse transitions to this characteristic state, Hence, it follows that under conditions of anomalous skin-effect, we can introduce, in a consistent way, the time of free flight ~ ( p )at any temperature. This leads to a sharp decrease of the electron-phonon free path time repby a factor (0 is the Debye temperature)’ and thus l/repM (he/&)(T/t1)3,and the electron-phonon collisions are essential even at very low temperatures T (so, for T M 5 O K , 8 M 100 O K the high-frequency repm leg s whereas the static tepM 5 x lO-’s). The reason for the sharp decrease of repphysically can be easily understood. In the static case 1/repis proportional to the number of phonons at low temperatures, i.e. ( T / I and ~ ) ~also t o (T/8)2because of the smallness of the displacements resulting from individual collisions with the low temperature phonons. In the anomalous skin-effect even a small displacement of an electron is essential and brings it out of the skin layer (and for D # const. also out of the “resonance” cross-section) ; therefore the factor (T/8)2disappears in the formula for l/rep. Apparently for fiw > kT in all formulae 6iolk enters instead of T . Now making use of the possibility of introducing the time of free flight we write down the current density j. We employ here a method developed in lo which is in principle the same as the method of CharnberszO. For simplicity we take ~ ( p = ) const. and we get:
D
296
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH.
VII,
5
1
where d E is the energy acquired by the electron in the electromagnetic field, reaching the point y at a time t, (we recall that the positive y-direction is the one along the inward directed normal); no(&)- tz0(e LIE) is the variation of the distribution function in which only the energy gain of the electrons is involved since the number of electrons is constant; t is the time in which the electron rotates in its orbit which we measure, to be exact, from the centre of the orbit (Fig. 5 ) . determine the position of the orbit ($J on The quantities E , the fermi surface ~ ( p = ) E ; the time t gives the position of the centre of the orbit y - r(t); integration over t from 0 to 2n/Q = T o corresponds to integration over the centres of all orbits going through the point y ;
+
Such a choice of the variables enables us near resonance to take the boundary condition at once into account. The reflection of electrons from the metal surface appears always to be diffuse or almost diffuse (see e.g. 30*31), however, even the much weaker assumption that the reflection is not too much of a mirror type is sufficient. In this case the electrons colliding with the surface do not make a periodic motion and hence do not take part in resonance. Therefore, it is sufficient to consider in the formula for the current density the deviation from the equilibrium distribution function for only those electrons which do not collide with the surface, i.e. for which y - r(t) - r > 0. This can be done in a formal way by substituting s(y - r(t) - r) into Eq. (1) where s(W) = 1 for W > 0 and s(W) = 0 for W < 0. Now the determination of LIE remains. Let t , be the instant corresponding to the time t of revolution in the orbit (i.e. at the time t, - t the electron was at the centre of the orbit). Then, with a probability given by the free path time z, the electron can obtain energy from t' to t i.e. during the time t - t ' ; t changes from t to - 00. During the time from t' until t' dt' the electron gains the energy ev(t')E(t') dt' where the field strength E should be taken at the point y - r(t) r(t') at a time t, - (t - t') (see Fig. 5 ) ;
+
E(t') exp (id,) = E(y - r(t)
hence References
p . 330
+
+ r(t'))exp [iwt, - iw(t - t')]
9 13
CH. VII,
ELECTRON RESONANCES IN METALS t
ds =
dt’ exp
[
-
(t - ‘‘I t
x exp [id,
297
] ev(t’) E(y - r(t) + r(t‘))
- iw(t
- t’)]
(2)
for simplicity t is considered to be constant. I
y
=
Fig. 5. The electron trajectory in
O
p,
=
const. plane.
Substituting Eq. (2) into Eq. (l),taking the boundary condition into account as described before and changing the limits of integration in Eq. (2) from - 00 and t to t - T o and t, we get a formula for the current density :
I=
---S,dcj[l 2e2 eH c
A3
x
ST’
an,
v ( t ) exp
0
(-
- e ~ p 2nio ( - T - ~ iwt
t
s(y - r(t) - r) dt
(3)
This formula was obtained in by immediate solution of the kinetic equation for the somewhat more general case t = t(p) (although the generalization of Eq. (3) to this case presents no difficulties-see also29). The kinetic equation
- en’(an,/ a&)exp (id,) in the variables E , t, p,, y taking for n = no(&) into account the possibility of introducing the relaxation time has the form : Refereizces p . 330
298
M. YA. AZBEL' AND I. M. LIFSHITZ
v,,a
(i m + - +ay- + -
a l t
at
[CH. VII,
1
n'=v.E. )
The boundary condition for diffuse reflection corresponds to n' I y - 0 , f'O - 0 72' I y - m , vff < o - 0. We return to Eq. (3). As is shown in the beginning of this section, in the formula for ill parallel to the metal surface, which is the only case of interest to us, E,, can be set equal to zero and v * E = v,Ep where the repetition of the greek subscript means summation over x and z. From formula (3) both the existence of cyclotron resonance at frequency 9 = w and frequencies Q = i w , gw, . . ., and the existence of the difference in resonance depth for a quadratic and a non-quadratic dispersion law can be seen. For a quadratic dispersion law Q does not depend on p,, as is shown before, and therefore j, cc 1 - exp (-2nim/Q - 2n/Qt), hence for z -+ 00 the current density at resonance (for w = SZ, 2Q, . . .) tends to infinity in proportion to t / T o . For a non-quadratic dispersion law it can be easily seen that the coincidence of w with one of the frequencies SZ which are different from the extreme value, leads to no characteristic property. When w equals or is a multiple of one of the extreme values of Q (w = qQo, q = 1, 2, . . .) resonance appears, however, in contrast to the case of a quadratic dispersion law, j cc dt/TOi.e. the height of the resonance is considerably smaller. We shall now proceed at once to calculate the surface impedance tensor Zap,given by the relation: ic2 Ea(O) = zu,~,= ( ~ ) z " ~ E ; (; o ) z a p
=z
Rap
t ix,,
(4)
where I, is the total current in the metal
I,
= 0
i A Y ) dY -
For the calculation of Z,, it is necessary to solve the Maxwell equations for El, together with Eq. (3). I t is convenient to extend the field E, to the region y < 0, outside the metal, in such a way that Ea is an even function of y : Ea(- y ) = E,(y). Since the fermi surface is centro-symmetric, a field that is symmetric Kcferences p . 330
CH. V I I ,
§ 11
ELECTROX RESONANCES I N METALS
299
in y is a solution of the equations if we replace s(y - r(t) - r) in Eq. (3) by the equal function for y > 0 ; s(1y - r(t) 1 - r ) . We shall simplify formula (3). Firstly, the largest parameter with tend to infinity in the inner integrals. the dimension of a time---can Secondly, we notice that the deviation of the factor s( I y - r(t) 1 - r ) from unity (which takes into account the “non-effectiveness’’ of the electrons colliding with the surface) cannot essentially affect the results, and leads only to a numerical factor before the impedance, of the order of magnitude of unity. This was proved in starting immediately from the equations; it was shown that this constant factor was almost unity (see also 44). The physical reasonfofor the smal leffect of the boundary condition under anomalous skin effect condition^^^,^^^^^,^ is the fact that at any rate the essential part is played by the “skimming” electrons in
’,
Fig.6. Electron trajectories near the metal surface.
<
the field which disappears at a depth 6, I , 1 even in the absence of a boundary (they traverse a maximum distance in the skin-in our case of the order 45, corresponding to the trajectories 1, 2 and 3 but not 4 in Fig. 6). The effect of the boundary consists in cutting off the trajectories of type 1 and 3, i.e. it diminishes somewhat the effective conductivity. This decrease is even more insignificant (the constant mentioned above is almost unity) since the impedance is propartional to the cube root of the conductivity G, because of which a change of the latter by a factor two will produce a change in the impedance by a factor of about Q. The relation 2 cc d is easily obtained, for example, from Pippard’s “ineffectiveness concept”, and it follows immediately even from a consideration of the dimensions when the relation between current and field has essentially an integral character. As the consideration of the boundary leads t o considerable mathematical difficulties and has little influence on the result we shall replace s(( 9’ - r(t) 1 - r) in Eq. (3) by unity. This corresponds formally to an infinite piece of metal in which the electric field is an even References p . 330
300
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
$1
function of y. Finally, we replace ano/ in Eq. (3) by a delta function S ( E - .so). Substituting j , into the Maxwell equations, we get: 4niw E,=--c2
x
To
v,(t)
2niw
eH h3 c
2e2
exp (- iwt) dt
0
1
To
v,(t')E,(y
- r(t)
+ r(t')) exp (iwt') dt' .
0
Formula (5) is derived under the condition that t ( p ) = const. In the general case8, it remains of the same form except that 1/r is now
To JTodt/r(p) 0 , the average number of collisions over a period. This result is quite reasonable since the orbits are characterized in momentum space only by the integrals of motion E and p,, and with respect to t (corresponding to p,) degeneracy occurs, also leading to an average. In order to simplify the treatment we shall consider in advance that t in Eq. (5) is constant. Eq. (5) can be solved easily by making a Fourier transform which immediately yields a relation between E, (0) and E,'(O) and which also gives the impedance (see Eq. (4)). Close to resonance the surface impedance tensor can always be reduced to principal axes, where the difference between quadratic and non-quadratic dispersion laws manifest itself also. We consider both cases separately : a) Quadratic dispersion law. Using the inequality r / d , 1, which is always nicely fulfilled in metals as is shown in $ 1 . 1 , we may show that, first, only v,, v o is essential, which has already been assumed, and, secondly, the tensor Z,, can be reduced to principal axes together with the real tensor B,,
>
<
V
n = - = (sin 8 cos rp, sin 8 sin rp, cos 8) V
(6)
where K is the gaussian curvature at the point E = c0, 9, 8 = in. Both principal values of the surface impedance tensor have resonant properties, and both R, and X , have a minimum at resonance. Since References p . 330
CH. VII,
3
11
30 1
ELECTRON RESONANCES IN METALS
in measurements a given field strength is used, this refers particularly to a minimum of absorption. Omitting the unwieldy calculations we give the final result :
b) Non-quadratic disfiersion law. In this case both inequalities 07 1 and r/d, 1 lead to the fact, which has also been pointed out before on the basis of physical arguments, that only the electrons with v,, m 0 and fix m p , (Q(p,) = Qext) are essential, i.e. the electrons moving almost parallel to the metal surface, and rotating in their orbit with a frequency close to the extreme value. The impedance can be reduced to principal axes together with the tensor A,, and the principal values of the impedance 2, can be expressed in terms of the principal values A , of the tensor A,, by the formula :
>
A,,
>
16e2
=-
3h3
9 ((3) f," [ ( 2nio 1 - e x p ----
i=l
B1K
Q = pi
+
1"
52
Qt
e-+x
The tensor aa8 rr 1 is of a rather complicated form7, which is of little interest to us; the reason for the maintenance of the non-resonant
JI 4 Fig.7. Closed fermi surface. Refevemes p. 330
302
M. YA. AZBEL’ AND I . M . LIFSHITZ
[CH.
VII,
9
1
quantity a,b becomes clear in § 1 . 3 ; the variables are the same as in Eq. (6) and the integration extends over the angles corresponding to the “girdle” u, = 0 on the fermi surface (see Fig. 7) ; ql,y 2 . . . rppl are points were D has a given extreme value with respect to vzriation of y ; when taking the cube root of Eqs. (8) and (7) the root corresponding to K, > 0 should be chosen (such a root always exists). IMPEDAXCE 1.3. ANALYSISOF THE SURFACE
An analysis of the surface impedance near resonance is most conveniently made for either dispersion law separately. In experiments it is often the derivatives of R and X (e.g. dRldH, dlnX/dH) instead of R and X that are measured and thus we shall also investigate the derivatives of the impedance, according to &. a) Quadratic disfiersion law. In this case the entire resonant curve can be constructed for any SL! (for values close to the resonance values as well as for those far from them) ; the shape of the curves R(H)/R(O), X ( H ) / X ( O )and X/(RdT) for w z = 1, 10, 50 is depicted in Fig. 8. The small maxima of R and X for w m (q 4)Q are not related to resonance and for Qt -+ 03 the value of the impedance at these points tends to a constant value different from zero. It is important to note that for finite t the depth of the resonance minimum and the frequency shift of the minimum relative to o/q are much different for R and X .
+
The frequency shifts for R and X have different causes. The shift for X is simply related to the fact that a small increase of the magnetic field, which hardly changes the resonance conditions, leads to an advantageous increase in the number of revolutions made by the electrons between collisions. To understand the frequency shift for R we consider how the variation with depth of the phase of the electric field manifests itself. A change in phase destroys the resonance synchronism, diminishes the energy acquired by the electron and thus impairs the resonance. As formula (7)and the graph of X / ( R d 3 )as a function of the magnetic field show, even a small variation of the magnetic field leads to X R,
>
References p . 330
CH. VII,
5
11
ELECTRON RESONANCES IN METALS
303
i.e. the phase of the attenuated field is alsmost unchanged over a depth 6. This proves to be favourable in spite of the fact that after about 1 cu - qQEa I t revolutions the electron appears near the surface when the phase of the field has been changed considerably. We proceed to analyze dRldH and dX/dH. At first sight it may look
XrH) R(H)33
4
3
2
1
Fig. 8. Theoretical resonance curves for WT = 1, 10 and 60 for a quadratic dispersion law. a) R(H)/R(O)vs m/Q;b) X ( H ) /X ( O )vs o/Q; C) X ( H ) / R ( H ~) % v w/Q. S
as if at resonance, where R and X are minimal, their derivatives are equal to zero. However, in fact, resonance does not correspond to zero values of dR/dH and dX/dH but to their maximum values. This is caused by the fact that for cot = 00 the functions R(H) and X(H) do not have a minimum at resonance, but a smallest value (equal to zero) which References p. 330
304
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
5
1
corresponds to a kink in these functions. Thus for w t = 00 we find from formula (7) that in the region where H < HEa, R(H) cc (HE8- H ) t
and dR/dH+O
when approaching resonance from this region, and in the region where H >HgS, R(H) cc (H - HPR,)aand dR/dH --f 00 when approaching resonance from this side (see Fig, 9). The functions X ( H ) and dX/dH behave in an analogous way. Therefore (dZ/dH),,, = Z(0) (q2wz)t/Hie,and for t -+ 00 (dZ/dH),,, -+ 00 (and not to zero) and (w - qQres)/wm (wt)-l; HLa is the magnetic
--H ffres
Fig. 9. Behaviour of R and dRldH as a function of H near resonance for a quadratic
dispersion law, and for a non-quadratic dispersion law for m&.
field corresponding to resonance on the principal harmonic (q = 1). The relative heights of the maxima of dR/dH and dX/dH are significantly larger than the reciprocal value of the relative depth of the minimum for X(H), and the resonance frequency shift is the same as for X and much smaller than for R. b) Non-quadratic dispersion law. In formula (8) for a non-quadratic dispersion law Q(y) enters instead of Q(P,) considered until now, Since the function p,(q) obviously has extreme values only at the points of support of the surface (see Fig. 7) and dQ/dq = (dQ/dp,) (dp,/dy), Q(q) must have extreme values first at the place where Q(p,)]Ealso I\'efcrciiccs
p. 850
13
CH. VII,
ELECTRON RESONANCES I N METALS
305
has them, and secondly at the extreme values of p,(v) i.e. at the elliptical points of support of the surface (which we discussed without proof in § 1.1) (at the hyperbolic points of support m* = 00, 52 = 0 and resonance is impossible). One may think that, since at the elliptical points of support I = 0 , the basic condition for cyclotron resonance, r/6, 1, is not satisfied. However, this condition ( 1 / 6 ~ l), in fact, places no severe restriction on the frequency, as near the point of support I M ( c / e H ) A g , / d R (Fig. 7 ) , and dg, w (ox)-*, and the condition r M ~,(wt)-*7 6, is needed. In fact if we take into account that z 2 rep2 (fi/kO) ( k O / f i ~ ) ~ ; I rn v / w , 6, m c/wo m &c/E,, and EOv/c rn ke, we can easily see that 1/6, 5 d w y Thus, resonance occurs on the central section (where Q(p,) always has an extremum), a t frequencies corresponding to the elliptical points of support (where, as can be shown7, m* = ( v d K ) - l , v and K are the velocity and the Gaussian curvature in the point of support) and at the non-central values of 52 which are extreme with respect to variation of p,. One should notice also that resonance on the central cross-section and on the points of support is definitely different from resonance at the extreme values of $2. In the first case, because of the central symmetry of the fermi-surface, the resonant term in A,, is proportional to the tensor n,(vl)np(yl) one of the principal values of which is equal to zero and the other equals unity. Consequently, only one of the principal values of the impedance has a resonant character. Since at resonance R and X have a minimum, instead of a maximum, for arbitrary polarization of the incident wave on the metal, the impedance will be determined in principle by the large non-resonant principal term, and the resonant part will represent only a small increase in the impedance. A substantial resonance will occur only in the case where the incident wave is polarized along the velocity v o at the point E = e0, 8 = in, v = y1 [for a point of support this direction coincides with the magnetic field direction (Fig. 7 ) ] . The current density corresponding to the electric field perpendicular to vo has a non-resonant character because in this case the electric field in the skin is almost perpendicular to the velocity and correspondingly performs almost no work. The derivative of the impedance with respect to the magnetic field is, in principle, determined by functions strongly depending on H and 4b960
>
>
RcJereiicts
p . 330
306
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
5
1
has therefore a resonant character for all directions of E with the exception of the direction perpendicular to yo. However, in agreement with what has been said before, the range of angles in which a strong anisotropy must be observed for R and X as well as for dR/dH and dX/dH, is very large. Thus, for absorption P = R,,(Ep)2 Rpp(EF)2, the range of angles is given by the quantity (R,,,/R(O))*,which is usually not too much different from unity (for the value of Rre8/R(0) see below). I n the case of resonance on the non-central cross-sections, there are at least two centrally symmetric cross-sections on which resonance occurs, and two points (C and D) which contribute appreciably to the current density. The velocities at the points C and D are, in general, not parallel, and for any direction of the electric field the current density (and thus also both principal values of the impedance) has a resonant character. Mathematically this is related to the fact that, in general, none of the principal values of the tensor n,(q~,)n~(p,)n a ( ~ , ) n p ( yis2 ) equal to zero. We shall further analyze only the resonance values of 2, (see formula (8)).The formulae for the resonance values of the impedance have a different form depending on whether the given section has a minimum or a maximum for SZ = eH/m*c. In the formulae for X , this affects the numerical constants only; in both cases XF/X,(O) rn ( q2/wt) *; 1 - qSZ&/o w (or)-'and the tensor aaPdoes not enter. The formulae for R,, however, differ qualitatively for m& and mzaX. Thisdifference canbe explained by the fact that the case of minimum effecis analogous to the case of a quadratic tive mass mzin = (I/gz) ( as/ dispersion law in the respect that a small change in H near resonance leads t o X R and to a significant increase of the resonance depth for R. In the case of maximum m* it is impossible to attain X > R for a small shift of H [all this can be proved starting from Eq. (S)]. Only for the sake of definiteness shall we consider electrons: as/a s > 0 and not holes: as/a& < 0. [All arguments are the same for holes and the result contains I as/8s I ; the case of equal numbers of holes and electrons does not lead to some special property which is reasonable since the Hall-field E, does not enter the formulae at all.] For maximum m*, R, = X , and (w - qSZ&)/w M (wt)-l, the tensor agP,just as in the case of Xu, does not enter. For minimum m*, the resonance depth of R, and the shift of the
+
+
>
References p . 330
CH. VII,
5 11
ELECTRON RESONANCES I N METALS
307
resonance frequency substantially depend on the non-resonant term as in the case of quadratic dispersion, in the given case, on aaD(since for aaP= 0 Rp = 0 would be attained), which for this reason was retained in Eq. (8). For aaP rn 1,
The curves of dR,/dH are also substantially different for the case of maximum or minimum m* 34a. For m& the curves are analogous to those for a quadratic dispersion law (Fig. 9). For m&x close to resonance on the low field side ( H = H,,, - 0) dR,/dH becomes -m and on 0 ) + 00 and the curves are of the the high field side ( H = H,,, form shown in Fig. 10. In either case (mzin and m Z X ) (dRa/dH),a, rn HylR(O)q*((ot)tand the frequency shift (w - q Q m a x ) / W w (wz)-l; H, is the magnetic field
+
Fig. 10. Behaviour of R and dR/dH as a function of H near resonance for a nonquadratic dispersion law, for m;,,.
corresponding to resonance on the principal harmonic (q = 1). It is essential that the relative resonance height for dR,/dH be significantly larger than the reciprocal of the relative resonance depth both for R, and X,, which makes measurements of dRJdH more attractive than those of R, or X,. As is clear from the above, the behaviour a t resonance of R, and Rejeretzces p . 330
308
M.
YA.AZBEL’
AND
I.
M. LIFSHITZ
[CH. VII,
5
1
dRJdH is very sensitive to the properties of the fermi surface (resonances at the points of support, at the central cross-section and a t the non-central sections for minimum and maximum m* are substantially different). Thus cyclotron resonance enables us not only to determine at once the effective mass of the electrons but also to obtain important additional information about the nature of the fermi surface, being an effective tool (together with other methods-De Haas-Van Alphen effect, Shubnikov-De Haas effect, quantum oscillations in high frequency fields, galvano magnetic phenomena, the study of the surface impedance under anomalous skin effect conditions) for constructing the shape of the fermi surface (see also the next section). I . 4. FURTHER DEVELOPMENT OF THE THEORY OF CYCLOTRONRESONANCE
The theory of cyclotron resonance displayed in 9 1 . 2 and 8 1 . 3 is in its essence a pure classical theory in which only the equilibrium function corresponding to a degenerate fermi gas has been introduced. Such a theory is, of course, completely sufficient for the study of the basic effect, since the distance &Q between the levels in a magnetic field is considerably smaller than the fermi boundary energy E~ (for the fundamental groups &ais about E ~ only , for H of the order lo8-lo9 Oe) and the quantum corrections for the classical formulae are quite small, usually even much smaller than the next unwritten anomalous terms. The latter have a relative magnitude of the order (s,/r)+.Nevertheless, the consideration of quantum effects is very significant. I n the first place the quantum effects have an oscillatory character with respect to variation of the magnetic field, with a period of oscillation in terms of the reciprocal magnetic field A ( H - l ) of the order ~ T C , U / E ~( p is the Bohr magneton) which is considerably smaller than the “periods” of cyclotron resonance (i.e. the distance between the harmonics) which are of the order e/mcw. This suggests the possibility of separating the quantum oscillations from the classical phenomenon. The knowledge of the periods of quantum oscillations can basically simplify the problem of the establishment of the form of the fermi surface. The quantum theory of the surface impedance for an arbitrary direction of the magnetic field was established for a general dispersion law E = ~ ( pand ) collision integral in 36 and, independently, somewhat Refeferciaces p. 330
CH. VII,
11
ELECTRON RESONANCES I N METALS
309
later for the special dispersion law E = p2/2m, constant relaxation time z and the magnetic field parallel to the metal surface in 3'. It has been shown in 36 that the periodicity of the oscillations caused by the magnetic field is the same in the h.f. case as in the static effects of De Haas-Van Alphen and Shubnikov-De Haas: O(H-l) = eh/cS, where S, is the extreme cross-sectional area of the fermi surface in a magnetic field parallel to the metal surface and the area of the central cross-section in an oblique field. This makes the experimental analysis of the impedance in a magnetic field very useful for establishing the shape of the fermi surface, and gives in one sample a very precise determination of Sex, from the from the resonant periods1of the quantum oscillations, and of ( as/ frequencies. [The amplitude of the quantum oscillations also includes ( as/a&),,,, but it is very sensitive to mosaic structure, impurities, deformations, etc. and makes the result at the least inaccurate.] The knowledge of Sex,and ( as/a&),,, makes it possible, at least for a convex fermi surface, to establish its form as well as the electron velocity on it (according to the method of Lifshitz-Pog~relov~~), which is one of the fundamental problems of the electron theory of metals. The fact is that because of the fermi statistics, as was indicated above, the form of the boundary fermi surface and the velocity of the electrons on it, in any case fully determine the dynamic characteristics of the conduction electron. In an oblique field the amplitude of the quantum oscillations has a resonance character, whereby the magnitude of this quantum resonant increment is considerably larger than the classical resonant increment. We consider this problem in somewhat more detail. Chambers39 noticed that cyclotron resonance can occur in an oblique magnetic field, for example, on the central cross-section where the electrons do not travel inside the metal (since the average value of their velocity on the central cross-section is equal to zero). However, the relative number of electrons which do not leave the skin layer after the nth revolution is of the order o~d,,/m 1. Their relative contribution to the impedance is of the order d,/r and it is always small (as long as the anomalous skin effect is present i.e. up to frequencies w 2 10%-1, corresponding to a magnetic field H 5 lo6 Oe)', 44. The difference between the central and non-central cross-sections involves only the order of tilting of H , which suppresses the resonance: in the first case it is v'OZ times larger (see also 62).
<
I?-fwemes p . 330
310
M. YA. AZBEL’ AND I. M. LIFSHITZ
[CH. VII,
9
1
The resonance term of Blount for open trajectories has an analogous character. All these conclusions do not refer to diamagnetic resonance; the latter may occur on anomalously small effective masses. The quantum oscillations in an oblique fieldS6,however, are determined by the electrons close to the central cross-section only, for 9, M P o ( p H / ~ , ) ’which , (even for the case of the principal zones) return to the skin. Because of this the amplitudes of quantum oscillations have a resonance character, increasing at resonance by a factor M (r/a0)( , u H / ~ ~Their ) t . ultimate contribution to the impedance becomes of the order (r/8,)(,uH/cO).Formally the classical and quantum resonance increments in an oblique field correspond to the expansion of the impedance in terms of different small parameters : classically in So/r and quantum-mechanically in ( , ~ H / E ~ ) * . The further development of the theory of cyclotron resonance is related to a consideration of the Landau theory of the fermi liquid40 which shows itself to be very fundamental for the understanding of the electronic properties of metals, We should recall that the concept starting from the representation of the electron conductivity as a gas of fermi particles with arbitrary dispersion law does not necessarily mean a Bloch one-electron model of the particles in a fixed periodic field. The system of electrons, on low excitation levels, interacting with each other and with the field of the ionic residues, represents in a dynamic respect a gas of charged quasi-particles i.e. of “elementary excitations” of the fermi type with a dispersion law corresponding to the crystal symmetry. In fact these quasi-particles are considered as “conduction electrons”. However, it is more exact to use a theory in which the dependence of the energy of the quasi-particle not only on its momentum, but also on the state of the other quasi-particles is taken into account, i.e. on the distribution function of the conduction electrons. Physically this can be represented as the dependence of the self-consistent field, in which the electron moves, on the state of all electrons. This theory takes into account that, at absolute zero for example, the variation of the energy of the system of strongly interacting fermiparticles on addition of one particle (i.e. by definition, the energy of an elementary excitation of such a system) is not only related to the occupation of the next energy level of the system, but also to the References p . 330
CH. VII,
5
11
ELECTRON RESONANCES I N METALS
311
change of all levels of the system, because of the strong interaction between the particles. As has already been said, this means that the energy of the quasiparticle does not depend only on its momentum, but also on the distribution function n in the system. This makes up the contents of the theory of the fermi-liquid of Landau, which can be developed in a phenomenological manner, starting from the assumption of a nonlinear dependence between the energy of the whole system and the number of elementary excitations. If for n = n,(~,)the energy of the system is E,, for n = tz0(&,) 6n the energy E of the system will be equal to
+
) the fermi-liquid differs and the energy of the quasi-particle ~ ( p in from the energy ~,,(p) of the quasi-particle in the gas:
and is related to the variation 6n of the distribution function. A consideration of this relation apparently leads to the fact that it is impossible to consider a particle with given ~ , ( p )in external fields, and the variation of the dispersion law itself in these fields should be taken into account. Together with the dispersion law ~ , ( p )the correlation function f(p, p’) is the most important characteristic of the electrons in a metal. It proves to be35that in the unique case when the difference between the properties of an electron of the fermi-liquid and those of a n electron of the fermi-gas is very essential, cyclotron resonance appears at very high frequencies, when 07 M ( ~ / 6 , ) ~60, ?. kT and the effective conductivity ueil M a , ( S , / r ) Z / ~is of the order of the static conductivity 0,. In35 a method is developed for a general reduction of the whole problem to the solution of the equations for the tangential field components, automatically satisfying the equation j,, = 0. There it is shown that the terms of the kinetic equation resulting from the fermi-liquid and from the field E, (neglected in the previous equations), have the same character; E , and the fermi-liquid demonstrate their effect under the same conditions (condition that E , be essential, see 5 1.2) and lead to similar results. References p . 330
312
M. YA.
AZBEL’ AND
I. M. LIFSHITZ
[CH. VII, $ 2
The fermi-liquid terms change somewhat the formulae for the impedance and lead to an additional broadening of the resonance curve. This additional broadening is of the same order of magnitude as the line width produced by electron-phonon collisions for 6w 7 kT. The study of ZJH) makes it possible to determine the character of the function f(p, p’). In the entire discussion, just as in 3 1.2 and $ 1.3, only the case of a negligible displacement current has been considered. I t should be emphasized, that in metals this is always permissible up to frequencies w 7 4no corresponding to magnetic fields H ? 4nmca/e. Even for bismuth this corresponds to enormously large o and H , and unattainable fields are necessary for good metals: w 7 1020s-1,H 7 10130e. It is reasonable that these frequencies are much larger than the frequency of degeneracy eo/fi and correspond to wave lengths for which the whole consideration as given before loses its meaning. From other theoretical investigations on cyclotron resonance those of Heine41,Rodriguez42,Phillips43,Mattis and Dresselhaus 37 should be mentioned where the investigators obtain by different methods the results of *, for a quadratic dispersion law and analyze them. The error in 41, 42, 37 consists of incorrectly accounting for the boundary condition (for more d e t a i l ~ ~ ~ ywhich ~ * ) , leads to the difference between the formulae of 37, 42 and those of 7, 36. Fortunately, in both the classical and the quantum-mechanical case the reflection of electrons from the surface as discussed in 1 . 2 , does not appreciably affect the results.
2. A Special type of damping (Non-Skin) in a Metal 2 . 1 . PHYSICAL BASISFOR NON-SKINDAMPING
It is well known that the attenuation of an alternating electric field in a metal is more rapid the higher the frequency of the field, so that the field and the current are large only in a small skin at the surface of the metal. However, it was shown in an article by one of the authorslo, that if the dispersion law of the electrons is substantially different from a quadratic one, this leads, under certain conditions (to which belongs resonance), to a totally different character of penetration of the field into a metal, which has never before been observed as iar as is known to us and which is apparently impossible in any other case than the one Rcjerrmrs
p . 330
CH. V I I ,
9 21
ELECTRON RESONANCES I N METALS
313
analyzed. For wt 7 r / 6 , (6, m c/wo), i.e. for w 7 (S,t/v)-k under resonance conditions on the central section of the fermi surface, and for wt M ( ~ / 6 , ) 2 , i.e. for o M (6tt/v)-* under resonance conditions on a non-central section of the fermi surface (but not a t a point of support !) the field and the current density in a metal vary with depth in the way shown in Fig. 11. [In all inequalities at high frequencies t should be considered as dependent on the frequency w-see 0 1 . 1 and, for exam€
Fig. 11. Attenuation with depth of the electric field E when field-splashes occur.
>
plc, because of this oz is, in general, impossible - see 5 1.3.1 In order to understand the origin of the gradually attenuated field “splashes” inside the metal, which change the monotonic decrease of the field, we consider in general how a field and a current penetrate into the interior of a metal when a magnetic field is parallel to its surf ace. Therefore, we return first to the motion of an electron in one of its orbits (orbit 1 in Fig. la), which passes through the ordinary skin near the surface, where in any case the electric field is not small (the constant magnetic field is perpendicular to the plane of the figure). In a layer of the order 6 the electrons acquire a directed velocity over an arc of length d/rs(r 6) and produce a current I of density j w IjS. As the electrons move down along the orbit the velocity parallel to the metal surface changes (along which only a current flows), and correspondingly the current changes by a factor cos p, and, secondly, the electrons will spread out through the bulk of the metal finding themselves in a layer of the order of V% sinp instead of 6 (for
>
p?
dV).
Referelices
p . 330
314
M. YA. AZBEL’ AND I. M. LIFSHITZ
[CH. VII,
92
Thus, the current density produced by electrons of a given orbit proves to be of the order (I/d%) cot ‘p, i.e. decreases sharply with the depth and for ‘p m 1 is a factor d/rls smaller than in the layer 6. At a depth y > Y the current density changes sign, remaining small in absolute value compared to I/S until the angle ‘p approaches the value ‘po of the lowest point of the orbit, so that 1 y o - ‘p I 2/6/r. At a depth d the current density increases again sharply and in this case differs only in sign from the current density at the surface. Apparently such a picture is valid for all trajectories of given radius passing through the narrow skin layer parallel to the surface, Therefore,
Fig. 19. Electron orbits passing through the “ordinary” skin.
if all electrons moved along orbits of the same radius (i.e. had the same value of the velocity in the xy-plane), then the current density produced by the electrons skimming along the metal surface in the layer 6, would produce “splashes” of current, and associated with them an electric field at a depth y = d. Such splashes would in turn lead to acceleration of new electrons, skimming along a layer at a depth d , causing the picture to reproduce itself at depths 2d, 3d, etc. The result would be the solution of the problem concerning the selfconsistent system of currents and fields. However, the picture drawn in Fig. 11 is physically clear. The phenomenon changes essentially if orbits of different radii (Grbits 2 and 3), which correspond to different sections of the ferrni surface, are considered (we remember that, for example, for free electrons Y = plc/eH = ( c l e w d 2 m s - $:, and I varies from 0 to (c/eH)1/2mE).The spread in radii leads to the fact that at any depth in the layer of the order 6 only a small part of the electrons collects (of the order 6 / ~ )and , the field, carried through into the metal will of course decrease rapidly, especially in the following “loops”. In the case References p. 330
CH. VII,
$ !
21
ELECTRON RESONANCES I N METALS
315
where the orbital cyclotron frequency f2 does not depend on the section (i.e. in the case of an ellipsoidal fermi-surface), this spread in the radii cannot be eliminated. However, if the frequency depends on the section of the fermisurface, i.e. on ps, then cyclotron resonance can be used to eliminate the large spread in radii, as only electrons near the extreme cyclotron frequencies take part, with a spread in p , of the order M $,dot (since w - SZ w 1/t),p , being of the order of the limiting fermi momentum). The spread in radii given by a certain w t is totally different for each of the following cases : a) for a central cross-section, where, as it is clear from a symmetry consideration, both f2 and d have an extreme value at the same time; b) for points of support, where, as it is shown in $ 1 . 3 d oc dAT; c) for other non-central cross-sections. For points of support Ad M d w d,/dz so that (see 1.3) Ad m d ? 6, always and splashes of current cannot occur. At other non-central cross-sections Ad m d / l / o z and for splashes of field strength and current density to occur it is necessary that Ad M do i.e. w t m ( ~ / 6 , (a ) ~more stringent inequality, as discussed in section 1, is, generally speaking, impossible). For a central section where d’(0) = 0, the spread Ad = +d”(O)$; M M d(wt)-l, and the structure shown in Fig. 11 occurs for c o t T ~ / 6 , , w ? v/d/ldo, which for I M 10-1 cm, 6 , M cm, ZI fw lo8 cm/sec corresponds to a wave length of the order of 1 cm and H M lo4 Oe i.e. values which have already been attained in experiments on cyclotron resonances. It can be easily understood that the negation of this inequality ( w t < ~/6,) leads to a decrease of the magnitude of the consecutive splashes in a geometric progression, proportional to the degree of diffusion of the current at the bottom 6/Ad M ( 6 / r ) o t ,and the splashes are rapidly attenuated. A more precise estimate shows that the splashes for m t < ~ / 6 , , y = ad, are of the order E(0) (u)-+ ( S , / Y ) ~ / (~w t ) s a / l z ;(do = d(O)), that the relative increase of the field near y = do is of the order (wt)’, and that the relative contribution to the impedance in from the splashes is of the order ( S O / ~ ) t ( w zandlfrom )* the following “impeculiar” term of the expansion of the order (dO/~)+(wt)-A. Thus, under certain conditions a characteristic chain of layers of field splashes inside the metal is obtained. References p . 330
316
M. YA. AZBEL’ AND I . M . LIFSHITZ
[CH. VII,
92
It is apparent that the arguments given are not a proof of the existence of such a structure. I n the simplest case of resonance on the central section and when r/do w t (r/80)2,the occurrence of such a structure can be proved by solving equation (5) by the Fourier method and determining E,(y) in the neighbourhood of the points y = ad, ( a is an integer) and in the intervals between them. The basic formal difference from the solution in the “ordinary” case of w t r/do, considered in 7, consists of the fact that the largest parameter is not the parameter of anomaly r/6, but the resonance parameter at.Since the analysis is rather complicated, we refer the reader to lo, 35 for details, where an investigation is given for arbitrary w t and the mathematical reason for such a field structure is explained, namely the proximity of an infinitely degenerate eigen value of the Maxwell equations. We give the results of this consideration only for the central section and w z ? r/So. 1. For y = do(a M-’(’), C’ M , where M m ( ~ / d ~ ) z ( w and ~ ) i is determined by the properties of the ferrni surface a t the point
< <
<
<
+
E
=
p , = 0,
E ~ ,
v, = 0, a being an integer.
I
M
E,
M -
2n-l (-
l)cc doE’(0)M-l
x(g2 - I)-$ [g
+ V’F- 11-a
0
x cos ([‘Mx 4-ins) dx g
=
1
+ x3 exp (-∈ + ii7co) ;
dT--ip-,di = + 1. 2 . In the neighbourhocd oi y = ad,, a = 1, 2 . , . , do is the diameter of the central section, the field is 4% times larger than the field 0
= sign (Q,/Q;
dg2
-1
g
between the maxima. 3. The distance from the maximum, where the about maximal value of the field is attained, is of the order doM-l. 4. When the number a increases the height of the maximum decreases for a 1 in proportion to a+ and the width increases. At distances y @/do the field between the maxima is almost unattenuated. The maxima gradually disappear; at large distances ( y 7 r2/6,) the field oscillates as cos (py/r)where p is of the order 1. The field attenuates at distances of the order rills M r2/d,.
>
licferc+lcts p . 330
<
CH. VII,
5 21
317
ELECTRON RESONANCES I N METALS
5. For y,, = 2bd, (b = 0, 1, 2) there are single extreme values of the field the sign of which alternate as (-l)b. 6. For y,, m ( 2 b l ) d , (b = 0, 1, 2) there are two extrema different only in sign; the field in the neighbourhood of these points is antisymmetric; E J y , y’) = - E,(y, - y ‘ ) . The signs of the first extrema of two adjacent pairs of extrema alternate as ( - 1 ) b f J ; the first extremum for y = d o has a sign opposite to that of the field for y = 0.
+
+
+
2 . 2 . NEW RESONANCE EFFECTS RELATED TO FIELD “SPLASHES” 1. The variation of the impedance of the bulk metal or its derivative with respect to the magnetic field (dR/dH,dXldH, d In XIdH) is most easily observed. One should only remember that the impedance, as was shown in the finite case, has a resonant character only for selected polarization directions of the electromagnetic wave. However, only the non-monotonic variation of the impedance (the splintering of the resonance curve) associated with the phenomenon of “splashes” can be clearly observed (which is related to the fact that the approach of the magnetic field to resonance leads to a resonant decrease of R and X and at the same time to an increase of R and X by a factor 1.84 because of the occurrence of “splashes” decreasing the total current for a given field strength), which dbes not always occur (see note at the end of 35). ( H - H,,,) d In X/dH is more conveniently measured (H,,, is the resonance value of H). The phenomenon of splashes causes the hump on the straight line out of resonance [since for dwz = I o - Qres I t l/t we have ( H - H,,,) d l n X / d H M - 1/6 for (w/dw) (d,/r) 1 as well 13. Such a criterion for the occurrence of splashes as for ( w / A w ) (d,/r) is obviously not very convenient. 2 . Discontinuities in the resonant values of the impedance and in its derivative in plates of the order of 10-3-10-1 cm thick for an increase of the frequency w are most promising in uniquely proving the existence of “splashes”. These discontinuities will be observed for a cut-off of a splash series, i.e. in the case of resonance for w = Q, and D = ad,. In addition to this the number of harmonics will increase by one (from a to a 1)-see Fig. 13, hence the number of harmonics observed also enables us to determine a and do = D/a = 2cp,magleH i.e. to obtain at once the diameter of the fermi surface. [The use of
<
+
Riferences
p . 330
>
>
318
M. YA. AZBEL’ AND I. M. LIFSHITZ
[CH. VII,
32
plates enables us also38ato obtain at once (aS/a~),,.. and Sex,for each section]. Apparently the effect will be noticed even for a < MS rn 1000. An analogous effect, which can, however, be observed much more easily takes place at fixed o when the d.c. magnetic field is rotated in the plane of a monocrystalline plate. For angles 4,where because of the anisotropy of do there will be discontinuities, and harmonics will
Fig. 13. Field splashes for different magnetic field directions parallel t o the surface, or for different frequencies.
appear or disappear, can be determined for the corresponding directions. It is obvious that for a direct construction of the fermi surface experiments are necessary on plates of different thickness and with different orientations of the surface with respect to the crystal axes. One may easily understand that there will be observed about a = D/d discontinuities of width of the order a-4M-l. When a cc M t the oscillations appear instead of the discontinuities. (The oscillations are connected with the periodic change of the field in the interior of the metal.) The amplitude of discontinuities or oscillations d - (d In Z/d In H) is about wzMa-+ exp (-aM)-a.
dP
3. A third effect produced by “splashes”, is the selective transparency of plates at resonance, when D = ad, cc H (to observe this it is necessary to change the frequency also, or to rotate H in the plane of the plate), and the observation of a field “leaking” through a depth of ( ~ z / l s , )(or)*. The observation of these effects is of considerable difficulty because of the almost ideal specular property of the metal layer; the reflection from the two surfaces of the plate (which is the only one essential in the case of “splashes” transparency) already attenuates the field by a factor (lOlso/il)z, where il is the length of the incident Refcpelzccs
p . 330
CH. VII,
31
ELECTRON RESONANCES IN METALS
319
wave. The field carried through, when it is not connected with splashes (i.e., when D w d,Mg), is attenuated again by a factor r/do e v0A/2ncd,. 4. The following effect is a “spatial” electron echo in some respect analogous to the well-known spin-echo. If we apply a pulse of length At 2n/Q, to the metal located in a d.c. magnetic field, and Q,t 1, after the intervals of time 2n/Q0,4n/Q0,. . . . when most of the accelerated electrons collect again near the metal surface (in the case of a quadratic dispersion law all electrons), the response splashes are observed. 5. The fluctuations in the metal have, for Q0z> 1, a particular character, however, this problem lies outside the scope of this review.
<
>
3. Paramagnetic Resonance 3 . 1 . PHYSICAL PICTURE OF PARAMAGNETIC RESONANCE
It is well known that in a d.c. magnetic field electron spins are oriented along or opposite to the direction of the magnetic field. The energy of the spins is -poH in the first case and poH in the second one, where po is the magnetic moment of the free electron. In an alternating electromagnetic field a transition probability for the electron spin exists. This probability has a resonant character, as is known from quantum mechanics, and has a maximum when the energy of the electromagnetic quantum equals the transition energy of the electron (i.e. the distance between the energy levels corresponding to different spin orientations) : &iw
= 2p0H,
iw
= SZ, = 2pu,H/&.
The transition probability is, of course, the same for both directions of the transition. However, the number of electrons oriented in the direction of the d.c. magnetic fields (N+)an&the number opposite to the field (N-) are not equal, because the direction with the field is more favourable; the difference N + - N - in equilibrium is given by the equality of the chemical potentials (see Fig. 14). Hence, since the number of spin flips caused by the alternating field is proportional to the number in the initial state, the alternating field will lead to an increase of the number of spins pointing opposite to the field direction and hence to a decrease of the difference AN = N + - N-, i.e. to depolarization of the electron gas. Since collisions producing a spin flip tend to establish a state Refereizces
p . 330
320
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
93
corresponding to thermal equilibrium and
AN,
+ poH)
= N O ( E 0 - p,H) - No(&,
=
-
2poH 8N,/a&,,
a differenceA N will be established in a homogeneous field corresponding to d N a = ( A N , - AN)/T,, i.e. AN = dN,/(l aT,) where T , is the time between collisions producing a spin flip; a is the probability per unit time for a spin flip, at resonance a = 4p2H:T,/!i:, 2 H , is thc amplitude of the a.c. magnetic field. The change of magnetic moment of the electron gas has a maximum at resonance and consequently the corresponding contribution to the impedance will also have a resonant character. In the non-linear approximation not only the time-dependent components of the magnetic moment but also the constant component
+
NC
I
N-l
I
I I
Fig. 14. Number of electrons oriented in the direction of the magnetic field and opposite to it as a function of their energy.
along the field direction will vary, - whereby, since T , is very long (up to values of the order 10-s-10-7 s 46), the non-linear effects occur at entirely attainable values of the a.c. magnetic field strength. Overhauser4' noticed that this can be used to polarize nuclei, since an electron spin flip resulting from a collision with the nucleus produces a nuclear spin flip (therefore, in particular, T , is high) and, as is shown above, by thermal collisions more spins are turned over at resonance along the field than in the opposite direction. However, all formulae which use directly the well-known quantum mechanical transition probability per unit time a, are not correct in the case of a metal. In fact, the transition probability per unit time can be introduced only in the case when an electron is in an almost homogeneous field Rrfdreiices
p. 330
CH. VII,
9
31
ELECTRON RESONANCES I N METALS
321
during a time much longer than the field period. Since the skin depth 6 w c [ ( l / w z)/(2no)]+ ( o is the conductivity of the metal), and since during a cycle the field traverses under free motion conditions a distance vT rn v / w , it is necessary that
+
> (v/c)(2no/z)*5 1013s-l and the magnetic field H = fiw/2p0 > lo6 Oe. This means that at v/w
T)*, i.e. w
normal magnetic field strengths and normal frequencies the electron leaves the skin in a time much smaller than a period of the a.c. field. One may think that this makes the observation of paramagnetic resonance in metals impossible, as the electron does not even succeed in “feeling” the frequency of the field. This is, however, not the case because of the following reason: As has already been discussed, the free path time of an electron corresponding to a spin flip is very long and much longer than the ordinary free path time z associated with the momentum. Therefore, the electron collides many times and diffuses slowly into the metal, because of diffusion travelling a distance d,,, = vl/tT,/S in a time T , . This means that it succeeds in returning to the skin many times and stays there in total a time of the order T,d/d,,, w Bl//T,lz/v. Hence, the condition for resonance receives the form: ( d / v ) d T , f z? 2n/w, or
w
7 (2nav2/c2) (T/T,)
corresponding to entirely attainable conditions. Thus paramagnetic resonance in metals can be observed, and for the compounding of a theory of this phenomenon it is necessary to consider the diffusion of spins into the metal. Comparison of theory with experiment must give the possibility of determining the fundamental parameters of the theory: the spin-relaxation time T , and the g-factor (or, rather g - 2). The arguments given show also that the high-frequency field carried by the spins will slowly attenuate, at least over a depth 6,,, = v d z T , / 3 . This means that the polarization of nuclei produced by paramagnetic resonance will also vanish at this depth. 3 . 2 . THEORY AND EXPERIMENTAL OBSERVATION
OF PARAMAGNETIC
RESONANCE As is clear from 3 3 . 1 the compounding of a theory of paramagnetic resonance separates in fact into two independent problems: the deterReferences p. 330
322
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
9
3
mination of the spin-relaxation time T,, and for a given T , the calculation of the surface impedance of the metal, The relaxation time T, was first calculated by Overhauser47. However, only Elliot48discussed the basic cause of relaxation (apart from the obvious collisions with paramagnetic impurities), namely the spinorbit coupling of electrons with the lattice, and obtained agreement between the calculated T , and the observed T,. The possibility of introducing consistently the spin-relaxation time was shown by Andreyev and G e r a ~ i m e n k ousing ~ ~ the method of statistical operators for quantized systems 50. They proved the equality of the transverse and longitudinal spin relaxation times. Since in this way it has been rigorously proved that in the theory of paramagnetic resonance the time T , enters as a parameter, we shall not consider the method of evaluation for T,, giving only the result. At a temperature T which is low with respect to the Debye temperature 8, it proves to be that T , M a(kT)-l In (kTv/yHs) and for T 8 T, w M a(kT)-l ln(kOv/pHs) where s is the velocity of sound and a w M es2[won(g - 2)2]1-1,w0 is the frequency of degeneracy, e the density of the metal, n the number of electrons per unit volume. The first theory of the absorption for a given T , at paramagnetic resonance in a d.c. magnetic field perpendicular to the metal surface, both in the case of sheets of finite thickness and in the case of bulk metal, was given by Dyson51. Formulae for the case of anomalous skin-effect were obtained by Kittel on the basis of the theory of Dyson. Dyson determined the magnetic moment, solving the equation of motion for the electron spin operator (see e.g.52)taking the spin diffusion in an inhomogeneous a.c. field into account. Diffusion was allowed for in the following way. The alternating magnetic field was taken at a point where the diffusing electron was located at a given instant, and after this the average was taken over all trajectories of the randomly moving electron. This method of calculating the moment is very complicated. We exhibit therefore, only the principal deductions and formulae of the theory of Dyson61 in the case where the surface relaxation can be neglected. 1. For:samples which are thin with respect to the skin 6, SO that D < 46 (D, as before, is the thickness of the sheet) the absorption line P per unit volume has the usual symmetric shape: 1 (W - Qo)T,; Q, = 2,~0H/?i, P = $w'HYxT,(~+ c~3-l; ~ (=
>
References
p . 330
CH. VII,
$ 31
323
ELECTRON RESONANCES I N METALS
x is the magnetic susceptibility; 2 H , is the amplitude of the alternating magnetic field. 2. For thick samples the line has a central structure of width equal to the natural width 1/T, with wings extending over a band l/Tg(Se,,/S)z if 6 < d,,,. 3. The central structure is always markedly more intense than the wings so that under normal experimental conditions the apparent width of the line is of the order Ti1. 4. The characteristic effect of electron diffusion is not t o broaden the line but to make a radical change in its shape. 5. For thick samples with narrow natural line width: D 6,,, d, the absorption line P per unit surface area has an asymmetric form and is given by the formula:
> >
P
rn - ( 4 2 / ~ ) - 1 0 2 H ~ ~ T g ( 6 2sign / 6 e ,a, ), [ d l
+ a: - l]*(l+ a:)-*.
> >
6. For thick samples with broad natural line width D 6 a, the absorption line per unit surface area is described by the formula:
P Fw :w2H;XTg6(1
- al)/(l
+ @?).
7. For thick samples the intensity of the line in the centre is reduced Be,, the integrated intensity of the line comes mainly from the diffuse wings and not from the centre. The theory is in good agreement with the experiments of Feher and Kip 46, who observed paramagnetic resonance in lithium, sodium, beryllium between 4"K and 296" K and in potassium at 4 O K . [The first electron paramagnetic resonance in metals was observed by Griswold, Kip and Kitte153;from the later experiments it should be mentionedsP, that resonance has been observed in solutions of Na, Li, K, Cs, Rb, Ca and in ammonia.] The agreement between experiment and theory becomes obvious from Fig. 15 and from a comparison of the theoretical curves of Fig. 16 and the experimental curves of Fig. 17. All figures are taken fromd6; T , in the figures corresponds to T , of the text :P is the absorbed power and x the magnetic susceptibility of the electron gas. From a comparison of the experiments with theory it is possible to determine two basic parameters characterizing the paramagnetism of the electron gas: the spin-relaxation time together with its dependence on temperature (which in the case of ignorable impurities a factor rn 6/6,,, by the diffusion effect. When 6
References
p . 330
<
324
M . YA. AZBEL' AND I . M. LIFSHITZ
[CH. VII,
5
3
coincides with the one calculated by Elliot 48) and the g-factor (or more exactly g - 2) for metals in which resonance is observed. A further development of the theory is given in articles by the authors and Gerasimenko 55, 5'3. This development was necessary in order to remove a number of
oi
THEORETICAL
I I I I 1 I I I I I Fig. 15. Comparison between theory and experiment of electron spin resonance ab sorption under completely anomalous skin-effect conditions.
Fig. 16. Derivative of the power absorption due to electron spin resonance in thick metal plates for different ratios of diffusion time TD to relaxation time T, (theoretical curves). Xeferejices p . 330
CH. VII,
3
31
325
ELECTRON RESONANCES I N METALS
restrictions in the theory of Dyson. First, free electrons with dispersion law E = fi2/2mare replaced by particles with an arbitrary dispersion law, and also, effects which are non-linear in the field HI are con-
LI-THIK PLATE FREQ = 320 Mc/rec
FREQ
=
320 Mc/sec
REQ = 320 Mc/sec
Fig 1 7 Electron spin resonance in thick plates for different ratios of TD/T,. T Dis the time it takrs an electron to diffuse through the skin-depth (expenmental curves).
sidered (in particular resonance saturation, important in the problem of nuclear polarization, which is not considered by Dyson). Moreover, the character of the penetration of the field into the metal has not been References
p . 330
326
M. Y A . AZBEL’ AND I. M. LIFSHITZ
[CH. V II,
93
studied; as is shown in 55 the nature of this penetration leads to selective transparancy of metal sheets under resonant conditions. Finally, Dyson considered only the case of a d.c. magnetic field perpendicular to the metal surface. All these limitations were avoided in 55. The use of the kinetic equation makes it also possible to consider the boundary conditions consistently and to estimate the character of the approximation in the diffusion theory. In order to calculate the magnetic moment M use is made of its relation to the density matrix f:
1
M = p 0 k 3 Tr@) dp
7
where 3 is the spin operator. The density matrix is found by means of the kinetic equation, the presentation of which is considerably facilitated by the fact that near to resonance f^ must be considered as operating on the spins but not on the coordinates and the momenta, and the hamiltonian consists of a classical part, giving the ordinary total time derivative, and a quantum term p$B, so that
( B is the magnetic induction). Here it should be noted that, strictly speaking, the commutator itself [.^ B,fi = [.^,A B + G . [Bf] consists of a first, quantum-mechanical, term responsible for the equalizing by the alternating field of the numbers of spins, which are oriented parallel and anti-parallel to the d.c. magnetic field, and a second, classical term (reduced to ?i/i multiplied by the Poisson brackets for B andfi, which is reponsible for the force acting on the spin in the inhomogeneous magnetic field. The first term results in ;t resonant flip of the spin and, in fact, this term determines the extent of depolarization of the electrons at resonance; the second term results only in orienting the spins along B and in the determination of the depolarization of the nuclei it can be omitted with a high degree of accuracy. The collision integral (afiat),,,, also consists of two terms, as it is related, first, to the rapid relaxation associated with energy and momentum (with a relaxation time t)and secondly, t o slow relaxation of the spins (with a relaxation time T J . Since T , z both types of collisions can be considered separately. The first type cannot change the
>
Refevences p. 330
CH. VII,
3 31
327
ELECTRON RESONANCES I N METALS
operator of the total spin of the system and leads to a partial equilibrium with matrix f^eq depending on the energy only, and for a fixed value of the spin moment: dp = Tr
Tr
f d p ; Tr
s
s
Pq6 dp = Tr fa dp.
For sufficiently low temperatures and in sufficiently weak fields (PH
<w ,
The second type of collisions leads to the establishment of a total equilibrium function both for the momenta and for the spins
The form of the kinetic equation enables us to determine in the equilibrium functions the parts giving the current and the magnetic moment, putting f^ = fir^ + f . 6 (3 is the unit matrix). The solution of the equation forfis very complicated and, as in the case of the work of Dyson, we formulate the results only. 1. The total magnetic moment equals M = ~ ( 5 b) where, choosing the z-axis along the d.c. magnetic field direction as we have consistently done b,(5) = Re [40).u$(5)1 (1 Re[~(0).u$(O)ll-l; iHob G b,
+ ib,
+
= u(C) (1
+ Re[~(O)u$(0)]}-~;y cc [zGI.
(10)
The idirection coincides with the inward normal to the surface of the metal; the subscript -“0” in u$ means that u* is taken at resonance, o = SZ,; the bar, as in 1 ,means the average over the fermi surface. The function u([)near to resonance equals, apart from a numerical factor of the order 1 (for 6 d,,,):
<
U(C)
= c[E,(o) + ~E,(O)IIS,,,Z(VH,~~~~)-~ exp (-c/&,,):
;,,
=
auT,1{3[l/t
+ i(o - SZO)T,/t]}-*.
(11)
The quantity a, giving both the magnitude of the magnetic moment and the attenuation depth is, in a strong magnetic field, essentially different for a magnetic field inclined t o the metal surface and a field strictly parallel to it. Physically this is related to the fact that in a References 9. 330
328
M. YA. AZBEL' AND I . M. LIFSHITZ
LCH. VII,
53
strictly parallel field the electrons disappear into the bulk of the metal because of collisions only, and between collisions they all have (in the case of closed orbits) zero average velocity in the t-direction, for Y 1 diffusion during a time T , takes place over a distance which is not -1 but over a distance which is a factor llr smaller i.e.
<
rdT,/3z.
>
For a field making an angle y8 '11 with the surface the quantity rll the quantity u m r / l . In a weak magnetic field (Y 5 I ) a is always of the order 1. The field E,(O) is given by the amplitude of the incident field by means of the usual formulae for lo, 33a. the surface impedance calculated for different Formulae (10) and (11) allow us to determine the dependence of the effect (e.g. the shape of the absorption line) on the angle y 6 , as well as the non-linear effects. Moreover, by means of these formulae the surface impedance56 and the transparency of the sheets55 can be determined easily. For the calculation of the impedance the linear approximation should of course be considered. I n the approximation which is linear in the a x . field, formulae (10) and (11) enable us to relate the projection of the magnetic moment on the metal surface to the projection E,, (using the relation between E , and E , , found for 6 = H from the equation ic= 0) : M , ( O ) = g,,E,(O). Since, in agreement with formulae (4) a is of the order 1, for y8
339
ic2
E,(O) = 4nw -L,E;(O) ~
where ZEDand
=
' 9
357
C;',,B,(O)
capare related t o each other in an obvious manner, and B,(O) = H,(O)
hence
Ea(O) = SrxoHp(O)
+ 4nM,(O)
+ 4n@~r)gypEp(O)*
(In the second term the ordinary &'$ is taken.) On the other hand, according to the definition of EL!$ (in absence of resonance, when 6 = H), E,(O) = [$)H,(O). Therefore = [$ - 472[~~%,,,,,[$?~ which solves the problem. From formulae (11) it is seen that, when approaching resonance, the depth over which M is attenuated increases sharply, and consequently a selective transparency of the sheet occurs a t paramagnetic resonance, since through a plate of thickness D, 6 D 5 6,,, a small part of the field, related to M, passes through almost without attenuation. How-
[,,
<
Refermcea p. 330
CH. VII,
9 31
ELECTRON RESONA4NCES 1s METALS
329
ever, also in this case, just as in point 3 of 8 2 . 2 , the effect is considerably diminished by reflection at the boundaries of the sheet, and in this case it is practically unobservable. In the simplest case of a d.c. magnetic field perpendicular to the metal surface, the transmission coefficient through such a sheet (defined as the ratio between the intensities of the transmitted and the incident wave), apart from a numerical factor of the order one, equals at resonance 55 :
The power of the transmitted wave W ~will Z have a maximum if the strength of the magnetic field of the incident wave H P equals
does not depend on the thickness of the sheet (2- is the wave length of the a.c. field). I n 51, 55, 56 a rigorous but very complicated solution of the problem of paramagnetic resonance in metals is given. Recently Kaplan5' proposed a simple method of obtaining the basic qualitive results of 51, 56. Kaplan used Torrey's equation 5 8 which represents the well-known Bloch equation with the addition of relaxation and diffusion terms. In a magnetic field perpendicular to the surface for E = p2/2rn the equation for the alternating part of the magnetic moment has the form : 8M
M H--+DAM;
-=yMA at
Ts e mc
= -,
or in an approximation linear in H ,
References p. 330
D
= 13
~
2
~
330
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
93
Instead of exhibiting the boundary condition (as is done in 57) which only weakly affects the character of the solution, in (13) we extend H, as well as M, to the region z < 0 in such a way that they are even functions of z (the magnetic field, following our convention, is along the z - axis). Putting in (13)
I
m
M
=
dkM, exp (iot
- ikz),
-m
1
m
H, =
dkH,, exp (icot - ikz)
-m
we get aM, - yM, A H, = yxH, A Hlk; a = iw
whence M, = X ( Q 2
Q;)-'{@H1k
+ DK2 + l/T*:
+ CrQo A
Hlk)*
Taking into account that M, = M-, we may get from this a formula corresponding to the basic formulae (10) and (11) for the impedance when H, is perpendicular to the surface, if we consider that 6
<
1
00
deft,
Hlk
dk
=
ZHl(0).
0
The whole theory presented corresponds to the normal state of the metal; in superconductors, as is shown by the authors69,paramagnetic resonance is impossible (a consideration of the theory of superconductivity does not alter this result). Thus, the results given in this section refer practically to all cases of paramagnetic resonance and, as is indicated above, allow us to determine experimentally the spin-relaxation time and the g-factor for the conduction electrons. REFERENCES Ya. G. Dorfman, Dokl. Akad. Nauk SSSR, 81, 765 (1951). :R. €3. Dingle, Proc. Roy. SOC.A 212, 38 (1952). 3 G. Dresselhaus, A. Kip and C. Klttel, Phys. Rev. 92, 827 (1953). B. Lax, H. Zeiger, R. Ikxtcr and F. Rosenblum, Phys. Rev. 93, 1418 (1954). W. Shockley, Phys. Rev. 79, 191 (1950); 90, 491 (1953). 4* I. M. Lifshitz, M. Ya, Azbel' and M. I. Kaganov, J. Exp. Teor. Phys. 30, 220; 31, 63 (1956), Soviet Physics J E T P 3, 143 (1956); 4, 41 (1957). ~ I JI. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 69, 419 (1959). 1
CH. VII]
ELECTRON RESONANCES I N METALS
331
M. Ya. Azbel’ and M. I. Kaganov, Dokl. Akad. Nauk SSSR 95 4 1 (1954). R. G. Chambers, Phil. Mag. 1, 459 (1956). 7 M. Ya. Azbel’ and E. A. Kaner, J . Exp. Teor. Phys. 32, 896 (1957); Soviet Physics J E T P 5, 730 (1957); Phys. Chem. Solids 6, 113 (1958). 8 M. Ya. Azbel’ and E. A. Kaner, J. Exp. Teor. Phys. 30, 811 (1956); Soviet Physics J E T P 3, 772 (1956). 9 M. Ya. Azbel’, Dokl. Akad. Nauk 100, 437 (1955). lo M. Ya. Azbel’, J. Exp. Teor. Phys. 39, 400 (1960). 11 B. Lax, Rev. Mod. Phys. 30, 122 (1958). I 2 A. B. Pippard, Phil. Trans. A 250, 325, (1957). 1 3 E. Fawcett. Phys. Rev. 103, 1582 (1956). 1 4 P. A. Bezuglyi and A. A. Galkin, J. Exp. Teor. Phys. 33, 1076 (1957); Soviet Physics J E T T 6, 831 (1958). 1 5 P. A. Bezuglyi and A. A. Galkin, J. Exp. Teor. Phys. 34, 237 (1958) Soviet Physics J E T P 7, 164 (1958). 18 A. F. Kip, D. N. Langenberg, B. Rosenblum and G. Wagoner, Phys. Rev. 108, 494 (1958). 1 7 M. S. Khaikin, J. Exp. Teor. Phys. 37, 1473 (1959); Soviet Physics JETP 10, 1044 (1960). Is D. N. Langenberg, A. F. Kip and B. Rosenblum, Bull. Amer. Phys. SOC.3, 416 (1958). D. N. Langenberg and T. W. Moore, Phys. Rev. Letters 3, 328 (1959). 2 o P. A. Bezuglyi and A. A. Galkin, J. Exp. Teor. Phys. 34, 236 (1958), Soviet Physics J E T P 7, 163 (1958). 21 P. A. Bezuglyi and A. A. Galkin, J. Exp. Teor. Phys. 37, 1480 (1959), Soviet Physics J E T P 10, 1049 (1960). 22 I. K. Galt, F. R. Merrit, W. A. Yager and H. W. Dail, Phys. Rev. Letters 2, 292 (1959). D. N. Langenberg and T. W. Moore, Phys. Rev. Letters 3, 137 (1959). er E. Fawcett, Phys. Rev. Letters 3, 139 (1959). 25 S. Foner, H. J. Zeiger, R. L. Powell, W. M. Walsh Jr. and B. Lax, Bull Am. Phys. SOC.1, 117 (1956). 28 B. Lax, K. J . Button, H. J. Zeiger and L. M. Roth, Phys. Rev. 102, 715 (1956). 27 I. E. Aubrey and R. G. Chambers, Phys. Chem. Solids 3, 128 (1957). Is I. II. Galt, W. A. Yager, F. R. Merrit, B. B. Cetlin and A. D. Brailsford, Phys. Rev. 114, 1396 (1959). 28 R. G. Chambers, Proc. Phys. SOC.(London) A 65, 458 (1952). so R. G. Chambers, Proc. Roy. SOC.A 215, 481 (1952). 31 I(. Fuchs, Proc. Cambr. Phil. SOC.34, 100 (1938). s2 G. E. Reuter and E. H. Sondheimer, Proc. Roy. SOC.195, 336 (1949). 38 A. B. Pippard, Proc. Roy. SOC.A 224, 273 (1954). M. I. Kaganov and M. Ya. Azbel’, Dokl. Akad. Nauk. SSSR 102, 49 (1955). 34 E. A. Kaner and M. Ya. Azbel’, J. Exp. Teor. Phys. 33, 1461 (1957); Soviet Physics J E T P 6, 1126 (1958). 34a M. Ya. Azbel’ and E. A. Kaner, J. Exp. Teor. Phys. 39, 80 (1960). 35 M. Ya. Azbel’, J. Exp. Teor. Phys. 39, 1138 (1960). a6 M. Ya. Azbel’, J. Exp. Teor. Phys. 34, 969; 1158 (1958); Soviet Physics J E T P 7, 669; 7,801 (1958); Phys. Chem. Solids 7, 105 (1958). 37 D. C. Mattis and G. Dresselhaus, Phys. Rev. 111, 403 (1958). 38 I. M. Lifshitz and A. V. Pogorelov, Dokl. Akad. Nauk SSSR 96, 1143 (1954). 38riM. Ya. Azbel’, J . Exp. Teor. Phys. 34, 754 (1958); Soviet Physics J E T P 7, 518 (1958). 5
6
332 38 40
41 42 43
44
45
47
4B
5u
51
53
63 54 55
5E
57 58
6o
$1
Ez
M. YA. AZBEL’ A N D I. M. LIFSHITZ
[CH. V I I
R. G. Chambers, Canad. J. Phys. 34, 1395 (1956). L. D. Landau. J . Exp. Teor. Phys. 30, 1059 (1956); 35, 97 (1958) Soviet Physics J E T P 3, 920 (1956): 8, 70 (1959). V. Heine, Phys. Rev. 107, 431 (1957). S. Rodriguez, Phys. Rev. 112, 1616 (1958). I. C. Phillips, Phys. Rev. Letters 3, 327 (1959). E. A. Kaner, J. Exp. Teor. Phys. 33, 1472 (1957); Soviet Physics J E T P 6, 1135 (1958). M. Ya. Azbel’, Dokl. Akad. Nauk 99, 519 (1954). G. Fehcr and -4.F. Kip, Phys. Rev. 98, 337 (1955). A. W. Overhauser, Phys. Rev. 89, 689 (1953); 92, 411 (1953). R. I. Elliott, Phys. Rev. 96, 266 (1954). B. B. Andreyev and V. I. Gerasimenko, J. Exp. Teor. Phys. 35, 1210 (1958); Soviet Physics J E T P 8, 846 (1959). I. I. Bogolyubov and K. P. Gurov, J. Exp. Teor. Phys. 17, 614 (1947). F. I. Dyson, Phys. Rev. 98, 349 (1955). D. I. Blokhintsev, Principles of quantum mechanics (Osnovy kvantovoi mekhaniki) 5 62, GITTL 1949. T. W. Griswold, A. F.Kip and C . Kittel, Phys. Rev. 88, 951 (1952). R. A. Levy, Phys. Rev. 102, 31 (1956). M. Ya. Azbel’, Gerasimenko, V. I. and I. M. Lifshitz, J. Exp. Teor. Phys. 31, 357 (1956), Soviet Physics J E T P 4, 276 (1957); I. M. Lifshitz, M. Ya. Azbel’ and V. I. Gerasimenko, Phys. Chem. Sol. 1, 164 (1956): M. Ya. Azbel‘, V. I. Gerasimenko and 1. M. Lifshitz, J. Exp. Teor. Phys. 32, 1212 (1957); Soviet Physics J E T P 5, 986 (1957). M. Y a . Azbel’. V. I. Gerasimenko and I. M. Lifshitz, J. Exp. Teor. Phys. 35, 691 (1958): Soviet Physics J E T P 8, 480 (1959). I. Kaplan, Phys. Rev. 115, 575 (1959). H. C. Torrey, Phys. Rev. 104, 563 (1956). M. Ya. Azbel’ and I. M. Lifshitz, J . Exp. Tcor. Phys. 33, 792 (1957), Soviet Physics J E T P 6, 609 (1958). M. Ya. Azbel’, J. Exp. Teor. Phys. 39, 876, 1276 (1960). E. I. Blount, Phys. Rev. Letters 4, 114 (1960). &I.S. Khaikin, J . Exp. Theor. Phys. 39, 513 (1960).
CHAPTER VIII
ORIENTATION OF ATOMIC NUCLEI AT LOW TEMPERATURES I1 BY
W. J. HUISKAMP KAMERLINGH ONNESLABORATORIUM, LEIDEN AND
H. A. TOLHOEK NATUURKUNDIG LABORATORIUM, GRONIKGEN
CONTENTS: Introduction, 333. - 1. Theoretical results on nuclear effects with oriented nuclei ; general theory; alpha and gamma radiation, 333. - 2. Theoretical results concerning beta radiation emitted from oriented nuclei, 335. - 3. Experimental results on alpha particle emission, 346. - 4. Beta asymmetry experiments, 352. 5 . Experimental results on gamma radiation, 365. - 6. Methods of nuclear orientation, 373. - 7. Nuclear orientation in ferromagnetic and antiferromagnetic substances, 374. - 8. Dynamic methods of nuclear orientation, 380. - 9. Concluding remarks, 300.
Introduction In this chapter the developments concerning orientation of radioactive nuclei since a previous review-paper written in 1956l (which will be indicated as I) will be surveyed. The most striking development in this period has been the discovery of the non-conservation of parity by means of the asymmetry of @-raysemitted from polarized nuclei. Further the introduction of dynamic methods of orientation has widened the field of application of this method of investigation of radioactive nuclei. Also the methods developed hitherto have been applied to many new nuclei. We refer to I for a general introduction and for notations.
1. Theoretical Results on Nuclear Effects with Oriented Nuclei; General Theory; Alpha and Gamma Radiation The purpose of this review is to present the experimental developments of the last few years and their theoretical interpretation. How1Zeferelzccs p . 391
334
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
I
ever, the scope of this review does not allow to deal with the details of the theoretical methods which are used. This is not necessary either as three books have now appeared dealing with the general methods which are used, such as the theory of coupling and recoupling of angular momenta, (Edmonds2, Rose3, Fano and Racah4). An account of the theory of radiations from oriented nuclei was given by Rosenfeld6. Further we may mention here some earlier review papers (theoretical and experimental), which have appeared on oriented nuclei &ll. Concerning the general theory a paper by Rose12 has appeared on the statistical tensors (orientation parameters) characterizing an ensemble of oriented nuclei. He discusses their expressions in terms of multipole operators with the nuclear angular momentum and also makes some remarks concerning their temperature dependence (for not too low temperatures). Fano13 has given a general proof for a property of the directional distribution for a chain of emission processes with parallel angular momenta, i.e., emission processes in which the decrease of nuclear spin (I’ - I”) is equal to the multipolarity (L) of the emitted radiation: I’ - I” = L. Such sequences are rather frequent in nature. We then have Nkfk= constant (see I, eq. (9)). Fano discusses a number of general features of such chains. The theory of gamma radiation from oriented nuclei is mostly completed; however, in case of intermediate states with lifetimes which are longer than, say lo-%, attenuation effects may appear, which depend on the complicated solid state interactions which occur. In the case of y-y correlations they were extensively studied, see the reviews by Steffen14 and Frauenfelder16. For certain cases the discussion of such effects for oriented nuclei should not be neglected. Further it may be of importance that the orientation of nuclei in certain crystals has no precise axial symmetry. Danielsls has given general formulae for this case, which might contain an explanation for certain anomalies of y-ray distributions observed at the lowest temperatures. As to the theory of a-emission from oriented nuclei, we do not want to add to the account given in I ; however, we mention that since that review the papers giving full details by Brussaard and Tolhoek” and by Rose (sees, 5 34) have appeared.
Refcremes p. 391
CH. VIII,
9 21
ORIENTATION OF ATOMIC NUCLEI I1
335
2. Theoretical Results Concerning Beta Radiation Emitted from Oriented Nuclei The experimental discovery of the non-conservation of parity in weak interactions has been followed by a rapid development and many papers on this subject have appeared. In I the consequences of the non-conservation of parity for the directional distribution of /I-rays from oriented nuclei have been concisely discussed (however, before the experimental confirmation). As the general subject of non-conservation of parity in weak interactions has now been reviewed by different authors1a27 we shall confine ourselves here as much as possible to those theoretical results, which are relevant to the p-emission from oriented nuclei (we also refer to these reviews for complete references). For the interaction Hamiltonian for the theory of ,+radioactivity, one has considered for many years (see e.g.28) the most general linear combination of the scalar (S), vector (V), tensor (T), axial vector (A} and pseudoscalar (P) interactions, specified by H,
= gp
7 Cj
(Wp
rjyn)
(ce
I;wY)
+ berm. conj.
where the I’, are the operators I’, = 1, r, = yp,
(1)
336
W. J . HUISKAMP AND H. A. TOLHOEK
[CH. V I I I ,
§2
The calculations based on (2) or (3) are generally made taking the possibility of complex values of C, and C; into account. It is of fundamental importance to state under what conditions the generalized Hamiltonian is invariant for the discrete operations : space inversion P , charge conjugation C and time reversal T . This may be summarized as follows Operation :
Invariance condition :
space inversion P,
c: = 0,
(4)
charge conjugation C,
C, real; C; pure imaginary,
(5)
time reversal T ,
C,, C; real.
(6)
1
Hence the expression (1) with real C, is invariant under P , C and T . However the C, and C; may have such values that H , according to (3) is not invariant for either P or C or T . One may calculate a number of effects on the basis of (3) and look in particular for such effects, which could show non-invariance for P, C or T . However, before discussing such effects, we want to indicate concisely some theoretical developments, that have proved t o be of importance in this field. The CPT-theorem. Under certain conditions (local fields with local interactions or even somewhat more general conditions), which one expects (or hopes) to be fulfilled in nature, one can prove that the theory is invariant under the operation CPT (or any permuted operator PCT, TCP etc.) even if the theory is not separately invariant under P , C or T. It follows that if no invariance exists under one of the operations P , C or T , the theory cannot be invariant for both the other operations. Further one sees that invariance for T is equivalent to invariance for PC. The theorem is based on considerations by Pauli30, Schwinger3' and Luders32 (see also J ~ s t ~The ~ ) theory . based on the interaction Hamiltonian (3) satisfies the CPT-theorem. The Two-component Nezltrino Theory. If one does not require invariance under space inversion, a possibility exists to describe a neutral massless spin-4 particle by a two component equation - iu
- Vy, =
iqV
(units such that 6 = 1, c = I),
(7)
instead of by the four component Dirac equation. In this formulation the following points should be stressed: IZejerrrices p . 391
CH. VIII,
3 21
ORIENTATION O F ATOMIC NUCLEI I1
337
(a) The spin of the neutrino is parallel to its momentum; the spin of the anti-neutrino is antiparallel to its momentum (or the opposite of these statements for the lower sign in (7)). (b) The masses of neutrino and antineutrino in (7) are exactly equal to zero. (c) The theory of #?-radioactivitywith the two-component neutrino is equivalent to the theory based on the Hamiltonian (3) if the following relations are imposed on the constants:
cj = c’.
(for left-handed neutrinos),
(84
CJ --
(for right-handed neutrinos).
(8b)
- C;
A choice between (8a) and (8b) can only be made on the basis of experimental data. We define as right handed neutrinos : neutrinos with spin angular momentum parallel to their momentum. One often uses the word helicity 8 for the component of the spin in the direction 1 for a pure state with of the momentum (normalizing to % = spin parallel to momentum). The use of the two-component neutrino theory for phenomena in which parity is not conserved was proposed by Landau35and by Salam36. by Lee and Lepton Conservation. One can investigate whether the “conservation of leptons” is a law which holds in nature. One attributes the leptonnumber +1 to e-, p-, Y and -1 to e4, pf, Y (the antiparticles). If one studies, eg., the processes
+
n-+p- +Y, p* --f e+ Y n +p e-
+ + Y, + + Y,
lepton conservation and two component neutrino theory together lead to a number of specific predictions which can be tested experimentally. In the preceding we have not considered a Hamiltonian, which is more general than (3) and which allows as well emission of neutrinos as antineutrinos in ,&decay (and both with any degree of panty violation). Such a theory (cf. e.g. l9 and 25 App.) contains 20 instead of 10 complex constants, (hence 40 and 20 parameters respectively) of which, however, only 35 are physically observable in #?-processes.Pauli37 and Pursey3* have discussed transformations which do not change the physically observable results of such a Hamiltonian, from which one can conclude which combinations of the constants can occur in the References p . 391
338
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
3
2
expressions for such results. One also finds a discussion in 3 7 , 3 8 of the way in which (4). . (6) have to be understood : The invariance conditions in (4), . . , (6) are sufficient, but one may make certain transformations with arbitrary phases which leave the physically observable results invariant and the necessary conditions for invariance for C, P or T have a somewhat generalized form. However, one may say that one can always come back to the form (a), . ., (6) by a suitable transformation. It can be remarked that the two-component neutrino theory with lepton conservation and T-invariance has the same number of constants (5) as the “old” parity conserving ,!?-decay theory. Universal V--A interaction. Feynman and Gell-MannaV,Sudarshan and Marshak*O and Sakurai41 have proposed that the form of all 4-fermion interactions should be such that all fermion fields occur only in the combination +(1+ y J y . This has as a consequence for ,&radioactivity that only V- and A-interaction would occur with mixing ratio 1 : 1 and with relative sign minus. The Preserct Situation. We do not want to reproduce all detailed arguments used to deduce the maximum possible information from experiments on the weak interactions (for this we refer to the summaries mentioned before). Summarizing we may say that all existing experimental data are compatible with the following : (1) The two-component neutrino theory is adequate for the description of weak interaction processes with neutrinos. (2) Lepton-conservation holds. (3) The theory is invariant under time reversal T (hence under PC) . (4) The 4-fermion interactions are adequately described by universal V-A interaction. However, the restriction should be made that the deviation of C,/C, = - 1.22 f 0.03 (cf. e.g2’) from unity should be attributed to a renormalization of the weak interactions by the strong interactions, although one cannot yet give a quantitative description of these effects. The non-renormalization of the Fermi part of the interaction was proposed by Feynman and Gell-Mann39n43 and by Gershtein and Zeldovitch43 using a conserved nucleon-pion current for this part. However, the Gamow-Teller part could not be formed with such a conserved current. Although no data exist which are in disagreement with (l),. ., (4), a number of further checks of (l),. . , (4) are very much desirable. For example more precise tests of time reversal invariance and precise investigations of the muon-capture interaction are very much needed. h’cferences
p . 391
CH. VIII,
9 21
ORIENTATION O F ATOMIC NUCLEI I1
339
However, we shall confine ourselves in the following mostly to effects related t o oriented nuclei. Allowed /3-transitions f r o m Oriented Nuclei. We consider allowed 1-decay of nuclei with spin I,, oriented with respect to an axis j (unit vector, which indicates a direction of rotational symmetry for the orientation, so that has the direction j) and orientation parameters fi and f 2 (see I) ?.The transition probability depends further on the momentum p of the emitted electron and the momentum q of the emitted neutrino; it was calculated in 2 9 , 4 4 . 4 5 , 4 6 . We give the expression taking Coulomb corrections into account (units such that &=l,c=l)
Ee and E , are the energies of electron and neutrino (including rest mass). E , is the maximum energy of the @-spectrum.E , = E,- E , = q ; m is the electron mass. In the formulae the upper signs refer to negaton decay and the lower signs to positon decay. F ( 2 , Ee) is the wellknown Fermi function describing the influence of the nuclear Coulomb field on the 8-spectrum. Averages are made for the electron and neutrino polarization. Formula (9) contains many effects: shape of the allowed 8-spectrum (average over directions p and q and over the nuclear orientation) ; electron-neutrino directional correlation (term with a%p q, observable by means of recoil experiments ; an average over nuclear orientation should be taken). The directional distribution of @-rays from oriented nuclei is obtained by averaging over the neutrino momentum q (6 is the angle between the vectors j and p)
t fi = (l/Z)Xmmum;f a sublevel nz = I,. Referelices
p . 391
=
( 1 / 1 2 )[C,m2u,
-
+I(I + I)];
a,
is the population of
W.
340
J . HUISKAMP A N D H . A . TOLHOEK
[CH. VIII,
92
We now give the expressions of the constants, Q*, b,, A,, D , in terms of the constants, which determine the Hamiltonian H p (for the expressions for ck and B , we refer to44).We denote the Fermi and GamowTeller nuclear matrix elements by MF and M,, (1 AtF l2 = I Jl 12, I M,, 12 = 1 Its 12; the product M,M& in the cross-term is real, ~ f APP.1
5
=
(
I cs 12 + I c; l2 + I c v l2 + I c; 12) I M , l2 + (1 c T i2 + 1 ck 1' + I c.4 l2 + I c; 12) I M G T
+
df1"
aZm
f-
P
TRefereiices
aZm
P
p . 391
d&
[ 2 Re (CsCk*
2 Im (CsCa*
2 Re (CJX
+ C;C:
-
+ C$,*
CvCk*
+ CLCk* - CvC;
(104
- C,Ci* - C;CX)
M,M&,
(13)
CcCh*)] M,M&
(14)
- C;C,*)]
-
12,
.
~
~
CH.
VIII,
9 21
ORIENTATION OF ATOMIC NUCLEI I1
341
In these expressions (I’ is the nuclear spin after the ,&emission) y = (1 - a222)&,
1
, b
(15)
if
1’= I ,
if if
I’ = I,, I’ # I,.
- 1,
We want to give a concise discussion of some features of these formulae : It is of importance to note in which combinations the observable vectors and axial vectors occur in this formula and the analogous formula for electron polarization, which we do not reproduce here. p and q are momentum vectors which change sign for a space inversion. j (direction of the polarization of the nuclei) and (electron spin direction) are axial vectors, connected with angular momentum and do not change sign for a space inversion. If invariance for space inversion exists, the transition probability may only depend on true scalars, such as p . q, but not on pseudoscalars (quantities changing sign for space inversion), such as j . p or p . <. The term p . q describes the directional correlation between electrons and neutrinos. j * p is a term characterizing a directional distribution of p-rays with a cos &term from oriented nuclei; p . is a term describing the longitudinal polarization of @-rays.Hence these latter effects can occur only in case no invariance for P exists. One sees this also immediately from the explicit formulae (9),. ., (14) and from (4). A similar argument is possible concerning the non-occurrence of certain terms in case invariance for time reversal exists. The argument relies on the relation of the time reversed free particle states to the original states. It is therefore only valid if the influence of the Coulomb field is neglected, for otherwise the final state interactions cause certain additional phase factors in the expression of the transition probability in terms of the matrix elements, which change the argument. One can say that for time reversal the momentum and angular momentum vectors ( p , q, j, <) change sign. It can then be concluded
<
<
References p . 391
342
W. J. HUISKAMP A N D H . A. TOLHOEK
[CH. V I I I ,
9
2
that observables which are odd (change sign for time reversal) can only occur in a transition probability, if no invariance for time reversal exists, when neglecting the influence of the Coulomb field of the nucleus. An example consists of the term j . (p A q) in (9). If the influence of the Coulomb field (final state interaction) is taken into account, such terms could even exist for a Hamiltonian invariant under T . The explicit formulae (11),. ., (14) demonstrate these statements for this special case. For a complete explanation of the general arguments we have to refer to e.g.18. As is seen from (9) and (1l),the experiments on the electron neutrino directional correlation (nuclear recoil in p-processes) allow in principle t o determine which relative weight the different interactions, S, V, T and A have. Only in 1958, after the discovery of non-conservation of parity, these “classical” experiments gave a sufficiently reliable result, namely that one has mostly V and A229 23. No S and T could be shown to exist ; during a number of years before 1957 one had assumed mainly S and T on the basis of insufficiently reliable data. The experiment on the directional distribution of ,%rays from polarized 6oCon ~ c l e i showed * ~ ~ ~that ~ the value of A M - 1 in this case, from which one sees (cf. (13)). (a) Parity is not conserved. (b) No invariance for charge conjugation exists (this would allow only a coefficient A- of the order aZm/fi, which is not sufficiently large). (c) The result is compatible with C’T
-
T
- C,
in case of T interaction,
(17)
C i = C, in case of A interaction. (18) These cases correspond to a different helicity for the neutrino, namely right-handed neutrinos in case of T interaction, left handed in case of A interaction. The decision between the two cases could be taken on the basis of recoil experiments in 1958, but historically this was preceded by an elegant, more direct experiment on the helicity of the neutrino in K-capture by Goldhaber, Grodzins and Sunyar4’. The choice for (18) is now well-established. The quantitative results which are now most precise, supporting the two-component neutrino description, are the measurements of the degree of polarization of B-rays from non-oriented nuclei, which agrees Refeferences p . 391
CH. VIII,
9 21
ORIENTATION O F ATOMIC NUCLEI I1
343
within the error of about 30/, with the theoretical value (-v/c) for the two-component theory. An experiment on the recoils in the decay of polarized neutrons48 has tested the occurrence of the j . (p A q)-term of (9) with the result that no deviation from time reversal invariance was found (the phase of C , and C, differing not more than 8" from 180"; further it was found that C , = (- 1.25 & 0.05) C,). From the preceding theoretical formulae and the summary of the experimental situation it follows that measurements on /?-radiation from oriented nuclei can now be most useful in two respects: (a) as a tool in nuclear spectroscopy, (b) possibly for further experiments testing invariance for time reversal. We want to give some further discussion of possible experiments on time reversal invariance by means of oriented nuclei. It was shown by Curtis and Lewis49and by M. Morita and R. S. M ~ r i t and a ~by~ ~ ~ ~ Dolginov51 that a test for time reversal invariance can be made by measuring the /3 - y directional correlation from polarized nuclei in allowed /?-transitions, if an M,M& interference term occurs. The term which has to be measured is j . (p A k) ( j . k)" where k is the momentum of the photon. The coefficient of this term has a factor Im (C&&*
+ CgC; - C,Ca*
- C;C;)M,M&.
(19) Hence the feasibility of this experiment requires a /3-transition of a nucleus, which can be well polarized and which has a substantial interference term M,M$,. From (13) it is seen that such an interference term can exist in principle in ,!I-transitions with I' = I , and that it can be quantitatively determined by measuring the magnitude of A . Another type of experiment, which also shows non-conservation of parity, can equally provide results on the existence of this type of interference terms : p-7 directional correlation with measurement of the circular polarization of the y-rays. For the purpose of comparison with (13) we give the formula for the directional correlation which occurs in this experiment 2,s 46, 50, 53. 527
Q, is a function of 1', I" and the multipolarity L of the y-radiation, (cf. I, eq. (1.7), (1.8)) e.g. References p . 391
344
W. J. H U I S K A M P A N D H. A. TOLHOEK
Q, =
F,
if
I’
- I” = L = 1,
1, I‘
if
I’
-
i
z = $1 for photons with helicity
I” = L
fl and
[CH. V I I I ,
= 2.
t=
9
2
(21)
-1 for the opposite
polarization. Wy?v)(B,t) is the same y-distribution as for polarized nuclei with degree of polarization
(andf,(Z’)
Fig.
=
0 for k
2).
Nuclear spins in case of @mission with subsequer 2L-pole y-radiation, for which the circular polarization is measured.
The value of A$ is given by
c,c;*
-
Cl.C+)] M,MET)
(23)
with if
I’
= I , - 1,
if
I‘
= I,,
if
I’ = I ,
(24)
+ 1.
CH. V I I I ,
9 21
O R I E N T A T I O N O F ATOMIC N U C L E I I1
345
It is seen that this is very similar to (13), however, the relative sign of the terms with I M,, j2 and M,MgT is different in both formulae. The magnitude of the Fermi matrix elements has been estimated quantitatively on the basis of the 1-1 coupling shell model for certain nuclei (52Mn, “Sc, Z4Na) by Bouchiat 54. For this purpose the deviation from the isotopic spin selection rule d T = 0 for M , due to the Coulomb interaction was calculated. The present experimental values do not all agree with the theoretical results obtained. This matter is of importance: a) in connection with the hypothesis of a conserved vector current 39, 42, 43, b) in connection with possible tests of time reversal invariance, as explained before.
FURTHER THEORETICAL RESULTS ON ,&TRANSITIONS FROM ORIENTED NUCLEI Many authors have made calculations of phenomena in connection with non-conservation of parity, /?-decay and oriented nuclei. Hitherto we have only discussed the simplest phenomena in this connection, on which also experiments have been performed. It is beyond the scope of this review to discuss all detailed results for directional distribution of /?-rays, P-y directional correlation, /?-polarization from oriented nuclei in allowed and forbidden ,&transitions which have been published. We give references to the relevant papers in the form of Table 1 (early references to calculations based on parity conserving /?-interactions are also included). The observation of the polarization of p-rays from oriented nuclei would provide independent information e.g. on the MFMgT interference term. The transverse polarization is already a “classical effect” : t h e t -7-term exists also for a parity conserving H a m i l t ~ n i a n Dol~~. ginov and Popov have pointed that the transverse polarization of p-rays from oriented RaE nuclei could provide a sensitive test for time reversal invariance. If nuclear recoil is taken into account (this means: if one does not average over the directions of the neutrino momentum) many different effects result from the non-conservation of parity also if the initial nuclei are polarized. They were studied in particular by Treiman72, Frauenfelder et u Z . ~ ~ B , i n ~ e and r ~ ~Bouchiat75 but many references of Table 1contain also results which are relevant in this respect. Although observation of nuclear recoil is very difficult in general, it has proved to be feasible in the case of polarized neutrons (recoiling protons Keferetzccs p . 391
346
W. J . HUISKAMP A N D H. A. TOLHOEK
[CH. V I I I ,
9
3
TABLE 1 References to the literature on phenomena on B-radiation from oriented nuclei.
Allowed transitions
Forbidden transitions
~
Lee et aLaO Alder el ~ 1 . ' ~ Jackson ct U Z . * ~ Berestetsky et a1.W Dolginovaz
directional distribution of 8-rays
,!I-y angular correlation (with or without polarization)
61
De Groot et aL8 KhutsishviliSs.67 Morita et a1.68 Alder et ~ 1 . ~ ~ Morita 6a Dolginovea Morita et aLs7 Lee-Whiting 65 Berestetsky et aLeo961 Mahmoud 7 0 Postma80
Morita et a1.6D Curtis et al.49 Dolginov5lS 6 4
Dolginov61962 Dolginov et aZ.63, Morita at al.6B
Tolhoek et aLS5 Jackson et a[." Berestetsky et al.60. DolginovBa Good et aLBO
Berestetsky et aL60. Dolginovfls Lec-WhitingGS Mahmoud 7 O
~
8-'"Y
polarization
observed) and by means of the properties of resonance scattering of y-rays from the recoiling nucleus.
3. Experimental Results on Alpha Particle Emission Roberts et al.' in Oak Ridge showed that alpha particle emission from aligned uranium and neptunium nuclei has a very anisotropic directional distribution, as was already discussed in I. Since then, using the same apparatus for measurements in the liquid helium temperature range, more detailed results were pubIished76~77. It was found that 233U,235U and 23'Np nuclei, when aligned by electric h.f.s. coupling in single crystals of uranyl or neptunyl rubidium nitrate, emitted alpha particles preferentially in directions perpendicular to the crystalline c-axis. This axis also was believed to be the preferred axis of nuclear alignment on the basis of known h.f.s. splittings. The results for 233Uthen indicated the existence of interference of L = 0 (S) and L = 2 (D) waves in the alpha particle intensity, such that a strong preference for alpha particle emission from the equator of the U nuclei results. Since these nuclei presumably have a prolate shape, Referewes p . 391
CH. VIII,
3 31
ORIENTATION O F ATOMIC NUCLEI I1
347
the result was contrary to the predictions of Hill and Wheeler78,who argued that the Coulomb barrier would allow alpha emission preferentially from the tips of prolate nuclei. Similar results were obtained for 235Uand 237Np,whereas no anisotropy was found for 234Uwith spin zero. Recently Roberts et al. extended their measurements, particularly on 237Np, t o temperatures below 1 OK, which necessitated considerable changes in the experimental procedure, of which the following aspects may be mentioned. a) The radioactive crystal consisted of a base crystal of uranyl rubidium nitrate with a 0.75 mg/cm2 coating of neptunyl rubidium nitrate. The237Npin the surface was estimated to give about the same heat production as the entire base crystal (6 ergs/min). The inhomogeneous heating probably does not result in temperature inhomogeneity at the temperatures used, see also c) and 5 4 . 2 . X-ray studies ascertained that the neptunyl coating grew in the same lattice structure and orientation as the uranyl base crystal. b) The a-particle intensity was measured inside the He-cryostat by a specially developed germanium counter79.A 1 x 1 cm large, 0.045 cm thick plate of pure Ge was etched and gold plated on both sides and a small voltage was applied across the electrodes. The electron hole pairs created in the Ge crystal by impact of an alpha particle give rise to a pulsed discharge ; the pulse height voltage is inversely proportional to the capacitance of the counter. It was found that the capacitance of the plate changed from about 600 p F at 77 OK to a value smaller than 100 pF at liquid helium temperatures, therefore giving an appreciably larger pulse height at low temperatures, which is in contradistinction to the behaviour of most scintillation detectors. At 77 OK a 200 keV resolution in the pulse height spectrum of 233Ua-particles was obtained (at 1 O K the resolution was somewhat lower). c) The extension of the measurements to temperatures of 0.2 OK required adiabatic demagnetization, hence a thermal switch between the sample and the 4He-bath. By means of copper and 3He exchange gas a thermal connection was established between the radioactive crystal and the coolant, which consisted of pressed Mn-NH,-sulfate, adiabatically demagnetized from 5000 Oe. A schematic drawing of this part of the experimental arrangement is given in Fig. 2. Not shown is that the assembly of Fig. 2 is mounted with cotton threads inside a metal cased vacuum space, immersed in liquid 4He. The heat contact with the 4He References
p . 391
348
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
3
bath is provided by 4He exchange gas or, more quickly, by means of a mechanical switch (upper part Fig. 2). The temperature was determined by a carbon resistance and by susceptibility measurements. d) The radioactive crystal was rotated by the magnetic force on a small permanent magnet inside the cryostat, induced by coil magnets outside the cryostat. The angle of rotation could be changed contin-
ROTATING OPPER CAN
Fig. 2. Part of the apparatus for the measurement of the directional distribution of alpha particles, used by Roberts et al.80.The sample is shown with the c-axis perpendicular to the plane of the drawing; it can rotate about a vertical axis. Only approximately one-fourth of the Mn-NHr-sulfate coolant is shown : the surrounding copper can is suspended from cotton threads inside a vacuum space, which in turn is surrounded b y helium and nitrogen dewars. RefiJvetzccs p . 391
CH. VIII,
9 31
ORIENTATION O F ATOMIC NUCLEI I1
349
uously between 0" and 90" and measured with an accuracy of fl" by measuring the capacitance of a variable condenser connected to the crystal. This procedure avoided a number of thermal connections to the refrigeration assembly. The directional distribution of a-particles from oriented nuclei may, quite generally, be expressed by
+
+
w(8,T ) = 1 B Z f 2 P 2 (cos 8) B,f,P,(cos 8) (25) where B, and B, are factors, which contain in particular the essentially nuclear factors for the amplitudes of emission of L = 0 and L = 2 a-particle waves ;f i and f4 describe the nuclear alignment as a function P
A - 0.238'K P=+0.0433°K I = 512 s=112
A ,P
i"""'
t5/2
I
~0945
0 f
1/2
-0116
-0153
* 512
-0207
?
112
f 312
Fig. 3. H.f.s. levels of the 237NpOf+-ion. A and P are magnetic and electric h.f.s. splitting parameters respectively; the energy scale is given in degrees Kelvin times k, the Boltzmann constant.
of temperature, T ; and P,, P, are Legendre polynomials in cos 8, where 8 is the angle of emission with respect to the axis of rotational symmetry, for this case the crystalline c-axis. Paramagnetic resonance datas1 of 237Npin uranyl rubidium nitrate have yielded the magnitude of the magnetic and electric h.f.s. splitting constants, A and P respectively, in the h.f.s. hamiltonian =ASJ, References p . 391
+ B(S,I, + SJJ + P{I?
-
+I(I+ 1))
360
W. J . HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
93
+
from which the small term B(S,I, SyIy) may be omitted for purposes of discussion (S = t).Also, it was showns1 that A and P have opposite signs; for P < 0 < A a state with I , = f p will be lowest, whereas for A < 0 < P the h.f.s. level scheme will be inverted and the lowest states have I , = f $ and I , = -I: ;. The level scheme for A < 0 < P is given in Fig. 3. If one calculates the functionsf, and f4 I
I
I A
I
>O
02
01
NORMALIZATION POINT 182
1
Fig. 4. Absolute magnitude of the coefficient, BPfB,of the P,(cos 0) term in the directional distributions of zs7Npalpha particles, given as a function of 1/T, where T is the temperature. Both the curves for P< 0 < A and for A < 0 < P are normalized t o the experimentally observed value of 1 Blfs 1 at 1 "K (normalization point). The upper curve is nearly a straight line up t o 1/T = 5'K-l.
from the resonance data, the results are, as regards both sign and temperature dependence, for the case P < 0 < A very different from the results for A < 0 < P. The experimental resultsso for as7Npare: a) the a-particles are preferentially emitted in directions perpendicular to the c-axis, as found earlier1.7s~77. b) The directional distribution does not show a P, (cos 0) term. c) The temperature dependence of the coefficient of P , (cos 6) is shown in Fig. 4 and definitely rejects P < 0 < A . References p . 391
ORIENTATION O F ATOMIC N U CLEI I1 35 1 9 31 A < 0 < P, although the lowest state has predominantly a
CH. VIII,
For
character, Roberts et al. calculate that a t 0.2 “K 40% of the nuclei are in I , = & 4 states, whereas 36% and 24% are in I, = & and & !-states respectively. Consequently the experimental result is that the a-particles are preferentially emitted in the direction of the nuclear spin, contrary to the interpretation in refs.76.77. The result A < 0 was contrary to the expectations of Eisenstein and Pryce82;Pryce showed late1-8, that the resonance data should have been interpreted by negative gz and A values; since these data require A P < 0 , the sign of P also had to be reversed. If Q denotes the nuclear electric quadrupole moment, PQ should be negative according to the theory of Eisenstein and Pryces2,which is based on the crystallographic evidence that the linear 0-U-0 uranyl ion lies along the crystal c-axis, surrounded by trigonal NO,-groups. If P > 0, the conclusion of Pryce is that Q < 0 ; the experimental results of Roberts et aLso should then lead to the interpretation of a preference for alpha particle emission from the flat surfaces of oblate nuclei. Roberts et al., however, suggest that Q > 0 on the basis of other nuclear data in the heavy element regiox~*~~J~. The conclusion of Roberts that cc-particles are emitted preferentially from the tips of the nuclei 237Np,233Uand 235U, which have prolate spheroidal deformation, is satisfactory from a theoretical point of view. If the formation probability for cc-particles is uniform on the surface of a deformed nucleus, one should indeed expect a preferential emission at the tips because of the larger barrier penetrability at these tips. And conversely a preferential emission of a-particles in equatorial directions should point to a strongly enhanced a-particle formation probability at the equator, so large that the higher barrier penetrability at the tips is overcompensated. Theoretically this would not be easy to understand except for a number of somewhat exceptional individual cases. However, the anisotropies for the three nuclei which have been measured apparently have the same sign. Hence, it is most satisfactory from the point of view of nuclear theory that the conclusion of the preliminary experiments and analysis had to be changed, because of the change of sign in A , hence in the sign of the orientation parameter fi.
Iz = f
+
References p . 391
352
W. J . HUISKAMP A N D H. A. TOLHOEK
[CH. VIII,
$4
4. Beta Asymmetry Experiments The first demonstration of parity non-conservation in weak interactions was given in Washington by Wu and Ambler et alF4, who showed that p--emission from polarized 6oConuclei is more intense in the direction opposite to the nuclear spin than parallel to the spin. lOcm LUCITE
UMPING TUBE
v d
b
Fig. 5. Experimental arrangements for measuring asymmetry of @-particle emission from polarized nuclei. a) Light guide cryostat for counting beta particles inside the helium dewar; the gamma counters measure the gamma ray anisotropy and hence the temperature ; b) Arrangement for counting positons by the intermediary of annihilation quanta, emitted from a glass cone below the sample and counted in coincidence by two diametrically positioned gamma counters.
Part of the experimental arrangement is shown in Fig. 5a; the 6oCo decay scheme may be found in Fig. 6. The s°Co source consisted of a thin crystalline layer of activated Ce-Mg-nitrate grown on a large surface perpendicular to the c-axis of a single crystal of non-activated Ce-Mg-nitrate. This crystal was cooled by adiabatic demagnetization, together with its surrounding Ce-Mg-nitrate housing, which served as Referewes
p . 391
CH. VIII,
3 41
ORIENTATION OF ATOMIC NUCLEI 11
353
an absorber of heat that may otherwise reach the sample by means of radiation, conduction or condensation of He-exchange gas. After cooling by demagnetization had been obtained, a solenoid was raised around the cryostat such that a vertical magnetic field was applied in the direction of small g-value of Ce-Mg-nitrate. That a sufficient degree of nuclear polarization was actually attained, was ascertained by 58
6oC0 5.3y
co 71d
162 117
'4
1.33
047
081
143
6oN1
5
e
v lo+ I
52
Ct
Fig. 6. Decay schemes of 6oCo, 5sC0, 52Mn(5.7 d) and 5zmMn (21 min).
measuring the anisotropy of the 6oCo gamma radiations by means of two NaI (Tl)-scintillation counters, the positions of which may be seen in Fig. 5a. The p-particles were counted by an anthracene scintillation crystal, viewed through a lucite light guide by a photomultiplier tube outside the cryostat. The ,%particle intensity was counted with the magnetic field alternately in the upward and in the downward direction. The results are shown in Fig. 7. I t is seen that, when the direction of the polarizing magnetic field is reversed, the gamma ray intensities remain unaffected, but the beta particle counting rate is drastically changed. This shows clearly that the probability for j3-emission does not possess mirror symmetry with respect to the plane perpendicular to the nuclear spin. As discussed in I, the absence of mirror symmetry with respect to that plane, signifies that parity is not conserved in p-decay. Henceforth we shall indicate by asymmetry the absence of this mirror symReferences p. 391
354
W. J. HUISKAMP AND H. A. TOLHOEK
[CH.VIII,
94
metry. The ,%particle intensity is, according to Fig. 7, smaller in the direction of the polarizing magnetic field than in the opposite direction. Since the spins of s°Co nuclei were, at low temperatures, found to be parallel (I, ref.lZ7)to the magnetic field, the emission of negatons is apparently favoured in the direction opposite to the nuclear spin. GAMMA RAY INTENSITY
1J
10
-
o
e
0
e r m
tol-
n12
,
,
I
,
P-ASYMMETRY
-
on
n.
tw
U I
EXCHANGE GAS IN
-
'0.7'
1'
2
I
'
I
4 6 8 TIME IN MINUTES
I
10
'
12
14
I
16
8
Fig. 7. Results of B°Conegaton-asymmetry experiment by Wu, Ambler et da4. The upper figure gives the gamma ray intensity as a function of time after demagnetization of the sample. The smoothed data are represented in the middle as the anisotropy E . The lower figure gives the beta particle counting rate with magnetic field anti-parallel ( 4 ) and parallel ( t ) respectively to the direction of emission of the p-particles. The nuclear spins are polarized in the direction of the magnetic field.
The emission of positons from polarized 5 8 C nuclei ~ was found in Leiden by Postma et aZ.85 and in Washington by Ambler et aZ.B6to be asymmetric as well. The apparatus used by Postma et al. is shown schematically in Fig. 5b. The nuclear polarization was obtained in the usual way with the aid of a polarizing field, applied to Ce-Mg-nitrate single crystals, of which the lowest was activated by 5 8 Con ~ its downward surface. The positons were first annihilated in a glass cone inside the cryostat and the annihilation quanta were counted in coincidence Refermces p . 391
CH. VIII,
41
ORIENTATION O F ATOMIC NUCLEI I1
355
by two scintillation counters outside the cryostat. This method of counting @-particlesis simple in the sense that introduction of a light guide into the cryostat with inevitable heat influx, is avoided. However, this constructional advantage is gained at the expense of a smaller counting rate because of an extra solid angle and the background of undesirable coincidences may be prohibitively large when the source emits high energy gamma rays. In the above 5 8 C experiment, ~ where some shielding was provided by the solenoid, the coincidence background amounted to a t least 25%. A separate scintillation counter was used for measuring the gamma ray intensity, from which the degree of nuclear alignment could be derived. It was found that positons are preferably emitted in the direction of the magnetic field, therefore in the direction of the nuclear spin of the 5 8 C nucleus. ~ Although the j3+-decay from 58C0 is, particularly a t the time when the experiment was performed (see $5), less accessible to simple interpretation than the 6oCo case, the experimental result shows that parity is not conserved in positon decay. Further the asymmetry effect has the opposite sign with respect to that of negaton emission. Similar results were obtained by Ambler et dB6, using the same techniques as described for the 6oCoexperiment, for positon emitting 5 8 C and ~ 5 6 C nuclei. ~ Grace et aLE7demonstrated the existence of a j3-asymmetry by means of photographic emulsions, using otherwise the same techniques for obtaining nuclear polarization of 60Coin Ce-Mg-nitrate. Most available films are, at low temperatures, almost insensitive to @-rays,but it was found that the sensitivity of Ilford Industrial GX-ray film was reduced by only a factor two when cooling from room temperature t o 1°K. A two minute exposure at 1 "K to a 10 pc source of 6oCoat a distance of 5 mm produced sufficient blackening to detect a change greater than 4% in the @-rayintensity. The error was largely due to the lack of reproducibility of the blackening over various parts of the film and to fogging by gamma rays. Before the results of these and of other @-asymmetryexperiments are discussed quantitatively and compared with theory, in 1-5 a comparison will be made with older gamma ray anisotropy measurements. 1. In gamma ray intensity measurements the radioactive nuclei are homogeneously distributed over the entire paramagnetic crystal. On the other hand, the ,&particle emitting source should, ideally, be very References p . 391
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W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
$4
thin in order to prevent energy losses and scattering in the source backing material. Very thin crystals, however, are hard t o grow and to mount in a cryostat ; also a sample of high specific activity is difficult to cool and maintain at low temperatures for a sufficiently long time. The compromise, adopted in the forementioned experiments, is to grow a radioactive layer as thin as possible on top of a nonradioactive single crystal. The crystal helps the layer to crystallize in a well defined tructure and serves as a sink for the heat produced by radioactive warm up, condensed He-exchange gas, absorbed radiation etc. A considerable number of back scattered j3-particles are thereby introduced in the mc Tsured intensity, requiring corrections to the observed asymmetry effect. 2 . Closely related to the foregoing remarks is the problem of temperature measurements. Experiments with Ce-Mg-nitrate have shownss that a thin, radioactive surface layer rapidly warms up to 0.01 to 0.02 OK after adiabatic demagnetization. At these temperatures apparently the radioactive heat production is carried off into the interior parts of the crystal, since the warm-up rate of the surface is roughly equal to that of the base crystal. This can be reasonably explained, assuming a T3-dependence of the heat conductivity. Since the surface temperature is different from that of the base crystal, susceptibility measurements are of little use; also, the temperature in the radioactive layer is expected to be inhomogeneous and only an average temperature is determined by a gamma anisotropy measurement. 3. A polarizing magnetic field is essential for obtaining nuclear polarization and @-asymmetryeffects. Apart from measuring the change in counting rate when heating the sample, one can also reverse the direction of the magnetic field in order to cause a variation in counting rate. 4. For the determination of the degree of nuclear polarization, fi G < I, > /I,the gamma ray intensity (if normalized to 1 at high temperatures), W(e),has to be compared e.g. in the case of s°Co with spin I,, = 5, with the formula
ye)
=
I
-
;-+ f2p2(cos e) - gg+ f4p4(cos e).
(26)
Measurements of " ( 0 ) for two values of 0 may lead to a determination of both f2 and f4, the latter often being negligible in /3-asymmetry experiments. If the h.f.s. level scheme is completely known, the temperreferences p. 391
CH. VIII,
3 41
ORIENTATION OF ATOMIC NUCLEI 11
357
ature dependence of both fl and f, can be calculated and hence the relation betweenf, andf, is known. For an unknown nuclear magnetic moment but equidistant h.f.s. levels, the relation between fl and fi is known as well, as may be seen from Fig. 8 where both functions are plotted nersus the parameter , ! I , which is the h.f.s. energy splitting divided by kT. However, in more complicated cases only an estimate of fl may be possible.
Fig. 8. a) Dependence of v/c on kinetic energy of beta particles, where ZI is the velocity of the particle and c is the velocity of light. b) Orientation parameters fi and f, as a function of the parameter ,9, which is the energy difference between successive h.f.s. levels, divided by k T . The curves I, = 2 and I, = 5 apply to 68Coand e°Co respectively; the asymptotic values offi andf, for complete polarization have been indicated by arrows. c) Magnitude of the asymmetry parameter, A+, as a function of the ratio, = cVMF/cAMGT
showing the effect of the Fermi-Gamow Teller interference term on the beta asymmetry, calculated for the case of W o .
5 . The analysis of the results of beta asymmetry measurements is complicated by the circumstance that the temperature determination from W(0) depends on the beta decay characteristics. The problem may be more precisely stated as follows. The beta particle intensity, W,(0), is linearly dependent on the degree of nuclear polarization, fl, of the initial nuclei. The gamma ray intensity, W(0),depends on the degree of nuclear alignment, fi, of the nuclear spin system after the beta decay has occurred. The relation between fi and the degree of nuclear alignment of the initial spin system, f,, is determined by the coupling of angular momenta I,, I’ (Fig. 1) and J , the angular momentum carried off in the beta transition. If I , = I’ Gamow-Teller ( J = 1) matrix elements, MGT,and Fermi ( J = 0) matrix elements, M,, may simultaneously differ from zero (and hence interfere in the References
p . 391
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W. J . HUISKAMP AND H. A. TOLHOEK
34
[CH. VIII,
directional distribution WJ. Then the relation between f 2 and depends on the ratio X = C,M,/C,MG, e.g. for
j ; = +(l
+ A)f2
L = X2/(1+ X 2 )
0
< A < 1.
ti’
(27)
Most beta asymmetry experiments are preceded by measurements of “(6) as a function of T . Then, if bothf2(T) and the relation between W(0) and fi are known (i.e. if respectively, h.f.s. levels and gamma decay characteristics are known), X and A may be determined from (27), and according to the preceding subsectionf, also can be calculated. As a result, the beta asymmetry experiment may be used as a check and will presumably give no new information. Conversely, however, if for example I, is unknown, a measurement of the beta asymmetry may give very useful complementary information. However, uncertainties on the T - T @relation, the h.f.s. interaction, reorientation of the nuclear spin 1‘,mixed multipolarity of gamma radiations etc. may provide complications which make the determination of I’ - I, from one @-asymmetryexperiment impossible.
RESULTS Hereafter the experimental results for various nuclei will be discussed on basis of the four assumptions of 3 2, namely assuming twocomponent neutrino theory, lepton conservation, time reversal invariance and V-A interaction. Then eq. (10) gives
where the upper sign refers to positons and the lower sign to negatons; D / C is the beta particle velocity compared with the velocity of light and 8 is the angle of emission with respect to the axis of nuclear orientation. The experimentally observed quantity is usually dexp
= (W t ) - W (J. )>/W t 1 + W (.1)I,
where W (t ) is the beta particle intensity in the direction of the polarizing magnetic field and W (4.) is the intensity opposite to the magnetic field, both normalized to unity at high temperatures. dexp will henceforth be called “experimental asymmetry” and should not be confused either with A , or with the concept of anisotropy of the directional distribution of /?-particlesin forbidden decays from aligned nuclei, which is discussed in I and later in this section. deXp should be corrected for finite solid angle, backscattering of References p . 391
CH. VIII,
$ 41
359
ORIENTATION OF ATOMIC NUCLEI I1
beta particles, background of compton electrons in the scintillator, influence of the magnetic field reversal on the beta particles and photomultipliers etc. I t is seen that the corrected value of dexp, when divided by v/c and by f l , yields an experimental value for A,, to be compared with eq. (13). A graphic illustration of the three factors in the product ( v / c ) f l A , is given in Fig. 8, A , representing the theoretical expression for the allowed 68Codecay as a function of the Fermi-Gamow Teller mixing ratio parameter, X. It may be stressed that the sign of f l is not always amenable to direct experimental verification, such as measurement of circular polarization of gamma radiation, but instead is often deduced from paramagnetic resonance data by theoretical reasoning.
-
6OCo deXp was founda4.86 to be -0.4 at vfc m 0.6 and fl m 0.6, as may be partly checked in Fig. 7. Since f l > 0 , A - = - 0.40/0.36 - 1 and this was the first indication that the interference between parity conserving and parity non-conserving terms in the beta interaction hamiltonian was close to maximal, or C,’ = C,. This result is just what one should expect for a two-component theory of the neutrino or antineutrino, in conjunction with the known fact that the 5+(,9-)4+ decay of 6oCo(Fig. 6) is a pure Gamow Teller transition. Furthermore, Ambler et dsa showed that d e xwas p proportional to v/c in the range from v/c = 0.4 to v/c m 0.8, as predicted by theory; the accuracy of the measurement was, as in the other experiments discussed hereafter, not high enough to establish the possible existence of a 2-dependent term, such as appears in formula (13). 5 8 ~ 0
Two sets of measurements by Postma et
+
aLs57
89
+
gave dexp = 0.12
f 0.05 and d e x=p 0.08 f 0.03 for f l = 0.68 and f l = 0.50 respec-
tively, and for v/c = 0.72. The averaged, corrected value of A+ is 0.32 f 0.09. Similarly, Ambler et aZ.86 found A+ = 0.33 f 0.05. These values of A+ agree with the theoretical prediction for a pure Gamow Teller transition (see $ 5) and a 2+(/3+)2+transition, for which case formulae (13) and (16) give A+ = l/(Io 1) = +.A small value for the ratio MF/MGThas been derived from the measurementsloOof gamma radiation from 5 8 C (I~ C,M,/C,M,, [ < 0.09) so that the experimental values found for A+ and M,/M,, are in good agreement.
+
References fi. 391
+
360
W. J. HUISKAMP A N D H. A. TOLHOEK
[CH. VIII,
34
Also in the 5 s Cresults, ~ d e xwas p found to be linearly dependent on v/cs6s89. 52Mn
+
Measurements by Ambler et ~ 1gave . A+ ~ =~ 0.23 -& 0.01, whereas Postma et ~ 1 .8 9~ found ~ . A+ = + 0.20 & 0.01. An experimental runs9 which is the positon emission counter part of Fig. 7, may be found in
%? 5800C
5600C
.t
5400C
880C
830C
4 780C
I
.1.
T
I
600C
I
dI 5 5oc
'
-+--!+-$ t. o t , s
I I I
I
10
15
8- asymmetry 4 3% vlt=054 I
20
I
25m
Fig. 9. Results of 62Mn positon asymmetry experimental run. The gamma ray intensity and beta particle intensities are plotted as functions of time after demagnetization. The gamma counters measured intensities in directions &c and &z with respect to the direction of nuclear polarization, which is opposite to the direction of the magnetic field, H. H(J. ) denotes the situation where the magnetic field is opposite to the direction of positon emission.
Fig. 9. It is seen that both in Fig. 7 and 9 the @-particlecounting rate is largest in the direction opposite to H. However, both the direction of preferential /%emissionwith respect to I and the preferred direction of I with respect t o H are for 52Mnopposite to that for 6oCo. Rrfereiices p . 391
CH. VIII,
9 41
ORIENTATION OF ATOMIC NUCLEI 11
361
Both experimental A+ values are higher than the theoretically calculated result for a pure Gamow Teller transition in the positon decay between an initial and a final state with spin 6 ( I , = I' = 6), leading to A+ = l/(I, 1) = 3. This discrepancy may be entirely due to errors in the estimate of f l from the gamma ray anisotropy, which was found to deviate from theoretical calculations. If one assumes the estimate of fl to be correct, Postma et al. give
+
X
= - 0.05 or - 9.3 ; Ambler et ul. find X = - 0.05 or X = - 8
f 1.
Except from the beta asymmetry from oriented nuclei, the determination of X can also be made from p-y-circular polarization correlation, as was pointed out in tj 2. Here also the remarkable fact was mentioned that the interference term has a different sign in both cases (cf. the formula (13) and (23)). The experimental result of Boehmg3for the ,!+-circular polarization correlation gave a result
A: = -0.48
& 0.15
which is larger than the value A; = - 0.166 for a pure Gamow Teller transition ( X = 0 ) , thus leading to a positive value of X , namely either X = 0.05 to 0.20 or X M 5. Hence one has contradictory experimental evidence for the value of X.One can reject the values of X with large I X 1 (namely X = - 8 & 1 and X m 5), but further one can not say more than that I X I 0.2. It is to be regretted that one can not decide to a definite value of X , since as was indicated in 3 2, cases with well established M,M& interference terms are interesting in several respects. Further experimental evidence is here most desirable. The experimental results have definitely excluded the possibilities I , = 5 or I , = 7. Hence one has here a case that a useful nuclear spin determination was made on the basis of a beta asymmetry experiment ; see also ref.132.
<
56C0
+
Ambler et a1.86 found A+ = 0.22 f 0.02, which is compatible with the theoretical prediction A+ = l/(Io 1) = when assuming a pure Gamow Teller transition in the allowed positon decay between the states with known spin 4+ --f 4f. Again some interference between Gamow Teller and Fermi matrix elements may be possible, particularly since Ambler et al. considered the estimate of fi from the gamma anisoReferences p . 391
+
+,
362
W. J . HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
94
tropy unreliable and prefer to calculate fi from the temperature, obtained in similar experiments with s°Co and 5 s C ~ .
W o , 56Mn Preliminary results on these short lived nuclides (18 h and 2.6 h respectively, see Fig, 10) were reportedg1. For 55C0dexp = 0.23 & 0.07, appreciably smaller than the esti-
+
56 25Mn3,
n-
3+
'-27 2.98 2.65
28
0.845 0.645 9 9 OIo
Fig. 10. Decay schemes of 66Co(18 h) and 66Mn(2.6 h).
mated value fi v/c rn 0.40 for a pure Gamow Teller transition in the decay from a state I , = {- to a I' = Q state (A+ = + 1). In the case of 56Mn,d e x=p 0.046 f 0.003, which is an order of magnitude smaller than the value f l v/c M 0.5, expected for the case of a pure Gamow Teller transition in the 3+(,9-)2+ decay (A- = - 1, fl < 0). The discrepancies may be of experimental origin and need further clarification.
+
l66mH0
Postma et ~ 1 . ~ ~ 9when ~ 2 , polarizing lssrnHo (> 30y) in Nd-ethyl sulfate, found d e x=p- 0.08 if fi M 0.6, or A- M - 0.14. This would nicely agree with the formulae for allowed decay between two states with spin 6 ( I , = I' = 6) and pure Gamow Teller transition: A - = - l/(Io + 1) = - 3 (Fig. 11). I t was also shown that the energy dependence of dexp does not fit the 'theoretical formulae for
+
References
p . 391
CH. VIII,
5 41
363
ORIENTATION O F ATOMIC NUCLEI I1
+
first forbidden beta decay with I , = I' 1 = 7. A few other possibilities could be eliminated from the spin and parity analysis, which is, however, complicated since the order of forbiddenness of the beta transition is unknown.
laoTb Experimentssg? 92 by Postma et al. on leoTb (Fig. ll), polarized in Nd-ethylsulfate single crystals, gave an average value of dex,/(v/c) = - 0.394 f 0.005 when fl = 0.42. Using another 919
+
..
166m..
175. 70' D10572-
67n099
-1.OMc" 90?lo 0.870
\\
%f.790
36%
w 0 . 3 9 6
I
0.817 800lo E2 I
I
0.877
\
0.706 800/0
I
E?
0.087 160 baDY94
0.35 Eltp2
0.267 0.187 2~loo~lo
*o
0.080 0 r9 8
175 O 71 L'104
Fig. 11. Decay schemes of lsoTb(72d). lesmHo (>30 y) and 176Yb(4 d). The lsoTbdecay scheme is much more complicated than is shown in the figure. I n particular a gamma radiation of 1.18 MeV between the 1.26 and 0.087 MeV levels and a radiation from the 1.26 MeV level to the ground state of lsoDy are not shown. The lssmHo decay scheme as proposed by Grace and modified by Boskma (see ref.Io7b),is somewhat simplified.
crystal, W-(e) was measured under an angle of 0 = 55" with respect to the crystal c-axis, giving an asymmetry effect of - 0.360 & 0.025 upon reversal of the polarizing field. From this it is calculated that dexP(O = 0) = ( - 0.70 f 0.05)vlc for fl = 0.65 and for beta-particle energies between 0.2 and 0.8 MeV. Essentially these results give A - = - v/c as might have been expected for an allowed decay between states I , = 3 and I' = 2. However, according to the value of logft = 8.8 for the predominant branch in the high energy end of the beta spectrum, i.e. for the 0.87 MeV radiation, this is a first forbidden decay. P ~ s t m ashowed ~ ~ from the theoretical formulae of M ~ r i t a ~ ~ that A & for first forbidden decays with allowed spectral shape and References p . 391
364
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
94
with A I = 0 or & 1, is in good approximation equal to the A , value for allowed decay. The conclusion then is that laoTbhas spin 3. The measurements under an angle of 55” with the c-axis were performed in order to observe a P,(cos 0) term in the directional distribution of beta particles emitted from aligned nuclei. No such effect was found within the experimental uncertainty of f 1%.
14Pr dexp M 0.08 was found for the first forbidden negaton decay of 143Prnuclei, polarized in Ce-Mg-nitrate. Since no gamma rays are emitted, fi can only be roughly estimated; assuming ~ ( l ~ ~ wP r ) ,u(l4lPr)= 4pN and I , = -2, a considerably larger dexp should have been expected. The measurements showed that +{W(t ) W ( J.)}was temperature dependent, unlike all other cases discussed above; when the high temperature value is again normalizedto 1 , the low temperature value may be designated by W(O = O,fl = 0 ) , and the data give W (0 = 0 , f i = 0) = 0.028 f 0.002. Measurements of the beta particle intensity in directions perpendicular to the nuclear orientation axis showed correspondingly W(O = in,fl = 0) = - 0.009 & 0.002. This experiment therefore, has shown for the first time the presence of a P,(cos 0) term in the directional distribution of beta particles emitted from oriented nuclei 1 4 6 ~ .This effect will be called “anisotropy” because of the similarity to the anisotropy of directional distributions of alpha and gamma emission from aligned nuclei and also in order t o distinguish it from the mirror asymmetry related to parity nonconservation. The term B , in the theoretical expression for the directional distribution W(6,f i = 0) = 1 B,(v/c) f,P,(cos 0) depends on the various first forbidden nuclear matrix elements and, in general, on the energy. B , would become independent of unknown ratios of nuclear matrix elements only in the case of unique first forbidden transitions ( d l = 2 , cf. I, eq. (1.11)).
+
+
+
+
TIMEREVERSAL INVARIANCE. A measurement of the j3-y-directional correlation for polarized 62Mnnuclei has been reportedg0, in which a search was made for terms which change sign under time reversal. Such a term is A k ) ( j . k ) , having a coefficient (19), as discussed in 3 2. The positons (momentum p) were observed in a direction perpendi-
l*(g
Referelaces p . 391
CH. VIII,
4 51
ORIENTATION O F ATOMIC NUCLEI I1
365
cular to the nuclear polarization vector, j, in coincidence with gamma rays (momentum k ) emitted perpendicular to p and under angles of 1-45' and -45" with respect to j. The difference between the coincidences of the -45" gamma counter and the coincidences of the +45" gamma counter, normalized to 0 at high temperatures, measures the term of interest; this difference was shown to be (0.012 & 0.022) (v/c)fi (apart from a trivial factor). The approximately zero result may be due to time reversal invariance but may also be due t o lack of sufficient interference between Fermi and Gamow Teller matrix elements. From the upper limit for the effect and from the measured ,&asymmetry Ambler et aLgodraw the conclusion that the phase 8 is restricted by 140" < 8 < 250", where 8 is given by
(8 = 0" or 8 = 180" if time reversal invariance holds). However, this conclusion is based on a value of 0.05 < 1 X 1 = I C,M,/CAMG, I < 0.1. It was discussed under 52Mnthat this seems not yet established beyond doubt so that further experimental evidence concerning time reversal invariance seems desirable. CONCLUSION a) Large asymmetry effects have been observed both in allowed and in forbidden beta transitions; b) the presence of Fermi-Gamow Teller interference has not been definitely established in beta asymmetry experiments with polarized nuclei ; c) spins of beta decaying nuclei could be determined in a few cases e.g. 52Mn,160Tb;the number of possible assignments could be reduced in some other cases; and d) a P,(cos 8) term was found in the directional distribution of a first forbidden decay.
5. Experimental Results on Gamma Radiation The experiments described in this section did not lead to discoveries of such fundamental importance as the beta asymmetry experiments discussed in the preceding section. On the other hand, a considerable amount of experimental data have been gathered in the field of gamma spectroscopy. It is the purpose of our discussion to give a short catalogue of the work done since I was written. A theoretical introducReferences
p. 391
366
W. J . HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
95
tion to the subject may be found, e.g. in I ; the experimental methods are well known and in the following we will only mention a few hitherto unused techniques. Practically all experiments were performed by measuring the gamma ray intensity in directions parallel and perpendicular to the axis of nuclear orientation, which intensities will be denoted by W(0) and W(+) respectively if normalized to 1 at high temperatures. In most cases we will quote the magnitude of the anisotropy, E
-{W(b)
- W(O)l/W(:n)
as a, albeit rather arbitrary, figure of merit, thereby complying to the usage of publishing E instead of W(0)and W(&c)separately. The results will be listed according to increasing 2 and A and not in chronological order. Since this' section is complementary to I, recent results will be given relatively more attention.
5zmMn 52Mn in the 21 min isomeric state (Fig. 6) was aligned by Bauer and Deutsch94 at M.I.T. growing 8 h 5aFeinto magnetically diluted nickel fluosilicate crystals, cooled to low temperatures. After Pf-decay of 62Fe the alignment of the 21 min state was measured by means of the gamma ray anisotropy of the 1.43 MeV radiation of 52Cr,populated by P+-decay of 52mMn. Comparison of the gamma anisotropy of the 1.43 MeV radiation with that of the 0.84 MeV E2 radiation from S4Mnin'the same crystal, gives for the nuclear gyromagnetic ratio = 0.36 i 0.08 g(52mMn)
for a pure Fermi p-transition and 0.52 Teller transition.
0.08 for a pure Gamow
52Mn From gamma ray anisotropy measurements of 52Mn(5.7 d) in CeMg-nitrate the nuclear magnetic moment was founde5to be =2 . 8 ~ ~ . ,~4(~~Mn)
Dynamic polarization methods (9 8) have since132 given the more reliable value ~ ( ~ 2 M= n )3.00 f 0.15 pN.This shows that in sufficiently large polarizing fields ( > 500 Oe) the degree of nuclear orientation of Mn nuclei in Ce-Mg-nitrate can, to a reasonable approximation, be References p . 391
CH. VIII,
5 51
ORTENTATION O F ATOMIC NUCLEI I1
367
described by the paramagnetic resonance data of Trenam on Mn in Bi-Mg-nitrate146f. Bauerg4points to the interesting fact that the nuclear g-values for 52mMnand 52Mnare nearly equal, whereas the spins are widely different, I, = 2 and I, = 6 respectively. 54Mn
Bauerg4oriented 54Mn(310 d) both in Ce-Mg-nitrate and in (10%Ni, 90%Zn) SiF,, 6H20. From the comparison between W(&z)and W(0) for the 0.84 MeV E2 radiation it was concluded that no change in orientation occurs in the preceding 0-decay, and therefore 54Mnhas probably spin 3, in agreement with Oxford results1 and the resonance data of Kedzie et ~ 1 . l ~Whereas ~. Bauer finds p (54Mn)= 2.55 f 0.21 pN, Kedzie et ~ 1 . report ~ ~ 2 p("Mn) = 3.29 f 0.06 pN. From the measurement of the circular polarization of the 0.84 MeV gamma ray, p was found to be positiveg4. 56Mn
Dagley et ~ 1 . 9 6in Oxford aligned S6Mn (2.6 h) in a fluosilicate single crystal, containing Zn, Ni and Mn in the proportions 90 : 10 : 1. This crystal was irradiated by about 5 x 1014 thermal neutrons/cm2, producing 100 pc 66Mnat the start of the experiment. The temperatureentropy relation for this crystal was determined from nuclear alignment experiments with %Coin a crystal of the same composition. The known nuclear magnetic moment and decay characteristics of 58C0 (Fig. 6), in conjunction with paramagnetic resonance data on stable Co, make such a determination feasible. From the known p(56Mn)and paramagnetic resonance data one can calculate the theoretical gamma ray anisotropy as a function of T , e.g. for the 0.845 MeV radiation (Fig. 10) with known multipolarity and spin change. It was found that the degree of nuclear alignment obtained was about 50% of the expected value, which discrepancy is attributed to radiation damage, since annealing could produce an increase of alignment up to 75% of the calculated value. Comparison of the 0.845 MeV anisotropy with the 1.81 and 2.13 MeV radiations gave values for the mixing ratio d(E2/M1) of + 0.19 f 0.02 and - 0.28 f0.02 respectively. Bauer et dg7 have grown 2.6 h 56Mnin Ce-Mg-nitrate and measured directional distributions of various gamma rays. The largest anisotropies, E , obtained with a polarizing field of 450 Oe, were for the 2.13 MeV, 1.81 MeV and the 0.845 MeV radiations E = + 0.40, E = + 0.18 References
p . 391
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W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII.
95
and E = + 0.27 respectively. By comparison of the last two anisotropies with E = 0.40 for the €22 radiation of 0.845 MeV, the mixing ratios of the 1.81 MeV and 2.13 MeV radiations were found to be d(EZ/Ml) = 0.11 f 0.06 and d(EZ/Ml) = -0.27 0.03 respectively. In a separate experiment 52Mn was added to the source and comparison of the anisotropies of the 52Mn and 56Mn radiations gave g(52Mn)/g(56Mn) = 0.47 f 0.05 or ~ u ( ~ ~ M = n3.35 ) f 0.25 ,uN.Measurements of the circular polarization of the 0.845 MeV radiation showed ,u to be positive. Again, as in 52MnexperimentsQ5,a substantially lower anisotropy was found if the polarizing magnetic field was removed.
+
5 5 ~ 0
Bauer et aLg8polarized 55C0 (18 h) nuclei (Fig. 10) in Ce-Mg-nitrate and measured the angular distribution of the 1.41 and 0.935 MeV gamma rays, simultaneously with the 0.80 MeV E2-radiation from 5 8 C nuclei, ~ which were grown into the same crystals. The largest anisotropies obtained were ~ ( 1 . 4 1MeV) =
+ 0.24 and ~(0.935MeV) = + 0.16 +
compared to ~ ( 0 . 8 0MeV) = 0.13, if a polarizing field of 450 Oe was applied. Using crystals with 55C0 alone, the linear polarization of the 1.41 and 0.935 MeV radiations also were measured simultaneously with the anisotropy. Simultaneous measurements of P , and E may, a t least in principle, lead to a determination of the mixing ratio of a gamma radiation even if the temperature or nuclear magnetic moment is unknown. This originates from the fact that interference terms between dipole and quadrupole radiation contribute very differently to the anisotropy E and to the degree of linear polarization P,, whereas for pure radiations for instance 1 P,(B = in) [ = [ E However, the experimentally measured quantity is not P,, but P,Q instead, where Q is the quality of the analyzer for the determination of the linear polarization; more precisely, Q is the ratio of the linear polarization dependent part of the Compton scattering differential cross section to the differential cross section for unpolarized radiation. Bauer et al. do not evaluate Q but follow a more complicated analysis, leading to the conclusion that the 1.41 MeV radiation is a pure quadrupole radiation and that the spin sequence --f 8 3 C which had not been ruled out by previous experiments, is definitely impossible.
I.
+
References p . 391
CH. VIII,
9 51
ORIENTATION OF ATOMIC NUCLEI 11
369
5 6 ~ 0
The results, reported in I, were extended by Diddens et aLg9 to measurements of the anisotropy and linear polarization of gamma rays of 56C0 nuclei, which were polarized in Ce-Mg-nitrate. The mixing ratios? S’(E2/Ml) were determined to have the following values: 0.15, S’(2.02 MeV) M + 10 S’(1.75MeV) M - 0.03, S’(l.81 MeV) M and S‘(2.13 MeV) M - 0.3. It was further found that the 0.845 MeV, 1.24 MeV, 2.61 MeV and 3.25 MeV radiations are pure E2 transitions, whereas the 1.75 MeV radiation is predominantly M1. The largest values of E and P,, obtained at 1/T = 250°K-1 and for the 0.845 MeV radiation, were: E = 0.32 f 0.01 and P,(&z)= - 0.32 f 0.015. The decay scheme and the assignments of spin and parity given by Diddens et aZ. agree with other investigations, except that there is doubt about the 3+ assignment of the 3.84 MeV level.
+
+
~*CO
Precise measurements of the anisotropy of the 0.81 MeV gamma radiation (Fig. 6) emitted from aligned 5 8 C nuclei, ~ were performed in Oxford100 in order to obtain an accurate value of the ratio I MF/MGTl, of Fermi to Gamow Teller matrix elements in the allowed (A1 = 0) 8+-decay of 58C~. Earlier experiments of Griffing and Wheatleyl had been made in biaxial Co-Tutton salt with the intention of measuring , u ( ~ ~ C Ousing ), the gamma ray anisotropy of s°Co as a thermometer. These experiments also yielded a value I C,MF/C,MG, l2 = 0.12 f 0.04, which was, however, incompatible with the later observed apparent absence of interference effects in the 8-asymmetry of polarized 5 8 C ~ nuclei. Dagley et ~ 1 .chose l ~ the uniaxial Ni-fluosilicate as a cooling salt in which 5 8 C was ~ grown. Only one single crystal was used in order t o avoid inhomogeneous temperatures. After a few small corrections were applied, I C,MF/CAMGT = - 0.003 f 0.005 was found, which agrees with the 8-asymmetry experiments. Later experiments of Wheatley et aZ.ll revealed that, although the gamma anisotropy of 6oCoin Co-Rb-tutton salt does fit the theoretical expectations nicely, a combination of %Co and 6oCoactivities does not give agreement with theoretical curves to the accuracy previously
t 6’ = 1/6,where 6 is defined according to I, ref.e1; elsewhere i n this chapter the convention of e.g. Blin-Stoyle and Gracelo is followed, where 6 has the opposite sign of (and is also the inverse of) the 6 of I, ref.*l. References p . 391
370
W. J. HUISKAMP AND H . A . TOLHOEK
[CH. VIII,
95
stated for I M,/M,, 12, systematic errors being as large as 0.1 - 0.2 and varying with temperature.
'UPr Grace et aZ.lo2oriented 142Pr(19 h) nuclei in Ce-Mg-nitrate, both in zero external field and under influence of polarizing fields of 300 and 600 Oe along the trigonal c-axis. At T = 0.004 OK in a 300 Oe polarizing field, a gamma anisotropy E = 0.14 was obtained, which is appre-
-kt2+
,
1.57
2.15
9f0b
b+.0.091 2.4 nr +M1+E2 6--Q2*02
*LO '46:Prn66
Fig. 12. Decay schemes of
14*Pr and 147Nd.
ciably larger than the gamma anisotropy of nuclei, aligned in zero field by the anisotropic h.f.s. coupling (gL = B = 0 ) : E = + 0.06 at T = 0.004 "K. The positive sign of E is only compatible with spin 2 for the 1.57 MeV state (Fig. 12). No additional anisotropy increase was found when using a 600 Oe magnetic field. From these results it was concluded that the nuclear magnetic moment p = 0.11 pN, assuming that no angular momentum is carried off in the b-transition 2- --f 2+. If the electron antineutrino field contained one unit of angular momentum, the interpretation of the results would have been ,u = 0.17 pN. Similar results were obtained by Daniels et aZ.lo3; although the highest values of E were somewhat lower (both in a 320 Oe polarizing field and in zero field E w 0.04) and correspondingly the temperatures somewhat higher, practically the same values for p were calculated. Daniels uses only the data at relatively high temperatures and calculates p from the experimentally observed relation E = a/T2,where a = 18.7 x lO-' OK2. References p . 391
CH. VIIL,
9 51
371
ORIENTATION OF ATOMIC NUCLEI II
The experimentally observedlo2-l o 3 dependence of E on the magnitude of the polarizing magnetic field shows that crystalline electric field effects are not negligible, but on the other hand, are a factor 10 smaller than expected on basis of paramagnetic resonance data on stable 141Pr. '47Nd
Most of the results were quoted in I, but paramagnetic resonance have since shown the spin of la7Nd (11 d) to be instead of p, and p = 0.56 pN (Fig. 12). Bishop et ~ 1 . 1 0 5 smeasured the anisotropies, E , of the 0.53 MeV and 0.090 MeV gamma radiations as well as the degree of linear polarization in the 8 = +n direction, Pl(+n).At T = 0.04 OK in Nd-ethylsulfate they find E = 0.20, P,(+n)= - 0.136 for the 0.53 MeV radiation and E = - 0.093, P1(+n)= - 0.055 for the 0.090 MeV radiation; a calculation of p from the 0.53 MeV data and 0.090 MeV data separately gives p = 0.44 & 0.06 pN and p = 0.27 & 0.02 pN respectively. From these results it may be concluded that the nuclear orientation is appreciably disturbed during the 2.4 x s lifetime of the 0.090 MeV level; the discrepancy between p = 0.56 ,uNand the p-value from the 0.53 MeV data may possibly be attributed to other causes than disturbance of nuclear orientation.
+
+
"Prn 149Pm(45d)was aligned by Grace et aZ.1°5bboth in the ethylsulfate and in the double nitrate. Preliminary data could be interpreted by assuming the Pmw ground state to be a non-Kramers doublet in the ethylsulfate and a singlet, influenced by a nearby doublet, in the double nitrate146d.The gamma radiation is probably nearly pure M1. 1WTb Johnson et a1.1O6 aligned 160Tb-nuclei (73 d) in Nd-ethyl sulfate (Fig. 11). The angular distributions at low temperatures could be expressed as W(O)= 1 A,P,(cos O), where A , > 0 for the 0.875 MeV radiation, which is mainly a quadrupole transition, A , < 0 for the 0.30, 1.18 and 1.27 MeV radiations, which are probably dipole radiations with Oyo,1% and 16% quadrupole admixture. Recent meas~rementsinLeiden~~7*gave, at temperature T=0.017 OK, ~ ( 0 . 3 MeV) 0 = 0.40, ~ ( 0 . 8 8MeV) = - 0.14, ~ ( 0 . 9 MeV) 6 = + 0.28,
+
+
References p . 391
372
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
+
35
= 0.25 and ~ ( 1 . 2 7MeV) = +0.17. From these anisotro~ ( 1 . 1 MeV) 8 piesit isconcluded that the 0.30 MeV, 1.18 MeVand 1.27 MeV radiations are dipole radiations with O%, 1% and 10% quadrupole admixturerespectively, assuming the 0.96 MeV radiation (whichis a superposition of a 0.964 MeV 2+ + O+ transition and a 0.960 MeV 3+ --f 2+ transition) to be pure quadrupole. Measurements of the linear polarization of the 0.88 MeV 0.96 MeV and of the 1.18 MeV 1.27 MeV radiations yielded P, = - 0.1 and P, = 0.3 respectively. The P , data agree approximately with the F data, but a more significant result is that the 1.18 MeV and 1.27 MeV gamma rays have a predominantly El M2 character and affirm the negative parity for the 1.262 and 1.359MeVlevels. The temperature dependence of the 0.96 MeV gamma ray intensity does not fit the calculations based on paramagnetic resonance data on Tb3f in Y-ethylsulfateI""". An increase of 50% in ~ ( 0 . 9 MeV) 6 was observed when a polarizing field of 200 Oe was applied in the direction of the large g-value (17.7) of Tb3+. The conclusion is that the nuclear orientation data are internally consistent when adopting the spin and parity assignments of Fig. 11, which assignments are also corroborated by other nuclear spectroscopic data.
+
+
+
166mH0
Measurements of the anisotropy and linear polarization of the 0.817 MeV and 0.706 MeV gamma radiations from aligned 166mH~ ( >30 y) nuclei were performed by Postma et ~ 2 1 . l ~ ' ~A. single crystal of Ndethylsulfate was used at temperatures down to 0.025 OK, but the anisotropies were found to be nearly independent of temperature for 0.025' < T* < 0.050 OK and the degree of nuclear alignment must have been nearly maximal (f2/fZmax > 0.95) in that temperature range. From the results it follows that the 0.817 MeV and 0.706 MeV radiations (Fig. 11) are nearly pure E2 and E l radiations respectively, as can be seen from the comparison of ~ ( 0 . 8 1 7 = ) - 0.44 and P,(&c) = + 0.36 resp. of ~ ( 0 . 7 0 6 = ) - 0.56 andP,(&z) = - 0.62. Consequently the 0.973 MeV and 1.79 MeV levels should be characterized as 5- and 6respectively, reducing the possibilities for the spin of 1 6 6 m H to~6 or 7 (see 3 4). The nuclear magnetic moment was estimated as 3.5 f 0 . 5 , ~ ~ . References p . 391
CH. VIII,
9 61
373
ORIENTATION O F ATOMIC NUCLEI 11
175Yb The anisotropies E of the 0.282 MeV and 0.396 MeV radiations of 175Yb (4.2 d) were measured by Grace et uZ.lOs to be 0.08 and - 0.03 respectively at the lowest temperatures obtained in ytterbium ethylsulfate (0.014 OK). From these results, in combination with angular correlation data, it is concluded that the 0.396 MeV level has spin and negative parity and that the 0.396 MeV radiation has a mixing ratio 6(E2/M1) = 0.10 0.03 l(Fig. 11). The nuclear moment was then estimated to be p = 0.15 & 0.04 pN, but it remains possible that the 3.4 x s lifetime of the 0.396 MeV state could be long enough so as to reduce the anisotropies; this might increase the value of p. Daniels et ~ 1 . 1 0 3did not find anisotropies of the gamma radiation if the 175Ybnuclei were incorporated in Ce-Mg-nitrate. Apart from the possibility that the Yb-ion did not enter the crystal lattice in a rare earth position, they suggest another interesting possibility. If the 0.396 MeV state belongs to an electronic configuration 01 the Yb-ion and has 3.4 x s lifetime, then the nuclear orientation in Ce-Mg-nitrate may be strongly disturbed. This is due to the fact that g, for Yb* in Ce-Mg-nitrate may be estimated to be roughly g, 3, whereas g, = 0 in the ethylsulfate.
+
+
+
-
*39Np A small alignment of 239Npnuclei was achieved in uranyl rubidium nitrate in zero external field at liquid helium t e m p e r a t ~ r e s lA ~ ~gamma . anisotropy of about 0.005 at 1.5 OK, compared to the normalization point at 7 OK, was observed. This shows that I > 4. 6. Methods of Nuclear Orientation In I a considerable part of the discussion was devoted to the methods of nuclear orientation; here we want to confine ourselves to those methods in which substantial progress was made since 1956 and in which nuclear orientation was observed by means of radiations from radioactive decay. These results are practically confined to alignment in ferromagnetic and antiferromagnetic substances and to dynamic orientation methods which are discussed mainly from an experimmtal point of view in 9 7 and 3 8 respectively. The other methods will be briefly mentioned here in the same order of succession as in I. References p 391
374
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
$7
As to the external field polarization or brute force method no new results have been published14". The Oxford group reported logs nuclear demagnetization experiments which are of considerable interest for the study of solid state physics and we will leave this subject open to a separate discussion*lo. The orientation of nuclei by magnetic h.f.s. alignment and polarization has been extended to many new nuclei as discussed in $ 5, but surprisingly few new paramagnetic salts have been introduced for this purpose. The latter stateof affairsisrelated to thelackof recentmagnetic data below 1 OK; new paramagnetic resonance data may be found inl03b, 1090. Little is to be reported146aon electric h.f.s. alignment; although the method is in principle applicable t o a large number of nuclei, nuclear electric quadrupole moments or the inhomogeneous electric fields are in most cases too small. Dynamic orientation methods based on paramagnetic resonance saturation have been developed considerably and have so many ramifications in other fields such as maser researchlog*that a separate review article would be appropriate. In 5 7 only some specific achievements are discussed in detail. As to the optical method of polarization the reader is referred t o a recent review article, where the steady and considerable progress in this field is reportedloge. A special case of nuclear polarization is found in the various experiments with polarized neutrons, which we mention, although this is outside the realm of low temperature physics. We want to call attention to two interesting subjects: a) The study of beta decay from polarized neutronsz6,which has set an upper limit to time reversal non-invariant terms in eq. (9). b) The polarization of *Li nuclei by means of irradiation of 'Li by polarized neutrons, which proved to have interesting solid state aspects lo9*. 7. Nuclear Orientation in Ferromagnetic and Antiferromagnetic Substances Whereas the orientation of radioactive nuclei at low temperatures was originally considered to be primarily of interest for studying nuclear properties, it was soon realized that for some nuclei the decay characteristics are known to a much higher precision than the process References p. 391
CH. VIII,
5 71
ORIENTATION OF ATOMIC NUCLEI I1
375
of nuclear orientation itself. Conversely, therefore, the measured gamma ray anisotropy may provide valuable information regarding the solid state environment of the decaying nucleus. We will discuss some specific examples.
FERROMAGNETIC METALS A. Co-metal. As reported in I, Kurti et al.lll*112aligned 6oConuclei in hexagonal Co-metal single crystals, cooled by heat contact with a Cr-alum-glycerol slurry. Ce-Mg-nitrate powder, having a low specific heat, was used as a thermometer by connecting it thermally with the Co-crystal and measuringitssusceptibility. At 1 / T M 30 OK-l thegamma ray anisotropy was E = 0.07; E could be expressed by 6 = a/T2, where a = 0.79 x lP4OK2, leading to an effective field at the position of the Co-nuclei of 193 & 20 kOe. This may be compared to specific heat measurements by Heer et aZ.ll3 and Arp et al.ll& which gave 183 kOe and 200 kOe resp. These results, however, disagree with measurements of Khutsishvilil, who gave E = a / T 2with u = 5 x OK2. In this experiment the nuclei were polarized by a large external magnetic field, applied in the direction of the hexagonal crystal axis, which is also the direction of easy and spontaneous magnetization and of nuclear alignment. Similar experiments in zero field by Daniels OK2, but when the metal crystal was et ~ 1 . yielded ~ ~ 5 a=3 x heated a few minutes to red heat and the experiments repeated, a O K 2 was found. Since Co changes from the value a = 1.5 x hexagonal to the cubic phase above 417 "C, the heat treatment may increase the proportion of cubic regions in the Co-crystal. The degree of nuclear alignment may in the cubic phase be appreciably smaller than in the hexagonal phase, which could explain the various data. However, recent lZ1nuclear resonance measurements of 59C0 in cubic Co metal have yielded a value of 217.5 kOe (extrapolated to 0 OK), which is equal within experimental error to the value of 219 kOe for hexagonal Co, obtained in recent specific heat measurements by Arp et aZ.114b. The large U-values obtained by Khutsishvili and Daniels remain unexplained and, if not due to lack in accuracy in the temperature determination, present an unsolved problem. In this connection calculations of the effective field by Marshall 116should be mentioned as well as recent m e a s u r e m e n t ~ ~of~ 'the effective field in iron metal, which proved to be opposite in direction to what was expected. References
p. 391
376
$7 Still more recently, many new results have been published, which are related to this subject146kJ7m. B. Samoilov et uZ.l18 in Moscow activated a (50% Co, 50% Fe) permendur alloy in a pile. The specimen, containing 3-4 pc 60C0, was soldered to copper, which was then cooled to low temperatures by heat contact with pressed Cr-alum. A small permanent magnet was mounted in the cryostat, such that the alloy was placed between the poles in a field of 1000 Oe. The gamma ray anisotropy was found to be E = 1.2 x 10-4/T2which results in an effective field of 250 kOef14b. C. A 0.1 mm thick disk of a gold-iron alloy, containing 0.3% Au, was irradiated by thermal neutrons, producing 4 pc 198Au. Samoilov et aZ.l19 cooled this sample to a temperature of about 0.015 "K in the same way as mentioned under B. From the magnitude of E and the known value of the nuclear magnetic moment p = 0.5 0.04 pN, the value of the effective magnetic field at the Au-nucleus was calculated to be 600 kOe. I). Samoilov et ~ 1 . also l ~ obtained ~ ~ nuclear polarization of lzzSb and 114mInnuclei, inserted as impurities in Sb-Fe-alloy (0.6% Sb) and In-Fe-alloy (0.5yo In) respectively. Magnetization of the sample was obtained by an electromagnet in the helium dewar, capable of producing a 2000 Oe field. The 0.56 MeV quadrupole radiation (2+ + O+) of lz2Sbshowed an anisotropy E = 0.025 at T = 0.03 OK, from which an effective field of about 190 kOe was derived. Similarly, the anisotropyof the 0.192 MeVgammarays of 114mInwas E = 0.08at T = 0.04 O K , leading to an effective field of 150 kOe if the nuclear magnetic moment is 4.7 pN. So far, anisotropies of gamma radiations of such high multipole order (E4) had not been observed. More recent results1Z0b gave lower limits to the effective fields in Fe at the position of Sb, In and Au nuclei of 280 kOe, 250 kOe and 1000 kOe respectively. The discrepancies with the former results were due to incorrect temperature estimates ; this again illustrates the need for accurate thermometers in indirect cooling experiments below 0.1 OK. E. Kogan et aZ.120C in Leningrad polarized 46Scnuclei solved in iron metal. The concentration of Sc was smaller than 0.6% which was found to be somewhat below the maximum solubility of Sc in Fe metal. Part of the experimental arrangement is shown in Fig. 13. The iron sample is magnetized by the residual field of a superconducting Nb-cylinder (900 Oe), which field was concentrated at the position of the sample by armco iron wedges up to a value of about 1700 Oe. The References p . 391
W. J . HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
CH.
VIII,
9 71
ORIENTATION OF ATOMIC NUCLEI I1
377
sample was externally cooled to T w 0.04 OK and a gamma anisotropy E = 0.01 was obtained. This anisotropy is insufficient to make a beta gamma correlation experiment feasible. A j3-y-directional correlation experiment for polarized 4 % nuclei ~ might give very interesting results because of the reported presence of considerable interference between Fermi and Gamow-Teller matrix elements in the 46Scbeta decay and consequently, the experiment might serve as a test for the validity of time reversal invariance in beta decay4915 0 , 51. With the same apparatus but 60Coinstead of 46Sc,a gamma aniso-
CONNECTED WITH MIXTURE OF Cr-ALUM AND FQOPANOL
Fig. 13. Central part of apparatus of Kogan et a2.lz0C for polarizing iron samples which contain diamagnetic impurities. The polarizing field is the residual field of a superconducting Nb-cylinder, concentrated a t the position of the sample by armco iron. The Cr-alum serves as a guard against heat conduction t o the sample; the sample is cooled by a Cr-alum-propanol mixture and the heat contact is provided by a metal rod.
tropy E = 0.2 was obtained, from which an internal field of 300 to 400 kOe at the position of the 6oConucleus was derived (Co concentration of 0.02%) 114b. The results quoted under A-E are of considerable interest for the theory of ferromagnetism in metals particularly as to the question whether the effective field predominantly originates from the influence of conduction electrons, 3d electrons, s electrons from inner shells, or from the interplay between these various electrons. See als0146g. *, 1. ANTIFERROMAGNETIC SALTS It was suggested by Daunt122& and Gorter122bthat it might be possible to align nuclear spins in antiferromagnetic single crystals. Proton References p . 391
VIII, 3 7 resonance experiments by Poulis et aZ.12shad shown that at low temperatures the magnetization of either sublattice remains constant during times ( l W 4 s) which are much longer than the proton precession time M lo-' s). Classically speaking, therefore, the magnetizations of the two sublattices do not interchange frequently enough so as to give a zero average field at the proton position during a single precession. Since the nuclear precession frequency caused by h.f.s. interaction inside a paramagnetic ion is in many cases one or two orders of magnitude higher, the above conclusion would therefore also apply to these nuclei. Antiferromagnetic alignment was first realized by Daniels et aZ.124912s for 64Mnnuclei in MnCl,, 4H,O and MnBr,, 4H,O, which have Nee1 temperatures, T,, of 1.6 OK and 2.2 "K respectively. Cooling of such salts to temperatures below 0.1 "K, as required t o obtain sufficient nuclear alignment, cannot be realized by direct adiabatic demagnetization starting from 1 OK.At 1 "K these salts still have a considerable specific heat and indirect cooling requires a large amount of heat transfer to a coolant paramagnetic salt. Daniels et al. used K-Cr-alum and obtained temperatures of 0.1 "K, in the MnC1, after about half an hour and in MnBr, after several hours. The gamma anisotropy increased with the above time constants to maximal values of respectively E = 0.07 and E = 0.06. Possibly the spin-lattice relaxation time is the dominant factor in the long time needed for cooling. The results also showed that the axis of nuclear alignment coincides with the axis of preference for the electronic angular momenta. No anisotropies of g°Co radiations were found in these salts at 0.1 "K. Miedema et ~ 1 . ~ 2aligned 6 both s°Co and UMn in the Co-NH,-tutton as well as in the Mn-NH,-tutton salt, These salts have transition temperatures of 0.084 "K and 0.14 OK respectively. When starting demagnetizations from 1 OK, final temperatures somewhat below T , can be reached; to obtain temperatures lower than +TN,however, indirect cooling again is needed. Miedema et aZ.laeused supercooled solutions of Paramagnetic salts in propyl alcohol as cooling agents ; in order to increase the heat capacity potassium chrome alum crystals were embedded in the solution as well. A large number of copper wires were assembled in the solution, providing a large contact area with the coolant. The radioactive crystal was glued onto a copper plate, soldered to the wires. 378
References p . 391
W.
J . HUISKAMP AND H. A. TOLHOEK
[CH.
CH, V I I I ,
3 71
379
ORIENTATION O F ATOMIC NUCLEI I1
The gamma anisotropies measured in the paramagnetic region are consistent with nuclear alignment along the two tetragonal axes of the tutton salts, as found earlier1 in the diluted salts. This is indicated in Fig. 14 by 8, m 0 for both 54Mn and 6oCoin Co-NH4-tutton salt above T m 0.08 OK. At lower temperatures there is a change in counting rates, which is interpreted as a rotation of the nuclear spins of 8, w 5" and 8, w 7" for 6OCo and 54Mn respectively towards the bisector, K,, of the tetragonal axes. Susceptibility measurements have shown 127 that below T , the susceptibility strongly decreases along the K,-axis, but increases to 1c
C
eNl -.
I )
0.06
T
j"
OK
I 0.12
Fig. 14. Preferred direction of 64Mn and e°Co nuclear spins in Co-NH,-tutton salt single crystals as a function of temperature. ON represents the angle (in degrees) between the preferred direction of nuclear alignment and the direction of the tetragonal axes, ON being zero in the paramagnetic region. 0: 64Mn A : E°Co.
high values in the K3 direction (K, 1K,). This behaviour is fairly well explained by assuming antiparallel alignment along staggering axes, somewhere between the two tetragonal axes and their bisector, K,. Whereas below T , apparently both electronic and nuclear spins tend towards the K,-axis, quantitative agreement of the nuclear alignment data with a molecular field model could not be obtained. Daniels et ~ 1 . 1 2 5find almost no change in the results for 6oCo in CoNH,-Tutton salt when decreasing T beyond TN. In CoC1, they find an axis of nuclear alignment, which differs from the axis of preference for the electronic angular momenta; here the anisotropies of 54Mnand 60Coare very different: E = 0.18 and E = 0.015 respectively, at T = 0.055 "K ( T , = 3°K). In MnSiF,, 6H,O, for which T , = 0.1 OK, Daniels et aE.125 report References p . 391
380
W.
J . HUISKAMP AND H . A. TOLHOEK
[CH. VIII,
58
large gamma ray anisotropies for both 54Mn ( E = 0.20) and s°Co (8 = 0.19) at about 0.05 O K , which cannot be accounted for by paramagnetic resonance data. Reviewing the progress made since the situation was briefly discussed in I, it may be stated that the feasibility of nuclear alignment in antiferromagnetic salts has been clearly demonstrated. This was also shown to be the case for Co-NH,-tutton salt, for which the previously reported1 negative result was possibly due to the small magnitude of the anisotropy E (which is of the order of 0.01). On the other hand the mechanism of nuclear alignment in the antiferromagnetic state remains a t present largely unknown and probably is fairly complicated and subject to considerable variation with the individual salts. The study of the problem is complicated by the circumstance that strongly antiferromagnetic crystals are difficult to cool as a whole and their nuclear spin system in particular, whereas in feebly paramagnetic substances, for example crystalline fields complicate the situation. It is therefore not possible to conclude definitely whether the antiferromagnetic interaction enhances or, oppositely, decreases the degree of nuclear alignment. Similarly, it is not clear whether the nuclei align along the direction of magnetization of the sublattices or whether they prefer other directions. Probably the differences between the various salts are too great to allow such general statements. 8. Dynamic Methods of Nuclear Orientation It was shown by Jeffries128* 129 that saturation of transitions characterized by A (S, I,) = 0 in paramagnetic resonance spectra can produce appreciably larger nuclear polarization than by saturation of the ordinary A S , = & 1, dI, = 0 transitions. Many of the problems involved had already previously been discussed by Abragani 130; here we follow the discussion of Jeffries131. Consider the system of a paramagnetic ion and a nuclear magnetic moment in an external magnetic field, H,, described by the spinHamiltonian :
+
<
Neglecting the last term and assuming B g,,BH,, we may consider XI r + B ( S + I - S J + ) as a small perturbation on the hamiltoniari
+
Refcfcnces p. 391
CH. VIII,
5 81
382
ORIENTATION OF ATOMIC NUCLEI I1
+
X,, ~ g , , # l H L S , AI,S, and the perturbed eigenfunctions are Y(S,>1,) =
where .!$ &+
=
E-
=
Ell
I s,,I , > + E+ I s, + 1, I , - 1 > +
+ + E! E:
+
=
1,
E,,
E-
[ S,
-
1, I ,
+1>
(30)
w1
+ s, + 1) (1+ 1,)( I I , + 1)B/2g,,BH, (31) 4 s + S,) (S - s, + 1) (1- 1,) ( I + I , + 1)B/2g,,BH,. (32) - d ( S - S,) (S
-
Next it may be shown that introduction of a radiofrequency (r.f.) field H , along the z-axis, considered as a time dependent perturbation 8 1 . 1 . = g1,BHrzSz cos mt induces transitions between the states y(S,, I,) and
y(S,
+ 1 , I , - 1) or y ( S , - 1, I, + 1) g:lB2H;z
since for instance
< Y(S2 + 1, I , - 1) I s, I Y(SP 13
w = ( S - S,) ( S s, 1) ( I 1,) (1- I , 1)B2H&/H2,. This result may be compared to the ordinary transitions
+ +
+
+
>2
(33)
Y(S,, 1,)+ d S 2 i 1, I,), which are induced by r.f. components perpendicular to H,, i.e. Z1.f.
W&B2
-=E Y(S,
+ S-Hrz)
= Sg#(S+Hrx
+ 1>I,)I s+1 d S , , 1,) =
(S
- S,)
>2
(S
(34)
=
+ s, +I) Hk34H;gfl.
(35)
The ratio of transition probabilities of the two types of transitions is, apart from a factor containing I and I,,
which may be orders of magnitude smaller than 1 and consequently, the A(S, I,) = 0 transitions are designated as forbidden transitions. It should be noted that in a microwave cavity the ratio H,,/H,,, averaged over the sample, is neither zero nor infinite. Similar results are obtained if the nuclear spin and electron spin do not belong to the same paramagnetic ion; such cases arise in semiconductors, where a scalar type of coupling AS * / may exist. Also the coupling between S and I may further be anisotropic h.f.s. coupling,
+
References p . 391
382
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
5
8
magnetic dipole-dipole coupling or interaction of the nuclear quadrupole moment. In the latter cases one may have other forbidden transitions like A S , = 0, 4 1, dl, = i 2 etc., which simultaneously provide relaxation mechanisms and may make the attainment of the steady state saturation very complicated. Whereas in the Overhauser method of nuclear p ~ l a r i z a t i o none l ~ ~saturates the A S , = f 1,A I , = (4 -2
1 -1 0 a a a Q T I - ,
_
I
I
I
I
I
L
L
1
-
-
I I
L
-
-
1
-2
-1
,
I
I
-
-
1
1
0
1
2
a I
I
I
L
2
I
I
1
I
I I
,
I I
I
, I
I
--I-IIII
1
I N I T I A L STATE
INITIAL STATE
I
I
I
I
I
I
L-2.-
'
--L
9
1
1 1 IMMEDIATELYAFTER APPLICATION OF RF FIELD
I
1
&
%
FINAL EClUILIBRIUM
A
1
IMMEDIATELY AFTER APPLICATION OF RF FIELD
' STATE
SATURATION OF FORBIDDEN TWNSITIONS
FINAL EOUILBRIUM STATE
B
OVERHAUSER EFFECT
Fig. 15. Diagrams showing the production of nuclear alignment by saturation of forbidden transitions (A) and by the Overhauser effect (B). The drawn arrows represent I,) = 0 relaxation the action of the microwave field, the dotted arrows indicate A ( S , transitions and the dotted lines are the AS, = - 1 relaxations. I n A relaxation by AS, = - 1 only is assumed, in B by A S , = - 1 and d(S, + I,) = 0 only. Since the h.f.s. energy is neglected in the figure, the energy difference between the upper (S, = 4)and lower (S, = - 4) levels is gll/?Hs.Further, a = exp (-gll/?H,/kT) and therefore, as in most dynamic methods, the degree of nuclear polarization does not, i n first approximation, depend on the h.f.s. energy splittings.
+
+
transitions and expects the d(S, + I,) = 0 relaxation to produce nuclear polarization, saturation of forbidden d(S, I,) = 0 transitions gives nuclear polarization directly. A schematic comparison between the two methods is given in Fig. 15 for S = 4 and I = 2; in A the I, = 0 level is filled at the expense of the I , = 1level by the action of the microwave field and subsequently the degree of nuclear polarization is increased by the establishment of thermal equilibrium by A S , = - 1 relaxation; in B there is no nuclear polarization immedi-
+
References 9.391
CH. VIII, §
81
ORIENTATION OF ATOMIC NUCLEI I1
383
ately after application of the microwave field. Other relaxation phenomena, e.g. 01, f 1 may appreciably reduce the nuclear polarization in both A and B. A more detailed analysis shows that such a reduction occurs in A if d I , = & 1competes with A S , = -I, in B if d1,= i.1 competes with d(S, 13 = 0, respectively. Since AS, = - 1 relaxation prevails in many substances, the Overhauser effect may be more difficult to achieve than the saturation of forbidden transitions. Further it can be shown that in cases where the nuclear polarization vanishes (fi = 0) by relaxation mechanisms, nevertheless nuclear alignment (f2)may be appreciable if tc (Fig. 15) is considerably smaller than 1, as is usually the case for HZ w lo4 Oe and T m 1 OK. It is to be noted that the sign of the nuclear polarization obtained by saturation of forbidden lines is opposite to that in the Overhauser method, which makes it experimentally possible to distinguish between the two phenomena, e.g. in cases where the forbidden lines are not resolved from the ordinary lines or if there is only one broad line. 5
+
RESULTS A. Abraham et ~ 1 . lin~Berkeley ~ produced a 10% polarization of 6oCo nuclei in deuterated La-Mg-nitrate with the isotopic abundance ratio Mg : 59C0: W o = lo4 : 60 : 1 and with a 2 mc 6oCoactivity. The 9400 MHz microwave field of approximately 0.1 Oe was applied along the external field of roughly 1500 Oe in the direction of the c-axis of La-Mg-nitrate. Two counters were located closely to the source and under angles 0 and in with [respect to the magnetic field; the difference between the two integrated photomultiplier outputs was directly recorded as a function of the magnetic field, which was swept over the59960 Co++ resonance. The result may be seen in Fig. 16. The magnitude of the observed anisotropy, E m 0.01, was 40% of what could be expected if neither d(S, + I,) = f 2 nor d1, = f 1 relaxation phenomena were active. B. Similar results132 were obtained for 52Mn and 54Mn,for which both I and p could be measured: for 52MnI = 6, p = (3.00 i 0.15) pN and for UMn I = 3, p = (3.29 & 0.06) pN. The number of overlapping paramagnetic resonance lines is very large for Mn, which has S = Q and in particular for 52Mn with I = 6. Therefore, it was impossible here simply to count the lines in the gamma anisotropy as a function of the magnetic field and a more indirect approach was followed. References 9. 391
w.
384
J , HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
38
Fig. 16. Dynamic polarization of SOCo in La-Mg-nitrate. The upper curve represents the paramagnetic resonance spectrum of 6@Co;the calculated peak positions, fitted to a certain extent to the experimental data, are indicated by vertical lines. From the W o resonance peaks the positions of allowed and forbidden resonance peaks of B°Co were calculated and are given in the lower figure by drawn and dashed lines respectively. Also shown is the observed gamma ray anisotropy, E ; the expected values of the anisotropy for saturation of allowed and forbidden lines are indicated by the height of the vertical lines.
Fig. 17. Decay schemes of 7OAs and laaSb.Some weak beta and gamma decay branches have been omitted from the figure. References
p . 391
CH. VIII,
3 81
ORIENTATION OF ATOMIC NUCLEI 11
385
C. Pipkin et ~ 1 . lS5 ~ used ~ 9 the saturation of forbidden transitions for obtaining nuclear polarization in 27 h 76As, incorporated as a donor impurity in Si crystals. The source was obtained by doping a few times l0ls As atoms in about 1 cm3 Si and subsequently exposing the crystal to a flux of l O l B neutrons/cm2 s. The sample was mounted inside a microwave cavity, immersed in liquid helium at 1.25 OK; simultaneous with the saturation of forbidden transitions by sweeping a magnetic field at about 8500 Oe, the 0.56 MeV quadrupole radiation was measured in a direction perpendicular to the magnetic field. To simplify the discussion, we assume that the first forbidden /I-decay preceding the most intense 0.56 MeV gamma radiation does not carry off angular momentum and is characterized by 2-(/?-) 2+, J = 0. Then the /?-decay does not change the populations of the h.f.s. levels and as a result the gamma ray intensity in the perpendicular direction is solely determined by the I , = 0 h.f.s. level population, since the 01,= 1 (I,= f 1 3 I, = 0) and A I , = f 2 ( I , = = rf 2 -+ I, = 0 ) gamma transitions have equal intensities in the &n direction. The A I , = 0 ( I , = 0 -+ I , = 0) gamma transition has zero intensity in the &n direction. The experimentally observed change in counting rate was about 2% at approximately 8225 and 8260 Oe (Fig. 18a, b), which must have been due to (S, = - +, I , = 0)-+ (S, = + Q, I , = - 1) and (-4, 1) + (++,0) transitions. This is more clearly demonstrated in Fig. 19; the sudden increase in counting rate when the microwave field is turned on corresponds to the saturation of forbidden transitions (Fig. 15A) and not to the Overhauser effect (Fig. 15B). Sweeping through the high field resonance resulted in a gamma ray intensity decrease, and therefore was caused by filling the I , = 0 level by the transition (-&, 1) 3 (+&, 0). From the fact that this transition is the high field resonance, one easily finds that the (-4, 1) level has a higher energy than the (-+, 0) level, and consequently S and I are preferably parallel, which leads to a negative sign for the h.f.s. constant A and a negative sign for the nuclear magnetic moment. The accuracy for A is determined by the precision with which one can measure the differencein magnetic fields in Fig. 18a. Double resonance experiments were performed, in which r.f. power was applied to the sample along with microwave power. A special cavity consisting partly of ,thinly silvered lucite was required for admitting the r.f. fields, generated by a variable r.f. oscillator. One of the References p . 391
386
W. J. HUISKAMP AND H. A. TOLHOEK
T
8300
'?
Q
8260
8220
[CH. VIII,
38
a80
MKNETIC FIELD IN OERSTEDS
v
5903
t
C 56 52 50 46 LOW FREOUENCY IN MHz
43
Fig. 18. Gamma ray intensity in a direction perpendicular to the magnetic field versus the magnitude of the magnetic field, showing the production of nuclear alignment of "As when the field is swept through forbidden resonances in the electronic paramagnetic resonance spectrum. The field sweep is from high t o low field in the upper curve (a), and conversely in the middle curve (b). During the time between the low field and the high field resonance, the 1, = 0 level is relatively densely populated in (a)while it is scarcely populated in (b). The lower curve (c) shows a t the left the saturation of a forbidden electronic resonance (-4, I)+($, O),thereby filling the 1, = 0 level, which is thereafter emptied through AT, = 1 and A I , = - 1 nuclear resonances consecutively.
experimental procedures is the following (Fig. 18c). After saturation of a forbidden transition d(S, + I,) = 0 at the appropriate value of the magnetic field, the r.f. power is swept in the region of 50 MHz over a A I z = 3 1 transition and at resonance the gamma anisotropy is destroyed (Fig. 18c). It is seen that there are, in fact, two resonances: (-4,0) 3 (-3, 1) and (-+, 0) + (-Q, -1). In more refined experiments the resonance frequencies could be determined with an accuracy of about 0.1 MHz giving A(76As)= - 93.66 f 0.06 MHz. If the preceding B-transition, which has an allowed shape, does carry away angular momentum, J , the analysis is more complicated. ConReferences #. 391
CH. VIII,
5 81
387
ORIENTATION O F ATOMIC NUCLEI I1
versely, however, the results can give information about J . Double resonance measurements in conjunction with measurements of the gamma ray intensity i n the direction of the magnetic field showed that the ,$-decay is described by a mixture of 50% J = 0, 20% J = 1 and 30% J = 2 matrix elements. The nuclear orientation was shown to persist for 2-3 hours, which agrees with relaxation time measurements in which the A (S , I,) = 0 transitions were found to have a relaxation time longer than 75 min, even though the AS, = f 1 transitions have a relaxation time of 4 minutes.
+
640
1
I
I
I
I
t
TIME IN MINUTES
Fig. 19. Gamma ray intensity of 7'IAs with the magnitude of the magnetic field fixed at a forbidden electronic resonance. After some counts without microwave field, a prompt change in counting rate is observed when the microwave field is switched on; the slower increase thereafter is due to AS, = & 1 relaxation, corresponding t o the attainment of the final equilibrium state in Fig. 15A.
Double resonance experiments on 75As, combined with the known nuclear magnetic moment of 75As, provided the value , u ( ~ ~ A = s) - - 0.903 f 0.005 pN, the error being mainly due to the unknown magnitude of the h.f.s. anomaly. D. Similar e ~ p e r i m e n t s were l ~ ~ performed with lz2Sbin Si; the important difference was that A ( S , I,) = 0 transitions had a much shorter relaxation time for Sb, one of the possible reasons being the higher concentration of l22Sb compared with 76As.The relevant part of the decay scheme is basically the same as in 76As:2- (8) 2+ (E 2) O+ where the p-transition has allowed shape though a non-isotropic betagamma directional correlation was observed. The lzzSb-experimentprovided the interesting situation that nuclear
+
References
p . 391
388
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
98
polarization could be obtained both by saturation of forbidden transitions as well as by the Overhauser effect i.e. saturating a A S , = 1, dI, = 0 transition and waiting for the d(S, I,) = 0 relaxation t o occur. The observed counting rate as a function of time was found to be different in the two cases: in the Overhauser effect a constant counting
+
+
36501
3490
I
[
I
8.45
MAGNETIC
FIELD
I
8.47
I
I
1
8.49
I
8.51
I
I
8.53
IN KILOOERSTED
Fig. 20. Saturation of two forbidden lines in the electronic resonance spectrum of Sb impurities in Si and corresponding changes in counting rate of lp2Sbgamma radiation. During the time between the two resonances the degree of nuclear alignment is practically reduced to zero due to relaxation phenomena.
rate is approached with a time constant of about 10 min. whereas the counting rate changes instantaneously when saturating a forbidden line. Fig. 20 shows the gamma ray intensity as a function of the magnetic field which is swept over the forbidden resonances. It is seen that the nuclear polarization decreases strongly between two resonances, which may be due to a combination of relaxation phenomena. Since no more resonances were found, it could be concluded that the nuclear spin of lzzSb equals 2. By double resonance again the h.f.s. splitting constant, A , was determined as A(lZ2Sb)= - 132.59 f 0.10 MHz from which it follows, with the aid of known A and p for lzlSb, that p('"Sb) = - 1.904 f 0.020 pN. E. Dynamic polarization of non-radioactive 29Si nuclei was ob, 139 at Saclay by saturation of the electained by Abraham et U Z . ' ~ ~ 138, tron resonance in P-doped Si, which was somewhat too broad for showing a 29Si h.f. structure. Microwave power was used for saturation and r.f. fields for measuring the intensity of the 2 5 resonance; the Si-polarization was shown t o be enhanced by a factor of about 100 (fl M 0.001) after saturation of the electron resonance and the sign of the polarization indicated that forbidden transitions A S , = 1, A I , = & 1 were responsible for the 2% polarization and not the Overhauser effect. Refer em es p 931
CH. VIII,
9 81
ORIENTATION OF ATOMIC NUCLEI I1
389
POLARIZED PROTON SAMPLES An important extension of this method was indicated by Borghini and Abragam 140, who obtained polarization of protons by saturation of electronic resonance in the free radical DPPH (diphenyl picryl hydrazyl). Here the electronic spin is coupled to a large number ( m lo2) of proton spins presumably by a dipolar coupling although scalar coupling may simultaneously be present. Whereas in the latter case the forbidden transitions are d(S, + I,) = 0, in the dipolar case more forbidden lines arise like d(S, I,) = f 2 , so that for nuclei with spin 4 (proton, 29Si)more than one forbidden line exists. The sample consisted of 10 mm3 proton rich polystyrene which contained 10% DPPH. 60 mW of 36000 MHz microwave power was applied to the sample at a magnetic field strength of 12000 Oe, whereas the proton resonance was observed with a r.f. spectrometer. It was found that the proton resonance peak became 50 times more intense when saturating the free electron resonance (fl m 0.02). Similar results were obtained by Uebersfeld et aZ.141and by Abraham et ~ 1 . l ~ ~ . It is tempting to speculate on the various applications in nuclear physics, which may be devised with the use of polarized protons in samples which are rich in protons and when degrees of polarization fl of the order of 20% could be reached. One could expect as well many applications in low energy nuclear physics as in high energy physics. A technical difficulty of applications to reactions or scattering with charged particles of energies of a few MeV is that the polarized sample should be a very thin layer. At energies of a few hundred MeV the situation is easier as the samples may be thicker. The beam intensity with charged particles must of course be rather limited because the dissipated heat should not warm up the sample too much. The experiments, which may become feasible in this way, concern polarization effects in proton-proton, proton-neutron, proton-nucleus or protonelectron collisions at lower and higher energies. J e f f r i e ~ let~ ~at. used hydrated paramagnetic crystals like tutton salts and La-Mg-nitrate to obtain proton polarization; at 1.7 OK an enhancement factor of about 20 was observed in the proton resonance intensity compared to non-saturation of forbidden lines AS, = 1, d I z = f 1. This is an order of magnitude less than predicted. More recently144 a proton polarization fl = 0.19 was reported.
+
References Q . 391
390
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
99
9. Concluding Remarks
We may conclude with a number of remarks concerning the present position of nuclear orientation at low temperatures (stressing the new points which have arisen since I, tj 4): (1) The discovery of non-conservation of parity (and non-invariance for charge conjugation) in @-radioactivity has been of fundamental importance. As to the fundamental aspects of @-interaction further experiments of importance could be tests of the invariance for time reversal (cf. 5 2). (2) The preceding discovery provides a new tool, useful for nuclear spectroscopy : the asymmetries of ,&rays from polarized nuclei can provide information such as the change of nuclear spin in p-decay and relative magnitude of nuclear matrix elements in /I-decay. (3) Thc “brute force” method of nuclear polarization (which could be applied to all nuclei with a not too small magnetic moment) might become more fcasible because of the technical developments concerning indirect cooling and strong magnetic fields. Further the asymmetries of @-raydistributions should enable one to detect degrees of polarization fi of a few percent only. (4) It has been shown that large internal fields exist at the position of nuclei of diamagnetic atoms when incorporated in iron metal; this has provided a possibility of polarizing Sb, In, Au and Sc nuclei. It may be expected that this method of nuclear polarization in ferromagnets can be extended to more nuclei; progress in this respect is largely determined by problems of metallurgy and radiochemistry. For small concentrations of diamagnetic elements, when other methods for probing the internal magnetic fields like specific heat measurements or nuclear resonance do not seem promising, polarization of radioactive nuclei may be particularly useful. (5) Nuclear alignment was found in some antiferromagnetic single crystals but the relation between the degree and the preferred direction of nuclear alignment on the one hand and the magnetic properties of the crystals below the Nee1 temperature on the other hand, remains largely unsolved. (6) Dynamic methods have produced polarization of some radioactive nuclei to such a degree that the anisotropies of gamma radiations could be measured. The combination of magnetic resonance measurements, both a t microwave and radio frequencies, with gamma References Q. 391
CH. VIII]
ORIENTATION OF ATOMIC NUCLEI I1
391
anisotropy measurements have provided accurate values for h.f.s. splittings and nuclear magnetic moments respectively. (7) The development of certain dynamic methods of orientation may provide samples with, e.g. protons (or deuterons) with a substantial degree of polarization (e.g. fi = 0.10 to 0.20) in a near future. Such samples could be very useful as targets for nuclear reactions (or scattering processes) as well in low energy nuclear physics as in high energy physics (cf. 5 8). REFERENCES
I. THEORY M. J. Steenland and H. A. Tolhoek, Progr. in Low Temp. Phys., Vol. 2, ed. by C. J. Gorter (North-Holl. Publ. Co., Amsterdam, 1957) p. 292. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, 1957). M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). U. Fano and G. Rscah, Irreducible tensorial sets (Acad. Press, New York, 1959). L. Roscnfeld, Lectures on Oriented Nuclei, Nordita, Copenhagen (1959) (mimeographed) R. J. Blin-Stoyle, M. A. Grace and H. Halban, Progr. Nucl. Phys. 3, 63 (1953). G. R. Khutsishvilt, Orientation of nuclei, Usp. Fiz. Nauk 53, 381 (1954) (in Russian). R. J. Blin-Stoyle, M. A. Grace and H. Halban, Beta- and Gamma-Ray Spectroscopy, ed. by I<. Siegbahn (North-Holl. Publ. Co., Amsterdam, 1955) p. 600. S. R. de Groot and H. A. Tolhoek, Beta- and Gamma-Ray Spectroscopy, ed. by K. Siegbahn (North-Holl. Publ. Co., Amsterdam, 1955) p. 613. lo R. J . Blin-Stoyle and M. A. Grace, Handbuch der Physik, Band 42 (Springer, Berlln, 1957) p. 555. l1 E. Ambler, Progress in Cryogenics Vol. 2, ed. by K. Mendelsohn (Heywood and CO., London, 1960) p. 233. la M. E. Rose, Phys. Rev. 108, 362 (1957). Is U. Fano, Nuovo Cim. 5, 1358 (1957). u R. M. Steffen, Advances in Physics 4, 293 (1955). l6 H. Frauenfelder, Beta and Gamma-ray Spectroscopy, ed. by K. Siegbahn (NorthHoll. Publ. Co., Amsterdam, 1955) p. 531. l8 J. M. Daniels, Can. J. Phys. 35, 1133 (1957). l' P. J. Brussaard and H. A. Tolhoek, Physica 24, 233 (1958); see also Physica 24, 263 (1958), and Physica 23, 955 (1957). Is T. D. Lee and C. N. Yang, Elementary particles and weak interactions, Brookhaven Nat. Lab. (1957) (BNL 443; T-91). la T. D. Lee, Proc. Rehovoth Conf. 1957, (North-Holl. Publ. Co Amsterdam, 1958) p. 346. C. S. Wu, Proc. Rehovoth Conf. 1957 (North-Holl. Publ. Co. Amsterdam, 1958)p. 346. a1 0. R. Frisch and T. H. R. Skyrme, Progr. Nucl. Phys. 6 , 267 (1957). Proc. 1958 Ann. Int. Conf. High Energy Physics CERN, GenBve; Session 8; M. Goldhaber and others, p. 233. z8 Congrhs International de Physique Nuclbaire, Paris, 1958 (Dunod, Paris, 1959); R. Nataf and others, p. 271.
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L. Grodzins, Progr. Nucl. Phys. 7, 163 (1959). J. J. Sakurai, Progr. Nucl. Phys. 7, 243 (1959). z6 E. J. Konopinski, Ann. Rev. Nucl. Sci. 9, 99 (1959). 21 A. Lundby, Progr. Elem. Part. and Cosm. Ray. Phys. 5 (North-Holl. Publ. Co., Amsterdam, 1960) p. 1. 28 S. R. de Groot and H. A. Tolhoek, Physica 16, 456 (1951). 29 T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956). 30 W. Pauli, Niels Bohr and the Development of Physics (Pergamon Press, London, 1955) p. 30. 31 J. Schwinger, Phys. Rev. 82, 914 (1951); 91, 713 (1953). 32 G. Luders, Kgl. Danske Vidensk. Selsk. Medd. 28, no. 5 (1954); Ann. Phys. (New York) 2, 1 (1957). 33 R. Jost, Helv. Phys. Acta 30, 409 (1957). 34 T. D. Lee and C. N. Yang, Phys. Rev. 105, 1671 (1957). 35 L. Landau, Nucl. Phys. 3, 127 (1957). 36 A. Salam, Nuovo Cim. 5, 299 (1957). 37 W. Pauli, Nuovo Cim. 6, 204 (1957). 88 D. L. Pursey, Nuovo Cim. 6, 266 (1957). a@ R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). 40 E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958); Proc. PaduaVenice Gonf. 1957. 4 1 J. J . Sakurai, Nuovo Cim. 7, 649 (1958). 42 M. Gell-Mann, Phys. Rev. 111, 362 (1958). 4a S. S. Gershtein, J. B. Zeldovitch, Zhur. Eksp. Teor. Fiz. S.S.S.R., 29, 698 (1955). (transl. Soviet Physics J E T P 2, 576 (1957)). 44 J . D. Jackson, S. B. Treiman and H. W. Wyld, Phys. Rev. 106, 517, (1957); Nucl. Phys. 4, 206 (1957). 4 5 M. E. Ebel and G. Feldman, Nucl. Phys. 4, 213 (1957). 46 I<. Alder, B. Stech and A. Winther, Phys. Rev. 107, 728 (1957). 47 M. Goldhaber, L. Grodzins and A. W. Sunyar, Phys. Rev. 109, 1015 (1958). 48 M. T. Burgy, V. E. Krohn, T. B. Novey, G. R. Ringo and V. L. Telegdi, Phys. Rev. Letters 1, 324 (1958); Phys. Rev. 120, 1829 (1960). 49 R. B. Curtis and R. R. Lewis, Phys. Rev. 107, 1381 (1957). 50 M. Morita and R. Saito Morita, Phys. Rev. 107, 1316 (1957). 5 1 A. 2. Dolginov, Zhur. Eksp. Teor. Fiz. SSSR 33, 1363 (1957) (transl. Soviet Phys. JETP 6, 1047 (1958)). 5 1 Yu. V. Gaponov and V. S. Popov, Nucl. Phys. 4, 453 (1957). 63 M. Morita, Phys. Rev. 107, 1729 (1957). 54 C. C. Bouchiat, Phys. Rev. 118, 540 (1960). 56 H. A. Tolhoek and S. R. de Groot, Physica 17, 81 (1951). 56 G . R. Khutsishvili, Zhur. Eksp. Teor. Fiz. SSSR 25, 763 (1953). 57 G . R. Khutsishvili, Zhur. Eksp. Teor. Fiz. SSSR 28, 370 (1955) (transl. Soviet Phys. JETP 1, 376 (1955)). 58 Morita, Ogata, Sakai, Bull. Kobayasi Inst. Phys. Research 6. 69 (1956). 69 I. M. Shmushkevich, Zhur. Eksp. Teor. Fiz. SSSR 33, 1477 (1957) (transl. Soviet Phys. JETP 6, 1139 (1958)). 60 v. B. Berestetsky, R. L. Ioffe, A. P. Rudik and K. A. Ter-Martirosyan, Nucl. Phys. 5, 464 (1958). 81 V. B. Berestetsky, B. L. Ioffe, A. P. Rudik and K. A. Ter-Martirosyan, Phys. Rev. 111, 522 (1958). 62 A. 2. Dolginov, Nucl. Phys. 5, 612 (1958). 8% A. Z. Dolginov and N. P. Popov, Nucl. Phys. 7, 591 (1958).
z4 25
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66 66
67 68
60 70
71 72 73
74
75
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A. Z. Dolginov, Zhur. Eksp. Teor. Fiz. SSSR 35, 178 (1958) (transl. Soviet Phys. JETP 8, 123 (1959)). G. E. Lee-Whiting, Can. J . Phys. 36, 1199 (1958). M. Morita, R. Saito Morita, Phys. Rev. 110, 461 (1958). M. Morita and R. Saito Morita, Phys. Rev. 109, 2048 (1958). A. 2. Dolginov and N. P. Popov, Zhur. Eksp. Teor. Fiz. SSSR 36, 529 (1959) (transl. Soviet Phys. J E T P 9, 368 (1959)). R. H. Good and M. E. Rose, Nuovo Cim. 14, 872 (1959). H. Mahmoud, Ann. Phys. (New York) 7, 429 (1959). A . 2. Dolginov and N. P. Popov, Zhur. Eksp. Teor. Fiz SSSR 38, 1518 (1900). S . B. Treiman, Phys. Rev. 110, 448 (1958). H. Frauenfelder, J. D. Jackson, H. W. Wyld, Phys. Rev. 110, 451 (1958). A. M. Bincer, Phys. Rev. 112, 244 (1958). C. C. Bouchiat, Phys. Rev. 112, 877 (1958). 11. EXPERIMENTS-NUCLEAR PHYSICS
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77 78 78
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82 85
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J. W. T. Dabbs, L. D. Robcrts and G. W. Parker, Bull. Am. Phys. SOC.,11, 2, 31 (1957). J. W. T. Dabbs, L. D. Roberts and G. W. Parker, Physica 24,6 9 (1958). D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953). F. J. Walter, J. W. T. Dabbs, L. D. Roberts, and H. W. Wright, O.R.N.L. 2877, Bull. Am. Phys. SOC.11, 3, 304 (1958). S. H. Hanauer, J. W. T. Dabbs, L. D. Roberts and G. W. Parker, Oak Ridge National Laboratory 2919. B. Bleaney, P. M. Llewellyn, M. H. L. Pryce and G. R. Hall, Phil. Mag. 45, 992 (1954). J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. SOC.London A 229, 20 (1955). M. H. L. Pryce, Phys. Rev. Letters 3. 375 (1959). C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105, 1413 (1957). H. Postma, W. J. Huiskamp, A. R. Miedema, M. J. Steenland, H. A. Tolhoek and C. J. Gorter, Physica 23, 259 L (1957) and 24, 157 (1958), and Communications
Kamerlingh Onnes Lab., Leiden, 310b. E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 106, 1361 L (1957) and 108. 503 L (1957). 87 M. A. Grace, C . E. Johnson, R. G. Scurlock and C. V. Sowter, Phil. Mag. 2, 1050 (1957). 88 E. Ambler, Proc. Int. Conf. Refrigeration, Copenhagen, 1958. (Unpublished). 89 H. Postma, Thesis, J. B. Wolters, Groningen (1960). 00 E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Physica 24, S 64 (1968) ; and Comptes Rendus du Congrhs International de Physique NuclBaire, Paris (1958) p. 831; and Phys. Rev. 110, 787 L (1958). 01 H. Postma and W. J. Huiskamp, Proceedings Int. Conf. Low Temp. physics, Toronto (1960). aa H. Postma, M. C. Eversdijk Smulders and W. J. Huiskamp, Physica, to be published. aa F. Boehm, Phys. Rev. 109, 1018 (1958). ~ 3 4 R. W. Bauer and M. Deutsch, Phys. Rev. 120, 946 (1960). O K W. J. Huiskamp, A. N. Diddens, J. C. Severiens, A. R. Miedema and M. J. Steenland, Physica 23, 605 (1957). P. Dagley, M. A. Grace, J . M. Gregory and J. S. Hill, Proc. Roy. SOC.London A 250, 550 (1959). 86
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R. W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960). R. W. Bauer and M. Deutsch, Nucl. Phys. 16, 264 (1960). eB A. N. Diddens, W. J. Huiskamp, J . C. Severiens, A. R. Miedema and M. J. Steenland, Nucl. Phys. 5, 68 (1958). looP. Dagley, M. A. Grace, J. S. Hill and C. V. Sowter. Phil. Mag. 3, 489 (1958). lol R. I. Hulsizer. W. J. Huiskamp, J. C. Wheatley and A. C. Anderson, Physica 24, S. 165 (1958). lo2 M.A. Grace, C. E. Johnson, R. G. Scurlock, Phil. Mag. 3, 456 (1958). Io3 J. M. Daniels, J . L. G. Lamarche and M. A. R. Le Blanc, Can. J. Phys. 36, 997 (1958). lo4 R. W. Kedzie, M. Abraham and C. D. Jeffries, Phys. Rev. 108, 54 (1957). lo5a G. R. Bishop, M. A . Grace, C. E . Johnson, H. R. Lemmer and J. Perez y Jorba, Phil. Mag. 2, 534 (1957). lo5b M. A. Grace, private communication. loB C. E. Johnson and D. A. Shirley, Bull. Am. Phys. SOC.11, 4, 373 (1959). lo’s To he published in Physica, see also ref. 89, 91. 107b H. Postma, A. R. Miedema and Miss M. C . Eversdijk Smulders, Physica 25, 671 (1059). 108 bf. A. Grace, C. E. Johnson, R. G. Scurlock and R. T. Taylor, Phil. Mag. 2, 1079 (1957). Q7
111. EXPERIMENTS-SOLID STATE ASPECTS N. Kurti, F. N. H. Robinson, F. E. Simon and D. A. Spohr, Nature 178, 450 (1956); M. V. Hobden and N. Kurti, Phil. Mag. 4, 1092 (1959). lUDb W.Low, Solid State Physics, edited by F. Seitz and D. Turnbull, Academic Press, New York, Suppl. 2 (1960). l o w J. W.Orton, Rep. Progr. Phys. 22, 204 (1959). low C . H. Townes, Quantum electronics (Columbia University Press, New York, 1960). loee G. W. Series, Rep. Progr. Phys. 22, 280 (1959). loQf D.Connor, Phys. Rev. Letters 3, 429 (1959). N. Kurti, Cryogenics, Int. Journal of LowTemp. Engineeringand Research, 1 (1960) 2. ll1 N.Kurti, J. Phys. Rad. 20, 141 (1959). lla M.A. Grace, C. E. Johnson, N. Kurti, R. G. Scurlock and R. T. Taylor, Phil. Mag. 4, 948 (1959). lI3 C. V. Heer and R. A. Erickson, Phys. Rev. 108, 896 (1957). 1146 V. Arp, N. Kurti and R. Petersen, Bull. Am. Phys. SOC.2, 388 (1957). 1lPb V. Arp, D.Edmonds and R. Petersen, Phys. Rev. Letters 3, 212 (1959). 115 J . M.Daniels and M. A. R. Le Blanc, Can. J. Phys. 37, 1321 (1959). 118 W. Marshall, Phys. Rev. 110, 1280 (1958). 1 1 7 S. S. Hanna, J. Heberle, G . J. Perlow, R. S. Preston and D. H. Vincent, Phys. Rev. Letters 4, 513 (1959). 118 B. N. Samoilov, V. V. Sklyarevskii and E. P. Stepanov, Zhur. Eksp. Teor. Fiz. SSSR 36, 1366 (1959); Soviet Physics J E T P 36(9), 972 (1959). 11Q B. N. Samoilov, V. V. Sklyarevskii and E. P. Stepanov, Zhur. Eksp. Teor. Fiz. 36, 644 (1959); Soviet Physics J E T P 36(9), 448 (1959). 120s B. N. Samoilov, V. V. Sklyarevskii and E. P. Stepanov, Zhur. Eksp. Teor. Fiz. 36, 1944 (1959); Soviet Physics J E T P 36(9), 1383 (1959). leob B. N. Samoilov, V. V. Sklyarevskii and E. P. Stepanov, Preprint (in Russian). 120C V. Kogan, V. D. Kulkov, L. P. Nikitin, N. M. Reinov, I. A. Sokolov and M. F. Stelmah, preprint (in Russian). 121 A. M.Portis and A. C. Gossard, J. Appl. Phys. 31, 205 S (1960). 1088
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J. C. Daunt, Proc. Int. Conf. on Low Temp. Oxford p. 157 (1952). C. J. Gorter, Proc. Int. Conf. Low Temp. Oxford p. 158 (1952). 123 N. J . Poulis and G. E. G. Hardeman, Physica 18, 201 (1952); C. J. Gorter, Rev. Mod. Phys. 25, 332 (1953). 124 J. M. Daniels and M. A. R. Le Blanc, Can. J. Phys. 36, 638 (1958). 325 J. M. Daniels, J. C. Giles and M. A. R. Le Blanc, Private communication, also presented a t the Int. Conf. on Low Temp. Toronto (1960). 826 A. R. Miedema, H. Postma and W. J. Huiskamp, R o c . Int. Conf. Refrigeration Copenhagen (1959). 1 2 7 A. R. Miedema, J. van den Broek, H. Postma and W. J. Huiskamp, Physica 25, 1177 (1959). 128 C. D. Jeffries, Phys. Rev. 106, 164 (1957). 1 2 9 M. Abraham, R. W. Kedzie and C. D. Jeffries, Phys. Rev. 106, 165 (1957); and Phys. Rev. 117, 1070 (1960). 130 A. Abragam, Phys. Rev. 98, 1729 (1955). 133 C. D. Jcffries, Phys. Rev. 117, 1056 (1960). 1 3 2 R. W. Kedzie and C. D. Jeffries, Preprint, to be submitted to the Phys. Rev. 1 3 3 M. Abraham, C. D. Jeffries, R. W. Kedzie and J. C. Wallmann, Phys. Rev. 112, 553 (1958). 1 3 4 F. M. Pipkin and J . W. Culvahouse, Phys. Rev. 106, 1102 (1957). 185 F. M. Pipkin and J. W. Culvahouse, Phys. Rev. 109, 1423 (1958). 136 F. M. Pipkin, Phys. Rev. 112, 935 (1958). -4.Abragam, J . Combrisson and I. Solomon, C. R. Paris 246, 1035 (1958); 247, 2337 (1958). 138 J. Combrisson and I. Solomon, J. Phys. Rad. 20, 683 (1959). z 3 O A. Abragam and W. G. Proctor, C. R. Paris 246, 2253 (1958). 340 M. Borghini and A. Abragam, C. R. Paris 248, 1803 (1959). 1 4 1 J. Uebersfeld, J. L. Motchane and E. Erb, C. R. Paris 246, 2129, 3050 (1958); J . Phys. Rad. 19, 843 (1958). 1 4 2 M. Abraham, M. A. H. MC Causland and F. N. H. Robinson, Phys. Rev. Letters 2, 449 (1959). 1 4 3 P. L. Scott, 0. S. Leifson and C. D. Jeffries, Bull. Am. Phys. SOC.I1 4, 453 (1959). (I, 1 1 and I, 12). 1 4 4 IITPAP Symposium on Polarization Phenomena of Nucleons, 1960, to be published in Helv. Phys. Acta. 1 4 5 For a review on the Overhauser effect the reader is referred to G. R. Khutsishvili, Usp. Fiz. Nauk. 71, 9 (1960). 146 References added an pyaof: a C. E. Johnson, J. F. Schooley and D. A. Shirley, Phys. Rev. 120, 1777 (1960). b A. Stolovy, Phys. Rev. 118, 211 (1960) c D. D. Hoppes, Proc. Int. Conf. Low Temp. Physics, (University of Toronto Press, 1960). d J . F. Schooley, D. A. Shirley and J . 0. Rasmussen, ibid. e C. A . Lovejoy and D. A. Shirley, ibid. j M. W. Levi, R. C. Sapp and J. W. Culvahouse, ibid. g G. R. Khutsishvili, ibid. h L. D. Roberts, F. J. Walter, J. W. T. Dabbs and G. W. Parker, ibid. i A. J. Freeman and R. E. Watson, Phys. Rev. Letters 5, 498 (1960). i D. A. Goodings and V. Heine, Phys. Rev. Letters 5, 370 (1960). k G. K. Wertheim, (Phys. Rev. Letters 4, 403 (1960). I V. Jaccarino, Phys. Rev. Letters 5, 251 (1960). m V. Jaccarino, M. Peter and J. H. Wernick, Phys. Rev. Letters 5, 53 (1960). 12%
322b
CHAPTER I X
SOLID STATE MASERS BY
N. BLOEMBERGEN HARVARD UNIVERSITY, CAMBRIDGE,MASSACHUSETTS C O N T E N T S :1. Introduction, 396. - 2. Paramagnetic resonance in maser materials. 400. - 3. Paramagnetic relaxation, 406. - 4. Maser circuits, 414. - 5. Noise, 420. 8. Millimeter and infrared solid state masers, 424.
1. Introduction
The three basic processes of interaction between electromagnetic radiation and matter are absorption, stimulated emission and spontaneous emission. They were first formulated in a general form by Einstein1. In an absorption process an oscillator of frequency v of the electromagnetic field changes its quantum number nuby 1, and the energy of the material system is increased by hv. In an emission process the reverse happens. In this formulation with quantized fields there is strictly speaking no fundamental distinction between stimulated and spontaneous emission. The latter may be regarded as emission induced by the zero point vibrations. For our purposes it is, however, advantageous to use the semiclassical language. A coherent microwave field constitutes the incident signal. One field oscillator corresponding to the signal mode is very highly excited. Consider two energy states in an assembly of particles. The separation between the upper and lower state is given by E2 - E l = hv21. (1) The transition probability per unit time for absorption and stimulated emission is equal to W12. This quantity is of course proportional to the square of field strength and the square of the matrix element connecting the two states. The net power absorbed from the electromagnetic field by the assembly of particles is (2) P = (n, - n2)hv2,W12.
+
References p. 427
CH. IX,
5
11
SOLID STATE MASERS
397
The population of the upper and lower state satisfy the MaxwellBoltzmann distribution law, if the system is originally in thermal equilibrium, n2/nl= exp (-hv,,/kT).
(3)
In this case the absorption processes dominate the stimulated emission processes. If, however, by one means or another, the population in a state with higher energy were made larger than the population in a state with lower energy, a negative absorption would result according to eq. (2). The material system would give off energy to the radiation field. The possibility of negative absorption was already noted by Lamb2, and was observed experimentally for a nuclear spin system by Pound and Purcell ,. Bloch had already observed an inverted resonance curve in his classic nuclear induction experiments with the technique of adiabatic rapid passage, but he did not interpret this in terms of a negative absorption. The practical implication to obtain amplification of microwaves by stimulated emission of radiation was clearly rerecognized by Prokhorov5, Townese and Weber’. Gordon, Zeiger and Townes6 constructed the first successful maser t, the ammonia beam atomic clock. In this device the molecular beam is separated in space. The NH, molecules in the upper state of an inversion doublet are focussed into a microwave cavity, while the molecules in the lower state are removed from the beam by an electrostatic focusser. Spin systems in solids offer the possibility to create an emissive or “maser” condition. The thermal equilibrium of the spin levels can be upset in a variety of ways. This is particularly easy at low temperatures where the spin-lattice relaxation times are long. The following methods to create a larger population in a state with higher energy are not restricted in principle to spin systems, but in practice these are the only systems in condensed matter to which they have been applied successfully thus far. 1. Sudden change in external parameter 2. Adiabatic rapid passage 3. The 180” pulse 4. Pumping in multiple level systems. In the first method an external parameter is suddenly changed to a new value, for which the state with the originally higher energy t This acronym stands for M(icrowave) A(mp1ification)by S(timu1ated) E(mission) Qf R(adiation). Kelerrnces p . 127
398
[CH.IX,$
N. BLOEMBERGEN
I
becomes the lower level while the populations remain unchanged. This method has only been applied to a nuclear spin systemS. The sudden reversal of magnetic field is not of practical importance, because in a paramagnetic material a field of several hundred gauss should be reversed in less than a millimicrosecond. The adiabatic rapid passage has been discussed extensively in the case of magnetic resonance4. The description in terms of an effective magnetic field in a frame of references rotating with the frequency of TRANSIENT METHODS FOR INVERTED DlSTRlBUTION
I
:u* NON-ADIABATIC REVERSAL
I I
I ADIABATIC RAPID PASSAGE
18Do PULSE
Fig. 1. Methods for the inversion of the population of two spin levels.
the applied high frequency field, is simple and is illustrated in Figure 1. It should be noted that the quantum mechanical equation of motion for any pair levels assumes the same form as the motion of a spin 4. If non-adiabatic terms are neglected, the population of any pair of levels can be inverted by applying a strong periodic perturbation, 2'12(t) of which the frequency is slowly varied through the resonant frequencyh-l(E, - El).If the strong perturbation %'12 is switched on. during a time t exactly at the resonant frequency such that &-1i@,,t
= n,
the population of the levels 1 and 2 is also inverted. In the case of magnetic spin resonance this can again be described very simply in a rotating coordinate system and corresponds to the 180" pulse also shown in Fig. 1. These three methods were all first invented in nuclear magnetic resonance. The last two can be and have been applied to electron spin resonance. They lead to a transient maser conditiong*lo.After the pulse or the adiabatic passage the inverted magnetization will tend to return to thermal equilibrium in a characteristic time T,, the spin lattice relaxation time. References
p. 427
CH. IX,5 11
399
SOLID STATE MASERS
The pump method leads to a steady state maser condition. This method is also quite general in nature. Consider a system of more than two energy levels out of which three levels are chosen, with energies E , > E , > E l , shown in Fig. 2. A strong resonant perturbation x13 is applied at the frequency h-l(E, - E l ) . The absorption at this pump frequency produces a saturation of this transition, n, = n3. In this saturated steady state condition the absorption and stimulated emis'
STIMULATED EMISSION
hYZ3
I I
I
I
I
I
I
I
I AESORPTION
I
9=90" , n , = n 3
I
n3)nz
hSe
bp
I
I
I
LARGE DRIVING FIELD H(Y3,) CAUSES SATURATION
n,(n,
n,
Fig. 2. Steady state inversion by three level pumping.
sion are balanced. Now consider the population of level E,. If n, > n3 = n, maser action will occur at the frequency
If n ,
< n3 = n, maser action will occur at the frequency v3, = h-l(E3 - E J .
Only if n1 = n, = n, no maser action will occur. The steady state population n2 in the presence of the pump power depends of course on the details of the relaxation mechanism between the various levels. This will be discussed more fully in the next section. The general nature of the argument should be emphasized. No assumptions have been made about the frequency separations or the nature of the transitions. It is only required that a strong perturbation is applied which only connects a pair of non-adjacent states. There may be more than one intermediate level. Often the transitions at v I 2and vZ3will be sufficiently separated in frequency from ~ 1 so 3 that the perturbation by the pump has a very small oscillatory character at the other transitions. Selective atomic collisions in a gaseous discharge may also fulfill the pumping function. In the case of optical pumping the frequencies Y,, and vI3may be very close. The separation References p. 427
400
N. BLOEMBERGEN
[CH, IX,
92
of the transitions can sometimes be achieved by the sense of polarization. The polarization or angular momentum pump was first proposed by Kastler 11, although not in connection with masers. The Overhauser effect12 is another example of the pump method. Bassov and Prokhorov13 proposed a pumping scheme for atoms in a beam, while the present author l4 suggested the three level solid state maser. This device which utilizes a paramagnetic material at liquid helium temperature serves as a very low noise microwave amplifier. It will be discussed in more detail in the remainder of this paper. The spontaneous emission which was mentioned at the beginning of this chapter should not be forgotten. I t means that some electromagnetic power w ill be emitted even in the absence of the incident signal. It can be shown that this corresponds precisely to thermal noise of the amplifiers. Its smallness is the very “raison d’etre” of the solid state maser. It will be discussed in more detail in section 5 . In a final section optical pumping in solids at low temperature and some other devices are mentioned which are closely related to masers. A device based on stimulated emission of radiation is defined as maser. Whether the device operates in the microwave region or in another part of the electromagnetic spectrum is less important. Nomenclature such as laser, iraser etc. to designate operation in the visible or infrared region will not be adopted. The term molecular amplifier or generator is also frequently used, especially in the non-english literature, but is less specific. There is a good recent review’ and an introduction in book form on masers 16. They give rather complete references to the literature until early 1959. The most recent contributions may be found in a conference report 16. Many references to recent work also appear at the end of this paper, although no effort has been made to achieve completeness.
2. Paramagnetic Resonance in Maser Materials The splitting of the spectroscopic ground state of paramagnetic ions in crystals can be described by a spin hamiltonian. Excellent reviews of the wealth of theoretical and experimental data have appeared ”. I t should be kept in mind that new data on spin hamiltonians are continually added, especially of ions as impurities in other host lattices and of splittings in the millimeter wave region. Consider the simple spin hamiltonian 2 = gs g * H, 0s; E(S: - s;) (4)
+
Refeyences p. 427
+
CH. IX,
5
21
SOLID STATE MASERS
401
which is adequate if the effective spin S < 2 and nuclear hyperfine interactions are absent. The first term represents the Zeeman energy, the second and third term the crystalline field splitting of the spin levels, E = 0 if a symmetry axis is present, D = 0 in cubic symmetry. The coordinate system has been chosen in such a fashion to diagonalize the crystal field tensor at the position of the ion. Often there are several crystallographically non-equivalent ions in the unit cell. For higher spin values fourth order and sixth order polynomials in the spin components should be added. Nuclear spin interactions present only an undesirable complication from the point of view of maser operation. At the microwave frequencies of interest the nuclear spin is effectively decoupled from the electron spin. Since simultaneous changes in electron and nuclear spin quantization are forbidden in first order, the effect of a nuclear spin I is to increase the number of nonequivalent ions by a factor 21 + 1. Diagonalization of the spin hamiltonian gives 2.5 + 1 energy levels. Transitions between a pair of these energy levels can be induced by a time varying electromagnetic field. A microwave field H,, exp (2nivt) near the resonant frequency k l ( E j - Ei)will induce transitions with a probability per unit time
where g(v - Y,) is a normalized shape function centered around the resonant frequency v $ ~ ,
1,
g ( ~ ~i,) dv = 1.
I t takes into account the distribution of initial and final states by variation of local fields. In very dilute magnetic materials the local variations in the crystalline field splitting parameters and magnetic field variations due to nuclear spin arrangements are most important in determining g(v). In more concentrated materials one has to add the dipole and exchange interactions between neighboring magnetic spins observed at microwave frequencies. If H,, is taken in a principal ( x - ) direction of the g-tensor and can be considered as a small perturbation, the population of the levels per unit volume niand n, will be given by the equilibrium Maxwell-Boltzmann distribution (3). An imaginary part of the susceptibility can be defined by equating the power absorbed to +wx”H$ Combination of eqs. ( 2 ) and (5) then yields Kefcrences
p . 427
402
[CH. IX,8
N. B L O E M B E R G E N W t , ) =
@-lP2g:
I (i I s* I i) l2 gw(4 (% - 4.
2
(6)
The corresponding real part of the susceptibility may be found from the Kramers-Kronig relations. This treatment can be generalized to other modes of polarization of Hrf. The energy levels have been determined for paramagnetic ions in a large number of substances by paramagnetic resonance. As an example the four spin levels of the Cr+++ion in ruby are shown as a function of
FIELD PARAMETER, GI0
r? , J
209
I
1 00 s g
~~~
2
2w
-:
-10
-20
-4
w
g
3
-30
-6
-40
-15 l
-8 0
-50 2 4 6 8 1 0 APPLIED MAGNETIC FIELD,/#. IN kOI
Pig. 3. The energy levels of the Cr+++ion in ruby with the magnetic field parallel t o the trigonal axis.
-20
0
0
l
z
F
2
F
!
J
3
MAGNETIC FIELD I N
KOe
Fig. 4. The energy levels of the Cr+++ion in ruby with the magnetic field perpendicular to the trigonal axis.
magnetic field in the direction parallel and perpendicular to the trigonal axis of the crystal (ruby is A120, with Cr as an impurity). The spin hamiltonian is described1* by eq. (4) with g,, = 1.9840 f 0.0006, gL = 1.9867 f 0.0006, D = - 0.3831 f 0.0002 cm-1. Tables of the matrix elements of the spin operator for arbitrary values of the external magnetic field have been published'. If the field is parallel to the trigonal axis, m, is a good quantum number and transitions with I d m I > 1 are strictly forbidden. This is the case for the straight lines in Fig. 3. Mixing of m,- states occurs for curved characteristics shown in Fig. 4. Ruby with a Cr : A1 atom ratio between 2 : lo4 and 1 : lo2 is an References p. 427
CH. IX,$
21
SOLID STATE MASERS
403
excellent maser material. Large single crystals can be grown and are commercially available. It can be cut and polished with ease. It is chemically and physically stable. It has negligible dielectric losses and a good thermal conductivity. The spin-lattice relaxation time is sufficiently long so that a rather small pump power can produce saturation. The full width of the resonance between points of half maximum intensity is of the order of dv = 6 x lo7 Hz. This width is partly due to the magnetic interactions with A1 nuclei and partly to variations in the crystalline field tensor. The line width will vary with resonant frequency and the pair of energy levels involved. For each transition the line width in gauss is related to Av by dv = ( av/ 8H)dH. If k T is large compared to the overall splitting of the spin quartet, the spin populations in thermal equilibrium with the lattice at temperature T are approximately given by n, - nf = $nohvu/kT,where no is the total number of Cr+++ions per cc. For a relative concentration Cr : A1 = 1 : lo3 corresponding to no = 4.7 x 1019, one finds from eq. (6) X " m 0.01 for vu = 6 kMHz at T = 2" K, if the spin matrix element is put equal to unity. In practice this element will vary widely with geometry. If the Zeeman energy becomes large compared to the crystalline field energy, transitions between non-adjacent states will have a very small matrix element, because they correspond to "forbidden" transitions with I Am, 2 2. The characteristics are then nearly straight. Adequate pumping between non-adj acent levels becomes difficult. For maser operation in the millimeter wave region one needs therefore salts with larger crystalline field splittings. The Cr+++ ion in emeraldlg may be suitable with h-lD = - 26 & 2 kMHz. The Fe+++ ion in Al,03 is another example. This ion has a spin S = 512. Its microwave spectrum has been measured by Prokhorov19, Bogle20 and others2I. In a cubic field with a trigonal component there are three Kramers doublets in zero magnetic field. The intervals between these three doublets are 12.07 kMHz and 19.13 kMHz. This permits the operation of zero field masers which have been discussed by Bogle22. The Fe+++ion in rutile (TiO,) has three doublets separated by 43 kMHz and 81 kMHz. If the magnetic field is applied parallel to the trigonal axis, one finds that the m, = -+- 4 transition is narrower than the others. This indicates that there is a distribution of crystalline field splittings. The frequency of the m8 = 4 --f - 3 transition is independent of the crystal field in first approximation.
I
References p. 427
404
N. BLOEMBERGEN
[CH. IX,
52
The random distribution of magnetic ions, if nothing else, will produce strains and vacancies in the lattice which leads to line broadening. At higher concentrations the magnetic interactions between ions become important. If the magnetic concentration is neither too high nor too low, additional resonance lines due to neighboring ion pairs have been observedzs.z4, The Ni* ion often has excessively broad lines which are hard to saturate. The breadth is undoubtedly due to variations in crystalline field splittings. The Cu++with S = has only two spin levels and an undesirable hyperfine structure. The Mn++has rather small crystalline fields splittings, also complicated by a large hyperfine interaction. The Ti+++is chemically not stable and has a very short relaxation time. The most promising materials for masers in the microwave region therefore contain Cr+++, Fe+++,V++ or Gd+++ions, preferably as impurities in simple oxide structures. These are physically and chemically stable (gems ), and all ions are often in equivalent lattice positions. I t should be mentioned, however, that the first experimental maser utilized dilute gadolinium ethylsulphate 25. Potassium cobalticyanide with Cr as the active magnetic materials has also been used successfully. Recent observations of the microwave spectrum have revealed that this material exhibits polytypism. There are several slightly different unit cells and consequently there are a large number of nonequivalent positions of the Cr atom in a single piece of materialz0. So far an elementary perturbation treatment has been adopted to describe the electromagnetic transitions between spin states. This procedure is not quite satisfactory to describe the operation of a pumped maser, because the pump field necessarily constitutes a large perturbation. The Kramers-Kronig relations are also no longer valid, since the magnetization is a non-linear function of the field under conditions of saturation. The rigorous approach is to start with the equation of motion for the density matrix @
= - i6-1(&@
- @A?)
where the hamiltonian now also contains the time dependent perturbation. The components of spin vector 5 are Tr(S@). The time dependent terms give the microwave susceptibility. Clogston27 has given a detailed algebraic analysis of a three level spin system with an applied microwave field at the pump and at the References p. 427
CH. IX,$
31
SOLID STATE MASERS
405
signal frequency. I t turns out that the diagonal components of the density matrix, corresponding to the populations in the various spin states, are given correctly by the relaxation rate equations from perturbation theory, if the applied pump field is small compared to the line width, H,, < A H . This is the case of interest for solid state masers. The result is not surprising. The phases of the off-diagonal elements are scrambled by the distribution of resonant frequencies g(v). Similarly, perturbation theory would continue to give valid results even in the case H,, A H , if H,, were given a random frequency modulation. Then Hrppresents a hot “black-body’’ radiation field over the width of the resonance, which gives rise to the ordinary saturation phenomenon. Interesting coherence effects, which might give rise to sideband resonances or structure in the microwave susceptibility and which are undesirable in maser applications, can thus be avoided. The following procedure is therefore justifiable. Calculate the populations of the spin levels from the rate equations which utilize transition probabilities per unit time derived from perturbation theory. Then determine X” from eq. (6). The determination of the real part X’ at high power levels is somewhat more involvedz8.Usually the interest is in weak signals at the maser frequency. Then the Kramers-Kronig relations are obeyed near this frequency. Transitions can also be induced by time varying crystal fields rather than Zeeman terms. Ultrasonically induced transitions are well known in nuclear spin systems. Mattuck and Strandberg29have given a discussion of them for electron spins. A periodic variation of D in the Hamiltonian (1) will induce quadrupole transitions. Ultrasonic saturation has recently been achieved at microwave frequencies30. The advent of microwave ultrasonics makes an acoustically pumped maser possible.
>
3. Paramagnetic Relaxation Transitions between the spin levels are not only induced by externally applied electromagnetic or acoustic fields. The modulation of the g-tensor and crystalline fields by the thermal motion of the lattice also causes transitions. They constitute a contact between the spins and lattice, and in the absence of external driving fields are responsible for the establishment of a Maxwell-Boltzmann distribution over the spin levels. The probability per unit time for these spin-lattice transitions will be described by quantities wij,which have the magnitude of References
p. 427
406
N. BLOEMBERGEN
[CH. IX,
53
inverse spin-lattice relaxation times. They satisfy the detailed balancing condition w f I = wg exp (--hv,/kT).
(7)
This relation also follows from the properties of the matrix element of a harmonic oscillator. A lattice quantum or phonon is absorbed when the spin energy increases and emitted when the spin energy decreases. Note that Ep,/(Gp, 1) = exp (--hv/kT), where Z,, is the average excitation quantum number of a lattice oscillator at frequency v and temperature T . Besides the emission or absorption of a single lattice quantum, Raman processes in which two lattice quanta take part may also occur. At liquid helium temperatures, however, the theory of Van Vleck31 predicts that single phonon processes should dominate and the spin-lattice relaxation time should be inversely proportional to T . This feature has been verified experimentally for the C W ion in a number of compounds, but other ions often have a more complicated temperature dependence. The dependence on the frequency of the transition v, which should be w, a v$ in the simplest case has been verified once. The order of magnitude of the transition probabilities wtl at 4’ K is in the range lot2 to sec-’ for Cr++t, F e w and Gd* ions, It varies of course with the matrix element for the particular transition, the nature of the chemical compound etc. Another type of relaxation process takes place entirely within the spin system. No energy is transferred to or from the lattice or a radiation field. A trivial case is the flip-flop between two pairs of equidistant spin levels. I t is possible, however, that processes in the spin system change the population differences between non-equidistant levels, i.e. affect the intensity of well-resolved resonances. The term “crossrelaxation” has been introduced 32 to describe this phenomenon. Higher order processes in which three, four or more spins take part simultaneously are quite important. For example, two downward transitions at frequency vu may be accompanied by one upward transition between a pair of levels with twice the spacing, Y , ~ = = 277,. The transition probability for a cross-relaxation process in which one spin makes the transition from level E, -+E,, a second one from level E , + E l , a third one from level Em --f E , will be denoted by w ~ , ~ ~ , , , ,It, . ,is. not necessary that the energy of the unperturbed levels of the spin hamiltonian be exactly conserved, but a small balance may be absorbed by the dipole-dipole or
+
ipZic
References $. 427
CH. IX,
9 31
407
SOLID STATE MASERS
exchange interaction between electronic spins and/or nuclear spins. These interactions which are not included in the unperturbed spin hamiltonian also determine the line shape functions g(v). For a two spin process the following approximate relationships holds m
wtj.kL
g(v - v k l ) g ( v - y$5> dv.
(T2)-1
(8)
0
The spin-spin phase memory time I', is defined as gmsx(v).The simple physical interpretation of eq. (8) is that the cross-relaxation time is inversely proportional to the overlap of the two resonances. Higher order spin processes are by no means negligible. Although the .characteristic time for a four-spin-flip w;: kz,,,, o p will be much longer than T,, it may still be much shorter than TI. Even higher order processes should sometimes be considered between well-resolved resonances. The population the i t h spin level is thus governed by the following rate equation
Higher order cross-relaxation terms should be added. Higher order spin-lattice terms, which also exist in principle, are usually not important. The equations can be linearized in the n, in the high temperature approximation, hv, k T . The steady state solution of (9), obtained by putting the left hand side equal to zero, describes the balance between the pump action of externally applied field(s), spinlattice and cross-relaxation mechanisms. An important question is whether the lattice vibrations really constitute a thermal bath for the spins. The specific heat of the lattice oscillators in the same frequency range as the spin resonances is very small. It may not be justified to assume that the wu are constants satisfying the relationship (7), where T is the temperature of the helium bath. A set of rate equations for the number of phonons in each lattice oscillator may be juxtaposed to eq. (9)>
<
dnph(vL)/dt
= - 'Ij
[Nt{Nph(3'Z)
+ '>
- ?zjNph(YZ)l - Cnph('Z)
References
p . 427
-
' L I ~ ( ~ Z ) I ~ 1'&
(lo)
408
N. BLOEMBERGEX
[CH. IX,
33
Assume that the density of final states in the spin system has a stationary distribution g(v) due to rapid cross-relaxation. If the density of lattice oscillators is e ( v ) , then cg and wu are related by w,g(v) = ‘ljnph@ (’) ’ The phonon relaxation time rphgives a phenomenological description of phonon-phonon interactions and interaction with the helium bath. The actual situation will be a great deal more complicated than this simple description. There will be a non-thermal distribution in the excitation of lattice oscillators. One should consider the interaction between each pair of modes with their actual degree of excitation. In practice the oscillators are divided in discrete ranges, those that are “on speaking terms” with spin transitions and those that are not. The relaxation of the hot phonon range is then represented by the last term of eq. (10). If rphis assumed to be very short, then nphM nPh,the lattice serves indeed as a heat reservoir for the spins. Several authors have questioned this assumption 33, 34. All experimental data on masers can however be explained by the set of equations (9) alone, with the assumption that nphis approximately equal to the equilibrium value ii,, corresponding to the bath temperature. Excessive heating of phonons in the region of a “hot” spin resonant frequency would even suppress the establishment of an inverted distribution necessary for maser operation32,35. The interplay of the power transfer balance between spins, lattice oscillators and bath36could in principle be very involved. It is fortunate that this complication in the magnetic materials used in masers above 1 ° K does not appear to arise. The advent of masers has however renewed the research activity in spin-lattice relaxation at low temperatures. It is possible that cross-relaxation effects in the system help to distribute the power flow over a wider frequency range3’. Since more lattice oscillators can take part in the heat conduction, the lattice does not constitute a bottleneck. It has been suggested16that the spins could be used as a generator for phonons. The stimulated emission of phonons in such a phonon maser (“maseph)’) would be described essentially by equation (10). If the phonon relaxation time is long and the spins are hot, r;: -+0 and ni = n j , the excitation of the lattice oscillator will increase linearly with time, and for an inverted spin distribution ni > nr,an exponential increase in the excitation of the lattice oscillators would result. They can never attain a negative temperature, because their energy levels Krfercnces p . 427
CH. IX,
5
31
409
SOLID STATE MASERS
have no upper bound. A limitation is of course set by the rate at which the spins can supply energy to the acoustic resonator and the latter loses energy through acoustical coupling with the environment or damping mechanisms. With careful control of this acoustic coupling an amplifier or coherent generator for phonons would result. I n practice, however, the phonons serve as a reservoir for the spins rather than vice versa. The hot spins can build up the energy density in the electromagnetic field rather than in the phonon field. The coherent excitation of lattice oscillators should be accomplished by use of the piezo-electric effect rather than the spins ,O. Conversely, such acoustical microwaves can then be used to study the interactions with spins and other phonons and determine the relative magnitudes of the quantities w~ and rph. The solution of three equations of the type (9) for a three level spin system (S = 1) becomes
hN w21v21 - w32v32 (11) 3kT w32 w21 w32 in the absence of cross-relaxation, in the limit of very large pump power at the highest frequency v3, and in the high temperature limit, with an arbitrary signal at the frequency v , ~ . Therefore, eq. (11) is valid only for W,, --f 00, W,, = 0, wtf,k L = 0, hv,, kT. Fig. 5 shows how the effective susceptibility at the pump frequency approaches zero with increasing pump power. This is the usual saturation of a spin resonance. If appreciable phonon heating occurred, the saturation curve would not have the simple theoretical form39. At the maser frequency the susceptibility, measured with small signal power W32 m 0, approaches a negative limiting value for W13 00. The curves were taken for Cr+++ions which have four levels, but if cross-relaxation is avoided the results are very similar to those for a three level system. If the middle level in a three level system comes nearly midway between levels E l and E,, cross-relaxation mechanism w,,,32 becomes dominant. If the middle level moves close to either the upper level or lower level cross relaxations w , , , , ~ or wS1, 12 become important. In either case there will be overlap between closely spaced resonances and the steady state condition becomes n, = n2 = n3 for W13 -+00 and w12,32 wi,. With sufficiently fast cross-relaxation, no maser action is possible in this case. The pump power heats up the spin system as a n,
-
n2 = n,
- n2 = -
+ +
<
--f
>
References
9. 427
410
[CH.IX,5 3
N. BLOEMBERGEN
whole. The cross-relaxation processes are responsible for the establishment of thermodynamic equilibrium within the spin system, as postulated by Casimir and du Press. This situation is incompatible with a steady state maser operation. At higher concentrations of magnetic ions higher order spin process gain rapidly in importance. In concentrated magnetic salts the Casimir-du Pre hypothesis is always well satisfied. Only dilute materials can be used in continuous wave masers, at least in the conventional microwave band. It is conceivable that higher magnetic concentrations are permissible with larger spacings
001
aio POWER
10
10 AT
VZ4
IN
100
MILLIWATTS
Fig. 5. The imaginary part of the susceptibility for two transitions between the four levels of Cr in K/(0.996 Co, 0.006 Cr) (CN), as a function of pump power at the frequency va4. The crystal was kept a t 2.6" K. The magnetic field H , = 1176 oersteds made a n angle of 10"with the a-axis in the ac-plane. The drawn curves have the theoretical form given by eq. (9) in the absence of phonon heating.
between the spin levels for ions which have resonances in the millimeter wave region of the spectrum. Various effects of cross-relaxationbetween the four levels of Cr+++ions have been noted by a large number of authors. The solution of the rate equations becomes algebraically involved. A large variety of situations can occur in which one or more cross-relaxation terms are important. S h a p i r ~Maiman40 ~~, and BOlger4lhave shown how cross-relaxationmay inhibit maser action in a certain specimen at low temperature, but not at a higher temperature. What counts is the relative magnitude of the spin-lattice and cross-relaxation time. At high temperatures the former are shorter and cross-relaxation effects become relatively less imporReferences p . 427
CH. IX,
3 31
SOLID STATE MASERS
41 1
tant. As a corollary higher magnetic concentrations may be used in maser materials at higher temperature. Mims42 has noted that sharp variations in the steady state populations occur, whenever the orientation and magnitude of the magnetic field is such that one transition frequency is close or equal to a multiple of another. Several other authors have made similar observation^^^^ 41 and have called the phenomenon harmonic cross coupling. Clearly it is an important special case of the multiple spin transitions discussed in the general framework of cross-relaxation. Harmonic cross-relaxation effects have been reportedls up to the eleventh harmonic! The collaboration of at least twelve spins is of course a highly concentration dependent process. If the concentration is chosen too high, however, the required resolution is not obtainable and everything is washed out in the thermodynamic spin pool of Casimir and du Pre. The observation of these very high order spin processes also requires a long spinlattice relaxation time, T , > 106T2.Cross relaxation effects can also account for certain anomalies in spin-spin relaxation reviewed by G ~ r t e r Additional ~~. dispersive regions in the susceptibility should be expected32near frequencies of the magnitude of each of the wv, The discussion to this point may have given the impression that crossrelaxation is usually detrimental to maser operation. This is not at all true. Cross relaxation is responsible for homogeneous saturation of the pump transition and homogeneous inversion of the population at the maser frequency32.Since only a small fraction of the total number 01 ions have a configuration of surrounding nuclear spins and crystalline field to be at resonance with the pump frequency, one might think that only a small fraction of ions would get pumped and become emissive. In that case there would either be only inversion in a very narrow frequency range, or more likely no maser action at all since there is no one to one correspondence between the pump and signal frequency distributions. Fortunately cross-relaxation effects saturate transitions on both sides of the pump frequency as well, and maser action is obtained over the full width of the signal frequency resonance. It should be noted that homogeneous cross-saturation can only occur if spins in nearby unit cells have different resonant frequencies. At extremely low concentrations of magnetic ions a “hole” can be burned. The same is also true if the line is broadened by large scale inhomogeneities in the external magnetic field or crystalline configurations. Appropriate maser action would still be obtainable, even in the Referenus p . 427
412
N. BLOEMBERGEN
[CH. IX,
3
presence of large scale inhomogeneities, if the pump is frequencymodulated to “cover” the entire resonance. Cross-relaxation between a. pair of levels of a Gd+++ion and an equidistant pair in a Ce+++ion was utilized in the first solid state maser25to obtain a faster relaxation rate for that particular pair of Gd+++levels through coupling with Ce+++which has a much shorter spin-lattice relaxation time. Maser operation in ruby is sometimes also dependent on a fast cross-relaxation process43. Cross-relaxation makes it possible to have a maser frequency higher
€3
1
Pump
Signal
vlj= 9595 MHz
v,J‘10590 M M
€2
6
Big. ti. The energy levels of Cr+++in ruby in a magnetic field of 1675 oersteds perpeudicular to the trigonal axis. Cross-relaxation between the transitions vI4 and vaSsaturatrs the former, n, = n,, when pump power is applied at vps. Maser action results at a higher frequency vlgthan the pump frequency4j.
than the pump frequency. This has been emphasized by Minis and and has been demonstrated experimentally by Geu~ic*~, Higa and A r a m ~Consider ~~. the particular arrangement of energy levels of the C++ ion in ruby shown in Fig. 6. Pump power is applied at the frequency ~ 2 3 ,which is equal to one-half the splitting ~ 1 4 .Harmonic cross-relaxation w , ~ 3, 2 , 1 4 is responsible for saturation of the v I Q transition also, n , = n4. Maser action was observed at Y,, > v12, implying that n3 > n,. Thercis no violation of conservation of energy or other thermodynamic considerations. For the emission of one quantum kv,,, the pump has to supply at least two quanta 2hv3,. The cross coupling converts these into one quantum kv,, > k ~ 1 3 .The difference 2hv,, - hv,, is taken up by the lattice through spin-lattice relaxation. This type of mechanism would be of particular importance in extending the application of masers into the millimeter wave region. This conversion of quanta absorbed by the spin system into larger Kefcrnzccs
p . 427
CH. IX,
5 31
SOLID STATE MASERS
413
quanta -larger level spacings may also be provided by exchange interactions between neighboring spins -is perhaps also helpful in permitting a larger fraction of lattice oscillators to carry away absorbed power. The rate equations (9) clearly also include a description of the case in which more than one transition is pumped. Fig. 7 shows the energy levels of Cr+++in ruby when the magnetic field makes an angle 6 = arc cos (3-Y2)with the trigonal axis. Such push-pull pumping4' has been used to enlarge the population difference at the maser frequency Y,,.
bl
,I
,
--
The rate equations also describe saturation of maser amplification if the signal power becomes too large. The occurrence of W,, in the denominator of eq. (11)is an example of this effect. They also govern the return to equilibrium in a pulsed maser after inversion of the level population. They do not include the effect of a larger coherent transverse time-dependent component of magnetization. The off-diagonal components of the spin density matrix should then be taken into account as was already mentioned before. In this case the energy of the spin system may be radiated away very rapidly. This radiation damping effect 48 is, however, negligible, if the spinspin phase memory time T , is very short compared to the radiation time,
T,
< (~ZM'@O)-~.
(12)
Here M is the transverse magnetization, 7 the filling factor and Qo the References p . 427
414
[CH. IX,3 4
N. BLOEMBERGEN
quality factor of the electric circuit. In pulsed masers care should be taken that this condition is satisfied. Otherwise the material cannot be used as an amplifying medium, since its stored energy is radiated away in a short burst. The combination of the solution of the relaxation rate equations (9) with the equations (2) and ( 5 ) give the (negative) absorbed power of the maser material. They provide the basis for a discussion of the circuit aspects of this type of solid state maser. 4. Maser Circuits
Consider a cavity with a resonance mode at the pump and signal frequency, corresponding to two spin resonances of the paramagnetic RECEIVER
I HELIUM LEWAfi
PARAMAGNETIC
CRYSTAL
Fig. 8. Schematic coupling diagram of a reflection cavity maser.
salt inside the cavity. Fig. 8 shows a reflection type cavity. If the incoming signal stimulates the emission of more power in the salt than the absorption losses in the walls of the cavity, more power will leave the cavity than is incident on it. The action of the paramagnetic salt may be described in terms of a magnetic quality factor QM which is equal to 252 times the electromagnetic energy stored in the cavity divided by the power absorbed per cycle at the paramagnetic resonance. If the whole cavity is filled with the paramagnetic material and if the mode of polarization and the value of X ” for this mode were uniform over the cavity, then the simple relation Q, = (4nX”)-lwould exist. Actually the direction of H,, with respect to the crystal axes will vary over the cavity and also the population difference n, - n, will be a References p. 427
CH. IX, 9 41
415
SOLID STATE MASERS
function of position, because the pump field is not uniform. Therefore the expression for the absorbed power has to be integrated numerically over the volume of the cavity in practical cases. If the pumping field has nodes in the magnetic material, there will always be absorbing regions which are not pumped. If the energy density at the pump frequency is chosen high enough, saturation can still be achieved in nearly the whole specimen with the exception of narrow regions around the pump nodes. It can be shown that spin diffusion is not sufficiently rapid to give saturation in these regions. The suggestion49that hot phonons would produce saturation at the nodes has been disproved by BOlger4l.Maser action is possible because stimulated emission from
\ dM ‘
AVO
Fig. 9. Equivalent circuit of a reflection cavity maser, when both cavity and paramagnetic spins are tuned t o resonance. Reactive parts have to be added off resonance. The equivalent noise current generators associated with the antenna conductance Q,-l at temperature TO.the cavity conductance Qo-l a t temperature T o and the magnetic spin conductance QM-1 a t spin temperature TM are indicated, together with incident and outgoing power.
the rest of the crystal dominates the absorption at the signal frequency near the pump nodes. The reflection cavity maser acts as a negative resistance amplifier with an equivalent circuit as shown in Fig. 9. The reactive parts have been omitted in this diagram. They are, of course, important in determining the band width of the amplifier. The unloaded Qo of the cavity describes all losses in the cavity, other than the spin resonance, such as eddy current and dielectric losses. The latter are usually negligible at liquid helium temperature. The external load coupling is described by Q,. A stable amplifier results for
Q, < 0 and Qol
+ Q,* > I Q,
I-l
> &l.
An oscillator results for
Q, < 0 and Q;’
+ Qi
1-l.
(15)
The power gain, if both the cavity and the spin system are tuned for resonance, is References p . 427
416
N. BLOEMBERGEN
PoIPi = G = (QG'
+ Qil-
[CH. IX,
Q;')'/(Q;l
+ QG + Qi1I2.
94
(16)
The band width B between points of half maximum gain, if the spin resonance is much wider than the cavity resonance, is given by
I
(Gf - l)B = 2v( Qi;:
1-1
- Q;')
(17)
which in the limit of high power gain reduces to a constant voltage gain-band-width product G*B FV 2v 1 QM 1-l. (18) The product is somewhat more favorable for a reflection-cavity than for a transmission cavity arrangement. It is essential to have a nonreciprocal coupling element to prevent instability of the amplifier by load variations and to prevent noise generated in subsequent stages to reenter the maser. The circulator in Fig. 8 serves this function. Fig. 10 shows a cavity developed by Cooper and Jelley50 for the PUMP POWER
SIGNAL COAX.
RESONATOR
RUBY
-
F- &'+ \
\
C AXIS
DC
Fig. 10. Diagram of cavity used in the 91-cm maser of the Harvard radiotelescope. Iieflwnces
p . 427
N. BLOEMBERGEN
Fig. 11. Packaged 21-cm maser assembly. The antenna feed, the helium dewar and permanent magnet with adjustment coils are clearly visible.
N. BLOEMBERGEN ~_________~_______ _..
Fig. 12. The maser a t the focus of the Harvard 60" radiotelescope.
CH. IX,
9 41
SOLID STATE MASERS
417
21 cm maser of the Harvard radiotelescope. The maser material is ruby with 0.05% chronium, placed on both sides of a 21 strip resonator at 1420 MHz. The pump frequency is 11.27 kMHz. A magnetic field of 2000 oersted is applied at right angles to the trigonal axis. A value QM = - 140 is readily attained at 4.2" K, corresponding to a gainbandwidth product of 20 MHz. The external coupling is adjusted to operate with 20 db gain at 2MHz bandwidth. The L-band circulator, developed by Davis et a1.51, has an insertion loss of 0.35 db and an isolation > 20 db. The small permanent magnet and helium dewar are shown in Fig. 11. The magnet weighs 60 lbs and has a corrective winding to tune the field to the spin resonance. The current in this auxiliary winding needs to be adjusted about twice during a day as the field in the magnet gap varies with environmental changes in temperature. A large fraction of the weight and all variations in field strength could be eliminated by using a superconducting solenoid52 of niobium wire. Fluctuations in maser gain due to small variations in magnetic field, temperature of the helium bath and microwave coupling of the antenna to the maser are reduced by an automatic gain control system. The incident pump power is regulated by a feedback to a ferrite modulator. This system has reduced the drift of the base line to an acceptable level as may be seen from the recording in Fig. 14. A recent development has eliminated the need for gain control. A comparator switch makes it possible to compare the antenna temperature directly with the noise temperature of a calibrated reference load. The whole detection system, including the maser, acts as a null instrument. The switch consists of another L-band circulator 51. Reversal of the magnetic field changes the sence of circulation. This arrangement has proved very satisfactory for radiometric purposes. The packaged assembly weighs 200 lbs and is shown at the focus of the 60' Harvard radiotelescope in Fig. 12, The dewar may tilt by 45". The filling of 3 liters of liquid helium lasts for fifteen hours. An X-band radio-telescope maser has been described by Townes and coworkers 53. A similar X-band radiometer has also been developed by Kikuchi and coworkers for the University of Michigan. The gain-bandwidth product is essentially determined by the number of spins available for stimulated emission. Non-resonant travelling wave structures permit larger gain-bandwidth products. If the stimulated emission per unit length of transmission line exceeds the losses References p . 427
41 8
94
[CH. IX,
N. BLOEMBERGEN
per unit length, an exponentially increasing wave results. The power gain is
G =~
I
X P( X i n g QM
I)
(19)
where 1, is the wave length of the slow wave structure or transmission line. The gain-bandwidth product in the limit of large gain is given by
BGablJ2w 3l12Av
(20)
where the gain is expressed in decibels; B is the bandwidth of the amplifier and A v is the width of the spin resonance assumed to be of Lorentz shape, both measured between the half-power points. DeGrasse, Schulz-duBois and Scovil54 have developed a travelling wave maser in which the non-reciprocal element has been built in. The microwave field has opposite sense of polarization in different parts of the cross section of the slow wave comb structure shown in Fig. 13. The forward wave is amplified by the maser material (ruby with 0.1yo chromium). The backward wave is attenuated by ferrite material or a
I ALUMINA I SPACER I
170 RUBY ISOLATOR MATERIAL
Fig. 13. Cutaway view of a ruby travelling wave maser. (After deGrasse, Schulz-duBois and Scovil.) References p . 487
CH. IX,
41
SOLID STATE MASERS
419
dark ruby with 2% chromium. These are placed in the other part of the cross section and are therefore only at resonance for the backward wave polarization. The dark ruby acts as an absorber in spite of its exposure to the pump field, because powerful cross-relaxation prevents saturation. A gain of 23db with a bandwith of 25 MHz has been attained at 59 kHz in a structure 3” long at 1.5” K. The backward attenuation is 29 db and the device is “short-circuit-stable”. The pump power requirement in the non-resonant structure is 10 milliwatts at 19 kMHz. The tuning range of 350 MHz is determined by the properties of the slow wave structure. It is achieved by varying the magnetic field and also making some adjustment in the pump frequency. In the cavity maser the cavity would also have to track the tuning which adds considerable complication. Wider bandwidth with a corresponding reduction in gain can be obtained by stagger tuning ruby crystals in slightly different orientations. Alternatively the gain-band width product can, of course, be increased by a longer structure with more active material. The rather long relaxation time of 0.1 sec limits the maximum signal strength that can be handled or the dynamic range. This saturation effect is expressed by transition probability at the signal frequency occurring in the denominator of eq. (11).Each surplus spin can emit at most one quantum in a relaxation time. The ruby maser begins to saturate at an output power level of six microwatts. This restriction is not so severe as might appear a t first, because the primary function of the solid state maser is as a very low-noise pre-amplifier. For this reason alone considerable effort has been devoted to the development of the devices described above which need liquid helium for their operation. The long relaxation time also determines the recovery after saturation e.g. during a radar transmitter pulse. This puts very high demands on a TR-switch. Attempts to use germanium diode and ferrite switches and also to switch the magnetic field off-resonance during the transmitter pulse have been made55. On the other hand, a long relaxation time permits the use of pulsed pump power and the peak pulse power of a transient signal can be much larger than the steady state dynamic range would indicate. During a pulse short compared to the relaxation time the heat capacity of the spin system can act as a storage box for the incoming energy. References p . 427
420
N. BLOEMBERGEN
[CH. IX,
05
The gain stability presents no serious difficulties in practice, provided a non-reciprocal element of good quality is available. Since the maser itself operates with a stable crystal and microwave hardware at low temperature, it has very good inherent stability. Fluctuations in the pump power have relatively little effect on the gain, if the power is well above the saturation level. This is evident from the horizontal asymptote in Fig. 5. On the other hand, a slight remaining dependence of gain on pump power may be used for a gain stabilization feed-back50. 5. Noise
Noise characteristics are the most fundamental aspect of solid state masers. This question has been discussed by several authors56. Here a very simple treatment shall be outlined based on a very general thermodynamic argument used by Nyquist to describe thermal Johnson noise and the physical interpretation that this noise corresponds to spontaneous emission5' from thermally excited levels. Consider ;t lossy paramagnetic salt in thermal equilibrium at temperature T . Its circuit characteristics can be described by an effective conductance QG. According to Nyquist's thermodynamic argument the mean square value in a frequency interval Av of a current noise generator in parallel to this conductance should bet
because the salt could form a matched termination for a lossless transmission line, the other end of which is also matched by a resistance at temperature T . I n a maser this thermodynamic argument breaks down because the emissive salt can never be matched by a lossless line to a radiation field at temperature T . In the case of the maser QG is negative and the temperature of the spins at the maser transition can be defined in terms of the populations of upper level nu and lower level n,. In the problem under discussion, as in many other microwave circuit problems, only the relative values of the conductances are of importance. The correct absolute level of impedance is obtained, if a factor QeZo-l is added to the right hand side of cq. (21). Here Z 0 is the characteristic impedance of the matched transmission line which through coupling with the cavity represents an external load describable by Qe. The same factor QeZ,-' has been omitted from the conductances in Fig. 9. A n excellent discussion of thermal noise and eq. (21) may be found e.g. in Lawson and Uhlenbeck, Threshold Signals, MIT Radiation Laboratory Handbook, Vol. 24, p. 64 ff (McGraw-Hill, New York, 1960). References p . 427
CH. IX,
5 51
421
SOLID STATE MASERS
T,
= hvUlk-l In (nl/nu).
(22)
I t is also negative. The temperature concept may be used, because there is no large transverse component of magnetization if condition (12) is satisfied. Since the spontaneously emitted power is proportional to nu, and Q g is proportional to n, - fi.u, the following proportionality may be written __
Ai2
K nu =
n1 - n u
(nl/nu)- 1
Q2 cc ehvIkTH - 1
Comparison with eq. (21) gives for the proportionality factor 4hvAv. Therefore, the Nyquist formula remains valid for negative QMand TBr. Note that these quantities reverse sign simultaneously. The equivalent current noise generators of the input conductance Q,' and the unloaded cavity conductance Qol are also shown in the circuit diagram of Fig. 9. They are respectively at the antenna temperature and the temperature of the helium bath. The ratio of the signal to noise power available at the input to that available at the output is the definition of noise figure. One obtains for the reflection cavity maser of Figs. 8 and 9
F=l+-
4
(Q"'
(eeeM)-l
+Q
i ' Y W
exp (hv/kT,) - 1 [ex, (hv/kT,) - 1
+
QM
~ X P(hvlkTcJ -
Qo
~ X P(hv/kTo)
t--
In the limit of high gain and high temperature, Qo hv k T , this reduces to
<
=
+I
TM
l/Ta*
-
1 1
> Q, w I Q,
I and (23)
Since T , is the input temperature, this relation may be simply expressed by stating that the effective noise temperature of the reflection cavity maser with circulator is I T , I. A similar result obtains in the case of the travelling wave maser. This means that the maser is equivalent to a noiseless amplifier plus an additional input noise power of k [ T , 1 Av. In practice the maser noise temperature is so low that the limit is set by other parts of the circuit, such as antenna spill-over, losses in transmission lines and circulator, and noise generated in stages following the maser. The former can be reduced by special antenna design, Refeyences p . 427
422
N. BLOEMBERGEN
[CH. IX,
95
the latter by cooling the components where losses occur and increasing the maser gain. The best overall system noise temperature of 18" I< has been reported58 for a system of a horn antenna directly coupled into the ruby traveling maser at 5.9 kMHz. A typical breakdown for Harvard L-band radiotelescope system 50 is Antenna spill-over (pointing t o zenith) Input coaxial cable from antenna feed horn Input directional coupler L-band circulator (0.35db insertion loss) Maser input coaxial line Maser spontaneous emission Second stage contribution (1000" K with 20 d b maser gain) Total effective noise temperature
20" K 15" K 5' K
25" K 7' K 2" K 10" K 84' K
There is clearly still room for considerable improvement over the present system. Nevertheless the maser has already improved the signal to noise ratio for a fixed observation time by a factor 1000 : 84. The drift scan signal of the nebula M33 at the hydrogen line using a bandwidth of 200 kHz and an integration time of 10 seconds with and without maser is shown in Fig. 14. It is gratifying that the theoretical
corresponds to a rms deviation of 0.2" K. References p. 427
CH. IX,
9 51
423
SOLID STATE MASERS
predictions have been verified by experimental noise measurements. There is negligible shot noise in the solid state masers. Its origin and effect is similar to shot noise in semiconductors, where the number of charge carriers fluctuates with resultant fluctuations in power gain. Statistical fluctuations occur in principle in the difference in spin populations between two levels which results in power gain fluctuations at the maser frequency. If 3, = PN is the average number in the upper state, fil = qN in the lower state out of a total number of N spins, the relative root mean square fluctuations in the maser conductance are (-1 + (9 q ) / @ - q ) 2 } 1 / 2 N-1/2.Since N M lo1*, this is a very small fraction even at high temperatures when
+
(9 + q ) / ( $ - q)2 M (kT/hv)2M lo4. The corresponding relative gain fluctuations are completely negligible. The spectral density of these fluctuations will fall off beyond frequencies wzI Ti1. In principle the complete power spectral density of the shot noise can be calculated from the linearized rate equations, t o which a fluctuating term has been added according to techniques familar from the theory of Brownian motion. The best maser with the lowest possible noise figure according to eq. ( 2 2 ) is one in which only the upper state is populated, T , -+ - 0 and T o = 0. In this case the equivalent input noise power is hvdv, and the minimum equivalent noise temperature of the maser is hv/k. This minimum noise power corresponds to precisely one quantum in the time for observation (Av)-l. It can be shown that this incertainty is compatible with the fundamental uncertainty relation between number of photons and phase ArtAp, M 1, which a maser as a phase sensitive detector has to satisfy59. A maser could e.g. be used in a balanced bridge circuit to measure both the in-phase and out of phase component of a signal. A device operating at the temperature T , = 0 would, of course, have no spontaneous emission noise. Since only the lower state is populated it could not be a maser either. Quantum counter^^^^^^ which have no output counting rate unless a signal quantum is incident are essentially operating at T , = + O . If a signal is becoming weaker and weaker, e.g. by increasing the distance between source and detector, eventually less than one signal quantum will arrive on the average during the time set for observation. A maser could never detect such a weak signal, but a quantum counter could. It would, of course, leave N
+
References p . 427
424
N. BLOEMBERGEN
[CH. IX,
36
the phase completely undetermined. In principle a microwave quantum counter can be constructed. The question is, however, purely academic, since one will never have a situation with such a small background a t microwave frequencies. The antenna noise temperature will always be so high that the present solid state masers are more than adequate. The question will become of importance, if one moves to higher frequencies into the infrared. Another class of low-noise receivers are the parametric amplifiersel, in which a reactive element is modulated. A very simple form is the modulation of the capacitance of a semiconductor diode. Modulation of ferromagnetic devices has also been achieved, but offers practical disadvantages. The varistor has an enormous advantage in simplicity, cost and bandwidth over the solid state masers. There are, however, also disadvantages. The high frequency limit is determined by the geometry of the diode. The gain stability presents a serious problem because of its sensitivity to power fluctuations at the modulation frequency. Resistive losses cannot be eliminated completely. Elimination of the concomitant noise would necessitate cooling which in turn affects the semiconductor properties. For many applications in which the background noise will not be very far below 300" K, parametric devices will be the practical solution for low-noise amplification. Solid state masers will be restricted to certain special applications, where the ultimate in noise reduction and stability is required. The study of the hydrogen line in radioastronomy offers an example (as shown in Fig. 14). A transoceanic microwave link via reflection from a satellite balloon has been proposed using a maser in the receiving antenna6,. The first radar echo from the planet Venus was received with a maser operating at 380 M H Z ~The ~ . low-frequency limit of masers is set by cross-relaxation between overlapping resonances. Below 500 MHz parametric devices will generally be more useful.
6. Millimeter and Infrared Solid State Masers The extension of solid state masers into the region of millimeter wavelengths offers considerable promise. Foners4 has recently operated a maser between 26 and 39 kMHz utilizing three levels of Fe+++ in TiO,. A pump power of 2-10 milliwatts was applied at 70 kMHz. The knowledge of energy levels of paramagnetic ions in this frequency range is scanty. Millimeter wave spectroscopy should be an active field of investigation for several years to come. This may lead to the disRcfereltces
p . 427
CH. IX, $ 61
SOLID STATE MASERS
425
covery of new suitable high frequency maser materials. It has already been mentioned that harmonic cross-relaxation may lead t o maser action at a higher frequency than the pump power. Availability of pump power at high frequencies still appears to be the limiting factor for continuous millimeter maser operation. Several transient schemes have been proposed to obtain intermittent power at a higher frequency than is fed in. After inversion of the population of a pair of levels at a lower frequency the magnetic field may be varied in strength or the crystal may be rotated in a time short compared to the spin-lattice relaxation time. In the new geometry the resonant frequency of the transition is higher. The population is still inverted. Additional energy has been fed into the spin system during the variation of the magnetic field F ~ n e has r ~recently ~ obtained several milliwatts of peak output power at 70 kMHZin pulsed magnetic fields up to 29000 oersted. In another proposals5 successive adiabatic rapid passages in a multilevel spinsystem would transfer the population of the lowest spin level to the highest spin level. To push masers into the far infrared region pumping at optical frequencies would be required. This would provide a coherent oscillator in the infrareds6. At the same time one would have a maser amplifier to use in conjunction with the coherent infrared oscillator. Spontaneous emission becomes more objectionable at higher frequencies. The proposed infrared quantum counter5g has the same requirements on optical pump power as the maser and might be preferable as a detector. Although theoretical estimates show that optical pumping of certain ions of transition group elements should be feasible, no successful optically pumped maser has yet been reported. The requirements on optical energy density to compete successfully with spontaneous emission and lattice-relaxation mechanisms are severe. The experimental difficulties are enhanced by the requirement that this pump power be concentrated in a solid at low temperature in a resonant structure capable of selecting a definite short wavelength mode. Many investigations are being conducted, however, into the combined microwave, infrared and optical properties of crystals containing paramagnetic ionss7,68. Laxs9 has reviewed the possibilities to utilize the energy levels of electrons and holes in semiconductors for maser operation. Although the orbital levels in a magnetic field, the Landau levels, are not as sharp as spin levels, they are resolved and can be separately excited References p . 427
426
N. BLOEMBERGEN
[CH. IX,
96
at millimeter and far infrared frequencies. They are not exactly equidistant, if the energy surfaces are sufficiently warped. One may by selective optical excitation across the gap from the valence band preferentially populate a particular Landau level in the conduction band. If a resonant structure is tuned to the cyclotron resonance corresponding to the transition to the next lower Landau level, maser action may result. It is estimated that a population of lo8electrons could be established with available light sources. This should be barely sufficient to produce a coherent oscillation in the resonant structure. Many experimental investigations are in progress, but a practical device is still rather remote. Since a relatively small number of carriers is involved, shot effect would become significant in this type of maser. The cyclotron resonance of cariers with negative mass70 gives rise to a negative absorption even for equidistant Landau levels. It is still necessary that an inverted distribution exists in that part of momentum ~ p a c e ~ which l - ~ ~ contains the electron states contributing dominantly to the resonance. The excitation by light of carriers across the gap provides a mechanism, in combination with collision processes, to establish such a distribution. The observed effect is too small for a practical maser application. An early suggestion by Aigrain involved the injection of a large density of carriers through a p-n junction. It was hoped that the resulting recombination radiation would attain such high energy density that stimulated emission would dominate the spontaneous emission and a coherent oscillator would result. The recombination radiation could not ionize new electron-hole pairs, because the simultaneous absorption of a phonon would be required for energy balance. A quantitative calculation shows however that this scheme is not realizable in practice. Bound donor or acceptor states may also be used to achieve inversion of population between a pair of levels. In fact, a successful two level maser used the two spin levels of a P-donor center in [email protected] excited bound states of impurities could possibly be excited preferentially by selective infrared absorption. Such schemes are, of course, very similar to the considered optical excitation of paramagnetic ions67F68. Another type of system of energy levels in solids which might lead to a successful application of optical pumping is provided by the vibrational bands of fluorescent complexes such as uranyl compounds. References p . 427
CH.
1x1
REFERENCES
427
Low temperatures would again be required to avoid excessive broadening of the vibrational levels. In conclusion, the general nature of the arguments leading to a maser, outlined in the introduction, should once more be emphasized. If an inverted distribution between any pair of energy levels can be established -by microwave pumping, by preferential optical excitation, by carrier injection, by optical decay into an excited metastable state or by any other means-a maser may result. It is necessary that stimulated emission is stronger than inevitable losses from other causes. The energy level systems of paramagnetic ions in insulating crystals, of electrons and holes in semiconductors, and of fluorescent complexes at low temperatures appear suitable in this respect.
Note added in $roof: Maiman (Nature 187, 493 (1960)) has recently obtained an optical maser by pumping ruby in the broad green absorption band with a powerful Hg flash discharge. Stimulated emission occurs at the fluorescent R-line near 6800 hgstrom (Compare also Phys. Rev. Lett. 5, 303 (1960)). REFERENCES A. Einstein, Phys. ZS. 18, 121 (1917). W. E. Lamb and R. C. Retherford, Phys. Rev. 79, 570 (1950). a E. M. Purcell, Physica 17, 282 (1951). F. Bloch, Phys. Rev. 70, 460 (1946). N. G. Basov and A. iLI. Prokhorov, J.E.T.P., 27, 431 (1954). 6 J. P. Gordon, H. J . Zeiger and C. H. Townes, Phys. Rev. 99, 1264 (1955). 7 J. Weber, Revs. Mod. Phys. 31, 681 (1959). 8 I. I. Rabi, N. F. Ramsey and J. Schwinger, Phys. Rev. 26, 167 (1954). G. Feher, J. P. Gordon, E. Buekler, E. A. Gere and C. D. Thurmond, Phys. Rev. 109, 221 (1958). 10 P. F. Chester, P. E. Wagner and J. G. Castle, Phys. Rev. 110, 281 (1958). l1 A. Kastler, J. Phys. Rad. 11, 255 (1950). l2 A. W. Overhauser. Phys. Rev. 92, 411 (1953). l 3 N. G. Basov and A. M. Prokhorov, J.E.T.P. 28, 249 (1955). l4 N. Bloembergen, Phys. Rev. 104, 324 (1956). l6 J. R. Singer, Masers (J. Wiley and Sons, New York, 1959). l8 Conference on Quantum Electronics (Columbia University Press, New York, 1960). 17 B. Bleaney and K. H. W. Stevens, Rep. Progr. Phys. 16, 108 (1953); K. D. Bowers and J. Owen, Rep. Progr. Phys. 18, 304 (1955); J. W. Orton, Rep. Progr. Phys. 22, 204 (1959). W. Low, Solid State Physics, Supplement 2 (Academic Press, New York, 1960). E. 0. Schulz-duBois, B.S.T.J. 38, 271 (1959);J. E. Geusic, M. Peterand E. 0. SchulzduBois, B.S.T.J. 38, 291 (1959). L. S. Kornienko and A. M. Prokhorov, J.E.T.P. 33, 805 (1957). go G. S. Bogle and H. P, Symmons, Proc. Phys. SOC.73, 531 (1959). 1
2
428
REFERENCES
[CH.IX
V. M. Vinokurov, Zaripov and Yafaev, J.E.T.P. 37 (10)-220 (1960). G . S. Bogle and H. P. Symmons, Aust. J . of Phys. 12, 1 (1959). H. A. Coles, J. W. Orton and J . Owen, Phys. Rev. Lett. 4, 116 (1960). p4 L. Rimai, H. Statz, hl. J. Weber, G. A. deMars and G. F. Koster, Phys. Rev. Lett. 4, 125 (1960). 25 H. E. D. Scovil, G. Feher and H. Seidel, Phys. Rev. 105, 762 (1957). 26 J. 0. Artman and J. C. Murphy, Bull. Am. Phys. SOC.I1 5, 73 (1960). 27 A. M. Clogston, J. Chem. Phys. Solids 4, 271 (1958). A. G. Redfield, Phys. Rev. 98, 1787 (1955). 2s R. D. Mattuck and M. W. P. Strandberg, Phys. Rev. Lett. 3, 369 and 550 (1959). so E. H. Jacobsen, N. S. Shiren and E. B. Tucker, Phys. Rev. Lett. 3, 81 (1959). J . H. van Vleck, Phys. Rev. 57, 426 (1940). 32 N. Bloembergen, S. Shapiro, P. S. Pershan, and J. 0. Artman, Phys. Rev. 114, 445 (1959). 3s Giordmaine, Alsop, Nash and Townes, Phys. Rev. 109, 302 (1958). 34 Strandberg, Davis, Faughnan, Kyhl and Wolga, Phys. Rev. 109, 1988 (1958). y5 N. Bloembergen, Phys. Rev. 109, 2209 (1958). 36 B. Bolger, Proc. Royal Dutch. Ac. 62 B, 315 ff (1959). See e.g. p. 392 of ref.16. 38 H. B. G. Casimir and F. K. du Pre, Physica 5, 507 (1938). 3 8 S. Shapiro and N. Bloembergen, Phys. Rev. 116, 1453 (1959). 40 T. H. Maiman, J . App. Phys. 31, 222 (1960). 41 B. Bolger, Thesis, Leiden (1959). 4 2 W. Mims and J . D. McGee Proc. I.R.E. 47, 2120 (1959). 43 W. S. C. Chang, J . Cromack and A. E. Siegman, J. Electronics and Control 6, 508 (1959). 44 J . E. Geusic, Phys. Rev. 118, 129 (1960). 45 C. J. Gorter, Progress in Low Temperature Physics, Ed. C. J. Gorter, Vol. 11, 266. (North-Holland Publishing Co., Amsterdam, 1957). 46 F. R. Arams, Proc. I.R.E. 48, 108 (1960). 47 C. Makhov, C. Kikuchi, J . Lambe and R. W. Terhune, Phys. Rev. 109, 1399 (1958). 48 N. Bloembergen and R. V. Pound, Phys. Rev. 95, 8 (1954). 48 R. J . Morris, R. L. Kyhl and M. W. P. Strandberg, Proc. I.R.E. 47, 80 (1959). 50 J. V. Jelley and B. F. C. Cooper, Rev. Sc. Inst. to be published (1960). 51 L. Davis, U. Milano and J . Saunders, Proc. I.R.E. 48, 115 (1960). 52 S. Autler, Rev. Sc. Inst. 31, 369 (1960). 53 J . A . Giordmaine, L. E. Alsop, C. H. Mager and C. H. Townes, Proc. I.R.E. 47, 1062 (1959). 54 R. W. de Grasse, E. 0. Schulz-duBois and H. E. D. Scovil, B.S.T. J. 38, 305 (1959). 5 5 F. E. Goodwin, Proc. I.R.E. 48, 113 (1960). 56 R. V. Pound, Annals of Physics 1, 24 (1957) ; M. W. Muller, Phys. Rev. 106, 8 (1957) ; M. W. P. Strandberg, Phys. Rev. 107, 1483 (1957); J. Weber, Phys. Rev. 108, 537 (1957). 57 H . B. Callen and T. A . Welton Phys. Rev. 83, 34 (1951); H. Nyquist, Phys. Rev. 32, 110 (1928); I. R. Senitzky, Phys. Rev. 111, 3 (1958). 58 R. W. deGrasse, D. C. Hogg, E. A. Ohm and H. E. D. Scovil, J. App. Phys. 30, 2013 (1959); ibid 31, 443 (1960). 5s H. Friedburg, ref.16. 60 N. Bloembergen, Phys. Rev. Lett. 2, 84 (1959). 6 1 See e.g. H. Heffner, ref.16. 62 J. R. Pierce and R. Kornpfer, Proc. I.R.E. 47, 372 (1959). 6 3 R. H. Kingston, Proc. I.R.E. 46, 916 (1958). 21
22
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1x1
REFERENCES
429
S. Foner and L. R. Momo, J. App. Phys. 31, 443 (1960); S. Foner, L. R. Mom0 and A. Meyer, Phys. Rev. Lett. 3, 36 (1959). 85 A. E. Siegman and R. J . Morris, Phys. Rev. Lett. 2, 302 (1959). 66 A . L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958). I. Wieder, Phys. Rev. Lett. 3, 468 (1959). 6* F. Varsanyi, D. L. Wood and A. L. Schawlow, Phys. Rev. Lett. 3, 544 (1959); S. Geschwind, R. J. Collins and A. L. Schawlow, Phys. Rev. Lett. 3, 545 (1959); J . Brossel, S. Geschwind and A. L. Schawlow, Phys. Rev. Lett. 3, 547 (1959). eg B. Lax, ref.16. io G. C. Dousmanis, R. C. Duncan, J. J. Thomas and R. C. Williams, Phys. Rev. Lett. 1, 404 (1958). :1 D. C. Mattis and M. J . Stevenson, Phys. Rev. Lett. 3, 18 (1959). 72 P. Kaus, Phys. Rev. Lett. 3, 18 (1959). 73 C. Kittel, Proc. Nat. Ac. Science, Washington, D.C. 45, 744 (1959). e4
CHAPTER X
THE EQUATION OF STATE AND THE TRANSPORT PROPERTIES OF THE HYDROGENIC MOLECULES BY
J. J. M. BEENAKKER ONNESLABORATORY, LEIDEN KAMERLINGH C O N T E N T S1. Introduction, 430. - 2. The second virial coefficient, 431. - 3. Thermal conductivity, 436.- 4.Viscosity, 437. - 5. The diffusion coefficient, 439.- 6.Thermal diffusion, 440. - 7. The influence of the total nuclear spin on the properties of H, and D,,442. - 8. Theoretical calculations, 445.
1. Introduction In the first volume of this series de Boer1 gave a survey of the transport-properties of gaseous helium at low temperatures. In his introduction he pointed out that in gases a t low temperatures interesting quantum effects can be observed. He limited his discussion to the helium isotopes, since for the hydrogenic molecules H,, HD, D, etc., too little data were available. I n recent years this gap has been partially filled by new experiments on the viscosity, the diffusion and the equation of state. Furthermore, successful attempts have been made to measure the influence of the ortho- and para-modifications on some of these properties. Hence, it now seems justified to give a comprehensive review of the data regarding the properties of the gaseous hydrogenic molecuIes at low temperatures. As the older data have been given in a very systematic way in a study by Woolley, Scott and Brickwedde, we will limit ourselves to the data which became available after the completion of their paper. As the data at higher densities remain very scarce, we will furthermore restrict ourselves to low densities, i.e. to the density-independent part of the transport-properties, and for the equation of state to linear deviations from the ideal gas law, the socalled second virial coefficient. First of all we will treat briefly the different experiments and then we will compare these data with results of theoretical calculations. For References
p . 453
CH.
x, $ 21
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
431
convenience we will discuss the influence of the ortho- and paramodifications in a separate section.
2. The Second Virial Coefficient 2 . 1 . INTRODUCTION
As is well known the equation of state of a gas a t moderately low densities may be written in terms of the pressure, 9, the density, d , and the temperature, T , as
p
= ATd(1
+ Bd + C d 2 ) .
(1)
Here A is a constant that depends on the units used for p and d, on the value of the ice-point on the absolute temperature scale and also slightly on the gas. B and C are the second and third virial coefficients. If one writes the equation of state in this way the virial coefficients are directly related to the contributions of binary and ternary collisions respectively. It is also possible to write an expression similar to (I) in terms of the pressure. In this case, however, the coefficient of the term in fl2 also contains a contribution from binary collisions. Hence, a density development is to be preferred if one likes to compare experimental data with theoretical predictions. Furthermore, the series expansion in d can, in general, be broken off at lower powers than would be the case for a development in p . For experimental convenience one often uses amagat units, i.e. the pressure is expressed in standard atmospheres and the density is taken relative to the density of the gas at the ice point under a pressure of one atmosphere. I n these units A from eq. (1) is given by A = [(1+B0+C0)273.15]-1. HereB, and C, are respectively the values of the second and third virial coefficients at the ice-point, and 273.15 is the absolute temperature of this point. In comparing data with theory, and in many thermodynamic calculations, it is more practical to express the density in mole/cm3, the second virial coefficient then being given in cm3/mole. The conversion factor from amagat units to cm3/mole is slightly dependent on the gas-because of the values of B , and C,-but it is for all practical purposes equal to 22.43 x lo3, this being the volume of one mole of gas under standard conditions. Since in a series development in the density the second virial coefficient arises from binary collisions, it is clear that in the case of a mixture, the second virial coefficient B, is given by References
p . 153
43.2
J. J. M. BEENAKKER
B,
= X'B,,
+ 2X(1
-
X)B,,
[CH. X,
32
+ (1 - X)'B,,.
In this expression X is the mole fraction of component 1,B,, and B,, are the virial coefficients of the pure constituents and B,, is the contribution arising from the interaction between molecules of species 1 and 2 . In some cases one assumes: B,, = fr(B,, B,,). This assumption has, however, little a priori foundation t. The advantage of this assumption is that B, becomes linear in the concentration, its physical significance being that the mixing as far as B is concerned becomes ideal i.e. that there is no pressure change on mixing at constant volume. In the case of non-ideal mixing one can write B,, = &(Bll B,,) E. This expression serves then as a definition for the excess quantity E . We will see later how it is possible to measure E directly. From the experimental point of view the main difficulty in determining the value of B is that one has to perform the measurements a t densities low enough to avoid an important influence of the higher virial coefficients. The accuracy, however, is proportional to the square of the density, Therefore one has to look for an optimum density range. Furthermore, in the range below the critical temperature, the maximum pressure is limited by the saturation pressure, and this makes the measurements rather difficult, especially well below the boiling point.
+
+
+
2 . 2 . ABSOLUTEDETERMINATIONS OF pV ISOTHERMS
The standard technique in measuring the second virial coefficient is to determine a pV isotherm i.e. to determine the pressure as a function of the amount oi gas in a reservoir at a fixed temperature. In the low pressure range one generally uses a mercury manometer, while at higher pressures (above 2 atmospheres) dead-weight gauges are used. In recent years pV isotherms have been determined by Johnston and White3 from 20.5" K upwards for normal hydrogen, and by Beenakker, Varekamp and Van Itterbeek4 for H,, HD, D, and mixtures of these gases with helium at 20.4' K. From the measurements of Johnston and White only a preliminary publication of smoothed pV f For a Lennard- Jones potential, neglecting quantum-effects, one can prove that below the Boyle temperature, whereB cc T K , the assumption for the interaction between
the different molecules (ell x This expression reduces to R,, very much alike. 2
2
Xefcvences p. 453
t o B,, = (B,,B,,)* if ul, = ( U l l U z z ) ~ . only in the case that the molecules are
E , ~ ) & leads
@,, + B,,)
CH. X,
3 21
EQUATION OF STATE AND T H E TRANSPORT PROPERTIES
433
data is at present available. From these data we derived values for the second virial coefficient, neglecting the small influence of the C coefficient (cf. Fig. 2). 2.3. RELATIVE DETERMINATIONS
As stated earlier, the accuracy of the second virial coefficient determinations in the saturation region is seriously limited by the low
Fig. 1. The apparatus for relative determination of the second virial coefficient.
pressures one can use. For the hydrogen isotopes the accuracy in the isotherm determinations, which is of the order of lo-* amagat at the hydrogen boiling point, is only between and amagat around 15" K. This problem can be overcome by taking advantage of the fact that the temperature dependence of the second virial coefficient of helium is rather well known at these temperatures. Thus it is possible to measure the change in the non-ideality of the hydrogen isotopes by Referdnces p . 453
434 J. J. M. BEENAKKER [CH. X, 5 2 comparing their behaviour with helium using a differential method. In this way it is possible to obtain data with a reasonable accuracy even at pressures of about 10 cm Hg. Fig. 1 gives a schematic diagram of the apparatus developed by the
-25
cm3 mole
.50
.t 00
T. -200
125
T
k
Fig. 2. The second virial coefficient of H, as a function of temperature. 0 Varekamp et al. Curve I - W. S. Brickwedde Curve I1 - calc. of De Boer et al. 0 Knaap et al. Johnston and White
a
Leiden group5. It consists of two reservoirs, R, and R,, of nearly equal volume. These reservoirs are connected to a differential oil manometer D by narrow capillaries. The apparatus is made symmetric. At the start of an experiment one of the reservoirs is filled with the hydrogen References p. 453
CH.
x, 9 21
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
436
isotope and the other with helium at the same pressure. During this procedure the temperature of the surrounding bath is kept constant. Now the temperature is raised by about one degree. As the hydrogen isotope behaves more like an ideal gas at the higher temperature, a pressure difference between the reservoirs will result. This difference is directly related to the change in the second virial coefficient as a function of temperature. The symmetry of the apparatus cancels out all small temperature changes. With this type of set-up measurements were performed on H,, HD, D, and on mixtures of these gases with helium. The same experiments were also performed on isotopic mixtures. In this case it was found that the change in the earlier mentioned excess E , from the hydrogen boiling point to lower temperatures] is zero. This suggests a zero excess as was substantiated later on by direct measurements, of which we shall speak in a subsequent section. The absolute value of the second virial coefficient a t the hydrogen boiling point is taken from the isotherm determinations. Recently, Knaap 13 e.a. extended these measurements to the temperature range between 20 and 80" K. In Fig. 2a comparison is given for H, with the data of Johnston and White3 and with the smoothed values given by Wooley, Scott and Brickweddes. In comparing the different results one has to bear in mind that those of W.S.B., with the exception of the values at the hydrogen boiling point, are based on rather high density measurements. Hence the influence of the higher virial coefficients is rather large. In this way the value of B becomes more or less an extrapolated one. From Fig. 2 we see that there remains a rather large uncertainty in the absolute value of the second virial coefficient. Taking a third virial coefficient into account in the data of Johnston and White and in the data of the Leiden group would make the difference between the several authors even somewhat larger, of the order of 1% in the low temperature range. 2.4. DIRECTEXCESS DETERMINATIONS
The rather large discrepancies between the data for the second virial coefficient at the hydrogen boiling point stimulated the search for a possibility of checking the recent Leiden isotherm determinations. This is possible by measuring the excess' directly. In the introduction we pointed out that for a zero excess the pressure change on mixing at constant volume is zero. By measuring this change, direct informaReferences
p.
453
436
J. J. M. BEENAKKER
[CH. X,
93
tion is obtained on the value of the excess. For this purpose the apparatus described in Fig. 1 was modified by introducing a third reservoir at low temperatures and a large Topler pump at room temperature. Two of the reservoirs are filled to the same pressure with the pure gases one desires t o mix. The third reservoir is also filled to the same pressure with one of these gases. This reservoir serves as a pressure reference. Now the two gases are mixed by means of the Topler pump and the pressure change with respect to the initial situation is measured on an oil manometer, as the pressure difference between the now connected reservoirs and the reference volume. Any small temperature fluctuations are balanced out in this way, as the pressure reference reservoir is maintained under the same conditions as the others. I n this way rapid and accurate determinations of the excess can be made. Knaap et aLs measured the excess for the stable hydrogen isotopes and for the mixtures with helium at the hydrogen boiling point. The agreement with the isotherm determinations is reasonable. Hence, one may conclude that large systematic errors in the Leiden isotherm determinations seem improbable, although it remains difficult to give a quantitative estimate of the largest possible error that could still give agreement with the excess determinations.
3. Thermal Conductivity 3.1. EXPERIMENTAL DATA
While relative thermal conductivity measurements are easily performed it is rather difficult to obtain accurate absolute values. The main difficulty arises from heat losses at the edge of the heater and from a temperature jump between the gas and the wall. Ubbinkg and de Haas solved this problem by constructing the apparatus pictured in Fig. 3. It consists of two plates A and B that are parallel to each other inside a vacuum chamber, W. A heater H, and a resistance thermometer Th, are placed in the upper plate. In the lower plate there is only a thermometer. A shield, C, which is kept at the same temperature as B to avoid heat losses in other directions, is mounted above the upper plate. By heating the upper plate the heat current through the gas layer between A and B can be measured as a function of the temperature difference. To correct for temperature jumps between the surface and the gas, plate B can be moved in a direction perReferences p . 453
CH.
x,tj 41
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
437
pendicular to A. In this way the temperature jump can be eliminated by measuring at different depths of the gas layer. Ubbink performed measurements on H, and D,, both at liquid hydrogen and at higher temperatures.
n Th
C
I4
B Th
A
Th
W
Fig. 3. The apparatus of Ubbink and De Haas for measuring thermal conductivity a t low temperatures.
4. Viscosity
4.1. EXPERIMENTS IN THE LIQUIDHYDROGEN TEMPERATURE RANGE
Although there are many older data available on H, and D, at liquid hydrogen temperatures, we will only include in our discussion the recent work of Becker and MisentalO and of Rietveld et aZ.ll, who performed measurements on H,, HD and D, under exactly the same conditions. The first group of authors used an oscillating cylinder as a viscometer. The amplitude of the oscillation as a function of time was measured by means of a variable condensor system attached to the cylinder. From the logarithmic decrement of the amplitude of oscillation measured in this way, the viscosity of the gas can be calculated. The apparatus is calibrated with a gas of known viscosity. For this purpose Becker and Misenta used helium at 77 2" K. For the viscosity they took the value calculated with Keesom's formula12. Rietveld et al. used an oscillating disk viscometer and calibrated their apparatus at different temperatures with helium gas. For the viscosity of helium they took 35.0 and 28.5 ,UPat 20.4 and 14.4" K respectively. Rejeyences
p . 453
438 J. J. M. BEENAKKER [CH. X, 5 4 4.2. EXPERIMENTS BETWEEN 20 AND 80" K I n order to compare the viscosity data with different models for the molecular interaction it is necessary to have data available over an extended range of temperatures. Until recently there was a rather large gap between liquid hydrogen and liquid oxygen temperatures. To fill this gap Coremans13et al. constructed a thermostat for temperatures
Fig. 4. The viscosity of H,, HD and D, as a function of temperature. 0 Rietveld et al. 0 Becker el al., Coremans et al.,
a
between 20 and 80" K and measured the viscosity of several gases, among which are the stable hydrogen isotopes. The measurements were performed with an oscillating disk viscometer. The apparatus was calibrated with helium gas. Between 20 and 80" K the viscosity values as derived from the Keesom formula were confirmed. Fig. 4 gives a survey of their data together with those of the authors described in the preceding section. In general the agreement between the different results is rather good if one takes into account the experimental uncertainties of 2 to 3%. References p. 453
CH. X,
9 51
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
439
4 . 3 . EXPERIMENTS ON BINARY MIXTURES
Rietveld, van Itterbeek and Veldsll performed an elaborate investigation on the viscosity of gas mixtures at low temperatures. Among other combinations they measured mixtures of the hydrogenic
Fig. 5. The viscosity of mixtures of H, and D, as a function of concentration at several temperatures.
molecules using an oscillating disk viscometer. Fig. 5 shows their results for H, - D, mixtures as function of concentration a t different temperatures. As can be seen from this plot the curvature of the viscosity versus concentration curve is always very small.
5. The Diffusion Coefficient The data on diffusion coefficients in gases a t low temperatures are very scarce. Recently some measurements have been made by Bendt l4, Refsrences
p. 453
440 J. J. M. BEENAKKER [CH. X, 5 6 who constructed a diffusion bridge, a schematic diagram of which is given in Fig. 6. The two bulbs 1 and 2 are filled with H, and D, respectively. The gas from these bulbs flows to a vacuum system through capillaries A and B. The pressures in 1 and 2 are adjusted in such a way that the pressure
p'
Fig. 6. Schematic diagram of Bendt's apparatus for the determination of the diffusion coefficient.
drop over the diffusion capillary C is as small as possible. The concentrations of the gas arriving in the vacuum system can be measured with a mass-spectrometer. In this way Bendt performed measurements at liquid hydrogen, liquid oxygen and room temperature.
6. Thermal Diffusion 6 . 1 . INTRODUCTION
If a temperature gradient is applied to a gas mixture the average velocities of the molecules of the two components will be different. This will give rise to a separation process that will be counterbalanced after some time by normal diffusion. In the stationary state one has
D,,grad X
1 + D, grad T = 0. T
(2)
Here D,, and D, are the normal and the thermal diffusion coefficients respectively, X is the mole fraction of component one and T is the temperature. An important quantity is k,, defined as k , = D,/Dlz. In terms of k , expression (2) becomes grad
X
=
-
1 T
k , - grad T.
The advantage of introducing the thermal diffusion ratio k,, is that Re/erences
p . 453
CH.
x,$
61
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
441
this quantity is fairly constant over a rather large temperature range. Hence, to a first approximation one obtains by integrating for the separation, AX
AX
=K,
In -.TI
T2
One sometimes uses cc = &/(l- X)X,a quantity that is also practically independent of concentration. In general, the separations remain rather small; of the order of a few percent even for large ratios of T . Hence, the main experimental difficulty is in analysing the concentration change with a sufficiently high sensitivity. 6 . 2 . EXPERIMENTAL RESULTS
A survey of many of the existing thermal diffusion data in gases is given in a book on this subject by Grew and Ibbs15, who performed many measurements at higher temperatures. Data below 90" K are still rather scarce. Van Itterbeek and collaborators l6 extended some of their measurements to liquid hydrogen temperatures. They used the two bulb method. An upper and a lower reservoir, connected by not too wide a tube, are kept at room temperature and a lower temperature respectively. The change in the concentration in the upper bulb is measured. As most of the gas is in the low temperature reservoir, this change in concentration is nearly equal to the total separation. A correction for this difference can easily be applied. For the analysis they used the change in the viscosity as a function of concentration. The viscosity was measured by an oscillating disk viscometer placed in the high temperature reservoir. The sensitivity of this method is, a t the best, of the order of 0.1 yo.In this way they measured the thermal diffusion in H, - D,, H, - Ne, D, - Ne and H, - He mixtures, a t several concentrations. From their results for the hydrogen isotopes they observed that at lower temperatures the thermal diffusion ratio is no longer constant. These measurements were later extended by Grew el a1.l'. They too used the two bulb methcd, but performed the analysis with a heat conductivity cell. The sensitivity of this method is of the order of 0.05 yofor the hydrogen isotopes. They also performed measurements at temperatures between 20 and 60" K. In agreement with the results of References p . 453
442
J. J . M. BEENAKKER
[CH. X,
57
Van Itterbeek el al. they found that the thermal diffusion ratio is no longer constant. For the absolute value of the separation, however, their results differ widely from those of the other authors. They found a much smaller separation.
7. The influence of the Total Nuclear Spin on the Properties of H, and D, 7 . 1 . INTRODUCTION
It is well known that hydrogen and deuterium occur in two modifications depending on the total nuclear spin. The symmetry conditions imposed on the total wave function of the molecule give rise to a different set of eigenvalues for the rotational states of the molecule in accordance with the symmetry properties of the total nuclear spin. Therefore, by combining two spins of one half, we get for H, a total spin zero with even rotational quantum numbers, and a total spin one with the odd values. At the high temperature limit the even and the odd rotational states occur in the ratio 1 : 3, according to their statistical weights, the so-called normal hydrogen. The most abundant modification at high temperatures is called ortho the other parahydrogen. Because of the even rotational quantum numbers which allow zero rotational energy, the para form is the stable one at low temperatures. For D, one gets a similar situation by combining two spins of one; total spin zero and two correspond to even rotational quantum numbers, total spin one to odd values. At high temperatures the ratio of abundance of even to odd is 2 : 1. The most abundant form at high temperatures is again called the ortho modification. In contrast to the case of hydrogen the ortho form is the stable one at low temperatures, because of its possible zero rotational energy. As the transition from ortho to para is forbidden, the equilibrium between the modifications is reached only very slowly. For many purposes one can treat the two modifications as different gases. It is clear that the differencein rotational states gives rise to a rather large difference in the caloric properties in the temperature range where the rotational levels become excited. This will also show up in the thermal conductivity in this temperature range, giving a method of determining the composition in a gaseous mixture. At the lowest temperatures, where all the molecules are in the lowest rotational state, References p . 453
CH. X,
5 71
EQUATION OF STATE AND THE TRANSPORT PROPERTIES
443
this effect disappears. Only rather recently it was shown that both in the transport properties and in the equation of state, small differences between the two modifications exist at lower temperatures. Older investigations by SchHferls and Long and Brown19 on the equation of state and by Waldmann and Becker20 on the thermal conductivity had already shown that these differences were certainly smaller than 2%, this being the order of magnitude of accuracy of their measurements. From these investigations it was clear that much more sensitive methods had t o be applied. Such work was done by Becker and Steh121 for the viscosity and rather recently for the equation of state by the Leiden groupZ2. 7 . 2 . THE DIFFERENCE IN THE SECOND VIRIAL COEFFICIENT To measure the difference in the second virial coefficient between para- and normal-hydrogen the Leiden group developed a differential method. For this purpose the apparatus described in section 1 . 3 was slightly modified by adding a Topler pump in which a platinum wire was mounted. The gas in the pump reservoir could be converted to the high temperature equilibrium composition by heating the platinum Cm)
0.1 0
lOlL
/
amagat
0.05
A T E
m 1-
A
2 0
/
100
75
50
~
2 5 & pGa H, 0
Fig. 7. The difference in the sccond virial coefficient between a mixture of ortho- and para-hydrogen and para-hydrogen as a function of the para concentration at the hydrogen boiling point. Rcferesces p. 453
444
J . J. M. BEENAKKER
[CH. X,
7
wire. The experimental procedure is as follows. At the bczinning of an experiment both vessels R, and R, are filled to the same pressure with para-hydrogen. Then a part of the gas in R, was pumped to the conversion apparatus and converted to normal hydrogen. Subsequently it was pumped back into R, and the change in the second virial coefficient was detected by means of the resulting pressure difference with respect to the initial situation as given by the pressure in the other reservoir. The second virial coefficient of para-hydrogen appeared to be about 1% less negative than that of normal hydrogen. The change is linear in the para concentration, as can be seen in Fig. 7. Furthermore, the temperature dependence of the effect using the relative method described earlier was measured. The difference in B increased slightly at lower temperatures. The linear concentration dependence was confirmed later by excess measurements8. For D, no effect was found. But in this case the method is about 20 times less sensitive because of the lower pressures that can be used and because the possible concentration change is smaller than in the case of H,. The largest possible difference in the second virial coefficient between normal and ortho-deuterium was given as 5 x l O W amagat.
VISCOSITY To measure the difference in the viscosity between the two modifications Becker et aL21 made use of a capillary Wheatstone bridge as shown in Fig. 8. The incoming gas stream is divided into two branches passing through the capillaries C, and C, which are kept at low temperatures and then passes through capillaries C, and C, which are a t 7 . 3 . THE DIFFERENCE IN
THE
Fig. 8. Schematictliagram of the apparatus of Becker et al. for measuring the difference in viscosity between ortho- and para-hydrogen. References
p . 453
CH. X,
3 81
EQUATION O F STATE AND THE TRANSPORT PROPERTIES
445
room temperature. By varying the resistance of C , the pressure difference over the bridge center can be made zero. This pressure difference is measured with a differential capacitance manometer, M. The bridge is balanced with para-hydrogen as the streaming gas, then one of the platinum wire converters, U, or U, is switched on, converting parahydrogen to normal hydrogen. As the viscosity differencebetween the two modifications is practically zero at room temperature, the resulting
25
PH,
50
75
Ol0
100
Fig. 9. The difference in the viscosity between a mixture of ortho- and para-hydrogen, and para-hydrogen as a function of the para-hydrogen concentration at different temperatures.
unbalance of the bridge gives the change of the viscosity at low temperatures. In this way measurements were performed on H, and D,. In both cases the non-rotating modification, i.e. para-H, and ortho-D,, has the largest viscosity. The effect was larger for H, than for D,. As can be seen irom Fig. 9, where some of the results are given, it is clear that the concentration dependence is not exactly linear.
8. Theoretical Calculations 8.1. INTRODUCTION A survey of the problems relative to the calculation of the transport properties at low temperatures has been given by de Boer1 in the first volume of this series. For the case of the second virial coefficient the reader is referred to a chapter by de Boer and Bird in the book by References p . 453
446
J . J. M. BEENAKKER
[CH.
x, 5 8
Hirschfelder Curtiss and Bird23. Here we will only give a brief outline of the related problems. At low temperatures there are two deviations from classical theory. The first type, the diffraction effects, becomes of importance if the De Broglie wavelength, 3, = h ( d 4 - l is of the order of magnitude of the molecular dimensions. A second type of deviations arises from the fact that for a system of identical particles the wave function has to be symmetrized. As long as only the diffraction effects are of importance and remain small, it is possible to describe the deviations from the classical behaviour with a series expansion in A*, = h2(( ~ ~ r n e as ) - was ~, proposed by de Boer and Birdz4,in good agreement with experimental results13.At lower temperatures, however, one has to treat this problem completely quantum mechanically. Until now the theory has been limited to spherical intermolecular potentials. This, of course, restricts the applicability in the case of the hydrogenic molecules, as in a spherical potential transitions between the various rotational states do not occur. There are reasons to believe that these limitations are not too serious. First of all, it is well known that several high temperature properties of hydrogen can be described rather well by a spherical model. On the other hand the occupation of the higher rotational levels is negligible at low temperatures, the excitation energy for para H, being of the order of 500" K. Calculations for the hydrogenic molecules were performed by de Boer et aLZ5and by Buckingham et ~ 1 . The ~ ~ first . group of authors performed the most extensive program covering the second virial coefficient and the transport properties for H,, D, and mixtures of these gases f . Furthermore, they treated the ortho-para effect for both gases. They used a Lennard-Jones potential of the form
where elk, the depth of the potential well, is 37.00' K and (T, the "diameter" of the molecule, is 2.928 A for both H, and D,. These values were obtained from high temperature second virial coefficient data. Buckingham et al. performed some calculations on the viscosity and the influence of the ortho-para modifications on this property for H,. They used a potential of the following type t We like to express our sincere thanks to Prof. de Boer and collaborators for putting their unpublished data on Dz and Hs-Dz Rejerences p . 453
mixtures at our disposal.
CH. X,
3 81
EQUATION OF STATE AND T H E TRANSPORT PROPERTIES
where fi =
a[a(l
447
+ b) + 6 - 8b]-1 and f, * - 1 + (1 + b)f,.
The values of u and b are 13.5 and 0.2 respectively. For values of < I , the attractive part of the potential is modified by a factor exp - ~ [ ( I J I )- l I 3 to avoid complications arising from the decrease in ? ( I ) that would otherwise show up. In this type of potential one uses, in general, rm as the characteristic length i.e. the value of I at the minimum of the potential. For comparison with the Lennard-Jones potential one has to bear in mind that r, = 21/60. The general procedure is to calculate the phase shifts ql (K*) for wave number K* and the orbital angular several values of the reduced -__ momentum (h/2n)dZ(Z 1) from the Schrodinger equation. The calculation must be extended to very large values of 1. For H, de Boer et al. used values up to 23. To be able to extend the calculations to higher temperatures, Buckingham el al. made use of the theory of de Boer and Bird and developed the cross-sections for higher values of K* in a series expansion in terms of ,I*,.They limited their calculations to the case of hydrogen using two values of A* very near to each other and performing their calculations for several values of c = 1.570 I& at a constant value of ,I*.This was done so as to be able to decide from experiment the best value for c, which is directly related to the van der Waals dispersion forces. If the phase shifts are calculated the second virial coefficient and the transport properties can be directly expressed in these quantities. One has to be careful, however, to symmetrize the wave functions in the proper way, for depending on the symmetrization, different sums over the phase shifts must be made. This problem was treated in extenso in a paper by Cohen and de Boer et ul. We will outline here briefly the situation for H,. For para-hydrogen, where the total nuclear spin is zero and the molecules are in the zero rotational state, the spatial wave function for two colliding molecules has to be symmetric. Hence, only the phase shifts for even values of 1 have to be taken into account. For a collision between an ortho- and a para-hydrogen molecule no symmetrization is necessary as these are distinguishable particles. For orthohydrogen the situation is more complicated as these molecules occur in nine almost degenerate states, corresponding to the three values of I
+
References p. 153
448
J . J. M. BEENAKKER
[CH.
x, 3 8
the z component of the total nuclear spin and the three values of the z component of the rotational moment. Hence, in the system of two colliding molecules there are 81 possible internal states. Of these states 5/9 have symmetric and 4/9 antisymmetric internal wave functions. As the total Gave function has to be symmetric (Bose statistics) the wave function for the relative motion is also distributed over the symmetric and the antisymmetric states in the ratio 5 : 4. Hence, the cross-section for the ortho-ortho collision and also the second virial coefficient are made up of 5/9 of the symmetric and 4/9 of the antisymmetric contribution. A similar situation occurs for D,. It is this symmetrization that gives rise to a difference between the ortho- and para-modifications in the case of a spherical potential. In the following section we will compare the experimental data with these calculations. 8.2.
COMPARISON WITH
EXPERIMENTAL DATA
8 . 2 . 1 . T h e Second Virial Coeficient The calculations are limited to H,, D, and H,-D, mixtures up to 37" K. Data are available only for the Lennard-Jones potential. Fig. 2 gives a comparison with the experimental data in the case of H,. The general temperature dependence is given rather well, but there remains a rather large discrepancy in the absolute values. The same is true for the case of D,. For the mixtures the excess is smaller than 0.06 X amagat, in agreement with experiment. 8 . 2 . 2 . T h e Viscosity and the Thermal Conductivity
To a first approximation the thermal conductivity is related to the viscosity by the relation [A], = 5/2C,[q], where C, is the specific heat. This relation appears to hold fairly well. Thus only the viscosity data will be compared directly with theory. Calculations are available for H,, D, and mixtures of H,-D, for a Lennard-Jones potential up to 22" K. Fig. 10 gives a comparison of {he experimental viscosity data with these calculations. The agreement is very good. The curvature of the viscosity versus concentration curves for the mixtures as predicted by the Lennard-Jones potential is in qualitative agreement with experiment, the experimental curvature being somewhat smaller. The accuracy of the experiments is such that the exact form of the curvature can not be determined. For the case References p. 453
1-
I
12
T
18
2 'K
Fig. 10. The viscosity of H, and D, as a function of temperature. 0 Rietveld et al. - calc. of De Boer et al.
0 Becker et al.
Fig. 11. The viscosity of H, as a function of temperature. 0 Coremans et al. - calc. of Buckingham ef al. References p . 453
450
J. J. M. BEENAKKER
[CH.
x, 9 8
of hydrogen there are also calculations available for the Buckingham potential up to very high temperatures. Fig. 11 gives the results for one value of the constant, c, as discussed earlier. The agreement with experiment is very good. 8 . 2 . 3 . The Ditfusion
Fig. 12 gives a comparison of the existing diffusion data with the calculations. The agreement is reasonable if one takes into account the experimental difficulties. 10
9 -
ern..
5
/
A
O '
T
15
20
OK
25
Fig. 12. The diffusion constant for H,-D, as a function of temperature. - calc. of De Boer et al. Bendt
a
8 . 2 . 4 . The Thermal Diffusion
As was pointed out earlier there is a rather large discrepancy between the various experimental data. In Fig. 13 we give the values of M as derived from the measurements by van Itterbeek et al. and by Grew et al. together with the theoretical predictions for a Lennard-Jones potential. There is qualitative agreement. More accurate measurements are necessary, however, before further comparison with theory can be made. 8 . 2 . 5 . The Influence of the Total Nuclear Spin
For the second virial coefficient data are available only for the Lennard-Jones potential. At liquid hydrogen temperatures the relative difference between the ortho- and para-modifications should be smaller than 10-3, while the experimental data give a difference of the order of 1% in B. For the viscosity there are calculations for the Lennard-Jones and the Buckingham potential for H, and for the Lennard-Jones potential only in the case of D,. The experimental differences in the viscosity References p . 453
CH. X,
9 81
EQUATION O F STATE AND THE TRANSPORT PROPERTIES
0
T
*
50
1 0
451
14 ,OK
Fig. 13. The thermal diffusion constant a as a function of temperature for a H,-D, mixture. 3 Lennard- Jones Classical calc. 1 Van Itterbeek et al. 2 Grew et al. 4 calc. of De Boer et al.
Fig. 14. The difference in the viscosity between normal and para-hydrogen as a function of temperature. 1 calc. of De Boer et al. Becker et al. 2 calc. of Buckingham el al. References
p. 453
452
J. J. M. B E E N A K K E R
[CH.X,
38
are appreciably larger than would be expected from theory (cf. Fig. 14). Furthermore, the curvature of the viscosity versus concentration curve as found experimentally is negative, in opposition to theoretical predictions (cf. Fig. 9). 8 . 3 . GENERALCONCLUSIONS
From the available data it is clear that the general behavior of the transport properties can be described rather well with the LennardJones potential. The situation for the second virial coefficient is less satisfactory. It does not seem impossible to obtain a better fit by adapting the potential parameters to the low temperature results. This approach seems reasonable if one bears in mind that the population of the rotational states of hydrogen at high temperatures is completely different from the low temperature situation. The discrepancies in the case of the ortho- and para-modifications, however, cannot be clarified in the same way, as these effects are only slightly dependent on the potential parameters. Furthermore, the curvature found for the case of the concentration dependence of the difference in the viscosity is also in qualitative disagreement with the results from the statistical effects for a spherical potential. Disagreement with theory is also indicated by the linear dependence on concentration for the second virial coefficient, although this effect is at the limits of the experimental accuracy. Hence, one is inclined to conclude that a spherical potential is too rough an approximation to describe the intermolecular potential at these temperatures. A reliable treatment of a non-spherical potential is not yet available for these low temperatures. Note added i n proof Rccently Michels et al. 27 showed that the data on the second virial coefficient for H, and D, above 90"K pointed to a difference in the potential parameters; E/k for D, being about 1.6"K smaller then for H,. We have succeeded in explaining this difference2s as arising from the difference in the Van der Waals-London dispersion energy, because of the difference in the polarisability between the two isotopes 29, 30, As there is also a difference in polarisability between the ortho and para modifications31 we could in the same way explain the experimental data for the second virial coefficient and the viscosity of these modifications. References p . 453
C H . X]
REFERENCES
453
REFERENCES J. de Boer, Progress in Low Temperature Physics, Ed. C. J. Gorter, 1, ch. 18, p. 381 (North-Holland Publishing Co., Amsterdam, 1965). 2 H. W. Woolley, R. B. Scott and C. G. Brickwedde, J. Res. Nat. Bur. Stand 41, 379 (1948). 8 H. L . Johnston and D. White, Trans. Am. SOC.Mec. Eng. 72, 785 (1950). 4 J. J. M. Beenakker, F. H. Varekamp and A. van Itterbeek, Communications Kamerlingh Onnes Lab., Leiden, 313a; Physica 25, 9 (1959). 5 F. H. Varekamp and J. J. M. Beenakker, Commun. 316c; Physica 25, 889 (1959). 6 H. F. P. Knaap, M. Knoester, C. M. Knobler and J . J. M. Beenakker, in progress. 7 F. H. Varekamp and J. J. M. Beenakker, Physica 24, 167 (1958). (Proc. Kamerlingh Onnes Conf. Leiden, 1958); A. van Itterbeek and J. J. M. Beenakker, Proc. 10 Int. Congress on Refrigeration, Copenhagen, 1959; C. M. Knobler, J . J. M. Beenakker and H. F. P. Knaap, Commun. 317a; Physica 25, 909 (1959). 8 H. F. P. Knaap, M. Knoester, F. H. Varekamp and J. J. M. Beenakker, Physica26, 633 (1960). J . B. Ubbink and W. J. de Haas, Commun. 266c; Physica 10,451(1943): J. B. Ubbink, Commun. 273b; Physica 14, 165 (1948). 10 E. W. Becker and R. Misenta, 2. Phys. 140, 535 (1955); E. W. Becker, R. Misenta and F. Schmeissner, Z. Phys. 137, 126 (1954). 11 A. 0. Rietveld, A. van Itterbeek and C. A. Velds, Commun. 314b; Physica 25, 205 (1959). la W. H. Keesom, Helium, 108 (Elsevier, Amsterdam, 1942). 13 J. M. J. Coremans, A. van Itterbeek, J. J. M. Beenakker, H. F. P. Knaap and P. Zandbergen, Commun. 311a; Physica 24,557 (1958),and Comm. 312d; Physica 24, 1102 (1958). l4 P. J. Bendt, Phys. Rev. 110, 85 (1958). l6 I(. E. Grew and T. L. Ibbs, Thermal diffusion in gases (Cambridge Univ. Press, 1952). l6 A. de Troyer, A. van Itterbeek and G. J. van der Berg, Commun. 282b; Physica 16, 669 (1950); A. de Troyer, A. van Itterbeek and A. 0. Rietveld, Commun. 285a; Physica 17, 938 (1951). l7 K. E. Grew, F. A . Johnson and W. E. J. Neal, Proc. Roy. SOC.A 224, 513 (1954). K. Schafer, Z. Physik. Chem. B 66, 85 (1937). lo E. A. Long and 0. L. J . Brown, J. Am. Chem. SOC.5 9 , I I , 1922 (1937). zo L. Waldmann and E. W. Becker, 2. Naturf. 3a, 180 (1948). *l E. W. Becker and 0 Stehl, 2. f . Physik 136, 615 (1952); E. W. Becker, R. Misenta and 0. Stehl, Z. f . Physik 136, 457 (1953). 22 J . J. M. Beenakker, F. H. Varekamp and H. F. P. Knaap, Commun. 319a; Physica 46, 43 (1960). 23 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954). 24 J. de Boer and R. B. Bird, Physica 20, 185 (1954). 26 E. G. D. Cohen, AT. J. Offerhaus, J. M. J. van Leeuwen, B. W. Roos and J. de Boer, Physica 22, 791 (1956). 26 R. A. Buckingham, A . R. Davies and D. C. Gilles, Proc. Phys. SOC.71, 457 (1958); R. A. Buckingham, A. E. Davies and A. R. Davies, Proc. conf. Therm. and Transp. Prop. Fluids, London, 111 (1957). A. Michels, W. de Graaff and C. H. ten Seldam, Physica 26, 393 (1960). 28 H. F. P. Knaap and J. J. M. Beenakker, Physica, in print. 2o E. Ishiguro, T. Arai, M. Kotani and M. Mizushima, Proc. Phys. SOC. A 65, 178 (1952). 30 R. P. Bell, Trans. Farad. SOC. 38, 422 (1942). a1 A. Babloyantz and A. Bellemans, Mol. Phys. 3, 313 (1960). 1
CHAPTER XI
SOME SOLID - GAS EQUILIBRIA AT LOW TEMPERATURES BY
2. DOKOUPIL
KAMERLINGH ONNESLABORATORY, LEIDEN
CONTENTS:1. Introduction, 454. - 2. Some properties of the phase equilibrium in a binary system, 456. - 3. Experimental methods for the determination of equilibrium curves, 459. - 4. Survey of Leiden results, 463. - 5. Theoretical determination of equilibrium curves, 477.
1. Introduction
The problem of the equilibrium between the solid and the gaseous phases of a system consisting of two components does not occupy exclusively low temperature physicists. Even before low temperature physics started its rapid development the effect of pressure upon the solubility of solids in compressed fluids at room temperature and at high pressures was considered by geologists and mineralogists1. If the pressure of the system is raised above its critical value one gets the supercritical fluid phase which is capable of dissolving solids to about the same extent as liquids with the same density. A review of such equilibria between the solid and gaseous phases in the critical region of interest to geologists was given by Booth and Bidwel12. In the past century the influence of pressure on the solubility of solids in various solvents at room temperature has been considered by chemists and the first theoretical attempts at explaining this effect have been made: trying to express the pressure dependence of the solubility by means of other physical constants using semi-empirical equations or thermodynamic relationsd. For the graphical representation of the solubility curves one usually draws the corresponding equilibrium diagrams where the influence of both variables, the pressure and the temperature can easily be followed. In the standard book by Bakhuis Roozeboom concerning the equilibria of heteroReferences 9. 480
CH. XI,
$ 11
SOME SOLID
- GAS EQUILIBRIA
455
geneous systems6 we find not only the respective qualitative phase diagrams but also some practical examples of interest. The application of low temperatures opened a new field for the study of the solubility of condensed solid gases in the so-called permanent gases. Pollitzer and Strebels examined the influence of noble gases on the concentration of the vapour phase of compressed liquid nitrogen, and Verschoyle7measured some isotherms corresponding to the system hydrogen and nitrogen just below the triple point temperature of nitrogen. Nevertherless, it was not until after the end of the second world war that the experimenters undertook more extensive and systematic investigations at low temperatures. In most cases the gassolid equilibria of a system of two components were measured in a characteristic temperature range between the critical temperature of the solvent and the triple point temperature of the solute. In such cases the solid is supposed to be pure and the gaseous (or fluid) phase generally contains dissolved solid in small quantities. A recent excellent review by Rowlinson and Richardsons on the solubility of solids in ccmpressed gases contains not only the low temperature investigations cn this subject up to 1957, but also gives a general survey for all temperatures of the experiments performed and the more or less successful theoretical attempts to understand the complicated matter. The first set of measurements by the Leiden group on equilibrium between solid and gas were published in 19559. These investigations concerned hydrogen systems with nitrogen and carbon monoxide. Recently measurements have been made on the systems heliumnitrogen, hydrogen-methane, and hydrogen-oxygen. The measurements on hydrogen-nitrogen have been extended to below the critical temperature of hydrogen at pressures slightly above the corresponding vapour pressure of pure liquid hydrogen. There are a number of reasons for the work on solid-gas systems at low temperatures. The standard experimental methods for obtaining representative samples of a mixture were adapted for use at low temperatures and new techniques for analysing very small quantities have been developed. The knowledge of solubility data on dissolved impurities in some technical gases is invaluable in low temperature engineering for purification, liquefaction and distillation purposes. Finally, the experimental results provide new data for the testing of some theories of mathematical physics. We shall first discuss some qualitative properties of the equilibrium References p . 480
456
2. DOKOUPIL
[CH. XI,
92
curves of a typical binary system under consideration. A survey of applied experimental methods for the determination of the equilibrium curves will be made, and analytical procedures for the determination of small concentrations will be described. Finally, the recent Leiden results will be reviewed and compared with the theoretical predictions.
2. Some Properties of the Phase Equilibrium in a Binary System Before one makes a choice from the different experimental methods for the determination of the equilibrium curves it is necessary to look at the corresponding phase diagram of a typical binary system under consideration. Since only binary systems will be considered it is obvious that all mixtures here investigated refer to the same kind of equi-
Fig. 1.
p,
7'-diagram of a binary system.
librium; therefore, qualitatively the properties are the same for all measured systems in the corresponding pressure and temperature interval. The complete binary system can be represented in the whole pressure and temperature range by means of a three dimensional space model5 which is referred to the three coordinate axes p , T, and x. p and T are the pressure and the temperature, while x is the mole concentration. I n the discussions we shall make use of some two dimensional equilibrium diagrams derived from the above model keeping one of the two parameters, p or T , constant. The choice of such a crosssection can be seen in the orthogonal projection of the space figure onto the p , T-plane as given in Fig. 1. For the sake of brevity the References p . 450
CH. XI,
5 21
457
SOME SOLID - GAS EQUILIBRIA
symbol G shall be used for the gaseous phase, L for the liquid phase and S for the solid phase. The subscript 1 refers to solute, the subscript 2 to the solvent. The fundamental points of both pure components are the triple points T,, T , and the critical points C,, C,. The two-phase equilibria of both pure components are given by the sublimation lines T , - S,, T , - S,,the vapour pressure lines T , - C,, T , - C, and the melting lines T , - R,, T , - R,. I n this special choice of both constituents of the mixture the critical line of the mixture C, - C, is cut into two discontinuous parts C, - A and B - C, by the three-phase line Q - B , A - T I giving rise to the supercritical region between the lower and upper critical points A and B. This three-phase line corre-
1
x
Fig. 2a, 2b. T , x-diagram and
2 ' X p , x-diagram of a binary mixture in the supercritical
-
region.
+ +
sponds to the equilibrium S , L G between the gaseous phase G and the liquid L , saturated by the solid S,. It is unnecessary to consider other three three-phase lines, Q - T,, Q - E and Q - F because the measurements were mainly performed in the supercritical region. Figs. 2a and 2b give typical examples of T , x- and p , x-cross-sections of the space figure in the supercritical region. The fixed temperature thus lies somewhere between T , and T,, where the subscripts refer to the lower and upper critical endpoints A and B in Fig. 1. The isotherm (see Fig. 2b) starts from the point corresponding to the vapour pressure of pure solid S, at the fixed temperature T,, passes through a minimum concentration of dissolved component 1 in the gaseous phase and rises with increasing pressure until it reaches the S , S , region at some very high pressure. The T , x- isobar for the constant pressure p
+
References p. 480
458
2. DOKOUPIL
[CH. XI,
2
above p , has a form similar to the isobar depicted in Fig. 2a at temperatures below T,. The detailed behaviour of such an isobar for temperatures above T , depends, of course, upon the pressure with respect to the maximum value of the three-phase line, and this is not of interest because the liquid phase rich in component 1 would be formed. If the pressure is decreased below the critical value of the solvent 2, then at a sufficiently low temperature, of course, the formation of liquid consisting mainly of component 2 will take place as is shown in Fig. 3a. Similarly, the isotherm touches the coordinate axis of the pure component 2 at its corresponding vapour pressure as given in Fig. 3b.
T
L+G d
_---Fig. 3a, 3b. T, x-diagram and p , x-diagram of a binary mixture below the supercritical region.
In both figures the connodal line S, + L + G locates the equilibrium between the gaseous mixture G and liquid mixture L saturated by the solid S,. All other areas of the diagrams are explained by the incorporated symbols. The diagrams belonging to the states intermediate between those discussed in Figs. 2 and 3 can easily be drawn. The main feature is that the inhomogeneous region L + G is still present but it no longer touches the axis 2, and the homogeneous liquid region is extended as a compressed fluid over the G-region. This digression on the rather complicated matter of heterogeneous equilibria of binary systems should be sufficient for understanding phenomenologically the behaviour of the systems measured. References p . 480
CH. XI,
3 31
SOME SOLID - GAS EQUILIBRIA
459
3. Experimental Methods for the Determination of Equilibrium Curves The standard experimental methods which are used at room temperature for the determination of the equilibrium curves can in principle be adapted for low temperature work. Indeed, a search of the literature reveals that the static method, the flow method and the circulation method have been put into practice by various investigators at low temperatures. These different procedures shall be described successively and the merits of each under the conditions of the solubility experiments for the S , G equilibrium shall be discussed. The static method is schematically shown in Fig. 4a. While working in the inhomogeneous region S , G, a gaseous mixture of an appropri-
+
+
Fig. 4a, 4b. Experimental methods for the determination of equilibrium curves.
ate initial composition is introduced into the equilibrium vessel V at constant temperature T through the inlet capillary I and the pressure is built up to the required value p. When the solid-gas equilibrium is reached the gaseous sample is quickly withdrawn through the outlet tube 0 by opening the valve K ; the sample is afterwards analysed. In Fig. 2b the filling process is represented at constant temperature of the equilibrium vessel by the path a - b. At equilibrium the deposited solid is given by the point d and the gaseous phase by the point c. The rapid sampling is characterized by the path c - s. The equilibrium state under the same p , T-conditions can, of course, be attained by first condensing the pure, less volatile component in excess in V , and successively adding the other component. When using such a modified static method, only a very small quantity of the solute is required. The sampling valve K is preferably placed in the cryostat and its vacuum tightness is essential. The classical type of References
p . 450
4 60
2. DOKOUPIL
[CH. XI,
93
stuffing box is not fully reliable at low temperature as noted by Verschoyle'. In the Leiden apparatus the stainless bellows-type valve proved to be superior to the tombac bellows which caused troubles in the beginning of the experiments on the solubility of oxygen in hydrogen. When sampling the gaseous phase (or the fluid one) the equilibrium conditions must not be appreciably disturbed due to the rapid pressure decrease. Therefore, the static method is applicable for high densities of the solvent only, i.e. at high pressures or at low teniperatures with respect to the critical state of the solvent. In Fig. 4b a sketch is given of the flow method. The gas mixture of an appropriate composition is passed at constant pressure through the inlet capillary I into the equilibrium vessel V kept at the constant temperature T. The gaseous phase leaves continuously through the outlet 0. When the equilibrium state is reached the gas is withdrawn and subsequently analysed. On the isobaric diagram (see Fig. 2a) the flow process is represented by the path a - b - c - s. d stands for the deposited solute. The flow rate must be carefully regulated in order to get a representative sample. The circulation method requires the reintroduction of the outcoming gas into the equilibrium vessel by means of some type of circulation device; for example, a bellows type pump or a mercury column circulating pump. In an ideal case this process corresponds to the path a - b - c - s - c - s . . .. When the stationary state is established the sample is drawn off and subsequently analysed. Although the volumetric method has been applied many times with great success for L G equilibria it can not be used in the investigation of S G equilibria because of the very low concentrations in the gaseous phase. The initial mixture of known composition is introduced in known portions discontinuously into the equilibrium vessel which is kept at constant temperature T , and the corresponding pressure is recorded. The precipitation of the solute (point f in Fig. 2b) or the retrograde sublimation of the solute (point g in Fig. 2b) appears in the pressure recording as a kink, which in this case is hardly detectable. The composition of the mixture at both points f and g is equal to that in the initial state a. Therefore it is not at all necessary to analyse any samples. It is evident that the two dynamic methods can be applied only in the regions S , G or L G (see Figs. 2a, 3a). At the three-phase equilibrium S , L G during the stationary flow the sample can only $J
+
+
+ + +
Rcfereiices p . 480
+
CH. XI,
9 31
SOME SOLID - GAS EQUILIBRIA
461
be withdrawn from the gaseous phase following the path a - b - c - s in Fig. 3a. If the temperature of the system in the equilibrium vessel be decreased below that of the three-phase equilibrium S, + L G, one gets the new phase equilibrium S, L , and the homogeneous liquid region L. If we are interested in the composition of the liquid along the equilibrium line g - Iz (see Fig. 3a) the withdrawal of the sample corresponding to point m must be done very quickly or otherwise a wrong sample belonging to some point on the boundary line c - 12 would be gotten. Therefore, the static method was used for the investigation of the solubility of solid nitrogen in liquid hydrogen. If a dynamic method is used one first develops the corresponding equilibrium state and then tries to remove the sample so quickly that the composition of the evaporated liquid does not change lo. As the solubility of S in L in the systems of our interest is of the order of some parts per million (= ppm) neither the standard calorimetric analysis nor the method based on vapour pressure measurements can be used for the S, + L G equilibrium. The gaseous sample usually contains only small quantity of the solute. If the standard chemical methods for the analysis are not sensitive enough the microanalysis requires special attention with respect to both species. Before explaining the Leiden method of analysis some techniques used by various investigators will be listed. For the analysis of CO, in H, during the purification process of hydrogen, Denton, Shaw and Wardll used the infra-red gas analyser. They could detect 0.25 ppm of CO,. The same sensitivity in air was reported by Webster 1 2 who trapped CO, at liquid oxygen temperature and subsequently by weighing determined the quantity adsorbed on an adsorbent. By means of adsorption of N, or 0, on charcoal at liquid nitrogen temperature Zinovjeval3 analysed the purity of hydrogen and helium; when using 1 m3 of gas it was possible to detect mole fraction with an accuracy of about 10-30%. In the case of solubility of p-chloroiodobenzene in compressed ethylene the radioactive tracer method was applied with success by Evaldl*. As he used the static method he analysed the gaseous phase directly in the equilibrium vessel. The method of analysis developed in Leiden was adapted for the determination of small condensable impurities in He or in H,. The simplified sketch of the analysis apparatus is given in Fig. 5. It consists of a closed system of a known calibrated volume which is filled
+
+
Referetaces fi. 480
+
462
Z. DOKOUPIL
[CH. XI,
33
by a known quantity of gas mixture. By means of a mercury circulating pumpM the gas is forced through the capillary C immersed in liquid hydrogen. The commutating valve V serves for the circulation of the gas flow in the same direction when the stroke of the pump is reversed. In this way all the condensable component is frozen out in C with a negligible partial pressure at 20" K. Subsequently the system is evacuated, and valve V is set in the neutral closed position in order to reduce the volume. The liquid hydrogen is removed and the quantity of the condensable component determined by reading the pressure on a manometer connected to F. The concentration down to 0.6 ppm could be determined for samples of about 0.5 1 NTP. This method was used for the analysis of the samples H, - N,, H, - CO, He - N,, H, - CH, V
Fig. 5. Analysis apparatus for a Ha-N,-gas mixture.
and H, - 0,. Petitlo applied the same principle of analysis to the mixtures H, - N,, H, - A and H, - 0,. The gaseous mixture of about 1 1NTP was circulated by means of a bellows reciprocating pump. The lowest concentration measured was 2 ppm. Briliantov and Fradkov15 did not circulate the hydrogen mixture to be analysed in their two-stage analysing system, schematically shown in Fig. 6. The relatively large sample of about 5-20 m3 passes through S , into the first condensation loop L,, kept at liquid hydrogen temperature. The amount of nitrogen-free hydrogen gas is measured by the flowmeter V before it enters the gas holder (S, was open and S, closed). The hydrogen mixture trapped in L , is about 1000 times enriched with nitrogen. When the first trap L , is warmed up to room temperature (valves S,, S,, S, closed) it contains enough pressurized gas to flow by References p . 480
CH. XI,
5 41
SOME SOLID - GAS EQUILIBRIA
463
itself through the second loop L , cooled to liquid hydrogen temperature. The amount of deposited solid nitrogen in L , is subsequently determined in the same way as in the above depicted single stage method: S , and S, are closed, hydrogen gas is pumped off through S,, S , is closed, liquid hydrogen removed from the Dewar glass and the partial pressure of gaseous nitrogen measured at F. Because of the single flow through the traps L , and L , it is necessary to first calibrate the
Fig. 6. Analysis apparatus for a very low concentration of condensable impurities in H,.
system with a mixture of known composition. The authors claimed that the lower limit of the mole concentration measurable was lO-lO, i,e. 1 mm3 in 10 m3 was detectable. 4. Survey of Leiden Results
For the reasons mentioned in the introductory section the number of binary systems discussed has been limited to the experiments performed by the Leiden group. The first solubility measurements on hydrogen-nitrogen and hydrogen-carbon monoxide are also mentioned in the survey article by Rowlinson and Richardson*. The full details of these experiments were published in 19559. The results concerning the system helium-nitrogen were submitted at the Paris conference of the International Institute of Refrigeration in 195531. The experiments on the system hydrogen-methane were communicated at the Kamerlingh Onnes Conference at Leiden in 19583a. The measurements on hydrogen-oxygen have not as yet been published. Refevences p. 480
464
Z. DOKOUPIL
[CH. XI,
94
- MIXTURE AND H,-CO - MIXTURE The Leiden measurements were performed using the flow method ; the analysis was carried out by freezing out the condensable component as described in section 3. The measured points were published earlierg; the corresponding tables listed the partial pressure fix of nitrogen in hydrogen and the composition of the vapour phase in equilibrium with the pure solid at 50, 25, 15, 10, 5 and 1.3 atm respectively. The temperature range was from the triple point of nitrogen down to the critical temperature of hydrogen. Petit lo measured the isobars of the same system at 35, 30, 25, 20, 15 and 12.5 atm only at temperatures below the critical temperature of hydrogen. As the pressures were almost above the critical temperature of hydrogen he actually again investigated the S, + G region discussed in Fig. 2a. He used the circulation method (see section 3) in this pressure region; the molecules of the circulating mixture, therefore, did not pass through a liquid-gas interface during the flow through the closed circuit. The results of both investigations are presented in Fig. 7 where the mole concentration is logarithmically given as a function of the linear temperature; the pressure is constant. In the region above the critical state the amount of the dissolved solid nitrogen in the gaseous phase decreases monotonically with decreasing temperature. I n the part of the supercritical region where the densities of the mixture approach the density of the liquid hydrogen at about 32" K, typical horizontal sections of the isobaric lines are observed. This behaviour corresponds to the convex boundary line of the S, + G region with respect to the axis 2. For p > perit(see Fig. 3a) the line c - n does not touch the hydrogen axis any more and would be given by the broken line c' - n'. The possible retrograde increase of the concentration cannot be detected using the dynamic method of the experiment. With further decrease of the temperature the compressed gas behaves like an ordinary liquid, i.e. the selubility of nitrogen is less dependent upon the pressure. By interchanging both axes of Fig. 7 the graph corresponding exactly to the type of isobars in Fig. 2a is gotten. From the same family of curves by cross plotting the isotherms which are related to the qualitative Fig. 2b can easily be derived. This is illustrated in Fig. 8 with results for the H, - CO system. The behaviour of the H, - COmixture is indeed quite similar to the system H, - N,g). 4.1. H,-N,
RefErci~cesp . 480
5 41
CH. XI,
465
SOME SOLID - GAS EQUILIBRIA
10
% 1
01
I T
20
I
1 60
I
30 .
50
7 'K
Fig. 7. n, T-isobars of H,-N,-mixture:
0 50 atm,
0 25 25
0 V
10 atm, 5 atm, atm, 8 1.3 atm, measured by Dokoupil e l
atm,
x 35 @ 30
d
25
+
atm, 20 atm, A 15 atm, atm, atm, y 12.5 atm. measured by Petit lo.
X
Fig. 8. Ktfcrcizccs
p . 480
p , x-isotherms
of H,-CO-mixture.
466 2. DOKOUPIL [CH. XI, 9 4 4.2. He-N, - MIXTURE The solubility of solid nitrogen in gaseous helium was investigated between 60" K and 35" K at pressures of 15, 10 and 5 atm31. The flow method again provided the samples which were analysed by freezing out the nitrogen at liquid hydrogen temperature. The experimental N+ He
I \ \\
Fig. 9. Partial pressure of N, in He as a function of temperature a t constant pressures: 15 atm, 0 10 atm, A 5 atm, - theoretical curves.
data are collected in Table I. Fig. 9 gives plot of the partial pressure against the reciprocal temperature. The advantage of such a diagram is that the isobars do not cross each other in contrast to Fig. 10 where the molar concentration is shown as a function of the temperature. In the illustrative Fig. 9 one sees how the partial pressure of solid nitrogen increases under the influence of a compressed real gas. References p . 480
CH. XI,
41
467
SOME SOLID - GAS EQUILIBRIA TABLE I
Nitrogen-helium isobar a t 15 atm. Temperature (OK)
Partial pressure (mm Hg)
65.4 59.7 55.9 50.2 45.4 42.0 39.9 38.8 35.4
156 50.2 20.9 3.55 1.03 0.185 0.0497 0.0387 0.0108
Concentration
(%) 1.37 0.440
0.184 0.0120 0.009 17 0.001 62 0.000437 0.000352 0.0000948
Nitrogen-helium isobar a t 10 atm Temperature (OK)
Partial pressure (mm Hg)
65.1 59.7 55.4 50.2 45.1 42.0 41.7 37.2
124 46.3 18.5 3.66 0.756 0.153 0.102 0.0153
Concentration
(%) 1.62 0.613 0.243 0.0482 0.00965 0.00204 0.001 34 0.000 201
Nitrogen-helium isobar at 5 a t m
References p . 480
Temperature (OK)
Partial pressure (mm Hg)
Concentration
65.0 59.6 54.9 50.4 44.8 42.0 35.2
139 41.3 15.2 3.02 0.42 0.131 0.0027
3.67 1.09 0.398 0.0816 0.011 1 0.003 55 0.000070
(%)
468
2. DOKOUPIL
[CH. XI,
34
0.01
0.001
.T 0.0001 I
30
T
40
! 1
60
7(
Fig. 10. x , T-isobars of He-N,-mixture: 15 atm, 0 10 atm, 5 atm, _- theoretical curves.
4 . 3 . H,-CH,
- MIXTURE
The equilibrium between the gaseous and the solid phase of the system H, - CH, has been examined in the temperature region from 90 to 55" K at pressures of 15, 10 and 5 atm32. In this temperature range the solid phase consists only of methane. For the determination of the concentrations of methane in gaseous hydrogen the flow method was applied; the analysis was performed by freezing out the methane at 20" I<. As the triple point of methane (90.5' K) is appreciably higher than the triple point of nitrogen and oxygen, and as those gases which may be impurities in methane are much more soluble in gaseous hydrogen under the same conditions, special attention was taken with respect References
p . 480
CH. XI,
5 41
469
SOME SOLID - GAS EQUILIBRIA
to the purity of methane and hydrogen. Hydrogen was decontaminated by means of a charcoal trap at liquid nitrogen temperature. According to Brilliantov and Fradkov15 hydrogen gas so treated has less than 2 x mole fraction impurity. The methane was prepared by the Shell Laboratory in Amsterdam and it contained less than 0.006 mole% N, and less than 0.003 mole% 0,. The results are listed in Table I1 and are presented graphically in Figs. 11 and 12. T A B L E I1
Methane-hydrogen isobar a t 15 atm Temperature (OK)
Partial pressure (mm Hg)
90.15 83.0 77.8 69.8 62.0 57.8 57.0
124 43.4 18.8 3.30 0.732 0.276 0.235
Concentration
(%) 1.09 0.382 0.165 0.0289 0.00642 0.00242 0.002 03
Methane-hydrogen isobar a t 10 a t m Temperature
("I<)
Partial pressure (mm Hg)
90.15 83.6 77.8 70.0 64.6 62.8 57.4 57.4
100 40.3 14.3 2.84 1.01 0.540 0.196 0.193
Concentration
(%) 1.32 0.528 0.188 0.036 3 0.0133 0.007 10 0.00258 0.002 54
Methane-hydrogen isobar a t 5 atm
-
Temperature ("K) ~~
90.15 84.4 70.2 65.1 64.7 64.1 62.7 57.1 57.0 Referelices
p . 480
Partial pressure (mm Hg) 92.6 39.2 2.42 0.655 0.726 0.596 0.327 0.104 0.0797
Concentration
(%) __2.44 1.03 0.063 7 0.0173 0.0190 0.0157 0.00867 0.002 74 0.002 10
a
t P
$ 7
CI
a
4 0
7- 9 k i '4- n2
I
0
1 0-
1-
0.1
0.1 -
PI 0.01OK 5
'
80
:
6 0 _T 5 0
Fig. 11. Partial pressure of CH, in H, as a function of tempcrature a t constant pressures : 15 atm, 0 10 atm, 5 atm. _ _ theoretical curves.
Fig. 12. x , T-isobars of H,-CH,-mixture. 15atm, 0 10 atm, 'J 5 a t m , __ theoretical curves.
a
N
CH. XI,
9 41
4.4. H,-0,
SOME SOLID - GAS EQUILIBRIA
471
- MIXTURE
In order to avoid work with large quantities of a mixture of oxygen and hydrogen needed for the flow method with the standard large equilibrium vessel, the first series of experiments was begun with a modified static method as described in section 3. A small equilibrium vessel which was primarily designed for the investigation of the solubility of solids in liquids was used. The bellows-type valve K operating at low temperatures (see Fig. 4a) limited the work to pressures below 20 atm. The results obtained by the static method gave values which were approximately 10 times smaller than the calculated solubilities below the triple point of oxygen. The explanation of this effect was indeed as suspected. Due to the small volume of the equilibrium vessel the expansion of the gas under pressure, when sampling, caused a decrease of the temperature and as a consequence a decrease of the oxygen concentration in the sample ; the instanteneous temperature decrease was simultaneously observed on the thermometer and the pressure dropped far below the value maintained constant during the period of time required for the attainment of equilibrium. It was necessary, therefore, to change the method of sampling and the whole procedure can be described as follows: pure oxygen is first solidified in the equilibrium vessel V with valve K closed; the pure hydrogen is added until the static equilibrium condition is attained; valve K is fully opened while the pressure is reduced continuously to atmospheric pressure by means of another valve outside the cryostat, the equilibrium vessel being refilled by new pure hydrogen which becomes saturated by solid oxygen in the equilibrium vessel, and leaves the apparatus as in the flow method. The combination of the static and the flow method depicted above gave satisfactory results from which it was learned that the solubility of oxygen is nearly 10 times less than the solubility of nitrogen in gaseous hydrogen. It was, therefore, concluded that it would be sufficient to use a mixture of a very low initial concentration of oxygen for the application of the standard flow method. The experimental results found by both methods were indeed in good agreement. The determination of the concentration of the samples was again performed by the method of freezing out the condensable component at 20" K. In some cases a check was made that this condensable component in the gaseous sample consisted only of oxygen. This was achieved by means of the explosion method in a McLeod gauge where a small Refeveizces Q. 4S0
472
[CH. XI,
2. DOKOUPIL
$4
T A B L E I11
Oxygen-hydrogen isobar a t 15 atm Temperature (OK)
Partial pressure (mm Hg) 1.4 0.91 0.75 0.83 0.47 0.31 0.16 0.090 0.013
53.0 52.0 50.9 49.7 48.9 48.0 46.0 44.0 40.0
Concentration
(%) 0.013 0.0080 0.0065 0.007 3 0.0047 0.002 8 0.0014 0.00079 0.000 11
Oxygen-hydrogen isobar a t 10 atm Temperature (OK)
Partial pressure (mm Hg) 2.6 0.85 0.29 0.15 0.093 0.049
54.7 52.0 49.1 47.1 45.8 44.0
Concentration
(%) 0.035 0.011 0.0038 0.002 0 0.0012 0.000 63
Oxygen-hydrogen isobar a t 5 atm Temperature
("I0 54.7 52.2 49.1 47.1 45.8 44.0 42.0 41.4 40.0
Partial pressure (mm Hg) 1.8 0.65 0.021 0.083 0.066 0.036 0.0086 0.0076 0.0036
Concentration
(%I 0.048 0.016 0.005 6 0.002 2 0.001 7 0.00097 0.00022 0.000 20 0.00012
quantity of pure hydrogen as added to the oxygen separated from the sample. Therefore it is believed that the results were not affected by the presence of some parasitic nitrogen. The isobars at 15, 10 and 5 atm were measured in the temperature Refevcitces p . 480
CH. XI,
§ 41
SOME SOLID - GAS EQUILIBRIA
473
1oc
mmHc
10
1
03
0.0'
0.00'
0.000'
61
Fig. 13. Partial pressure of 0, in H, as a function of temperature a t constant pressures: 0 lOatm, 5 atm, -- theoretical curves. 15 atm,
a
region 55 - 40" K. The corresponding points are given in Table 111 and are plotted in Figs. 13 and 14.
The x - T-diagrams reveal that for every fixed temperature there exists a definite pressure corresponding to a minimum concentration of the solute in the gaseous solvent. For all mixtures investigated by the Leiden group these approximately linear plots shown in Fig. 15 have been derived. Refereiiccs p . 480
474
[CH. XI, $
Z . DOKOUPIL
Fig. 14. x , T-isobars of H,-0,-mixture: 15 atm,
0 IOatm,
-- theoretical
5 atm,
curves.
'K
Fig. 15. Optimunl pressure as a function of temperature. Referciices p. 480
4
CH. XI,
$ 41
475
SOME SOLID - GAS EQUILIBRIA
4.5. SOMEREMARKS ON S + L EQUILIBRIUM In the experiments on S , + G equilibirum of H, - N, the threephase line was approached from the high pressure region along the gas boundary. The lower critical end point B (see Fig. 1) is not far from the critical point of pure hydrogen C,, which means that the pressure corresponding to this state is of the order of magnitude of 1 3 atm. In Fig. 3a the three-phase equilibrium S, + L + G is given by the connodal line d - g - c where g stands for the concentration of the liquid phase containing more dissolved nitrogen than the gaseous phase given by the point c. If the temperature is further decreased, the binary equilibrium
'K T
*
Fig. 16. Solubility of solid N, in liquid H, between 26-33" K.
between solid k and liquid rn is established as presented in Figs. 3a and 3b. This equilibrium has been investigated by means of the modified static method (see section 3) at pressures slightly higher than the corresponding vapour pressure of liquid hydrogen. In contrast to the difficulties when the same method was used at temperatures above the critical temperature of hydrogen (see the section on H, - 0, system), the sample could be rapidly withdrawn without appreciably disturbing the equilibrium state. This is due to the sufficiently high density of References p . 480
47 6
[CH.XI,
Z. DOKOUPIL
94
the fluid. A t temperatures below 27" K, however, it was not possible to perform the sampling rapidly enough because the liquid hydrogen behaved as a rather incompressible hard liquid. The measurements for the system H, - N, are listed in Table I V T A B L E IV
Solubility of nitrogen in liquid hydrogen Temperature
32.5 32.0 32.0 32.0 31.5 30.3 30.3 30.0 29.5 29.2 29.05 29.0 29.0 28.5 28.5 28.5 28.0 27.5 27.5 27.0 27.0 27.0 27.0 27.0 27.0 26.5 26.5 26.5 26.3 26.3
Mole concentration
7.0 1.6 2.6 6.8 8.0 9.4 7.4
x x x
x
x x 10-6 x 10-8 5.8 x 10-6 4.0 x 4.2 x 6.6 x 7.4 x 3.8 x 3.5 x 4.0 x
10W 10-6 10V 10-6
4.9 x 10-6
5.4 x 1.7 x 1.1 x 1.1 x 1.1 x 2.5 x 1.8 x 2.5 x 2.7 x 1.4 x 4.0 x 9.5 x 4.4 x 5.4 x
10-6 10-6 10-6 10-6 10-6
10-7 10-7 10-7 10-7
15.3 17.0 14.5 13.0 13.5 12.2 12.0 10.0 11.0 12.0 11.0 11.0 9.5 11.7 10.0 -
11.0 13.2 11.0 8.9 7.5 7.4 7.0 6.8 6.0 13.5 10.0 8.2 16.5 6.0
and are graphically given in Fig. 16. A rough extrapolation of the curve down to 20.4' K would result in a solubility of about mole fraction of nitrogen in the liquid. Using the above described analysis apparatus no measurable solubility of oxygen in liquid hydrogen in the same 9, T-region could be found. Refereizces
p . 480
CH. XI,
5 51
SOME SOLID - GAS EQUILIBRIA
477
5. Theoretical Determination of Equilibrium Curves We consider equilibrium between the pure solid substance 1 and the gaseous phase consisting of the compressed gas 2 and the evaporated molecules of the solid. The solvent 2 is supposed to be insoluble in the condensed gas. At the fixed pressure p and temperature T the chemical potential of the component 1 is the same in both phases. Equating thus the chemical potential of the dissolved component pl(p, T , xl) to t h a t of pure solid p,(p, T ) subjected to the hydrostatic pressure p, gives implicitly the fundamental relation for the concentration x1 at the pressure p and temperature T . ,us@,T ) can be determined from the chemical potential ps(pol, T ) of the pure solid at its vapour pressure .pol by adding the integral of the molar volume of the solid vol. ,us(pol,T ) in turn, is equal to the chemical potential of the same component in the vapour state pl(pol, T ) .Making these substitutions we get eq. (1) :
E"l(P, T , XI) = Ps(P, T ) rv
= h(pOlJT , $-
1 0 '
dp'
Both chemical potentials in eq. (1) refer to the gas phase and can be determined from the corresponding Gibbs functions for pure gas and for the gas mixture. These substitutions result in the following eq. (2) : PXl RT log __
= RT
log f
Po1
The first integral refers to the gas mixture, the second to the pure .component 1 and the third to the solid. The enhancement factor f arbitrarily defined here in terms of pressure shows how much greater the partial pressure is than the one computed by Dalton's law alone. In case of ideal behaviour of gases only the Poynting hydrostatic term remains to be evaluated*. Such simplification, however, cannot be allowed when working with real gases, as the enhancement factor f References p . 480
478
2. DOKOUPIL
[CH. XI, (3
5
can attain rather high values. In extreme cases, factors f of the order of magnitude of about 25000 have been reported in the literaturefR. Eq. (2) derived in terms of p , T variables can be rewritten in terms of integrals with volume or density as the independent variables. This depends, of course, on the choice of one of the numerous equations of state for the gas mixture which are usually given as an analytical relation between the pressure, volume, temperature, mole fractions and the constants of the equation of state of the pure constituent gases. Usually it has the same form as the equation of state of a pure component, while the constants of the equation of state for the gas mixture are essentially given by a set of combination rules for the constants of the pure components. Analytically one can expand any equation of state in a power series in 9, V or d and the result of the integration of the formula (2) would again be a series expansion. Such a procedure is not always necessary and various attempts at evaluating eq. (2) have been made in earlier work. The Van der Wads equation of state has been used by McHaffief7 and by Perkinsf*. Robin, Vodar and BergeonfSz1 interpreted their measurements with help of an equation of state limited t o the second virial coefficient using the Lennard- Jones potential for the calculation of the force constants. Evald, Jepson and RowIinsonz3gave the expansion in reciprocal volumes for the virial form of the equation of state. The 9, T-expansion and the V , T-expansion have been derived by Dokoupil et aZ.9 for the special choice of the Beattie-Bridgman equation of state. Reuss and BeenakkerZ4used a virial expansion in the density up to the B-virial coefficient considering the quantum corrections. In most of the above cited papers an important simplification has been made by putting x1 w 0 and x 2 m 1, if the solubility was sufficiently low. For large x1 Evald26 used an iteration method in order to solve for x1 from eq. (2) written in p , T variables. The evaluation of x1 for fixed fi, T-values allows us to predict the solubility figures or to make the comparison with the experimental points as shown in Figs. 9-14. Conversely when using the experimentally determined concentrations one can calculate any parameter from eq. (2) and compare its value with the value determined independently by another experiment or theory. Reuss and BeenakkerZ4,and Rowlinpointed out that the linear term in the density expansion of eq. (2), i.e. - Bl,dz, can be easily determined from the known x1 and the known equation of state for the pure solvent (or from the known Referetices p . 480
CH. XI,
9 51
SOME SOLID - GAS EQUILIBRIA
479
isotherm data). This important B,,-coefficient of the virial equation of state for the mixture can be compared with values calculated by different models of forces acting between unlike molecules. Moreover, such an indirect experimental determination of B,, can be achieved in the low temperature region which is not accessible by the usual standard experimental methods for determining B,, (see Fig. 17).
Fig. 17. The coefficient B,, for Ha-N,.
0 calculated by Reuss and Beenakker from the results of Dokoupil et d.*.
A
calculated by Rowlinson and Richardson from the same results8, z6, 0 measurements of Michels and Wassenaar measurements of Michels and BoerboomZ8, measurements of Verschoyleag, - calculated line*$26.
v
Another possible use of eq. (2) would be the determination of the vapour pressure Po, of the solute30. As a conclusion of these theoretical remarks it may be stated that the agreement with the experiments would be better if a more elaborate equation of state would be used. This is due to the limited range of validity of every equation of state. Usually the agreement between theory and experiment is better at higher temperatures and lower pressures. The error becomes greater with decreasing temperature and higher pressures and is sufficient to account completely for the discrepancy between the observed and the calculated results.
Refercnces p . 480
480
REFERENCES
[CH.XI
REFERENCES
H. C. Sorby, Proc. Roy. SOC.12, 538 (1863). H. S. Booth and R. M. Bidwell, Chem. Rev. 44, 477 (1949). F. Braun, Ann. Phys. Chem. (2) 30, 250 (1887). J. H. Poynting, Phil. Mag. 12, 32 (1881). H. W. Bakhuis Roozeboom, Die heterogenen Gleichgewichte, 2. Heft (Friedrich Vieweg u. Sohn, Braunschweig, 1904). F. Pollitzer and E. Strebel, 2. phys. Chem. 110, 768 (1924). 7 T. T. H. Verschoyle, Trans. Roy. SOC.London, A 230, 189 (1931). * J. S. Rowlinson and M. J. Richardson, Advances in Chemical Physics, vol. 11, edited by I. Prigogine (Interscience Publishers Ltd., New York, London, 1959). Z. Dokoupil, G. van Soest and M. D. P. Swenker, Appl. Sci. Res., A 5, 182 (1955); Commun. Leiden, No. 297. lo P. Petit, Compt. rend. 297, 1171 (1958). l1 W. H. Denton, B. Shaw and D. E. Ward, Trans. Instn. Chem. Engrs. 36, 179 (1958). l2 T. J. Webster, Proc. Roy. SOC.London, A 214, 61 (1952). l 3 K.N. Zinovjeva, Ind. Lab. 1. 30 (1955). l4 A. H. Evald, Trans. Farad. SOC.49, 1401 (1953). l5 N. A. Brilliantov and A. B. Fradkov, J. Techn. Phys. 27, 2405 (1957). l8 G. A. M. Diepen and F. E. C. Scheffer, J. Amer. Chem. SOC. 70, 4085 (1948). l7 J . R. McHaffie, Phil. Mag. 1, 561 (1926). l8 A. J. Perkins, J. Chem. Phys. 5, 180 (1937). l Q S. Robin and B. Vodar, Discussions Farad. SOC.15, 233 (1953). 2o S. Robin, J. chim. Phys. 48, 501 (1951). 21 S. Robin, B. Vodar and R. Bergeon, Compt. rend. 232, 2189 (1951). 22 S. Robin and B. Vodar, J. Phys. Rad. 13, 264 (1952). 23 A . H. Evald, W. B. Jepson and J. S. Rowlinson, Discussions Farad. SOC.15, 238 ( 1953). 24 J . Reuss and J . J. M. Beenakker, Physica 22, 869 (1956); Commun. Leiden, Suppl. No. 110e. 25 A. H. Evald, Trans. Farad. SOC.51, 347 (1955). 26 J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworths, London, 1959). 27 A. Michels and T. Wassenaar, Appl. Sci. Res. A 1, 258 (1949). 28 A. Michels and A. J. H. Boerboom, Bull. SOC. chim. Belg. 62, 119 (1953). 29 T. T. H. Verschoyle, Proc. Roy. SOC.,London A 111, 552 (1926). 30 H.T. Gerry and L. J. Gillespie, Phys. Rev. 40, 269 (1932). 31 M.D. P.Swenker and 2. Dokoupil, Suppl. a u Bull. Inst. internat. Froid, Annexe 1955-3, 339. 32 W.A . J. Versteegen and Z. Dokoupil, Physica 24, S168 (1958) (paper presented a t the Kamerlingh Onnes Conference, Leiden, 1958).
ERRATA OF THE VOLUMES I AND I1 VOLUME I Chapter V , E . F. Hammel, The low temperature profierties of helium three p . 103, subscript Of Fig. 13 Legend
4He a t 4°K.
3He at 2.5"K.
A 3He at 3°K.
VOLUME I1 Chapter V I I I , D. Shoenberg, The De Haas-Van Alphen e#ect, 9. 238 (21) n.=- 8 v
t(PlP2P3)-*.
3Nnk (L)
p . 261 (30) m0 u1 = -,
u2 =
m1
m2m3-mP2 '
a3 =
u4 =
-
m2m3-m42'
1m0m4
m2m3-m42' ( 30)
Chapter I X , C. J . Gorter, Paramagnetic relaxation p. 278, subscript of Fig. 6 Fig. 6 . eav = 2n zav for iron ammonium alum and a number of dilute alums as a function of the parallel field H ,
A. Feo.ol,A10.,,4NH4(S04)2 * 12H20;T 0 Feo.o,2A10.,4,NH,(S0,),
*
=
1.70"K
12H20;T = 1.72"K
8 Feo.26Alo.,6NH4(S04)2 * 12H20 ; T = 1.64"K
F e NH4(S04),* 12H20
; T = 2.05"K
Chapter X , M . J . Steenland and H . A . Tolkoek, Orientation of atomic nuclei at low temfieratures
p . 303, last line of
+
(2.4) eQ(azV/8z2)[31,2-1(1
+ 1)]/[41(2I- l)]-gNpNH - I + G(S,S').
Chapter X I V , H . van Dijk and M . Durieux, The tem.@eraturescale in the liquid helium region
p . 432, line 10 E
= l n { ( p V J / ( R T )} (2B/V,) - ( 3 C / 2 V 2 )
482
ERRATA OF THE VOLUMES I AND I1
p . 443, (7) lologficmHg,OOC = 1.197 - (3.018/T)
+ 2.484 lolog T - 0.002971"'.
P . 456, (1) In P = i - {L,/(RT))+ #ln T - {l/(RT))(J,TS,dT - J;Vtd+) Chapter X I V
p . 457, line 1 where
E
= In {(#V")/(RT)} -(2B/V,) - (3C/2Vi)
+
E
(7)
(1)
AUTHOR I N D E X Abraham, B. M., 119, 123, 129, 131, 132, 135, 139, 150, 151 Abraham, M., 371,373, 380,383,389,394, 395 Abrahams, E., 231, 277, 285, 287 Abragam, A., 380, 388, 389, 395 Abrikosov, A. A,, 113, 114, 133, 136, 145, 150, 190, 224, 231, 250, 251, 262, 284, 285, 286 Adams, E. D., 149, 152 Akhiezer, A. I., 277, 278, 279, 287 Alder, K., 339, 343, 346, 392 Aldrich, L. T., 63, 79 Alsop. L. E., 408, 417, 428 Ambler, E., 334, 342, 352, 354, 355, 356, 359, 360, 361, 364, 365, 391, 393 Amer, J.. 478, 480 Anderson, A., 173, 222, 223, 283, Anderson, A. C. 136, 149,152, 369, 394 Anderson, P. W., 117, 132, 147, 150, 152, 171, 230, 253, 256, 259, 262, 276, 277, 282, 283, 285, 286, 287 Andreyev, B. B., 322, 332 Androes, G. M., 173, 261, 282, 283 Andronikashvili, E. L., 10, 14, 25, 56, 57 Arai, T., 452, 453 Ard, W. B., 139, 151 Ardenne, G. M. V. van, 99, 111 Arkhipov, R. G., 75, 79 Arnold, G. P., 108, 112 Arp, V., 375, 376, 377, 394 Artman, J . O., 404, 405, 408, 411, 428 Atkins, K. R.,37, 56, 59, 66, 79, 81, 95, 97, 98, 99, 100, 101, 111, 112, 115, 121, 123, 143, 144, 145, 150, 152, 266, 286 Aubrey, I. E., 292, 293, 331 Autler, S., 417, 428 Azbel’, M. Ya., 288, 290, 291, 292, 293, 295, 291, 299, 300, 301, 302, 305, 307, 309, 310, 312, 315, 316, 317, 318, 324, 326, 328, 329, 330, 331, 332 Babloyantz, A,, 452, 453
Bakhuis Roozeboom, H. W., 454,456, 480 Bardasis, A., 253, 258, 259, 282, 286 Bardeen, J., 116, 117, 150, 171, 176, 191, 195, 196, 202, 213, 216, 224, 227, 230, 256,203,272,276,282,284,285,286,287 Basov, N. G., 397, 400, 427 Bauer, R. W., 366, 367, 368, 393, 394 Baum, J. L., 123, 124, 151 Becker, E. W., 437, 438, 443, 444, 449, 451, 453 Beenakker, J. J. M., 432, 434, 435, 436, 438, 443, 444, 449, 452, 453, 478, 480 Bekker, 178 Beliaev, S. T., 281, 287 Bell, R. P., 452, 453 Bellemans. A , , 452, 453 Bendt, P. J., 439, 450, 453 Berestetsky, V. B., 346, 392 Bergeon, R., 478, 480 Bernardes, N., 118, 124, 126, 150 Bezuglyi, P. A., 219, 285, 292, 331 Bhagat, S.M., 45, 56, 57 Bidwell, R. M., 454, 480 Bincer, A. M., 345, 393 Biondi, M. A,, 172, 176, 223, 243, 247, 248, 249, 282, 283 Bird, R. B., 446, 453 Bishop, G. R., 371, 394 Blakewood, C. H., 13, 58 Blatt, J . M., 81, 111, 183, 184, 262, 284, 286 Bleaney, B., 349, 350, 393, 400, 427 Blin-Stoyle, R. J., 334, 369, 391 Bloch, F., 397, 398, 427 Bloembergen, N., 139, 151, 400, 405, 408, 409, 410, 411, 413, 423, 427, 428 Blokhintsev, D. I., 322, 332 Blount, E. I., 310, 332 Boehm, F., 361, 393 Boerboom, A. J. H., 479, 480 Bogle, G. S., 403, 427, 428 Bogoliubov, N. N., 171, 200, 202, 203, 214. 253, 256, 270, 282, 284, 285
484
AUTHOR INDEX
Bogolyubov, I. I., 322, 332 Bohn, D., 253, 254, 286 Bohm, H. V., 172, 187,219, 220, 221,283, 285 Bohr, A., 281, 287 Bohr, N., 181, 284, 336, 392 Btjlger, B., 408, 410, 415, 428 B6mme1, H. E., 218, 285 Boorse, H. A., 194, 207, 249, 284, 286 Booth, H. S . , 464, 480, Borghini, M., 389, 395 Bouchiat. C. C., 345, 392, 393 Bowers, K. D., 400, 427 Brailsford, A. D., 292, 331 Braun, F., 454, 480 Brendt, P. J., 108, 112 Brewer, D. F., 37, 42, 44, 45, 46, 57, 122, 123, 124, 130, 131, 132, 133, 151 Brickwedde, F. G., 84, 111, 430, 435, 453 Brilliantov, N. A., 462, 469, 480 Broek, J. van den, 354, 379, 395 Broese van Groenou, A , , 37, 57 Brout, R., 149, 152, 254, 286 Brown, A., 194, 207, 284 Brown, 0. L. J., 443, 453 Brueckner, K. A., 115, 117, 123, 130, 132, 133, 150, 254, 259, 282, 286, 287 Brussaard, P. J., 334, 391 Buckingham, M. J., 86, 87, 111 Buckingham, R. A., 446, 447. 449, 460, 451, 453 Buekler, E., 398, 426, 427 Burgy, M. T., 343, 392 Burnham, J. B., 95, I l l Butler, S. T., 81, 111, 183, 184, 284 Button, K. J., 292, 331 Callen, H. B., 420, 428 Careri, G., 58, 69, 60, 61, 63, 64, 71, 74, 77, 79, 139, 140, 141, 151 Casimir, H. B. G., 175, 207, 219, 240, 283, 410, 428 Castle, J. G., 398, 427 Cetlin, B. B.,292, 331 Chalis, L. J., 134, 151 Chambers, R. G., 179, 243, 284, 290, 292, 293, 295, 296, 297, 299, 309, 331, 332 Chang, W. S . C., 411, 412, 428 Chanin, G., 276, 277, 287 Chase, C. E., 81, 95, 96, 97, 99, 100, 111, 112, 183, 284 Chester, G. V.. 45, 57
Chester, P. F., 398, 421 Chou, C . , 208, 284 Clement. J. R., 84, 111, 150, 243 Clogston, A. M., 404, 428 Clusiiis, K., 87, 95, 111 Cohen, E. G. D., 446, 447, 449, 451, 453, 460
Cohen, M., 10, 56, 187, 255, 284, 286 Coles, B. A., 404, 428 Combrisson, J.. 388, 395 Compton, V. B., 277, 278, 279, 287 Connor, D., 374, 394 Consolo, 72, 79 Conwell, E. M., 75, 79 Cooper, B. F. C., 416, 420, 422, 428 Cooper, L. N., 117, 150, 171, 176, 195, 196, 206, 213, 216, 224, 256, 259, 282, 284. 287 Corak, W. S., 208, 284 Coremans, J. M. J., 438, 449, 453 Corenzwit, E., 277, 278, 279, 281 Craig, P. P., 31, 56 Critchlow, P. R., 45, 56, 57 Cromack, J., 411, 412, 428 Culvahouse, J. W., 367, 385, 395 Curtis, R. B., 343, 346, 377, 392 Curtiss, C. F., 446, 453 Dabbs, J. W. T., 346, 347, 348, 350, 351, 393, 395 Dagley, P., 359, 367, 369, 393, 394 Dail, H. W., 292, 331 Daniels, J. M., 334,370, 371,372, 375, 378, 379, 391, 394, 395 Dash, J. G., 135, 139, 151 Daunt, J. G., 123, 124, 129, 130, 131, 132, 133, 151, 195, 284, 377, 395 Davis, L., 408, 417, 428 Davies, A. E., 446, 447, 449, 450, 451, 453 Davies. A. R., 446, 447, 449, 450, 461, 453 De Boer, J., 113, 115, 123, 152, 430, 435, 446, 447, 449, 450, 451, 453 De Bruijn Ouboter, R., 158 Debye, P., 78, 79 Decker, D., 178, 210, 243, 284, 285 De Graaff, W.. 452, 453 De Haas, W. J., 436, 437, 453 Dehmelt, €3. G., 139, I51 De Mars, G. A., 404, 428 Denton, W. H., 461, 480 De Troyer, A., 441, 450, 451, 453 Deutsch, M., 366, 367, 368, 393, 394
AUTHOR INDEX Devlin, G. E., 243, 244, 286 Dexter, R., 289, 330 Diddens, A. N., 366, 368, 369, 393, 394 Diepen, G. A. M., 478, 480 Dijk, H . van, 84, 111 Dillinger, J. R., 87, 111 Dingle, R. B., 11, 56, 265, 266, 286, 289, 330 Doidge, P. R., 179, 180, 284 Dokoupil, Z . , 455, 463, 464, 465, 466, 478, 479, 480 Dolginov, A. Z., 343, 345, 346, 377, 392, 393 Donnelly, R. J., 36, 45, 46, 56, 57 Dorfman, G. Ya., 289, 330 Dousmanis, G. C., 426, 429 Dresselhaus, G., 282, 289, 309, 312, 330, 331 Dresselhaus, M., 282 du Pr6, F. K., 410, 428 DuprB, 71 Dyson, F. J., 200, 284, 322, 329, 332 Ebel, M. E., 339, 392 Edeskuty, F. J., 120, 121, 122, 123, 125, 126, 132, 150, 151 Edmonds, D., 375, 376, 377, 394 Edmonds, A. R., 334, 391 Edwards, D. O . , 37, 42, 44, 45, 46, 57, 123, 124, 151 Edwards,M.H., 81,95, 97,98,99,111 Edwards, S. F., 231, 285 Ehrenfest, P., 94, 98, 111 Ehrenreich, H., 255, 286 Eisenstein, J. C., 351, 393 Einstein, A , , 396, 427 Elliot, R. I., 324, 332 Emery, V. J., 117, 132, 150 Ende, J . N. van den, 175, 283 Erb, E., 389, 395 Erickson, R. A , , 375, 394 Erlbach, E., 241, 246, 282, 285 Evald, A. H., 461, 478, 480 Eversdijk Smulders, Miss M. C., 362, 363, 372, 393, 394 Faber, T. E., 179, 237, 242, 243, 284 Fairbank, H. A., 63, 79, 122, 123, 133, 134, 135, 136, 137, 151 Fairbank, W. M., 86, 87, 110, I l l , 112, 124, 139, 142, 143, 149,151,152 Falge, R. L., 277, 278, 279, 287
485
Fano, U., 334, 391 Fasoli, U., 60, 61, 79 Faughnan, 408, 428 Fawcett, E., 292, 293, 331 Feher, G., 320, 323, 332, 398, 404, 412, 426, 427, 428 Feldman, G., 339, 392 Ferrell, R. A., 60, 79, 227, 228, 229, 237, 256, 262, 285, 286 Feynman, R. P., 2, 8, 9, 10, 11, 30, 56, 62, 79, 183, 187, 284, 338, 345, 392 Filimonov, A. J., 164, 169 Findlay, J. C., 99, 111 Finnemore, D. K., 210, 243, 285 Fisher, J . C., 253, 258, 259, 281, 286 Flicker, H., 121, 143, 144, 145, 150, 152 Foner, S., 292, 331, 424, 425, 429 Forrester, A. T., 176, 284 Forrez, G., 99, 111, 143, 152 Fradkov, A. B., 462, 469, 480 Frauenfelder, H., 334, 345, 391, 393 Freeman, A. J., 377, 395 Friedberg, S. A., 102, 112 Friedburg, H., 423, 425, 428 Frisch, 0. R., 335, 369, 391 Frohlich, H., 176, 191, 283 Fuchs, K., 296, 331 Fukuda, N., 254, 286 Gaeta, F. S., 60, 61, 76, 77, 79 Galkin, A. A , , 219, 285, 292, 331 Galt, I. K., 292, 331 Gammel, J. L., 115, 130, 132, 133, 150 Gaponov. Yu. V., 343, 392 Garfunkel, M. P., 172, 176, 223, 243, 247, 248, 249, 282, 283 Garwin, R. L., 136, 138, 139, 141, 151, 160, 162, 163, 168 Garwin, R. W., 241, 246, 282, 285 Gavenda, J. D., 172, 209, 219, 220, 283, 285 Geilikman, B. T., 272, 287 Gell-Mann, M., 254, 286, 338, 345, 392 Gerasimenko, V. I., 322, 324, 326, 328, 329, 332 Gere, E. A., 398, 426, 427 Gerritsen, A. N., 58, 76, 79 Gerry, H. T.. 479, 480 Gershtein, S. S., 338, 345, 392 Geusic, J. E., 402, 403, 411, 412, 427, 428 Giles, J. C., 378, 379, 395 Gilles, D. C., 440, 447, 449, 450, 451, 453
486
AUTHOR INDEX
Gillespie, L. J., 479, 480 Ginsberg, D. M., 172, 195, 218, 236, 237, 238, 239, 251, 258, 282, 283 Ginzburg, N. I., 267, 270, 286 Ginzburg, V. L., 2, 12, 41, 56, 176, 183, 184, 195, 268, 283 Giordmaine, J . A., 47, 408, 428 Glover 111, R. E.. 172, 195, 218, 227, 228, 229,236,237,238,239,251.258,283,285 Goldhaber. M., 342, 392 Goldstein, L., 116, 118, 123, 126, 130, 132, 133, 145, 150 Goldstone, J., 255, 286 Good, R. H., 346, 393 Goodings, D. A,, 377, 395 Goodkind, J. M., 142, 143, 152 Goodman, B. B., 194, 195, 207, 208, 282, 284 Goodwin, F. E., 419, 428 Gordon, J . P., 397, 398, 426, 427 Gordy, W., 139, 151 Gor’kov, L. P., 224, 231, 251, 269, 285, 286, 287 Gorter, C. J., 37, 45, 57, 86, 87, 111, 113, 152, 175, 176, 207, 219, 263, 264, 283, 286, 354, 359, 360, 377, 378, 393, 395, 411, 412, 428 Gossard, A. C., 375, 394 Gottfried, K.. 255, 286 Grace, M. A., 334, 355, 359, 363, 367, 369, 370, 371, 373, 375, 391, 393, 394 Grasse, R. W. de, 418, 422, 428 Grayson-Smith, H., 99, 111 Gregory, J. M., 367, 393 Grew, K. E., 441, 450, 451, 453 Griffing. 369, 393 Grilly. E. R., 119, 120, 122, 123, 124. 125, 126, 127, 128, 129, 136, 150, 151 Griswold, T. W., 323, 392 Grodzins, L., 335, 342, 892 Groot, S. R. de, 334, 335, 345, 346, 391, 392 Guenault, A.M., 173, 273, 282, 283, 287 Gurov, K. P., 322, 332 Hake, R. R., 178, 284 Halban. H., 334, 391 Hall, G. R., 349, 350, 393 Hall, H. E., 10, 12, 15, 16, 17, 19, 20, 21, 24, 25, 54, 55, 56, 57 Hammel, E. F., 95, 111, 119, 130, 132, 150, 151
Hammond, R. H., 223, 282, 285 Hanauer, S. H., 347, 348, 350, 351, 393 Hanna, S., 375, 394 Hardeman, G. E. G., 378, 395 Hart, Jr., H. R., 136, 137, 138, 142, 149, 151, 152 Hayward, R. W., 342, 352, 354, 355, 359, 360, 361, 364, 365, 393 Hebel, L. C., 172, 217, 222, 283 Heberle, J., 375, 394 Heer, C. V., 375, 394 Hein, R. A., 277, 278, 279, 287 Heine, V., 262, 286, 312, 332, 377, 395 Heller, G., 178 Henshaw, D. G., 104, 108, 112 Hercus, G., 87, 111 Herfkens, J . H. J., 99, 111 Herring, C., 279, 287 Higa, 412 Hill, D. L., 347, 393 Hill, J . S., 359, 367, 369, 393, 394 Hill, R. W.. 85, 86, 87, 111 Hirschfelder, J. O., 446, 453 Hogg, D. C., 422, 428 Hollis-Hallett, A. C., 36, 56 Hoppes, D. D., 342, 352, 354, 355, 359, 360, 361, 364, 365, 393, 394 Hiickel, 78, 79 Hudson, R. P., 342, 352, 354, 355, 359, 360, 361, 364, 365, 393 Huiskamp, W. J., 354, 359, 360, 362, 363, 366, 368, 369, 378, 379, 393, 394, 395 Hulm, J. K., 271, 273, 287 Hulsizer, R. I., 369, 394 Hung, C. S., 37, 56 Hunt, B.. 37, 56 Ibbs, T. L., 441, 450, 451, 4.53 Ioffe, B. L., 346, 392 Ishiguro, E., 452, 453 Jaccarino. V., 376, 395 Jackson, J. D., 339, 340, 345, 346, 392, 393 Jacobsen, E. H., 404, 409, 428 JaffB, G., 76, 79 Jeffries, C. D., 361, 366, 367, 371, 373, 380, 383, 389, 394, 395 Jelley, J. V., 416, 420, 422, 428 Jepson, W. B., 478, 480 Jeener, J., 134
AUTHOR INDEX
Johnson, C. E., 355, 370, 371, 373, 374, 375, 393, 394, 395 Johnson, F. A., 441, 453 Johnston, H. L., 208, 284, 432, 434, 435, 453 Jost, R., 336, 392 Kadanoff, L. P., 262, 275, 286, 287 Kaganov, M. I., 290, 291, 292, 299, 305, 328, 330, 331 Kamerlingh Onnes, H., 175, 283 Kaner, E. A , , 290, 293, 295, 297, 300, 301, 302, 305, 307, 312, 315, 316, 328, 331, 332 Kaplan, I., 329, 330, 332 Kaplan, R., 249, 286 Karolyuk, A. P., 219, 285 Kastler, A., 400, 427 Kaunisto, Leila, 95, 102, 111 Kaus, P., 426, 429 Kavakin, I. P . , 10, 14, 56 Kedzie, R. W . , 361, 366, 367, 371, 373, 380, 383, 394, 395 Keesom, Miss A. P., 87, 111 Keesom, W. H., 44, 57, 87, 95, 111, 154, 157, 161, 168, 175, 283, 437, 453 Keller, W. E., 95, 111, 120, 150 Kellers, C. F . , 80, 82, 83, 86, 110, 111, 112 Kerr, E . C., 81, 93, 96, 97, 98, 99, 111, 120, 122, 123, 150 Kerry, E. C., 108, 112 Khaikin, M. S., 243, 250, 251, 286, 292, 294, 309, 331, 332 Khalatnikov, I. M., 74, 79, 113, 114, 133, 136, 150, 190, 224. 251, 284, 285 Khutsishvili, G. R., 334, 346, 375, 377, 382, 391, 392, 394, 395 Kikuchi, C., 413, 417, 428 Kilpatrick, J. E., 132, 151 Kingston, R. H., 424, 428 Kip, A., 289, 292, 293, 320, 323, 330, 331, 332 Kittel, C., 222, 285, 289, 323, 330, 332, 426, 429 Klein, O., 285 Klemens, P. G., 270, 287 Knaap, H. F . P., 435, 436, 438, 443, 444, 449, 452, 453 Knight, W. D., 173, 261, 282, 283 Knobler, C. M . , 435, 453 Knoester, M., 435, 436, 444, 453 Kogan, V., 376, 377, 394
487
Kojo, E., 86, 87, 93, 95, 100, I l l Kok, J. A,, 175, 283 Kompfer, R., 424, 428 Konopinski, E. J., 335, 374, 392 Konstantinov, 0. V . ,226, 285 Koolhaas, J., 76, 79 Koppe, H., 211, 285 Kornienko, L. S., 403, 427 Koster, G. F . , 404, 428 Kotani, M., 452, 453 Kramers, H. A., 58, 76, 77, 79, 86, 87, 211, 130, 132, 151 Kresin, V. Z., 219, 272, 275, 285, 287 Krohn, V . E . , 343, 392 Kubo, R., 139, 152 Kulkov, V. D., 376, 377, 394 Kuper, C. G., 42, 57 Kurti, N., 374, 375, 394 Kyhl, R. L., 115, 408, 428 Lamarche, J. L., 370, 371, 372, 394 Lamb, H., 38, 57 Lamb, W. E., 397, 427 Lambe, J,, 413, 428 Landau, L. D., 1, 2, 3, 11, 38, 56, 58, 62, 63, 79, 113, 114, 130, 133, 138, 145, 146, 150, 176, 190, 263, 268, 283, 286, 310, 332, 337, 392 Lane, C. T., 10, 13, 45, 56, 57, 63, 79 Langenberg, D. N., 292, 293, 331 Laquer, H. L., 121, 143, 144, 150, 165, 166, 169 Laredo. S . J., 271, 287 Larsson, K. E., 108, 112 Laue, M. von, 175, 283 Laurmann, E., 175, 240, 243, 283 Lawson, 420 Lax, B., 289, 291, 292, 328, 330, 331, 425, 429 Le Blanc, M. A. R., 370, 371, 372, 375, 378, 379, 394, 395 Lee, D. M., 122, 123, 133, 134, 135, 136, 151 Lee, T. D., 335,337,339, 342,346,391,392 Lee-Whiting, G. E., 346, 393 Leifson, 0. S., 389, 395 Lemmer, H. R., 371, 394 Le Pair, 166, 167 Levgold, S., 277, 287 Levi, M. W., 367, 395 Levy, R. A., 323, 332 Lewis, H. W., 242, 286
488
AUTHOR INDEX
Lewis, R. R., 343, 346, 377, 392 Lifshitz, I. M.. 291, 292, 305, 309, 324, 326, 328, 329, 330, 331, 332 Limburg, W . , 99, 111 Lindhard, 226, 285 Llewellyn, P. M., 349, 350, 393 Lock, J. M., 175, 240, 283 Logan, J. K., 84, 111 London, F., 2, 56, 104, 112, 116, 150, 170, 175, 181, 199, 280, 282, 283 London, H., 2, 11, 56, 175, 176, 178, 247, 280, 283 Long, E. A., 443, 453 Lounasmaa, 0. V . , 85, 86, 87, 93, 95, 100, 102,111 Lovejoy, C. A., 372, 395 Low, F. J., 139, 141, 149, 151, 152 Low, W . , 374, 394, 400, 427 Liiders, G., 336, 392 Lundby, A., 335, 338, 392 Lutes, 0. S . , 210, 285 Lynton, E. A., 243, 275, 276, 277, 287 MacKinnon, L., 218, 285 Mager, C. H., 417, 428 Mahmoud, H., 346, 393 Maiman, T. H., 410, 427, 428 Makhov, C., 413, 428 Mamaladze, Yu. G., 25, 57 Mapother, D. E., 178, 210, 243, 282, 284, 285 Markham, A. H., 87, 111 Marshak, R. E., 338, 392 Marshall, W., 375, 394 Martin, P. C., 262, 275, 286, 287 Masuda, Y . , 173, 222, 223, 283 Mathur, V. S., 285 Matinyan, S . G., 25, 57 Matsuban. T., 183, 184, 284 Matthias, B. T., 173, 176, 277, 278, 279, 281, 284, 287 Mattis, D. C., 224, 230, 276, 285, 287, 309, 312, 331, 426, 429 Mattuck, R. D., 404, 428 Maxwell, E., 81, 95, 96, 97, 111, 176, 210, 283, 285 May, R. M., 285 Mc Causland, M. A. H., 389, 395 Mc Cormick, W., 66, 79 McGree, J. D., 411, 412, 428 McHaffie, J. R., 478, 480 Meissner, H., 180
Meissner, W., 175, 176, 283 Mellink, J. H., 45, 57 Mendelssohn, K., 37, 42, 46, 57, 195, 270, 284. 287 Merrit, F. R., 292, 331 Meshkovsky, A. G., 176, 283 Meyer, H., 149, 152 Meyer, L., 44, 57, 58, 61, 65, 71, 72, 73, 75, 79 Michels, A., 452, 453, 479, 480 Miedema, A. R., 354, 359, 360, 363, 366, 368, 369, 272, 378, 379, 393, 394, 395 Migdal, A. B., 256, 281, 286, 287 Milano, U . , 417, 428 Miller, P. B., 217, 236, 243, 245, 246. 248, 249, 282, 285, 286 Millett, W . E., 81, 96, 111 Mills, R. L., 117, 120, 122, 123, 124, 125, 126, 127, 128, 129, 136, 150, 151, 259, 281, 287 Mims, W., 411, 412, 428 Misenta, R., 437, 438, 443, 444, 449, 451, 453 Mizushima, M., 452, 453 Modena. I., 71, 79, 139, 140, 141, 151 Montroll, E. W . , 81, 102, I l l Moore, T. W., 292, 293, 331 Morel, P., 117, 132, 150, 259, 282, 287 Morita, M., 343, 346, 363, 377, 392, 393 Morrel, P., 192, 204, 284 Morris, R. J.. 415, 428 Morse, R. W., 172, 209, 219, 220, 221, 283, 284, 285 Moszkowski, S. A., 281, 287 Motchane, J. L., 389, 395 Mott, N. F., 2, 41. 56 Mottelson, B. R., 281, 287 Miihlschlegel, B., 232, 235 Muller, M. W., 420, 423, 428 Murphy, J. C., 404, 428 Nakajima, S., 226, 285 Nakamura, K., 200, 284, 287 Nambu, Y., 171, 282, 283, 285, 287 Nash. F . R., 408, 428 Neal, W . E . J., 441, 453 Nesbitt, 176, 283 Nethercot, Jr., A. H., 249, 286 Netzel, R. G., 87, 111 Newell, G. F., 81, 102, 111 Nier, A. O., 63, 79 Nikitin, L. P., 376, 377, 394
AUTHOR INDEX Novey, T. B., 343, 392 Nyquist, H., 420, 428 Ochsenfeld, R., 175, 176, 283 Offerhaus, M. J., 446, 447, 449, 450, 451, 453 Ogata, 346, 392 Ohm, E. A,, 422, 428 Olsen, T., 209. 219, 220, 285 Onsager, L., 2, 11, 56, 69, 78, 79, 81, 102, 105, 111 Orton, J. W., 374, 394, 400, 404, 427, 428 Osborne, D. V., 10, 35, 56, 56 Osborne, D. W., 119, 123, 129, 131, 132, 135, 139, 151 Otnes, K., 108, 112 Overhauser, A. W., 320, 322,332,400,427 Owen, J., 400, 404, 427, 428 Palevsky, H., 108, 112 Paris, C. R., 388, 389, 395 Parker, G. W., 346, 347, 348, 350, 351, 393, 395 Pauli, W., 336, 337, 338, 392 Pearce, D. C., 87, 111 Pearson, J. B., 95, 111 Pellam, J . R., 31, 56, 99, 111, 145, 152 Penrose, O., 45, 46, 57 Perel, V. I., 226, 285 Perez y Jorba, J., 371, 394 Perkins, A. J.. 478, 480 Perlow, G. J., 375, 394 Pershan, P. S., 405, 408, 411, 428 Peshkov, V. P., 120, 121, 135, 136, 149, 150, 151, 161, 162, 163, 164, 168, 169 Peter, M., 241, 246, 286, 376, 395,402,403, 427 Petersen, R., 375, 376, 377, 394 Petit, P., 461, 462, 464, 465, 480 Phillips, I. C., 312, 332 Phillips, N. E., 208, 243, 285 Pierce, J. R., 424, 428 Pines, D., 171, 187, 191, 192, 204, 253, 254, 281, 283, 284, 285, 286, 287 Pipkin, 385, 387, 395 Pippard, A. B., 91, 94, 100, 111, 176, 178, 179, 180, 237, 241, 242, 243, 245, 247, 249, 262, 269, 280, 283, 284, 286, 292, 328, 331 Pitayevskii, L. P., 12,15, 16,18,31,56,117, 152 Pitt, A., 99, 111
489
Pogorelov, A. V., 309, 331 Pollitzer, F., 455, 480 Pomeranchuk, I., 58, 62, 63, 79, 114, 118, 124, 138, 145, 150 Pontius, R. B., 175, 283 Popov, N. P., 345, 346, 393 Popov, V. S., 343, 346, 392 Portis, A. M., 375, 394 Postma, H.,346, 354, 359, 360, 362, 363, 364, 372, 378, 379, 393, 394, 395 Poulis, N. J., 378, 395 Pound, R. V., 139,151,397, 413,420,423, 428 Powell, R. L., 71, 79, 292 Poynting, J. H., 454, 480 Preston, R. S., 375, 394 Primakoff, H., 118, 124, 126, 150 Proctor, W. G., 388, 395 Prokhorov, A. M., 397, 400, 403, 427 Prozorova, A., 252, 286 Pryce, M. H. L., 349, 350, 351, 393 Ptukha, T. P., 120, 150 PurcelI, E. M., 139, 151, 397, 398, 427 Pursey, D. L., 337, 338, 392 Quinn, J. J., 256, 286 Rabi, I. I., 398, 427 Racah, G., 334, 391 Ramsey, N. F., 398, 427 Rasmussen, J. O., 371, 395 Redfield, A. G., 173, 222,223,283,404,428 Reich, H. A., 136, 138, 139, 141, 142, 143, 151, 152, 160, 162, 163, 168 Reif, F., 58, 61,65. 71, 72, 73, 75, 79, 173, 261, 283 Reinov, N. M., 376, 377, 394 Retherford, R. C., 397, 427 Reuss, J., 58, 63, 71, 79, 478, 480 Reuter, G. E., 179, 242, 299, 328, 331 Reynolds, C. A,, 95, 111, 176, 283 Richards, P. L., 172, 195, 218, 236, 237, 238, 239, 243, 251. 258, 282, 283 Richardson, M. J., 455, 463, 478, 479, 480 Rickayzen, G., 171, 202, 225, 226, 253, 257, 272, 283, 284, 285, 287 Rietveld, A. 0.. 437, 438, 439, 441, 449, 450, 451, 453 Rimai, L., 404, 428 Ringo, G. R., 343, 392 Roberts, L. D., 346, 347, 348, 350, 351, 393, 395
490
AUTHOR INDEX
Roberts, T. R., 155, 156, 161, 165, 166, 168, 169 Robin, S., 478, 480 Robinson, F. N. H., 374, 389, 394, 395 Robinson, W. K., 102, 112 Rodriguez, S., 312, 332 Rogers, K. T., 268, 286 Romer, R. H., 139, 140, 141, 151 Roos, B. W., 446, 447, 449, 450, 451, 453 Rorschach, H. E.. 139, 141, 149, 151, 152 Rose, M. E., 334, 346, 391, 393 Rosenberg, H. M., 274 Rosenhlum, B., 292, 293, 331 Rosenhlum, F., 289, 330 Rosenfeld, L., 334, 391 Roth, L. M., 292, 331 Rowlinson, J. S., 455, 463, 478, 479, 480 Rudik, A. P., 346, 392 Rutgers, A. J., 175, 283 Sakai, 346, 392 Sakurai, J. J., 335, 337, 338, 392 Salam, A, , 337, 392 Samoilov, B. N., 243, 376, 394 Santini, F., 139, 140, 141, 151 Santini, M., 139, 140, 141, 151 Sapp, R . C., 367, 395 Sarachik, M. P., 241, 246, 282, 285 Saris, B. F., 44, 57 , Satterthwaite, C. B., 176, 208, 243, 272, 282, 284, 287 Saunders, J., 417, 428 Sauter, F., 178 Sawada. K.. 254, 286 Scaramuzzi, F., 58, 59, 63, 64, 66, 71, 74, 79, 79 Schafer, K., 443, 453 Schafroth, M. R., 81, 111, 183, 184, 201, 284 Schawlow, A. L., 241, 243, 244, 246, 285, 286 Scheffer, F. E. C., 478, 480 Schmeissner, F., 437, 438, 453 Schneider, W. G., 115, 150 Schooley, J. F., 371, 374, 395 Schrieffer, J . R., 117, 150, 171, 187, 196, 213, 216, 224, 253, 256, 258, 259, 262, 282, 285, 286 Schuch, A. F., 127, 151 Schulz-duBois, E. 0.. 402, 403, 418, 427, 428 Schwettman, H. A , , 141, 152
Schwinger, J., 336, 392, 398, 427 Scott, P. L., 389, 395 Scott, R. B., 430, 435, 453 Scovil, H. E. D., 404, 412, 422, 428 Scurlock, R. G., 355, 370, 371, 373, 393, 394 Seidel, G., 135, 157, 161, 168, 208, 243, 285 Seidel, H., 404, 412, 428 Seitz, F., 374, 394 Senitzky, I. R., 420, 428 Series, G. W., 374, 394 Serin, B., 176, 243, 275, 276, 277, 282, 283, 287 Sessler, A . M . , 117, 132, 150, 259, 281, 287 Severiens, J. C., 366, 368, 369, 393, 394 Shalnikov, A. I., 176, 267, 283, 286 Shankland. D. G., 281, 287 Shapiro, S., 405, 408, 409, 410, 411, 428 Shaw, B., 461, 480 Shaw, R. W., 210, 243, 282, 285 Sherman, R. H., 120, 121, 122, 123, 125, 126, 132, 150, 151 Shirkov, D. V., 171, 203, 214, 253, 256, 282 Shirley, D. A,, 371, 372, 374, 394, 395 Shockley, W., 291, 292, 330 Shoenberg, D., 175, 240, 243, 283 Siegman, A. E.. 411, 412, 428 Simon, F. E., 374, 394 Singer, J. R., 400, 427 Sklyarevskii, V. V., 376, 394 Skyrme, T. H. R., 335, 369, 391 Sladek, R. J., 271, 287 Slater, J. C., 181, 284 Slichter, C. P., 172, 217, 222, 283 Soda, T., 117, 132, 150. 259, 282, 287 Soest, G. van, 455, 463, 464, 465, 478, 479, 480 Sokolov, I. A., 376, 377, 394 Solomon, I., 388, 395 Sondheimer, E. H., 179,242,299,328,331 Sorby, H. C., 454, 480 Sowter, C. V., 355, 359, 369, 393, 394 Spedding, F. H., 277, 287 Spees, A. H., 95, 111 Spiewak, M., 249, 252, 286 Spohr. D. A., 374, 394 Squire, C. F., 99, 111, 145, 152 Sreedhar, A. K., 130, 131, 132, 151 Stacey, F. D., 76, 79 Stasior, R. A., 143, 152
AUTHOR INDEX Statz, H., 404, 428 Stech, B., 339, 343, 346, 382 Steele, W . A , , 37, 42, 57 Steenland, M. J., 333, 350, 354, 359, 360, 366, 368, 369, 379, 380, 391, 393, 394 Steffen, R. M., 334, 391 Stehl, O., 443, 444, 449, 451, 453 Stelmah, M. F., 376, 377, 394 Stepanov, E. P., 376, 394 Stevens, K. H. W., 400, 427 Stevenson, M. J., 426, 429 Stolovy, A , , 374, 395 Strandberg, M. W. P., 404, 408, 415, 420, 423, 428 Strebel, E., 455, 480 Sudarshan, E. C. G., 338, 392 Suhl, H., 277, 278, 279, 281, 287 Sunyar, A. W., 342, 392 Swenker, M. D. P., 455,463, 464, 465, 466, 478, 479, 480 Swihart, J . C., 203, 284 Sydoriak, S. G., 119, 120, 121, 123, 124, 125, 129, 130, 131, 132, 136, 143, 144, 150, 151, 155, 156, 161, 165, 166, 168, 169 Symmons, H. P., 403, 427
491
427, 428 Townsend, 71, 79 Treiman, S. B., 339, 340, 345, 346, 392, 293 Trenam, R. S., 367, 395 Tsakadze, D. S . , 25, 57 Tserkovnikov, Iu. A., 171, 202, 203, 214. 256, 282, 284 Tsuneto, T., 219, 240, 258, 282, 285 Tucker, E. B., 404, 409, 428 Turnbull, D., 374, 394 Tyablikov, S. V., 171, 203, 214, 256, 282 Tyndal, 71, 79 Ubbink, J. B., 436, 437, 453 Uebersfeld, J.. 389, 395 Uhlenbeck, G. E., 420 Usui, T., 115, 152 Valatin, J. G., 171, 200, 214, 283 Van den Berg, G. J., 99, 111, 143, 152, 441, 450, 451, 453 Van Itterbeek, A., 432, 435, 437, 438, 439, 441, 449, 450, 451, 453 Van Leeuwen, J. H., 181, 284 Van Leeuwen, J. M. J., 446, 447, 449, 450, 451, 453 Van Vleck, J . H., 405, 428 Varekamp, F. H., 432, 434, 435, 436, 443, 444, 453 Velds, C. A., 437, 438, 439, 449, 453 Verschoyle, T. T. H., 455, 460, 479, 480 Versteegen, W. A. J., 463, 468, 480 Vincent, D. H., 375, 394 Vinen, W. F., 10, 12, 15, 16, 17, 20, 25, 32, 33, 37, 39, 40, 41, 43, 44, 46, 49, 50, 51, 52, 56, 57, 66, 69, 70, 79, 183, 284 Vinokurov, V. M., 403, 428 Vodar, B., 478, 480 Vries, G. de, 129, 131, 132, 151
Taconis, K. W., 158, 166, 167 Taylor, R. D., 81, 93, 96, 97, 98, 99, 111, 120, 122, 123, 135, 139, 150, 151 Taylor, R. T., 373, 394 Telegdi, V. L., 343, 392 Ten Seldam, C. H., 452, 453 Terhune, R. W., 413, 428 Ter-Martirosyan, K. A , , 346, 392 Tewordt, L., 272, 287 Thomas, J. J., 426, 429 Thomson. J . C., 58, 59, 63, 64, 66, 71, 74, 79 Thomson, Sir W., 18, 19, 56 Thurmond, C. D., 398, 426, 427 Tinkham, M., 172, 195, 218, 227, 228, 229, Wagner, P. E., 398, 427 236, 237, 238, 239, 243, 251, 258, 282, Wagoner, G., 292, 293, 331 283, 285 Waldmann, L., 443, 453 Tisza, L., 81, 111 Walker, E. J., 122, 123, 133, 136, 137, 151 Tolhoek, H. A., 333,334,335,345,346,350, Wallmann, J. C., 373, 395 354, 359, 360, 379, 380, 391, 392, 393 Walmsley, R. H., 10, 45, 56, 57 Tolmachev, V. V., 171, 203, 214, 253, 256, Walsh, Jr.. W. M., 292, 331 282 Walter, F. J., 347, 351, 393, 395 Tomita, K., 139, 152 Walters, G. K., 124, 139, 151 Torrey, H. C . , 139, 142, 152, 329, 332 Wannier, G. H., 73, 79 Townes, C. H., 374, 394, 397, 408, 417, Ward, D. E., 461, 480
492
AUTHOR INDEX
Wasscher, J. D., 86, 87, 111 Wassenaar. T., 479, 480 Watson, R. E., 377, 395 Weber, J., 397, 400, 402, 404, 420, 423, 427, 428 Webster, T. J., 461, 480 Weinstock, B., 119, 123, 129, 131, 132, 135, 139, 150, 151 Weiss, P. R., 231, 277, 285, 287 Welker, H., 195, 284 Welton, T. A , , 420, 428 Wentzel, G., 207, 254, 284, 285, 286 Wernick, J. H., 376, 395 Wertheim, G. K., 376, 395 Wexler, A., 208, 284 Wheatley, J. C., 136, 137, 138, 142, 149, 151, 152, 369, 393, 394 Wheeler, R. G., 13, 56 Wheeler, J. A., 347, 393 White, D., 208, 284, 432, 434, 435, 453 Wilhelm, J. O., 99, 111 Williams, R. L., 58, 60,61. 69, 71, 75, 76, 79, 426, 429 Williams, S. R., 139, 151 Wilks, J., 87, 111, 134, 151 Winkel, P., 37, 56, 57 Winther, A,. 339, 343, 346, 392 Wolga, 408, 428
Woods, A. D. B., 108, 112 Woolley, H. W., 430, 435, 453 Wright, H. W., 176, 283, 347, 393 Wu, C . S . , 335, 342, 343, 352, 354, 359, 391, 393 Wyld, H. W., 339, 340, 345, 346, 392, 393 Yafaev, 403, 428 Yager, W. A,, 292, 331 Yang, C . N., 335, 337, 339, 342, 346, 391, 392 Yarnell, J. L., 108, 112 Yntema, J. L., 115, 150 Yosida, K.. 261, 287 Zachariasen, W. H., 277, 278, 279, 287 Zandbergen, P., 438, 449, 453 Zaripov, 403, 428 Zavaritskii, N. V., 208, 272, 285, 287 Zeiger, H. J., 289, 292, 330, 331, 397, 427 Zeldovitch, J. B., 338, 345, 392 Zemansky, M. W., 194, 207, 284 Zinov’eva, K. N., 135, 136, 139, 149, 151, 160,161,162,163,184,168,169,461,480 Zubarev, D. N., 171, 202, 203, 214, 256, 282, 284 Zucker, M., 243, 275, 277, 287
SUBJECT I N D E X Absorption, infrared 236, 251 - of microwaves 248. 322 adiabatic rapid passage 398 amagat units 431 anisotropy of a-radiation 347 - of ,%radiation 352ff, 357 - of y-radiation 355 antiferromagnetism 374, 377 atomic clock 397 attenuation of sound 12, 145 - of ultrasound 218 Backflow 186 binary systems, equilibrium of 456ff, 463ff Bose-Einstein condensation 201 boundary energy 180, 268 Calorimeter 83, 150 clusters 105 coherence properties 172, 179, 199, 212ff compressibility of the He isotopes 89, 92 121, 128 conductivity in superconductors 233 correlation distance 117 - in p-y-emission 364 CPT-theorem 336 critical current 267 - field 177, 209 - velocity 37, 47 cryostat 153, 165 cyclotron resonance 289ff Damping in metals 312 density of 4He 96 diamagnetism 181 diffusion coefficient 439, 450 Electron orbits 291, 296, 313 - pairs 195ff - phonon interaction 191 energy gap 108, 171ff entropy of 3He 132 equation of state 430ff
equilibrium curves 459ff - in solid-gas systems 454ff excitations, collective 252ff -, elementary 3, 185ff, 204, 252 Fermi-Gamow-Teller interference 369 Fermi gas 115 Fermi liquid properties 113ff. 310 ferromagnetism 375 filmflow 153, 166 flow, irrotational 4 -, non turbulent 03 -, superfluid 36 -, turbulent 66, 68 friction in 12, 17
357.
Groundstate of liquid *He 2 - of superconductors 195ff Heat flow 43, 63ff helium ions 58ff -, structure of 59 We, liquid 113ff -, solid 118, 142 Impedance of metals 317 infrared maser 424 - transmission and absorption 236, 251 interaction between elementary excitations 189ff, 271, 408 - between vortex lines 20 -between vortex lines and rotons 18 -, weak 335, 342 invariance for space inversion 341 - for time reversal 343, 364 ion clusters in helium 59 ionic currents in helium 60 - mobility 62ff ionization 75 Ising model 81, 102 isotope effect 177, 203
494
SUBJECT INDEX
Knight shift 261 Lambda transition 8,80ff, 94, 163
-, thermodynamics of 89ff lepton conservation 337 life-time of excitations 204 line shape function 401, 407 Magnus effect 9ff maser 396ff - circuits 414ff - materials 402ff -, three level 399, 404 -, transient 398 -, travelling wave 418 Meissner effect 171, 224 melting curve of *He 123 mixing excess measurements 435 mixtures of SHe and 4He 162 -, analysis of 461 mobility of ions 62ff momentum ordering 104 mutual friction in 4He 12, 17 Neutrino 336, 359 nuclear deformation 351 - magnetic moment 366 - magnetic susceptibility 118, 124, 148 - spin relaxation 138, 172, 222 - spin system 116 - spin, its influence on other properties 442ff noise 420ff Ordering, long range 182 orientation of nuclei 333ff, 356, 373ff Overhauser effect 320, 382, 388, 400 Pair correlations 116, 195ff - excitations 200 - interactions 191, 204, 281 paramagnetic relaxation 405 - resonance 319ff, 374, 389, 400 parametric amplifier 424 parity, non-conservation of 335ff penetration depth 240, 313, 325 persistent currents 181, 263 phase diagram 127, 454ff phonon emission 408 phonon-phonon interaction 408 plasmons 188, 252 polarization of p-radiation 345, 36i
- of y-radiation 361 positon emission 354 proton polarization 389 PVT relations 119, 148, 432 Quantization of circulation 5, 25ff quantum oscillations 308 quasi-particles 115, 184ff, 197, 291, 310 Radiation, a- 333, 346
-, 8- 335,339 -, y-333, 365
recombination 75ff refrigerator 160ff relaxation, cross- 406ff -, spin-lattice 138, 405 resonance, double 385 - saturation 374, 389, 413ff resonator 13, 43 rotating superfluid 10 Second sound resonator 43 selfdiffusion in sHe 136 sheets, transmission through 329 skin depth 289 skin impedance 174, 179, 246, 258, 298ff. 309 solubility of solids in gases 454ff sound attenuation 12, 145 - resonator 13 - velocity 9Off, 143, 165 specific heat of antiferromagnetics 103 -, electronic 207, 211 - of SHe and "e 82ff, 129, 148, 158 spin relaxation time 321 superconducting alloys 275 superconductivity 109, 170ff superconductors, electromagnetic properties of 178ff, 224e superfluid flow 36 - turbulence 43, 49 - behaviour of fermions 116 susceptibility, electron spin 261 Thermal conductivity 134, 173, 270, 436. ' 448 - diffusion 440, 450 - expansion coefficient 89ff, 120ff time of free flight 295 transition, co-operative 81, 103 - probability 212ff. 396ff
SUBJECT I N D E X
transport properties 134ff, 149, 430ff turbulence 43,49,66 - threshold 66ff two fluid model 183, 263
495
-.
Ultrasound 409 attenuation of 218
virial coefficient, second 431ff 443ff, 478 viscosimeter 161 viscosity of SHe 134, 161, 43713 vortex line 6ff - sheet 7 -waves 18, 22, 31 vorticity in liquid He I1 Iff. 69
Vapour pressure 119, 154 vibrating wire experiments 25ff
Zero point energy 55 zero sound 114, 146
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