PROGRESS I N LOW TEMPERATURE PHYSICS
IV
CONTENTS O F V O L U M E S 1-111
VOLUME I
c. J. GORTER, The two fluid model for superconductors and helium I1 (16 pages) R. P. FEYNMAN,
A.
Application of quantum mechanics to liquid helium (37 pages)
Rayleigh disks in liquid helium I1 (10 pages)
J. R. PELLAM,
c. HOLLIS HALLET, Oscillating disks and rotating cylinders in liquid helium I1 (14 pages)
E. F.
HAMMEL, The low temperature properties of helium three (30 pages)
J. I. M. BEENAKKER and K. B. SERIN,
c. F.
w. TACONIS, Liquid mixtures of helium three and four (30 pages)
The magnetic threshold curve of superconductors (13 pages) The effect of pressure and of stress on superconductivity (8 pages)
SQUIRE,
T. E. FABER and A. B. PIPPARD, K.
The electronic specific heats in metals (22 pages)
J. G. DAUNT, A. H.
COOKE, Paramagnetic crystals in use for low temperature research (21 pages)
N. J. POULIS
and c. I. GORTER, Antiferromagnetic crystals (28 pages)
D. DE KLERK
L.
Kinetics of the phase transition insuperconductors (25 pages)
MENDELSSOHN, Heat conduction in superconductors (18 pages)
and
M. J.
STEENLAND, Adiabatic demagnetization (63 pages)
NBEL, Theoretical remarks on ferromagnetism at low temperatures (8 pages)
L. WEE,
Experimental research on ferromagnetism at very low temperatures (1 1 pages)
A. VAN ITTERBEEK,
I. DE BOER
Velocity and attenuation of sound at low temperatures (26 pages)
Transport properties of gaseous helium at low temperatures (26 pages)
VOLUME I1
Quantum effects and exchange effects on the thermodynamic properties of liquid helium (58 pages)
J. DE BOER,
H.
c. KRAMERS, Liquid helium below 1 "K (24 pages) and D. H. N. WANSINK, Transport phenomena of liquid helium I1 in slits and capillaries (22 pages)
P. WINKEL
K. R. ATKINS, B. T.
Helium films (33 pages)
MATTHIAS, Superconductivity in the periodic system (13 pages)
C O N T E N T S O F V O L U M E S 1-111
VOLUME I1 (continued)
E. H.
SONDHEIMER, Electron transport phenomena in metals (36 pages)
v.
A. JOHNSON
D.
SHOENBERG, The De Haas-van Alphen effect (40 pages)
and
K . LARK-HOROVITZ,
Semiconductors at low temperatures (39 pages)
c. J. GORTER, Paramagnetic relaxation (26 pages) M. J. STEENLAND and H. A.
TOLHOEK, Orientation of atomic nuclei at low temperatures
(46 pages)
c. DOMB and J. s. DUGDALE, Solid helium (30 pages) F.
H. SPEDDING, s. LEGVOLD,A. H. DAANE and the rare earth metals (27 pages)
D.
BIJL,The representation of specific heat and thermal expansion data of simple solids (36 pages)
H. VAN DIJK
L. L). JENNINGS, Some
physical properties of
and M.DURIEUX, The temperature scale in the liquid helium region (34 pages)
VOLUME I11
w. F. G.
VINEN,
Vortex lines in liquid helium 11 (57 pages)
CARER!,Helium ions in liquid helium I1 (22 pages)
M. J. BUCKINGHAM
and w.
M.
FAIRBANK, The nature of the %-transition in liquid helium
(33 pages) E. R. GRILLY K.
and
E. F.
HAMMEL, Liquid and solid 3He (40 pages)
w. TACONIS,3He cryostats (17 pages)
J. BARDEEN and J. R.
M. YA. AZBEL’
w. I.
and
and (63 pages)
HUISKAMP
SCHRIEFFER, Recent developments in superconductivity (118 pages)
I. M. LIFSHITZ, Electron H. A. TOLHOEK,
N. BLOEMBERGEN, Solid J. J. M.
resonances in metals (45 pages)
Orientation of atomic nuclei at low temperatures I1
state masers (34 pages)
BEENAKKER, The equation of state and the transport properties of the hydrogenic molecules (24 pages)
z. DOKOUPL, Some solid-gas equilibria at low temperatures (27 pages)
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P R O G R E S S I N LOW TEMPERATURE PHYSICS EDITED B Y
C.J. G O R T E R Professor of Experimental Physics Director of the Kamerlingh Onnes Laboratory, Leiden
VOLUME IV
1964 N O R T H - H O L L A N D PUBLISHING C O M P A N Y AMSTERDAM
0 1964
NORTH-HOLLAND PUBLISHING COMPANY
-
AMSTERDAM
No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher
PUBLISHERS:
NORTH-HOLLAND PUBLISHING CO. - AMSTERDAM SOLE DISTRIBUTORS FOR U.S.A.: INTERSCIENCE PUBLISHERS, A DIVISION OF
JOHN WILEY & SONS, INC. - NEW YORK
PRINTED I N THE NETHERLANDS
PREFACE T O FOURTH VOLUME
Since this series was started in 1954 a new volume has come out about every third year. The number of pages per volume has increased slowly but surely. While the number of chapters per volume has decreased considerably, their average length has more than doubled. Longer contributions have been invited deliberately but an increasing number of authors have considerably exceeded the space reserved for them. It would seem that, if the character of the series is to be preserved, the average chapter should not grow much longer. The chapters devoted to liquid helium, to metals including superconductors, and to magnetism together with nuclear orientation, each generally take up about 30 percent of the total space. Other subjects, such as molecular physics, thermometry and semiconductors complete the remaining 10 percent. In the present volume liquid helium occupies less space than in the preceding ones. This was accidental; one main contribution to this volume was invited but has not been forthcoming. I feel that after 56 years, liquid helium remains the most enigmatic and, at the same time, one of the most investigated substances; its large share of space in the series is justified. It has often been stated that low temperature physics is overlapping more and more with other fields of research. This increases the interest in review papers on low temperature problems. It happens repeatedly that an invitation to write a review paper is refused because the prospective author is writing or has been engaged to write a review in another series. 1 feel, however, that a collection of expert reviews gathered into an anthology on low temperature physics remains of great interest to many. It is up to the reader to judge. C . J. GORTER
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CONTENTS
Chapter I V. P. PESHKOV, CRITICAL VELOCITIES
AND VORTICES IN SUPERFLUID HELIUM
Page 1
1. Critical velocities in narrow channels, slits and films, 1 . - 2. Critical velocities in wide channels, 11. - 3. Oscillating discs and spheres, 20 - 4. Persistent currents, 23. - 5. Summary of experimental data, 24. - 6. The Landau criterion and excitations in the superfluid, 26. - 7. Quantised vortex lines, 29. - 8. Concluding remarks, 35. PROPERTIES I1 K. W. TACONIS and R. DE BRUYN OUBOTER, EQUILIBRIUM OF LIQUID A N D SOLID MIXTURESOFHELIUM THREE AND FOUR
.
.. . . .. .
1. Introduction, 38. - 2. The equilibrium between vapour and liquid mixtures (dew- and boiling-curve), 49. - 2.1. General survey of the experimental data, 49. - 2.2. Calculation of the excess chemical potentials and the excess Gibbs function, 51. - 2.3. The equilibrium between vapour and a dilute liquid mixture of 3He in 4He 11; equilibrium with respect to the solute, 57. - 3. The equilibrium between the He I and the He I1 phase (Mine) and between two liquid mixtures in the stratification region (the phase separation diagram), 59. - 3.1. The lambda transition, 59. - 3.2. The Keesom-Ehrenfest relations for a liquid mixture and the discontinuity in the slope of the first order equilibrium curves at the junction with the second order lambda-curve, 61. - 3.3. The phase separation diagram, 64. - 4. The osmotic equilibrium in He I1 (pseudo thermostatic equilibrium) and some applications, 72. - 4.1. The osmosis in He I1 derived from the equilibrium between pure liquid 4He on one side and a mixture on the other side of a superleak (semipermeable wall); equilibrium with respect to the solvent, 72. - 4.2. The quasi-equilibrium between the osmotic and fountain force in the liquid mixture, when a heat current is present (the heat flush effect), yielding the diffusion coefficient and the heat conductivity of the mixtures, 74. - 4.3. The quasi-equilibrium between two mixtures of slightly different concentration in the He I1 region separated by a capillary, yielding the viscosity of the mixtures, 77. - 4.4. The quasiequilibrium between two liquid mixtures of different concentration in the He I1 region, connected by the helium surface film, yieldingthe isothermal flow rate of these films, 80. - 4.5. A refrigeration cycle, as an application of the osmotic pressure and the heat of mixing, 81. - 5. The equilibrium between liquid and solid mixtures (freezing- and meltingcurve) and the phase-separation of solid mixtures, 84. - 5.1. The freezing pressures of 3He-4He mixtures, 84. - 5.2. The equilibrium between two solid mixtures in the phase separation region, 87. - 5.3. The minima in the freezing curves of 3He-4Hemixtures and the minima in the melting curves of pure 3He and pure 4He, 90.
38
X
CONTENTS
I11 D. H. DOUGLASS, Jr. and L. M. FALICOV, THESUPERCONDUCTING ENERGY GAP.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
1. Historical introduction, 97. - 2. The physical significance of an energy gap, 99. - 3. The theory of the superconducting energy gap, 102. - 3.1. Derivation of the energy gap equation, 102. - 3.2. Elementary excitations and density of states, 107. - 3.3. Temperature dependence of the energy gap, 111. - 3.4. Mechanisms for the effective electron interaction and solutions of the energy gap equation, 113. - 3.5. Magnetic field dependence of the gap, 123. - 3.6. Dependence of the energy gap on impurities and spatial inhomogeneities, 135. - 4. Theory of superconductive tunnelling, 140. - 4.1. Semiphenomenological theory of electron tunnelling in superconductors, 140. - 4.2. Microscopic theory of electron tunnelling in superconductors, 146. - 5. Experimental determinations of the superconducting energy gap, 153. - 5.1. Specific heat and thermal conductivity, 153. - 5.2. Photon excitations across the gap, 157. - 5.3. Energy gap acoustic attenuation measurements, 167. - 5.4. Energy gap from electron tunnelling experiments, 172.
IV G. J. VAN DEN BERG, ANOMALIES IN DILUTE SITION ELEMENTS
METALLIC SOLUTIONS OF TRAN-
. . . . . . . . . . . . . . . . . . . . . . . . . . .
194
1. Historical remarks, 194. - 2. Electrical resistance and magnetoresistance, 198. - 2.1. “Diluted” alloys of transition metals of the first long period, 200. - 2.2. “Diluted” alloys of transition metals of the second long period, 208. - 2.3. “Diluted” alloys of transition metals of the third long period, 209. - 2.4. Transition metals also as solvents, 210. - 3. Thermal resistance, also in a magnetic field, 211. - 4. Thermoelectric power, 215. - 4.1. Normal metals as solvent, 216. - 4.2. Transition metals as solvent, 221.-5. Magnetic properties, 222. - 5.1. The magnetic susceptibility, 222. - 5.2. Electron spin resonance, 227. - 5.3. Nuclear magnetic resonance; Knight shift, 228. - 5.4. The De Haas-Van Alphen effect, 230. - 5.5. Magnetic remanence, 231. - 6. Hall effect, 233. - 6.1. Noblemetal based alloys, 233. - 6.2. Transition-metal based alloys, 237. - 7. Specific heat, 238. - 8. Optical properties, 244. - 9. Theoretical considerations, 245. - 9.1. Resonance hypothesis, 245. - 9.2. Molecular field model, 246. - 9.3. Criticism of the molecular field treatment, 251. - 9.4. The “ion pair and isolated ions” treatments, 251. - 9.5. The virtual bound state treatment, 253. - 9.6. Conclusion, 259. STRUCTURES OF HEAVY RARE-EARTH METALS.. . . 265 V KEI YOSIDA, MAGNETIC 1. Introduction, 265. - 2. Survey of experimental results, 268. - 3. Theoretical consideration, 275. - 4. Relation between the Fermi surface and the screw structure, 285. - 5. Summary, 292.
TRANSITIONS . . . . . . . . 296 VI C. DOMB and A. R. MIEDEMA, MAGNETIC I . Introduction, 296. - 2. The king model, 299. - 2.1. General remarks. Thermodynamic properties, 299. - 2.2. Magnetic properties, 302. - 3. The Heisenberg model, 304. - 3.1. General remarks. Thermodynamic properties, 304. - 3.2. Magnetic properties, 305. - 4. Remarks on the analysis of experimental data, 307. - 4.1. Thermal data, 307. - 4.2. The exchange constant, 309. - 5. Experimental data, 310. - 5.1. Ferromagnets, 310. 5.2. The rare earth metals, 318. - 5.3. Combined ferro- and antiferro-
XI
CONTENTS
magnetism, 319. - 5.4. Antiferrornagnetism, 322. - 5.5. Layer type antiferromagnets, 325. - 5.6. Antiferromagnetism in special lattices, 326. 6. Comparison between experiment and theory, 333. - 6.1. Ferromagnets, 333. - 6.2. Cobalt tutton salts, 335. - 6.3. Antiferromagnets, 336. - 7. Conclusions, 339. VII L. NEEL, R. PAUTHENET and B. DREYFUS, THE RARE EARTH GARNETS 344 ? 1. Introduction, 344. - 2. The magnetic properties of the rare earth garnets, 346. - 2.1. Preparation of the rare earth garnet-type ferrites, 346. - 2.2. Crystal structure, 348. - 2.3. Magnetostatic properties, 349. 2.4. Magnetic interactions, 353. - 2.5. Interpretation of experimental results, 354. - 2.6. Temperature variation of the inverse of the paramagnetic susceptibility above the Curie point and the spontaneous magnetisation of yttrium ferrite, 358. - 2.7. Magnetic interactions between rare earth ions, 361. - 2.8. Thermal variation of the spontaneous magnetization and of the inverse of the paramagnetic susceptibility in the rare earth ferrites, 361. - 2.9. The rare earth gallates, 362. - 2.10. Europium and samarium ferrites, 365. - 2.11. Substitutions in the rare earth ferrites, 366. - 3. The levels of rare earth ions in the garnets, 369. - 3. I . Crystalline fields, 369. - 3.2. Specific heats, 370. - 3.3. Thermal conduction, 373. - 3.4. Spectroscopic investigations, 373. - 3.5. The visible and near infra-red spectrum, 374. - 3.6. The far infra-red spectrum, 376. - 3.7. Inelastic scattering of neutrons, 378. - 3.8. Giant anisotropy, 379. - 3.9. An example of the “reconstruction” of a garnet, 381. VIII A. ABRAGAM and M. BORCHINI, DYNAMIC POLARIZATION TARGETS.
. . . . . .
. .
. . . . . .
. .
. .
. . .
OF NUCLEAR
. . . . .
. . . 384
Introduction, 384. - 1. Dynamic polarization at low temperatures, 385. 1.1. Electronic and nuclear paramagnetism: generalities, 385. - I .2. Dynamic polarization: generalities, 395. - 2. Spin temperature theories of dynamic polarization, 400. - 2.1. Limit of very strong r.f. fields. Homogeneous spin systems, 401. - 2.2. Arbitrary r.f. field strengths. Homogeneous spin systems, 405. - 2.3. Homogeneous spin systems with nuclear spin diffusion. Solid effect theory, 41 1 . - 2.4. Relaxation and polarization by unlike electronic spins in the case of nuclear spin diffusion (leakage), 412. - 2.5. Inhomogeneous electronic spin systems, 414. - 3. Experimental arrangements and results, 415. - 3.1. Experimental results, 415. - 3.2. Laboratory apparatus and large targets, 426. - 3.3. Thin targets, 433. 4. Future developments, 440. - 4.1. Nuclei other than protons, 440. - 4.2. Target size and non-resonant dynamic methods, 441. - 4.3. Target materials, 445.
IX J. G. COLLINS and G. K. WHITE, THERMAL EXPANSION OF SOLIDS . . . . 450 1. Introduction, 450. - 2. Theory, 451. - 2.1. The Griineisen y, 451. - 2.2. Theoretical models, 453. - 2.3. Anisotropic materials, 455. - 2.4. Electronic and magnetic contributions to the thermal expansion, 456. - 3. Experimental methods, 457. - 3.1. Experimental techniques, 457. - 3.2. Analysis, 458. - 4. Dielectric solids, 460. - 4.1. Ionic solids, 460. - 4.2. Discussion of ionic solids, 463. - 4.3. Inert-gas solids, 464. - 5. Metals, 464. - 5.1. Isotropic elements, 464. - 5.2. Anisotropic metals, 467. - 5.3. Unusual metals, 469. - 6. Glasses and diamond-structure solids, 471, 6.1. Glasses, 471. - 6.2. Diamond-structure solids, 472. - 6.3. Ice, 473. - 7. Superconductors, 473. - 8. Summary, 476.
XII
CONTENTS
X T. R. ROBERTS, R. H. SHERMAN, S. G. SYDORIAK and F. G. BRICK-
WEDDE, THE1962 3He SCALE OF TEMPERATURES. . . . . . . . . . . . . 480 1. Introduction, 480. - 2. Brief historical review of earlier determinations of the 3He vapor pressure-temperature relation, 482. - 3. Plan for development of the 1962 3He scale, 488. - 4. The 1961 L.A.S.L. intercomparisons of the 3He and 4He vapor pressures, 489. - 5. The determination of the critical pressure and temperature of 3He, 491. - 6. Calculation of a vapor-pressure equation below 2 "K using a theoretical equation from statistical mechanics, 492. - 7. Extension of the vaporpressure equation to the critical point, 493. - 8. The 1962 3He vaporpressure scale, 494. - 9. An evaluation of the 1962 3He scale, 495. - 9.1. Fit of the input 1961 L.A.S.L. vapor-pressure data, 495. - 9.2. Fit of the experimental thermodynamic equation (ETE) scale, 496. - 9.3. Fit of the Argonne Laboratory vapor-pressure data, 499. - 9.4. Fit of the heat-ofvaporization data, 500. - 9.5. Fit of gas thermometer, isotherm and acoustic interferometer measurements, 503. - 10. Thermodynamic properties of 3He consistent with the 1962 3He scale, 505. - 11. Corrections to the measured pressure of a 3He vapor-pressure thermometer, 509. 11.1. Correction for the 4He impurity in 3He, 509. 11.2. Correction for the thermomolecular pressure ratio, 510. - 11.3. Hydrostatic pressure corrections, 511. - 12. Conclusion and considerations for the future, 51 1.
-
........................
515
.........................
527
AUTHORINDEX. SUBJECT INDEX
CHAPTER I
CRITICAL VELOCITIES AND VORTICES IN SUPERFLUID HELIUM BY
V. P. PESHKOV
INSTITUTE FOR PHYSICAL PROBLEMS, Moscow
CONTENTS: 1. Critical velocities in narrow channels, slits and films, 1. - 2. Critical velocities in wide channels, 11. - 3. Oscillating discs and spheres, 20. - 4. Persistent currents, 23. - 5. Summary of experimental data, 24. - 6. The Landau criterion and excitations in the superfluid, 26. - 7. Quantised vortex lines, 29. - 8. Concluding remarks, 35.
1. Critical Velocities in Narrow Channels, Slits and Films Immediately after the discovery of the superfluidity of helium, arose the question of critical velocities. For the first time Kapitzal and Allen and Misenerz found that below the I-point helium becomes superfluid, i.e. it flows through narrow slits and capillaries without experiencing any frictional force. However, the first detailed experiments of Kapitza already showed, that the reversible superfluid flow of helium through the slit occurs only at velocities not exceeding some critical velocity. Kapitza immersed a vacuum-jacketed vessel in liquid helium. This vessel communicated with a helium bath via an annular channel formed by two optically polished quartz discs of 3 cm diameter. The disc which communicated with the inner vessel had a central orifice of 1 cm diameter. The outer solid disc was pressed to the inside one by a special device, so that it was possible to set the annular slit thickness at will and to observe the interference fringes to determine its dimensions. The dependence of the volume rate of helium flow on heat input to the vessel (full curve) and temperature difference (dotted curve) are shown on Fig. 1. The transfer took place owing to the thermomechanical effect at level differences less than 3 mm. As can be seen from the figure, the volume velocity rises linearly with the heat input up to 4 x 10-3 cm3/sec at a temperReferences p . 36
2
V. P. PESKOV
[CH.I,
81
ature of 1.935" K and average annular slit width 0 =0.14 p, without any temperature difference between the vessels. At larger heat inputs temperature differences appear quite distinctly, while the velocity rises at a less than linear rate. Having carefully analysed his experiments, Kapitza came to the conclusion that, taking the average value of the slit thickness, the critical velocity at which breakdown of reversibility of the process begins increases with decreasing thickness of the slit. Thus at 1.64" K and slit width 3 p the critical velocity is 14 cm/sec, but when the slit width is only 0.3 p it becomes 40 cmlsec at the same temperature. Kapitza did not observe any dependence of critical velocity on temperature
Fig. 1. Rate of flow u and temperature difference A T in thermo-mechanical transport through a narrow channel as function of the heat input Q.
with the same slit width. We must keep in mind that in the experiments carried out by Kapitza the slit was not homogeneous: the thickness of the narrowest slit varied from 0 up to 2.6 x 10-5 cm. In calculations the slit width was taken equal to 1.3 x 10-5 cm, so the value of critical velocity reached approximately 1 m/sec. In their experiments on film transfer of superfluid helium, Daunt and Mendelssohn4 have observed the same phenomena. The rate of transfer was very high and practically independent of level differences. The results of the experiment on film transfer of superfluid helium flowing from a glass beaker raised above the helium level, carried out by Daunt and Mendelssohn, are given in Fig, 2. As can be seen from the figure, during the first six minutes the level inside the beaker dropped quickly, but afterwards a constant transfer velocity was established independent of the level differReferences p . 36
CH. I,
4 11
CRITICAL VELOCITIES AND VORTICES
3
Fig. 2. The film transfer of superfluid helium.
ence. The reduction by 65 % of the level difference after 33 minutes did not cause any change in the film transfer velocity of superfluid helium. The dependence of film transfer velocity on temperature was also determined by the same authors. According to their data, the volume velocity of transfer was practically independent of temperature from 1" K up to 1.5" K, but then dropped quickly at the I-point. The linear velocity of the film transfer at 1.5" K was determined by the authors to be 20 cm/sec after measuring the film thickness, which was 3.5 x 10-6 cm. Thus the increase of pressure head across a slit at first causes an increase in the transfer velocity of superfluid helium; then, above some critical value, although the transfer velocity continues to increase, irreversibility appears with gradual disturbance of superfluid flow.
140
.-. ? 0
L"
'.
- 30 -u2 TI
Q
b
s! 20 c
a
C
0
u 5
n
2 10
Fig. 3. Atkins' method for observation of the film flow oscillations.
References p . 36
4
[CH.I,
V. P. PESKOV
51
In films, the transfer occurs at the critical velocity, and the increase of level differences practically does not change the transfer velocity. This is explained by the fact that the transfer path length is many orders of magnitude greater than the film thickness, so that any small friction force leads to considerable losses. The existence of a distinct critical velocity in films was confirmed by the result of Atkins’ experiment 5. A sketch of Atkins’ apparatus is given in Fig. 3. An empty beaker dipped into a bath of liquid helium was filled and emptied alternately while fdm flow took place. The change of helium level in the
/:I;-------
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Y 0 4
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I
1
2
I
I
-
3 4 H e a t input (W/seC)
I
5
I ~10-4
Fig. 4. Film flow under thermal potential.
capillary was observed. The results of these observations are given in the figure. As can be seen the flow of helium proceeded automatically after the levels became equal, and created a level difference of opposite sign. Then flow in the opposite direction began, and some practically undamped oscillations arose. The amplitude of oscillation being invariable, it can be seen that at velocities that are below the critical one, superfluid flow in the film really occurs, i.e. without viscosity losses. Thus Atkins’ experiment clearly shows the precise boundary between flow in the subcritical region where undamped oscillations occur and flow in the supercritical region, where the losses in the film are so great that order-of-magnitude pressure variations do not affect the volume transport. The experiment of Chandrasekhar and Mendelssohn6 is another manifestation of critical velocities in films. A heater was placed at the bottom of a References p . 36
CH. I,
0I]
5
CRITICAL VELOCITIES AND VORTICES
Dewar vessel, represented in Fig. 4. An optically polished glass cap covered the top of the vessel, which was approximately half immersed in liquid helium. When the helium levels inside the vessel and outside it became equal, and equilibrium had been established, heat was introduced through the heater. The film flow rate at small heat inputs was less than the critical one and directly proportional to the power output of the heater. However, as can be seen from Fig. 4, the flow rate having reached some critical value, remained constant in spite of a large increase in heat input. It is worth mentioning that
-
18
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E
p '4
; 9 10
u
c
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-2
6
U
4
2
m-Mendelssohn and Whitefl950) 0.4
0.8
12 I6 Temperature(%)
20
Fig. 5. Helium film flow rates as a function of temperature.
the temperature dependence of the flow rate of the superfluid film is very small. The results of film flow measurements along a glass wall at temperatures from the 1-point down to 0.3' K carried out by Hebert, Chopra and Brown7 are given in Fig. 5. The variations of helium flow rate at temperatures greater than 1' K are explained by a change of density in the superfluid component p s . The product of the velocity of the superfluid part by the film thickness, v,d, is practically independent of temperature, and in absolute value coincides with measurements carried out by Daunt and Mendelssohn. An increase of flow rate is observed at temperatures below 0.9"K, which cannot be explained by the increase of the superfluid part, as its change is less than 1 %. It is possible that this flow rate increase is connected with an increase in film thickness, This coincides with the measurements of film thickness carried out References p . 36
6
V. P. PESKOV
[CH.I,
81
by Burge and Jackson* and Bowersg. The film flow rate as a function of temperature deduced by Ambler and Kurti10 and Waring11 is given at the top of Fig. 5. In both these experiments, as well as in many others, the flow rate is considerably greater than that indicated by Herbert, Chopra and Brown7 and by Daunt and Mendelssohn4. This is explained by the fact that the helium used by these authors had impurities (air or hydrogen) which, when deposited on the surface, created an absorbed layer that increased the effective thickness of the film. Experiments carried out on unsaturated a m flow are contradictory and were conducted in conditions where it was not possible to guarantee that the surface on which the film was formed was really smooth, at least to within 10-7 cm. It is quite clear that for a definite conclusion it is necessary to carry out tests on the transfer and simultaneous determination of film thickness, using films formed on crystal facets which are really flat with an accuracy up to atomic dimensions, for example fine mica films. A very interesting peculiarity during film flow was observed by Eselson and LazarevI2. Using a calibrated glass beaker with thick walls of 2.54 mm inner diameter and 50 mm length, the film flow rate for different initial conditions was observed. In the first case the beaker was completely plunged into helium; then it was lifted and the flow rate was observed. In the second case, after helium had flowed out off the beaker, it was immersed almost completely and filled with helium by flow of the film. Then it was lifted, as in the first case, and again the flow rate was observed. At a temperature of 1.52" K in the first case, the film flow rate at first decreased rapidly from cm3/cm sec, while the level of the helium in the 2.4 x 10-4 to 1.1 x beaker dropped from 2 to 6 mm below the rim. Then the flow rate stabilized, diminishing to 1 x cm3/cm sec for a level drop in the beaker of a further 28 mm. At the same temperature, in the second case the flow began when the level was 12 mm below the rim of the beaker and the flow rate decreased slightly upon further drop in level, but the absolute value of the flow rate was 10 % less than in the first case. At a temperature of 1.69"K this difference was only 5 %, while at a temperature of 1.88" K it was impossible to observe a difference between the two cases of flow. A difference of 30 % in flow rate at a temperature of 1.3" K under the same conditions was observed by Allen The helium flow through very fine capillaries has much in common with film flow. Such flow under pressure heads was studied by Seki and Dickson14. They have used millipore - a fine organic filter, commonly used for biological References p. 36
CH. I,
5 I]
CRITICAL VELOCITIES AND VORTICES
7
Fig. 6. A schematic drawing of the apparatus with Millipore.
purposes. This filter is simply a set of a great quantity of fine straight channels. The Seki and Dickson apparatus is shown in Fig. 6. The authors were certain that the superfluid flow took place only along millipores, and that no gaps had been formed in the seals between the millipores and the glass tube. The results of these experiments are given in Fig. 7. As can be seen, for pores of 0.45 u , diameter the flow rate reaches its critical value when the level difference is 5 mm and subsequently increases very little with pressure. The dependence of flow rate on temperature under a pressure of a 1 cm column of helium for the same pores is shown in Fig. 8. The temperature dependence of the critical velocity of the superfluid com-
I
I
I
0
5
10
P(mm liquid He)
Fig. 7. Flow rate us. pressure in liquid He level difference for 0.45 p pores. References p . 36
8
V. P. PESKOV
[CH.I,
51
ponent, calculated according to the formula v, =vp/ps, p is represented by the upper curve. The critical velocity for very fine pores depends very little upon the temperature, as is the case in film flow. Some measurements of flow rate under the pressure of a 1 cm liquid helium column for millipore channels of
1.0
2.o
1.5
T ( W
Fig. 8. Flow rate for 1 cm level difference us. temperature for 0.45 p pores.
0.1 p to 5 p diameter were made by Seki and Dickson. The results are given in Fig. 9. The increase in flow rate when the pore diameter is a few microns may be explained by the fact that the flow of the normal component restrained by viscous forces, becomes of great importance. That is why when flow occurs through wide capillaries and slits a question arises: how to conduct an
201 1(1.16"K
References p . 36
CH. 1,
5 11
CRITICAL VELOCITIES AND VORTICES
9
experiment to determine the critical velocity more accurately. For this purpose the overshoot method was used by Winkel, Delsing and Poll15. This method works in the following way. If two isolated vessels filled with superfluid helium communicate through a fine slit, equilibrium is established quickly on account of superfluid flow. This equilibrium is determined by the thermomechanical effect, i.e. 7-2
V * A p= J S d T , TI
where V is the specific volume of liquid helium, A p is the pressure difference
in helium between two vessels on the same level, S is the entropy, and TI and T, are the temperatures of the first and second vessels. This equilibrium will change very slowly due to parasitic heat input, viscous flow of the normal component, and thermal conductivity. If in one of these vessels, heat is generated in order to maintain equilibrium the helium flows in a superfluid manner from one vessel to another. If the critical velocity is not surpassed, immediately after stopping the heat input the level difference remains constant to within the oscillation amplitude, which is determined by the kinetic energy of superfluid flow in the slit. But if the critical velocity is surpassed, the dynamic equilibrium due to the thermomechanical effect is disturbed and the temperature of the vessel where heat is generated, rises in comparison with the equilibrium one by a n amount AT. It is clear that after heat generation is stopped the flow of helium will not cease, but will continue until a sufficient quantity of the superfluid helium component flows in to establish equilibrium. This quantity of helium, measured by the rise in level in the vessel, the authors call the overshoot c]. The apparatus to make observations by the “overshoot method” used by Winkel, Delsing and Poll, was of the same construction as the Kapitza apparatus. Two optically polished glass disks formed the slit in the apparatus. For a long slit, disks of inner diameter ri = 0.590 cm and outer diameter ro= 1.024 cm (ro--ri=0.434) were used; for a short slit, disks of inner diameter ri = 0.871 cm and outer diameter ro = 1.005 cm (ro-ri = 0.134) were used. Three wires placed between the glass disks along a radius and clamped by springs, determined the slit thickness. The long slit was not so perfect and its thickness varied from 1.2 p to 5 p. The shorter one was more homogeneous. In experiments its thickness varied between 0.4 p and 6 p . An example of overshoot observation is given in Fig. 10. Here the relative velocity is given on the abscissa u=us-v,, i.e. the difference between the References p . 36
10
[CH. I,
V. P. PESKOV
1
Fig. 10. The overshoot as a function of the relative velocity 8 .
superfluid velocity us and normal velocity u,. The overshoot is given on the ordinate axis, measured in cm of rise in level in the inside vessel. Thecritical velocity appears distinctly, as can be seen on the figure. However the picture becomes less clear at lower temperatures and with wide slits. The velocity of the normal part is considerably smaller than the velocity of the superfluid part, so that the relative velocity does not differ greatly from the superfluid velocity, but the authors suppose that us is the critical one and not u =us-v,. Fig. 11 shows the critical velocity as a function of temperature for different 30
-I
//
, /
I
\
\
\ \
\
To'\\, \
\
20 -
\
-
+ : h = 3.1 p (long slit)
y7 : h = 2.4 p
f d
(short slit)
A : h = 1.5 p (short slit)
: h = 0.43 p (short slit)
E
1
10 -
r I
I
0 13
References p . 36
I
I
16
19
Fig. 1 I.
The critical velocity usb as a func-
CH.I,
8 21
CRITICAL VELOCITIES AND VORTICES
11
channel widths (h) according to Winkel, Delsing and Poll. The dotted curve here is the velocity vf in the film, calculated from the equation R =vfdps/p according to Daunt and Mendelssohn4. Here R is the volume transport per cm and d = 3 x 10-6 cm, the thickness of the film. It should be noted that the fact that at the I-point vf is in the vicinity of zero, does not correspond to Daunt and Mendelssohn’s data. If we take the experimental value of the fraction p,/p obtained by Andronikashvilile, then in the immediate vicinity of the I-point, of M 20 cm/sec.
2. Critical Velocities in Wide Channels An original method for determining whether the regime in wide capillaries is subcritical or supercritical was proposed by Vinen179 18. The method is based on the fact that in the subcritical regime the superfluid helium component behaves like an ideal fluid. Thus the second-sound attenuation, representing 100
75
*- 5 0
-5
c, .e 2.5 I
: 0
Fig. 12. The excess attenuation a’of secondsound propagated perpendicularly to a heat current W.
relative oscillations of the normal and superfluid components, is low and is determined by viscosity and thermal conductivity losses in the normal component only. In supercritical conditions, when breakdown of superfluidity is observed, the second-sound attenuation increases greatly due to extra losses caused by interaction between the normal and superfluid components. Vinen’s apparatus consisted of a tube 10 cm long with a rectangular cross section either 0.240 x 0.645 cm (case 1) or 0.400 x 0.783 cm (case 2). One end of the tube was closed and equipped with a heater, the other end was connected to the liquid helium bath. The heater emitting second-sound was placed along the longitudinal axis of the tube on the smaller side of the rectaiigle; a thermometer was placed on the opposite side. The tube was used as a second-sound resonator with the direction of References p . 36
12
V.
[CH.I,
P. PESKOV
2
propagation perpendicular to the heat flow. The attenuation of the secondsound was deduced from the bandwidth of the resonance curve or from the resonant amplitude. The most spectacular critical phenomena were observed at a temperature of 1.4" K. Fig. 12 shows Vinen's results for the smaller tube (case 1). However such spectacular results were obtained only in some isolated cases. Therefore Vinen devised a more sensitive method. It is based on the fact that in the supercritical regime the time required to reach a stationary state depends to a great extent on history. If we switch on a heat current W,
0
2
4
0
1
2
Fig. 13. Typical plots t against W I .
which is several times more than the critical one, the second-sound attenuation does not increase instantly but changes gradually in the direction of the nominal stationary value. If z is the time during which the attenuation increases to half of its equilibrium value, one can see that z depends on conditions existing before the supercritical heating current W, was switched on. Vinen first switched on a weak heat current W,, then, after waiting till equilibrium conditions were established, which took more than 200 sec, he switched on W, and determined 5. Fig. 13 shows the results of the experiment. As may be seen, with the exception of case (d), the critical value of W , is characterized by a sharp decrease of z. Vinen made his measurements at different temperatures and obtained the results given in Table 1. References p . 36
CH. I,
8 21
13
CRITICAL VELOCITIES AND VORTICES
TABLE 1 Critical heat currents and velocities ~
~~
apparatus no. 1
~
temperature 1.295 1.400 1SO0
no. 2
1.304 1.400
~~
wcrit.
~
10 (US - ~ n )crit.
(W cm-2)
lo2 (US) wit. (cm s-l)
9.70 i 0.30 15.9 i 0.3 21.2 & 0.4 7.70 f 0.3 11.5 0.3
3.02 4.75 6.30 2.40 3.44
6.53 6.40 5.55 4.95 4.64
103
+
(cm s-1)
Similar experiments were performed by Chase19. He used nine identical capillaries connected in parallel. These stainless steel capillaries were 5.16 cm long, had an inner diameter of 0.8 mni and a wall thickness of 0.25 mm. One end of the capillaries was connected to the liquid helium bath and the other to a chamber containing a heater and a thermometer. The capillaries and the chamber were enclosed in a vacuum jacket. Chase measured most accurately the dependence of grad T on heat current density at different temperatures. The results of his measurements are given in Fig. 14. The critical heat current Wcrit.was determined from the breaks in these curves and the velocity of the superfluid component calculated from the
Fig. 14. Grad T as a function of heat current density.
Referencesp . 36
14
V. P. PESKOV
[CH.I,
82
equation v, =- pn W,,,Jp,pST, where v, is the superfluid component velocity; p, pn, p, are respectively the total density of helium, of its normal component and of its superfluid component; S is the entropy and T the temperature. Chase obtained the curve shown in Fig. 15 and representing the iemperature dependence of the critical superfluid velocity v,. At temperatures below 1.6" K, besides the break at Wcrit.on the curves ofthe dependence of grad T o n W, Chase discovered in the supercritical region a second small break, which corresponds to the value of 0, marked in Fig. 15 by the sign 0. Chase assumed that this phenomenon is accounted for, by the onset of turbulence in the normal component.
Fig. 15. Temperature dependence of the critical superfluid velocity.
Judging from Vinen's data it is difficult to say what is critical: the superfluid velocity in relation to the walls us, or the relative velocity of superfluid and normal components o,--v,. An answer to this question was given by Peshkov and Struckov's20 experiment, when critical velocities were observed in one capillary, first in superfluid flow only, and then in a countercurrent like in Vinen's experiment. For the measurements there was used the technique of second-sound attenuation which was previously used by Vinen18. The measuring apparatus is shown in Fig. 16. It is made of glass. The critical velocity was observed when helium flowed through the tube 1, 10 cm long, having the diameter 0.385 cm. Counterflow of the superfluid and normal components was produced by a References p . 36
CH.I,
5 21
15
CRITICAL VELOCITIES AND VORTICES
heater 2. Tube 1 was separated from the upper part of the apparatus by means of a 50 p mesh silk net and a compact layer of powder 3 consisting of Fe,O, particles, the dimensions of which did not exceed a few microns (jewellers’ rouge). Flow of the superfluid component was caused by the thermomechanical effect through the rouge powder excited by the heater 4. The superfluid component velocity was measured to an accuracy of 3 % using a cathetometer and a stopwatch for determining the rise of level in
Fig. 16. The apparatus for critical velocity observations.
tubes 5 and 6. To provide adiabatic conditions the apparatus was enclosed in a vacuum-jacket 7 and a copper shield 8. Plug 9, which can be lowered or raised, controlled thermal contact between the helium inside the apparatus and the outer helium bath. Tube 1 and the cap 11 which was separated from it by the wide gap 10 (h rn 1.2 mm) formed a second-sound resonator. The length of the cap h ’ w 4 . 5 mm. To provide maximum quality of resonator, the cap length was calculated from the equation $1= h’ +h. The wave length of second-sound 1-2.3 cm; the gap width is chosen so that the gap area would be a little greater than the cross sectional area of tube 1. The second-sound emitter 12 was placed on the bottom of the cap. The
+
References p . 36
16
[CH.I,
V. P. PESKOV
82
second-sound receiver 13, made of 40 /* phosphor-bronze, was glued to a thin paper strip and fixed on the side of the gap as close to its bottom as possible. The quality of the resonator proved to be of the order of 140. To determine the critical velocity in the case of superfluid flow only, the following operations were carried out. After the establishment of stationary conditions in the helium, flow of the superfluid component was caused by means of heater 4 situated above the rouge; then, after a period of time sufficient for establishment of equilibrium conditions, heater 2 situated beneath the rouge was put into operation. Heater 2 caused a counterflow W, in what was obviously the supercritical region (7-8 times over). Then the time z, during which the second-sound amplitude fell half-way to its full
-" 3
:r'..-2
b
1
1
2
4
6
8
1
0 IO'V,
2
4
6
8
1
0
(crn/sec)
Fig. 17. Dependence of r upon the velocity us.
value, was measured. Such measurements were undertaken at different initial velocities of the superfluid component 0:. To determine the critical value of ur in case there is a counterflow in steady state helium a small heat curIent W was generated by heater 2. Then, after a period of time sufficient for establishment of stationary conditions, a heat current W, obviously over the critical value was generated by heater 2 and finally the time z was determined. The results of experiments made at T = 1.44OK and W,= 5.5 x lod2W/cm2 are given in Fig. 17a. The crosses refer to a case of counterflow and the circles to the case when only the superfluid component flows. As one can see from the figure, the value of us is the same in both cases. At T= 1.32' K (Fig. 17b) the critical value of us also remained equal in both cases. At the same time, considering the mean values psus = pn u, in counterflow, us - u, at T = 1.44" K will differ 9 times and at T= 1.32" K 20 times. Thus, judging from the experiment, one can see that the breakdown of superfluidity occurs due to the velocity us relative to the walls, exceeding some critical value but not due to the value of us- u,. References p. 36
CH.I,
0 21
17
CRITICAL VELOCITIES A N D VORTICES
To find out whether pressure drop along a capillary, and a critical velocity exist when the flow is purely superfluid, Kidder and Fairbank21 devised a rather sensitive method. Their apparatus is shown in Fig. 18. The difference of levels in superconductive tin resonators R was determined by the beating of two klystron generators stabilized by these resonators. The equipment used was able to sense level differences corresponding to grad p = 3 x dynes/cm3. A superfluid flow was set up in the 1.1 mm capillary F between two super-
V, krn/sec) Fig. 18. Kidder and Fairbank's apparatus.
Fig. 19. Pressure gradient vs. superfluid velocity.
fluid filters S by means of a thermomechanical effect produced by the heater H. The measurements taken at T = 1.30"K are given in Fig. 19. As can be seen from the figure this method clearly demonstrates the existence of a critical velocity 0., la overcritical conditions grad p=ap,u,(o,-u,). Values of u, and a at different temperatures are given in Table 2. It is characteristic that upon TABLE 2 Critical superfluid velocities uc, and the proportionality constants a, at various temperatures Temp. (" K)
vc (mm/sec)
a (cm-1)
1.26 1.30 1.48 1.57
1.4 I .3 1.1 0.95
1.5 1 .o 0.95 0.80
References p 36.
18
[CH.I, 0 2
V. P. PESKOV
lowering the temperature, the value of the critical velocity increases;but does not decrease as in experiments performed by Vinenls and Chasele. During experiments when superfluidity was destroyed, it was observed that at velocities a little above critical, stationary conditions were established rather slowly. Mendelssohn and Steele22 used for their experiment a glass capillary 150 cm long with an inner diameter of 1 mm. One end of the capillary was connected to the helium bath, where the temperature was stabilized to an accuracy of 10-5 OK; the other end was soldered up and was equipped with a heater and a thermometer. When a heat current a little above the critical one was switched on, the temperature difference between the ends of the capillary increased slowly and linearly with time. After some time the increase of temperatrue difference stopped and thereafter remained constant. The authors explained this phenomenon by the gradual development of turbulence, the front of which moved with a velocity of about 4 cm/sec or some multiple thereof. The kinetics of the breakdown of superfluidity were investigated in more detail by Peshkov and TkachenkoZ3. They used a coiled German silver capillary 8 m long and having an inner diameter of 1.4 mm enclosed in a vacuum-jacket. One end of the capillary was connected to a liquid helium bath, the temperature of which was stabilized to an accuracy of 10-5 OK; the other end was soldered up and a heater was wound around it. On the capillary, at approximately equal distances from each other, 12 thermometers were placed and measurements of any three of them could be taken simultaneously on one record. Fig. 20 shows the time dependence of thermometer temperatures at a
0
20
40
60
Krnin)
Fig. 20. Time dependence of thermometer temperatures. References p . 36
CH. I, 4 21
CRITICAL VEMCITIES AND VORTICES
19
bath temperature T = 1.34" K. The heat current along the capillary was W=4.4 x w/cm2, i.e. W = 1.19 W, (W, is the critical heat current). Thick lines correspond to recorded data and thin ones to interpolated recordings. The numbers on the curves are the corresponding thermometer numbers. Thermometer RI2is nearest to the heater. Analysis of these curves shows that turbulence, increasing the thermal resistance of the helium develops from both ends-cold and hot. Clearly-seen breaks in the timedependence of thermometer temperatures for Klo, R8,R, and R5, following at quite definite periods of time, testify to the fact that development of turbulence from the hot end occurs at constant velocity vH=2.2 mm/sec. The constant rate of temperature rise of R4 and the cessation of the rise at the 40-th minute, before the onset of steady conditions in the whole capillary,
2Liii?3-
z!
€2
E
. d
>k
1
3
5
i02w(w/crn2,
Fig. 21. Dependence of front velocity upon thermal current density.
serves as evidence that there exists a second turbulent front moving from the cold end with velocity u, = 1 mmlsec. Fig. 21 shows the dependence of the front velocity upon thermal current density. It should be noted that this simple picture is characteristic only of the case when, before the switching-on of a heater, the helium inside the capillary has settled down, which required 20 minutes waiting at W = 0. The probability of a centre of turbulence occurring inside the capillary depends on overcriticality, but in settled-down helium even at W= 1.5 W, turbulence centres inside the capillary were not observed. If, inside the capillary there were regions of residual turbulence, they served as centres from which two turbulent fronts spread with velocity 0, and 0., It is obvious that exactly these cases led Mendelssohn and Steele to the conclusion that the velocities of the fronts may be multiple, as they observed only the rate of temperature rise at the hot end. The data given in Fig. 21 draw serious attention to the question of whether the critical velocity corresponds to zero velocity of the cold front u,, hot front uH or to equality of uH to the velocity of retreat of the cold front. References p . 36
20
V. P. PESKOV
[CH. I,
83
However all these conditions do not differ greatly and give at T = 1.34" K, 10-2 w/cm3, which corresponds to us=0.11 cm/sec and u, = 1.9 cm/sec.
W,(=3.7 f0.1) x
3. Oscillating Discs and Spheres Critical velocity may also be observed rather clearly during the oscillation of discs and spheres in superfluid helium. Hollis-Hallet z4 performed experiments aimed at viscosity determination by means of an oscillating disc. He used two variants: one duraluminum disc, 0.073 cm thick and 1.52 cm radius, and a
~
+
*
+
c t+-+ + t-+-+-+-
+at 2675% Oati362%
0
xot 2.091'K Period 01 oscillotion
mot t948-K TE 250sec I
0
1
2
3
5
0 Rodions Fig. 22. The variation of the logarithmic decrement with amplitude.
pile of 18 mica discs 0.003 cm thick and 1.727 cm radius damped between discs 0.0109 cm thick and 0.695 cm radius. The discs were mounted on a duraluminum spind. Both the single disc and the pile of discs were connected to a long quartz rod suspended on a thin suspension. Mirrors weie fixed at the upper end of the rod to observe the amplitude of the torsional oscillations of the discs immersed in liquid helium. Apart from determing viscosity, Hollis-Hallett has noticed that the logarithmic decrement 6 of oscillations remains constant and equal to a certain value A at small amplitudes of oscillations. However, at amplitudes exceeding a certain critical pm, the decrement depends greatly on the aniplitude and noticeably exceeds A. References p . 36
CH. I,
8 31
CRlTiCAL VELOCITIES AND VORTICES
21
Hollis-Hallett has determined the values y,,, for different periods in the range 2.6-18.7 sec at temperatures of 1.28"-2.16" K. He expected that at the same temperature the linear critical velocity should be identical: i.e. p,,,/T, where Tis the period of oscillation, should be constant, but it has not proved to be so. Later on Benson and Hollis-Hallett 25 carried out investigations using the same apparatus but substituting for the discs an oscillating sphere of radius
-
10
12
,4
16
e0K
l.B
2~
22
Fig. 23. The variation of &T* temperature.
with
1.294 cm to avoid edge effects. Fig. 22 shows the reported data for a period of oscillation of 25 sec. Measurements were also undertaken at T= 10.4 sec and T = 6.5 sec. On the basis of data of these and earlier measurements they deduce that at a given temperature pc/T*,but not (oc/T,remains constant. Fig. 23 shows these data. It can be seen from this figure that qc/T* does remain constant to within the accuracy of the experiment. Since the penetration depth of the viscous oscillations I =(qT/np)*, it can be said that the product of the linear velocity and the penetration depth of the viscous oscillations remains constant at a given temperature. However, the results of measurements by Benson and Hollis-Hallett are Referencesp . 36
22
V. P. PESKOV
[CH.I,
83
not in agreement with data of Andronikashvili26 and Gamzemlidze27. The latter has performed a special experiment to clear up the conditions necessary for critical phenomena to occur. He has also observed constant damping at small amplitudes of oscillation and an increase of damping beginning with a certain critical amplitude. However, in his experiments yJT, but not y,/T*, is identical for the two different periods T, i.e. the critical velocity u, exists. His results and the data of Hollis-Hallett are giveninFig. 24, where O.o-T= 3.42 sec, +-6.85 sec,
,4
+-8.95
,,6
2B
2o
2L
Fig. 24. The critical velocity us. tempera-
sec, x -14.45 sec, A -is defined by the equation: vk =0.1O5/d/ps cm/sec; data of Hollis-Hallett for the radius R = 1.572 cm, W-3.15 sec, 0 -3.78 sec and 0-11 sec. Gamzemlidze has also performed experiments with a disc to which 250 grains per square cm are stuck (grain sizes: 0.05, 0.1 and 0.2 mm). For oscillations of the disc with the grains, the critical velocity decreases by a factor of 3 and is less than in the experiments by Hollis-Hallett. On the basis of his investigations Gamzemlidze has concluded that, assuming that the elongation of the lines of the current is due to the flow round the grains, then the critical velocity defined by the equation v k = References p . 36
CH.I,
P 41
CRITICAL VELOCITIES A N D VORTICES
23
+
(1 1.75 7c d d ? , where Dk0 is the peripheral velocity of the disc, dis the linear size of the grain and c the concentration of the grains, is constant and independent of c. However, there still exists a difference in principle between these experiments. In measurements by Hollis-Hallett the critical velocity changes with the period of oscillation, yJ Tt remaining constant. In Gamzemlidze's experiments the critical velocity is independent of the oscillation period. A more accurate experiment needs to be performed to settle the question. uko
4. Persistent Currents
An interesting experiment has been performed by Bendt 28. He has observed the critical velocity in a rotating ring. A ring 2.2 mm in width, 1.8 cm in depth and 8.83 cm in average diameter is filled with liquid helium and set into rotation. After the whole liquid has been dragged into motion, the ring is brought to rest, waiting for a time necessary for the normal component to stop. The existence of superfluid circular flow is then determined. To detect the superfluid flow Bendt has used the ability of an ideal liquid flowing round a plate to create a noticeable rotational moment, maximum at 45", which tends to turn the plate toward the plane perpendicular to the flow, i.e. a modified Rayleigh disc. He immersed such a plate in helium at an angle of 45' to the flow. The plate, made from a thin foil, is suspended almost vertically and is held by adhesive forces with one side stuck to a wire stretched along the radius. One can easiIy notice that the plate springs back from the wire when the rotational moment is obtained. Bendt observed that if the rate of the rotation is less than a certain critical velocity the superfluid helium remains at rest after the ring has been stopped, but if the rate of the rotation is great an undamped superfluid flow remains in the ring for at least 20 min. The critical velocity depends greatly on the prehistory and varies from 0.55 cm/sec (1.19 rpm) to 0.81 cm/sec (1.75 rpm). The greatest critical velocity is obtained if the helium has not previously been brought into rotation. "Memory" of the preceding rotation of the superfluid flow lasts from 50 to 75 min. Since the plunging of the plate always led to a full stop of the superfluid flow, Bendt concluded that the distribution of the velocity along the radius should take the form u w l/r. A jump of the velocity of the superfluid component should take place at the ring walls. On the basis of these investigations it can be said that such a jump does exist; however, it is impossible to determine its value. References p . 36
24
V. P. PESKOV
[CH. I,
85
5. Summary of Experimental Data Atkins 29 has summarized the results of measurements of the critical velocities at T = 1.4" K as a function of the characteristic length for different cases. His data are given in Table 3 and in Fig. 25. Data of the most illustrative and TABLE 3 Critical velocities at 1.4"K
Type of experiment Unsaturated films Saturated films Narrow channels (overshoot procedure) Glass capillaries Oscillations in a U-tube Oscillating disc Oscillating sphere Rotating cylinder viscometer Second sound in a heat current
Channel width or characteristic length
Critical velocity
d (cm)
(cm sec-l) 46 46 25 12
5
x 10-8
3 x 4 x 3 x 2.6 x 8.1 x 2.1 x 4.2 x 7.1 x 5.5 x 6.9 x 10.7 x 10.6 x
24.0 x 40.0 x
10-6 10-6 10-5 10-4
10-3 10-3 10- 2
10-2 10-2
10-2 10-2
u0.c
08,C
d
8 3 1 0.62 0.21 0.14 0.30 0.26 0.15 0.07
(cm3 sec-I cm-1) 2.3 x 4.6 x 10-5 7.5 x 10-5 4.8 x 10-4 2.4 x 10-3 7.8 x 10-3 8.1 x 10-3 1 3 x 10-3 8.8 x 10 x 10-3 16.5 x 1 0 - 3 17.9 x 16.1 x 7.4 x 10-3
0.051 0.033
12.5 x 10-3 13.2 x 10-3
characteristicexperiments reported can be summarized in the following way: 1. For volume transport of superfluid helium through the film and narrow (less than 1 p) slits and capillaries there exists: a) a critical velocity; the volume transport at a lower rate takes place without losses; b) a hydrostatically or thermomechanically increased pressure head leads to the breakdown of superfluidity and such large losses that the rate of flow hardly increases, remaining practically equal to the critical velocity; c) the critical velocity of the superfluid component u,, from the I-point to 0.8"K depends slightly on temperature and increases steadily at lowet temperatures; at 0.3"K it increases by 25 %. Perhaps this is an apparent rise associated with the increased film thickness; References p . 36
CH. I,
0 51
CRITICAL VELOCITIES A N D VORTICES
25
d) since the volume transport through the film possesses inertia, undamped oscillations occur a t small heads; e) the volume velocity of transport through the film can remain higher than the steady value for a long time (more than 20 min). 2. For heat flow giving rise to flow through capillaries, provided psvs= = pnun,there exists: a) a critical velocity below which the superfluid component moves without interaction with the walls and the normal component; the normal component
Fig. 25. The critical velocity as a function of d.
moves laminarly, with a corresponding thermal gradient along the capillary ; b) pressure head increase leads to the breakdown of the superfluid motion and a sharp increase of the thermal gradient; however, the velocity of the superfluid component flow can be higher than the critical velocity; c) in narrow capillaries (of the order of 1 p or less) the temperature dependence of the critical velocity us, of the superfluid component is negligible. In wide capillaries (of the order of 1 mm) us, at T > 1.7" K at first depends slightly on temperature, but then starts to increase towards the A-point. If Tc 1.7" K, us, sharply decreases with decreasing temperature. There may be References p . 36
26
V. P. PESKOV
[CH.I, 0 6
two relationships, one from T to 1.7" K and another above; at 1.7" K one can observe a break in the temperature dependence curve; d) in long capillaries at low supercritical velocities the region of brokendown superfluidity,associated with higher thermal gradients, fills the capillary slowly (with velocities of the order of some millimeters per sec). The velocity of the motion of the fronts of the broken-down superfluidity depends on the supercritical velocity. There exist fronts moving from the hot end towards the cold end and there are fronts moving from the cold end towards the hot end. 3. When the superfluid component moves through a capillary (0, = 0) one can observe the critical velocity above which the pressure gradient along the capillary occurs. The critical velocity decreases slightly (instead of increasing as in the case of o, ps onpn =0) with increasing temperature. 4. In capillaries of several millimeters diameter the critical velocity o, is identical at on =0 and on M 200, i.e. us, but not o,-u,, is critical. 5. When a disc oscillates in liquid helium, one can also observe a critical velocity which manifests itself in the fact that the damping decrement begins to increase rapidly after a certain amplitude has been obtained. 6. When a ring Wed with superfluid helium is brought into rotation, one can observe the critical velocity. If the ring rotates with a rate exceeding the critical velocity, helium continues to rotate for a long period of time ( M 50-70 min) after the ring has been stopped. 7. The rate of breakdown of the superfluid motion depends on the prehistory, i.e. nuclei are essential for the breakdown of superfluidity. 8. Fig. 25 shows the dependence of the critical velocity on the characteristic length at T = 1.4" K. What is the reason for the breakdown of superfluidity? What takes place in helium after the critical velocity has been exceeded?
+
6. The Landau Criterion and Excitations in the Superfluid
Landau in his work on superfluidity of helium30 favours the view that the reason for the critical velocity is the breakdown of superfluidity due to the creation of excitations. If in the superfluid liquid flowing through a capillary with velocity V a certain excitation is created possessing impulse p and energy E , then in the reference frame in which the liquid was initially at rest, the liquid energy Eo is equal to the excitation energy. In conformity with the well-known formulae of classical mechanics on the References p . 36
CH.I,
4 61
21
CRITICAL VELOCITIES A N D VORTICES
transformation of the energy and impulse in the reference frame in which the capillary is at rest, the liquid energy is E, = E, +Po fi 3Mo2, and the impulse j =po--Mfi, where M is the mass of the liquid, or since Eo = E and
+
&=I?.,
E l = &+ p a + & M ~ ~ .
Since the excitation can be created only due to a decrease in the kinetic eneigy of the liquid +Mo2,then e + j f i should be less than 0, i.e. E +jU< 0. The smallest fi correspondingto this relationshipis obtained for antiparallel 13 and p or E-PG c 0, i.e. o >&/p. In accordance with this condition the creation of a phonon required that o should be greater than the velocity of sound, as for the phonon E = cp (c is the velocity of sound). To create a roton from the shape of the spectrum suggested by Landau3l E =A +(p-~,)~/(2 1 ~ one ) ~ obtains o, = = ( ~ / p )m ~ 70 ~” m/sec. In practice the observed critical velocities are two orders of magnitude less; therefore the breakdown of superfluidity is not due to the creation of phonons or rotons. Dash32 has suggested an idea that atomic “clusteIs”shou1d in common generate.a quantum excitation. The size of the cluster V m 03, where 0 is the cooperative distance within which the superfluid liquid moves as a whole. In this case the excitation can occur only if P
V-
X
2
> I/-P J , 2 = & . 2
Assuming that E and V are independent of temperature, Dash obtains the temperature dependence o(, T ) m l / f i . Comparing lc, 2c and 3 one can see that the temperature dependence of the critical velocity varies under different conditions. Therefore, at least the condition that u and E are constant is not fulfilled and this assumption does not reveal the physical nature of the breakdown of superfluidity. Ginsburg33 has assumed that when the superfluid component is set into motion a tangential discontinuity of the velocity occurs at the walls. It is associated with the surface energy per unit area 0 = h Na/2ma2 = h/2nza4 w w5 x erg/cm2 where N is the concentration of atoms in liquid helium ( N = 2.2 x 1028), a- N-* = 3.5 x 10-8 cm, m is the mass of the helium atom and h is Planck’s constant. Proceeding from dimensional considerations, Landau and LifshitzS4have assumed that this energy is equal to a-p,(kTAU4/p)*,where p and ps are the densities of helium and its superfluid component, k is Boltzmann’s constant, TA=2.19” K and U is the velocity of second-sound. References p . 36
28
[CH.I,
V. P. PESKOV
06
According to this evaluation 0 - 5 x 10-2 erg/cm2, i.e. both evaluations coincide. Assuming that a region with volume u and surface s, can be formed separately from the rest of the liquid, Ginsburg writes for it ~ ( p=) 3Mu2 os, where M = p,u is the mass of the liquid and u is the velocity of the motion in the coordinate system associated with the liquid. For p = Mu, then the criterion u, = ~ / p=(+Mu2 cm)/Mu min. Since u = u,, then v, = 2/20s/p, v ; but s/u l/d, where d is the size of the slit or the capillary therefore,
+
+
-
v, = 42d/psd
N
Am
JG (0.1 + l)/&cm/sec N
i.e. the value is close to the magnitude obtained in experiments with narrow capillaries. erg/cm2 had to be However, if surface energy of the order of 0 irreversibly used for the formation of the velocity jump in the subcritical regime, then each passing of the velocity through zero would lead to considerable losses and condition Id would not be fulfilled. The amount of kinetic energy in the film +psu,2= 10-4 erg/cm2. From the experiment by Atkins (Fig. 3) it is seen that the irreversibly lost energy of the surface discontinuity 0 < 10-6 erg/cmz. Furthermore Gamzemlidze35 has carried out direct investigations of the dry friction in superfluid helium. He has determined the minimum momentum necessary for a pile of discs suspended in helium to be set into motion and revealed that the rotation starts at arbitrarily small momentum. In any case, the surface energy should be less than 10-lo erg/cm2. Thus, the idea suggested by Ginsburg is untenable. An interesting idea has been advanced by Kuper36. He has assumed that ripplon excitations can occur representing quanta of surface waves. The surface wave velocity can be expressed as
where a is the surface tension; g is the force acting upon a unit mass at the liquid surface; d is the thickness of the film; p is the helium density and K the wave number. @ . When K= 0, the minimum phase velocity umin = I Evaluating “g” for a film thickness of d- 3.5 x 10-6 cm, Kuper has References p . 36
CH. I,
8 71
CRITICAL VELOCITIES AND VORTICES
29
derived 0, m 70 cmlsec. This critical velocity in the film corresponding to the appearance of “ripplons” has the same order of magnitude as the experimental value of the velocity. However critical velocities in thin slits and capillaries with thickness similar to that of the film are approximately of the same value; no surface waves can occur there.
7. Quantised Vortex Lines A concept originated by Onsager37 of the existence of vortex lines, that is quantization of the superfluid helium component circulation, provides quite a new opportunity of explaining the nature of critical velocities. The integral round such a line has to be
f
h ijsdP = n - = 2x11 1.5 X 10-4Cm2/SeC Jn where A is Planck’s constant; m is the He* atomic mass; n is an integer. This concept makes it possible to consider the superfluid component vortex lines, which are like whirlwinds, as a new type of excitation and apply to them Landau’s relation (us, > B/p). The question arises - does such a phenomenon really exist in superfluid helium? The most effective justification of this concept was provided by Vinen’s experiments 38. Vinen proceeded from the following consideration. If a quantum vortex with a circulation that is a multiple of film occurs round a thin oscillating wire it will deflect the wire, due to the Magnus effect, in a direction perpendicular to the motion, thus gradually turning the plane of oscillation. The rate of rotation of the plane of oscillation will be proportional to the circulation and consequently a multiple of some minimum value. Fig. 26 shows a diagram of Viaen’s apparatus. A thin beryllium-copper wire W 0.025 mm diameter and 5 cni long, was stretched by a spring S between two pistons A and B. By moving piston B with the help of tube X the tension was set in such a manner that the oscillation frequency became approximately 500 c/s. The whole device could be rotated by a separate synchronous motor at a rate of 0.1-2 rpm. The wire was placed in a perpendicular 3 kG magnetic field. The upper and lower parts of the cylinder C glued together by bakelite F were electrically isolated. Each of them was connected through an amplifier to an oscillograph. The wire oscillations across the magnetic field caused a potential difference at the wire ends that was registered by the oscillograph while oscillation along the magnetic field did not produce a potential difference. Wire oscillations were excited by a current pulse travelling through the wire. The procedure of the experiment was the following: References p . 36
30
V. P. PESKOV
[CH.I, f 7
The apparatus was put into rotation at a temperature above the kpoint and rotated at a constant speed for 20 minutes, then in the course of 30 minutes the temperature was slowly reduced to 1.3" K. When the rotation was stopped the wire oscillation was excited and the oscillograph recorded
U Fig. 26. Diagram of Vinen's apparatus.
the oscillation damping. Typical records are shown in Fig. 27 - a) without a vortex on the wire; b) circulation * A / m and c) circulation M 2+1. Besides simple attenuation (a) an apparent stopping of the wire and than resumption of oscillation are observed (b), (c), as shown on these photos. This means that the wire oscillation plane rotates and on turning 90" the oscillations are not registered. The circulation quantization is confirmed by this experiment as the stopping point in case (b) is approximately twice as late as the one in case (c). References p . 36
$71
CH. I,
CRITICAL VELOCITES AND VORTICES
31
It is noteworthy that in some experiments the circulation was not a multiple 1.4 h/m. This circumstance indicates that one vortex can of A/m, i.e. completely surround the wire while the other one remains in the helium and touches the wire only at one end. But in most cases the vortex with circulation A/m, remained stably on the wire and the multiple oscillation of the wire did not cause its destruction. The existence of a stable vortex around a wire is confirmed by the Vinen experiment but it does not give any answer about the existence of free vortex lines. This is not a trivial question, as the velocity field v, = +h/nrM =h/rm has at the point r=Oasingularity and obviously a minimum radius “a” should exist starting from which the equation would be correct. Developing Onsager’s idea Feynman39 came to the conclusion that vortex lines should appear during helium rotation in the superfluid component, being parallel to the axis of rotation, but in an equilibrium state while rotating with an angular velocity w in 1 cm2 of a plane normal to the axis of rotation there should be 2mw/h = 2.1 x lo3 w vortex lines. The kinetic energy per unit length of line will be N
b n
J
3
pS(A/mr)’2nrdr = p,h’n~n-’In(b/a)
N
10-8p,p-’ ln(b/a)erg/cm.
a
Here ‘cu” is a minimum and “b” is a maximum vortex radius. As lengthening the line requires energy expenditui e, the vortex lines possess the property of elasticity - they try to reduce their length. To prove the existence of vortex lines in free rotating helium, Hall40 took advantage of this property. If we take a rough disc and hang it on a thread into the helium rotating vessel and then force it to oscillate, we can expect that vortex lines are attached to the disc’s irregularities. The disc oscillations will draw in ends of the vortex lines and a wave will run on them which will be like that which arises during oscillation of a rope. Vortex wave resonances can be observed if the vortex line length is continuously changed as well as the helium level over the disc. This phenomenon was observed by Hall. The dependence of the oscillation period T(sec) of the disc, roughened on the upper side, upon the height I (mm) of the helium layer above the disc is shown as the result of Hall’s experiment in Fig. 28. Helium was film fed into the rotating vessel. Hall could estimate the vortex core diameter a = 6.8 f1.6 A by determining the wave length on vortices at a given oscillation frequency. References p . 36
32
V. P. PESKOV
[CH.I,
$7
Andronikashvili and Zakadze41 have carried out the same experiments. Thus the existence of a new type of excitation in superfluid helium vortexlines, was confirmed by experiments. Feynman 39 in his article on application of quantum mechanics to liquid helium, assumed that the critical velocity in capillaries may be explained by formation of a Karman voitex path consisting of quantum vortices in free helium when a jet issues from the capillary. However experiments show that a field of destroyed superfluidity gradually fills the whole capillary, not remaining outside it. For vortex rings
Fig. 28. Vortex waves.
in free helium, Atkins29 obtained the following equation for the energy: E = 2 n R2ps A m-1 In (Rfa) and for the impulse : p = 2 n2 R2p, A m-1, where R is the vortex ring radius, ps is the superfluid component density, A is Planck's constant, nz is the helium atomic mass and a is the core radius. Believing the Landau ratio to be correct: us, =E/P = hm-1 In (R/a) for wide (d > 10-3 cm) capillaries and substituting one quarter of a diameter instead R, Atkins obtained at T= 1.4" K perfect concordance of the equation with the experimental results for a =2 A. This equation does not agree with experiment for capillaries and gaps whose dimensions are d < 10-3 cm, as shown in Fig. 25. Some possible processes of superfluid disturbance in helium are examined by Vinen in his review about vortex lines in liquid helium 11. The existence of some quantity of vortex lines in helium, the ends of which are attached to projections on the walls, is the first and most obvious reason in Vinen's opinion. Then, at rather greater relative velocities, u =us - u,, thermal movement will lengthen the vortex lines until some equilibrium is established corresponding to a homogeneous filling by vortex lines of the whole capillary. References p. 36
Fig. 27. Typical records.
This Page Intentionally Left Blank
CH.I,
9 71
CRITICAL VELOCITIES AND VORTICES
33
For this process Vinen gives the equation dzldt = x1 pn vz'jp
+ y v'
- x2 hz2/ni - 3 x3 Bp, vzjpd,
where z is the total length of vortex line per unit volume, u =us- u,, d is the capillary width, xl, xz and x3 are constants of the order of 1. y = 1.1Tllcm-fsec*.
The first term here corresponds to increase of z due to thermal motion. The second one is introduced on the basis of empirical data and represents possible production of vortex lines due to relative movement of us and u,. The introduction of this term seems unwarranted and contradictory to the existence of a critical velocity. The third term corresponds to a decrease of the vortex lines due to interaction between them and the fourth one corresponds to a decrease of the vortex lines due to their interaction with the wall. As Vinen's equation is introduced on the basis of statistics it is possible, using properly selected parameters, to bring it into correspondence with experimental data for regions of fully-developed quantum turbulence, that is when velocities greatly exceed the critical one. But it is hardly possible to assume it as a basis for studying phenomena connected with the critical velocity and the nature of the breakdown of superfluidity. Nevertheless the idea of the possibility of the development of quantum turbulence due to interaction of moving vortex lines with rotons and phonons seems rather plausible. The second possibility Vinen considers to be the development of vortex lines on wall protuberances. For example, when a protuberance in the form of a knife-edge, has a height of H = 10-3 cm and the length of the vortex line is 6 = 10-6 cm, critical velocity might be of the order of us= (him) (2/H6)+In (Sjao) w 30 cmjsec. However, we should consider direct development of such a line unlikely. The energy per unit length will be equal to 81 = 10-8 In (S/ao) = 3 x 10-8 erg/cm and the superfluid component kinetic energy of this volume 82 = 3 psusS2= 5 x 10-l1 erg/cm. The third possibility Vinen like Ginsburg33 considers to be the creation of vortex sheets. Believing that in this case likewise, slip develops on protuberances, for H = 10-3 cm Vinen obtains for the critical velocity a value, us= =l/(2S/ps H ) w 30 cmjsec. Later these vortex sheets, being unstable, may break up into vortex lines. Finally, the fourth possibility Vinen considers to be development of vortex References p . 36
34
V. P. PESKOV
[CH. I,
57
rings of very small radius, believing that conditions in the vicinity of the axis of the vortex line differ greatly from the simplest case so that the energy for development of such lines may be much less than the theoretical one. The last assumption cannot be considered convincing, since for free vortex lines v, =e/p = (A/2mR) ln(R/ao) that is v, should increase with decrease of R and can begin decreasing only at R m ao = 3 A. But under these conditions, one should expect transformation of a vortex ring into a roton, for which (E/p)minm 70 m/sec. Assuming that the breakdown of superfluidity in capillaries is caused by the formation of vortex rings of the size of the order of the capillary radius R, then at T= 1.3" K over a wide range of radius of some five orders of magnitude, the critical velocity us fits the equation43
4psus, an:R2 ( R + u , , ~ ) = 4p , RA2
In ( R / a ).
This equation represents a rough estimate of equality of the vortex ring energy and the superfluid helium energy in the volume mnR2 (R-to,, T), where 0:
= 0.12; T = 3 x 1 0 -~ sec;a = 3 x lo-* ciii.
It can be seen from this equation that at R w 10-3 cm, us, = 3 cm/sec and v,z x 10-3 cm, i.e. at a radius less than 10-3 cm, the kinetic energy necessary for creation of the voItex ring should be gathered from a length of the capillary markedly exceeding its radius. It is clear that the immediate formation of vortex rings under these circumstances is hardly probable. Meservey44, developing the first variant suggested by Vinen, considers that the critical velocity will be exceeded and instability of the superfluid motion will occur if the Gorter number G (similar to the Reynold's number) is greater than 1 (in the case of isothermal flow) or 4 (in the case of counterflow). The main point of Meservey's suggestion consists in comparison of the energy transferred by the normal component to the superfluid component with the energy dissipated by viscous losses. Assuming that the force of interaction (suggested by Gorter and Mellink45 for the greatly over-critical regime) between the normal component and the superfluid component is given by
where A is a temperature-dependent constant, this gives the energy transferred References p . 36
CH. I,
8 81
CRITICAL VELOCITIES A N D VORTICES
35
to the superfluid component as
The energy lost due to viscosity is En= qj(rot ijn)2dz. The Gorter number is:
s
s
G 2 E ES,,/& = p s p , A (us - uJ4 dz/q (rot&)’ dz. Meservey has managed to choose a series of experimental data which in the range from 1.1” to 2” K lead to agreement in order of magnitude with the temperature-dependence of us, d at G = 4. However this is not surprising inasmuch as use( 7‘)variations greatly depend upon “d” as shown in Fig. 8 and in Fig. 15. Almost each law can be applied at the corresponding value of d. In the case of counterflow when on the average PS
6s
+ P n fin = 0,
G = (Ps/Pn)fl (TI us, d
and the expression ( p , / p , ) f i (7‘)is almost independent of temperature. Thus, at constant Gorter’s number G, the product of critical velocity us, and the characteristic value “d” does not change with temperature. Such a conclusion could be true for films and thin capillaries. However the relation us, = const/d is not true in this case. The latter relation is true for wide capillaries but u,,d greatly depends on temperature. Thus the adoption of the criterion suggested by Meservey is hardly possible.
8. Concluding Remarks The comparison of experimental data and individual authors’ opinions has led to the following as the most realistic picture of the breakdown of superfluidity. At small subcritical velocities, helium in capillaries and film begins to move creating vortex sheets in the superfluid component, as the start of superfluid flow is not accompanied by dry friction. This can be established within the accuracy of experimental error. This superfluid flow with vortex sheets becomes unstable at velocities higher than the critical one, and quantum vortex rings and lines are formed. In thin capillaries and in a film this is connected with very great losses, because to create one vortex line it is necessary to concentrate kinetic energy in a length of helium a hundred times larger than the film and capillary thickness. References p . 36
36
V. P. PESKOV
[CH.1
In large capillaries and with oscillating discs the critical phenomena are connected not only with the appearance of the first vortex lines, but also with the kinetics of the development of quantum turbulence. The flow of the superfluid component relative to the walls, i.e. us is responsible for the creation of the first vortex sheets and lines. It was impossible to observe this process directly owing to the poor detectability of the latter. The counterflow of the superfluid and normal parts, i.e. us-uu,, governs the kinetics of the development of quantum turbulence. For a more complete revelation of the nature of the critical velocities it is necessary to get more precise values of the surface energy of vortex sheets in superfluid helium, as well as to clarify the kinetics of the transformation of vortex sheets into vortex lines and the further development of quantum turbulence. The first question requires further experimental work, while the second one involves both theoretical calculations and further experimental checking. REFERENCES P. L. Kapitza, Nature 141, 74 (1938). J. F. Allen, A. D. Misener, Nature 141, 75 (1938). P. L. Kapitza, Zhur. Eksp. Teor. Fiz. SSSR 11, 581 (1941). 4 J. G . Daunt, K. Mendelssohn, Proc. Roy. SOC.A 170,423,439 (1939). K. R. Atkins, Proc. Roy. SOC.A 203, 119 (1950). B. S. Chandrasekhar, K. Mendelssohn, Proc. Phys. SOC.A 64,512 (1951). 7 G. R. Hebert, K. L. Chopra, J. B. Brown, Phys. Rev. 106, 391 (1957). E. J. Burge, L. C. Jackson, Proc. Roy. SOC.A 205,270(1951); Phil. Mag. 41,205 (1950) R. Bowers, Phil. Mag. 44, 1309 (1953). lo E. Ambler, N. Kurti, Phil. Mag. 43, 260 (1952). l1 R. K. Waring, Phys. Rev. 99, 1704 (1955). I2 B. N. Eselson, B. G. Lazarev, Zhur. Eksp. Teor. Fiz. SSSR 23, 552 (1952). l3 J. F. Allen, Nature 185, 831 (1960). l4 H. Seki, C. C. Dickson, Proc. VII Int. Conf. Low Temp. Phys., Toronto, p. 569 (1960) l5 P. Winkel, A. M. J. Delsing, J. D. Poll, Physica 21, 331 (1955). l8 E. L. Andronikashvili. Zhur. Eksp. Teor. Fiz. SSSR 16, 780 (1946). l7 W. F. Vinen, Proc. Roy. SOC.A 240, 114 (1957). la W. F. Vinen, Proc. Roy. SOC.A 243, 400 (1957). l8 C. E. Chase, Phys. Rev. 127, 361 (1962). 2o V. P. Peshkov, V. B. Strukov, Zhur. Eksp. Teor. Fiz. SSSR 41, 1443 (1961). J. N. Kidder, W. M. Fairbank, Proc. VII Int. Conf. Low Temp. Phys.,Toronto, p. 560 (1960). 22 K. Mendelssohn, W. A. Stele, Proc. Phys. SOC.73, 144 (1959). 23 V. P. Peshkov, V. K. Tkachenko, Zhur. Eksp. Teor. Fiz. SSSR 41, 1427 (1961). a4 A. C. Hollis-Hallett, Proc. Roy. SOC.A 210, 404 (1952). 25 C. B. Benson, A. C. Hollis-Hallett, Can. J. Phys. 34, 668 (1956). 2a E. L. Andronikashvili, Thesis, Moscow (1948). 27 G. A. J. Gamzemlidze, Zhur. Eksp. Teor. Fiz. SSSR 37,950 (1959). 2
CH. I]
CRITICAL. VELoclTIEs AND VORTICES
37
P. J. Bendt, Phys. Rev. 127, 1441 (1962). K. R. Atkins, Liquid Helium (Cambridge, 1959) p. 199. L. D. Landau, Zhur. Eksp. Teor. Fiz. SSSR 11, 592 (1941). 3 l L. D. Landau, Zhur. Eksp. Teor. Fiz. SSSR 14, 112 (1944). 32 J. G. Dash, Phys. Rev. 94, 825 (1954). 33 V. L. Ginsburg, Zhur. Eksp. Teor. Fiz. SSSR 29, 254 (1955). 34 L. D. Landau, E. M. Lifshitz, Dokl. Akad. Nauk SSSR 100,669 (1955). 96 G. A. Gamzemlidze, Zhur. Eksp. Teor. Fiz. SSSR 34, 1434 (1958). 36 C. G. Kuper, Physica 22, 1291 (1956); Physica 24, 1009 (1958). 37 L. Onsager, Nuovo Cimento 6 (Suppl. 2), 249 (1949). 38 W. F. Vinen, Proc. Roy. SOC.A 260, 218 (1961). 39 R. P. Feynman, Progress in Low Temperature Physics, Ed. C. J. Gorter, Vol. I, p. 17 (North-Holland Publishing Co., Amsterdam, 1955). 40 H. E. Hall, Phil. Mag. 9 (Suppl.) N 33, 89 (1960). 41 E. L. Andronikashvili, D. S. Zakadze, Zhur. Eksp. Teor. Fiz. SSSR37,322,562(1959). 42 W. F. Vinen, Progress in Low Temperature Physics, Ed. C. J. Gorter, Vol. 111, p. 1 (North-Holland Publishing Co., Amsterdam, 1961). 43 V. P. Peshkov, Proc. VII Int. Conf. Low Temp. Phys., Toronto, p. 555 (1960). 44 R. Meservey, Phys. Rev. 227, 955 (1962). 46 C. J. Gorter, J. H. Mellink, Physica 15, 285 (1949). 28
Z8
CHAPTER I1
EQUILIBRIUM PROPERTIES OF LIQUID AND
SOLID MIXTURES OF HELIUM THREE AND FOUR BY
K. W. TACONIS KAMERLINGH ONNES
AND
R. DE BRUYN OUBOTER
LABORATORIUM, LEIDEN, NEDERLAND
CONTENTS:1. Introduction, 38. - 2. The equilibrium between vapour and liquid mixtures (dew- and boiling-curve), 49. - 2.1. General survey of the experimental data, 49. - 2.2 Calculation of the excess chemical potentials and the excess Gibbs function, 51. - 2.3. The equilibrium between vapour and a dilute liquid mixture of 3He in 4HeII; equilibrium with respect to the solute, 57. - 3. The equilibrium between the He I and the He I1 phase (I-line) and between two liquid mixtures in the stratification region (the phase separation diagram), 59. - 3.1. The lambda transition, 59. - 3.2. The Keesom-Ehrenfest relations for a liquid mixture and the discontinuity in the slope of the first order equilibrium curves at the junction with the second order lambda curve, 61. - 3.3. The phase separation diagram, 64. - 4. The osmotic equilibrium in He I1 (pseudo thermostatic equilibrium) and some applications, 72. - 4.1. The osmosis in He I1 derived from the equilibrium between pure liquid 4He on one side and a mixture on the other side of a superleak (semipermeable wall); equilibrium with respect to the solvent, 72. - 4.2. The quasi-equilibrium between the osmotic and the fountain force in the liquid mixture, when a heat current is present (the heat flush effect), yielding the diffusion coefficient and the heat conductivity of the mixtures, 74. - 4.3. The quasi-equilibrium between two mixtures of slightly different concentration in the He I1 region separated by a capillary, yielding the viscosity of the mixtures, 77. - 4.4. The quasi-equilibrium between two liquid mixtures of different concentration in the He11 region, connected by the helium surface ~, yielding the isothermal flow rate of these films, 80. - 4.5. A refrigeration cycle, as an application of the osmotic pressure and the heat of mixing, 81. 5. The equilibrium between liquid and solid mixtures (freezing- and melting-curve) and the phase separation of solid mixtures, 84. - 5.1. The freezing pressures of 3He-4He mixtures, 84. - 5.2. The equilibrium between two solid mixtures in the phase separation region, 87. - 5.3. The minima in the freezing curves of 3He-4He mixtures and the minima in the melting curves of pure 3He and pure 4He, 90.
1. Introduction The isotopes 3He and 4He are the only two stable substances which remain liquid down to absolute zero, in contradiction to the predictions of classical References p . 93
38
CH. 11,
5 11
39
EQUILIBRIUM PROPERTIES
statistics that all substances ought to be solid at absolute zero. It is a consequence of the weakness of the interatomic attractive forces (LondonVan der Waals forces) and the large zero-point energy of the atoms in the liquid, which corresponds to an extra repulsive force between the atoms. The interatomic attractive potentials, as a function of the distance, are almost the same for 3He and 4He, but the zero-point energy varies inversely proportional to the atomic mass. Due to this difference in zero point energy, pure liquid 'He has a larger molar volume V and a larger vapour pressure P , and the absolute value of the internal energy U is smaller than in pure liquid 4He. Both isotopes have the same Lennard-Jones parameters: Elk = 10.2"K and = 2.56 A. Some of the properties of the liquid isotopes at absolute zero are summarized in the following Table 1 : TABLE l1 ~
__
3He
36.6 Molar volume VL.O(crn3/mole) Mean distance d ( A )between the atoms in the liquid 3.9 (joule/mole) is minus the heat of vaporiInternal energy UL,O zation at absolute zero LL,O - 21.2 Compressibility ,yL,o(cm3/joule) 0.361
4He 27.5 3.6 - 59.62
0.1203
Liquid 'He-4He mixtures show strong positive deviations from an ideal liquid mixture (positive heat of mixing HE,positive excess Gibbs function GE),which are in the first instance caused by the large difference in the internal energy U and the molar volume V of the pure components. Neglecting entropy effects, one may expect, according to Prigogine, that the mixing becomes ideal if one has first brought the pure isotopes to the same molar volume as they have in the mixture. The excess Gibbs function GE and also the heat of mixing HEis then equal to the work done by compressing the lightest isotope, pure liquid 3He, and by expanding the heavier isotope, pure liquid 4He, to the right mixture volume. If a t a certain critical temperature T, the heat of mixing H E becomes larger than the product of the temperature T and the entropy of mixing
the tendency to find a situation of minimum energy overcompensates for References p . 93
40
K.W.TACONIS A N D R. DE BRUYN OUBOTER
[CH.II, 8 1
the tendency of maximum entropy, or maximum disorder. Hence the mixture becomes unstable below the critical temperature T,= GE/(+R) =0.8"K and the mixture separates below this temperature into two phases of different composition. Another consequence of the large positive excess is the fact that the solubility of 3He in liquid 4He is small and the vapour phase is much richer in 3He than the liquid phase, particularly at lower temperatures. In addition to their difference in mass, the isotopes 3He and 4He also obey different statistics. The specific heat C, of liquid 4He, first studied by Keesom, shows a sharp peak at 2.17"K,which corresponds to a &transition from the superfluid state, called He I1 by Keesom, to the normal state, He I. The superfluid properties of liquid 4He have been attributed by F. London to the pecularities of Bose-Einstein statistics, since the 4He atom contains an even number of fundamental particles, and condensation in momentum space to zero momentum occurs together with a scarcity of low energy excitations. However the 3He atom contains an odd number of fundamental particles and obeys Fermi-Dirac statistics. Pure liquid 3He shows no indication for superfluidity (or ktransition) down to 0.0I"K. Near absolute zero liquid 3He has an abundance of low energy excitations in contrast to liquid 4He. The Idransition plays an important role in the thermodynamic properties of liquid 3He-4He mixtures. The addition of 3He to liquid 4He lowers its &temperature and the height of the peak in the specific heat at the Atemperature falls rapidly with increasing concentration. Below 1°K the excess entropy SE becomes negative and the total entropy of mixing starts to decrease to zero with decreasing temperature in accordance with Nernst's heat theorem. In dilute mixtures of 3He in liquid 4He I1 the 3He atoms can, according to Pomeranchuk, be treated as free particles which move through the superfluid as an ideal gas. The specific heat below 1°K is constant and equal to that of an ideal monatomic gas. In the two fluid model, as proposed by Landau and Tisza, the 3He forms a part of the normal component and does not participate in the flow of the superfluid component. The acceleration of the superfluid is equal to the negative of the gradient of the chemical potential of the 4He per unit of mass. This means that the superfluid moves towards regions of high temperature or high concentration, when temperature and concentration gradients are present in the fluid. If two quantities of liquid of different 3He concentration are separated in space by a narrow constriction (superleak), an osmotic pressure difference will be observed, References p . 93
CH. ll, 8
11
EQUILIBRIUM PROPERTIES
41
since only superfluid 4He can pass frictionless through the superleak (semipermeable wall). The osmotic pressure is connected to the ideal gaspressure of the solute (3He) in the solvent (4He 11). Solid helium can be produced only by applying a pressure, as proved by Keesom. This is a consequence, as Simon first pointed out, of the large zero-point energy. At absolute zero both liquid and solid have zero entropy and the latent heat of melting is zero. Melting is then a pure mechanical process. The internal energy of the solid is greater than that of the liquid. At low pressures the liquid is the stable phase. The solid is only formed at pressures where its smaller volume causes a sufficient decrease of the p V , term in the Gibbs function Go = Ho = Uo + p Vo to compensate the increase in the internal energy Uo. At absolute zero the change in internal energy on melting A uo 1
is equal to the negative of the product of the pressure p times the change in volume on melting d Uo = p A Vo .
-
1
1
Solid mixtures are unstable below a critical temperature T, = Gt/(+R)= Ws/(2R)=0.38"K and have a regular behaviour. The heat of mixing is temperature independent and equal to El: = G: = Xs(1 - X s )Ws, and there is no excess specific heat CE and no excess entropy Si down to 0.05"K. This means that above this temperature the entropy of mixing of the solid is given by the temperature independent classical expression:
On the other hand in the liquid quantum degeneracy of the entropy of mixing already starts below 1°K and we get the peculiar situation that below 1°K the entropy of the liquid becomes lower than the "constant" entropy of the solid. Consequently the freezing-pressure curves of 3He-4He mixtures have a minimum at a certain temperature and below this temperature a negative slope. Before reviewing the various features in more detail we want to present here a survey of the properties which were studied together with the many physicists who have contributed to the present state of affairs.
References p . 93
42
[CH.11, 8
K. W. TACONIS AND R. DE BRUYN OUBOTER
Measured by Sommers Esel'son, Berezniak Roberts, Sydoriak
Number of reference
T (OK)
'
1.3 -2.17 1.35-3.2 0.6 -2
1
X 0-0.13 0-1 0-1
Boilingcurve Peshkov, Kachinskii Sreedhar, Daunt
5
0.936 0-0.12
8
___-
Dewcurve
Sommers Esel'son, Berezniak
2
3
1.3 -2.17 1.4 -2.6
0-0.8 (M.84 ~~
Distribution coefficient CVICL
Osmotic pressure
Specific heat CL
Heat of mixing H L ~ Volume of contraction VLE
Mine ( Ta-X)
References p . 93
Wansink, Taconis, Staas
Daunt, Probst, Johnston, Aldrich, Nier Beenakker, Taconis, Dokoupil Wansink, Taconis London, Clarke, Mendoza Dokoupil, Van Soest, Wansink, Kapadnis, Sreeramamurthy, Taconis de Bruyn Ouboter, Taconis, le Pair, Beenakker Sommers, Keller, Dash de Bruyn Ouboter, Taconis, le Pair, Beenakker Kerr Ptucha Dash, Taylor Abraham, Osborne, Weinstock Daunt, Heer Kerr Dash, Taylor Elliot, Fairbank Esel'son, Berezniak, Kaganov Zenove'va, Peshkov Dokoupil, Van Soest, Wansink, Kapadnis, Sreeramamurthy, Taconis
7
1.2 -2
XL
0
--f
8 9
1.2 -2 0.8 -1.2
0-0.04
'1
l2
1.1 -3
0-0.417
0.4 -2.2
0-1
l4
1.02
0.086
l3
0.4 -2
0-1
l8
3.2 -1.2
0-1
lo
0-0.5
--f
19 20
21 22
18 20
24 25
26
0-XA.
ia
0-0.417
CH. 11,
'
$11
43
EQUILIBRIUM PROPERTIES
(" K)
23
1.2-2.2
0-0.07
0.5-2.2 1-2
0-1 0-1
23
1.2-2.2
0-0.07
16
1-2
0-4
0.4-2
0-1
Y4E
Calculation of GE,c(Pby: Wansink Sreedhar, Daunt de Bruyn Ouboter, Beenakker, Taconis Roberts, Swartz
CE
theoretical considerations on GE by: Prigogine, Bingen, Bellemans, Simon
17
application : cooling cycle based on H E and osmotic pressure proposed by: London, Clarke, Mendoza
11
Y3E
HE calculated from boiling curve by: Sommers Wansink HE, S E calculated from boiling curve by: Roberts, Swartz
T
Number of reference
X
6
'5 16
2
SE calculated from H Eand CE by SE
de Bruyn Ouboter, Taconis, le Pair, Beenakker
Keesom-Ehrenfest relations : Keesom Ehrenfest stout de Boer, Gorter Esel'son, Lazarev, Lifshitz, Kaganov de Bruyn Ouboter, Beenakker Roberts, Swartz Sadnikidze References p , 93
l3
30
LO4
31
A04
32
I I 1 I , I'
33
34
35 18 38
I 1*
44
K. W.TACO-
AND
[CH.ll, 8 1
R. DE BRUYN OUBOTER
~~~~~~~~~~
Measured by
Nine (Ta-X)
de Bruyn Ouboter, Taconis, le Pair, Beenakker Roberts, Sydoriak Esel'son, Ivantsov, Shvets I-line at different pressures: Fairbank le Pair, Taconis, Das, de Bruyn Ouboter
-
Phaseseparation curve liquid ( TP.S-X)
Velocity of second sound in dilute mixtures
Normal density pn measured by means of the Andronikashvilli method
Referencesp. 93
Number of reference
T (OK)
13
&xA*
4
O-XA*
27
28
29 ~
Walters, Fairbank Zenov'eva, Peshkov Roberts, Sydoriak de Bruyn Ouboter, Taconis, le Pair, Beenakker Brewer, Keyston phase-separation curve in the liquid at different pressures: Fairbank le Pair, Taconis, Das, de Bruyn Ouboter Zenov'eva
X
as 26 4
0.5-0.88 0.5-0.88 0.5-0.88
13
0.4-0.88
39
0.15
28
29 88
Lynton, Fairbank King, Fairbank Kraniers, Niels-Hakkenberg Sandiford, Fairbank
45
Pellam Berezniak, Esel'son Dash, Taylor
50
46 47 48
51 20
CH. Il, 9
I]
45
EQUILIBRIUM PROPERTIES ~~~~~
~
Number of reference theoretical considerations on the kline: Heer, Daunt
37
theoretical considerations on the phase separation curve: 17
Nernst’s heat theorem with respect to the entropy of mixing: Keesom H e r , Daunt Cohen, Van Leeuwen de Bruyn Ouboter, Beenakker
37
Dilute liquid mixtures: Landau, Pomeranchuk, Zharkov, Silin Feynman Linhart, Price de Bruyn Ouboter, Taconis, le Pair, Beenakker ’
2
Sommers Prigogine, Bingen, Bellemans, Simon Cohen, Van Leeuwen
theoretical considerations on the effective mass m*3; Feynman Chester
References p . 93
40
41
40 35
42 43
44
13
43
49
T (OK)
X
46
K. W. TACONIS AND R. DE BRUYN OUBOTER
Number T of (OK) reference
Measured by Viscosity vn
Diffusion; heat-conductivity
Staas, Taconis, Fokkens Dash, Taylor
52
20
Beenakker, Taconis, Lynton, Dokoupil, Van Soest Ptucha
53 54
Absorption first sound
Harding, Wilks Guptill, Van Iersel, David
55
Absorption second sound
Niels-Hakkenberg, Kramers
47
Surface film mobility
Van den Meijdenberg, Taconis, le Pair Esel'son, Shvets, Bablidze
57
Edwards, McWilliams, Daunt
59
Vignos, Fairbank le Pair, Taconis, Das, de Bruyn Ouboter Weinstock, Lipshultz, Lee, Kellers, Tedrow Berezniak, Bogoyavlenskii, Esel'son Zenov'eva
60
56
.-
Phase separation solid
Freezing-curve
Melting-curve
58
29
61 85 88
in progress
___ .-
Heat conductivity solid mixtures
Walker, Fairbank,
62
In this chapter the following notation is used:
N X
C S H
= number of moles = = = =
mole fraction (3He of a 3He-4He mixture); X i = N i / c i N i molar ratio; C = X / ( 1- X ) entropy enthalpy (heat function)
References p . 93
[CH.11,s 1
X
CH. 11, 8 11
47
EQUILIBRIUM PROPERTIES
.___
Zharkov, Khalatnikov
________ 64
-_____ Khalatnikov
85
Khalatnikov
65
Prigogine, Bingen, Bellemans Montroll, Potts theoretical considerations on the freezing-curve: Lifshitz, Sanikidze le Pair, Taconis, Das, de Bruyn Ouboter Beremiak, Bogoyavlenskii, Esel'son
Sheard, Ziman Klemens, Maradudin
L G p C, T T,
17 88
87
29 89
68
a7
= heat of vaporization = Gibbs function (thermodynamic potential); G = x i x i p i = partial chemical potential = heat capacity at constant pressure = absolute temperature =
critical temperature of mixing (upper consolution point), top of the phase separation curve
References p . 93
48
K.W. TACONIS A N D R. DE BRUYN OUBOTER
[CH. ,!I
81
Tp.s = phase separation temperature Tp.s(X);P.S. = the phase separation curve (stratification curve) TA = lambda temperature; I = lambda curve or second order phase transition line V = molar volume p = pressure Bil = second virial coefficient arising from the interactions between molecules of species i and j posm= osmotic pressure j3 = expansion coefficient, defined by j3 E (a V/aT),,/ V x = compressibility, defined by x = -(aV/ap)/V. All quantities will be expressed per mole. The subscript i refers to the ith component. Subscripts 3 and 4 refer to the component 3He and 4He respectively, e.g. Pi is the partial vapour pressure of the iihcomponent. The subscript L refers to the Liquid phase, the subscript S to the Solid phase and the subscript V to the Vapour phase. The superscript O is used to denote quantities referring to a pure substance. The subscript refers to the absolute zero temperature T = 0°K. In the case of phase separation of two liquid phases the subscripts u and 1 refer to the upper and lower phase respectively. The partial molar entropy, enthalpy and volume are indicated by Si, Hi and Vi respectively and we define the partial molar thermodynamic quantity Q,by means of the relation: Qi 3
Q
aQ + Xj-axi
aQ ax,
Q + (1 - Xi) -.
The superscript is used to denote the excess functions. The heat of mixing H E ( X ,T ) (the excess enthalpy) is defined as the heat required to keep the temperature T constant when X, mole of pure liquid 3He is added to (1 - X,) mole of pure liquid 4He; @(XL, T ) = HL(XL, T ) - (XLH,",(T)
+ (1 - XL)H,"L(T))
*
The excess entropy Sf is defined by means of the relation:
SL(T X,)
= XLS3OL(T )
+ (1 - XL)%(
- RIX,InXL
T )+
+ (I - X,)In(l-
x,)]
and the entropy of mixing is given by the expression :
References p. 93
+ S,"(TXL)
CH. 11, $21
EQUILIBRIUM PROPERTIES
49
The excess Gibbs function GE is given by the relation:
GE = HF - TSE = X,C(!L
+ (1 - XL)p:L,
JQ = QA - QB 1
is the difference between the equilibrium values of the quantity Q at both sides of the first order equiJibrium lines a and b.
AQ 2
= QkG + 0) -
QdG.- 0)
is the difference between values of the quantity Q at both sides of the second order transition curve (lambda curve) which separates the fluid phase 11 from the fluid phase I. p = density p n = density of the normal component (phonons, rotons, 3He particles) x = p , / p =normal fraction U, = velocity of first sound U2 = velocity of second sound m; = effective mass of the 3He particle in He I[ Eo3 = effective potential of the 3He particle in He 11. The subscript n refers to the normal component and the subscript s refers to the superfluid component. = viscosity K = effective heat conductivity.
2. The Equilibrium between Vapour and Liquid Mixtures (Dew- and Boilingcurve) SURVEY 2.1. GENERAL
OF THE
EXPERIMENTAL DATA
The dew- and boiling-points can be measured by introducing a 3He - 4He gas mixture of known composition (X)into an equilibrium vessel at constant temperature ( T ) (see fig. 2, T = 1.7”K). The dew-point (D) is determined by observing the pressure at which condensation commences and the first drop of liquid is formed, having a concentration XI. Continuing the condensation process, the liquid will finally reach the average concentration R when the equilibrium vessel is filled. Then the pressure of the boiling-point can be determined, and in the mean time the vapour has reached the concentration ( X 2 ) . Whereas the early measurements of the vapour pressure of mixtures were only dealing with “very dilute solutions”, and consequently often in error References p . 93
50
[CH.11, I 2
K. W. TACONIS A N D R. DE BRUYN OUBOTER
due to demixing effects from unavoidable heat flows, the first really reliable data on the boiling- and dew-curves were presented by Sommers2 (up to 12%); later on boiling point measurements were made by Wansink7 for dilute mixtures, studying especially the behaviour of the distribution coefficient C& and by Sreedhar and Daunte (up to 12%).Finally in 1960 the whole concentration range was investigated by Roberts and Sydoriak4 for the boiling curves, and a welcome set of dew-point data was given by Berezniak and Esel'son3, whose boiling point measurements were less successful when submitted to a consistency test originating from thermodynamic considerationslsv 6% 70. The data obtained by Roberts and Sydoriak4are shown in Fig. 1 and Table 2. The pressures in the equilibrium vessel are compared with a pure 'He vapour pressure thermometer and both are in an excellent heat contact with an extra 'He bath, used to cover the temperature range from 0.6 to
09-
08
-
07 -
0597
----e
Ob-
04-
I
I
12
I
I
16
I
I
20
\
I
I
2 A d
Fig. 1. The boiling point measurements of Roberts and Sydoriak4 are shown as a plot of the ratio of mixture vapour pressure, Px,to measured 3He vapour pressure P$. The lambda line is shown in dashes. Measurements below the stratification temperature are omitted in this figure. References p . 93
CH. 11,
5 21
51
EQUILIBRIUM PROPERTIES
2.4"K. Therefore in Fig. 1 the vapour pressure is plotted relative to the 'He vapour pressure. The boiling point measurements of Roberts and Sydoriak4 show a clear discontinuity in the temperature derivative of the vapour pressure at the &point (c.f. Fig. 1). Later on in Section 3.2, the magnitude of this discontinuity will be discussed and compared with the discontinuity in the specific heat a t the A-point. 2.2. CALCULATION OF THE EXCESS CHEMICAL POTENTIALS AND GIBBSFUNCTION
THE
EXCESS
The first person to interpret with success the vapour-liquid equilibrium data on 3He-4He mixtures by means of classical solution theories was Sommers 2. Wansink 23, Sreedhar and Daunt6, De Bruyn Ouboter, Beenakker and T a c o n i ~and ~ ~Roberts and Swartz16 calculated the excess chemical potentials pFL and the excess Gibbs-function GE from the vapour pressure using the following procedure. From the equilibrium condition between the liquid and vapour phase piL= piv, it follows that the excess chemical potential pFLcan be determined from the following relation:
+ RTlnX,, + p k = = plv + R T In Xi, + R T In (P,,,/PF) + (PI",- PF)Bii. piL = pL :
(1)
If we define the partial pressure Pi by Pi = XivPIo,and use the equilibrium condition for the pure component pfL= p&, we get the following expression for the chemical excess potential : p k = R T In (Xiv?,JXiLPF)
+ (Pt,, - P/)Bii =
+ (PI,,- P,")Bii.
= R T In (P,/XiLPF)
(2)
Hence it is in principle possible to calculate p k if the pure substances are known, and if one knows the total pressure as a function of the liquid concentration (boilingcurve) and the concentration of the vapour in equilibrium with this liquid (dewcurve). We can write the Gibbs-Duhem equation:
- Sd T + Vdp = 1X i dpi in the following form:
References p . 93
I
(3)
T A B L E2 VI N
2
2 p
t?
The smoothed boiling-point measurements of Roberts and Sydoriak 4. Smoothed values of R = lo00 Px/PsO, at stratification temperatures (Table 2a), at lambda temperatures (Table 2b) and at 0.1"K temperature intervals (Table 2c), for values of X , the He3 mole fraction. Parentheses enclose values of TSbelow the range of our measurements and the somewhat uncertain interpolations near the consolute temperature. In Table 2c asterisks separate He I1 and He I phases (See section 3.1) and the letter "S" designates the region of stratification, see Section 3.3. For purposes of extrapolation of Table 2c it should be remembered that for each value of X the curve of R versus T terminates at Ts. When interpolating near T Aone should also note that the slope changes abruptly at Ta, except for X = 0, for which there is an inflection at T?. (see Section 3.2)
X
Table 2a
1
Rs
1
Ts
P
Table 2b
Table 2c: Values of
~
TA
0.6" 0.7" 0.8" 0.9"
Ra
.E
R at the following temperatures: 1.0" 1.1" 1.2" 1.3" 1.4"
-3
~
1.5" 1.6" 1.7" 1.8" 1.9"
2.0" 2.1" 2.2" 2.3
'
~
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38
(0.00) (0.08) (0.16) (0.22) (0.27) (0.31) (0.35) (0.38) (0.42) (0.45) 0.477 0.507 0.536 0.563 92 0.588 90 0.614 89 0.636 88 0.658 881 0.679 87 0.700 86
2.1735 186 2.145 205 2.116 223 2.087 240 2.058 260 2.028 279 1.998 298 1.967 316 1.936 334 1.904 352 1.873 368 1.840 386 1.807 402 1.773 419 1.738 436 1.704 453 1.667 472 1.630 430 1.592 507 1.553 526
1 166 290 395 480 558 629 687 735 775 805 834 859 879 899
,
S S S S S
2 128 230 319 405 478 547 605 652 692 726 751 771 792 810 824 835 847 856 863
4 106 195 277 356 425 487 541 586 623 654 681 704 724 742 157 771 782 793 802
14 22 31 8 95 89 89 91 173 160 152 150 249 228 214 205 319 292 273 259 384 352 328 309 440 405 377 354 490 450 418 392 532 490 455 426 567 522 485 456 597 551 513 482 625 579 538 505 648 601 561 528 669 621 582 548 687 640 599 565 703 656 616 581 718 670 630 596 730 683 642 608 740 694 654 620 749 704 664 630
43 98 150 201 250 296 337 371 403 432 456 480 500 519 536 551 565 578 589 599
56 107 154 200 246 287 324 356 385 411 434 455 476 493 510 524 537 549 560 570
71 87 104 122 140 117 129 142 156 170 161 170 179 189 199 204 209 214 22 1 228 245 245 247 249 253 28 1 279 277 277 278 315 309 303 301 299 345 334 327 323 318" 371 360 350 343 336 * 395 381 369 360 352' 416 401 387 375*369 436 420 403 389'386 454 435 418 403 *403 471 452 431 * 418 420 486 465 443*435 435 500 417 454'451 452 512 485 '470 467 468 522 496*486 483 484 532*506 501 499 500 540'520 516 515 516
157 174' 190 184 199*211 210 221 * 232 234*241 252 257" 263 272 279 *285 293 298 305 313 319 324 331 337 342 348 355 359 365 373 378 384 390 393 400 406 410 416 423 427 433 439 443 448 455 459 464 472 476 479 487 492 496 503 507 511 519 523 527
205 225 245 264 284 302 322 339 357 374 391 407 423 438 454
6 3 0
b
0
'
470 485 501 516 532
&
& N
TABLE 2 (continued)
a
-
2
-
3A -
X
2
0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98
Table 2
Table 2b
1
Table 2c: Values of R at the following temperatures:
Ts
Rs
Ta
Ri
0.6" 0.7" 0.8' 0.9"
0.718 0.737 0.753 0.769 0.783 0.796 0.809 0.822 0.832 0.841 0.848 (0.86) (0.87) (0.87) (0.86) 0.836 0.811 0.781 0.754 0.724 0.695 0.667 0.636
854 846 839 833 827 821 817 811 806 803 800
1.514 1.472 1.430 1.387 1.342 1.297 1.250 1.203 1.154 1.105 1.055 1.005 0.954 0.904
545 564 581 600 617 635 653 671 690 709 728 748 766 786
S S S S S S S S S S S S S S S S S S S S S S S 905 917 930 942 956 970 983
805 815 828 839 852 863 877 890 0.604 903 0.570 920 0.531 (0.48) (0.43) (0.36) (0.26)
--
S S S S
S S S S S S S S S S S S
s
S S S 861 871 880 892 904
918 933 947 964 981
807 812 816 818 820 821 S S S S S S S S S S
s
825 832 840 849 860 871 884 898 912 928 943 961 980
1.0'
1.1'
1 577 1
1.2" 1.3" 1.4"
757 713 673 638 606 763 720 681 646 614 584' 768 726 687 652 620 590' 772 731 693 658 626 * 596 775 734 697 662 630'607 618 777 738 701 666'634 778 740 703 670*644 630 780 742 706 672" 654 643 781 744 708 *681 665 656 782 746 711 * 689 677 669 783 747 * 717 699 689 683 785 749'725 710 701 696 785 754 735 723 716 712 787 762 745 735 729 726 789 770 756 747 743 740 793 779 767 761 756 755 799 788 780 774 771 770 807 798 792 788 785 784 817 809 805 802 801 798 829 822 818 816 815 814 841 853 853 829 829 829 852 848 846 844 844 845 865 862 860 859 860 860 880 876 875 876 876 876 894 892 892 891 892 892 910 908 908 908 909 909 927 925 925 926 927 927 942 941 942 944 945 945 960 959 960 952 963 963 979 979 980 680 981 981
1.5" 1.6" 1.7" 1.8" 1.9"
2.0" 2.1" 2.2" 2.3"
547 * 536 560 550 572 565 584 578 597 593 610 607 624 621 638 636 652 650 667 666 68 1 681 696 696 71 1 71 1 726 726 740 741 755 756 770 770 785 786 800 800 815 816 831 831 846 848 862 863 878 879 893 895 910 91 1 927 928 946 946 964 964 981 981
536 551 567 583 598 612 627 642 656 67 1 684 699 715 729 745 759 774 789 804 820 834 850 866 881 897 914 931 947 964 982
530 545 561 576 592 607 62 1 636 651 666 682 697 71 1 727 742 756 771 787 802 817 832 848 864 879 896 912 929 946 964 981
530 546 562 578 593 608 623 638 653 667 682 697 713 728 743 757 772 787 802 818 833 848 864 879 896 912 929 946 964 982
532 548 564 579 595 609 625 640 654 668 683 698 713 728 743 758 773 788 803 819 834 849 865 880 896 913 929 946 964 982
--- 7
539 554 570 585 601 616
542 558 574 588 604 619
547 563 578 593 608 623
54
K.W. TACONlS A N D R. DE BRUYN OUBOTER
[CH. II,
02
Even when P is not strictly constant the terms on the right hand side of the above equation (4) are usually negligible compared with any one term on the left hand side. This equation (4) may be rewritten in the following integral form70: XL P 4EL
=
- J-d& X L 1
(const. T, P,,,).
-x,
(5)
0
J
I O - ~ ~ ~ T=1.70°K
mmH9
rnrnng It
15
10
100
1c
IC
C
5
i P
pt 1 " x
0 5
C
0
10-4t
nrn Hg 1
i P
/ / / 3
Fig. 2. Vapour-liquid equilibrium diagrams at 0.7, 0.8, 1.1 and 1.7' K. 17 Boiling-point measurements of Roberts and Sydoriak4. 0 Boiling- and dew-point measurements of Esel'son and Berezniak3. The dew-curves are calculated by De Bruyn Ouboter, Beenakker and Taconis15.
References p . 93
CH. 11,s
21
55
EQUILIBRIUM PROPERTIES
Formula ( 5 ) enables us to compute the excess chemical potential ptL at a concentration X,, provided we know the excess chemical potential ,uyL at all compositions intermediate between zero concentration and the concentration X,. Partial integration of (4) over the whole concentration range gives us the following expression:
j.. i
p 3 , dXL = ptLdX,
(const. T,P,,,) .
(6)
0
0
Satisfaction of this equation may be considered a necessary but not sufficient test of the thermodynamic consistency of experimental data. Extending this method, due to Redlich and Kister69.70, it is possible to write eq. (6) in a somewhat different form :
1
1
1
De Bruyn Ouboter, Beenakker and Taconisls used eq. (7), the method of Redlich and Kister69.70, as a test for the consistency of the experimental boiling- and dew-curves of Esel’son and Berezniak3. Their data was found to be not consistent. It appears that the very accurate boiling-point measurements of Roberts and Sydoriak4 are the most reliable’s. The calculation of the excess chemical potentials p: 15 are mainly based on the boilingpoint measurements of Roberts and Sydoriak4 (see Fig. 1 and Table 2), and the dew-point measurements of Esel’son and Berezniak3 are only used in a zero approximation, since these measurements do not have the required accuracy for a thermodynamic calculation. The excess chemical potentials are in principle calculated by the following method: As a zero approximation we use the dew-point measurements, from which we can calculate the partial vapour pressure P3= XVP,,,, since the total pressure Ptotis known from the boiling point measurements made by Sommersz and by Roberts and Sydoriak4. Hence &, is known to a zero approximation at different concentrations. Now it is possible to determine p t L to a first approximation by graphical integration of eq. (5). From p:, we derive the partial pressure P4 and then have Xv = 1 - (P4/Pt0,)in a first approximation. Once the first References p . 93
VI
a
T A B L E3 The mole fraction of 3He in the vapour, XV,calculated by Roberts and SwartzlG 1.o
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.O"K
0.04
0.9173
0.8620
0.7996
0.7217
0.6445
0.5702
0.5006
0.4328
0.3715
0.3154
0.2692
0.08
0.9560
0.9254
0.8872
0.8376
0.7834
0.7248
0.6622
0.5992
0.5349
0.4752
0.4186
0.1
0.9636
0.9388
0.9056
0.8647
0.8168
0.7631
0.7070
0.6470
0.5871
0.5284
0.4716
0.2
0.9782
0.9632
0.9431
0.9174
0.8859
0.8492
0.8078
0.7618
0.7124
0.6665
0.6320
0.3
0.9826
0.9709
0.9553
0.9352
0.9105
0.8813
0.8470
0.8083
0.7771
0.7487
0.7224
0.4
0.9846
0.9745
0.9611
0.9438
0.9226
0.8968
0.8726
0.8494
0.8278
0.8069
0.7874
0.5
0.9855
0.9763
0.9639
0.9483
0.9318
0.9152
0.8989
0.8830
0.8671
0.8510
0.8358
0.6
0.9859
0.9775
0.9677
0.9569
0.9454
0.9339
0.9222
0.9106
0.8982
0.8861
0.8741
0.7
0.9876
0.9814
0.9746
0.9671
0.9593
0.9511
0.9429
0.9344
0.9258
0.9172
0.9086
0.8
0.9906
0.9866
0.9822
0.9774
0.9724
0.9672
0.9618
0.9564
0.9508
0.9454
0.9398
0.9
0.9948
0.9928
0.9906
0.9883
0.9858
0.9831
0.9805
0.9779
0.9751
0.9725
0.9700
Y
XL
CH. I t , §
21
EQUILIBRIUM PROPERTIES
57
approximation is calculated, the whole procedure is repeated. The excess Gibbs function GE is calculated by means of the relation GE = XLptL + (1 - X,)pE. In Figure 7A is plotted GE as a function of the liquid concentrations, X,, at O.PK15. In the neighbourhood of 1°K the excess Gibbs function GE can be reasonably expressed by means of the relation: GE = XL(l - XL)WL, in which W J R = 1.54"K63 15, or by graphical fitting of Gf to polynomials of the form GE/R = XL(l - XL)[1.558 - 0.063(2XL- 1) - 0.215(2XL- l)'] at l"K16. Only in the neighbourhood of 1°K do we have a nearly symmetrical curve. This is caused by the fact that the peak in the specific heat at the ,I-point becomes thermodynamically of minor importance at higher concentrations (see Section 3.3) and that at this temperature region the entropy of mixing is still nearly ideal (see Section 3.3). At higher and lower temperatures the mixture starts to deviate considerably from this symmetrical behaviour. The vapour compositions calculated by Roberts and Swartz l6 (above 1°K) are given in Table 3 and the vapour-liquid equilibrium diagrams, constructed in Leidenl5, are presented in Fig. 2. 2.3. THEEQUILIBRIUM BETWEEN VAPOUR AND A DILUTELIQUIDMIXTURE OF 3HE IN 4HE 11; EQUILIBRIUM WITH RESPECT TO THE SOLUTE According to Pomeranchuk42 the 3He atoms can be treated as free particles which move through the superfluid as an ideal gas. The 'He atoms do not interact with one another and the assembly of 3He atoms is non degenerate. To describe the properties in more detail P0meranchuk4~proposed the energy spectrum associated with the 3He atom to be
E = EO3+ p2/2m;
(8)
where Eo3 is the effective potential and p2/2rn; is the kinetic energy associated with the translational motion through the superfluid of the 3He particle; which has an effective mass m;. The statistical mechanics of the 3He atom is nearly the same as for an ideal gas42 and the partial chemical potential p3Lis equal to:
g 3 is the statistical weight (degree of degeneracy). We like to remark that eq. (9) does not contain the chemical potential of pure liquid 3He. References p , 93
58
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH.11, 8 2
De Bruyn Ouboter, Taconis, Le Pair and Beenakkerl3 analysed the vapourliquid equilibrium data of the dilute mixtures below 1°K. The partial chemical potential of the vapour pgYis equal to
The equilibrium condition between the liquid and vapour phase p i L= piv gives the following equation:
if we assume the vapour phase to be ideal. From the smoothed vapour pressure measurements of Roberts and Sydoriak4 one can calculate the partial vapour pressure P3 which at temperatures below 1°K is nearly equal to the total vapour pressure Ptot13t15. With formula (1 1) De Bruyn Ouboter, Taconis, Le Pair and Beenakkerl3 calculated the potential energy NE,, per mole and found at constant concentration its value between 0.6“ K and 1”K to be independent of temperature. For the effective mass m; they used the experimentally determined value m; = 2.7 m3 2% 45-489501 51 (see Section 4.2) derived from the velocity of second sound and the normal density; the value of NEo, is rather insensitive to this choice. NEo3 has to be interpreted as the depth of the effective potential well, relative to the gas phase, in which the 3He particles moye. Comparing the value of NEoJ of - 23.5 j ~ u l e / m o l efor ~~ a 10% mixture, with the value of pure 3He, obtained from the heat of vaporization at absolute zero L:,, of - 21.2 joule/mole (see Table l), we see that the potential well of 3He in a 4He surrounding is only slightly larger, as one should expect in a “cell model” taking into account the zero point energy of the 3He and the large compressibility of the liquid 4He. Also the partial molar volume
derived from the molar volume experiments by Kerr 18, is slightly less than the molar volume of pure liquid ,He (VtLm36.6 cm3/mole; see Table 1). The molar volume experiments of Kerr18 satisfy in whole the He I1 region to the relation: (12) VL = (1 - XL) VlL + X L V 3 L 9
in which V,,
NN
References p . 93
35 cm3/mole.
CH.XI, 0 33
59
EQUILIBRIUM PROPERTIES
In general Kerr18, Ptuchal9 and Dash and Taylor20 find that the mixing of the helium isotopes in the liquid state is attended by a volume contraction (negative excess volume). For a 50% mixture the volume contraction I VE/VidcalI is 3.3% at 2.2"K and increases regular to 12.1% at 3.2"K.
3. The Equilibrium between the He I and the He I1 Phase (I-line) and between two Liquid Mixtures in the Stratification Region (the Phase Separation Diagram) 3.1. THELAMBDATRANSITION
Several investigators determined with various methods the shift of the I transition to lower temperatures of mixtures. We may mention the onset of superfluidity (Abraham, Weinstock and Osborne2l, Daunt 22), second sound propagation (King and Fairbank46, Fairbank and Elliot 24), discontinuity in the slope of the vapour pressure curve (Roberts and Sydoriakd), the peak in the specific heat (Dokoupil, Van Soest, Wansink, Kapadnis, Sreeramamurthy, Taconis12) and the discontinuity in the slope of the density (Kerrls). The earliest available data showed at low concentrations a too steep decrease, which later on was settled to - 1.5"K per mole fraction; less steep than the old theories expected. The full drawn line (Fig. 3) originates from Roberts and Sydoriak4 and covers most of the more recent reliable data, such as second sound24146 and specific heat 1%13 results over the complete concentration range. The variation of the lambda temperature T Awith concentration X has been calculated by Heer and Daunt 22. A 3He - 4He mixture is considered to be an ideal mixture of a degenerate Bose-Einstein gas of 4He atoms and a (non)-degenerate Fermi-Dirac gas of 3He atoms. According to London71 the transition temperature TAsof a pure ideal Bose-Einstein gas is equal to :
(the experimental value is TAa= 2.17"K). The change in the A-temperature is a consequence of the change in
Referencesp. 93
N4/VL =N4/(N3V:L
+N4V,0,), and so:
60
[CH. U,
K. W. TACONIS AND R. DE BRUYN OUBOTER
33
I 0 0 X
02
0.4
0.6
08
1
___)
Fig. 3. The lambda and stratification temperatures as a function of the liquid concentration. The smoothed line denotes the values of Roberts and Sydoriak4 (Table 2a and 2b). 0 Zenov'eva and Peshkov 26. 0 De Bruyn Ouboter, Taconis, Le Pair and Beenakker 13. a Dokoupil, Van Soest, Wansink, Kapadnis, Sreeramamurthy and Taconis 12. R curve of the rectilinear diamete+.
This simple treatment appears successful in describing the lambda line as a function of the concentration. However, a weak point is that the lambda temperature of pure liquid 4He decreases with increasing density (or increasing pressure), in contradiction with this reasoning. Neglecting the term due to the difference in molar volume {XL(V:L/V:L - 1) M 0}, one gets the simple relation
References p . 93
CH. II,
8 31
EQUILIBRIUM PROPERTIES
61
3.2. THE KEESOM-EHRENFEST RELATIONS FOR A LIQUIDMIXTURE AND THE DISCONTINUITY IN THE SLOPE OF THE FIRST ORDER EQUILIBRIUM CURVES AT THE JUNCTION WITH THE SECOND ORDER LAMBDA-CURVE Stout32, De Boer and Gorter33, Esel'son, Lazarev, Kaganov and Lifshitz34, De Bruyn Ouboter and Beenakker 35, Roberts and Swartz l6 and Sanikidze36 have developed relations between the thermodynamic properties at the lambda transition point by extending the Keesom30-Ehrenfest31 relations to the case of a mixture. The entropy S, the chemical potentials pi,the enthalpy H a n d the volume V are continuous along both sides of the lambda curve (A), but their derivatives with respect to the temperature T and the concentration Xi show discontinuities at the lambda curve which are related to the slope of the lambda curve33-35 in the following way :
indicates the difference between the values of the quantity Q at both sides of the lambda transition curve. The lambda line, in the P-Tplane, is connected at its ends to the vapour phase (boilingcurve) and the solid phase (freezingcurve), and, in the T-X plane, is connected to the phase separation region. We therefore first consider a system of two phases, A and B, in equilibrium with each other. In the space with coordinates p , T and X we have two first order equilibrium surfaces, a and b, one depicting phase A and the other phase B. The space between the two equilibrium surfaces is the inhomogeneous region. Each point on one equilibrium surface a is in equilibrium with a certain point on the other equilibrium surface b. The two phases A and B are in equilibrium with one another if the temperature T References p . 93
62
K. W. TACONlS AND R. D E BRUYN OUBOTER
[CH. 11,
53
and the pressure p are the same in both phases, and if the chemical potential pi of each component i is the same in both phases. By differentiating the equilibrium condition Pi.4 (PYT,X i A ) = P i B ( P , T,XiB) (17) along both equilibrium surfaces a and b in such a way that the infinitesimal displacement maintains equilibrium between the two phases, one can derive, using the Gibbs-Duhem relation and the definition of the partial thermodynamic quantities, the following modified Clausius-Clapeyron equations (expressions for the slope of the first order equilibrium curves 35372) :
AS 1
(3) AX, ax,,
AV 1 - (""-)AXi' axiA
1
AX,
ax;,
1
-T ( E )
axk (g)p,a= p,T
AH aHA ' 1 _ -__
AX^
ax,,
1
We can use these equations for the representation of the slope of the boilingand dew-curve, the melting- and freezing-curve, and the phase separation curve. If both phases have equal compositions,
AX, = 0 1
(azeotrope) eq. (18) has the same form as the Clausius-Clapeyron equation for a system of one component. For a system of one component eq. (18) is reduced to (Clausius-Clapeyron):
AS dP _- 1 dT-dV' 1
References p . 93
CH. n, 8 31
63
EQUILIBRIUM PROPERTIES
We consider the junction of the second order line (Mine), with the first order equilibrium curve (boilingcurve (Section 2), freezingcurve (Section 5) or upper branch of the phase separation curve (Section 3.3)). From the equations (18), (19) and (20) one sees immediately that there is a discontinuity in the slope of the first order equilibrium curve at the junction with the second order (lambda) line, since there is a discontinuity in (A = L):
as, ax,, ’ ~
#G, ~
ax;
and
a VL ax,, ~-
(see eq. (16)).
For a system of one component we see from eq. (21) that we have no discontinuity in the slope of the first order equilibrium curve. From the equations (16) and (20) it follows that the discontinuity in (dXiL/dT),. at the junction of the first order line (a) with the lambda line (A) is equal to 35 :
This equation determines the discontinuity in the slope of the phase separation curve at the junction with the lambda line. From the equations (16) and (18) it follows that the discontinuity in (dp/dT)x,,,I at the junction is equal to 35 :
1
If the mixture is azeotropic (AX, = 0) 1
it follows from eq. (23) that
Equation (23) gives us the expression for the discontinuity in the slope of the freezing curve and the boiling curve respectively at the junction with the lambda curve. The discontinuity in the slope of the freezing curve has been References p . 93
64
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH.11,
53
found experimentally by Le Pair, Taconis, De Bruyn Ouboter and Das29 and will be discussed in Section 5.1 (see Fig. 15). If one makes the following assumptions : V, = XLV;OL
+ (1 - XL)V‘&
;
2B34 = B,,
+ B4,
and
AP = 0 2
one gets from eq. (23) the expression of Roberts and Swartz16 for the discontinuity in the slope of the boiling curve: AC,
and if one further makes the assumption
av, vv - v, x vv ax, X v - x , (Xv-xL)
x
(RTIP) Xv-x,
follows from eq. (23) the expression of De Boer and Gorter33 and Esel’son, Kaganov and Lifshitz34: dlnP AdT The very precise boiling point measurements of Roberts and Sydoriak4 (see Fig. 1) show a clear discontinuity in the temperature derivative of the vapour pressure at the lambda point of the mixture. From the measured discontinuity Roberts and Swartzla calculated the change in the specific heat at constant pressure with the aid of eq. (24). Their results are in fair agreement with the specific heat measurements 1 2 ~ 1 3 . The rapid fall of the jump of the specific heat
at the lambda point with the concentration, indicates that the second order transition becomes thermodynamically of m h o r importance at lower temperatures and higher concentrations. 3.3. THEPHASE SEPARATION DIAGRAM The stratification of liquid mixtures was discovered by Walters and Fairbank38. At saturated vapour pressure below 0.9”K they detected two layers in their container, the upper light one contained more ’He and the lower References p . 93
CH. 11,s
31
EQUILIBRIUM PROPERTIES
65
heavier fraction was richer in 4He as followed from the change of the 3He nuclear magnetic resonance susceptibility over the length of the tube shaped vessel. The layers were separated by a sharp visible boundary as was first observed by Peshkov and Zenove'va26, who studied this equilibrium in a glass tube surrounded by a 'He cryostat. They also found that the 4He rich lower fraction always has superfluid properties. However, so also does the upper layer when very near to the top of the stratification curve at temperatures above the intersection (A') of the A-line and this curve (see Fig. 3). As a matter of fact Zenove'va and Peshkov26 observed that the upper phase was vigorously boiling around a small heating coil at temperatures below TI*,while the lower phase remained immobile. As the temperature was raised, there came a moment (2'2 Tat)at which the boiling of the upper phase ceased instanteneously, becoming just as quiet as the lower phase. The full stratification line in Fig. 3 (see also Table 2a and 2b) presents the smoothed values from Roberts and Sydoriak4 which are again obtained from discontinuities in the slope of the vapour pressure curves. They agree with specific heat data supplied by the work of De Bruyn Ouboter, Taconis, Le Pair and Beenakkerl3 using a calorimeter to study the demixing process down to 0.38"K. The calorimeter contains, in addition to the filling of a known quantity of mixture, a heater and a thermometer, an extra vessel in which some liquid 3He can be condensed. This 3He bath is used to bring the temperature of the calorimeter down to the lowest value (0.38"K), and at this temperature the 'He bath is evaporated by using the heater. After the last drop of liquid has left the calorimeter at 0.38"K its heat capacity is due only to the mixture, and standard specific heat measurements can be obtained up to temperatures well above the A-point, by measuring the temperature rise of the calorimeter after introducing a known amount of heat into it. The results are presented in a group of four diagrams in Fig. 4, A, B, C and D, which, for a proper discussion of the whole concentration range, is divided into four parts. A: The low 'He concentration region up to 15%: at all temperatures we are above the phase separation line. Besides the discontinuity at the Atransition important information is found in the almost constant contribution to the specific heat below 1°K due to the 3He, which is close to 4R X,. The contribution of the 4He to the specific heat is practically negligible at these temperatures. The 'He behaves like an ideal monatomic gas as explained by the theory of Pomeranchuk (see Section 2.3). As an example the 9.4% curve is reproduced; the horizontal line indicating the 3 R X, value. References p . 93
x a ! -
B
d l
1c
1c
PS I
C
T
as
I
1
I
15'
1c
1c
D
J
mdCl
5
d
C w
I
T
- Q 5
1
I
1.5
I
I.
1
,
2 OK
Fig. 4. The specific heat C as a function of the temperature T at different concentrations XLmeasured by De Bruyn Ouboter, Taconis, Le Pair and Beenakker13. For the pure components see ref. 73.74.
CH. II, 0
31
EQUILIBRIUM PROPERTIES
61
B: The concentration range between 15 and 51%: the specific heat measurements now start well below the phase separation line. A second discontinuity in the specific heat, AC,.,, is found at the phase separation curve, since only below this line does the heat of mixing contribute to the heat capacity of the mixture. The I-peak is found at still lower temperatures and falls rapidly with increasing concentration while broadening still more. There is no short range ordering observed just above the phase separation region. As an example the 39% curve is reproduced. C : The concentration range between 51 and 73% covering the top of the phase separation curve: the contribution of the heat of mixing is going through a maximum as the concentration of the two separated mixtures are coming gradually very close together. At the temperature of the intersection of the I-line and the phase separation line, a distinct irregularity I*is found in all three concentrations used, in this region at the same temperature T,. = 0.80”K.The border of the phase separation is clearly marked since from there on the I-transition is contributing to the specific heat. As an example the 63.8% curve is reproduced. We would like to mention that the stratification curves in general, obey a law of the rectilinear diameter72. This also appears to be the case for the 3He -4He stratification curve35. However, the slope of this rectilinear in agreement with eq. (22), as can be seen diameter changes abruptly at T,*, from Fig. 3. D: The concentration range above 73%: here we have no lambda peak. The heat of mixing is again strongly represented in the phase separation region. Outside the phase separation region a short range ordering in the helium I phase is observed which normally introduces the A-point transition when approaching it from above, however, it had here no chance to appear since the stratification intercepted its development. At higher concentrations this short range ordering becomes less pronounced because the “virtual lambda-line’’ is then further away from the phase separation curve. The heat of mixing H E can be derived from the specific heat measurements inside and outside the phase separation region13. The availability of the extra contribution to the specific heat originating from the heat of mixing H E offered the possibility to evaluate its complete behaviour. In Fig. 5 a picture is given of its magnitude as a function of temperature over the whole concentration range13. Earlier (1953) Sommers, Keller and Dash1* have measured at T = 1.05”K and A’, = 0.086 the heat of mixing directly. Two adjacent, thermally isolated chambers containing pure liquid 3He and liquid 4He were mixed together References p . 93
68
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH. 11,
93
to give an 8.6% solution by rupturing the membrane dividing them. During this procedure the temperature fell from 1.05"K to 0.78"K. The measured heat of mixing was estimated to be 0.71 joule/mole at 1.05"K,which is in satisfactory agreement with the values of De Bruyn Ouboter, Taconis, Le
+t% 7-
Fig. 5. The heat of mixing H E as a function of the concentration X at different temperatures T measured by De Bruyn Ouboter,Taconis, Le Pair and Beenakker 13. 0 Sommers, Keller and Dash 14, one point at XL = 0.086 and T = 1.02"K.
Pair and Beenakker13 determined from the specific heat measurements in the phase separation region (see Fig. 5). The other interesting thermodynamic function, the excess entropy SE, can be derived now with satisfactory precision by subtracting the excess Gibbs function GE as it follows from the vapour pressure data (Section 2.2) from the heat of mixing HE,using the relation GE= HE- TSE. As an example, this procedure is shown in Fig. 7A. Here is plotted the excess Gibbs function GE,the heat of mixing HE, and the excess entropy SEas a function of the concentration X , at O.YK13. In Fig. 6 a picture is given of the magnitude of the excess entropy SEas References p . 93
CH. 11,s 31
69
EQUILIBRIUM PROPERTIES
a function of the concentration X , at different temperatures, and in Fig. 7B is plotted the excess entropy SE as a function of the temperature T for a dilute mixture of 10% 3He in liquid 4He. The trend is that this quantity becomes negative at low temperaturesI3. Nernst's heat theorem states that the entropy at absolute zero should be equal to zero (third law of thermodynamics). In classical thermodynamics this is not the case for an ideal mixture, where there remains a n entropy of
Fig. 6. The excess entropy SE as a function of the concentration X at different temperatures T, according to De Bruyn Ouboter, Taconis, Le Pair and Beenakker 13.
mixing of - RIXLlnXL-t(1 - X,)ln(l - X,)] (positive). In quantum statistics this difficulty does not arise, since here this term goes to zero with decreasing temperature (because of the degeneracy 37, 41). Hence, if one describes the thermodynamic properties of a mixture in terms of classical excess functions one will obtain, in the case where there is no phase separation, the value of R[X,lnX, (1 - X,)ln(l - X,)] (negative) for SE at T = O"K35. In Fig. 7B we see that the total entropy of mixing goes to zero with decreasing temperature, in agreement with what Nernst's heat theorem suggests. Another consequence of the quantum degeneracy of the entropy of mixing in the liquid, is the finite slope of the phase separation curve at absolute zero. The logarithmic behaviour of the classical entropy of mixing would probably give rise to a vertical slope in the T - X diagram at absolute zero. In Section 5.3 will be discussed the minimum of the freezing curves of 3He -4He mixtures in connection with the negative excess entropy of the liquid. 353
+
References p . 93
+
70
K. W. TACONJS A N D R. DE BRUYN OUBOTER
[CH. 11,
53
For a dilute mixture of 3He in superfluid 4He one can discuss the thermodynamic excess properties by means of the theory of Pomeranchuk 4 2 (see Section 2.3). The heat of mixing HE and the excess Gibbs function GE are given by13:
where Xu and X , are the concentrations of the upper and lower phase that are in equilibrium with each other in the phase separation region. The values for HE derived from the specific heat measurements in the phase separation region and the values for G: derived from the phase separation curve are in agreement with the results obtained from eqs. (26) and (27), if one uses for NEo3 the value derived from the boiling point measurements (Section 2.3). Pomeranchuk's treatment is however not very satisfactory near the lambda point.
-1
Fig. 7A. H E ,GE and TSEas a function of the concentration X at 0.9" KI3. G E w X L ( ~- XL) WL; WL/R= =
1.54O
K13.15.
References p. 93
Fig. 7B. The excess entropy SE as a function of the temperature T for a dilute mixture of 3He in liquid 4He according to De Bruyn Ouboter and Beenakker35 (0). 0 lim SE = T+ 0 X L = 0.1
=+R[XLlnX~+(1-XL)In(l-X~)] (Nernst's theorem).
CH. 11,s
31
EQUILIBRIUM PROPERTIES
71
The thermodynamic excess functions are far more complicated in the intermediate region, as all types of interactions (i.e. 3He - 3He, 3He - 4He, 4He - 4He) are contributing. Furthermore, at higher temperatures the lambda phenomenon also plays an important role. A trial to interpret the experimental excess quantities meets several serious complications, since at the temperature of the 4He lambda point these excess functions must compensate the marked anomalies introduced into the “ideal” quantities by the 4He thermodynamic functions, such as the specific heat. Hence no simple description may be expected. At low enough temperatures one may perhaps negIect the lambda phenomenon and try to describe the properties by means of the model of Prigogine, Bingen, Bellemans and Sirnon17. They explained a positive excess Gibbs function by pointing out that, neglecting the collective motions, the difference between the pure isotopes is only due to the zero point energy, which gives rise to large differences in the molar volumes. In this picture the mixing becomes ideal if one has first brought the pure isotopes to the same molar volume as they have in the mixture. The excess Gibbs function is in this case equal to the work done by compressing the lightest isotope and by expanding the heaviest one. These results for GE give the right order of magnitude. However, as they pointed out themselves, this is all one may expect from such a model. We may remark that this model is also very successful in describing the properties of the hydrogen isotopes. Cohen and Van Leeuwen40 calculated the free energy of an isotopic mixture of a hard sphere Bose (4He) and a hard sphere Fermi (3He) gas in the first order as a function of the hard sphere diameter, using the pseudopotential method developed by Huang, Yang and Leedo. They found that above a critical density there exists a critical temperature T, below which an instability occurs, leading to a phase separation of the mixture into two coexisting phases. In particular, calculations were carried out for the case of liquid 3He - 4He mixtures using experimental values for the liquid density. In their theory the lambda line is given by eq. (15 ) of Daunt and Heer 22. The critical temperature T, (the top of the phase separation curve or upper consolution point) coincides in this model with the point where the lambda line intersects the phase separation curve at a concentration of X, = 67% and a t a reduced temperature of
T,(X = 0.67)/TA0(0)= T,(X = 0.67)/TA0(0)= 0.48 (or T, = 1.04”K) (hence, the “virtual lambda line” is located inside the phase separation region). References p . 93
12
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH. 11, $ 4
The experimental values, however, are: X , = 65%, T, = 0.88” K and X,. = 73%, T,. = 0.80”K, indicating that the upper consolution point (X,, T,) does not coincide with the junction of the lambda line and the phase separation curve (T,*,X,*).
4. The Osmotic Equilibrium in He I1 (Pseudo Thermostatic Equilibrium) and some Applications 4.1. THE OSMOSIS IN HE I1 DERIVED FROM THE EQUILIBRIUM BETWEEN PURE LIQUID4HE ON ONE SIDE AND A MIXTURE ON THE OTHER SIDE OF A SUPERLEAK (SEMIPERMEABLE WALL) ;EQUILIBRIUM WITH RESPECT TO THE SOLVENT In general this equilibrium can only be realized below the lambda-point of the mixture and therefore, for not too high concentrations. This is because of the fact that a superleak, a substance with microscopic pores, acting as a semipermeable wall, is permeable for the superfluid fraction of the liquid helium 11, which consists of helium four atoms only. The osmotic pressure exerted by the mixture on one side of the wall can be balanced by a fountain pressure on the other side, which can be produced by raising the temperature of the vessel that contains the pure helium liquid. The osmotic effect was discovered by Daunt, Probst, Johnston, Aldrich and Nier 8, first studied by Beenakker, Taconis and Dokoupi19 and explored extensively by Wansink, Taconis and Staas7 and by London, Clarke and Mendoza11. Its appearance opens the possibility of a number of applications which will be reviewed in Sections 4.24.5. Before we can do so, however, we should like to give an analysis of the behaviour of the chemical potential of the solvent p4Lin such an equilibrium. For a dilute mixture of 3He in superfluid 4He (XL+ 0) the chemical potential pSL for the solute 3He is given by eq. (9) and the chemical potential p4Lfor the solvent in the solution is equal to 75:
We may remark that the non-ideality of the dilute mixture does not influence the chemical potential p4Lfor the solvent. It is, however, the chemical potential pUJL for the solute which is influenced by the non-ideality of the dilute mixture (see eq. (9)). We derive the equation for the osmotic pressure by considering the equilibrium between a dilute mixture on one side and pure liquid 4He on the other side of a superleak, both liquids being at the same temperature. In order to obtain equilibrium a “negative” osmotic References p. 93
CH. 11,
0 41
EQUILIBRIUM PROPERTIES
73
pressure Ap,,, is applied to the pure solvent. The equilibrium condition requires that the chemical potential of the solvent must be the same in both liquids :
Thus we get for the osmotic pressure of a dilute mixture:
which is Van 't Hoff ' S law 75. In the case of a more concentrated solution the non-ideality also plays a role, and instead of eq. (28) we get: 703
P'4L(T, XL) = P:L(T) 4-R 7 ' h (1 - XL> + PL :
(31)
and the osmotic pressure is given by the relation'O: APosm =
+
RTln(1 - XL) pzL 1/4L
Fig. 8. The osmotic pressure measurements of Wansink, Taconis and Staas 7. The osmotic pressure posm/Tcompensated by a fountain pressure j f d T / T , divided by the bath temperature T, as a function of the liquid concentration X . A:T>1.8OK; B : 1 . 4 " K < T < 1.7"K; C:T<1.3"K. Van 't Hoffs law eq. (30), calculated from ~4~ by means of eq. (31). ~
___-__
References p . 93
74
K. W. TACONIS A N D R. DE BRUYN OUBOTER
[CH. XI,
04
In the limit XL+O we again get Van 't Hoff's law eq. (30), since ,u:L+O and ln(1- XL)+ - XLand P4 = (1 - XL)Pz. Wansink23 has calculated the excess chemical potential ,utLof the solvent from the small deviations from Van 't Hoff's law (at more concentrated solutions up to 5%). The order of magnitude of this effect and the deviation from ideality is demonstrated in Fig. 8. From eqs. (32) and (2), neglecting the influence of the second virial coefficient, one gets the following relation for the osmotic pressure: V2LApos,,,= RT In (P,/Pt) = RT In ((1 - X,)p,,/P~).
(33)
4.2. THEQUASI-EQUILIBRIUM BETWEEN THE OSMOTIC AND FOUNTAIN FORCE IN THE LIQUID MIXTURE, WHEN A HEATCURRENT IS PRESENT (THE HEAT FLUSHEFFECT),YIELDING THE DIFFUSION COEFFICIENT AND THE HEAT CONDUCTIVITY OF THE MIXTURES
A steady heat flow in a liquid mixture results in a displacement of 3He in the direction of this current and a temperature gradient in opposite direction. The equilibrium situation can be understood from a discussion of the forces acting on the superfluid. Let us start with the derivation of the equation of motion of the superfluid component, as given by Landau and Pomeranchuk42165. The change in energy U4, caused by the transport of a small amount of superfluid dN4, keeping the momentum p of the normal fluid (phonons, rotons and 3He-atoms) constant with respect to the superfluid, is equal to
Since the superfluid carries no entropy, one has
The change of energy by transport of 1 mole superfluid is , ~ 4 , ~ and by a superfluid transport of the unit of mass we have an energy change equal to , u , ~ / M ~This . quantity can be considered as the potential of the force acting on the superfluid; hence we get the following equation of motion for the superfluid : 1 dvs _ -- - -grad j ~ ~ ~ . (34) dt M4 Using equation (28) (pgL(T , P , X )= piL(T,P) - RTX,) valid for a non-ideal References p . 93
CH. II, 8 41
15
EQUILIBRIUM PROPERTIES
dilute mixture and the relation dp:, du, -= dt =
1
R
M4
M4
- -grad P:~(T, P ) + -grad L V:
- -gradp M4
or
“1$= dt
= - S:,dT+
--gradp+ 1 PL
L + G-grad
M4
T
V:,dp, TXL =
R + -grad
TX,,
M4
S-gradT+-gradT&. :L R
M4
eq. (34) becomes:
(35)
M4
The first term on the right hand side of this equation in grad p is similar to the one encountered in classical hydrodynamics. The second in grad T expresses the fact that the superfluid component moves toward regions of high temperature. The first two terms together are responsible for H. London’s thermomechanical relation: grad p = (pLS:,/M4) grad T (fountain effect). The third term is the osmotic pressure and expresses the fact that the superfluid component moves toward regions of high concentration. In stationary condition (ao,/dt = 0) and with no pressure gradient (grad p = 0) we get the following thermo-osmotic relation: R grad TX, = SxLgrad T. This means that the sum of the “fountain pressure” and “osmotic pressure” is the same everywhere and is even valid when no superleaks are present. The heat flush effect, in which the ’He migrates toward regions of lower temperature, is a consequence of this phenomenon. Heat flush experiments were of actual importance, at the time this method was regularly applied, in dilute solutions to concentrate in a simple way the 3He fraction at one special cold spot of a heat conduction cell. It can also be understood from the following reasoning: a heat flow through the vessel placed advantageously vertical, in which the heat is supplied a t the bottom and flows to the cooled top, will be accompanied by a normal fluid flow upward and a superfluid flow downward, according to the two fluid model, causing the high heat conductivity by internal convection. Now the ’He atoms will join the normal fluid flow, and therefore a concentration gradient will appear, since the ’He atom can not transform into superfluid at the cold spot in order to move backward. It can only diffuse backward against the normal fluid drift. Finally, internal equilibrium can only exist in a steady state when a temperature gradient is built up that can supply the necessary fountain pressure to balance the osmotic pressure corresponding to the concentration gradient. This internal pressure equality in dilute mixtures demands that: References p . 93
76
[CH.U, 0 4
K. W. TACONIS AND R. DE BRUYN OUBOTER
RTaX M~ az
S:LaT M~ az'
The steady state, in which no net 3He is transported, is given by:
ax
XV, = D -
(37)
az
and the heat current density W is given by:
w =~s:,Tv,.
(38)
This results in: 0 aT RXP s4L= - -. az &,D
The effective heat conductivity
Keff
(39)
is then equal to
It was this "diffusion" contribution to the conductivity, which was studied 6
' O F
t
16
132.10
t
Y
ndl
1k
t
zl 1
Fig. 9. The effective heat conductivity Keel as a function of the temperature T a t different concentrations X,measured by P t ~ c h a ~ ~ . References p . 93
CH.II,
4 41
EQUILIBRIUM PROPERTIES
77
by Beenakker, Taconis, Lynton, Dokoupil and Van Soest53, and it is the main part of the total conduction at temperatures above 1.3"K, as was pointed out by Ptucha54, who performed more precise and excellent work on this subject later on (see Fig. 9). He placed in a 6 mm high vessel, which was used as a conduction cell, four horizontally flat thermometers, to check the temperature distribution in the vessel. The heat conductivity for concentrations up to 1% shows a minimum just above 1°K and its behaviour was analysed on the basis of the theory de65.9 The fall of the conductivity veloped by Khalatnikov and Zh ark 0 v ~ ~ between the lambdapoint and the minimum is due to the diffusion part above. Below 1°K it rises again since the usual conduction, mainly due to phonon conduction, increases at lower temperatures (see Fig. 9). The diffusion coefficient governing the conduction region above the miat the lambda point to 50 x at 1.2"K, nimum, increases from l x in agreement with the work of Beenakker, Taconis, Lynton, Dokoupil and Van Soest 53, Garwin and ReichV6, Careri 86. 4.3. THEQUASI-EQUILIBRIUM BETWEENTWO MIXTURES OF SLIGHTLY DIFFERENT CONCENTRATION IN THE He I1 REGION SEPARATED BY A CAPILLARY, YIELDING THE VISCOSITY OF THE MIXTURES When two vessels in thermal equilibrium with a helium bath are connected by a capillary tube and partially filled with some liquid mixture, one containing a somewhat higher concentration than the other, which is below the lambdatemperature of the mixture, a flow of superfluid helium through the capillary will very soon establish a semi-equilibrium in such a way that the liquid level in the more concentrated vessel will reach a higher position. The height difference h between the levels will depend on the concentration difference A X corresponding to the osmosis between the two vessels, since the capillary acts as a semipermeable wall giving free passage to the superfluid, but permitting only a very small flow of the normal fluid which contains the 3He atoms. For a dilute solution, the height difference is given by
R
h = A X --
T (eq. (30); Van 't Hoff's law),
M4
whereas for higher concentrations a small correction (Section 4.1, eq. (32)) has to be applied. Starting from this semi-equilibrium, the final equilibrium, with levels at the same height and equal concentration, is reached by an References p . 93
I8
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH. 11,
54
isothermal viscous flow of the normal liquid through the capillary. The flow of the superfluid has almost the same speed, although, especially at very low concentrations, an appreciable correction has to be applied for its amount in order to derive the real speed of the normal fluid vn from the rate of change of the levels. This experiment was explored by Staas, Taconis and Fokkens52, using the observed v, and h to measure the viscosity of mixtures in a 75.6 p capillary tube. In order to insert in the Poiseuille equation the right pressure head, to h has to be added the vapour pressure difference between the two vessels containing mixtures of small concentration difference. Fig. 10 shows the viscosity q as a function of the temperature T at different concentrations. There is only qualitative agreement with the measurements of Dash and Taylorz0,who measured the viscosity with an oscillating disk method. The theoretical treatment of the viscosity by Zharkov63, in which three contributions to the viscosity are introduced to explain its complicated behaviour, is verified. From this flow behaviour it follows that the 3He moves together with the normal fluid part of the 4He52. As a whole the two fluid model is strongly confirmed. Consequently, one may conclude that the 3He particle can readily collide with the rotons and phonons and will nearly always participate in the motion of the normal component42~53. Hence the density of the normal component pn is equal to4?
n , , the phonons and rotons make Below 1°K one has ~ , , ~ < < r n ~ p X / rsince a negligible contribution, so that the normal fraction is almost equal to : x
=p,/p
=(rn;/m,)~.
From the Andronikashvilli-type experiments1 by Pellam 50, Berezniak and Esel'son51 and Dash and Taylor20 one finds x M 2X and m;/rn3 M 2.7. This rather large effective mass has as a consequence the fact that the Fermidegeneracy temperature :
is so low that the specific heat measurements are not influenced by the References p . 93
CH. II,
4 41
EQUILIBRIUM PROPERTIES
19
degeneracy. At not too low concentrations, To is of the order of the phase separation temperature. Another confirmation of Pomeranchuk's point of view comes from the velocity of second sound in dilute solutions. At low enough temperatures, and by neglecting the 4He contribution, one has the expression for the sound velocity of an ideal monatomic gas with mass m:, which is given by:
Fig. 10. The viscosity as a function of the temperature. Concentrations:0, 0.2,0.5,1, 5, 10, 15, 20, 25, 30, 100 % measured by Staas, Taconis and Fokkens 52.
From this expression one sees that the velocity of second sound becomes, at lower temperatures, independent of the 3He concentration and is proportional to J T . This is in agreement with the results of King and Fairbank46, who found a value of m;/m3 -2.8, and with those of Kramers and Niel~-Hakkenberg~~ that give m:/m3 =2.5. In Feynman's theory43 the motion of the 3He atom through the superReferences p. 93
80
K. W. TACONIS AND R . DE BRUYN OUBOTER
[CH.11,s
fluid is treated as a microscopic hydrodynamical problem and the effectivc mass m: is classically the sum of the true mass of the 3He atom and onc half the mass of the displaced fluid. Thus
m; = in3 + + ( $ ) m 4 , which gives m;/m, M 1.9. In this model the backflow of the superfluid arounc the 3He atom gives rise to the large effective mass m;. We like to stress thl fact that in this classical picture the large volume occupied by the ,HI in the solution raises the effective mass considerably (V3 > V:).
4.4. THEQUASI-EQUILIBRIUM BETWEEN TWO LIQUIDMIXTURES OF DIFFEREN CONCENTRATION IN THE He I1 REGION, CONNECTED BY THE HELIUM SUR FACE FILM,YIELDING THE ISOTHERMAL FLOW RATE OF THESE FILMS Another valuable application of the osmosis was introduced by Van dei Meijdenberg, Taconis and Le Pair57 in their study of the superfluid flov properties of the helium film in equilibrium with a liquid mixture of givei concentration. In the apparatus two vessels, A and B, filled with mixture of different concentrations, are separated by a superleak S (see fig. 11) Transport of superfluid helium can take place through ! from A to B by means of the helium film which covers th wall of vessel A if the concentration in B is higher. Th mobility of the superfluid fraction of the film facilitates th transfer of helium four. Transfer will occur till equilibriun is reached and the concentrations are practically equal. When the film is pure enough so that reliable and repro ducible results are obtained the flow rate is found to b independent of the osmotic pressure head and thus of th concentration difference, and only slightly dependent on th path length, H, just as in the case of pure 4He. There is, hoa ever, a strong dependence on the concentration in vessel A To illustrate this, some results are presented in Fig. 1: showing the flow behaviour as a function of temperature for two vessels of different length; a long and a short one Quantitatively the whole flow phenomenon is still not un derstood, for instance, the decrease of the flow rate is nc simply connected to the decrease in the available amour of superfluid helium in the film at higher concentration. References p . 93
CH. II,5 41
EQUILIBRIUM PROPERTIES
81
Fig. 12. The flow rate fi as a function of the temperature Tat different concentrations, measured by Van den Meidenberg, Taconis and le Pair5'. b) H = 21 rnm. a) H = 90 rnrn;
4.5. A REFRIGERATION CYCLE, AS AN APPLICATION OF THE OSMOTIC PRESSURE AND THE HEATOF MIXING
Finally H. London11 pointed out that an interesting cooling cycle can be followed making use of the unusual properties of He I1 mixtures, such as osmosis, heat of mixing, phase separation and very effective destillation possibilities. The cooling is produced by adiabatic dilution and is followed by destillation in order to reconcentrate the diluted solution. An apparatus according to this proposal is under construction by London, Clarke and Mendoza11. The idea is that in an equilibrium vessel at the lowest temperature T I of the cycle, perhaps even T , = O.l"K, a mixture of liquid helium is present. It is separated in an upper concentrated layer containing about Xu = 99.5% 3He and a lower one with about X,= 2.6% 3He (see Fig. 13, the refrigeration cycle is drawn in a T - X diagram). The equilibrium vessel is fed at the top by a stream of almost pure 3He (2+3). At the boundary of the two separated mixtures it is mixed with pure 4He (3+4), which is introduced through a superleak (S=O) at the bottom (3'), forming the X , = 2.6% mixture (4) which is leaving at the bottom of the vessel. The latter mixture is warmed up (4+ 1) to a temperature T , high enough ( T , m0.8'K) to evaporate (at 4), by means of a diffusion pump, the 3He out of the liquid mixture, making use of the very high'3He concentration of the vapour in equilibrium with a dilute liquid mixture at low temperatures (see Fig. 2). Finally this almost pure 3He fraction is condensed References p . 93
82
K.W. TACONIS A N D R. DE BRUYN OUBOTER
[CH.II, 5 4
(at 2) and after cooling ( 2 - + 3 )is fed in again at the top of the vessel, thus closing the cycle. An analysis of the process shows that it is the negative enthalpy difference between the liquid 3He entering the cycle and the dilute mixture leaving it at the same temperature which gives the cooling capacity. An appreciable amount of heat can be taken up by the solution when warmed up from the lowest to the highest temperature (4+ 1) of the cycle, whereas the heat of mixing at the lowest temperature (4) is comparatively small and almost equal to the heat to be extracted from the 3He, which has to be cooled
Fig. 13. A refrigeration cycle diagram.
down to this lowest cycle temperature (2+ 3). It is therefore essential that a heat exchanger be used in the process between the two streams to and from the vessel (heat exchanger between (2+3) and (4+ 1)). The flow upward (4+ 1) is most important as can be seen from the numbers in Fig. 13. Its behaviour is carefully discussed by Staas and Taconis77 in a similar flow experiment in which a mixture is destilled in a closed cycle working between two different temperatures. The change of the concentration accompanied by the temperature rise, which is of great importance in the cycle, is fully discussed 77. In Fig. 13 the refrigeration cycle is represented in the T - Xdiagram operReferences p . 93
CH. 11,s
41
83
EQUILIBRIUM PROPERTIES
ating between 0.1 and 0.8"K. All quantities drawn in this diagram refer to the case in which 1 mole of solute (3He) is taken around the whole cycle. ( Y / X ) is the volume of solution containing one mole of solute ('He). We will discuss the enthalpy balance of the whole cycle: (1 -t 2). T = 0.8"K. In (1) 3He is evaporated (c,/c,>> 1), (see Fig. 2) and in (2) the pure 3He vapour is condensed. The condensation energy is larger than the evaporation energy, and the system loses an amount of energy equal to
T=0.8 n
0
In addition work is done in (1) of amount - Jp,,,dV= - RT, M - 75 to keep the mixture in (1) at the fixed concentration of 0.3%. Thus, the total loss of energy is - 14 J. (2-t 3). The liquid 3He has to be cooled from 0.8"K to 0.1"Kby an amount of energy equal to 0.8
-
J C:dT=
-2.05
0.1
(this is done by a heat exchanger, using the amount of cold produced during (4)+ (1)). (3-t 4). T = 0.1"K. The 3He solution is diluted, by adding superfluid 4He ( S = 0) through a superleak (2'- 3'+ 4), and cooling can result from the external work done during the expansion of the solute "gas" 'He, ( wposm-V/X,M RT,) (see eqs. (30) and (35)), and from the heat of transition from the concentrated to the dilute phase ( w HE/&M $RT,) (eq. (26)). In total an amount of heat of m $RT, % 2.1 J can be absorbed at 0.1"K. (4- 1). We have seen in Section 4.2, eq. (35) that the steady-state condition requires that the osmotic pressurep,,, = RTX = constaiit along the capillary leading from (4) to (l), since one may neglect the fountain pressure at low enough temperatures. In (4) we have a 2.6% mixture at O.l"K, so at (1) at 0.8"K we have a mixture with a concentration of 2.6 x (0.1/0.8) M 0.3%. During this process an amount of heat
+
0.8
CposmdT= q f X d T x 0.1
+ 145
0.1
can be absorbed when 1 mole 3He is transported. References p , 93
84
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH. U,
85
5. The Quilibrium between Liquid and Solid Mixtures (Freezing- and Melting-curve) and the Phase Separation of Solid Mixtures 5.1. THE FREEZING PRESSURES OF 3He-4He MIXTURES The freezing-point of a mixture of known concentration at a certain temperature is determined by measuring the pressure at which a relatively large quantity of liquid mixture is in equilibrium with a negligibly small quantity of solid so that the concentration of the liquid is still known and not changed by the existence of crystals of different composition. The melting-point of this mixture is analogously determined by measuring the pressure at which the almost completely solid mixture is in equilibrium with a negligibly small quantity of liquid, so that the different concentration of the latter does not disturb the solid composition. Usually the investigation of the freezing-points is chosen out of two pos-
Fig. 14. The apparatus for measuring the freezing pressures of 3He - 4He mixtures 29. C1 = filling capillary B = Bourdon-gauge F = the piece of ferroxcube Ca = measuring coil 3He = 3He-cryostat.
sibilities: (first method) by introducing continuously, at constant temperature, a liquid of known concentration into a vessel, causing a continuous increase of the pressure till solidification starts (sometimes observable by the blocking of a capillary), or (second method) by detecting, at a constant filling of the vessel, by lowering continuously its temperature, the appearance of a kink in the pressure temperature curve originating from the difference in density between liquid and solid. The last method has the advantage that due to the volume change attending the solidification, the freezing curve can be followed rather accurately as the cooling proceeds if only the contraction is large enough. A complication still arises, however, when a minimum is present in the freezing curve as is the case of pure 3He. The first method then can not be used below References p . 93
CH.,!I
8 51
85
EQUILIBRIUM PROPERTIES
this minimum, because of the fact that the capillary through which the liquid mixture is fed into the vessel and which comes from a higher temperature is blocked at the moment the growing pressure passes the pressure of the minimum, since then somewhere the capillary will have the temper-
L
0
I
I
I
I
I l l
T . 0.5
I
I
I l l 10
Ill
I
I
(5
W %
Fig. 15. Freezing curves of 3He-4He mixtures measured by Le Pair, Taconis, Das and De Bruyn Ouboter 28. The dash-dot line respresents the onset of phaseseparation in the liquid of a 50.5 % mixture2B. The numbers denote the mole fractions 3He. A indicates one point of complete melting at 0.07" K of a 17.6 % mixture from Edwards, McWilliams and Dauntsg (see Fig. 17; Section 5.2), here we are, as also denotes the dashed curve, in the phase separation region. References p . 93
86
K. W. TACONIS A N D R. DE BRUYN OUBOTER
[CH.11,
85
ature corresponding to this minimum pressure. Fortunately the second method can be used here too. It is essential, however, that, firstly, the pressure of the enclosed quantity of the mixture is measured at the vessel itself and, secondly, that the cooling of the liquid is attended by a pressure increase so strong that this negative pressure coefficient of the liquid is larger than the negative slope of the freezing curve below the minimum. Weinstock, Lipshultz, Lee, Kellers and Tedrow61 applied a strain gauge
Fig. 16. T-X diagram of 3He-4He mixtures in the solid-liquid equilibrium regionzg; a represents a freezing curveat a pressure of 47.5 kg/cm2; b of 40.0; c of 35.0; d of 30.0:
e of 25.9; f of 24.76; g of 24.0; h of 23.62 and i of 23.55 kg/cmz. A and A' represent projections of the azeotropic line and M represents projection of the line through the minima in the P-T curves. hcp-bcc represents the projection of the line through the intersections of the extrapolated solid-solid transition line with the freezing curves. (So the kink in the freezing curves should not exactly coincide with this line.) References p . 93
CH. 11, 0 51
EQUILIBRIUM PROPERTIES
81
attached to a thin walled vessel by means of which the pressure could be measured. Le Pair, Taconis, Das and De Bruyn Ouboter29 gained sensitivity by using a more flexible vessel in the form of a Bourdon-gauge, the displacement of which was recorded by attaching to the free end of the Bourdongauge a piece of ferroxcube F surrounded by a coil C , and measuring the change of the inductance in this coil as a function of the position of the piece of ferroxcube F. The principle of the apparatus is shown in Fig. 14. In Figs. 15 and 16 results are presented. The curves in Fig. 15 show the freezing lines of five different concentrations (8.9; 22.8; 50.5; 64.5 and 80.1%) as a function of temperature. Fig. 16 is derived from these results and contains the isobars of freezing. The occurrence of the minima in the freezing curves (see Fig. 15) will be discussed in Section 5.3. In Section 3.2 (eq. (23)) we have seen that a discontinuity has to be expected in the slope of the freezing curves at the junction with the lambda curve35. The measured freezing curves29 in Fig. 15 clearly show such a discontinuity. In Fig. 16 a T - Xdiagram is given with freezing lines at constant pressure. A peculiar behaviour of the mixtures is shown below a pressure of 25.9 kg/cm2, where the freezing curve is a closed line, within which a region of solid must exist. The dashed line M is the projection on the T - Xplane of a line through the minima in the P - T curves. This line intersects the freezing curves at constant pressure at the points where (dT/dX,), = co , The dashed lines A and A‘ are projections of the line which connects the azeotropic points of the freezing curves. The intersection of these lines with the freezing curves occur where (dT/dXL), = 0. The drawn line A is the projection of a line connecting the “triple”-points of A-transition and freezing. Again the freezing curves show a kink at the intersection with the “A-line” 35.
5.2. THEEQUILIBRIUM BETWEEN TWO SOLID MIXTURES IN THEPHASE SEPARATION REGION
The phase separation (Fig. 17 and 18) was derived from specific heat data. Edwards, McWilliams and Daunt measured the specific heat of solid 3He -4He mixtures over the complete concentration range and in the temperature range from 0.05”K to 1°K. Their results show that the two isotopes are completely separated at absolute zero. Their data are given in Fig. 17, in which a diagram is represented of the specific heat CJR versus the temperature T at different concentrations Xs.Below the phase separation temperature an extra contribution to the specific heat, due to the heat of mixing, is observed. The set of specific heat curves for different average concentrations, has a common envelope on the low temperature Referencesp . 93
88
K. W. TACONIS AND R. DE BRUYN OUBOTER
[CH. II,
85
side of the discontinuity. The critical point of mixing (top of the phase separation curve) is found at a temperature Ts,crit = 0.38"K and a concentration Xs, = 3. Edwards, McWilliams and Daunt59 pointed out that from their specific-heat data it follows that the solid mixtures behave completely regular. Since the main property of a regular mixture is, although there exists a difference in the interaction energy between like and unlike pairs of atoms, which results in a heat of mixing, H t =&(1 - X s ) W,, that there is still a complete random mixing of the different atoms resulting in an excess entropy equal to zero, and hence also an excess specific heat equal t o zero, as has been observed. Thus we have the following equations in the case of a regular mixture:
Fig. 17. The specific heat of 3He-4He solid mixtures Cs/R, measured by Edwards, McWilliams and Daunt59, as a function of the temperature T. X X S = 0.9997, P = 35.8 atm A XS = 0.79, P = 35.8 atm 0 X S = 0.50, P = 35.8 atm V X S = 0.9989, P = 35.8 atm 0 X S = 0.9972, P = 35.8 atm X s = 0.50, P = 27 atm 0 XS = 0.953, P = 35.8 atm A XS = 0.176, P = 30.0 atm.
References p . 93
CH.I1,g 51
89
EQUILIBRIUM PROPERTIES
The critical point of mixing is determined by the following simultaneous equations : Ts,crit = Ws/2R = 0.38" K, XS,crit= Q ; yielding for Ws/R = 0.76" K. (45) The phase separation curve is given by
and for the specific heat in the phase separation region of the solid follows:
Thus, although a regular mixture has no excess specific heat: C: = 0, it still has the extra contribution in the phase separation region due to the temperature independent heat of mixing. Eqs. (46) and (47) with Ws/R= 0.76"K were experimentally completely verified by Edwards, McWilliams and Daunt69. This leads to the conclusion that the entropy of mixing of the solid is given by the temperature independent ideal classical term of Gibbs : - RIXslnXs (1 - Xs)ln(l - Xs)],
+
at least down to 0.05"K.
0.4-
0
I
I
X,
I
I
I
0.5
I
I
I
I
1
mixtures derived from the specific Fig. 18. The phase separation curve in solid heat measurements of Edwards, McWilliams and Daunt59. The drawn phase separation line is represented by eq. (46) with WEIR= 0.76" K. The circles indicate the concentrations and temperatures for which the transition has been directly observed. References p . 93
90
K.W. TACONIS AND R. DE BRUYN OUBOTER
[a. n, 5 5
5.3. THE M m IN THE FREEZING CURVES OF 'HePHe MIXTURES AND MINIMA IN THE MELTING CURVES OF 'HE AND PURE 4HE
THE
Let us first, as an introduction, discuss the minima in the melting curves of pure 'He and 4He and see whether they can lead us to the explanation of the minima in the freezing curves of the 'He 4He mixtures. For 'He the situation is as follows: The nucleus of 'He has a spin of +h, with an accompanying magnetic moment, and if each 3He nucleus in the liquid was free to orient its spin in two directions with equal probability, the entropy would be at least Rln2, or 5.75 joule/mole"K. Measurements1* show that the entropy of liquid 'He falls below Rln2 at about 0.4"K, when the nuclear spins exhibit some ordering. P o m e r a n ~ h u ksuggested ~~ in 1950 that this does not occur in solid 'He until much lower temperatures are reached, because the nuclei are then localized very close to lattice sites and their amplitude of oscillation is smaller than the interatomic distance, so that the wave functions do not overlap sufEciently to give the strong exchange effect needed to align the nuclear spins. We now consider the melting curve. The entropy of the freezing liquid falls below Rln2 at 0.33"K. At this temperature the entropy of the melting solid is roughly equal to the entropy of the freezing liquid (and equal to Rln2). The entropy of the solid remains Rln2, and is constant, to a very much lower temperature. Hence we get the situation that below 0.33"K the entropy of the solid is greater than that of the liquid. From the Clausius-Clapeyron equation (21)
-
'*
dp/dT = AS/d V = (SL 1
1
it follows, that if
- &)/(VL - &) ,
AS=&-& 1
is negative below 0.33"K and if AV = VL - Vs 1
remains positive, then the slope of the melting curve will be negative. For this reason Pomeranchuk78 expected in 1950 a minimum in the melting curve and in 1959 Edwards, Baum, Brewer, Daunt and McWilliams79 and Grilly, Sydoriak and Mills80 observed experimentally a minimum in the melting curve at 0.33"K. Their results are shown in Fig. 15, The melting pressure P! around this minimum obeys the relationE0:
P! References p . 93
= 28.91
+ 32.2 (T - 0.330)'
atm.
CH.
n, I 51
EQUILBNUM PROPERTIES
91
We remark that in 1957 Walters and Fairbank81 have observed a rise in temperature when the solid at T=0.2"K (below the temperature of the minimum, 0.33"K) was melted by a rapid decrease of pressure. At much lower temperatures (probably below 10-3"K) the melting curve will become horizontal again in accordance with Nernst's heat theorem. The same conclusion can be obtained starting at absolute zero and reasoning what happens when the temperature is raised. At absolute zero the entropy of the solid and liquid are equal to zero. If the temperature is raised, the entropy of the solid becomes greater than that of the liquid. The solid is the high temperature stable phase and the melting pressure falls as the temperature is raised. Increasing the temperature above 0.33"K we get a normal situation in which the entropy of the liquid exceeds that of the solid. Now the liquid is in the high temp,erature stable phase and, with further increasing temperature, the melting pressure increases. Also for 4He a minimum was predicted in the melting curve by Goldstein82 in 1960 as follows: solid 4He contains transverse as well as longitudinal phonons (three degrees of freedom) while transverse phonons are absent in the liquid (one degree of freedom). In a temperature region where only phonons contribute to the entropy, we expect the entropy of the solid to be larger than the liquid entropy. The estimated difference between the melting pressure at absolute zero and at the temperature of the minimum is of the order of of the melting pressure. This minimum was observed by Wiebes and Kramers83 by measuring the heat of melting. During melting, caused by lowering the pressure in their calorimeter, they observed a temperature rise below 0.8"K and a temperature fall above 03°K. This method is very sensitive because the heat of melting is of the same order of magnitude as the heat capacity of their calorimeter. This minimum was also observed directly by Le Pair, Das, Taconis and De Bruyn O ~ b o t e rusing ~~, the same detection techniques as mentioned in Section 5.1, at 0.77"K. Between 0.5"K and the temperature of the minimum (0.8"K) they measured a pressure difference of 0.008 kg/cm2 84. Following one of the authors (De B.O.) we can now try to attack the minimum in the freezing curves of 3He - 4He mixtures (see Section 5.1) by discussing again the entropy difference between liquid and solid. In Section 5.2 we have seen that the entropy of mixing in the solid is given by the ideal classical temperature independent value - R [XslnX, + (1 - Xs)ln(l - Xs)]down to very low temperatures (0.05"K). On the other hand, we have learned in section 3.3 that the entropy of mixing in the liquid is far from ideal13~36(see Section 3.3; Figs. 6 and 7B). Below about 1°K the
+
Referencesp . 93
92
K.W. TACONIS AND R. DE BRUYN OUBOTER
[CH.U,
5
excess entropy Sz becomes negative, and starts to compensate for the positive ideal entropy of mixing. See Fig. 7B in which SF is plotted versus the temperature for a 10% mixture. This means that in the liquid, quantum degeneracy of the entropy of mixing already starts below 1°K. Hence again we get below a certain temperature the peculiar situation in which the entropy of the liquid becomes lower than the ,,constant” entropy of the solid. In Section 3.2, eq. (18) it was derived that the slope of the freezing curveis given by the modified Clausius-Clapeyron equation :
From this equation it follows that if the partial entropies of the liquid are smaller than the ,,constant” partial entropies of the solid (S3, < SJS S,, <S4J below a certain temperature, and if the partial liquid volumes remain larger than the partial solid volume (V3, > V3s;v,, > v,,), then the slope becomes negative. For mole fractions of about 4, this effect should be the most pronounced. For the same reason we may expect that the melting curves of ’He -4He mixtures also have a minimum, since the slope of the melting curve is given by:
Since little experimental data are present at the moment, it is difficult to give a quantitative verification. However, this entropy effect gives the right order of magnitude. From Figs. 15 and 16 follows that the minima in the freezing pressure curves for the mixtures are considerably lower than the freezing pressure of pure ,He. This can be explained by the following reasoning: Let us start at constant temperature with pure ,He at its freezing pressure and see what happens when a small amount of 3He is added to respectively the liquid and solid. When at constant temperature a little amount of 3He is added, the entropy of the solid becomes greater than that of the liquid. References p . 9 3
CH. II, 8 51
m m m m PROPERTIES
93
Hence the solid is the increasing concentration stable phase and the freezing pressure falls with increasing concentration below the freezing pressure of pu;e 4He. We like to mention that one has also to consider the possibility that the difference between the internal energies of solid and fluid decreases for a mixture29, and therefore the freezing pressure is lowered for a mixture, since the fluid has a larger positive heat of mixing (Section 3.3) than the solid (Section 5.2). In Fig. 16 the dashed line M is the projection on the T - X plane of a line through the minima in the P - T curves (freezingcurves). We may remark finally that this apparent anomaly in the low concentration region is due to the different mechanisms which determine the minima for pure 4He and for the mixtures. The minimum in the freezing curve of pure 4He occurs at a lower temperature than, for example, the minimum of an 8.9% mixture. As a conclusion of this chapter we emphasize the fact that in a short period of time so much work has been performed in order to understand the properties of liquid helium mixtures, which are often at first sight very complicated. A picture has been developed on the basis of which more work is indicated, especially in the low temperature region down to 0.1" K. Also information of the He1 phase up to the critical temperature is far from complete and the research of the solid phase with its various crystal structures and its melting has still to be carried out. REFERENCES 1 2 3
4 6 6
7
* 9 10
11
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95 R. P. Feynman, Phys. Rev. 91, 1301 (1953); 94, 262 (1954). P. B. Linhart and P. J. Price, Physica 22, 57 (1956). 46 E. A. Lynton and H. A. Fairbank, Phys. Rev. 80,1043 (1950). 46 J. C. King and H. A. Fairbank, Phys. Rev. 93,21 (1954). 47 H. C. Kramers and C. G. Niels-Hakkenberg, International School of Physics “Enrico Fermi“, XXth Course, Varenna, 1961, to be published in Nuovo Cimento; Proc. 7th intern. conf. Low Temp. Phys. (Toronto, 1960) p. 644;8th intern. c o d , Low Temp. Phys. (London, 1962) p. 158. 48 D. J. Sandiford and H. A. Fairbank, Bull. Am. Phys. Soc. 11,4,291 (1960); Proc. sec. Symp. on liquid and solid 8He (Ohio State University Press, Columbus, 1960)p. 154. 49 G. V. Chester, Superfluidity, International School of Physics “Enrico Fermi”, XXth Course Liquid Helium, Varenna, 1961. 5a J. R. Pellam, Phys. Rev. 99,1327(1955). 51 N. G.Berezniak and B. N. Esel’son, Soviet Phys. ’JETP 4, 766 (1957). F. A. Staas, K. W. Taconis and K. Fokkens, Physica 26, 669 (1960). 63 J. J. M. Beenakker, K. W. Taconis, E. A. Lynton, Z. Dokoupil and G. Van Soest, Physica 18,433 (1952); J. J. M. Beenakker and K. W. Taconis, Progr. Low Temp. Phys. 1, ed. C. J. Gorter (North-Holl. Publ. Co., Amsterdam, 1955) Chap. 6. 54 T. P. Ptucha, Soviet Phys. JETP 39, 621 (1960);Phys. Abhandlungen aus der Sowjet Union, Band 5, Heft 5, 432. 55 G.0.Harding and J. Wilks, Proc. Roy. SOC.A268, 424 (1962). 513 E. W. Guptill, A. M. R.Van Iersel and R.David, Physica 24, 1018 (1958). 57 C. J. N. Van den Meijdenberg, K. W. Taconis and C. Le Pair, Physica 27, 117 (1961). 68 B. N.Esel’son, A. D. Shvets and R. A. Blablidze, Soviet Phys. JETP 7, 161 (1958); B. N. EseI’son and B. G. hsarew, Doklady Mad. Nauk S.S.S.R. 72, 265 (1950); Zhur. Eksp. Teor. Fiz. SSSR 20,742 (1950). 59 D. 0. Edwards, A.S. McWilliams and J. G. Daunt, Phys. Rev. Letters 9, 195 (1962); Physics Letters 1, 218 (1962). 6O J. Vignos and H. A. Fairbank, 8th Low Temp. Conf. (London, 1962); Bull. Am. Phys. Soc. 7, 77 (1962). H.Weinstock, F.P. Lipshultz, D. M. Lee, C.F. Kellers and P. M. Tedrow, 8th Low Temp. Conf. (London, 1962); Phys. Rev. Letters 9, 193 (1962). 62 E. J. Walker and H. A. Fairbank, Phys. Rev. 118, 913 (1960). 63 V. N. Zharkov, Soviet Phys. JETP 6, 714 (1958). 64 I. M. Khalatnikov and V. N. Zharkov, Soviet Phys. JETP 5, 905 (1957). 65 I. M.Khalatnikov, Zhur. Eksp. Teor. Fiz. SSSR 23, 169 and 265 (1952); Usp. Fiz. Nauk 59,673 (1956); 60,69 (1956). 66 F. W. Sheard and J. M. Ziman, Phys. Rev. Letters 5, 138 (1960). 67 P. G. Klemens and A. A. Marudin, Phys. Rev. 123, 804 (1961). 66 E. W.Montroll and R. B. Potts, Phys. Rev. 100, 525 (1955). 69 0.Redlich and A. T. Kister, Ind. Eng. Chem. 40, 345 (1948); Chem. Eng. Progr. Symp. Ser. 48, no. 2 (1952);49 (1952); E. F. G.Herington, Nature 160, 610 (1947). 70 E. A. Guggenheim, Thermodynamics(North-Holland Publ. Co. Amsterdam, third ed., 1957). 71 F. London, Superfluids 2 (Wiley and Sons, 1954). 72 J. S. Rowlinson, Liquids and liquid mixtures (Butterworth Scientific Publication, London, 1959) p. 162. 73 W.H. Keesom and Miss A. P. Keesom, Commun. Kamerlingh Onnes Laboratory No. 221d; Proc.Kon.Acad.Amsterdam,35,736(1932); H.C.Kramers,J.D. Wasscher and C. J. Gorter, Physica 18,329 (1952); J. Wiebes, C. G. Niels-Hakkenberg and H. C. Kramers, Physica 23, 625 (1957); M. J. Buckingham and W. M. Fairbank, Progr. Low Temp. Phys. 3 (North-Holland Publ. Co. Amsterdam, 1961) Chap. 11. 48
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T. R.Roberts and S. G. Sydoriak, Phys. Rev. 98,1672 (1955); D. W. Osborne, B. M. Abraham and B. Weinstock, Phys. Rev. 94,202 (1954); 98,551 (1955); D. F. Brewer, J. G . Daunt and A. K. Sreedhar, Phys. Rev. 110,282 (1958); 115,836 (1959); A. C. Anderson, G. L. Salinger, W. A. Steyert and J. C. Wheatly, Phys. Rev. Letters 6, 331 (1961).
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78
(1962). Ra
*4
3. Wiebes and H. C. Kramers, Physics Letters 4, 298 (1963). C. Le Pair, K. W. Taconis, P. Das and R. de Bruyn Ouboter, Letter in Physica 29,755 (1963).
85
66 87
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N. G. Berezniak, I. V. Bogoyavlenskii, B. N. Esel’son, Sov. Phys. JETP 16,1394 (1963). G. Careri, J. Reuss and 3. J. M. Beenakker, Nuovo Cmento X 13,148 (1959). I. M. Lifshitz and D. G. Sanikidze, Soviet Physics JETP 35,713 (1959). K. N. Zenov’eva, Soviet Physics JETP 17, 1235 (1963). N. G. Berezniak, I.V. Bogoyavlenskii and B. N. Esel’son Zhur. Eksp. Teor. Fiz. SSSR 45,487(1963).
Nofe added in pro08 The references 85, 87, and are added in proof on pag. 46-47. This too recent work is however not discussed in Section 5.
CHAPTER I11
THE SUPERCONDUCTING ENERGY GAP BY
D. H. DOUGLASS JR. AND L. M. FALICOV OF PHYSICSAND INSTITUTE FOR THE STUDYOF METALS, DEPARTMENT UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS
CONTENTS: 1. Historical introduction, 97. - 2. The physical significance of an energy
gap, 99. - 3. The theory of the superconducting energy gap, 102. - 4. Theory of superconductivetunnelling, 140. - 5. Experimental determination of the superconducting energy gap, 153.
1. Historical Introduction The idea that an energy gap could be responsible for the superconducting properties of metals goes back to the middle thirties when F. London1 (1935) suggested that from the microscopic point of view it was possible to explain his phenomenological equations by assuming that “the electrons be coupled by some form of interaction in such a way that the lowest state be separated by a k i t e interval from the excited states.” However, infrared absorption measurements down to 1014 cycles per second 293 showed no difference between the normal and the superconducting state appearing in this range. The first experimentalevidence that a gap could exist came from the microwave experiments of H. London4 (1940) who observed that as the temperature was decreased, no appreciable absorption of the electromagnetic radiation took place up to a frequency of lo9 cycles per second. These two independent experiments gave upper and lower limits to the value of a possible energy gap. Welker6 formulated in 1939an unsuccessful theory which attempted to explain the Meissner effects in terms of an energy gap. Ginzburg7-9 introduced a twofluid model in which the number of superconductingelectrons was proportional to exp(--E,/UcBT). This evidently suggestedthe existence of a characteristic energy gap eo NN kBT, across which the “superconducting” electrons had to be excited to become “normal”. Ginzburg went even further and, by References p. 189
97
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D. H. DOUGLAS3 JR. A N D L. M. FALICOV
[a. a8 1
analogy to the Landau theory of superfluid heliumlo, assumed that the elementary excitations in a superconductor consisted of a pair formed by an “electron” outside the Fermi sphere and a “hole” inside it. The energies of these excitations, referred to the ground state, were P2 P: E,(P) = - - - + A , 2m 2m
IPI>P,
P; P2 4I(P)=---+A, 2m 2m
IPI
so that the minimum energy of any pair was 24. Ginzburg did not attempt to explain either the nature and origin of 4,or its behavior as a function of temperature, and therefore he called his theory “quasi-microscopic”. Daunt and Mendelssohnll(1946) observed zero thermoelectric power in superconductors and postdated the existence of an energy gap which would explain the lack of an entropy flow associated with a supercurrent. In the early fifties the theoretical and experimental contributions which threw light on the problem of a superconducting energy gap began to multiPlYOn the experimental side, more accurate measurements of the electronic thermal conductivity by Goodman12 and specific heat by Brown et a1.15 showed that an exponential law gave a better fit to the data than the until then successful Gorter-Casimir cubic law. This was the first direct and unequivocal experimental evidence of the existence of a gap. Goodman explained his results in terms of Koppe’s14 microscopic theory of superconductivity, which was reinterpreted in terms of an energy gap. In 1957 Glover and Tinbarn15 were successfulin reaching the far infrared region of the electromagneticspectrum and could observe,in the case of lead, a sudden drop in the absorption as the frequency was decreased. This gave a first precise value for the gap, which turned out to be of the predicted order of magnitude and about 3 k ~ T ,Later . experiments in the far infrared region on tin and lead16 and in the microwave range on aluminum17 confirmed the earlier results and gave firm support to the then newly born theory of Bardeen, Cooper and Schriefferls. From the theoretical viewpoint, the early fifties showed a slow but steady progress towards the understanding of the microscopic details of superconductivity. The fact that the electron-phononinteractionwas known to play an important role in the mechanisms leading to superconductivity plus the then almost certainty of the existence of an energy gap produced at first unReferences p . 189
m,8 21
TEIE SUPBRCONDUCTING ENERGY GAP
99
successful but enlightening theoriesIOt20. In 1955 Bardeen21proved that the Meissner effect could be explained in terms of an energy gap, and that it was in fact a direct consequenceof the existenceof suchagap.Buckingham22(1956) showed that, because of the second order character of the superconducting transition, the square of the energy gap had to vary linearly with temperature in the neighborhood of the transition temperature. The work by Cooper23 (1956), which proved the possibility of obtaining bound pairs of electrons in a metal in the presence of attractive interactions, was the starting point from which Bardeen, Cooper and Schrieffer18constructed their enormously successful theory. This theory contains an energy gap of the right order of magnitude and was able to explain most of the experimental data available at the time. Soon after, B0goliubov2~and Gor’kov26 were able to rederive, using different formulations, the results of Bardeen, Cooper and Schrieffer. The new approaches, with their very elegant mathematical techniques, made the field of superconductivity extremely attractive to a great number of theoretical physicists, whose exponentially increasing number of contributions have now flooded the literature. Since the beginning of the present decade, one major experimental breakthrough in the study of the superconductors has opened many possibilities for determiningthe properties of the energy gap. The discovery by Giaever26-28 and by Nicol et al.29 that the current-voltage characteristics of “sandwiches” consisting of superconductors and/or normal metals separated by thin insulating layers were non-linear and that the non-linearity could be easily interpreted in terms of the energy gap provided a simple technique from which an enormous wealth of information was obtained. New effects and data as well as their subsequent or precedent interpretations and theories are being found now almost day by day. In this respect the problem is still open and active, and the present review must be considered only as a brief summary of the present state of a rapidly moving field.
2. The Physical Significance of an Energy Gap
In order to understand the meaning and implications of the superconductive energy gap, it is instructive to review the properties of some other systems which also exhibit energy gaps, and point out the similarities and differences between them and superconductors. Any of these systems consists in general of a very large number of particles, and it is possible to assume that in the absence of any external perturbation and in the limit of zero temperature it is in equilibrium in its state of minimum energy, the ground state. From this ground state the system can be excited by means of external References p . 189
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[CH.III, 4 2
D. H. DOUGLAS JR.AND L. M. PALICOV
agents into states of higher energies; these excited states may in general be decomposed in a superposition of independent or nearly independent elementary excitations, the so-called normal modes of excitation. The normal modes propagate through the system with a well-defined propagation vector k and a non-negative characteristic energy E(&). The total energy of an excited state, measured with respect to the ground state energy, can be fairly well represented by the sum of the energies of the excited normal modes. The elementary excitations can be of two general classes: a) Excitations of Bose-Einstein type, which can be created or destroyed individually. b) Excitations of Ferm-Dirac type, which can only be created or destroyed in pairs. In this case it is useful to call these normal modes quasiparticles and quasi-holes, so that in any state of the system the number of quasi-particles equals the number of quasi-holes. Many of the important properties of the elementary excitations are given by the functions E(&).It is said that a given class of normal modes has an energy gap E, if the minimum value of E(k) is not zero. In the case of Bose-Einstein excitations, two possible E(&) curves are shown schematically in Fig. 2.l(a) and (b); (a) may represent the energy
k
-kF
kF
A k
-k,?q
kF+q k
Fig. 2.1. Schematic representation of E(k) curves. (a) and (b) correspond to BoseEinstein excitations. Curves (c), (d), (e) and (f) represent Fermi-Dirac excitations; the full lines corresponds to quasi-particles and the dashed lines to quasi-holes. References p . 189
CH.III, 8 21
THE SUPERCONDUCTING Eh'ERffY QAP
101
spectrum of lattice vibrations (phonons) and does not exhibit a gap, while (b), which may represent the spectrum of plasma oscillations (plasmons), shows an energy gap q,, In the case of metals, the value of E,, for plasmons is in the range 5-30 eV. For F e d excitations, three possible E(k) curves are shown schematically in Fig. 2.l(c), (d) and (e). Curve (c) corresponds to a normal metal, in which an infinitesimal amount of energy is enough to excite a quasi-electron quasihole pair; the absence of an energy gap leads immediately to a finite conductivity at zero frequency and zero temperature. Fig. 2.1 (d) represents an insulator (or semiconductor); the energy gap, E , =24, that is the minimum energy required to excite a quasi-electron quasi-hole pair, leads naturally to zero conductivity at zero frequency and temperature. Fig. 2.1 (e) corresponds to a superconductor; it may be thought of as a normal metal in which the minimum energy of excitation, always at Ikl=kF (that is on the Fermi surface) has been changed from zero to 4. At this point it is worth mentioning the main difference between an insulator and a superconductor. In the insulators the minima of E(k) are determined primarily by the lattice and consequently fixed with respect to the lattice frame. A change in the distribution of the electrons can only be accomplished by creating quasi-electrons and quasi-holes. On the other hand the minima of E(k) in a superconductor are essentially determined by the electron distribution and the interactions among the electrons, and therefore it is possible to displace the electron distribution uniformly, displacing the minima of E(k)with it. Such a caseis shownin Fig. 2.1 (f) and corresponds to a superconductor carrying a supercurrent. So in the first case the energy gap, originated by a lattice potential external to the electrons, is responsible for the zero conductivity, while in the superconductingcase the energy gap, due to the internal electron interactions, may be considered responsible for the infinite conductivity of the system. In the case of the Fermi excitations, the existence of an energy gap is always related to some binding energy. In the insulator, the energy gap is due to the difference between the binding energies of two atomic states, strongly modified by the periodic arrangement of the atoms in a lattice. In a normal metal the lattice effects completely overwhelm the difference in binding energy of the atomic states and the energy gap therefore does not exist. In a similar fashion it may be expected that the superconducting energy gap is due to some kind of binding energy between particles. This binding energy requires an effective attractive interaction among the particles. If this References p . 189
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D.H.DOUGLASS JR.AND L. M. FALICOV
[a. m,I 3
is essentially a pairing interaction, it would lead to the formation of some kind of bound pairs. Moreover the binding energy of a pair will be modified by the presence of all the other particles and in some favorable cases will give rise to an energy gap in the excitation spectrum of the total system. This mechanism is absolutely general and applies equally to all the many-fermion systems with any kind of effective attractive interactions, e.g. electrons, nucleons and helium 3. Therefore, from a theoretical point of view, the problem of determining a “superconducting” energy gap in the excitation spectrum of a manyfermion system consists essentially of three steps : a) Determining the interactions which may cause the formation of bound pairs. b) Selection of the particles which form the bound pairs. c) Study of the interactions and scattering processes between pairs and determination of the energy spectrum of the excitations. These mechanisms will be studied in some detail in the following section. 3.
The Theory of the Superconducting Energy Gap
3.1. DERIVATION OF THE ENERGYGAPEQUATION
The first successful microscopic theory of superconductivity, that of Bardeen, Cooper and Schrieffer (BCS)l*, has its origin in the work of Coopera3. He proved that two electrons, excited slightly from the ground state of a Fermi distribution at zero temperature, could form, in the presence of any attractive interaction, a real bound state. The energy of this state lies below that of the ground state of the Fermi sea, and the wave function of the pair is localized. This evidently indicates the possibility of a new kind of ground state in which, by putting a macroscopic fraction of the electrons in such bound pairs, the total energy would be reduced from that of the “normal” state; in other words the usual ground state of the Fermi sea must be unstable against the formation of Cooper pairs*. In order to minimize the energy of the new (superconducting) ground state, the electron states from which the Cooper pairs are to be formed must be chosen so as to maximize the binding energy of each pair and to obtain the
* Recently Van Hove80 has pointed out that the normal ground state may be unstable even in the presence of repulsive interactions. This is certainly related to the observation of Bogoliubov, Tolmachev and Shirkoval (p. 91) that superconductivity may exist for some cases of repulsive interactions. Since it seems to be only of academic interest, this possibility will not be pursued further. References p . 189
CH. III,
8 31
THE SUPERCO~UCXINGENERGY GAP
103
greatest possible number of such pairs. Since in the scattering processes due to electron interactions the total momentum (or the total k-vector) is in general conserved, the maximum number of pairs that can be scattered coherently is obtained by pairing states such that their total momentum is a constant. In particular for the ground state, which carries no net current, the best possible pairing is between states of equal and opposite momentum. At the same time, since the exchange terms reduce the effective strength of the interaction, the pairing of states of opposite spin maximizes the binding energy. In this way the superconducting ground state may be formed by assuming that it can be expressed solely in terms of Cooper pairs, i.e. the states kf and -kJ are always occupied simultaneously; consequently, only the interaction terms which scatter a given pair of electrons from one such state into another of the same kind will be of paramount importance in determining the condensation energy of the superconducting state *. To include these considerations in a quantitative theory of the superconducting state it is convenient, following BCS, to use the formalism and notation of second quantization, expressing the band energy terms, i.e. kinetic energy plus lattice potential energy, and the electron interactions in terms of creation and destruction operators. These are indicated by c& and c&/#respectively, where k denotes a definite Bloch state in the crystal and cr is a spin index. These operators satisfy the usual Fermi anticommutationrules [ck/aklo, c$/a']
[Ck/a, ck'/a']
+
+=
= [ck:m
(3.1)
3
ckfja']
+
=0
(3.2)
and i
(3.3) is the number operator whose expectation value gives the probability of the state k/u being occupied. In this formulation, the tetal band energy of the electrons is equal to the sum over all possible states k/a of the product of the one-electron energy e, of the state and the probability of the state being occupied, i.e. = c:a * (3.4) nk/a
= ck/u ck/n
zek
La
By convention, the energy &k is measured from the Fermi level and includes
* The Cooper pairs kf, and -kJ.are energetically the most favorable, but they are not necessarily the only ones that may contribute to the superconducting ground state. In s ~ m ecases of magnetic or spin-dependent interactions, the partial pairing of electrons with parallel spin may play an important role. This problem has been discussed by Werthamer et a1.8* in connection with magnetic properties of superconductors. References p . 189
104
D. H. DOUGLASS JR. A N D L. M. FALICOV
[CH. III, 9
3
self-energy renormalizations due to interactions. The interaction energy between electrons is represented by a sum of scattering terms in which two electrons initially in the states kla, k'lu', are scattered into the states k +lc/a, k'-ic/u' by an interaction strength - V ( K ) ,assumed to be attractive
xi
=-
c v(K)
'k++K/a
'k'lr' ' k / a
C:-w/a'
kk'r
(3.5)
At this stage it is convenient to introduce the Cooper pairing assumption, in which the electrons occupy the states k t , -kJ simultaneously, and are scattered from k t , -kJ into another set of paired states k't, -k'J. The pair creation and destruction operators
satisfy the commutation relations
[bk,bk+']-= ( I [bR,bk']-
-
nk/$
- n-k/+)6kkq,
=o,
(3.7) (3.8)
[ b k , b r ] + = 2 bkbk'(1
- &<).
(3.9)
In terms of these operators,
& @ , = ~ 2 € k b b~k ,
(3.10)
k
&@*
=-
c
V(K)
b k = ~bk
kK
=
-
c
Vkk'
b: b k
(3.11)
kk'
where only the scattering among Cooper pairs has been retained in A?,.In setting up an explicit form for the ground state function Y, BCS assume that the probability with which a specific configuration of pairs appears can be expressed as a product of probabilities that each individual pair state is occupied, i.e. ~ = ~ [ u k + v k b t ] @ o , (3.12) k
where Gois the vacuum state, ok2the probability of having the pair k occupied, uk2 the probability of not having the pair k occupied. From the definitions of Uk and 0, 2 2 uk uk (3.13)
+
Evidently (3.12) describes a state in which the total number of electrons (or References p . 189
CH.III, 8 31
105
THE SUPERCONDUCTMG ENERGY GAP
pairs) is not well determined,since it mixes states with any number of particles, However if u k and u k are properly chosen, the distribution of probabilities is highly peaked in the neighborhood of the true number of particles and the fluctuations, which are small, cause no serious difficulties. The expectation value of &‘, +XI for the state (3.12) is computed to be
w
Ek v i
=2
( s o +&‘I)
-
k
vk#f
ukukuvvv
*
(3.14)
kk’
If now u k is considered a variational parameter and (3.13) is taken as a constraint, the value of u which minimizes Wis given by v:
=+-
(6:
+A:)* Ek
1
’
(3.15)
where Ak, the so-called energy gap parameter, is A, =
(3.16)
vkk’ uk,Vk,,
k’
where By inserting (3.13) and (3.15) into (3.16), it follows that
(3.17) where
(3.18)
This function corresponds, as it will be seen shortly, to the energy of the quasi-particles. Equation (3.17) is the fundamental result of the BCS theory. It is a nonlinear integral equation for the energy gap parameter. One possible trivial solution, d k = 0, corresponds to the normal metal. This is easily seen by looking at (3.15), (3.13) and (3.14). If d = 0, then u 2k
&!i=o,
&k
vi=o,
u:=1,
&k>O;
Ek
(3.19)
= lEkl 9
as shown in Fig. 3.1 (a); and
w, = 2 ck& k
(3.20)
eke0
is the total energy of the normal ground state. If on the other hand we assume Referencesp . 189
106
D. H.D O W O W JR. A N D
L. M. FALICOV
E A k A b
O
U
0
d
1
,
‘k
Fig. 3.1. The occupation of electrons in the ground state o k 2 , and the energy of the quasi-particles E k in the normal metal (a) and in the superconductor (b).
that a non-trivial solution dk# 0 exists for (3.17), which corresponds to the superconducting state, then (3.21) It can be easily proved that
ws < WN.
(3.22)
In particular, if the following assumptions are made : (3.23)
=O
where ODis a cut-off energy, and the density of statesN(e,) can be replaced by a constantN(0) in the interval ickl<@D, (3.21) turns out to be Ws = WN-M(O) 8, [(@;
+ A’)* -
(3.24)
For most superconductors to which the previous assumptions are applicable, it is also found that A<<@,, (3.25) and therefore (3.24) can be approximated by w S gW ~ - + J ( O ) A * .
(3.26)
The functions OR2 and Ekfor this case are plotted in Fig. 3.1 (b). These results then imply that whenever a non-trivial solution A k # 0 of References p . 189
CH.m, 5 31
107
TKE SUPERCONDUCTlNGENERGY GAP
(3.19) exists, the energy of the superconducting ground state is lower than that of the normal state, i.e. the system must be superconducting at low temperatures. At the same time the distribution of electrons is no longer a sharp step function, but has the shape shown in Fig. 3.1 (b) which strongly resembles the Fermi distribution at finite temperatures (k,T-d). 3.2. ELEMENTARY EXCITATIONS AND DENSITY OF STATES When the structure of the superconducting ground state is determined, as given by formulae (3.12), (3.13), (3.15) and (3.17), it is necessary to determine the nature and properties of the elementary excitations (quasi-particles). It is always convenient to assign to each quasi-particle a definite k-vector and spin state. In the case of a normal Fermi distribution, the creation of an excited state u (with one particle added to or substracted from the system) of definite “momentum” and spin-up can be accomplished in two ways: a) by adding one electron in the state kt (if unoccupied);b) by removing one electron from the state -k& (if already occupied). Therefore, if the occupation of particles at T = 0 is kept in mind, the creation operator ah+ for spin-up quasi-particles is + a: = c k / + , Ek > O(quasi-electron) ; (3.27) ak < 0 (quasi-hole) a: = c-klL,
.
-
The minus sign in the last line corresponds to a convenient choice of an arbitrary and irrelevant phase factor elq. In a similar fashion the creation operators j?; for spin-down quasiparticles are + = ck/L , ak > 0(quasi-electron) ; (3.28) where this time the phase factor is chosen to be plus one. In a superconductor these operators cannot be applied, since a given state k is partially occupied, with probability f&’, and partially empty, with probability uk2.Consequently if a given quasi-particle with definite k-vector and spin is to be created, this can only be accomplished by partially creating one electron and partially destroying one. This is done by defining the following set of creation operators: a:
- vkc-k/J.
uhc,+/+
3
(3.29) The system of equations (3.29) define a transformation from the usual electron References p . 189
10s
[a. m, I 3
D. El. DOUGLASS JR. AND L. M. FALICOV
creation and destruction operators (c+, c) to the new quasi-particle creation and destruction operators (a+$+ ,a$). It is generallyknown asthe Bogoliubov canonical transformation31 and a general theory of superconductivityexactly equivalent to BCS can be obtained from it by requiring a ground state with no quasi-particles and minimum energy with respect to the variational parameters zik and o k , which are subject to the constraint condition (3.13). The transformation is called canonical because it preserves the Fermi anticommutation rules, i.e. [a
l, @-k’] + = [b ip k ’ ] + = b;kk’,
[ak,
3
+ = [Pk, b k ’ ] + =
bkt]
+ =0
(3.30)
and consequently the quasi-particles may be regarded as independent fermions. It is also evident that in the normal metal, where zik and V k are given by (3.19), (3.29) reduce to (3.27) and (3.28), as expected. To determine the excitation energy of one quasi-particle, the expectation value of Zo ZI must be computed for the wave functions
+
yt = a:
y = C k+/ t k g i U k ‘
+
Vk’b:,) @o
3
(3.31) which after some algebraic manipulations turns out to be WEk -- W Bk -- s +(&:
+A:)*=
@+Ek*
(3.32)
The existence of an energy gap now becomes apparent, because for d k # 0 the minimum energy of the excitations is Ek,
=
3
(3.33)
do,
k, being one vector on the Fermi surface (&k, = 0). In addition, since the quasi-particles must be always created in pairs in order to conserve (at least in the average) the total number of electrons, the minimum energy of an excited state is (3.34) We,, L Ws 24, = Ws E,
+
+ .
One of the most important functions to be determined, and the one whichis most susceptible of experimental verification is the density of excited states as a function of the excitation energy Ek.For this purpose it is convenient to divide the quasi-particles into two different groups : those with k-vectors outside the Fermi surface (superscript l), and those with k-vectors inside the Fermi surface (superscript 2). In the normal metal group 1 corresponds to References p . 189
CH. m, g 31
THE SUPERCONDUCTING ENERGY GAP
109
quasi-electrons and group 2 to quasi-holes, but in the superconductor this correspondence is no longer exactly valid because of the smearing out of the electron distribution (Fig. 3. I (b)). For a general Ak which is an arbitrary function of the k-vector, the density of states for each group X s ( E ' ) is such that Ns(E')dE' is equal to Q/4 n3 times the total volume in k-space between the surface of constant energy E' and E' $- dB' (here Sa is the volume of the sample). The computation of the volume in k-space involves an integral in which the lattice effects and the superconductingeffects appear together through the dependence of Ek on &k and A,. However, all these complications appear only when Ak is anisotropic, i.e. is not constant over the energy surfaces &k = constant. If Ak can be assumed to be only a function of &k (i.e. isotropic energy gap), and the density of states in the normal metal "(8) is known (by measurements of the electronic specific heat, for instance) the calculation can be greatly simplified. In this case, since the total number of states is conserved ds =Ns(E') dE' ,
"(e)
(3.34)
then dvs(Ei) ="(s)
I-]:[
-
-- E ' N N(fJ(E')>"- dz) [(E')' - A']* + A? ds
(3.35)
Since E' cannot take values smaller than A, equation (3.35) is only valid for E' > A,, and Xs(Ei)= 0, E' < A,. (3.36) The second term in the denominator of (3.35) shows that any abrupt change in d(e) will be reflected in a corresponding irregularity of&.#). In (3.35) the argument of"(& J ~ 3 2 - 4 2 ) must be taken with positive sign for group 1 and with negative sign for group 2, In general, the expansion about the F e d energy s = 0 is "(&)
="(o)
&Jv&(O) + ...,
(3.37)
which indicates that the density of states of both groups of quasi-particles are not the same. This is shown schematically in Fig. 3.2 (a). However the range in E for whichXN(&)may vary appreciably is much larger than A, and for nearly all practical purposes only the first term in (3.37) must be retained. If "(&) and A(e) are assumed to be constant, the superconducting density of states takes the form shown in Fig. 3.2 (b). If on the other hand A(&)is References p . 189
110
D. H.DOUOLASS JR. A N D L. M. FALICOV
5ilL -JJ EZ
A
A
E’
Fig.3.2. The density of state function JYs(E). (a) general behavior; E’ (b) the constant Jf”N(e),
constant A model; (c) the BCS model.
assumed to have a cut-off at =8,,as in formula (3.23),MS(Ei)will display irregularities at E‘= 6J1, and E‘=(82 +A2)* as shown in Fig. 3.2 (c). Up to this point only quasi-particle excitations have been considered. There are, of course, other types of possible microscopic excitations in a superconductor, which can be classified in two separate categories:those also present in the normal state which are only very slightly modified by the superconductingproperties (e.g. phonons, plasmons), and those characteristic of superconductors which are a consequence of special properties of specific superconducting interactions. A feature of the superconducting state is the possibility of collective excitations with energies lying in the energy gap. These are called excitons and their possible existence was pointed out by Anderson33 and Bogoliubov31. They cannot exist in the normal state since the finite density of quasi-particle states near the Fermi energy would lead to rapid decay; in the superconductors they are likely to have rather long lifetimes. The nature of the exciton spectrum is closely related to the angular dependence of the interaction vkk,. For a spherically symmetric V& (which includes the special case V,, constant), the only collective mode which may exist is the plasma oscillation, and References p . I89
a. m, 31
111
TRE SUPERCONLXJCTINGENERGY QAP
since its maincontribution comes from the long range Coulomb interaction, the energies are several orders of magnitude larger than the energy gap (15 eV). If V,, has some anisotropy and could be expanded in spherical harmonics, the p, d, f, etc. parts, if negative, may lead to excitons of the correspondingangular characterwith energieswithin the energy gap. It is worth pointing out that the odd terms in the expansion (p,f, etc.) will tend to couple partially states of parallel spin, in opposition to the Cooper pairing which comes from the even terms in the series (mainly from the constant s part). Finally it is necessary to mention the possibility of macroscopic excitations in superconductors, namely the supercurrent carrying states in which the complete Fermi distribution is displaced from the ground state and the Cooper pairing takes place between states k q / r and -k q / $ , i.e. each Cooper pair has a net k-vector 2q.
+
+
3.3. TEMPERATURE DEPENDENCE OF THE ENERGYGAP The effect of a finite temperature on a normal metal is to excite some electrons from below the Fermi surface to states above it, so that the occupation of states is given by the Fermi function f ( Q ) = [eXp(Ek/kBT)
f I]-'.
(3.38)
This is equivalent to saying that some quasi-electrons and quasi-holes have been created, with an occupation probability given by a modified Fermi function (3.39) d E k ) =f ( l & k 1) = ceXdiek //kBT) + I]-' where Ek = lekl is the excitation energy of the quasi-particles. In a superconductor, the effects of a finite temperature are more complicated. In addition to the creation of some quasi-particles, the system must readjust itself to the new distribution of electrons, and this will be reflected in a change in the energy gap parameter. In order to obtain the occupation probability for quasi-particles and the temperature dependence of the energy gap dk( T), it is necessary to go back to the original BCS formalism, determine the free energy F and minimize it with respect to the parameters. The quasi-particles must have a given probability of occupation gk =gk(Ek,T ) which is assumed to be another variational parameter (in addition to u k ) . The free energy F is given by
F References p . 189
= <<*o>>
+
(<St>>
- TS,
(3.40)
112
[m.m,5 3
D. H.DOUGLASS JR.A N D L. M. FALICOV
where <( )) indicates a thermal average and S is the entropy of the system. For a normal metal, it is evident that =2cEk(1
<(2ci))N
k ek
- gk) + 2cEkgk* k
(3.41)
ek>O
For a superconductor it can immediately be generalized to 2
((so))S
= 2 c E k ( 1 - gkju; +2cEkgkUk= k
k
=
c
tEk
[(l - 2gk) u2k
+ gk]}
Y
(3.42)
k
that is, only the fraction (1-2gk)of Cooper pairs are effectively occupied. In a similar fashion
and the entropy term is given by the usual expression
- TS =2kBTz[gkkgk k
f
(l - gk)k(l
- gk)].
(3.44)
If (3.42), (3.43) and (3.44) are inserted in (3.40) and I; is minimized with respect to uk and gkkeeping (3.13) as a constraint once more, the equilibrium values are given by (3.45) gk = [exp(E&/kBT)+ l 1 - I Y (3.46)
(3.48) where (3.45) shows that the quasi-particles, as expected, satisfy the FermiDirac law; (3.46) and (3.47) are unchanged from the T=O expressions and (3.48) gives the new integral equation for the energy gap parameter as a function of temperature. The trivial solution A , =0 corresponds again to the normal state, and the superconducting solution dk # 0,when it exists, is such that Fs < F N . (3.49) References p . 189
f2I-I. myB
31
113
THE SUPERCONDUCTINO ENEROY OAP
I
D
I
T/ T,
Fig. 3.3. The temperature dependence of the energy gap in the BCS theory.
At this point it is worth mentioning that, as pointed out by equation (3.48) is such that if A , # 0 over some region of the Fermi surface, then Ak # 0 everywhere, except perhaps at isolated points or lines. This comes out from the structure of the integral equation. In addition A , can take on positive as well as negative values (or even complex values in a more general formulation) but the meaningful quantity is always l d k l and co=21d0lYexcept when lifetime effects are considered. However, in general, a superconducting solution exists only for values of Tless than a critical value T,, the transition temperature. For the BCS model of the potential, to be discussed shortly,
1.76 k,T, = A (T = 0)
(3.50)
and d ( T ) is shown in Fig. 3.3. This function A(T) has been tabulated by Muhlschlegel35.
3.4. MECHANISMS FOR THE EFFECTIVE ELECTRON INTERACTION SOLUTIONS OF THE ENERGY GAP EQUATION
AND
To determine the magnitude of the superconducting energy gap and its dependence on k-vector and temperature, it is necessary to solve the gap equation (3.17) or (3.48) which requires knowledge of the effective interaction VkkP between electrons. Several mechanisms of interaction are possible, all of which are essentially detailed modifications of two fundamental processes: a) Electron-lattice interactions, b) Coulomb interactions among the electrons. The isotope effect (discovered by Maxwell38 and by Reynolds et aL37) showed that, at least for tin, thallium and mercury, the superconducting transition temperature was a function of the isotopic mass : T, cc ~ - ( O . S O k 0 . 0 5 )
(3.51)
This evidently indicates that the lattice vibrations play a very important role hferenees p. 189
[a. m,B 3
D.n.DOUOLASS JR.AND L. M. FALICWV
114
in superconductivity, and consequently the electron interaction via phonon exchange must have a fundamental contribution to Vw. Bardeen and Pines38 showed that the effective matrix element - Vk,k + for scattering of two electrons with initial crystal momenta k and k' by exchange of a phonon of wave vector K into new electron states (k K) and (k'- K) could be written
+
(3.52) where hw, is the phonon energy and M, the matrix element for a single scattering of an electron by a phonon. This expression evidently gives an attractive interaction if I&k - &k + c tie, RJ,@ , , where eDis the Debye cutoff energy of the phonon spectrum. The Bardeen-Cooper-Schrieffer model makes the rather strong simplifying assumptions: a) (3.52) is replaced by an average (constant) value -2)MKIZ(hwK)-lin the region where lskl and Itk+ are less than ,@, and zero elsewhere; b) The Coulomb interaction V & is also taken as an average such that it is constant for and + less than @, and zero elsewhere. In this way
.I
- v,, = - (h$+G",)= - V
for I &k I,[
&k,
I both < 8,
otherwise,
=O
(3.53)
as illustrated in Fig. 3.4 (a). In addition, the summation in (3.17) is reduced to an integration over the energies &k,, and a constant density of statesM(0) is assumed over the range 181 < OD;the energy gap equation then becomes 833
1
where A = A
(&k),
<
f
de
(3.54)
eD,and
The integration of (3.54) is straightforward and gives A= References p . 189
@D
"28,exp(Sinh(l/Jcr(O) V )
l/J-(O)V),
(3.55)
O
1'.
0 O
1.
0
Fig. 3.4. A schematic representation of the various approximations for the kernel K(e, e') of formula (3.62) used in the solutions of the energy gap equation: (a) BCS model, (b) Tolmachev model, (c) Swihart model, (d) Morel and Anderson approximation and (e) Garland approximation.
where the latter expression is valid in the weak coupling limit N(0)V< 1. When the BCS assumptions are introduced into (3.48), d(e, T ) for lei < eD is given by Refirenets p . 189
116
D. H. DOUGLASS JR. A N D L. hi. FALICOV
rm. m 8 3 (3.56)
which reduces to (3.54) for T= 0. This expression gives a continuous decrease in A as T is raised; the transition temperature T, is obtained by setting A equal to zero: en 1 tanh (&/2kBTc) de (3.57)
xm=s 0
&
In the weak coupling limit (3.57) reduces to k& = 1.14BDexp(- l/N(O)V).
(3.58)
Since the Debye energy BD,which is proportional to the mean phonon frequency, varies with isotopic mass as M-*,(3.58) gives immediately the isotope effect (3.51). If (3.58) is combined with (3.59, it leads to 2d(T = 0) 2 3.52 kBTc.
(3.50)
This is called the law of corresponding states because it gives a universal relation between the energy gap at T= 0 and the transition temperature. An explicit evaluation of d ( T ) from (3.56) and (3.57) also shows that d(T)/d(O)is nearly a universal function of T/Tc. This is shown in Fig. 3.3. For the strong coupling limits, Thouless3g has shown that this function can be expressed in the simple form
(3.59) which agrees with the weak-coupling case within 0.1 percent. The BCS model, which contains rather strong simplifying assumptions and only one adjustable parameter N(0)V, gives surprisingly good agreement with most experimental data. However there are several discrepancies which require more sophisticated models. For instance the lack of isotope effect in some transition elements40 and the structure in the superconducting density of statesNs(E) in lead41.42 clearly show that the BCS model is not complete enough to include these phenomena. In addition, since the model is completely isotropic it cannot account for the anisotropy of the energy gap found experimentally in several metals. The first attempt to improve the BCS model was done by Tolmachev31, to be who assumed the effective Vkk, References p . 189
CH. III,I 31
TFIE SUPERCONDUCTIN0 ENERGY GAP
-
= - v for I &k 1,
+ vi for I
&k
117
I &k*I both < @,,
1) I &k’ [ both < &1
and I &k 1 and/or 1 &k’ 1 > O D
=O
(3.60)
otherwise.
That is, he replaced the function V(ek,ek’)by two plateaus in the &k ek,plane one attractive (electron-phonon interaction) with a cut-off at the Debye OD, and one repulsive (Coulomb interaction) with another cut-off at energies greater than ODbut less than the Fermi energy. This is shown in Fig. 3.4 (b). The solution for d(e, T ) is
=
- di(T),@D
= 0,
< I&< [ &I ;
It.l>El.
(3.61)
The BCS and Tolmachev models, as well as those to be discussed in the next few paragraphs can be expressed as special cases of a generalized isotropic model, i.e. energy gap constant on the surfaces of constant energy (Ak = A(&)). In the general isotropic case, the summation over k in (3.48) can be replaced by an integration over energies ( E or ~ Ek, depending on the model for the interaction) and two integrations over angular variables, which can be performed separately. In this way Vkk’ is replaced by a kernel Kand the energy gap equation becomes
or in terms of the quasi-particle energy E
E’ A(E,T) =
”I
- d(EiT) 7 dE’
E’[E’’ - dZ(E:T)]*
K(E,E’)tanh(-). E’ 2kBT
(3.63)
The kernel K must have some convergence properties at large values of the variables. Both equations are non-linear, and the non-linearity, as it is apparent from (3.62), occurs essentially at E’ % 0. An iteration method based on a quasi-linearization of (3.62) has been proposed by Zubarev43 and Garland4*. It simplifies enormously the mathematical difficulties which arise References p . 189
118
D. H.DOUGLASS JR.AND L. M. FALICOV
[CH.III, 8 3
in the numerical solution of the integral equation. Garland defines the normalized gap parameter p(s) and the normalized kernel I(&,&’)by
(3.64) and by using (3.64), (3.62) in the case T= 0 can be transformed into p(d) K(0,O)ds’ [fpZ(&’)A2(0)
+ &’Z]*
[I(E,E’)- I(s,O) I(O,s’)] .
(3.65)
This equation is still non-linear, but has the very important property of having a kernel which vanishes at the point of maximum non-linearity, E’ = 0. Consequently it is a very good first approximation to neglect pzA2 in the denominator of the integrand to obtain the linear integral equation
The energy gap at E =0 in this approximation is given by pl(s‘) K(0,s‘) ds’ [A;(O)p;(&’) s r Z ] + ’
(3.67)
+
This procedure is susceptible to iteration with (3.66) and (3.67) as first order solutions because, assuming that the n-th approximation to A(&) is known, the system of linear equations to be solved for the (n 1)”’ approximation is
+
(3.67~) This iteration procedure converges very rapidly. In the weak coupling limit (“0) V S 0.25 in the corresponding BCS model), the first approximation AI(E) is estimated to be correct to better than five percent; AZ(s) is estimated
to be accurate to better than one percent. Swihart45946 considers a particular solution of the generalized isotropic References p . 189
CH. HI,
8 31
119
THE SUPERCONDUCTING ENERGY GAP
gap equation (3.62) by using the Bardeen-Pines interaction (3.52) for “jellium” (free electron gas and uniform background of positive charge susceptible to carrying phonons) to which a screened Coulomb interaction is added. The resulting kernel is K(E,E‘) = V ( (E
1 a2x2
I
- E’ I), x2
-1
V(X) = -~ Ig 1+2x2 - 1 a2x2
1,
(3.68)
where E is measured in units of the plasma energy tzw, and a2 = kt/4kF2,k. being the inverse Fermi-Thomas screening length and kF the Fermi momentum. For the actual solution of the integral equation an approximation of V(x)by a two square-well function is taken. The kernel in this form is 0.4
03 02 I 0 0
-
‘ 0 I
Y
4-0.1 -0.2
-a3 -0.4 0
1
2
3
4
5
6
7
8
€/OD
Fig. 3.5. The energy gap parameter for two different temperatures as calculated by S ~ i h a r t The ~ ~ .kernel was a two square-well curve with an attractive part of strength N ( 0 ) V = 0.4 and a repulsive part of strength N ( 0 ) V = 0.6. The solid curves are for a cut-off of the repulsive interaction at 4 OD. The circles are the solution at T = 0 for a cut-off of 3.6 OD.
illustrated in Fig. 3.4 (c). Some results are shown in Fig. 3.5. The most important features that come out of this approximation are: a) The temperature dependence of the gap is nearly the same as that given by the BCS model, although the values of d(T)/d(O) are in general a few percent higher for the same TIT,; b) The electronic specific heat and the critical magnetic field curves can be brought into slightly better agreement with experiment by adjusting the strengths of the kernel and the position of the cut-offs; References p. 189
120
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH. IN,
83
c) The isotope effect is found to vary for different parameters. If the change in the energy gap parameter is related to the change in isotopic mass by
(3.69) where A is the energy gap at T = 0 and E = 0, the coefficient C is equal to zero in the BCS model. For various choices of parameters Swihart found C to vary between 0.005 (for Hg and Pb with a cut-off for the Coulomb interaction at 1000 0,)and 1.3 (for Ru and a Coulomb cut-off at 10 OD). A very important contribution to the understanding of the electronphonon interaction in superconductors was made by Elia~hberg4~ and reexamined from a different point of view by Liu48. They prove that the Bardeen and Pines formula (3.52),which gives the effective electron-electron interaction via phonon exchange in normal metals, is not correct when applied to superconductors. Formula (3.52) was obtained by Bardeen and Pines from a self-consistent calculation by considering the emission and absorption of a virtual phonon in a normal electron gas. If a similar calculation is carried out in a superconductor, taking the Cooper pairing into account from the very beginning, the effective superconducting electronelectron interaction via phonon exchange turns out to be
(3.70) where Ek and Ek+ ,are the superconducting quasi-particle energies and AO, a modified phonon energy which for most practical purposes can be replaced by the normal phonon energies Am,. The fact that Ek instead of &k appears in Vkk‘,and the different functional dependence on the energies result in some important changes in A @ ) , mainly the appearance of local maxima and minima at energies close to OD,2 0 D , etc. The kernel K(E,E’) of (3.63)is expected to exhibit a damped resonance near the Debye energy which must give rise to oscillations in A(E) stronger than those found by Swihart. Morel and Anderson49 have calculated the parameters of the superconducting state using the Eliashberg interaction and including the Coulomb repulsion, which is cut off at approximately the Fermi energy. The resulting kernel is schematically shown in Fig. 3.4 (d). Their results are in good orderof-magnitude agreement with the observed transition temperatures, but lead to isotope effects with exponents at least 15 % lower than the experimental values for non-transition metals. Culler et a1.50 have solved numerically the integral equation (3.63) for the References p . 189
CH. III,
0 31
THE SUPERCONDUCTING ENERGY GAP
121
s-wave (isotropic) part of the energy gap parameter d ( E ) . They used the Eliashberg interaction and a screened Coulomb repulsion, which resulted in a kernel K(E,E’) = f {F(E’ + E ) + F(E’ - E ) - C} , (3.71) where, measuring energies in terms of the Debye energy OD,
(3.72)
f is a coupling constant proportional to the square of the deformation potential constant (i.e. proportional to lMK/2)and f C represents the screened Coulomb interaction. Figure 3.6 shows some results of the calculation
cr-4
2z:E/ -4
2
4
6
8
1
0
\
?=o 9
0.7
1
2
E/B,
3
Fig. 3.6. Calculation of the energy gap and density of state curve by Culler el ~ 1 5 0 .(a) shows the normalized gap at T = 0 for € = 0.891, C = 0.5 and do = 0.15 OD. (b) shows the corresponding normalized density of states.
obtained numerically by means of an “on-line” computational scheme. It is worth noting that d ( E ) and the density of statesM,(E) show strong oscillations at E = OD and E = 20D. The role of the Coulomb interaction in superconductivity has been thoroughly studied by 5 l ; the dynamical screening of the interaction due to a self-consistent dielectric constant is fully taken into account. For the non-transition metals this more accurate treatment eliminates some of the discrepancies between experiment and the previous theory of Morel and Anderson. In particular the isotope effect exponents predicted by Garland are in excellent agreement with experiment. In the case of transition metals, Garland shows that the role of the Coulomb References p . 189
122
[CH.m, 8 3
D. H. DOUGLASS JR. AND L. m. FALICOV
interaction is of paramount importance. For “clean” transition metals in which d-band and s-band electrons can be clearly defined, the presence of the “heavier” d-electrons may dynamically anti-screen the s-s Coulomb interaction giving rise to a net attraction between s-electrons, i.e. an enhancement of the superconducting properties or, theoretically, even a different kind of superconducti-;ity which is caused solely by Coulomb effects. An example of
-
0.4
-
-
- -.
-2 -
0
I
o <
-0.4
I
-
.(
--2
-
-
I 0.4
I
I
I 10
I00
I 1000
a
I
Fig. 3.7. The kernel Kss(0, E ) and the normalized energy gap parameter used by Garland44 for the s-band of a “clean” transition metal.
a Coulomb induced kernel K,,for the s-electrons and the corresponding energy gap parameter is shown in Fig. 3.7. For “dirty” transition metals in which no electron can be clearly defined to be s-like or d-like, these effects are washed out and the usual kind of superconductivity induced mainly by electron-phonon effects is recovered. However, the screened Coulomb interaction contributes in this case much more strongly to the effective VkV,and very large deviations from the 3 exponent in the isotope effect are obtained. The kernel K(e,d)used by Garland in this case is shown in Fig. 3.4(e). He has approximated the kernel computed in ) a self-consistentfashion by a set of plateaus of constant value in the ( 8 , ~ ’plane. Pokrovskii52 and Pokrovskii and Ryvkin 53 have studied the properties of the energy gap equation (3.17), (3.48) for the anisotropic case. They assume a BCS cut-off-type model for the energy gap parameter A,, which is in this case only a function of position on the Fermi surface. By integrating over energies first, they reduce (3.17) to the form (3.73) where the unit vectors n, n’ indicate directions in k-space (i.e. positions on References p . 189
CH. Ill, 0 31
THE SUPERCONDUCTING ENERGY GAP
123
the Fermi surface) and do is an element of area of the Fermi surface. A similar non-linear equation is found for T# 0. Although an explicit solution of (3.73) was not attempted, several general results, mostly in the form of inequalities, have been obtained. The agreement with experimental results is in general very poor, which suggests that the model is very crude and not a substantial improvement on the BCS model.
3.5. MAGNETIC FIELDDEPENDENCE OF THE GAP If a magnetic field H, is applied parallel to the surface of a bulk superconductor, because of the Meissner effect this field decays “exponentially” to zero in the superconductor with a characteristic penetration length 1.Because of this spatial variation, any change in the gap parameter A with H would result in a spatial variation of A as well. The variation of A with distance, however, is characterized by the bulk coherence length to(5, x Av,/A) which is in general much longer than A. The variation of H and A in a bulk specimen are illustrated in Fig. 3.8. In specimens which are small compared with to
A‘!
H = 0)
Fig. 3.8. The spatial variation of H ( x ) and A 2 ( x ) in a semi-infinite superconductor.
(but not necessarily small compared with A), the variation of A with position is small and it is usually a good approximation to assume that A is a constant. Thus the calculation of the H, dependence of A is greatly simplified for such thin specimens. One should bear in mind, however, that the expressions for the gap from the microscopic theory are all for systems of infinite volume; for thin films the proper boundary conditions must be considered. In zero magnetic field this problem is not serious (gap for films = gap for bulk) as it will be seen in the next section in the discussion of the Anderson theory of the “dirty” superconductor, but in a magnetic field it is not obvious that boundary conditions are not important. References p , 189
124
D . H. DOUGLASS JR.AND L. M. FALICOV
[CH. In,
83
In addition to the possible difficulty mentioned above, there are at least three theoretical difficulties in calculating the response of a system to a magnetic field. 1 - In treating the magneticjeld Ha as a perturbation. Although Ha can be made arbitrarily small, the vector potential A which usually appears in the Hamiltonian goes as r x H which of course diverges for large r. This difficulty can be avoided by using the now standard techniques for dealing with transport properties of electrons in a magnetic field.
2 - The breakdown of the perturbation approach. It is well known that the magnetic field introduces a quantization of the one-electron energies (Landau levels). These quantized states cannot be obtained by treating H,, as a perturbation. The usual arguments to disregard this effect are that the discreteness of the levels only becomes important when the cyclotron energy h eH/mc, which gives the difference between two Landau levels, is greater than the average thermal energy k,T. At normal helium temperatures these would require fields of the order of 10 kilogauss, which is of course much greater than the superconducting critical field of the usual soft superconductors.
3 - The breakdown of time-reversal symmetry. Since the presence of a field destroys the time-reversal symmetry of the Hamiltonian, the correlation of the electrons in a Cooper pair will be greatly affected. The phase relations in the wave function may be drastically changed with the consequent destruction of some coherent terms. The importance of these effects can be illustrated by two experimental results : the disappearance of the Josephson effect in superconducting tunnelling in the presence of extremely low magnetic fields (Sections 4 and 5 ) ; and the approximate Hf dependence of the surface resistance in N b ~ s n ~ ~ , which obviously cannot be expanded in powers of H. Because of the difficulties mentioned above, no completely satisfactory theory of the magnetic field dependence of the energy gap has been developed yet. Most experimental results however, can be explained in terms of the phenomenological Ginzburg-Landau (GL) equations 55, introduced before the BCS theory. Theoretical contributions in the last few years have been mainly devoted to obtaining a microscopic derivation of those equations as well as their range of validity and their extension to more general conditions. References p . 189
CH.III,
4 31
125
THE SUPERCONDUCTING ENERGY GAP
3.5.1. The Ginzburg-Landau (GL) Theory55-60 In 1950 Ginzburg and Landau55 proposed a phenomenological theory of superconductivity which was a non-linear extension of the London theory. They introduced an order parameter y (which may be complex) which was allowed to vary with position and magnetic field; the original meaning of y/ was that IyI2 corresponded to the superconducting electron density but as Gor’kov617 G 2 has proved, t,u is proportional to the gap parameter A . The GL equations are only valid in the London limit (i.e. A>> C) and were originally derived for T S T,; this latter restriction is not necessary and the theory will be presented without it. The starting point is the introduction of a magnetic Helmholtz free energy for the superconductor, which was “derived” from plausibility arguments,
2m
C
superconductor
Here FNois the free energy density of the normal state in the absence of a field; AF is the difference of free energy densities of the superconducting and normal states also in the absence of a field and is a function of The third term is the gauge invariant “kinetic energy” and the last term is the field energy in the superconductor. A is the vector potential, H is the magnetic field and e* = 2e is the charge of a “Cooper” pair. In principle all of these terms are functions of position and magnetic field. Because the proper variables of the magnetic Helmholtz free energy are temperature T and magnetization A, where
1~(~.
A=
‘S
d3r[H(r) - H,] ,
(3.75)
all space
H, = appliedfield, this function is not continuous at the critical field. The function which is continuous at the critical field and whose proper variables are T and H, is the magnetic Gibbs free energy, 3’;it is related to S by
9 = F - A *H,. From equations (3.74), (3.75) and (3.76) References p . 189
(3.76)
126
D. H.DOUGLASS JR. AND L. M. PALICOV
[CH. UI,
e
- ihVv - 'Av C
53
1'1+
superconductor
d3r [H(Y) - H,]' ,
(3.77)
all space
+
where =Jd3r[FNo HO2/8n] ; note that the last term on the right hand side is to be integrated over all space. Ginzburg and Landau minimized 9 with respect to A and v*, which gives two differential equations and two boundary conditionst (variation of 3 leads to the same equations)
ie'h 471e*' V'A = - (v*Vv- w V ~ ' ) 7 I v l2 A , mc mc
+
vxA
= Ho
onsurface,
. Aa F + - ( - i h 1V - ; A ) aly 2m
(itivv
$=O,
e8
+ C A+) L =0
on surface,
(3.78) (3.79) (3.80) (3.81)
where the gauge is V -A = 0. In a singly-connected superconductor the phase of w can be chosen so that ly is real. For one-dimensional, singly-connected problems the above equations become
(3.82)
v x A = H,, 8AF e*' -+-Av=av mc2
-dly =o dx
onsurface,
(3.83)
k2d2ry m dx2 '
(3.84)
on surface,
(3.85)
which are of course easier to solve.
t The geometry is taken to be that of a cylinder of arbitrary cross section to eliminate demagnetization effects. It may or may not be multiply connected. References p . 189
CH.UI, 4 31
THE SUPERCONDUCTING ENERGY GAP
127
For the free energy differenceAF, Ginzburg and Landau used in the original paper a power series expansion in Iv/I2, which was expected only to be valid near T,
(3.86) where Hcbis the bulk critical field and wT is the value of ly in the absence of H . As stated above the theory is not restricted to the use of (3.86); if a AF which is good for all T were available it may be used in equations (3.80) or (3.84). The solutions of the GL equations for several particular cases will now be discussed in order to illustrate the field dependence of ty (or A).
Superconducting hay-space. H, parallel to surface55. I f the changes of w with field are assumed to be small, the integration of (3.82) and (3.84) with the assumption (3.86) yields
H(x) E H , exp (- x / I , ) ,
(3.88)
where I, is the penetration depth
(3.89) and K, is a dimensionless coupling constant KO = (J2e'/hc) 1; Hcb.
Gor'kovols
62
(3.90)
has shown that K, w 0.96 I ,
to-'.
(3.91)
Values of K, for the pure soft superconductors range from 0.01 (Al) to 0.3 (Pb). If K, is assumed to be small, (3.87) becomes
From this equation it is seen that the maximum change in t,y for H, < Hcb is of the order of 5 %, which justifies the initial assumption. Equation (3.88) References p . 189
128
[CH. 111, 5
D. H. DOUGLASS JR. AND L. M. FALICOV
3
and equation (3.92) show that the characteristic lengths over which H(x) and ~ ( xvary ) are A, and respectively. This is illustrated in Fig. 3.8.
<,
Superconducting plate of thickness d < to.H, parallel to both surfaces If y is assumed to be independent of position, the direct integration of (3.82) leads to 569 60, 631 64.
H,AL sinh (@x/AL) Ay(X) = __ @ cosh(@d/2AL)'
-+d<x<$d,
(3.93)
where
(3.94)
To evaluate (3.84) in a self-consistent way, A2 must be replaced by its value averaged over the film. This gives sinh(@d/A,) - (@din,) =0. coshZ(@d/2AL)
(3.95)
If a normalized free energy difference is introduced, f (@I= 8 zAFH,,-~, (3.95) reduces to
which for @ dAL-1 << 1 reduces to
Ht(@)
=( -
1 af g z ) 6 6 )
2
(3.97)
f&.
Equation (3.96) gives a generalized relationship between the applied external
(a)
( b)
(C)
Fig. 3.9. Schematic representation of the magnetic field dependence of the normalized gap 9. (a) corresponds to a second order phase transition, (b) and (c) represent first order phase transitions. References p. 189
CH. III,
0 31
THE SUPERCONDUCTING ENERGY OAP
129
field H, and the normalized “gap”. There are at least three classes of curves for @(H:) which are illustrated in Fig. 3.9; they of course depend on f(@). Curve (a) shows a finite intercept with the @ = 0 axis at a value of H which is the absolute maximum; curve (b) a finite intercept at a point where H is not the absolute maximum; curve (c) approaches @ = 0 at infinite values of H . The question of whether the normalized energy gap parameter @ drops discontinuously to zero (first order phase transition) at the critical field or goes continuously to zero (second order phase transition) can be answered by computing simultaneously @ ( H ) and the value H = H , for which 9,= 8, in equation (3.77). It is obvious, however, that curves (b) and (c) must correspond to first order phase transitions; in curve (b) the values of @ less than @I (value at the maximum H ) correspond to a region of thermodynamic instability and therefore cannot be reached. Curve (a) is the only one for which a second order phase transition is possible, and only if H, is not smaller than the value of Ho for which @ = 0. I f f i s given by (3.86) fGL = - 2a2+ (3.98) 0 4 9
the solution cP of (3.96) for small H,, is found to decrease quadratically with H O
(3.99) where
1 sinhx - x D(x) = 8 xcosh’x
(3.100)
Fig. 3.10. The reduced energy gap as a function of reduced magnetic field for films of various ratios d / l . References p. 189
130
[CH.111, 6 3
D. H. WUGLASS JR. A N D L. M. FALICOV
The solutions of (3.96) for arbitrary fields are shown in Fig. 3.10 for various ratios d/AL;the magnetic field for each case is normalized to the critical field H, as determined from (3.77). For d/,lL<J5 the curves are of type (a) in Fig. 3.9 and @ goes smoothly to zero. For d/A,> J5 the solutions are of type (b) in Fig. 3.9 with a resulting discontinuous drop in @ at the critical field. The minimum value @I of the normalized gap has also been calculated as a function of din, and is shown in Fig. 3.1 1. The transition from second-
&'--- - - - - - _ _ _ - -.- - -
.
0.41
/,/
?
Magnelic fleld equal to zero on one side of him
Magnetic field equal on both sides of Film
1 1-
order to first-order behavior at dfL, =J5 is seen to be very abrupt. The other curve in the same figure corresponds to the solution of (3.82), (3.84) and (3.98), assuming that the magnetic field is zero on one side of the film and equal to Ho on the other side*; the phase transition is always first order. This example proves that the boundary conditions are essential in determining the field dependence of the energy gap. Finally it must be mentioned that, when dealing with multiply-connected specimens, the quantization of magnetic flux must be taken into account. Under these conditions, for a thin superconducting cylindrical tube of radius r and thickness d< r and such that rd<<,lL2,theGLequations, withfgiven by (3.98), yield for the magnetic field dependence of the normalized gap65 (3.101)
* These boundary conditions can be achieved by passing a current in the axial direction of a long, hollow cylinder; on the inside surface the field is zero and on the outside surface it is equal to 2Z/r, where Z is current and r the outside radius 60. References p . 189
CH. xu, I 31
THE SUPERCONDUCTING ENERGY GAP
131
where
(3.102) is the magnetic field corresponding to one flux quantum and n is the quantum number defining the macroscopic state of the system. If n is such that the system is always in the state of minimum free energy, (3.101) is periodic in Ho with period H*. The GL theory as presented above does not contain mean free path effects which are important in thin films and alloys. Gor’kov62 has shown, however, that the dependence on mean free path I appears only through I and K. All the results quoted above may be taken over by making the replacements
where x is a function of loll, which can be approximated within an accuracy of about 20 % by x = (1 f
3.5.2. Microscopic Theories of the Magnetic Field Dependence of the Energy Gar, Because of the success of the GL equations in describing the experimental results for the response of a superconductor to a magnetic field, several theoretical contributions have been made attempting tojustify these equations in terms of the microscopic BCS or Bogoliubov theories. Gor’kov, who had been successful in rederiving the energy gap equation and the energy spectrum of a superconductor using the Green’s function formalism25, generalized his method to include a magnetic field61. He obtained two non-linear integral equations relating the so-called normal and anomalous Green functions, which depend on the spatially varying energy gap parameter A @ ) , the vector potential A(r) and the parameters of the metal (Fermi energy, Fermi momentum, etc.). For tempeartures T S T,, where the gap is small, an expansion in powers of A(r) is permissible. Gor’kov kept only terms up to the fourth power in A , assumed that d ( r ) and A(r) are slowly varying functions and used the fact that in the temperature range under consideration I>>co (London limit). In this way he obtained two equations which can be put in a one to one correspondence with (3.78) and (3.80) of the GL theory, assuming a AF given by equation (3.86), e‘=2e,
References p . I89
(3.103)
132
D. H. DOUGLASS JR. AND L. M. FALICOV
and that close to T, Hcbr
[CH.111, $ 3
= (Tc- T)
Y;
*
Rapoport and Krylovetskii66 claim to have extended the calculation of Gor’kov to all temperatures. Their theory is based on an expansion at temperatures close to T, which gives in first order a coefficient identical to the first order term in the expansion of B(T, H =0) for T S T,. They then replaced this coefficient, for arbitrary temperature, by 42(T). In this way they obtained equations similar to those of the GL theory (with the same correspondence of Gor’kov’s theory), but the coefficient in front of the [ 1 - @ 2 ] @ term, which is HCb2/2 z yg,is replaced by 3 A2(T)/cFWerthamer67 and Tewordt6* have obtained equations analogous in form to those of the GL theory. They contain, however, additional non-linear terms. The derivation is valid in the London limit and for arbitrary temperatures and magnetic field strengths. In the limit of A<< k,T (i.e. close to Tc), the equations reduce to those of Gor’kov. Gupta and Mathurs9 have computed the dependence of A( T = 0) on H. They find, by expanding in powers of H (or equivalently, in powers of A ) up to H2
d(H) - A(0) = -
d(H)
c
1 e2vi -- I A , 12, 3 A(0)c2
-
q << d/Au,(Londonlimit); (3.105)
4
- d(0) z -
IA , 12,
q
2 d/Av, (Pippard limit);
4
where vF is the Fermi velocity and A , is a Fourier component of the vector potential. Nambu and Tuan70 have made a similar calculation using a different formulation. Their results are
d(H) - d(0) = - c A - $ $ q 2
I A , 12,
4 << d/huF(Londonlimit), (3.106)
4
4
4 References p . 189
2
(Pippard limit).
CH. In, 4 31
133
THE SUPERCONDUCTING ENERGY GAP
It is evident that (3.105) and (3.106) differ qualitatively for small q ; the former depends on the vector potential A while the latter depends only on the field strength H. For large q both results agree qualitatively apart from different numerical constants. Both calculations have been done for an infinite medium, and it is dubious whether they can be applied to practical cases where in general thin films have been used. Bardeen 71 has made a very detailed calculation of critical fields and currents in superconductors. He computes the Helmholtz free energy difference between the superconducting and normal states, assuming that the energy gap is a variational parameter. His result is
where t = TIT, is the reduced temperature and g, and g, are two dimensionless functions of A/kBT.
(3.108)
211(3) A’ g , z-1671’ kiT2 ’
A << kBTc ,
A T, 1 2c(2)kiT2 Elg+ - - A’ ’ AoT 2
d
>> kBT
(3.109) ;
where C is Riemann’s function and A,, =A( T= 0, H = 0). In the region t 5 1 where A << k,T, (3.107) reduces to
when only terms to order A 4 are kept. For t x 1, (3.1 10) is exactly equivalent to the GL expression (3.86). For t -20 and finite A : AF = - M ( 0 ) A 2 { i
- ig A +Iy(O)V[lg~~). A0
(3.111)
With these expressions Bardeen constructed the Gibbs free energy difference References p . 189
134
D. H. DOUGLASS JR. AND L. M. FALICOV
[am, .9 3
by adding a magnetic energy term in which the total magnetic moment A ( H ) is given by A=
- f2aH~V(O)d,dtanh
(&).
(3.112)
valid for small fields, where D is the volume of the specimen and a is a numerical factor which depends on the shape of the specimen but is independent of H and A . He then considered small specimens in which A can be assumed constant in space, and derived an expression for A( T,H ) by minimizing the -gN).His results for a thin film geometry of total free energy difference (gS a material withM(0) V=O.3 (indium) are shown in Fig. 3.12. It is to be
A - H ~
Fig. 3.12. The normalized gap as a function of H 2 for thin films at various reduced temperatures according to Bardeen 71. The transition between first and second order behavior occurs at t = 0.325.
noted that for the high temperature range the expected linear dependence between 4 2 and H z i s found. At lower temperatures the curves begin to bend, and for t < 0.325 there is a discontinuity in d(H) at the critical field, which gives rise to a first-order phase transition. The curves in this case correspond to case (c) in Fig. 3.9. The predicted first order transition does not seem to agree with present experimental data. References p . 189
CH. 111,
5 3J
THE SUPERCONDUCTING ENERGY GAP
3 . 6 . DEPENDENCE OF THE, ENERGYGAP ON
135
IMPURITIES AND SPATIAL IN-
HOMOGENEITIES
3.6.1. The Anderson Theory of “Dirty” Superconductors The theory of superconductivity described so far dealt only with “clean” metals, i.e. metals in which the electronic wave functions can be well approximated by Bloch functions or plane waves. The theory of “dirty” superconductors formulated by Anderson72 explains the rather surprising fact that the superconducting properties of a metal are essentially insensitive to disorder and non-magnetic impurities. For instance, amorphous films at helium temperatures have transition temperatures not much different from carefully annealed bulk samples with long electronic mean free paths. In very dirty samples in which the collision time z is of the order of 10-14 sec the uncertainty in energy hz-l is of the order 0.1 eV; this value is much greater than the energy gap and all superconducting effects would be expected to disappear or to change in a drastic way. Experimentally only slight changes are found, which in general consist of a sharp initial drop in T, of the order of 0.1” K and a subsequent gradual change as impurities are added. The fundamental point of Anderson’s theory is the idea that when dealing with “dirty” metals in which scattering times are very short, the problem of superconductivitymust be studied by considering first the impurity scattering exactly, and the exact eigenfunctions of the electrons must then be used to compute the interactions and the superconducting state. These exact eigenfunctions In) are of course impossible to compute, but, since they must exists it is reasonable to assume that they can be expressed as normalized linear combinations of the Bloch states Ik/a) of the clean, perfect crystal
(3.1 1 3) (3.114)
If the scatterers are non-magnetic and there is no magnetic field present, the system must have time-reversal symmetry, i.e. the function (3.115) is another eigenfunction of the scattering Hamiltonian with the same energy E, of In). The states In) and I -n) are in this case the ones to be coupled in forming a Cooper pair. To determine the energy gap equation it is necessary to find the interaction References p . 189
136
[CH.III, 5 3
D. H.DOUGLASS JR. AND L. M. FALICOV
term in the new many-particle Hamiltonian, which must be written, similar to (3.5) and (3.1 l), 2P, = - ~ v n n ’ C n C+n C+- , . C n ? (3.116) nn’
where c i and c, are now the creation and destruction operators for the new electronic states In). Once the expression for V,,,,,is known the derivation of the energy gap equation follows step by step the corresponding derivation for the “clean” metal and the energy gap equation is (3.117) similar to (3.48), where En = (&’,
+ A’,)*.
(3.118)
The computation of V,,,,. is straightforward and it is given by a weighed average over V k k ’ , but with the correct energies E,, (or En)replacing the old energies &k (or E k ) . In this way, for instance, the Bardeen-Pines interaction and the Eliashberg interaction are respectively given by
- vnn, = - vnn, =-
c
c
I Anka 1’ 1 An‘k’a’ 1’ I M k ’ - k 1’
kk’aa‘
2
2
(&, - &n,) - h
I Anko 1’ I An‘k‘s’ 1’ (En,
hwk’-k
2 mkok.-k
I M k ‘ - k 1’
(En‘
ha,*-,)’- E’,
9
+hQk’-k),
(3.119)
(3.120)
kk’aa’
in close correspondence with (3.52) and (3.70). For very dirty specimens IAnka(’ will be a very smooth function of k (k on the Fermi surface) and (3.119) and (3.120) represent essentially an average of V k k ’ over the Fermi surface. Consequently, when the dirty superconductor approach becomes valid, the effective interaction and the energy gap parameter will be much more isotropic than in the clean superconductor case. The Anderson theory, with its consequent isotropic gap, must become applicable when the energy uncertainty hz-1 is of the order of the energy gap parameter A or equivalently when to.Here I = u F t is the mean free path and tox hz1~A-1is the coherence length. A different approach to the theory of dirty superconductors and superconducting alloys, based on a Green’s function analysis, was done by Abrikosov and Gor’kov 73*74 with essentially the same results.
Is
References p .
I89
CH. 111,
5 31
THE SUPERCONDUCTING ENERGY GAP
137
3.6.2. The Effect of Magnetic Impurities When magnetic impurities are added to a superconductor, the transition temperature T, drops rapidly with increasing concentration, and superconductivity is soon destroyed. This can be explained in terms of the magnetic scattering of the electrons, which by flipping the spins destroys the correlation of the Cooper pairs. Abrikosov and Gor’kov75 have extended their theory of non-magnetic alloys to include the effect of magnetic impurities. They assume a random distribution of impurities with magnetic moments oriented also at random. They find that there are two relevant scattering times 71 and z, such that TI-1 gives the total scattering rate and z,-l the scattering rate for those processes that flip spin. The transition temperature T, of the dirty sample is given by (3.121) where T,, is the transition temperature at zero concentration of impurities, y is the logarithmic derivative of the Gamma function and A =~ t , k g T C .
(3.122)
At low concentration of impurities p<< 1, (3.121) reduces to nh T, = T,,- -, 4k&
(3.123)
i.e. the temperature decreases proportionally to the concentration. For T,/Tco<<1, (3.121) takes the form (3.124) where the logarithm of y is Euler’s constant. The critical condition for the disappearance of superconductivity is (3.125) Abrikosov and Gor’kov also studied the behavior of the energy gap parameter A , ( T = 0) for a given concentration c of impurities. They find References p . 189
138
lg':
A A0
D. H. DOUGLASS JR. AND L. M. PALICOV
=
-Ig{
[CH.111,
53
1 + [l - (1 - h-'zsAC)']' h-'z,A, A
sin-'(h-'z,A,),
h-'r,A,
However, due to lifetime effects, the apparent energy gap 2Ac, but is given by E , = 24, [l - (A-'z,A,)-']'.
E,
< 1 . (3.126) is* no longer (3.127)
From (3.127) it is clear that the apparent gap vanishes when A ~ , - l = d , and from (3.126) this value corresponds to z, = h/A, = hA,
'exp (lr/4).
(3.128)
If (3.125) and (3.128) are considered together and the reasonable assumption is made that the spin scattering rate is proportional to the concentration of impurities, it is easily seen that the apparent energy gap will vanish at a concentration equal to 0.91 times the concentration at which superconductivity disappears. This would predict a range of concentrations of magnetic impurities in which an apparent gapless superconductivity exists. 3.6.3. Superimposed Films of Normal Metals and Superconductors When a normal metal is placed in metallic contact with a superconductor, the superconducting properties are expected to extend into the normal metal, and, conversely, the superconducting properties will be affected by the presence of the normal metal. These effects can be understood qualitatively in terms of the coherence length to,which gives the characteristic distance over which the energy gap parameter changes appreciably. However, all attempts made to extend a microscopic theory of superconductivity to encompass the case of metallic contacts have encountered non-trivial mathematical difficulties due to the intrinsic non-uniformity of the system and the non-linearity of the superconducting effects. Parmenter70 has made an extension of the BCS theory to the case of a position-dependent energy gap. He finds that a thin slice of normal material
* eo is here the lowest possible energy for which the density of quasi-particle states is not zero. References p. I89
CH. III,
8 31
THE SUPERCONDUCTING ENERGY GAP
139
sandwiched between bulk superconductors is superconducting for thicknesses less than about 10-5 cm. In this case the inverse of the energy gap varies quadratically with position in the normal metal layer. Conversely a thin film of a superconductor sandwiched between bulk normal material is non-superconducting for thicknesses less than about 10-5 cm. Cooper77 studied the case of superimposed films of normal and superconducting materials. He argues that the pairs sample an effective attractive potential which is simply the spatial average of the potential on both sides of the interface. Assuming for simplicity V = 0 on the normal metal side, he obtains for the effectivex (0) V parameter
(3.129) instead of the usual expression (3.55). In (3.129) ds and dNare the thicknesses of the superconducting and normal films, respectively. The formula can only be valid when both ds and dN are smaller than the coherence distance. In such a case reasonable qualitative agreement with experiment is obtained. Douglas 78 has developed a phenomenological theory of superimposed normal-superconducting films. He applied a method similar to that of the GL theory and defines a posirive free energy difference between the “superconducting” and normal state of a normal metaI. By minimizing the total free energy of the system with respect to the order parameter (energy gap) he obtained a non-linear differential equation for the spatial variation of the order parameter. De Gennes and Guyon79, Werthamer80 and De Gennessl use a completely different approach, similar to the Abrikosov and Gor’kov theory of dirty superconductors. In particular it is found that: a) The data depend critically on the mean free path; the case of the “very dirty” films, i.e. very short mean free path, can be easily understood, but cannot be derived from the GL equations, which are not valid in this regime. b) Neither the spatially dependent energy gap parameter A ( r ) (also called in this case the “pair potential”) nor the BCS parameterN (0) V, which is now local, are continuous across the boundary. However, for “dirty” metals, the ratio d ( r ) / X ( O )V is continuous. c) When d ( r ) depends only on one space coordinate, the energy gap E, of the excitation spectrum of the whole system is equal to twice the minimum of IA(r)l. d) The transition temperature of the superimposed normal-superconducting films may be used, under some well-defined experimental conditions References p . 189
140
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH. III,
$4
to determine the effective electron-electron interaction in the normal metal. e) For very thin films, in which dN and ds are much smaller than the coherence lengths, a result similar to (3.129) is found. The effective BCS parameter turns out to be
V , in this case may be attractive or repulsive.
4. Theory of Superconductive Tunnelling
One of the experimental techniques which has given more detailed and complete information on the superconducting energy gap is electron tunnelling between two metallic films separated by a thin insulating layer. Measurements of the current Z and differential conductance dZ/d V as a function of applied voltage V yield accurate values of the energy gap, the density of quasi-particle states, the dependence of these quantities on various parameters, and in general a great deal of information on the microscopic properties of the superconducting state. Consequently a detailed theory of superconductive tunnelling must be built up in order to explore the possibilities of the technique. The next few paragraphs will be concerned with various approaches to such a theory. A semi-phenomenological theory which contains the most important features of the results will be given first, followed by a microscopic quantum-mechanical approach which justifies the assumptions of the simple theory and opens the possibility of exploring new effects. 4.1. SEMI-PHENOMENOLOGICAL THEORY OF ELECTRON TUNNELLING IN SUPERCONDUCTORS
In all the tunnelling experiments the samples are prepared by starting with a given metal 1 (in the form of a thin film or a bulk specimen). A thin insulating layer, in general an oxide of the same metal, is prepared on the surface of metal 1 and a film of another metal (2) is finally deposited on top of the insulating layer. The resulting sandwich, metal 1-insulator-metal 2, constitutes a sample. The experiments are performed by applying a voltage difference between 1 and 2 and measuring the current or by passing a given current through the sample and measuring the voltage difference. Fisher and GiaeverBz and Giaever and Megerle83used the following assumptions in deriving the equations for the tunnelling current: 1) The “electrons” of both metals are independent particles satisfying Fermi-Dirac statistics, i.e. having an occupation probability given by References p . I89
CH. III,
4 41
141
THE SUPERCONDUCTING ENERGY GAP
f(8,T) = [exp(e/kgT)
+ 1]-’
(4.1)
where E is measured from the Fermi energy. 2) The difference in Fermi energies between the two metals is given by the applied voltage V (here the electronic charge e is implicit in V ) . 3) There is a matrix element Tkqconnecting states on both sides of the barrier. This matrix element is considered constant when the energy is varied in the region of interest. 4) The density of states of the “electrons” is considered constant when the metal is in the normal state MN(E) =“(O) > (44 and is given by =MN(o)nS(c) (4.3) when the metal is superconducting. In general =xiN(o)ni(E)
(4-4)
9
where n, is the reduced density of states (i refers to either metal 1 or metal 2) such that niN( E ) = 1 and
=O
9
(4.5)
161
if a constant energy gap model is considered. With these assumptions the probability of an “electron” k making a transition from 1 to 2 per unit time is
4
The total current is then given by
z =eCIPlkZ - P , ! ~ ] ,
(4.7)
k
which, by using the assumptions mentioned above, turns out to be
j:
+
I = C dE ~ ~ ( E ) I ~ Z ( E V )[ f ( ~ )-f(& -a
where
References p . 189
+ V ) ],
(4.8)
142
[CH.III, 5 4
D. H. DOUGLASS JR. AND L. M. PALICOV
The calculation of (4.8) can be divided into three cases of interest, corresponding to the combinations normal-normal I", superconductor-normal &N, and superconductor-superconductor 1 ,. 4.1.1. Normal-Normal Case I", In this case nlN= n 2 N = 1. Fig. 4.l(a) illustrates the reduced density of states of the two normal metals and the occupation of the levels at finite temperaI
1
I
I
Unoccupied
(a)
Voltage
Voltage
(b)
(C)
Fig. 4.1. Schematic representation of tunnelling between two normal metals. (a) Reduced density of states and occupation probability. (b) I-Vcurve at T = 0. (c) I-Vcurve at T # 0.
tures. Under the usual conditions V<<+ and kBT<<eFwhere,+ is the Fermi energy, (4.8) can be readily integrated to give
I"
=
cv.
(4.10)
It is to be noted that the normal-normal case shows an ohmic behavior and C can be immediately interpreted as being the conductance. It is also worth mentioning that the initial assumptions are such that no structure in the normal density of states, such as peaks due to d-band electrons, can be reflected in the current. Formula (4.10) is verified by experiment.
4.1.2. Superconductor-Normal Case ISN. Here n, = nls, n2 =n2N = 1. A schematic diagram of the reduced density of states and the occupation probabilities at finite temperatures is shown in Fig. 4.2(a). At T= 0, because of the cut-off in the Fermi factors (4.8) reduces to 0
ISN(T = 0) = CJnls(&)de -V
References p . 189
(4.11)
CH.111, 5 41
143
THE SUPERCONDUCTING ENERGY GAP
and the differential conductance (4.12) gives immediately the reduced density of states. This result is very general and independent of the particular model for n&). If a constant energy gap model (4.5) is assumed, (4.1 1) can be integrated directly
,o< V G d , ;
IsN(T = 0) = 0
=cJvz-d:, ISN(v,= - lSN(V)
V2.4,;
(4.13)
9
which is plotted in Fig. 4.2(b). From the figure it is seen that the energy gap parameter is identified in a more or less unambiguous way. When T # 0, the solution of (4.8) cannot be expressed in a closed form, even by assuming a simple constant energy gap model. The integral however
(0)
(b)
( C)
Fig. 4.2. Schematic representation of tunnelling between a superconductor and a normal metal. (a) Reduced density of states and occupation probability. (b) I-Vcurve at T = 0. (c) I-V curve at T # 0.
has been computed in the form of a series which converges rapidly for limiting cases 83.
m=l
where K1 is a modified Bessel function of the second kind. The complete solution for all values of V is illustrated in Fig. 4.2(c) for kBT/Al 0.15. Considerable thermal “smearing” at V dl is to be noted;
-
References p. 189
N
144
D.
n. DOUGLASS JR.AND L. M. FALICOV
[CH. IR,
04
this means that the experiments must be done at very low temperatures in order to observe a sharp edge. In the limit V -to, equation (4.14) becomes W
(4.15) In this expression the current ratio, which is easily measured, is only a function of T and dl( T ) . The exponential dependence on AI( T)/kBTat low T is the same as that of the specific heat and thermal conductivity (see
A I T ) /kT
Fig. 4.3. Plot of the ratio zSN/INN as a function of A ( T ) / ~ B The T . exponential behavior is apparent at the lower temperatures.
formula (5.10)) and is plotted in Fig. 4.3. Energy gaps determined by the use of (4.15) by Giaever and Megerle83 were found to agree quantitatively with the more direct methods. 4.1.3. Superconductor-Superconductor Case I,, Here ni = nls, n2 = Itzs. The schematic diagram for the reduced density of states in the constant energy gap model for the two superconductors is illustrated in Fig. 4.4(a). At T= 0 equation (4.8) reduces to References p . 189
CH. III,
Q 41
145
THE SUPERCONDUCTING ENERGY GAP
/1 '2
(a1
(cl
(bl
Fig. 4.4. Schematic representation of the tunnelling between two different superconductors. (a) Reduced density of states and occupation probability. (b) I-Vcurve at T = 0. (c) I- V curve at T # 0.
Iss( T = 0) = C
i
+ V)dE.
(4.16)
~z~S(E)IZ~S(E
-VtAi
If (4.5) is used for n,, and nzs, (4.16) can be integrated to give
o<
I,,(T=O)=O, =
V < A , +A,;
- 2 C d l A 2 [ V 2 - ( A 2 - A1)2]-f.K(~)+ +C[V2-(A2 -Al)2]'E(~),
where K and respectively,
E
V>A1
(4.17) +LIZ;
are complete elliptic integrals of the first and second kind
a = [ V 2 - ( A , i4,)2]f.[Vz- ( A ,
- Al)2]-+,
and &Ss( - J9= - I S S ( V It is to be noted that there is a discontinuity SI= + z C J A , A , of the current at V = A1 AZwhich is a direct consequence of the singularities in ns at the edge of the gap. Fig. 4.4(b) illustrates the properties of (4.17). It is easily verified that (4.17) reduces to (4.10) when d l +O, A Z +O; and it reduces to (4.13) when A2 +O. For the particular case of two identical superconductors A 1 = Az, (4.17) reduces to
+
Iss(T=O,A1 = A 2 ) = 0 , =
where a =[ V z - 4 A z ] + References p . 189
V-1
o<
V<2A;
- 2CA2V-' K(M) + C V E ( a ) ,
(4.18) I/ > 24
and the discontinuity at V = 2 A is 6Z=+nCA.
146
D. H. DOUGLASS JR.AND L. M. PALICOV
[CH.III,
54
Equation (4.18) is equivalent to that found by Giaever, Hart and Megerle84. In general for arbitrary nlS(c) and nzs(c) it can be shown that due to the piling up of states at the edges of the energy gap, a measurement of the differential conductance (dIss/dV),=, as a function of Vgives a much more sensitive way of determining the fine structure of as(&)than the corresponding measurements of (dISN/dV),=,. For T# 0 the integral for the tunnelling current must be done numerically. Shapiro et aZ.85 have made this calculation for a constant gap model and find semi-quantitative agreement with experiment ; they have shown that at V A , -A , the current displays a logarithmic singularity N
I
1
I s 0 # 0) a k
1 v -I
A2
-
1
I1
7
(4.19)
and at V = A , + A , there is a finite discontinuity
Between the two singular points there is a region of negative resistance. This solution is illustrated in Fig. 4,4(c). Although neither the logarithmic singularity nor the discontinuous jump are expected to be observed experimentally because of lifetime and anisotropy effects, a well defined maximum and a very abrupt rise are observed; these points are assumed to correspond to IA2-A, and A , + A , respectively from which both A , and A , are obtained. For the particular case of A , = A 2 = A and V < 2 4 , Taylor et aZ.86 have computed Issnumerically for a constant energy gap model and have proved that a negative resistance region appears at temperatures kBT5 0.3A . When A>> kBT, V < 24, the current can be very well approximated by
I
=
>>
kBT)
O G V<2A,
(4.21)
where KOis a modified Bessel function of the second kind. Formula (4.21) exhibits a slight negative resistance region.
4.2. MICROSCOPIC THEORY OF ELECTRON TUNNELLING IN SUPERCONDUCTORS The semi-phenomenologicaltheory of superconductive tunnelling described in the last paragraphs gives most of the features found experimentally. References p . 189
CH.111, 8 41
147
THE SUPERCONDUCTING ENERGY GAP
However some of the original assumptions on which the theory is based require a theoretical justification. In particular the use of the “golden rule” to obtain the expression for the current implies the use of a tunnelling Hamiltonian which is not usually employed in studying tunnelling processes. Moreover the assumptions of constant matrix elements r k q and constant density of normal states are certainly not correct. Various contributions to the theory of superconductive tunnelling have dealt with these questions, and it has been proved that the results of the simple semi-phenomenological theory are correct. In addition, the possibility of finding new effects was suggested during the course of these investigations. Bardeena7 has laid the foundations of a Hamiltonian formalism which gives equivalent results to those usually found in barrier penetration problems and consequently justifies the use of the “golden rule” in the computation of the current. But agreement with experiment, Bardeen says, is only obtained if the matrix element Tkqis the same for the corresponding transitions in the normal and superconducting cases. Cohen et ~ 1 . 8 8and PrangeaQhave also justified the use of the Hamiltonian formalism and have proved that the Hamiltonian of the system can be written in a very general form s =&Or + Z T (4.22)
+*,
where XI and Xz are the Hamiltonians of metals 1 and 2 respectively and ST is the tunnelling term responsible for the transfer of electrons from one side of the barrier to the other. HTcan be written *T
=
[TkqC;aCq/a
+ TqkCqfoCk/a]
3
(4.23)
where c’ and c are the creation and destruction operators used in Section 3, k indicates the one-electron states in metal 1 and q the one-electron states in metal 2. Harrison90 has computed the tunnelling matrix element Tkqfor the case of independent electrons. He finds that for the case of specular scattering at the boundaries, there is a selection rule which conserves the transverse components of the k and q vectors as well as the spin, and that, in addition
(4.24) where v k l and vql are the components of the group velocity of the electrons normal to the barrier in both metals and I,, is a tunnelling integral which depends exponentially on the thickness of the insulating layer. Since
References p . 189
148
[CH. 111,
84
it is evident that l T k q I 2 cannot be considered independent of the energies and eq. On the other hand it is easily seen that
&k
D. H. DOUGLASS JR. AND L. M. FALICOV
1 "k 1 CC [ N N ( E k ) ] -
(4.26)
and consequently the product v k l vqL"(&k) J V N ( E ~ ) is independent of the density of normal states.* This immediately leads to the exact linear relation between 1 " and Y (4.10) and the impossibility of obtaining any information on "(6) by measuring the tunnelling current. When one or both metals are superconducting, the behavior of the wave functions in the insulating layer is essentially unchanged, and the tunnelling matrix elements are also given by (4.24),in which the velocities still correspond to the group velocity of the normal electrons and not the group velocity of the superconducting quasi-particles. Consequently
1 Tk4 I 2 M N ( E k ) J V N ( E q )
CC
9
which immediately gives the results of the semi-phenomenological theory. However one question remains to be elucidated. When one or both metals in the specimen are superconducting, the independent "electrons" to be considered in the superconducting part of the sample are not electrons in fact, but quasi-particles, and it is well known from the Bogoliubov transformation (3.29) that the relations between normal electrons and quasi-particles depend on the coefficients ukand v k which are very strongly energy dependent. Do not these coefficients take part in the process and destroy the simple relation between Z and TZ,(E)? Cohen et ~ 1 . ~have 1 solved this problem by computing the current I S N in the case of a superconductor-normal specimen. They used the Bogoliubov Hamiltonian3l for the superconductor and the transformation (3.29) to express the tunnelling Hamiltonian (4.23) in terms of the quasi-particle operators. The current was computed from the usual quantum mechanical expressions
* Although 1 I k q 1 in equation (4.24) may depend on the energies &k and cq, this dependence is probably weak, and I k q can be considered constant. It must be mentioned in addition, that the specular scattering on the boundaries does not correspond to the actual experimental conditions, where diffuse scattering is more likely to occur. The fundamental result of cancellation of energy dependence is however still valid. References p. 189
CH.III,
8 41
149
THE SUPERCONDUCTING ENERGY GAP
ISN =
-e(N,)
ie
= e (NN) = -
A
(NNM
- XNN),
(4.28)
where NNis the total number of electrons in the normal metal, Ns the total number of electrons in the superconductor, the dot represents total derivatives with respect to time and the angular brackets indicate expectation values at a given temperature. In the calculation of (4.28), when looking at the process from the superconductor side of the barrier, complications due to the fluctuations in the number of particles inherent in the BCS and Bogoliubov theories may arise. When proper care is taken and all possible spurious terms are eliminated (this difficulty may be avoided by computing the rate of change of electrons on the normal metal side), it is found that
kq
+ v: [ f ( & k ) - (l - d E q ) ) ]
6(Eq
-k
Ek
+ v)>
Y
(4.29)
where f is the Fermi-Dirac function and g the modified Fermi-Dirac distribution (3.39). It is important to notice that in the tunnelling process there are two superconducting channels q and q' such that q is below the Fermi surface and q' above it, Eq = Eq. and
+ Uq' = 1, Vq + Uq' = 1.
ux
2
2
2
(4.30)
If (4.30) is taken into account and the relations g(E) = f ( c ) , g(-
E)
= 1 --f(E),
E
>0 ; E <0 ;
(4.31)
are kept in mind, (4.29) reduces immediately to (4.8), i.e. the results of the semi-phenomenological theory are obtained exactly with no additional energy dependent factors appearing in the expression for the current. Bardeeng2has obtained the same result by using a slightly different approach in the calculation. The problem of computing Z, is more complicated, since the fluctuations in the number of particles become in this case very critical and due care must be taken to eliminate all spurious terms. Josephsong3 has solved the problem by using a modified Bogoliubov transformation in which the quasi-particles contain exactly one electron or one hole. This formulation requires the introduction of a new operator S : defined in such a way that when acting on Refevences p. 189
150
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH. III,
04
the ground state of a superconductor with N electrons it creates a new superconducting ground state of the system with ( N 2) electrons. The equation of motion of S : is
+
- ihs: =pis: - S'Xi
= 2EFiSf,
(4.32)
where E~~ is the Fermi energy of the metal i. Equation (4.32) indicates that S: oscillates in time with a frequency v = 2 ~ ~ ~ h Josephson -1. computes the total current to order lTkqlzin the same fashion described above for the superconductor-normal case91 and he obtains
Is = I &
+ + JssS:S2 + 9J&S;S1,
(4.33)
where 1; reproduces the results of the semi-phenomenological theory (4.8), but the two additional terms represent a new effect. Equation (4.33) predicts (i) At finite voltages the usual d.c. current occurs, but there is also an a.c. current of amplitude lJssl and frequency 2Vh-1 (one microvolt corresponds to 483.6 Mc/s); (ii) At zero voltage Issis zero, but a d.c. current up to a maximum of JssI can occur. A quantum mechanical explanation of the additional effect can be given; (i) corresponds to the coherent transfer of an electron pair across the barrier with photon emission, leaving the quasi-particle distribution unchanged, (ii) corresponds to pair transfer without photon emission. The magnitude of Jss depends very critically on the presence of currents and magnetic fields in the films. In particular the effect may be destroyed by the presence of magnetic fields such that the amount of flux between the films, including that in the penetration regions, becomes of the order of a quantum of flux hc/2e. The use of the tunnelling Hamiltonian (4.22), (4.23) permits the calculation of higher order effects, i.e. terms in the current proportional to the fourth or higher power of the tunnelling matrix element Tkq.In the superconductorsuperconductor case, when (4.23) is expressed in terms of the quasi-particle operators, .3E"T = NTkqUkUq T - q - k w q > ~ k + @ q+
I
c kq
-
+ ( T - k - q ~ k u+qTqkuqvk)a:fi: + similarterms}.
(4.34)
Schrieffer and WilkinsB4,in attempting to explain the experimental results of Taylor and BursteinQ5,have computed second order effects (proportional to Tkq14)inwhich the aq+ pk+ term of (4.34) acts first, followed by a term of the kind ak+ clq. This results in a microscopic transfer of electrons from one side of the barrier to the other in which the quasi-particle distribution in one metal References p . I89
CH. III, 5 41
THE SUPERCONDUCTING ENEROY GAP
151
is left undisturbed and two quasi-particles Dk+ and ak+ are created in the second film. Conservation of energy requires this process to have thresholds at Y = d , and Y = d , at T=O. This is illustrated in Fig. 4.5 in which the “electrons” on one side of the barrier are assumed to be extracted from the Fermi level and are transferred to the quasi-particle levels on the other side. Many other microscopic processes are possible in second order. Schrieffer, Scalapino and Wilkins 96 have also used the tunnelling Hamiltonian formalism to investigate the structure in the density of super-
I1
’b
Fig. 4.5. Schematic diagram of the processes proposed by Schrieffer and Wilkins84 in attempting to explain the excess current found experimentally by Taylor and Bursteines. (a) At V = da two quasi-particles from Sa are transferred as a Cooper pair to Sb. (b) At V = db in addition to the previous process, a Cooper pair from Sa is transferred as two qUaSi-partiCleS to Sb. (C) d b < v < d a db (d) Sa = S b = s.
+
conducting states due to the details of the phonon spectrum. They find that the possibility of transferring electrons from one film to the other results in a complex energy gap parameter, and that the reduced density of states which appears in the tunnelling integral (4.23) is given by (4.35) instead of the usual expression of the BCS theory (4.36) derived from (3.35). The results of a calculation of the complex d ( E ) for a References p . 189
152
D.H.DOUGLASS IR. AND L. M. PALICOV
[CH.111, 8 4
phonon density of states consisting of two Lorentzian distributions whose parameters are chosen to fit the values in Pb are shown in Fig. 4.6. The corresponding tunnelling density of states (4.35) is shown in Fig. 4.7.
c
-0.6
-1R
1
Fig. 4.6. The real di and imaginary da parts of the energy gap parameter as computed by Schrieffer el al.S* for a two Lorentzian phonon distribution. hall is the energy of the peak of the less energetic (transverse) phonon. The parameters were adjusted so as to fit the values for Pb.
t
0.94 (x92
Fig, 4.7. The tunnelling density of states as computed by Schrieffer ef al.98 using the results of Fig. 4.6. The dashed line represents the corresponding density of states from the BCS model.
References p . 189
CH. III,
8 51
THE SUPERCONDUCTING ENERGY GAP
153
5. Experimental Determinations of the Superconducting Energy Gap
5.1. SPECIFIC HEATAND THERMAL CONDUCTIVITY 5.1.1. Specific Heat In general the thermal energy (measured from the ground state) of a system whose excited states can be expressed as a superposition of “independent” quasi-particles is m
Here Ej is the (positive) energy of the quasi-particle of class j (in general j = 1,2), N ( E ) the density of states of quasi-particles and g the modified
Fermi factor discussed in Section 3.3. The specific heat is
s m
dW C, = -- = - 2kiT dx x2JI’(x)g‘(x), l3T 0
where
and the factor of 2 comes from the summation over quasi-particles, assuming Y ( E 1 )= N ( E 2 ) . If a simple model with a gap is assumed forY(E), i.e.
E
N(E)=O,
&(E) = N o , E > A ;
(5.3)
the specific heat becomes
s m
C, = - 2kiTN,, dx x2g‘(x).
(5.4)
AlkRT
If A = 0, the integral becomes a constant ( - A ) and
CeN= 2 k i A N 0 T = yT,
(5.5)
which is linear in T as in the normal free electron gas. If A # 0 and (5.4) is evaluated at low temperatures (kBT<
154
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH.IU,0 5
Thus for this simple model at low temperatures, the specific heat displays, in addition to a slowly varying factor T-I, a strong exponential dependence on A/T. This exponential behavior is characteristic of any energy gap model. The electronic specific heat in a superconductor must have a more complicated expression than (5.6),because the density of statesNs(E) is not correctly given by (5.3) (see for instance Fig. 3.2) and in addition the energy gap parameter A is a function of temperature. The specific heat can be computed accurately from any theory once the entropy S (or the free energy F ) is known.
In the BCS theory, by inserting the value of S computed in Section 3.3 and making the approximation A( T = O)>> k&”, it is found that
where K , and K3 are modified Bessel functions of the second kind. For
0.2 < TIT, < 0.7 (5.8) can be fairly well approximated by L eS
- r 8.5 exp (-
1.44Tc/T),
Y Tc
(5.9)
while for TIT, < 0.1 it goes to the expected limiting exponential dependence exp (- A(O)/k,T) =
(5.10)
G)t
=2.37 -2 exp(- 1.76Tc/T). Specific heat measurement on superconducting vanadium by Corak et al.97 and on superconducting tin by Corak and Satterthwaiteg* are shown in Fig. 5.1. It is seen that a “law of corresponding states”
5 ‘= 9.17exp(Y Tc
1.5 T,JT),
similar in form to (5.6) and (5.9), can be fitted to the data very well. References p. 189
(5.11)
CH. III, 5 51
155
THE SUPERCONDUCTING ENERGY GAP
This simple exponential form gives a strong evidence of an energy gap, with 24 m 3.0 kBTc (5.12)
and tinso.
Fig. 5.1.
- BCS - - - 3(Tc/T)
-
X
0-
0
ALGOODMAN Al PHILLIPS A l ZAVARITSKII
0
ZN PHILLIPS
A A
V GOODMAN ZN ZAVARITSKII
0
-
SIC -1.0
0
...
-g -2.0-
-3.0
\
-
\
-4.0-
I
1 1
I
2
I
3
I
I
5
I 6
I
I
7
8
_-
TT /; Fig. 5.2. Reducedsuperconductingelectronic specificheat versus Tc/Tfor Al,Zn andV00.
References p . 189
156
D.
n. DOUGLASS JR. AND L. M. FALICOV
[CH. 111, 8
5
for both V and Sn. At lower temperatures the data no longer follow a simple exponential, as for instance the measurements on aluminum and zinc shown in Fig. 5.2loo,which means that more detailed theories must be considered. At this stage the specific heat measurements lose their claim as a method of measuring the gap which is more or less independent of the various theories. As formulae (5.9) and (5.1 1 ) show, the data agree fairly well with the BCS theory, although some significant discrepancies have been observed. Whereas BCS predicts a “downward” curvature in the lg C, versus T-l plots (i.e. “effective” gap increasing with T-1) the measurements on Al, Zn and other metals show a very definite upward curvature (“effective” gap decreasing with T-1). One possible explanation of this is based on the anisotropy of the energy gap34J01. If the gap were different at different points of the Fermi surface, then the “effective” gap appearing in the exponential at intermediate temperatures would be larger than the “effective” gap at lower temperatures. This is due to the fact that as T + 0, the contributions of the smaller gaps are much more important because of the Fermi factor. 5.1.2, Thermal Conductivity
Making “reasonable” assumptions, it can be shown that the thermal conductivity is roughly proportional to the specific heat and should therefore exhibit in superconductors the typical exponential behavior with temperature characteristic of the energy gap. The experiments of Goodman12 gave results which could be interpreted in terms of an exponential from which estimates of a gap were inferred. Later investigations showed significant deviations from the exponential law, probably due to the breakdown of one or more “reasonable” assumptions. Because of these ambiguities, the specific heat data are usually accepted as stronger evidence for the gap than the thermal conductivity data. Zavaritskii l o 2 has measured the thermal conductivity in a single crystal of zinc, parallel and perpendicular to the hexad axis. He finds a marked anisotropy which can be interpreted in terms of an anisotropic energy gap. For a heat flow parallel to the hexad axis d ( T = O ) r 1.75k,Tc is obtained, while for a perpendicular direction 4( T = 0) = 1.2kBTc. Morris and Tinkham’O3 and Morris104 have measured the magnetic field dependence of the thermal conductivity of thin superconducting films. If the electronic term K, dominates, and the scattering is essentially due to impurities, then the ratio of the thermal conductivity in the superconducting state to that in the normal state is105 References p . 189
CH. 111,
5 51
THE SUPERCONDUCTING ENERGY GAP
157
(5.13)
where g(E/kT) is the modified Fermi function (3.39). Morris and Tinkham assumed that (5.13) is applicable to film specimens, and that the effect of a magnetic field is adequately represented by a change in A . By inverting the thermal conductivity data, the magnetic field dependence of A as a function of temperature and thickness was determined. The results of their experiments will be discussed in Section 5.4 together with the measurements of the gap by the electron tunnelling technique.
5.2. PHOTON EXCITATIONS ACROSS THE GAP On the basis of any simple energy gap model, one would expect that at T = 0 only those photons for which hv 2 E ~ i.e. , those which carry enough energy to excite quasi-particles “across the gap”, could be absorbed in a superconductor. In this way the absorption spectrum at low temperatures must exhibit a well-defined cut-off which gives the most direct evidence of an energy gap. Prior to 1954, as mentioned in Section 1, radio-frequencyand microwave measurements up to v 1010 cps on miscellaneous superconductors showed essentially no absorption at low temperatures. On the other hand, infrared measurements down to v 1013 cps showed an absorption in the superconducting state equal to that in the normal state. Thus the superconducting energy gap was bracketed between 10-5 and 10-1 electron-volts. Photons of these energies lie in the far infrared, an extremely difficult region to reach. However Glover and Tinkham15 were successful in performing experiments in that region of the spectrum. Their technique has been used to study various properties of the energy gap. N
-
5.2.1. Energy Gap at Zero Temperature Richards and Tinkham lo6have measured the absorption edges of the superconducting elements In, Sn, Hg, Ta, V, Pb and Nb on bulk specimens. Their results are shown in Fig. 5.3 in which the difference between the power reflected from the metal in the superconducting and normal states divided by the normal state value are plotted versus photon energy. These measurements are essentially T = 0 measurements, since they were all made well below ( T < 1.4”K) the transition temperatures of the studied elements. They chose the onset of absorption as the characteristic frequency of the gap and plotted References p. 189
158
D. H. DOUGLASS JR.AM) L. M. FALICOV
[CH.m, 5 5
Fig. 5.3. Low temperature absorption curves for various bulk superconductors versus photon energy106.
Tc
(OK)
Fig. 5.4. Measured values of the energy gap at T = 0 in temperature units as a function of the critical temperature. The BCS law of corresponding states is represented by the straight line106.
References p. 189
CH. m, 8 51
THE SUPERCONDUCTING ENERGY GAP
159
kB-le,(T= 0) versus T,as shown in Fig. 5.4. It is seen that there are large deviations from the BCS law of corresponding states, which is represented by the solid line.
5.2.2. Temperature Dependence of the Gap Absorption curves on Pb and V as a function of temperature were also measured by Richards and Tinkham106. The curves for vanadium are shown in Fig. 5.5; Fig. 5.6 shows the temperature dependence of the gap for both V and Pb with a fit to the BCS theory (Fig. 3.3).
Fig. 5.5. Absorption curves for vanadium as a function of temperature106.
REDUCED TEMPERATURE ( I = T/T,)
Fig. 5.6. The temperature dependence of the energy gaps of Pb and V. The solid line is the BCS theoretical curve106.
Referencesp. 189
160
[CH.111, 8 5
D . H. DOUGLASS JR. AND L. M. FALICOV
Biondi and Garfunkello7 measured the absorption (surface resistance) of microwave photons on A1 as a function of temperature. From their data, shown in Fig. 5.7, the absorption edges can be clearly determined, at least at the lower temperatures. Fig. 5.8 shows the temperature dependence of the energy gap that they inferred from the data. It is seen that their results get closer to the transition temperature than Richards and Tinkham’s experiments do. Energy gaps determined by direct excitation with photons are listed in Table 5.1.
Energy (in units of kT,)
Fig. 5.7. Isotherms of surface resistance ratio versus photon energy in AI1O7. 4.0
I
I
I
----------
Reduced
Fig. 5.8. References p. I89
I
,-BCS
thec
I
temperature, t=T/Te
Energy gap in A1 versus reduced ternperature1O7
CH. III,
5 51
161
THE SUPERCONDUCTING ENERGY GAP
TABLE 5.1 Energy gap from photon absorption 2A(O)fknTc
References
A1
In
Pb
Hg
Nb
Ta
Sn
V
-
4.1 3.9
4.1 4.0
4.6
2.8
< 3.0 -
3.6 3.3 3.5
3.4
-
-
3.16
-
-
-
-
-
-
-
_ 4.31
_
-
3.0
-
-
-
-
-
-
Richards and Tinkham106 Ginsberg and Tinkharn108 Biondi et al. l o 9 Biondi and Garfunkell07 Richardslol Leslie and Ginsberg114
5.2.3. Anisotropy of the Energy Gap
Later measurements by Richards'l' of the far infrared absorption versus photon energy on single crystals of pure tin are shown in Fig. 5.9. These data were taken at T = 0.32Tc with unpolarized photons incident upon various crystallographic faces. For these pure specimens, estimates based on the theory of the anomalous skin effect indicate that the electrons which are effective in interacting with the incident photons lie in a band about 30" wide around the Fermi surface, parallel to the direction of the face exposed and displaced somewhat from the equator of the Fermi surface in the direction
hw/kT,
Fig. 5.9. Frequency dependence, for (110), (001) and (100) planes of pure tin, of the fractional difference between power reaching the bolometer in the superconducting and normal states
References p . I89
162
D. H. DOUGLASS JR.A N D L. M. PALICOV
[CH.III,
45
away from that face. From Fig. 5.9 it is seen that the average gap for the (001) and (100) faces are essentially the same, while the average gap for the (1 10) face is 20 % higher. Adkins 112 has measured the microwave surface resistance of tin and tinindium alloys in single-crystal and polycrystalline wires. For pure tin the energy gap was well defined only with specimens with the (001) direction oriented within 10’ of the wire axis. In that case, as well as in the polycrystalline or alloyed samples, the gap was found to be about 3.6 k,T,. Adkins inferred that the energy gap must be anisotropic, having values different from 3.6 k,T, at orientations far from the tetrad axis, but that these regions of different E, must be comparatively unimportant, since they essentially do not contribute to the average. 5.2.4. Mean Free Path EfJects on the Gap
Richards 111 has also tested Anderson’s 72 theory of the “dirty” superconductor. As discussed in Section 3.6, when the collision time in the sample is long enough (z>>tZ~,-‘), the energy gap is expected to be anisotropic around the Fermi surface (Fig. 5.9). In a “dirty” superconductor (z< the rapid scattering connects states all around the Fermi surface giving rise to an “isotropic” energy gap and therefore to a sharper absorption edge. Fig. 5.10 shows a comparison of the data for a pure single crystal of Sn (h-’ E, z x 250) with those for a single crystal specimen of Sn with 0.1 % In
hw/kT,
Fig. 5.10. Frequency dependence, for the (001) plane of pure and impure tin, of the fractional difference between the power reaching the bolometer in the superconducting and normal states lll. References p . 189
CH. IU, 8
51
163
THE SUPERCONDUCTING ENERGY GAP
/ F R E Q U E N C Yin cm-1
Fig. 5.11. Absorption curve for an alloy of 10.0 atomic percent thallium in lead113.
(A-'eo z w 0.9). It is clearly seen that the edge is sharper in the "dirty"
specimen, in support of Anderson's theory. Ginsberg and Leslie 113 have made similar photon absorption experiments on Pb-TI alloys as a function of T1 concentration. Fig. 5.1 1 shows absorption versus photon energy on a Pb (10 % T1) sample, where it is seen that the onset of absorption is still "sharp" at this concentration. Later measurements of the gap width c0114 and transition temperature115 as a function of concentration are shown in Table 5.2. The data from this table show that the transition temperature decreases linearly with concentration c, with a slope -dT,/dc = O.O25"K/%T1; this is shown in Fig. 5.12 by a solid line. If the BCS TABLE 5.2
Energy gap 114 and transition temperature115 of Pb(Tl) alloys At. %T1 0.0
0.2 1 .o 3.1 5.3 7.7 10.0
References p . 189
Gap width (cm-I) 21.8 21.65 21.5 21.6 21.2 19.6
k0.5 0.3 kO.3 f 0.4 &0.3 &0.3
TC (" K)
7.177 f0.005 7.167 7.140 7.077 6.923
164
[CH.Iu, 4 5
D. H.DOUGLASS JR. A N D L. M. FALICOV
law of corresponding states (i.e. linearity between A ( T = 0) and T,) is assumed, and the experimental ratio at 0 % of TI 24 ( T = o)/kBTc= 4.37 is used, 24 would be expected to change linearly with concentration with a slope d (2A)/k,dc = --O.lIoK/% T1; this is shown in Fig. 5.12 by a dotted line. It
I
(
,
(
,
(
I
,
i
30OTclO1-Tc(%T1) in degree Kelvin 0 Z[A(O)-A(%TBq
in degree
Kelvm
Tc (01= 7 177 OK
20-
2A(0)=314"K 2AlO)/ Tc = 4 37
*K
3Tc/3C =-.025
L
0
l
l
2
OK/%
l l t l l 4 6 Concentrotion of T I In Pb
\
l 8
l
l 10 %
Fig. 5.12. Plot of changes in T, and the energy gap in Pb-Tl a function of the TI concentration. The data points are taken from the experimental measurements of Leslie and Ginsberg114 and Finnemore and Mapother115.
is seen that the experimental measurements of 24 for c < 8 % scatter about this line, but that the 10 % value shows a marked deviation. 5.2.5. Photon Experiments on Films The first spectroscopic evidence of an energy gap in superconductors was obtained from the study of far infrared absorption in thin films of Pb and Sn by Glover and Tinkhaml5. Their measurements did not extend to low enough energies to see the beginning of the edge. Later and more accurate measurements by Ginsberg and TinkhamlO8 on Pb, Sn,In and Hg covered the entire range of interest. Data for Sn, In and Pb are shown in Fig. 5.13. It is seen that the onset of absorption occurs at hv 3-4 kBTc.Ginsberg and Tinkham quote the values 4.0 f0.5 k,Tc, 3.3 0.2 l kBTc -and 3.9 A= 0.3 kBTc for the energy gaps of Pb, Sn and In respectively. These are in good agreement with the
-
References p. 189
CH. III,
5 51
165
THE SUPERCONDUCTING ENERGY GAP
measurements of the energy gap in the bulk for the same metals. Measurements were also made on Hg, but the data were not entirely reproducible, although the anomalous "bump" in the absorption was observed in all cases. I
1
I
I
I
1
I
I
I
I
V
I
I
I
1
Fig. 5.13. Absorption experiments in thin films of Sn, In and Pblo8.
5.2.6. Structure in the Absorption Spectrum It is appropriate to discuss here the structure which is present in all the absorption curves for Pb, Hg and Sn. In the case of Pb and Hg (Figs. 5.3 and 5.13) an anomalous absorption at energies below that of the main edge is observed, whereas for Sn (Figs. 5.9 and 5.10) it occurs at an energy higher References p. 189
166
D. H.DOUGLASS JR. AND L. M. PALICOV
[am, .0 5
than that of the main edge. Richards and Tinkham remark that if such structure were present in the Nb and V curves, they probably would have seen it. The suggested explanations for this structure fall into two general categories: single-particle excitations and collective excitations. The singleparticle excitation approach assumes that the energy gap is anisotropic over the Fermi surface and that there are, as expected in general, several isolated pieces of Fermi surface in different bands. If average gaps were different for the different pieces, and also each gap were “sharpened“ by the Anderson mechanism, then it is conceivable that a superposition of two or more of these edges could duplicate the observed curves. The fact that these effects have been observed in thin films and in non-dilute alloys for which rapid scattering would be expected to connect the isolated pieces of Fermi surface and hence to smear out the individual edges, makes this explanation inconsistent. Collective excitations in the superconducting state, however, may account for the observed structure. These excitations (excitons), discussed briefly in Section 3.2, may have for anisotropic potentials Vkk, energies within the energy gap. TsunetoIlG has considered the s- and d-wave part of the potential to be nonzero, and has found a precursor absorption which in the case of lead and mercury is an order of magnitude smaller than those observed experimentally. However his calculation is valid only for weak coupling, and a more accurate treatment including strong coupling and lifetime effects may lead to better agreement with experiment. No calculation has been made for Sn,which shows an anomalous absorption for energies above the energy gap. It is noted in passing that if the upward deviations from the BCS theory in the specific heat measurements are accepted as evidence of an anisotropic energy gap, this anisotropy is strongly correlated with the appearance of this anomalous absorption. Vanadium and niobium for which no structure has been observed show little deviations in the specific heat curves; tin, which has some deviation, shows structure above the gap; and lead and mercury, which have significant deviations, show structure within the gap. It is also worth noticing that the BCS law of corresponding states, which gives a ratio of 3.5 between energy gap and transition temperature, gives much better agreement with the experimental values for the transition metals than for Sn, Pb and Hg. These facts strongly support the conclusion of Garland’s44 theory for the dirty transition metals and the suggestion of Schrieffer117 that, in the soft superconductors with strong coupling, the lifetime effects must play a very important role and, among other things, would increase the value of 2A( T= O)/k,T,. References p . 189
CH. 111,
5 51
THE SUPERCONDUCTING ENERGY GAP
167
5.2.7. Magnetic Field EfSects
No change in the superconducting absorption spectrum was observed with the application of a magnetic field of 8000 gauss on a 12A film of Pb108 or with a field very close to the critical value on a bulk specirnenlo6.The reason for the negative result in the thin film is that 8000 gauss are far below the film critical field: from the theoretical considerations of Section 3.5 the expected change is of the order of a few percent. The negative result for the bulk experiment can be interpreted in the same terms; the gap at the surface only decreases a few percent from the zero field value up to the critical field. 5.3. ENERGY GAPACOUSTIC ATTENUATION MEASUREMENTS The first measurements of the attenuation of longitudinal acoustic waves in superconductors were made by Bommel 118, followed soon after by MacKinnonl19. They observed a rapid monotonic decrease in the attenuation as the temperature was lowered below T,. This was expected on the basis of the two fluid models, in which the attenuation is due to the “normal” electrons. The observed decrease, however, was much more rapid than that predicted by the Gorter-Casmir two-fluid model. One of the major successes of the BCS theory is the simultaneous explanation of the rapid decrease in acoustic attenuation and the increase in the nuclear relaxation rate. According to the BCS theory120, the matrix elements which describe the coherent scattering of quasi-particlesby external agents contain contributions from three different effects: a) the temperature dependent energy gap, b) the increase in the density of states near the edge of the gap; and c) a coherence factor which depends on whether or not the spin of the quasi-particles is changed in the scattering. For the acoustic attenuation case, effects b) and c) exactly cancel each other. For processes which involve spin-flip,e.g. nuclear relaxation experiments, these two effects add. The theoretical expression for the attenuation of longitudinal acoustic waves resulting from the BCS (isotropic) model in the ql>> 1, Am<< A limit is ffS
- =2[exp(d(T)/li,T)
+ I]-’,
(5.14)
ffN
where q and w are the wave vector and frequency of the phonons respectively, I is the mean free path of the electrons, and us and gNthe attenuations in the superconductingand normal states. Tsuneto 1 2 1 has shown that the expression also holds when ql << 1. Thus it is seen that, in terms of the BCS model, the attenuation is a very simple function of the energy gap parameter. References p , 189
168
D. H. DOUGLASS JR. AND L. M. FALICOV
-
I
I
I
.
d N
10-
[CH. III,
55
I
m.
08-
06-
041
O,
i I
Fig. 5.14. Normalized longitudinal wave attenuation in Sn for two different orientationsl22.
2
I
I
I
I
I
1
2
I 3
"N 1
0
00
\
Fig. 5.15.
Tc/T
Plot of Ig ( o ~ s / ~ Nversus ) ( T e / T )for longitudinal waves along differentcrystalline axes in Sn lz2.
References p . 189
CH. 111, $
51
169
THE SUPERCONDUCTING ENERGY GAP
It is usually found that equation (5.14) fits the experimental data within a few percent, provided that some “adjustment” of the value of A( T = 0) is allowed. Figure 5.14 shows the measurements of Morse el a1.122 on a single crystal of tin, with q along two different crystal directions; considerable anisotropy is to be noted. At low temperature these data are easily fitted with an expression of the form (5.14); this is best illustrated in Fig. 5.15 in which Ig(as/aN) is plotted against TJT. An effective gap for each orientation derr(n)can be determined from the slope of the corresponding line. This procedure however, only points out the deficiency of the isotropic model, which cannot logically be used. Before taking the anisotropy of the energy gap into consideration, it is necessary to find out which electrons on the Fermi surface are effectively interacting with the sound wave. If a parabolic energy band model is used, from conservation of energy and momentum and a sharply defined Fermi surface one can show that for q l > 1 the phonons can scatter only those electrons which lie near the intersection of the Fermi surface and the plane A k = m* v,, where k is the wave vector of the electrons, m* their effective mass and v, the velocity of sound. In the usual case q >k,T> A(m*u,/k,)* which is the region usually attained experimentally, the attenuation goes as
’
(5.15)
where A‘”’ is the minimum value of A , on the “equator” of the Fermi surface mentioned above. Equation (5.15) has the same low temperature behavior as (5.14). For k,T<
where A, is the absolute minimum of A, over the Fermi surface;f(n) is a function of the direction of propagation of sound and takes on very large values when A , lies close to the relevant “equator” of the Fermi surface. When (5.16) is applicable the slope of Ig a, versus T-’ must be the same for all directions of q and the anisotropy of Ak cannot be obtained in the usual way. This case has not yet been found experimentally. References p . 189
170
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH. [II,
55
TABLE 5.3 The effective energy gap at T Direction of propagation
=
0 in tin from acoustic attenuation
24 e f d kTc ~
References
Polar Azimuth Indices angle from (100) from (001) 0" 90" 90" 90" 0"
90" 90" 90" 90" 90"
-
0" 18" 45" 0" 6" 12" 24" 30"
(001) (loo) (110) (001) (loo)
3.1 & 0.1 3.5 & 0.1 4.3 f 0.2 3.8 i 0.1 3.1 i 0.1 3.5 f 0.2 3.7 rt 0.2 4.0 3 0.2 4.1 40.2 4.0 j=0.2
Morse et al.
Bezuglyi et al.
White tin is the metal in which the anisotropy of the energy gap has been investigated most carefully. Because of its tetragonal symmetry, the anisotropy may be expected to be greater than in other superconductors with cubic or hexagonal lattices. Values of the effective energy gap parameter A,,,( T = 0) as a function of direction of propagation of sound are given in Table 5.3. They have been obtained by fitting the experimental data with curves of the form (5.14). The angular dependence of 2Aeff/k,T, when q is rotated perpendicular to the (001) axis is shown in Fig. 5.16. It is seen that there is a large
Fig. 5.16 The effective energy gap in Sn as a function of direction of propagation of the sound wave according to the measurements of Morse et al. lZ2 and Bezuglyi et al. lZs. References p , I89
CH.m, 8 51
171
THE SUPERCONDUCTING ENERGY GAP
variation of the effective gap in the basal plane (3.5 k~T,-4.3k,T,) and an even larger one over the whole Fermi surface (3.1 kBTc-4.3kBTc).If the Pokrovskii model is assumed to be correct, then deff=A(") and the variation TABLE 5.4 The effective energy gap at T = 0 from acoustic attenuation in various superconductors Element Direction of propagation
Purity
2de f !/k~ T c
References
~
In
polycrystal
-
3.5 & 0.2
Morse and Bohm124
A1
(100) (1 10)
8 p.p.m.
3.7 0.3 3.7 i 0.3
David and PoulisLZ5
c-axis Ga
3.5
R~~
b-axis
=
50000.
Hart and Roberts lZ8
'4.2
3.9
* 0.1
V
(110)
R300 - 150 __ R4.2
3.5
Ta
(110)
R300 - - 35
3.5 f 0.1
Levy et al. 127
R4.2
-
V
singlecrystal
0 3 '
Ta
single crystal
R300 - - 1s
~
R4.2
15
3.2 5 0.2 3.5
R4.2
V
(100) (110)
'300 -- -
130
R4.2
(111)
* 0.2
Levy and Rudnick128
3.1 5 0.2 3.4 f 0.2 3.2 0.2
+
Bohm and Horowitz1zo
Zn
(1210) (1010)
Nb
(100)
Mo
(loo)
99.9999%
R300 - - - 500
3.8 3.4
* 0.2 0.2
3.96 f 0.1
Dobbs and Perzlso
3.5 f 0.2
Horowitz and Bohrn131
R4,2
References p. 189
_ - 450
R300 -
R4.2
172
D. H. DOUGLASS JR. A N D L. M. FALICOV
[CH. 111,
55
of the energy gap over the Fermi surface can be partially mapped. The method gives information at most on one energy gap for each crystal orientation and the Fermi surface, in general, has several sheets with probably different energy gaps in each; in addition, depending on the location of the maxima and minima for A,, some "blind" areas may be found. Although the data of Table 5.3 are not complete, the following statements in terms of the Pokrovskii model can be made: (1) the minimum gap on the Fermi surface is probably in the (100) directions, and is less than 3.1 kBTc; (2) the maximum gap may be in the (001) direction and is greater than 4.3 kBTc;(3)2d10,, < 2dll, < 3.8 kBTc;(4) the energy gap in the basal plane, 18" off the (100) axis is greater than 2A1,, and smaller than 4.3 kBTc.The millimeter wave measurement of Adkins112of an average gap of 3.6 kBTcfor an angular "area" of 40" centered about a point 10" off the (001) axis is consistent with these statements if the gap drops rapidly from its value along (001) to values in the vicinity of 3.5 kBTcas one goes away from this axis. The measurements of Zavaritskii132 by the electron tunnelling technique of gaps of 3.5 kBTc,2.81 kBTcand 4.06 kBTcare in general also consistent with the model; the 2.81 k,Tc could be the (100) gap and 4.06 kBTcthe gap in the basal plane 18" from (100). The fact that for orientations close to the (001) direction a sharply defined gap of 3.5 kBTcis the only one found which does not seem to be consistent with the above identifications, It is quite probable that with a more refined anisotropic model and more complete experimental data the value of the energy gap can be mapped over most of the Fermi surface by a procedure similar to that above. Such information would presumably allow one to correlate the anisotropy of the energy gap with that of the Fermi surface and thus provide a starting point for a correct explanation. Table 5.4 contains a list of values of the effective energy gap for various superconductors, as determined by fitting the low temperature variation of the longitudinal acoustic attenuation to an expression of the form exp(-d,,,/kBT).
5.4. ENERGY GAPFROM ELECTRON TUNNELLING EXPERIMENTS From the theory of tunnelling between two metals developed in Section 4 and schematically illustrated in Figs. 4.1, 4.2 and 4.4 it is seen that the superconducting energy gap and the density of quasi-particle states can be identified in a more or less unambiguous way from the current-voltage curves. This, coupled with the relative ease in performing the experiments, has generated a flurry of experimental and theoretical activity in the field of superconductive tunnelling. The experiments give detailed information on : References p . 189
CH.ILI,
8 51
THE SUPERCONDUCTING ENERGY GAP
173
the magnitude of the energy gap and its dependence on temperature, magnetic field, position, crystal orientation and impurities ; the density of states ; the mechanisms of superconductivity, especially the influence of the phonon spectrum on the density of states ;the coherent tunnelling of superconducting pairs ; the interaction of quasi-particles and microwave photons ; the mechanisms of tunnelling, especially the appearance of higher order processes such as double tunnelling. These effects will now be considered separately. 5.4.1. Temperature Dependence of the Gap Current-voltage curves for Al-Pb samples taken by Giaever and Megerle s3 are shown in Fig. 5.17. All the features illustrated in Figs. 4.1,4.2 and 4.4 are
.Voltage h V )
Fig. 5.17. Current-voltagecharacteristic of a Pb-A1 sample at various temperatures83.
evident here; when both metals are in the normal state ( T > T,[Pb] x 7.2” K) a linear characteristic is obtained, as predicted by equation (4.10); for T,[Pb] > T > T, [All x 1.2”K the curve is similar to that shown in Fig. 4.2; and when T c T, [All the aluminum gap is clearly “seen”. By measuring the superconducting-normal tunnelling for Pb-A1, Sn-A1 and In-A1 and applying equation (4.15), Giaever and Megerle were able to References p. 189
174
D. H.DOUGLASS JR. A N D L. M. PALICOV
[CH. III,
$5
determine the temperature dependence of the energy gap, their results are shown in Fig. 5.18 together with a comparison to the BCS theory. Direct evidence of the temperature dependence of the energy gap is shown
T '.Ot
--
.
W006
w
0'41 a2
0 1
I 0.6
I
I
Q2
'
0.4
I
0.8
I
1.0
TIT,
Fig. 5.18. The energy gap of Pb, Sn and I n versus temperature, as determined from the application of equation (4.15) to experimental data88.
Fig. 5.19. Detailed current-voltage characteristic of an AI-In sample showing the change of the energy gap in A1 as a function of temperaturea3.
References p . 189
CH. III,
8 51
175
THE SUPERCONDUCTING ENERGY QAP
in Fig. 5.19 for an Al-In specimen; in this figure the In gap is essentially constant and the A1 gap can be seen “opening up” as the temperature is lowered through the aluminum critical temperature.
Fig. 5.20.
measureI
I
I
I
,3501
ilooE lA 50
0
0.1
0.2
0.3
0.4
1 0.5
1 0.6
I
0.7
0.8
0.9
i.0
Blt)
Fig. 5.21. The energy gap of A1 from tunnelling measurements on two AI-Pb samples versus the theoretical BCS function 13s. References p . 189
176
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH. 111,
85
More precise measurements of the temperature dependence of the Al gap done later by Douglass and Meservey on an Al-Pb specimen133 are shown in Fig. 5.20. These data, along with others from a different specimen, are shown in Fig. 5.21, plotted versus the BCS theoretical values in the weak coupling limit. Although the data for a given specimen can be very well fitted to the BCS curve, it is seen that the value of the zero temperature energy gap must be “adjusted” from sample to sample. For the two samples of Fig. 5.21 the ratio of the two energy gaps at zero temperature is different from the ratio of the transition temperatures. This discrepancy may be due to strains, impurities, mean free path effects, etc. For metals other than Al, such differences appear to be much smaller. Townsend and Suttonl34.135 have made measurements of the temperature dependence of the energy gap on bulk Ta and Sn films; they find that Ta agrees very well with the BCS theory but Sn has values of the gap which are about 5 % higher in the middle of the range. Values of 2A(T=0)/kBT, as determined by various investigators by the tunnelling technique are listed in Table 5.5. Although the measurements are usually good to three significant figures, only the first two are probably meaningful in the case of thin film experiments for the reasons mentioned above.
Al thickness-1600A
0
I
0.5
1.0 Voltage (mV)
1.5
Fig. 5.22. Detailed current-voltage characteristics of an AI-Pb sample showing the energy gap of Al as a function of applied magnetic fields3.
References p . 189
M
u
6
Y
P %
TABLE 5.5
h
Energy gap at zero temperature from tunnelling measurements References
2d(O)/kB Tc ~~~
~
Nb
Ta
~
V
Hg
4.33 f0.1 4.04 &0.1 4.30 f0.08* 4.67 & 0.08* 4.20 & 0.08 4.26 f0.08
3.84 f 0.06 3.60 f 0.1
3.6 3.6
3.5 3.65 f 0.1
Pb
A1
TI
In
Sn
3.46 0.1 3.63 f 0.1 3.10 f 0.05 3 3.51 f 0.18
4.2 f 0.6 2.5 f 0.3
N
B
5 0
0
3.47 f 0.07 3.45 f 0.07 2.817 3.50t 3.627 4.06+
3.1 - 3.4 3.37 i 0.1
3.4 3.57 i 0.05 4.5 f 0.1
* Two different gaps seen in same specimen. t
1
Giaever and Megerle 83 Shapiro et al. Townsend and Sutton 134,135
Multiple gaps seen in various orientation of single crystals.
Douglas and M e ~ e r v e y l ~ ~3 Zavarit~kiil~~
co
Zavaritskii 132
g 2 rn P
Sherrill and Edwards 137 Giaever 138 Dietrich139 Taylor and Burstein Q5 Bermon and Ginsberg140
2
$
178
[CH. IlI,
D. H. DOUGLASS JR. AND L. M. FALICOV
$5
5.4.2. Magnetic Field Dependence of the Gap* The first direct measurements of the magnetic field dependence of the energy gap by the electron-tunnelling technique were done by Giaever and Megerles3 on an Al-Pb sample. Their data are shown in Fig. 5.22, where it can be clearly seen that the A1 energy gap is decreasing continuously as the magnetic field is increased. (The I
I
I
I
I
I
I
I I I
1 -
- A \
I
I I
N
‘\A
-
\
-4
-
\\
20.6
\
X
N -
I I I -
Y
I
I
-
0
-
* e
-
4
g0.4
-0.2 -
I l
a
\\
II I
l -
Aluminum
E
w
I
Symbol thickness
Approx.
Reduced tamp
k
3000 B 4000 8
0.745 0.714
3.5
A
-
0
L
I
I
I
I
I
I
1.9
I
I
(Mognelic f i e l d 1 2 ( H / H c f
Fig. 5.23. Energy gap of aluminum as a function of magnetic field from tunnelling measurements on Al-Pb samples141.
Pb gap is also decreasing, but only slightly because of the high critical field.) Later tunnelling measurements by Douglass141 also on Al are shown in Figs. 5.23 and 5.24. These measurements show for TIT, > 0.75 that: 1) for a ratio of thickness d to penetration depth A less than 1.9 the field dependence of A can be fitted reasonably well by the function (5.17)
* As mentioned in Section 3, the change in the energy gap parameter with magnetic field depends on the boundary conditions at the surface; there can beno unique dependence on the field without specifying the boundary conditions. Unless otherwise stated, the terms “field dependence” in regard to superconducting plates, refer to equal fields on both surfaces, applied in a direction parallel to them. References p. I89
CH. III,5 51
179
THE SUPERCONDUCTING ENERGY GAP
2) for a ratio d/A greater than 2.4 the energy gap drops abruptly to zero at the critical field and the value of this drop versus d/L agrees qualitatively with the predictions of the GL theory. Additional measurements by Douglass and Me~ervey13~ on A1 films of intermediate d/A values are in good qualitative agreement with the GL theory. As mentioned in Section 5.1, Morris and TinkharnlO3 have measured the field dependence of the energy gap using values of d(H)/d(O) inferred from thermal conductivity measurements on films of Pb, Sn and In. They find that at the higher temperatures good agreement is obtained with the tunnelling 1.0,
I
I
0.8N
F - Q
>0.6
-
-
.
a a
A
500 2000 3000
0
4000A
0
I
1
Aluminum Approx Sym bo' thickness
I
I
I
I
I
1
-
Reduced temp
1
t I _
I
.I
I
1.0
0
I
I
30 Thickness/( penetration depth1
I 4.0
1
J 50
d/X
Fig. 5.24. Energy gap of A1 at the critical field versus film thickness over penetration depth from tunnelling experiments on ALPb samples l41.
Fig. 5.25. The magnetic field dependence of the energy gap of Pb from thermal conductivity measurements. The two values for each experimental point come from different factors of proportionality in the d ( T = 0), T, relation. The upper curve corresponds to the BCS 1.76 factor and the lower curve to the experimental value 2.3. The experiments were done on a 500 A film104.
u
'0
02
04
06
IH/ HtIe
References p . 189
08
10
D. n. DOUGLASS JR. AND L. M. FALICOV
180
[CH.Ill,
55
measurements and with the GL theory for the field and thickness dependences. At the lower temperatures the field dependence of d apparently takes on a different functional form. Fig. 5.25 shows A(H) for a 500 A film of lead inferred from the thermal conductivity measurements by Morris1°4. It is seen that at T= 4.5"K good agreement with the GL theory is obtained, whereas at the lower temperatures significant deviations are found ;the gap apparently goes continuously to zero at the critical field. Meservey and Douglass142 have measured the field dependence of A on a 1000 A film of lead using the electron tunnelling technique; the data were
-
o.2 10
20
3.0
4.0
5.0
Magnetic Field(gauss*)
Fig. 5.26. Energy gap of Pb versus magnetic field from tunnelling experiments on a Pb-Pb sample 134.
taken at T=0.87"K (T /T,=O.12) and are shown in Fig. 5.26. The energy gap appears to go continuously to zero with magnetic field, but can be better fitted with a function of the form
than with the GL expression. Although the GL theory can be extended to all temperatures according to Rapoport and Krylovetskii66, it appears to be in quantitative disagreement with experiment at the lower temperatures. For lead with N(O)V= 0.39, Bardeen71 predicts that for thin films, the energy gap will drop abruptly to zero at the critical field for TIT,< 0.27; at T/T,= 0.12 the value of the discontinuity in d(H,)/A(O)is 0.416. The data in Fig. 5.25 and 5.26 however do not show any abrupt drop. References p . 189
CH. III, 8 51
181
THE SUPERCONDUCTING ENERGY GAP
5.4.3. Density of States Giaever, Hart and Megerle** have made measurements of the differential conductance dZ/d V for superconductor-normal specimens at T = 0.3" K on Sn, Pb, In and Al. They find quite good agreement between these measurements and the density of states of the constant energy gap model; better agreement is obtained however if an energy breadth or smearing is introduced in the manner of Hebel and Slichter143. The measurements on a Pb-Mg junction at T=0.3' K are shown in Fig. 5.27. The energy gap and the increased density of states close to the edge of the gap are evident; it is worth noting the structure in the curve near the mean phonon frequency
PblMgQ I Mg W . 3 4 x10-3ev
T= 0.33'K
L
0
.4 Energy (in 8 units of 12 E)
16
Fig. 5.27. The differential conductance of a Pb-Mg sample versus voltage. showing the superconducting density of state of Pb8*.
( V OD x 64(0)). Such a structure is expected from the consideration of more general isotropic models. Equation (3.35) shows that any changes in A(&)(or in dd/d&)are reflected in the density of states; Morel and Anderson49 and Culler et a1.50 have shown that the Eliashberg form of the electronelectron interaction via phonon exchange, and an Einstein model for the phonon spectrum, give rise to oscillations in the density of states occurring at about the phonon energies and its multiples. The structure at multiples of the phonon energy has been observed experimentally by Rowell et al. 41. The density of states in lead at energies close to the mean Debye energy N
References p . 189
182
[CH.III, 0
D. € DOUGLASS I. JR. AND L. M. FALICOV
-
5
(V 64 in Fig. 5.27) has been examined with a more sensitive apparatus by Rowell, Anderson and Thomas42;their measurement of dI/dV on a Pb-A1 sample (Al in the normal state) is shown in Fig. 5.28. The structure present
-
nl (V): Cnlculotion of Schrieffer el 01
-
104-
102-
0.98-
-
0.96-
0
2
4
6
10
8 (V-A,,(O))
12
14
16
18
20
in mV
Fig. 5.28. The differential conductance of a Pb-A1 sample versus displaced voltage. The ~ dashed curve is the tunnelling density of states computed by Schrieffer e f U Z . ~(from Rowell et al. 42).
I
0
I
I
1
1
I
I
I
I
I
7
8
9
Pb-Pb
I
2
3
4 5 6 (V-2Apb10))in mV
10
Fig. 5.29. Plot of d2Z/dY2 versus displaced voltage on a Pb-Pb sample. The arrows indicate the values of the Van Hove singularitiesfrom the experimentaldata of Brockhouse et a/. 144 (from Rowell et al. 4 9 . Referencesp . 189
CH. III, 8 51
183
THE SUPERCONDUCTING ENERGY GAP
in the experimental curves is closely reproduced by a theoretical calculation of the tunnelling density of states n,( V )by Schrieffer, Scalapino and Wilkins96 shown in the same figure. Their calculation was based on equation (4.35) and made use of the complex energy gap whose real and imaginary parts are shown in Fig. 4.6. In the calculation the distribution of phonons was approximated by the sum of two Lorentzian distributions centered at 4.4 and 8.5 mV (indicated by t and 1 respectively in Fig. 5.28) and having half-widths of 0.75 and 0.5 mV respectively. Measurements of d21/dV2on a Pb-Pb sample by Rowell, Anderson and Thomas42 have yielded some remarkable results which are shown in Fig. 5.29. Because of the increased sensitivity of the “two superconductor” junction, much more structure is resolved, most of which can be correlated quantitatively with the neutron measurements of the phonon spectrum of Pb by Brockhouse et uZ.144. Scalapino and Anderson145 have shown that at voltages corresponding to Van Hove singularities (i.e. discontinuities in the derivative of the phonon density of states resulting from a stationary point in w(k)),d21/dV2 should show anomalies. Predicted energies for Van
20
25 I
V-(ACLpb(Ol+AA,(0))in mV 30 35 40 50 I I I A1 -Pb
U
I
050
l
l
070
0 90
~ ( 1 0 ’p3 s~ ~
Fig. 5.30. A plot of the differential conductance versusdisplaced voltage on a Pb-A1 sample together with the density of phonon states of A1 as a function of phonon frequency4a.
Hove singularities from the data of Brockhouse et ~ 1 . 1 ~indicated 4, by arrows in Fig. 5.29, occur at the following voltages: 3.68 mV (t 100 and t l l l ) , 4.58 mV ( t loo), 5.17 mV ( t llO), 6.00 mV (I loo), 7.68 mV (I loo), 8.35 mV ( t loo), 8.68 mV ( I 110), 8.93 mV ( I 100) and 9.03 mV (I l l l ) , where the iongitudinal or transverse character and the direction of propagation of the phonon are given after each energy. These stationary points are only those along the symmetry directions and are probably not a complete set. The References p . 189
184
D. H. WUGLASS JR. AND L. M. FALICOV
[CH. In,
85
structure appearing at V x 1.7 mV may correspond to the precursor absorption present in the far infrared data. Measurements of d21/dV 2for Sn-Sn samples also show a large amount of structure. Measurements of dI/dV over the range 20-45 mV on an AI-Pb junction at T = 0.35" K taken by Rowell, Anderson and Thomas42 are shown in Fig. 5.30 along with the density of phonon states of aluminum given by Walker146 and Phillips14'; the peak in the density of phonon states and the end of the phonon spectrum are clearly reflected in the dZ/dV curve.
5.4.4. Influence of Magnetic Impurities on the Gap Reif and W 0 0 l f ~ ~ have * studied the influence of magnetic impurities on the transition temperature (determined by measuring the resistance) and the energy gap (determined from tunnelling experiments) in quenched superconducting films. Fig. 5.31 shows the results of measurements on Al-In
0 0
I
I
0.2
I
I
I
I
I
0.4 06 c (atomic per cent of Fe)
1--,.
I
0.8
Fig. 5.31. Transition temperature and energy gap 24 of quenched In films as a function of Fe impurity c o n ~ e n t r a t i o n ~ ~ ~ .
samples, with indium containing controlled concentrations of iron. The transition temperature shows the expected linear decrease with increasing Fe concentration; the energy gap, however, is reduced faster and, beyond a concentration of 0.8 %, no evidence of the gap in the tunnelling characteristic is seen at all, although the In film is still superconducting (i.e. zero resistance). Phillips149 and Suhl and Fredkinl5O have attempted to explain this result by References p . 189
CH. 111,
5 51
THE SUPERCONDUCTING ENERGY GAP
185
studying the change in the apparent energy gap parameter due to the presence of magnetic (spin) impurities. In particular Phillips applies the theory of Abrikosov and Gor’kov75 to examine the influence of spin-flip scattering on A and on the density of states. The spin-flip scattering rate 7,’ gives all quasi-particle states a complex energy E= Re E+ ir, i.e. reduces them to decaying states with a Lorentzian broadening ;this in turn reduces the effective energy gap in the density of states, the minimum excited state being at +&,
= A [l
- (TsAh-’)-3]?
When t,= Ad- the apparent energy gap vanishes, although zero resistance is still found as long as A > O . The theory, according to the discussion of Section 4.6, predicts that superconductivity disappears at concentrations equal to 1.1 times the concentration at which the gap vanishes. The factor found experimentally is of the order of 2. Phillips argues that this may be due to oversimplifications in the Abrikosov and Gor’kov approach. 5.4.5. Spatial Variation of the Gap
Although a number of theories and experimental results imply spatial variations of the energy gap, the first direct measurement of such an effect was done by Smith et al. using the tunnelling technique. They prepared a composite film of lead and silver (a 5700 A Ag film in metallic contact with a 1500 A Pb film). The combined film was superconducting at helium temperatures. Measurements of the gap on the Ag side by the electron tunnelling technique showed an energy gap 24 =0.16 mV. This is to be compared with the gap 24 = 2.7 mV for pure Pb films.
5.4.6. Tunnelling in the Presence of Microwave Radiation Dayem and Martinis2 have performed tunnelling experiments with “twosuperconductor” samples (Al-Pb, A1-In and Al-Sn) placed in resonant microwave cavities of various frequencies (24.82 kMc/s, 38.83 kMc/s and 63.02 kMc/s). The measured differential conductance dI/dV as a function of V shows narrow and high peaks for values of the voltage V = A , + A , & fnhv, where n = 1, 2, . . ., 5 , which indicates the absorption or emission of microwave photons with the simultaneous tunnelling across the oxide layer. The appearance of the multiphoton peaks, which seems surprising at first sight, can be easily explained in terms of the modulation of the quasiparticle energies by the electric and magnetic microwave fields, and the applications of standard time-dependent perturbation theory in the presence References p . I89
186
D. H. DOUGLASS JR. AND L. M. IJALICOV
[CH. III,
85
of oscillating fieldsaa1153.The absolute magnitude of the peaks, however, is not so easily understood and the explanation may require a detailed knowledge of the geometry. The normal component of the microwave magnetic field is expected to play a very important role. 5.4.7. Anisotropy of the Gap from Tunnelling Zavaritskii 132 has measured the current-voltage characteristics for samples consisting of a single crystal of tin, an oxide layer and a tin film less than 1000 A thick. At T= 1.36” K the characteristics indicate the presence of anisotropy in the energy gaps which manifests itself through the appearance, for some crystal orientations, of additional structure near the threshold V = 2 4 . The experiment gives a value 4=0.56 mV for the Sn films and d1=0.58 mV, A2=0.45 mV and A3=0.65 mV for the single-crystal. d1 appears in every-orientation while A 2 and 43 are only evident in some particular orientations. Zavaritskii suggests that these values may correspond to energy gap parameters for different sheets of the Fermi surface. These data have been compared in Section 5.3 with other experiments which also give information on the anisotropy of the energy gap in Sn. Townsend and Sutton1349 135 have observed additional structure in the current-voltage characteristics of thick Pb-Ta samples. When analyzing the data in terms of Anderson’s theory of “dirty” superconductors they arrive at the conclusion that the Pb film satisfies the conditions necessary to exhibit an anisotropic energy gap. From the data two values for A[Pb] were inferred and these are A1 = 1.33mV and A 2 = 1.45mV. Dietrich139 investigated the energy gap of single crystals of Ta by means of the tunnelling technique using Ta-Pb samples. Measurements for the (100), (110), (210) and (211) orientations showed essentially no anisotropy of the energy gap. The experimental values range from A = 1.01 mV (for (1 10) and also for polycrystalline specimens) to A = 1.025 mV (for (100)). 5.4.8. Other Superconductive Tunnelling Effects As mentioned in Section 4, conventional tunnelling is a first order effect proportional to the square of the tunnelling matrix element which for two superconducting films at T= 0 has a threshold at V=dl + A 2 . The effects to be described in the next few paragraphs only exist in the case of two superconducting films, have different thresholds and may be of order higher than the first. The effect predicted by Josephsong3and found experimentally by Anderson and Rowell154 is a first order process (proportional to the square of the References p . 189
CH. m,5 51
THE SUPERCONDUCTING ENERGY GAP
187
tunnelling matrix element) and consists of the coherent tunnelling of Cooper pairs across the barrier giving rise to an a.c. current of frequency v = 2h- V , where V is the difference between the Fermi level of the two superconductors (i.e., the applied voltage). At zero voltage this reduces to a d.c. current. Figure 5.32 shows the current voltage characteristic of a Pb-Sn sample at T = 1.5" K taken by Anderson and Rowell. The possibility of observing the effect depends very critically on the preparation of a very thin oxide layer, for which the noise due to thermal fluctuations is greatly reduced, and the more or less complete shielding of magnetic fields around the sample. An external magnetic field would be expected to destroy the phase coherence of the pairs across the barrier when the field between the 8-
,
I
6 -
4
z 4 -
*
g , 0
-
2-
-
F--Td-_-----
-
-___c-_---
0I
1
I
I
Fig. 5.32. Current-voltagecharacteristicfor a Pb-Sn sample at T = 1.5"K showing a supercurrent a V = 0 which corresponds to the effect predicted by Josephson; (a) corresponds
superconductors including the penetration layer corresponds to a fraction of the flux quantum hc/2e; for a typical experiment with an area of 2A x 1 mm this is of the order of a few tenths of a gauss. Curve (a) in Fig. 5.32 shows a supercurrent of 0.65 mA in a field of 0.006 gauss ;curve (b) a supercurrent of 0.30 mA with a field of 0.4 gauss; a field of 20 gauss was observed to reduce the supercurrent to zero. Anderson and Rowell offer experimental evidence that the observed supercurrent is not due to superconducting bridges across the barrier. The a.c. effect has not been observed yet. Taylor and Burstein95 have observed, in the case of tunnelling between two superconductors, an excess current which is temperature independent, and exhibits thresholds at V = A l and V = d 2 . Figure 5.33 shows their measurements on a Sn-Sn junction at T = 1.2" K. Because of the sharpness of the thresholds, they reject the possibility of this excess current being caused by normal metal inclusions in the superconducting films : an excess current due References p . 189
188
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH.
m, 0 5
to normal-superconductor tunnelling would exhibit considerable thermal “smearing” as shown in Fig. 4.2 (c). They attribute this current to second order effects, i.e. processes which are proportional to the fourth power of the tunnelling matrix element. A possible process has been proposed by Schrieffer and Wilkins O4 and is described in Section 4; it consists of Cooper pairs being removed on one side of the barrier and transferred to the other side as quasi-particles. The observation of this effect, being second order, depends upon obtaining very thin oxide layers.
0
2
1
V/A
3
Sn(OJ
Fig. 5.33. Current-voltage characteristic for a Sn-Sn sample at T = 1.2” K showing an excess current with a threshold at V = A . The data were taken by Taylor and Burstein.
Acknowledgments The authors are grateful to P. W. Anderson, J. W. Garland, D. M. Ginsberg, P. L. Richards, J. M. Rowell and J. R. Schrieffer for communicating their results before publication. Discussion with M. H. Cohen, J. C. Phillips and R. Meservey are gratefully acknowledged. This work was partially supported by the Office of Naval Research, the National Science Foundation, the National Aeronautics and Space Administration, and the Army Research Office. The authors would also like to acknowledge the use of the general research facilities provided by the National Science Foundation, the Atomic Energy Commission and the Advanced Research Projects Agency.
Added in Proof The following recent developments are called to the reader’s attention. 1) Rowell [Phys. Rev. Letters 11, 200 (1963)] has observed the periodic cancellation of the Josephson current with magnetic field discussed in Section 5.4.8. 2) The influence of phonons on the tunnelling density of states has been References p . 189
CH. 111 ]
THE SUPERCONDUCTING ENERGY GAP
189
observed in In by Adler and Rogers [Phys. Rev. Letters 10,217 (1963)l and in Hg by Bermon and Ginsbergl40. 3) Interesting microwave measurements on tunnelling junctions have been reported by Shapiro [Phys. Rev. Letters 11, 80 (1963)l and interpreted in terms of the A. C. Josephson effect. 4) Further refinements in the theory of the Josephson effect have been reported by Ambegaokar and Baratoff [Phys. Rev. Letters 10, 486 (1963)J and by Ferrell and Prange [Phys. Rev. Letters 10, 479 (1963)l. 5) It appears that the field dependence of the gap in thin films may not be as simple as the analysis in Section 3.5. I implies. Maki [Prog. Theor. Phys. 29, 603 (1963)l has pointed out that there is a distinction between the actual gap in the excitation spectrum and the gap parameter. The actual gap goes to zero with magnetic field faster than the gap parameter, in much the same way as with increasing concentration of magnetic impurities, as described in Section 3.6.2; there is a range of fields for which “gapless” superconductivity appears. In addition, Meservey and Douglas have found quantitative disagreement with equation (5.17) for very thin films near T,. One possible explanation for this discrepancy is that the assumption of spatial invariance of the gap is incorrect. REFERENCES
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2
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1
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lo4 lo5
CH. 1111 l45 146
l47
I49 l50
151 152 153 154
THE SUPERCONDUCTING ENERGY GAP
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193
CHAPTER IV
ANOMALIES IN DILUTE METALLIC SOLUTIONS
OF TRANSITION ELEMENTS BY
G. J. VAN DEN BERG
KAMERLINGH ONNESLABORATORY, LEIDEN
CONTENTS: 1 . Historical remarks, 194. - 2. Electrical resistanceand magnetoresistance, 198. - 3. Thermal resistance, also in a magnetic field, 211. - 4. Thermoelectric power, 215. - 5. Magnetic properties, 222. - 6. Hall-effect,233. - 7. Specific heat, 238. - 8. Optical properties, 244. - 9. Theoretical considerations, 245.
1. Historical remarks When the temperature of a metal or an alloy is so low, that the electrical resistance due to the lattice vibrations (phonon scattering of the conduction electrons) has become negligible, the residual electrical resistance (impurity and defect scattering) is constant if due to independent electron scattering. During experiments on the behaviour of the electrical resistance as a function of temperature it was observed that for some less pure metals (diluted alloys) the description for low temperatures mentioned above was not valid. Meissner and Voigt 1 carried out systematic investigations on the electrical resistivity as a function of temperature in order to test Gruneisen’s formula for this function2 and to look for superconductive metals. Though data were obtained at a limited number of liquid helium temperatures, on the average three, an increase of resistivity between 0.5 and 2% was found upon decreasing the temperature in the case of Mg, Mo, Te, Co and Pd. As a result of measurements on Pd specimens of different purities the authors concluded that the behaviour of pure specimens should be normal. Consequently Keesom3 tried to use a technical Mg wire as a resistance thermometer for the liquid helium region some years later. After a preliminary survey of De Haas and Voogd4 concerning the data at their disposal in 1932 (for a test of the T5 law5 of the ideal resistivity below a twelfth of the Debije temperature) extensive experiments on the References p . 259
194
CH.TV, 4 11
ANOMALIES IN DILUTE METALLIC SOLUTIONS
195
resistance of rather pure metals as a function of temperature were started6. De Haas, De Boer and Van den Berg', measuring between 1 and 2 l o K , found a minimum in the resistance-temperature curve of pure gold from "Heraeus" (impurity c A close study of this anomaly8 showed that the temperature of the minimum was a function of the resistance ratio
"4
@
A
RUTGERS LEIDEN
Fig. 1. The resistance ratio at the minimum of the R-T curve, Rmin/R273, as a function of the temperature of the minimum. 0 Measurements at Rutgers University, 1950 O. A Measurements at Kamerlingh Onnes Laboratory, 1936-'37 *.
at the minimum (see Fig. I) and that a certain chemical impurity, probably iron, was the cause of it. The minimum was not due to contamination by the electrical welding, nor to size effects, nor to a heating by the measuring current related to the minimum in the thermal resistance (no difference between 10 and 40 mA current). In some of the silver wires prepared with less precautions than the other ones, the anomaly could also be demonstrated8. Measurements below one degree absolute of the resistance of a less pure gold wire showed an increase down to at least 0.3"KlO. Similar experiments were started again after 1945 by several authors911l-16, during which the temperature region was extended to 0.006OK17. I refer to Section 2 for further information. Lindela carried out a wide series of investigations on the specific atomic increase of the resistivity when small amounts of normal and transition metals were added to the noble metals (Cu, Ag and Au). Norbury's rulelD states that the increase of resistance per atomic per cent, A p, increases with the valence difference between solvent and solute. The scattering power of an impurity atom depends on its horizontal distance, in the periodic table, from the base metal (Fig. 2). Linde18 gave this rule the functional form Referencesp. 259
196
G . J. VAN DEN BERG
[CH. IV,
81
A p = a + b Z 2 , a and b being constants for a given solvent metal and a given row in the periodic table and 2 being the valence difference. For the
3.6
2.I
2.1
; i i
’
of various elements when dissolved in Cu according to ZimanZ0.
addition of the normal metals Linde’s rule turned out to be valid, but for addition of the transition metals and especially of Mn this rule was not obeyed. Other properties also showed deviations. Gerritsen, in cooperation with Linde, started experiments on the electrical resistivity of Ag- Mn alloys at low temperatures2I. The curve of the results for a concentration of about 0.1 at. % Mn in Ag showed a minimum as well as a maximum at decreasing temperatures (see Fig. 3). For smaller values of the concentration, only the minimum remained for measurements above 1.2”K. Here is the link with the investigations on rather “pure” metals described above (see also Section 2). Gerritsen and Linde22, from the point of view of an s-d-scattering, expected and also observed an anomaly in the magnetoresistance of Ag- Mn alloys, which was negative for the lowest temperatures and for not too small concentrations. The value amounted in some cases to - 30%. Other transition metals like Cr, Fe, Co and Ni were dissolved in small quantities in Cu, Ag and Au, when possible. More or less pronounced anomalies were observed 23-27 (see Section 2). In 1956 a series of investigations from still another point of view was published by Owen, Browne, Knight and Kittel 2* on an alloy of a transition metal (Mn) in Cu. The authors expected marked effects - on the electron and nuclear spin resonance and on the susceptibility - of the exchange Referencesp . 259
CH. IV,8 1J
191
ANOMALIES IN DILUTE METALLIC SOLUTIONS
interaction between 3d5 ion core-electrons of the Mn atoms and the 4-s conduction electrons of the crystal. They were particularly interested in observing the effect on the conduction electrons of the host metal when a low concentration of a second component was added. Owen et al. decided 2980 2920 2860 2800
47s 965
I
95s 945
P
600
4150 4050 3950
3850
-2
0
10
1s 20'K
OATS
10
15
2dK
Fig. 3. The resistance ratio rT = R ~ / R a 7 3as a function of T for Ag-Mn alloys with atomic concentrations as indicated in the graph 21.
at the beginning to emphasize the study of diluted solutions of Mn in Cu for several reasons concerning both the applicable methods and the data known about the two metals (see Section 5). MacDonald and Pearson 29 investigated the relation between the minimum in the electrical resistance and the thermo-electric power (see Section 4). Kemp, Sreedhar and White 30 and, nearly simultaneously, Rosenberg31 published heat conductivity experiments on impure Mg, while De Nobel and Chari32 measured the thermal conductivity of Ag-Mn alloys (see Section 3) in order to look for anomalies, both with and without a magnetic field. References p . 259
198
0.J. VAN DEN BERG
[aN, .8 2
De Nobel and Du Chatenier33 started an extensive investigation on the specific heat of such “diluted” alloys of transition metals in noble and other normal metals (see Section 7). The aim of this work was the determination of the “magnetic” influence on the specific heat and the measurement of the change in the density of states (change in electronic specific heat). Finally, experiments be mentioned on the Hall effect34, first started by Fukuroi and Ikeda35, on impure, unannealed gold (see Section 6).
2. Electrical resistance and magnetoresistance As mentioned in Section 1, several authors7-17 started experiments on the temperature dependence of the electrical resistivity of “pure” metals and found for some metals such as Au, Ag, Cu and Mg a minimum in the resistance 0s temperature curve. Though sometimes the laboratory number of the metal was the same, the results differed considerably36. This shows the large influence of the preparation (of the wires measured) as well in the laboratory of the manufacturer as in the laboratory of the investigator. As an example may be mentioned the temperature of the minimum in the R - T curve for different Mg specimens made from the same Johnson, Matthey and Co, Ltd (J.M.) material No. 1848: 0.07”, 2”, 5”, 5” and higher than 25°K for a resistance ratio R/R273 at the minimum of respectively 15.7, 8.0, 7.5, 16 and 100 times Rorschach and Herlin13measured the resistance of Mg magnetically, eliminating any end contamination from welding the current and potential leads. A minimum was still found at 16”K, while in the same material J.M. No. 1848 Yntema and Lane37 found the minimum above 4.2”K. The magnetic resistance of the Mg specimen was positive for small magnetic fields, as has also been found by MacDonald and Mendelssohnll and Thomas and Mendozal5 for fields of some thousands of oersteds. Only Kan and Lazarevla did not find a minimum in the curve for an annealed polycrystalline specimen with a residual resistance ratio R / R 2 7 3of 25.7 x when mounted in a way “which avoided end contamination”. This summing-up of conflicting results shows the importance of purity of the material and the careful treatment of the specimen for reliable results. Two aspects of the research on the “pure” metals with a minimum in the resistance-temperature curve are to be presented here because of their influence on the theoretical work of Section 9. 1. The tentative description of the temperature dependence in the region at or below the minimum. Van der Leeden38 proposed to represent the electrical resistance of gold wires between 0 and 12°K as follows: References p . 259
CH.IV,9 21
ANOMALIES IN DILUTE METALLIC SOLUTIONS
199
where rid(T) = the electrical resistance ratio for an ideal pure lattice, = the resistance ratio due to chemical impurities, zch zph = the resistance ratio due to lattice defects, C = approximately constant for all materials
-
f ('K)
I and 11) for gold wires of different .purity8. Curve 111 represents Fig. 4. R-T curves ( rid(T) of equation Dots with (2) represent identical results.
(see Fig. 4 and Fig. 5). Croft et a1.17 measured the resistance of gold wires in a two-stage demagnetization apparatus to as low as 0.006"K and proposed below 4°K: I ( T ) = constant - alog T,where a=---
lo3 dr r d(1ogT)'
Dugdale and McDonald39 found such a relationship for gold wires between 0.1 and 2°K. Alekseevskii and Gaidukow40 derived for the range above 0.3"K the relation
+
r ( T ) = AQog T / T ~ # " ) ( Blog I T/T,~~)-'.
(3)
2. The possibility of the absence of a minimum in polycristalline as well as monocrystalline material. This can be deduced from a lot of experiments. Referencesp . 259
200
G . J. VAN DEN BERG
[CH.IV,
$2
The influence of the crystalline state was also especially investigated by Schmitt and Fiske41, Alekseevskii and Gaidukov42 and Dugdale and Gugan43 but no effect could be found.
Fig. 5. The agreement between the curve represented by eq. (l), and the experimental results of 10 (Van der Leedenas).
2.1. “DILUTED’’ALLOYS OF TRANSITION METALS OF THE FIRST LONG PERIOD The measurements of “diluted” alloys of Cr, Mn, Fe, Co and Ni in Cu, Ag and Au (see Section l), as far as the solubility permitted, had as a result that anomalies were found in the electrical resistance-temperature curve ( R - T curve). Mn23144, Cr23.44 and Fe26944945as an impurity in Au gave a maximum and a minimum in the curve for a concentration of about 0.1 at.% along with a decrease of the resistance in a magnetic field (negative magnetoresistance 12*24, 26v 46,47) (Fig. 6 and Fig. 7). The last mentioned result was obtained for Mn in C U ~ ~ and V ~Agzz, ~ Pbut ~ not Q for Fe and Cr in Cu. The existence of a minimum in the R - T curve for alloys of the noble metals with a small amount of Ni is not certain, neither it is sure that Co will cause it in Au50. This may be deduced from the random behaviour of the R - T curve with an increasing Co content 27 (see also Section 4). The accidental introduction of Fe as an impurity during the treatment cannot be excluded. Schmitt5l demonstrated this also for a diluted alloy of Zn in Cu. Here the minimum in the R-T curve was due to an impurity already present or introduced in the base metal. References p . 259
CH. IV,
8 21
201
ANOMALIES IN DILUTE METALLIC SOLUTIONS
P 012 A u - F e
p =lo99 4.2
0
5
10
"K
Fig. 6. The electrical resistance as a function of temperature of Au-Fe 0.12 at.X45.
20
I0 0
10
50
i 0
0
-10
-2 0 IO'AR~ RH-O
t
5
10
15
2 0
2 5c
-T
Fig. 7. The magnetoresistanceof Ag-Mn alloys in a transverse magnetic field of 20 kOe2a. 0 c=o, c = 0.05, 0 c = 0.24, @ c = 0.02, c = 0.155, X c = 0.61. The ordinate scale at the right is only for c = 0.
+
References p . 259
202
[CH. N,4
0.J. VAN DEN BERG
2
68z*m-
A normal metal as an impurity in a pure noble metal does not cause a minimum in the R - T curve. Knook et ~1.52’53showed this directly by measuring the R - T curves of the “diluted” alloys Cu-Sn, Cu-Fe and Cu- Sn- Fe. The base metal Cu has to contain less than 0.8 ppm Fe in 265 [*cmwDIS
D 16
68
26
,;f, ,
io4p
, ,
0-1
2
3
7Tz-7-+3’
D‘4,0dpyf
P 4.K
3405
0-1
2
3
4’K
Fig. 8. The specific electrical resistance PT as a function of temperature T for Cu-Sn alloys with concentrations 0.0045, 0.018, 0.141 and 0.259 at. % SnS2.
order to be sure of the results (Fig. 8 and 9). The increase of the specific resistance of Cu at liquid helium temperatures was proportional to the Sn content and can be represented by
if pm,T = the specific resistance of the pure metal, 6 p , , , = the atomic increase of resistance due to the impurity (1 at.%), which is constant within 0.8% below 20°K. The value at 273.15”K is 10% higher (2.8 8pQcm/at.%). The system Cu-Fe is represented in Fig. 10. For this system formula (4) (see above) can be also used for the resistance at the minimum. According to experiments on Cu-Fe by White54 and by Dugdale and M a ~ D o n a l d ~ ~ the R-T curve below 1°K is horizontal. This permits a rather good definition of the absolute depth of the minimum (= pOoK-pmin).When this quantity is plotted on a double logarithmic scale as a function of concentration one obtains a straight line, which makes it possible for this special Cu- Fe system to determine the concentration of Fe in a Cu specimen of a manufacturer, if all the specimens have been treated in the same way. The relation between Tminand the concentration c of Fe in Cu could be given by: 1
Tmi,= 43.7 c ET. References p . 259
(5)
CH.IV, 8 21
203
ANOMALIES IN DILUTE METALLIC SOLUTIONS
d 25405
€36
€37
2505
I
0 2
Fig. 9.
5
(0
15
20
OK
0 2
E39
5
10
I5
p-T curves for a series of Cu-Sn wires. The pure Cu wire contains 0.8 ppm
FeS2.The concentration of Sn increases from 0.0015 to 0.947 at. %.
References p . 259
204
G.J. VAN DEN BERG
O
T
S
10
15
20'
[CH.IV, 5 2
K
Fig. 10. p-Tcurves for a series ofCu-Fe wires52. The concentration of Fe increases from0.0005to0.123 at. %. Note that the scales of the two ordinates are different. References p . 259
0 21
CH. N,
ANOMALIES IN DILUTE METALLIC SOLUTIONS
205
This agrees rather well with the formula Rmin/(R273 - RmiJ cc T;f;i2,,found by Pearson 55. For the alloys Cu-Sn-Fe it was possible to represent the specific
By increasing the Sn concentration at a fixed Fe concentration Knook could imitate the results of impure Cu-Sn alloys as measured e.g. by Pearson et al. 56. Domenicali and Christenson57 measured the resistivity of Cr, Mn, Fey Co and Ni in Cu and Mn, Fe and Co in Au as well as that of the pure base metals over a tremendous range of temperature, between 4 and 1200°K. The difference dp, between the resistivity of the alloy and the “pure” base metal plotted as a function of temperature showed a remarkable behaviour, inter alia a maximum at 65°K for Au-Cr 0.09 at.%. It is a pity that the authors did not carry out the same experiments on alloys of Cu or Au of the same origin with nonmagnetic soluted atoms in order to compare the deviations from Matthiessen’s rule in both cases. As far as can be deduced from the figures d p , at low temperatures is not proportional to the concentration of the transition metals in disagreement with Knook’s results 529 53 which may indicate the introduction of unidentified impurities. As mentioned above, Cr, Mn and Fe of the first long period produce as impurities in the noble metals Au, Ag and Cu, anomalies in the resistancetemperature curve. As an exception Linde58 found also an abnormally high temperature for the minimum in the R - T curve for V in Au and no maximum, the latter in agreement with magnetic measurements 59. But Mn and Cr in Zn49~60,Mn and Fe in Mg15$37p64and Mn in Cd61 also show anomalies in the R - T curve. Hedgcock et aI.62 investigated the alloys Mg-Mn, Mg- Fe, AI-F e and A1-Mn and succeeded in demonstrating maxima and minima in the first series of alloys, minima in the second one and a normal behaviour in the A1 alloys. This last-mentioned result is in agreement with experiments by Boorse and Niewodniczanski63 and by Thomas and Mendozal5. The results on the Mg-Mn alloys show e.g. the behaviour of the R - T curve some degrees below the maximum at 6°K in this curve for Mg - Mn 0.6 at.% (Fig. l l a ) and show the absence of a maximum, but still the presence of a minimum in this curve (Fig. l l b ) for a small concentration of 0.04 at.%64. For the alloy Cd-Mnxat.% a minimum at about 4.5”K was found by Muir61, while the R - T curve for the base metal Cd was normal! Concerning the alloys with Zn as the base metal, Muto et aI.49’60 inReferences p . 259
206
&I
Mg-Mn k060 at %)
7 6'
0.53
k*
[m.w,0 2
0. J. VAN DEN BERG
I
Mg-Mn (e0.35 at . '10) 0.52
0.33
f I% 0.32
Mg-Mn W0.56 a t . %) 0.50
Mg-Mn W0.22 at . %I 0.22 049
0.21
Mg-Mn W0.16 a t . %) 0.16
0.37
0.15 1
2
3
4
5
6
as 1
2
3
4
5
6
7
8
T PK)
Fig. lla. The electrical resistance ratio rT = & / R z ~ s as a function of temperature T for Mg-Mn alloys with concentrations between 0.16 and 0.60 at. X e z .
vestigated the alloys Z n - M n and Zn-Cr and compared the results with the normal ones for a Zn-Sn 0.05 at.% alloy and the abnormal ones for a Cu-Mn 5.4 at.% alloy. For the alloy Zn-Cr 0.10 at.% a negative temperature coefficient was found, while the magnetoresistance at 4.24"K was a positive one with a small hump in the temperature curve in the low field range. In the R - T curve for Zn-Mn 0.12 at.% there appeared a minimum, and the negative slope at temperatures below that minimum decreased when the temperature did the same, suggesting the approach of a maximum in the R - T curve49. This should have been in agreement with the negative magneto-resistance at magnetic fields below 30 kOe, which passed to a positive one at fields stronger than about 60 kOe. Hedgcock and Mui1-65, however, deny the possibility of the existence of a maximum, References p . 259
9
T PK
CH.IV, 6 21 85
P
0
\s
80-
15
-
70
-
0
207
ANOMALIES IN DILUTE METALLIC SOLUTlONS
-
*,“\
1
10
IS
20 25 5 (-K)
30
35
40
Fig. l l b . The electrical resistance ratio rT as a function of temperature T for Mg-Mn alloys with concentrations between 0.005 and 0.16 at. Xsn.
208
[CH.IV, 5 2
G.J. VAN DEN BERO
because measurements on a series of Zn - Mn alloys having compositions extending to (0.42 at.%) near the limit of solid solubility of Mn in Zn did not result in a maximum in the R - T curve down to 1.8”K. The resistance of the most concentrated alloy remained constant between 5 and 1.8”K, which is, be it understood, below the minimum in the curve. Before leaving the first long period of transition elements as solutes in the noble metals (Cu included), Mg, Zn and Cd, attention is drawn to Ti as a solvent and Mn, Cr, Fe, Co, Ni, Zr and Nb as solutes66-6* (see 2. 4).
2.2. “DILUTED” ALLOYS OF TRANSITION METALS OF THE SECOND LONG PERIOD The following series of alloys has been investigated: Au-Pd,
Ag-Pd, Cu-Pd, Au-Rh, AU- MO52,533 69. CU-Ru,
Cu-Rh,
Only the alloy Au-Mo showed anomalies in the electrical resistance and magnetoresistance. For these investigations it is again very important that the base metal be free from impurities arising from the first long period, especially free from Fe, Mn and Cr. Experiments have shown that some ppm Fe in Au is too much for normal behaviour of the residual resistance 7O. In order to test if an abnormal behaviour of the electrical resistance of such alloys is genuine, a series must show that the “depth” of the minimum increases proportionally to the amount of the solute. The same can be required for the residual resistance. Deviations from this correlation are indicative of a role played by unidentified impurities or by segregation. Melting and degassing of the base metal may e.g. increase the influence of Fe which was present as an oxide. Experiments of Coltman, Blewitt and Sekula71 have shown that small amounts of O2 may often influence the anomalies in the R-T curve at low temperatures. Here see also experiments by Domenicali and Christenson 72 concerning the influence of small amounts of 0, during high-temperature annealing on the cross sectional uniformity of the concentration of the solutes. The preparation of Au-Mo alloys was complicated. By a diffusion process at 1000°C in a rolled “sandwich” of molybdenum between gold strips, it was possible to dissolve some Mo. A negative magnetoresistance, as in the case of Ag-Mn alloys, was measured to an amount of - 12% at a temperature of 1.3”K and in a magnetic field of 21 kOe (Fig. 12). Above 1.2”K no maximum in the R-T curves was detected. The low effective concentration of Mo, between about 0.004 and 0.03 at.%, may be the reason for this absence. References p . 259
CH. IV,
8 21
209
ANOMALIES IN DILUTE METALLIC SOLUTIONS
,
+S
0
-5
--1O
H
___c
5
10
15
20
kOo
Fig. 12. The magnetoresistance of Au-Mo as a function of the transverse magnetic field for temperatures below 21” K69.
2.3. “DILUTED” ALLOYS OF TRANSITION METALS OF THE THIRD LONG PERIOD The alloys Au-Ta, Au-Ir, Au-Os, Au-Re, Ag-Re and Cu-Re have been i n v e ~ t i g a t e d s ~The . ~ ~ .existence of a minimum in Au-Os, due to Os, is almost sure. No doubt exists concerning anomalies in the curves of the resistance and the magnetoresistance us temperature for Au- Re alloys. Due to superconductive filaments (?) a sudden decrease of R occured at a temperature of about 1.8”K, in fact below that of the minimum in the R-T curve, simulating a sharp maximum in that curve. By means of a magnetic field of some hundreds of oersteds the superconductivity could be destroyed, and in stronger fields a decrease of the magnetoresistance at increasing field strength was f0und7~for the very “diluted” alloys. For the more concentrated ones (> 0.1 at.% Re) a negative magnetoresistance was found at temperatures below 2.5”K, being - 9% at 1.24”K. The Ag-Re and Cu-Re alloys contain only some ppm of the transition metals and the R -T curves show a minimum above liquid helium temperatures and show more or less an indication of the superconductivity of Re filaments. This is a proof for References p . 259
210
[aN, .5 2
G . J. VAN DEN BERG
the segregation of Re in agreement with the conclusion by Holland-Nell and Sauenvald74 about the insolubility of Re in Ag and Cu.
2.4.
TRANSITION METALS ALSO AS SOLVENTS
As mentioned in 2.1 measurements have been made on “diluted” alloys of transition metals in a transition metal as base metal. Thomas and Mendozal5 measured the electrical resistance of Mo, Co and W, which metals were not very pure. In the Mo wires there were as impurities Fe: 0.008%; N and Cr : traces. These wires had a minimum in the R-T curves and one of the wires had an abnormal magnetoresistive behaviour. At high fields and 1.2”K the magnetoresistance became negative. For Co and W no anomaly was found. MacDonald, Pearson and Templeton44 gave some information about the electrical resistance of alloys with Pt and Pd as base metals and Mn and Fe as solutes during experiments on the thermoelectricity (see Section 4) but details were left out. An extensive investigation was carried out on alloys with Ti as solvent and Mn, Cr, Fe, Co, Ni, Nb, A1 and Zr as solutes by Berlincourt et a1.66-e8. The 99.92 wt % Ti showed a minimum in the R - T curve at 14.1”K with a difference of 0.84% between the resistance at 14.1”K and 42°K (impurities a.0. Zr 0.05 wt %, Mn 0.003 wt %, Cr, Fe 0.002 wt %). The Ti-Mn alloys had a deeper minimum and a negative magnetoresistance which was of the
4.2 O K 0.95
0.95
I-
l’m ---__
0.95
0.900
0.95
20
40
60
80
100
I20 0.90
H (KILOGAUSS)
Fig. 13. The electrical resistance ratio p( T)/p4.2 as a function of magnetic field for Ti-Mn 0.101 at.% at 4.2 and 1.2” K. The solid and dashed curves correspond, respectively, to transverse and longitudinal fields. The black dots represent transverse steady field datae7.
Referencesp . 259
CH.IV, 5 31
ANOMALIES IN DILUTE METALLIC SOLUTIONS
21 1
order of - 30% at 1.2"K in a field of 120 kOe for the 1 at.% alloy, and - 5% for the 0.1 at.% alloy. The saturation for the last-mentioned alloy appeared to be complete below 100 kOe (Fig. 13). The results for the Ti-Mn alloys are quite comparable with those for Cu-Mn alloys (see above). The R - T curves for the alloys Ti-Cr 1.15 at.% and Ti-Fe 0.96 at.% showed a minimum of 0.84 and 1.66% respectively. The magnetoresistivity for these alloys, as well as for the alloys with Coy Ni, Nb, Zr and A1 as solutes, is small and positive. From the first-mentioned results one should conclude that there is a localized magnetic moment in the hcp phase of Ti-Mn alloys, which was confirmed by susceptibility experiments68. However, according to Matthias et al. 75, such a localized moment should lead to a decrease of the transition temperature of superconductivity in disagreement with these experiments. This may be correct, when it is assumed the superconductivity is due to retained enriched bcc inclusions, wherein the conduction electron concentration and hence the density of electronic states at the Fermi surface is raised. The hcp Ti-Mn may have a depressed T,. Experiments on single phase hcp specimen and two-phase specimens consisting of bcc filaments in an hcp matrix sustain the above mentioned hypothesis B8. Summarizing the results of this section, one can say that the largest abnormal effects are caused by the transition metals with the largest number of unpaired spins in the d-band: Cr: 3d54s; Mn: 3d54s2;Fe: 3d64s2; Mo: 4d55s; 0s: 5d66s2 and Re: 5d56s2; the effects, however, depend on the solvent. For the spectroscopic states of some of these solutes, see Section 5. 3. Thermal resistance, also in a magnetic field After the discovery of anomalies in the electrical resistivity, it was worthwile to pay attention to possible similar anomalies in the thermal resistivity. The problem, however, is to make the right splitting of the total thermal conductivity into the contribution from the lattice and that from the electrons. At low temperatures the lattice contribution may be a small part of the total conductivity, but this depends on the scattering of the electrons by the impurities or defects present. The electronic thermal resistivity can be represented by we = aT2 BIT. The results of measurements of the thermal resistivity for Mg have been interpreted in two ways : 1. Kemp, Sreedhar and White30 published a curve of wT us T for rather
+
References p . 259
212
[CH.IV, 5 3
G . I. VAN DEN BERG
pure Mg (J.M 1848; 99.98%with 0.013%Fe) which increased slightly below about 5°K as did the electrical resistivity curve (w =thermal resistivity). 2. Rosenberg31, however, published a curve of wT vs T 3 for less pure Mg (J.M 1703; 99.95% with 0.03%Mn) which was a straight line above 11°K. The plotted values of wT drop below this line, pass through a minimum at about 6°K and then rejoin it again. This should signify that the thermal resistivity decreases below the value for a normal metal (alloy) and that the electrical resistivity minimum should be an extra decrease instead of originating from an extra scattering at temperatures below the temperature of the minimum (see Section 9). Spohr and Webber76 investigated both sorts of Mg, with main impurity Fe (cold worked) and Mn (J.M 1848 and Dow No. 7286) respectively. They found that the relative deviations from “normal behaviour” dp/p, and d w/w, are completely equivalent for the
i 0
5
10
J
15
20
3
(OK1
Fig. 14. Percentage deviation of the observed electrical and thermal resistivities from the “normal)’ behaviour plotted as a function of temperature for a Mg specimen with Fe (0.013 %) and one with Mn (0.043 %) as predominant irnp~rity’~. 0 Aplpn X Awlwn.
Referencesp . 259
CH. IV,
8 31
ANOMALIES IN DILUTE METALLIC SOLUTIONS
21 3
electrical and thermal resistivity in the case of Mg with Mn as main impurity, but that for the other more pure but heavily cold-worked specimen (Fe impurity) there was a difference between the two quantities between 5 and 15°K (see Fig. 14). The curve for wT us T 3 (compare with Rosenberg31) deviated to higher values of wT below 15°K for the specimen with Mn as
T("K)'
Fig. 15. The product of thermal resistivity w and temperature T as a function of T3 for Mg- Mn 0.043 at. %77.
the main impurity (see Fig. 15). One possible explanation for the discrepancy between Rosenberg's results and those of Spohr and Webber lies in measurements of the specific heat 78. The variation of the Debye temperature in the region concerned may give the result measured by R0senberg7~.The Lorenz ratio L = p / w T had a constant value of 2.64 x lo-* watt ohm deg-' below 5"K, but this value of L is too high according to theory. Chari and De Nobel32 used the same Ag-Mn rods as Gerritsen and Linde21 with concentrations of 0.55, 0.32 and 0.14 at.% Mn for measurements of the heat conductivity in magnetic fields. Now a pronounced anomaly was found. The electronic thermal conductivity calculated by the method of Griineisen79 and De Haasso increased considerably in a magnetic field (see Fig. 16). There is in this respect a close parallelism between the relative thermal magnetoresistance and the electrical one (see Figs. 17 and 18). Charis1 analysed the thermal conductivity data afresh in 1961 and paid much attention to the Wiedemann-Franz-Lorenz parameters2. Starting from the justified assumptions that the lattice conductivity Lg (or resistivity w,) is unaffected by the magnetic field H and that the Lorenz parameter L e = -1,= -
oT
P w,T
is independent of H , justified by a linear dependence of 1on aT for various fields at a fixed temperature T (see e.g. Fig. 19), Chari derived the values of References p . 259
214
[CH. IV, 8
0. J. VAN DEN BERO
3
L, for various temperatures for the Ag-Mn alloys mentioned above (see Fig. 20). Because of the large (negative) values of the thermal and electrical magnetoresistance of these alloys, this analysis. could be carried out. Ac-
'I
T - 2
3
4
'Y
Fig. 16. Thermal conductivity of Ag-Mn alloys in different magnetic fields measured as a function of temperature by Chari and De Nobel32. 0 4 .:H=OkOe, A A:H=19kOe, 8 : H = 12 kOe, : H = 25.5 kOe. From top to bottom concentrations are 0.14, 0.32 and 0.55 at. % Mn respectively.
cording to MakinsonS3the electronic Lorenz parameter L, would fall significantly below the normal value in the presence of inelastic small-angle scattering because this type of scattering can contribute substantially to the heat resistivity. At sufficiently low temperatures, where the scattering of the electrons is predominantly due to impurity scattering, Chari expects a normal value for L,,because the scattering can be considered to be effectively Referencesp . 259
CH.IV, g 41
21 5
ANOMALIES IN DILUTE METALLIC SOLUTIONS
-
20
2 5 kOc
30
Fig. 17. The electrical magnetoresistance of Ag- Mn alloys as a function of field at 4 and
1.5°K32. Ag-Mn 0.14 at. % 0.32 at. % 0.55 at. %
0 : T = 4"
v :T = 1 9 K ;
K,
A : T = 4" K,
: T = 1.5" K;
0: T = 1.5" K.
@ : T = 4"K,
elastic. Following Schmitt and Jacobs' calculation of the contribution of the spin-flip scattering processes4s, Chari calculates a much larger contribution of these inelastic processes to the electronic thermal resistivity in the region of the maximum in the resistance-temperature curves (Fig. 21) or of the transition from the paramagnetic to the ,,antiferromagnetic" state. 4. Thermoelectric power At the beginning of this subject one must keep in mind that the thermoelectric power (T.P.) is a property which is very sensitive to impurities in metals and to the inhomogeneity of the specimen. Well known is the measurement of ,,pure" Cu against ,,pure" Cu, a thermo couple of the same pure wires, which gave results indicating the difference in behaviour of these two pure specimens. Another remark is that direct absolute measurements (against a superconductor of which the transition point is high enough) deserve preference 0
-01
-0.2 WH.0
-03
0
n
-5
10
15
20
2 5 kOC
30
Fig. 18. The thermal magnetoresistanceof Ag-Mn alloys as a function of field at 4 and 1.5"KS2.For meaning of symbols see Fig. 17.
References p . 259
216
G . I. VAN DEN BERG
[CH.IV, 8 4
to indirect ones and that measurements of T.P. at small temperature differences are preferable to the integral measurements, which yield E- T curves of which the gradients give the data for T.P. 4.1 NORMAL METALS AS SOLVENT
Early measurements by Borelius et aLs4 and by Keesom and Matthyss5
Fig. 19. The electronic thermal conductivity in a magnetic field HI as a function of the product of the electricalconductivityO[H]and the temperature T for a Ag- Mn 0.14 at. % specimen81.
L -
10,
2
3
T('K)
References p . 259
4
Fig. 20. The electronic Lorenz parameter L, as a function of temperature for three Ag- Mn alloysB1.
CH. IV,8 41
ANOMALIES IN DILUTE METALLIC SOLUTIONS
217
on “diluted” alloys of Au and Cu with small amounts of some of the transition metals showed very large characteristic thermoelectric powers at low temperatures against ‘‘silver normal”, consisting of Ag with 0.37 at.% Au, as reference. The term “very large” signifies of the order of - 1OpVPK (see a review by Borelius at the Lorentz-Kamerlingh Onnes Conference
a t 01. of Mn
Fig. 21. f(1ower curve) and f’ (upper curve), the maximum contributions of inelastic impurity scattering as fractions of the electrical resistivity PO and the product of electronic thermal resistivity at 0 ” K times temperature (w,T)o respectively as a function of the atomic percentage of Mn in AgS1.
195386),in comparison with the “normal” value, e.g. for Sn single crystalss’, which is of the order of some hundredth of a p V r K (Sommerfeld, Mott and Jones, Wilson inter alia88). MacDonald and Pearson 29 investigated the relation between the abnormal behaviour of the electrical resistance at low temperatures and that of the thermoelectric power at 15°K for the same alloys. They found an analogous curve for their “depth” of the minimum, (R4.2 - Rmin)/Rmin,as a function of the residual resistance as well as for the absolute thermoelectric power at 15°K as a function of this residual resistance. One must keep in mind now that the resistance minimum of the several diluted alloys was not due to Ge, Sn, Ga, Bi, In, Si, Pb etc., but to Fe89 (see also below: Gold et ~1.90)and for that impurity in Cu one finds a References p . 259
218
G . J. VAN DEN BERG
[CH.IV,
54
linear relation between S1, (T.P.) at 15°K and the “depth” of the minimum, that means, as confirmed by the experiments by Knook et a1.52163 (see Section 2.1), an approximately linear relation between SI5and the (limited) concentration. This relation is in disagreement with Sommerfeld’s88 theory of electrons according to which the absolute T.P. should be proportional to the absolute temperature and independent of the concentration of the impurity. As described by De Vroomen89 one cannot consider these impurity atoms like Fe as “static” in their exchange with the electrons because they have an internal degree of freedom as shown by the peculiar effect in the specific heat (see Section 7). The presence of some ppm of a transition metal in Cu will make application of the conventional theory risky or even unallowed. Gold et a1.90 have executed experiments on Cu in order to test the influence of the different impurities on the T.P., from which data could be derived on the Cu itself. In the scope of this section the influence of the transition metal is important. A very pure “natural” copper specimen with a residual electrical resistance ratio r of 3.1 x had a small (< 0.lpVrK) positive T.P. below 13°K. A “J.T.H.” commercial Cu, containing 0.002% Fe had a small negative value for S(T.P.) as an oxide, with an r = 22.3 x of - O.OSpV/O K. The same material melted under reducing conditions had a T.P. of - 9,uVpK, consistent with the supposition that the iron has now entered into solid solution. The addition of 0.054 wt % Fe gave the value 3200 x for r and - 16.OpV/”Kfor the T.P.. Conclusion: The negative value of the T.P. increases with the Fe concentration. Starting from a relation proposed by Kohlerg’, which was in essence given earlier by Nordheim and Gorter92, Gold et aZ.90 write:
where s:,Sb, pa, p b , are respectively the T.P.’s and the electrical resistivities or corresponding residual resistance ratios which would be obtained if each of the metals a, b were present. The conditions for the application of this rule are: a spherical Fermi surface, a transition probability only dependent on the angle of scattering of the electrons, mutually independent scattering by a and b. According to experiments on the electrical resistivity, the last condition seems to be fullfilled even in the case of a transition metal (Section 2.1). Adding the impurity b to the pure metal a and plotting S against l/p (inverse residual resistance ratio ( R 2 7 3 - R4.2)/R273) one should obtain a Referencesp . 259
CH.IV, 0 41
219
ANOMALIES IN DILUTE METALLIC SOLUTIONS
straight line with intercept S b as I/p-+O, this value being independent of S, and pa of the pure metal. Here it is assumed that the only role of the solute impurities is to scatter the electrons (see Fig. 22). The "pure" starting metals Cu 1 and Cu 2 have the values - 0.05 and - 3.2pV/"K respectively, 0
50 I
100
150
I
I
2
0
SIS'K
-4i -6F
Fig. 22. The thermoelectric power at 15" K, &5OX, as a function of the inverse residual resistance ratio for diluted Cu- Ni and Cu Fe alloys
-
but the two lines for Ni extrapolate to approximately the same value of s b w - 1.2pV/"K, the value of T.P. for Ni in Cu at 15°K. In Fig. 23 one can read the characteristic values for Sn, In, Ge and Ga in Cu, lying between 1 and - lpV/"K at 15°K. For Fe, however, one finds in Fig. 22 the value of - 16.2pV/"K at the same temperature. The transition metal Fe dominates undoubtedly. The conditions mentioned above are not all fulfilled, but the experiments seem to indicate that the description given can be accepted even at 15"K,where phonon scattering becomes of importance! In Fig. 24 are plotted the values of S,, (T.P.) of oxygen containing Cu (reduced and not reduced), the same Cu with Sn (reduced and not reduced) and Cu with Fe (reduced). The curve ABC gives the variation of the size
+
Referencesp. 259
220
54
G . J. VAN DEN BERG
[CH. N,
+2 0
-2
+2-
-2 0
20
-'-
40 R293*6-R4
60
80
2 'K
Fig. 23. The thermoelectric power at 15" K, SlS'K, as a function of the inverse residual resistance for diluted alloys of Cr in Asarco-Cu and Sn, In, Ge and Ga in J.T.H.-CU. The data for the latter four alloy series correspond to the region BC in Fig. 24
t -f
1
Fig. 24. The thermoelectric power at 15" K, S15'K, as a function of the inverse residual resistance for diluted alloys of Fe and Sn in J.T.H. (oxygen-containing) Cu 0 Cu Cu-Sn El Cu(reduced) Cu-Fe(red.) 0 Cu-Sn(red.) References p. 259
CH. rv,
8 4)
ANOMALIES IN DILUTE METALLIC SOLUTIONS
221
of the anomaly as the tin concentration is increased; the point C for l/p+ 0 then gives the value of SI5for Sn in Cu, being approximately 0. Under reducing conditions the addition of Sn reduces the size of the anomaly which is already present in the starting material (DBC). The line AD cuts the ordinate at E, the value for Fe in Cu, while the points for added Fe fit this line. This is consistent with the hypothesis that hydrogen reduction brings the iron present in the initial oxygen-containing copper into solid solution. This is also in agreement with the electrical resistance experiments by Coltman et al. 71 on copper, mentioned in Section 2, during which the minimum appeared after reduction and disappeared during annealing at 700°C in an atmosphere of air at 30 microns pressure. Surveying the results on the T.P. of “diluted” alloys of transition metals in the noble metals (MacDonald, Pearson, Templeton et al. 449 50, Tanuma 93 and others94) and in Zn, Mg and Al (Hedgcock et al.95) one concludes: 1. In many investigations the starting material was not pure enough or did not remain so after the treatment necessary for the mounting. 2. The temperature dependence is still questionable. 3. In A1 neither an electrical resistance anomaly nor a thermopower anomaly is present. The T.P. at 15°K for Al-Mn 0.005 and 0.046 at.% was respectively - 0.20 and - 0.26pV/”K against pure Al. This value can be compared with the partial T.P. of Mg in Al and of B group metals in Cu, and is not to be considered as semi “giant”. 4. Only for Mn in Au, Ag and Cu and for Cr in Au does there seem to be a correlation between the change-over when cooling to low temperatures from negative to positive thermoelectric power and the onset of cooperative magnetic spin alignment (the maximum in the electrical resistance). Au- Fe alloys show this maximum in the R-T curve, but no change-over at low temperatures! 5. The “giant” thermoelectric power is established for Cr, Mn and Fe in Au and Cu, for Mn and Fe in Ag and for Mn in Mg and Zn. As to the transition metals Co and Ni in the noble metals, one has to be cautious! A comparison with the behaviour of the electrical resistance seems to be allowed: a large number of unpaired spins in the d-shell of the transitional impurity is essential. 4.2. TRANSITION METALS AS SOLVENT
In Pt as solvent the addition of Mn and Fe produces a “giant” T.P.. Small amounts of Fe in Pt give a large negative T.P. that becomes temperatureindependent after its initial rapid increase in magnitude, but larger concenReferences p . 259
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G . J. VAN DEN BERO
[CH.I V ,
85
trations cause a positive T.P. which is also roughly temperature-independent. Pt and Au are qualitatively similar as parent metals, but Fe (0.5 at.%) in Pt gives a change-over from a negative to a positive T.P. at the low temperatures, which is not the case for Fe in Au. Pd presents a different picture in that no large thermoelectric powers are produced by the transition metal solutes Fe and Mn. MacDonald, Pearson and Templeton44 remarked that the change-over of sign for the “intermediate” concentrations (around 0.5 at.%) may be interpreted according to the suggestion of Bailynge as a change from “ferromagnetic’, to a dominantly “antiferromagnetic” alignment of the solute spins. These authors found that the more concentrated P t - Fe and P t - Mn alloys also exhibit directly some degree of ferromagnetism at 4.2”K, while Au-Fe alloys in the same concentration range do not. However, the more concentrated Pd-Fe and Pd-Mn alloys show the same ferromagnetism and neither a large T.P. nor a change-over of sign below 1°K. Here again a lot of questions remain to be solved.
5. Magnetic properties A lot of experiments on the magnetic properties of Cu-Mn, Ag-Mn, Au-Mn, Cu-Fe, Au-Fey Au-Ti, Au-V, Cu-Co, Cu-Ni and other alloys have been carried out in the last ten to twenty years, but in general only those investigations at low temperatures concerning the “diluted” alloys will be considered within the scope of this section.
5.1. THEMAGNETIC SUSCEPTIBILITY It would be expected that the total magnetic susceptibility of a pure metal with filled inner shells and paired electrons in the outer shells should be given by the sum of the diamagnetism of the core electrons, the paramagnetism of the conduction electrons, the diamagnetism due to the orbital motion of the conduction electrons and the paramagnetism of the nucleus. For a completely degenerate electron gas and neglecting a temperaturedependent electron-phonon interaction, this sum should be approximately temperature- and field-independent 07. Measurements by Bowers 98 on very pure Cu show that it is likely that the susceptibility of Cu is substantially independent of temperature. In the formula x = (- 0.083 + 0023/T) x cgs units about one fifth of the 1/T component results from the nuclear moment and it would only require 3 parts in lo7 of paramagnetic ions, such as Fe”, to explain the remainder of the 1/T dependence, which level of impurity is plausible for the Cu used. References p. 259
CH.IV, I 5 J
ANOMALIES IN DILUTE METALLIC SOLUTIONS
223
Sonder and Sekula99 measured the magnetic susceptibility of “pure” (99.999%) Cu after annealing at 900°C in different atmospheres. An anneal for 2 hours under 15 micron of CO created a resistance minimum. The susceptibility showed no field dependence. Below 100°K an increase in paramagnetism occurred which could be accounted for by assuming approximately 5 x I O l 7 magnetic centers per cm3 (= 6ppm) each having 2 Bohr magnetons. Such an anneal for 4 hours under 20 micron of air removed the resistance minimum (see Section 2). The results of the susceptibility measurements were such if 2 x 1017 Fe atoms were precipitated, corresponding to about 2 PPm. The same experiments were carried out on a Cu-Fe 0.1% specimen. Almost all the Fe was in a paramagnetic state. After an anneal in air the sample was strongly ferromagnetic. Nontransition elements as a solute should not cause a temperature dependence of the susceptibility and the results obtained by Hedgcock for “diluted” Cu-Sn alloys must consequently be considered in the light of the remarks made in Section 4 concerning the introduction of free Fe in the solvent Cu by adding Sn. For Cu-Fe alloys HedgcocklOO found an increase of the paramagnetism below the temperature of the minimum in the R - T curve. Two possible explanations were given: One based on the Korringa-Gerritsen model (Section 9) for the increase of the density of states at the Fermi level and one based on the suggestion made by Friedel101 and by Mottl02 of the existence of bound or localized electrons around the impurity with unpaired spins (Section 9). The groups Owen to Kip103 and Owen to Kittel 28 investigated the Cu- Mn system and one Ag-Mn 4.2 at.% and Mg-Mn 0.67 at.% (see Fig. 25). For T > 8, the positive Curie temperature, the Curie-Weiss law x = C/(T - 8) represents the results fairly accurately. As the temperature is lowered a gradual transition to a certain ordering appears, the maximum susceptibility occurring at a temperature slightIy higher than 8. Van Itterbeek et al. 104 extended the Ag-Mn and Cu-Mn research, while Hedgcock et a1.105 extended that on Mg- and Al-based alloys. In Table 1 a comprehensive survey of data is given collected from the publications of the above-mentioned authors. Van Itterbeek, Peelaers and Steffens also calculated the effective magneton number for Mn in Ag and in Cu and found an average number of 5.8 and 4.9 respectively, approximating the electronic configuration of the Mn atom most closely to 3d54s2and 3d64srespectively. In Figs. 26 and 27 also are plotted the References p . 259
224
G. J. VAN DEN BERG
[CH. IV,
$5
TABLE 1 Host metal
cu cu cu cu cu cu cu
cu cu Ag Ag Ag Ag Ag Ag Ag Ag Mg Mg Mg Mg Mg
*
Atomic percent of Mn 0.029 0.20 0.25 1.4 1.40 1.61 2.57 4.82 5.6 0.5 1.24 1.67 1.75 3.50 4.2 5.10 5.30 0.018 0.021 0.6 0.6 0.67
Curie temp.
Transition region
(OK)
TN(OK)
0 & 0.5 1-2
-
7 5 1 - 3.35 - 0.29 8.3 20 37 f.2 - 13.7 - 4.5 11.3 13 i 1 17.6 19
-
0 =t0.5 0 & 0.5 0 i 0.5
4 < 1.2? 10 - 15 4 1.9 - 4 10.6 11.5 30 - 60 < 1.2? < 1.2? < 1.2? 1.2 - 1.9 6 10 - 20 11.4 -
2 2 0-4
4.95 -
5.05 4.54 4.55 4.45 4.35 5.19
-
6.03 5.89 5.74 5.58 5.66 5.65 4.8* 5.5* 4.8' 4.1 3.55
room temperature.
data of Owen et al. 289103, Gustafsson'OB, Morris and Williams107, Valentiner and Becker108, NCel109, Scheil et ~1.110,Kronqvist et aZ.l11 and Myers112, also for higher concentrations of Mn. The theoretical value for a spin value 3 should be [b(s 1)]* or 5.92. The value 4.9 indicates a spin of about 2. Note also the spin value which could be deduced from specific-heat results by De Nobel and Du Chatenier33 (see Section 7). Starting from susceptibility measurements at temperatures higher than 90°K Scheil et aZ.113 calculated an average effective magneton number of 4.9 for Fe in Cu at concentrations between 0.57 and 2.16 at.%. This indicates a doubly charged Fe ion contrary to the low-temperature specific heat value of s = 4114 (see Section 7). The Curie temperature, found by extrapolation from the relatively high measuring temperatures (90" K), remained positive, which should indicate a "ferromagneticyy ordering for Cu- Fe 0.57 at.% below 18°K. In the case of Au-Fe the magneton number of Fe,
+
References p . 259
CH.IV,
0 51
ANOMALIES IN DILUTE METALLIC SOLUTIONS
225
deduced from measurements above 9WK, is also 4.9 for the smallest concentrations 115. Vogt and Gerstenberg59 measured the susceptibility of Au -Ti alloys with concentrations 1, 3 and 7 at.% Ti. A temperature-independent paramagnetism was found at 90°K and higher instead of a Curie-Weiss one. Several explanations have been given, but these appear to be incorrect or not applicable (see59p. 147 etc.). One may conclude that perhaps at 1 at.% Ti the 2 or 3 d-electrons per atom would already form a common band, while the d-electrons of solutes such as Cr, Mn or Fe represent localized magnetic moments. But here see Friedel's conceptionll6. Using his rule of thumb p * A E2 w , where p is the number of d-electrons or holes for Ti, A E is the energy difference between the up and down position of the spin of a delectron in the transition metal ion and w is the resonance width of the virtual bound state, it is clear that the elements with a small value for p (like Ti and Ni) may not satisfy this condition for a permanent magnetic moment, contrary to elements like Cr and Mn. For V in Au the possibility
1O-
yX 06
0:
0:
01
Fig. 25. Inverse of volume susceptibility x as a function of temperaturelo3.
References p . 259
226
[CH.IV, 5 5
G . J. VAN DEN BERG
A 103 0 104 -1.
108 109
0110 112
Fig. 26. The effective moment per Mn atom in Cu- Mn alloys as a function of atomic concentrationof Mn.
of satisfying the rule of thumb is greater. Vogt and Gerstenberg59 also carried out experiments on Au-V 1, 3, 7, 11 and 15 at.% alloys. The susceptibility increased with decreasing temperature and a part of it obeying a Curie-Weiss law was found with a negative value of the Curie temperature. The effective number of magnetons increased so steeply with decreasing concentration of V that Vogt and Gerstenberg expect the more diluted alloys (< 1at.x) to have 3 d-electrons responsible for the magnetic moment. This is comparable with the behaviour of Cr in Au, possessing probably 4 d-electrons. At higher concentrations of V, however, the behaviour can be compared with that of Ti in Au. An analogous transition from CurieWeiss paramagnetism to a temperature-independent one also occurred in
A 103 0 104 106 Q
107 109
0110 c) 111
Fig. 27. The effective moment per Mn atom in Ag-Mn alloys as a function of atomic concentration of Mn. Referencesp . 259
CH.
IV,§ 51
227
ANOMALIES IN DILUTE METALLIC SOLUTIONS
the Cu-Mn system above about 20 at.% Mn, when the probability of nearest neighbours for Mn ions strongly increases in the disordered cubic face-centered lattice Turning to Cu-Ni alloys one is tempted to make a comparison between these alloys and the mentioned Au-Ti ones. In the former the 3d-band of Ni is expected to have a small number of open places while in the latter it should contain a small number of unpaired spins. Experiments by Pugh et ~1.117,also on Ag-Pd, show that one has to be very careful with conclusions drawn from the representation of the susceptibility by aT b c/T. An impurity like Fe may play a significant role.
.
+ +
SPIN RESONANCE 5.2. ELECTRON
The Berkeley group Owen to Kip103 and Owen to KittelZ8 carried out electron-spin-resonance measurements in the temperature range 1.2- 300°K on Cu-Mn alloys with concentrations 0.5, 1.4, 5.6 and 11.1 at.%, on Ag-Mn 4.2 at.% and Mg-Mn 0.67 at.% using wavelengths of about 3.3 and 1.2 cm. In the temperature range above the ‘‘Nee1 temperature”, in the paramagnetic region, the intrinsic g-value is very close to the free spin value 2.00 and the single absorption line is very close to the free spin resonance field H, after correction for skin effects. Below the “NCel temperature” the position of the line can be described by H = ( H i - H:)*,but not for the very dilute alloy (clat.%), where H, is a parameter which increases as the temperature decreases. This condition resembles the form predicted by the antiferromagnetic resonance theory of Kittell18 and Nagayima et ~1.119,but more detailed experiments appeared to be inconsistent with this theory. The results are approximately consistent with the Mn having a 6 S , ground state, as would result from Mn2+, 3d5, though there are discrepancies. The effective spin per Mn ion appears to be rather less than S = 3. The mentioned 6S, ground state has been denied by Collings and Hedgcock105 for Mn in Mg, who based their objection on susceptibility measurements. These authors carried out resonance measurements on a Mg- Mn 0.60 at.% alloy of which the electrical resistance versus temperature curve was previously determined and showed a maximum between 6 and 7°K. A line broadening in the region of the resistance minimum was found. In the region of the maximum the electron spin resonance and magnetic susceptibility results indicated an antiferromagnetic transition. The apparent shift of the resonance field is zero above about 6.5”K, but increases steeply below this temperature, being inversely proportional to the temperature below 3.5”K. Above 6.5”K the g-value for E.S.R. was 2.00. References p . 259
228
0.J. VAN DEN BERG
[CH. IV,
55
Electron-spin-resonance measurements were also carried out on A1- Mn 1 at.% and on A1-Fe and Mg-Fe alloys, but no signal could be detected even at 4.2" K. The spectroscopic state of Mn in Al appeared to be different from that in Mg, as may be expected from experiments mentioned elsewhere in this chapter. This may also be so for Fe in Al, but Fe in Mg gave an extra complication. Collings, Hedgcock and Sakudo 120 did not succeed in
1
Fig. 28. The line width A
- ('K) T
2
O1
= A,,,,
v : H e = 8.40 kOe;
10
of Cu-Mn 0.011 at. % as a function of temperatureIz1. He = 5.83 kOe; 0 He = 3.56 kOe.
+
detecting an electron spin resonance in Au-Fe 7.06 at.% or Cu-Fe 4.55 and 0.55 at.% at 77 and 4.2"K, or in Cu-Co 0.57 at.% at 77°K. The absorption found for the Cu-Fe alloys at room temperature must be due to precipitated iron, as may be expected for those large concentrations. Keeping the similar behaviour of the electrical resistance of Mg- Mn and Au-Fe in mind, one may be astonished at the deviating behaviour of the alloy Au-Fe as regards spin resonance. MAGNETIC RESONANCE; KNIGHT SHIFT 5.3. NUCLEAR
A study of the Knight shift of the Cu nuclear spin resonance in these alloys was undertaken as an independent check on the magnitude of the conduction electron magnetization, this being proportional to the Knight shift. Starting from a simple molecular field model Owen ef at.28 expected to find a four times larger Knight shift of the Cu nuclear resonance at 1.2" K References p . 259
CH. IV,5 51
229
ANOMALIES IN DILUTE METALLIC SOLUTIONS
in a 0.03% Mn alloy than in pure Cu. Their experiments and those by Van der Lugt et al.'21 and by Sugawara122 did not give the expected results. Hart123 gave an explanation for this absence of an extra Knight shift, remembering of the remarks by Friedell01 concerning the shielding off of
x -
O1
4
2
T
10
20
(OK)
Fig. 29. The line width A = d m m of Cu-Mn0.026 at. % as a function of temperatureIz1. 0 : He = 3.56 kOe. : He = 8.40 kOe; : He = 5.83 kOe;
+
the Mn ions in the Cu solvent (see also Section 9). The line width A,, of the resonance line increases when the temperature decreases, and the negative slope of the curve A,, vs T is an increasing function of the Mnconcentration. For every concentration the slope depends on the static magnetic field used during the measurements (Fig. 28, 29 and 30). The broadened resonance line is somewhat asymmetrical and has approximately the Lorentz shape, in contrast to that for pure Cu which has nearly a Gaussian shape. As calculations by Behringerl24 and by Van der Lugt121 showed, it is impossible to attribute the large line widths only to dipolar interaction. From the calculations by Yosida125, who used a RudermanKittell26 indirect spin-spin interaction, one can deduce that a broadening and, in first approximation, no shift should occur. The broadening is apReferencesp . 259
230
[CH.IV, 4 5
G. J. VAN DEN BERG
projrimately one third of the experimental value l2I. Experimentally could be demonstrated (Sugaware122,Van der Lugt et al. 121,Chapman and SeyI I I O U ~ ~ ~that ’ no line broadening occurred as compared to pure Cu in
i 0 2 4 20
01
-
10
40
r ( 0 ~ )
Fig. 30. The line width d = dmmof 01Mn 0.066 at. % as a function of temperature 181. :He = 5.85 kOe; 0 :He = 3.59 kOe; : He = 8.53 kOe; -I- :He = 6.85 kOe; : He = 1.76 kOe.
+
“diluted” Cu-Ti and Cu-Ni alloys, just like in Cu-Zn, Cu-AI and Cu-Ag. To what extent the line broadening at low temperatures in Cu-Sn and Cu-Co is due to the less than 70 ppm Fe that may be present in these alloys is unknown. 5.4. THEDEHAAS-VAN ALPHEN EFFECT The amplitude of this effectis related to the relaxation time of the conduction electrons. In order to determine whether an energy-dependent relaxation time would account for any anomaly which might be found in the De HaasVan Alphen oscillations, Hedgcock and Muir65 made a study of this effect in a Zn-Mn alloy which exhibited a resistance minimurn49*60. Zn was References p . 259
CH. IV,
$51
ANOMALIES IN DILUTE METALLIC SOLUTIONS
23 1
known to exhibit a large-amplitude De Haas-van Alphen effect 128. The experiments were carried out in magnetic fields up to 8 kOe and in the temperature interval 1.6-4.2”K. The periods of the oscillations in pure Zn and in the alloy Zn-Mn 0.010 at.% were found to be the same within the 1.5% experimental error. However, the variation of the amplitude of the oscillations with field and temperature was found to be anomalous in the alloy. In order to explain this anomaly Hedgcock and Muir investigated the effect of an energy-dependent relaxation time on the De Haas-van Alphen oscillations. A relaxation time, that was assumed to be zero in the energy range E, - kT, < E < E, + kT, and to be constant elsewhere, was found to fit the experimental results, with Td = 0.75”K.The electrical resistance was calculated from the expression given by Korringa and Gerritsen l29 for a relaxation time having the above behaviour. This calculated resistance was found to agree within the experimental error with the measured one.
5.5. MAGNETIC REMANENCE The alloy Cu-Mn was the object of study concerning both the susceptibility and the remanence. The magnetization experiments by Schmitt and Jacobs48 on alloys with Mn concentrations of about 0.05, 0.2, 0.4, 1.0 and 1.8 at.% showed clearly that those samples with a concentration of 0.4 at.% and more exhibited hysteresis in the liquid helium range. The susceptibility of those samples goes through a maximum at a temperature near to the Curie point, in agreement with the findings of Owen et al. 103. The temperature of this maximum is near or slightly below that at which the hysteresis first appears. The existence of the maximum was interpreted as a gradual transition to antiferromagnetism (Berkeley group), but the evidence linking the maximum in the susceptibility to the presence of hysteresis and remanence also can be interpreted by the short range order picture of small “ferromagnetic” domains coupled “antiferromagnetically” 130. Lutes and Schmit 131 studied the magnetic remanence of Cu- Mn alloys with concentrations 0.47, 1.04, 4.6 and 10.0 at.% Mn by field-cooling. Only the first-mentioned concentrations should be considered (“diluted”). Roughly speaking, the isothermal saturation field is proportional to the concentration over the absolute temperature : HI x 3.5 x 103c/T, when HI in kOe. The temperature dependence of the saturated remanence is evidently different from that associated with spontaneous magnetization in a ferromagnet. It is similar, instead, to the temperature dependence of paramagReferences p . 259
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G. I. VAN DEN BERG
[CH. IV,
85
netic susceptibility. This suggests a comparison with the Brillouin function B,(gJp,H/kT) (see Fig. 31). The following relation is assumed: = BdgJpBHO/kT)
MS/MS,O
where Ms,o = saturated remanence at 0” K ; J = 2; g = 2; Ho = eff. internal field. Ms,o must somehow represent the unbalanced moment between the spins, oriented parallel and anti-parallel to the previously applied field. For the atomic concentrations of 0.47 and 1.04% the values of Ms,o in %
Fig. 31. The saturated remanence, M,, as a function of temperature for Cu-Mn 0.47 and 1.04 at. %. Lutes and SchrnitI3l gave M. in units of galvanometer deflection per gram. The solid curve is the paramagnetic relation for J = 2, MS,O= 0.808 mm/g and H = 5.30 kOe. The dashed curve is this relation for J = 2, MS,O= 0.251 mm/g and H = 3.40 kOe.
alignment are 1.9 and 2.7 respectively. The effective internal field values are 3.4 and 5.3 kOe. One possible explanation of the remanence might be the statistical variation of the population between domains, ferromagnetic domains “antiferromagnetically” coupled. Lutes and Schmit 131 calculated the domain size or better the linear dimension of a domain, ID, by means of: ID
= [~/NP&,Ol+
where A = atomic weight p = density N = Avogadro’s number Ms,o is expressed as fractional alignment. Referencesp . 259
CH.IV,
0 61
ANOMALIES IN DILUTE METALLIC SOLUTIONS
233
The domain size appears to become smaller at higher concentrations and cm for 1.0 at. % Mn in Cu. the linear dimension is of the order of Kouvel132 investigated Cu-Mn 5-30 at.% and Ag-Mn 10-25 at.% alloys. Magnetic hysteresis loops displaced from their symmetrical positions about the origin were observed when the specimens were cooled to 13°K in a magnetic field. The susceptibility x reached a maximum at a temperature T, which is much lower than the temperature T, of the anomaly in the R-T curve, indicative of a magnetic transition. At T, there appeared kinks in the I/x us T curves. A microscopic exchange anisotropy model is qualitatively consistent with the experimental results (see Section 9).
6. Hall effect Few low temperature experiments on this effect have been done on the alloys treated in this chapter: Blue133 has measured the Hall coefficient of “diluted” alloys of Mn, Fe, Co and Ni in Cu but only at room temperature and 77°K in magnetic fields up to 10 kOe. 6.1. NOBLE-METAL BASED ALLOYS
For Au-Cr 0.03 and 0.05 at.% Teutsch and Love1s4 found a monotonic increase of the coefficient at very low temperatures and they denied an anomaly corresponding to the resistance minimum in Au and the mentioned alloys. This is in contradiction to the interpretation by Fukuroi and Ikeda 35 of the results of their measurements on an unannealed gold wire. An explanation for this contradiction has been given by Gaidukov l35, starting from the results for a gold wire with a normal behaviour as to the electrical resistance together with a field-independent Hall coefficient and for a gold wire with an anomalous R - T curve whose Hall coefficient is field-independent below 8 kOe. Teutsch et al. measured in a field strength smaller than 8 kOe and Fukuroi et al. did it in a 20 kOe magnetic field. Franken et al. 34 investigated Ag- Mn and Cu- Mn alloys and for comparison pure Ag and Cu samples as well as Cu-Sn and Ag-Au alloys. For the pure metals the Hall coefficient was also field dependent in low fields with great variations from sample to sample, perhaps due to the differences in texture of the polycrystalline rolled material 136. However, below 10°K the field dependence disappeared. A large influence of the degassing (melting in high vacuum) was observed. Unannealed samples show a field dependence only at higher fields (10 kOe) for the Hall coefficient which is itself relatively large at low temperatures. Ag-Au and Cu-Sn alloys, measured for comparison with “diluted” alloys of Mn, did not show References p . 259
234
[a. IV, 4 6
G. J. VAN D E N BERG 1.4
12
1.0
I
I 100
OK
loo0
Fig. 32. The relative Hall coefficient R/R290 as a function of temperature for diluted alloys34. El Cu-Sn 0.04 at. % (annealed), v Ag-Au 0.76 at. % (not annealed), 0 Ag-Au0.27 at. % (not annealed), A Ag-Au 1.8 at. % (not annealed).
F 1.0
kOe
Fig. 33. The relative Hall coefficient, R/Rzoo as a function of magnetic field for the alloys Ag-Mn 1 .O and 4.2 at. % (annealed)34. 0
El References p. 259
T = 1.3"K,
4.2"K,
2.5"K,
A 1O0K,
c)
12'K,
0 14°K.
@ 20"K,
CH. IV,
8 61
235
ANOMALIES IN DILUTE METALLIC SOLUTIONS 1.3,
1.2
1.1
1.1
Fig. 34. The relative Hall coefficient, R I R m , as a function of I ~ f p / p l * H -for ~ the alloys Ag-Mn 1.0 and 4.2 at. %34. 0 I .3" K, decreasing field H, 4.2"K, decreasing field H, @ 1.3' K, increasing field H, V 4.2"K, increasing field H .
e l 1.01
A
I
I
5
10
kOc
I
15
Fig. 35. The relativeHalIcoefficient,R / R m as a function of magnetic field for diluted alloys Ag-Mn 0.01, 0.09 and 0.3 at. %s4. 0 T=1.3"K;.1.7'K; D2.5"K; v4.2"K; A 10"K ; 0 14" K and @ 20" K.
a field dependence. The temperature dependence of the Hall effect of such alloys is represented in Fig. 32. For Ag-Mn 1.0 and 4.2 at.% a field dependence has been observed together with hysteresis (see Fig. 33). This brought Franken to a splitting up of the Hall coefficient into a normal part and an extra part proportional to the magnetization. In formula :
R H = R,H
+ R,I
(8)
where R = measured Hall coefficient, R, = normal Hall coefficient, R , = extra Hall coefficient, I = magnetization and H = external magnetic field. When the Curie-Weiss law is still valid for the alloys, one dan write eq. (8) as R = R, R,x = R, R,,,Cf(T- O ) , where x = C/(T - 8). One can expect the R, (normal coefficient) to be approximately temperature-independent. The term in R due to the magnetization will be-Pome more important as the temperature decreases and will become more or less
+
References p.259
+
236
G. J . VAN DEN BERG
[CH.IV,
86
constant at the temperatures where the susceptibility of these alloys reaches its maximum as a function of T. This will depend on whether the maximum in the x us T-curve is flat or not (Mn concentration). Using the quadratic relation between the magnetoresistance and the magnetization predicted by Korringa and GerritsenI29 and observed by Schmitt and Jacobs for Cu-Mn4*, one can derive R = R, R' ( d p / p l * H - ' ; this relation was found to be obeyed by plotting R against Idp/p13H-' (see Fig. 34). An extrapolation of the straight line gives an acceptable value for the ratio of the normal Hall coefficient to that at room temperature. Experiments on Cu-Mn alloys, with concentrations between 0.01 and 1.1 at.% did not give results, which could be used for a representation as in the case of Ag-Mn 1.O and 4.2 at.%. Probably the mentioned concentrations in the Ag- Mn alloys were sufficient for some local long range ordering (see Section 9). The field dependence for the Ag- Mn alloys with smaller concentrations (0.01-0.3 at.%) has been plotted in Fig. 35, while the temperature dependence can be read from Fig. 36.
+
Fig. 36. The relative Hall coefficient as a function of temperature for annealed diluted alloys of Cu- Mn and Ag- Mn34. @ : Cu-Mn 0.1 at. % 0: Cu Mn 0.01 at. % 0 : Ag-Mn 0.01 at. % : Ag-Mn 0.3 at. % for H = 0 kOe c) : Ag-Mn 0.09 at. % :Ag-Mn 1 at. % A :Ag-Mn 4.2 at. % for H = 10 kOe.
-
References p. 259
CH. IV, 5
61
ANOMALIES m DILUTE METALLIC SOLUTIONS
231
The high field dependence has also been found by Alekseevskii et al.I37 in strongly diluted Au-Fe alloys, without however a pure metal field dependence. A remarkable simple form for the Hall voltage us H-relation
Fig. 37. The Hall field Ey as a function of magnetic field at T
=
1.45" K137.
was found by him and his co-workers (Fig. 37). For several temperatures between 0.07 and 295°K such a curve was found, while the change in the slope of the curve starts at about 8500 Oe. At the same field strength there is a change in slope of the curve for the magnetic force against the value of HdHldx. One sample, which does not show a minimum in the resistance-temperature curve does not show the change in slope at 8500 Oe in the curve of the Hall voltage Ey against the magnetic field H either, indicating a field independence of the Hall coefficient. The value of 8500 Oe is just that of a magnetic field needed to eliminate the minimum in the R us T-curve. 6.2.
TRANSITION-METAL BASED ALLOYS
For the alloys with the transition metal Ti as solvent field independence of the Hall coefficient was found66 for all those measured with Cr, Mn and Fe as solutes, if the temperature was 77" K or higher. The field independence broke down at lower temperatures for the alloys Ti-Mn 1.00 at.% and 2.01 at.%, but not for the Ti-Fe and Ti-Cr alloy, nor for a Ti-Mn Referencesp . 259
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[CH.IV,8 7
0.J. VAN DEN BERG
0.114 at.% alloy, for which however the Hall coefficient was much larger (negative) than that for the base metal Ti at all temperatures. Taking into account in Ti the ordinary “Pauli” susceptibility xp, one might write for the total magnetization, if a localized moment magnetization IL existed : I = I p + IL = X p H
+ a(14*l/df
(9)
or substituted in equation (8): R = R,
+ RJ,, + RA( Idp,l/p)*H-’.
(10)
Extrapolation from the high-field linear curves in a plot like Fig. 34 yields values for the sum of the first and second term of the right-hand side of equation (10) of approximately - 20 x lo-’ cm3/C for both Ti-Mn 1.00 and 2.01 at.%, being about double the value of that for the base metal Ti at T < 4.2”K.
7. Specific heat As mentioned in Section 1, measurements on the specific heat on the alloys concerned have been started for two reasons: a) the determination of the “magnetic” contribution to the specific heat; b) measurement of the change in the electronic specific heat as an indicator of the change in the density of states. The first measurements were carried out on a Ag-Mn 0.09 at.% specimen138. It was impossible to represent in the usual way the specific heat by C = a(T/B) + yT, because of an anomaly in the C - T curve. An extrapolation of the line for C/T vs T2could not give the value for y, the electronic rn rnJ mole dug2
24
12
20 16 12
8
$1 0
T
e ~ q - ~I on at Aq-Mn
-2
4
6
8
% 10 ‘K
Fig. 38. The ratio of specific heat to temperature, C/T, as a function of temperature for Ag-Mn 139.
References p . 259
CH.IV,
8 71
ANOMALIES IN DILUTE METALLIC SOLUTIONS
239
specific heat coefficient. The hump in the C/Tus T curve (Fig. 38) could be described roughly as of a Schottky type, assuming the 6 S level for Mn to be split up into equidistant levels. Fictitious internal fields calculated for the lower concentrations with the help of an adaption of Schottky curves appeared to be much higher than expected from the deviations from Curie’s law. The area between the curve in the alloy and that for pure Ag was approximately 0.09 Rln6 or cRln(2S l), indicating a spin value of 3 for the Mn ion. Crude extrapolation of the curves was needed for this calculation, but more recent results confirm the value 3. With increasing magnetic field the hump in the C/T vs T curve became lower while the just-mentioned area remained approximately constant. This resembles the results of the measurements by Meyer and Taglang140 on the specific heat of the compound Au,Mn. A series of “diluted” alloys has been investigated in the last years, and especially in more recent years the scheme published by Friedel and coworkerslle (Section 9) was taken as a guide. After the alloys Ag-Mn33, Cu- Mn33,141,142, Au-Mnl41,Ag-Crl41,Au-Cr141, C~-Cr141,C~-Fe114>13D, Cu-Co143, Au-Co144, Cu-Nil41.145, the alloys with the parent metals Mg, Znand A1 were measured: Mg-Fe146, Mg-Mn146*147, Zn-Cr141, Zn-Mn141, Zn- Fe 141, Al- Cr 141, Al- Mn148 and Al-Fe 141. Measurements of the temperature and composition dependence of the elastic constants of “diluted” alloys of Mn in Cu, carried out by Waldorf149, did not indicate an anomalous lattice contribution to the observed specific heat larger than 0.5% per at.% Mn. As can be read from Table 2, representing Friedel’s scheme (Section 9), the specific heat of A1 alloysl4l?148 does not show an anomaly as a function of temperature, in agreement with the normal behaviour of the electrical resistance of these alloys (see Section 2). In the case of Zn as a parent metal Cr and Mn dissolved in small quantities cause an anomaly but Fe does not. Martin147 investigated the Mg-Mn 0.025 and ca 0.15 at.% alloys between 0.4 and 1.5”K and found an abnormal behaviour. The maximum in the specific heat of the more concentrated alloy is probably in the same region as the maximum in the electrical resistance (see Fig. 11). The maximum of the specific-heat anomaly for the more diluted specimen (0.025 at.%) appears to lie below 0.4”K, and it is interesting to speculate whether the maximum in the electrical resistance might also be found in this temperature range. The measurements of Hein and Falge64 of the electrical resistance of Mg-Mn 0.043 at.% below 1°K to as low as 0.22”K did not indicate such
+
Referencesp. 259
240
G . J . VAN DEN BERG
[CH. IV,
57
T A B L E2 Friedel's scheme EF in eVI 5.5
I cu
1.5
Mg
9.5 Zn ~
2 3
4 3
sc Ti V Cr Mn Fe
2
co
1
Ni
4 5
13 A1 -~
+- resistance shows an anomaly
resistance shows no anomaly 0 specific heat anomaly observed by Du Chatenier and De Nobel 0 specific heat anomaly observed by other investigators A no specific heat anomaly observed by Du Chatenier and De Nobel A no specific heat anomaly observed by others ? dubious anomaly.
evidence. Logan, Clement and Jeffers l 4 6 measured a normal specific heat us temperature curve between 3 and 13°K for the same concentration of Mn and for the alloy of Mg with as the principal impurity 0.013 at.%Fe. Concerning the Cu-based alloys, it can be remarked that Cr 141, Mn 33~141,142 and Fe1149139certainly cause an abnormal specific heat us temperature curve in agreement with the electrical resistance anomalies, which are not quite the same. Franck, Manchester and Martin 114 calculated a magnetic entropy content per gramatom Fe of 1.3, 1.22 and 1.17 cal/"K, which values are close to Rln2, in agreement with Leiden results. Therefore they propose that the anomalies may result from a thermal excitation process between two energetically separate states of each soluted atom. Some confirmation for this assumption may be obtained from the results of magnetic susceptibility measurements on very diluted alloys of Fe in Cu150, which when reinterpreted in the light of more recent theory151 lead to a spin of for the Fe ions in diluted solutions at low temperatures (assuming complete quenching of the orbital magnetic moment). However, the results of these susceptibility measurements must be treated with some caution 98. Here see also experiments by Scheil et ~1.113resulting in an effective magnetic moment
+
References p . 259
CH. IV,
0 71
ANOMALIES IN DILUTE METALLIC SOLUTIONS
241
of 4.9 for Fe soluted in Cu or a spinm2 and to Mossbauer effect experiments giving a spin value larger than one. (For the impurity Cr such a difference in spin value was not found; the spin value for Mn in Cu was 2 contrary to 4 from the specific heat experiments.) The just-mentioned anomalies in the specific heat curve of Cu-Fe alloys cannot be represented by a simple Schottky anomaly, but by a sum of simple Schottky anomalies corresponding to a whole spectrum of energy gaps. Here should be remembered the rather good approximation for the Ag-Mn 0.28 at.% with
:
4 1
9
3
“I__________ 0
I
-p_.!?c”-
ec
- - -- - - -
2* ( D o g 2 )
Fig. 39. The ratio of specific heat to temperature, C/T, as a function of T 2for Cu-Mn alloys 142.
6 fixed energy levels of the split 6 S level for Mn (3). An “internal” field working on the Mn ions of the order of 25, 30 and 35 kOe could be calculated from the best fitting Schottky curve using data obtained in an external field of respectively 0, 4.5 and 14 kOe. As to the Cu- Mn alloys there are at present experiments by Zimmerman and Hoarel42 as well as by Du Chatenier and De Nobel33.141. The firstmentioned authors thought it possible to conclude from Fig. 39, that “the anomalous specific heat (that of the alloy minus that of pure Cu) at very low temperatures is nearly linear in T,and furthermore, is nearly independent of Mn concentration”. Overhauser 152 gave an explanation starting from his concept of spin density waves, but Marshall153 had serious objections to this concept of antiferromagnetism. The last-mentioned author gave another explanation based on Yosida’s interaction125. In order to decide if the conclusion of Zimmerman et al. from their experiments above about 2°K was correct. Du Chatenierl4l and Miedema measured the speReferences p . 259
242
[CH.IV,5 7
G . J. VAN DEN BERO
cific heat of some Cu-Mn alloys below 1°K. No exact conclusion could be obtained because of the steep rise of the specific heat due to the interaction of the electronic and nuclear moments of the Mn ions. The total specific heat could be represented approximately by:
C = 0.018/T2 + 3.06TmJ/mole0K for Cu-Mn 0.15 at.% and C = 0.13/TZ + 3.06TmJ/mole°K for Cu-Mn 1.0 at.%. From the first coefficients one can deduce a splitting parameter of 0.0238' K for Cu-Mn. A much better fit could be obtained for two Au-Mn alloys (Fig. 40) with 0.08 and 0.12 at.% Mn by the formulas C = 6.6 x 10-3/T2 + 7.8T + C,,mJ/mole'K C =10.5 x 10-3/T2-I-7.8T+C,,mJ/mole"K
for c = 0.08 at.% for c = 0.12 at.%.
Here the splitting parameter is 0.0173"K. In order to obtain a wider range for the test of the dependence on T, Du Chatenier and Miedema carried out some measurements of the specific heat of two Cu-Cr (0.1 and 0.6 at.%) alloys, only 1% of the nuclei of the solute having a magnetic moment. The hyperfine structure of the 1% of the nuclei gave a contribution wich was inobservably small. Here the specific heat can be represented by: C = 2.5T mJ/mole'K for both concentrations. 1oc
%-
mole K
10
1
0 '
I
/
-
Fig. 40. The specific heat of Au Mn as a function of temperature. 0,b:0.083at. % a:puregold. A,c:O.I6at. %
References p . 259
CH.IV,
71
ANOMALIES IN DILUTE METALLIC SOLUTIONS
243
The suggestion154, that there are two classes of alloys, one exhibiting only a minimum in the R-T curve together with a concentration-independent temperature of the maximum in the specific heat curve and another class showing a minimum as well as a maximum in the R -Tcurve together with a concentration-dependent temperature of the maximum in the specific heat curve, seems not to be quite correct (e.g. Zn - Mn alloys do not obey this rule). Continuing the discussion of Friedel’s scheme, one meets the normal behaviour of Cu-Ni alloys, while the Cu-Co alloys probably behave normally at low concentrations. Crane and Zimmermanl43 found anomalies for higher concentrations of Co, which may be due to impurities like Fe or, in comparison with experiments by Friedberg et ~1.145on Cu-Ni alloys and by Weil et aZ.155 on Cu-Co alloys, due to fluctuations156 in the concentration. The onset of the influence of hyperfine structure on the specific heat for these higher concentrations seems to be excluded by the results on a quenched specimen Cu-Co 2 at.%. Experiments on the electrical resistance of Cu-Ni alloys do not show any abnormal behaviour if the materials are pure enough neither do those on Cu-Co. The Ag and Aubased alloys with Mn and Cr behave anomalously because of the localized magnetic moment, but for Co as solute144 this is again uncertain. One might argue this among other things with the experiments of Le Guillerm, Tournier and Weill57 on the magnetization of Cu-Co 0.6, 1, 2 and 3 at.% and of Au-Co 2, 3 and 4 at.%. For concentrations of Co higher than 1% the ratio of the Curie constant to that value which this constant would eventually take if in the alloy the Co atoms were isolated and had a magnetic moment equal to that in massive Co ( p = 1.59 x lo-”) increases. This indicates that the Co atoms are associated. The experiment by Kobayashi158 and those by Becker159, by Bean and LivingstonelG0and by Mituilsl also sustain the opinion that Co in concentrations of more than 1 at.% in Cu precipitates during isothermal aging, while the particle size increases with increasing time, giving rise to a superparamagnetism. Gaunt and Silcoxl62 made an electron microscope study of the Co particles in a water quenched Au-Co 1.5% alloy and Cu-Co 2.4% alloy and observed an approximately spherical form in the former and a buttonlike one in the latter alloy. In order to have correct values of the specific heat of the parent metals for comparison with that of the “diluted” alloy some authors measured the value of this quantity for pure metals. The results may be represented by: C = yT u ( T / O ) ~ “[/?(T/O)’]’’ and the values of y and 0 are given in Table 3.
+
References p. 259
+
244
G. I. VAN DEN BERG
[CH. IV,
Q8
T A B L E3 Values of the electronic specific heat coefficient, y, and the Debije temperature at 0" K, 00, for some pure metals
N.B.S. Ford. Dearborn
Metal Au Ag
Cu Mg
y
eo y
eo y
eo y
0.74 165 0.610 225 0.687 344.5 1.32
0.740165 164.6 0.690 l 4 2 344
-
eo 406 Zn
A1
y
0.63
eo 300 y
1.36
0.658 lfi5 336 -
6'0 426
N.R.C., Ottawa 0.646 166 226.2 O.69O1l4 344 1.23 167 110 440- 56 -
+
-
1.34148*
K.O.L. Leiden O.74O1*l 165.2 O.682l4l 226.2 0.721 141 338.9 -
mJ/mole degz "K mJ/mole degz "K mJ/mole deg2 "K mJ/mole degz "K
0.65 299 1.39 * 420
mJ/mole deg2 "K mJ/mole degz "K
N.B.S. = N.B.S. Monograph 21, R. J. Corruccini and J. J. Gniewek, 1960, a compilation from the litterature up to 1960,* devired from measurements of alloys. According to Garland and Silverman lfi3the best fit for Zn can be obtained by C = yT e(T/e)3+ B (T/B)5 with y = 0.65 mJ/rnole deg2 and 60 = 322 O K: a = 1944 J/rnole deg. j? = 353.8 x 103 J/mole deg. Seidel and Keesom'sproposal issomewhat differentifi4.
+
+
8. Optical properties Abelks168 obtained the optical constants of Au- Ni alloys from measurements of the reflectance and transmittance of films of the order of 500 A thick, prepared by evaporation in vacuum. An absorption band in the infra red was found, which shifted towards shorter wavelengths as a function of the Ni concentration. The intensity of the band appeared to be a linear function of the concentration below 10 at.%. For Au-Sn and Au-Ge alloys the existence of absorption bands in the infrared could not be demonstrated. The extent to which the study of optical properties may provide some information regarding the virtual bound states (see Section 9) depends on a theoretical study of the relation between the existence of these states and that of the absorption band. For the Au-Ni alloy one should not expect these states to be effective. Caroli l69, however, calculated very recently the near infrared absorption for Au-Ni alloys with a concentration between 2 and 12 at.%, starting from Anderson's model (see Section 9). For the alloy Au-Ni 7 at.% Caroli finds agreement between the values for the width in energy of the virtual bound states derived from Referencesp. 259
CH. IV,
8 91
245
ANOMALIES IN DILUTE METALLIC SOLUTIONS
the optical data via the expression for the infrared absorption and the values derived from the residual resistance increase.
9. Theoretical considerations Reviewing the experimental results as to the different properties mentioned in the preceding sections, it is clear that a theory on the diluted alloys in question has to explain the following points: 1. the abnormal behaviour of the electrical resistance at low temperatures, the occurrence of a single minimum as well as that of a minimum followed by a maximum at lower temperatures, 2. the occurrence of a negative electrical and thermal magnetoresistance, for the alloys with a maximum in the R-T curve, 3. the occurrence of a “giant” thermopower at low temperatures, 4. the existence of a transition from a paramagnetic to a more or less antiferromagnetic state as indicated by the flat maximum in the susceptibility us temperature curve, 5. the broadening of the nuclear resonance line without a considerable Knight shift, 6 . the abnormal De Haas-Van Alphen effect as to the amplitude of the oscillations, 7. the variation of the Hall coefficient as a function of the temperature, 8. the occurrence of a magnetic entropy, manifesting itself in the specific heat data of certain alloys.
9.1.
RESONANCE HYPOTHESIS
Chronologically Korringa and Gerritsen 129 were the first who presented a theoretical explanation for the just-mentioned points 1 and 2. At that time the incorrect interpretation of the minimum in the R-T curve of “diluted” alloys with non-transition metals was still in use and was taken therefore into account by these authors. The model of independent conduction electrons was modified by admitting interactions between them. Normally the electrical resistance is described by a collision time z ( E ) , the time between two collisions, which is a function of the energy E measured from the Fermi level. One can then describe the abnormal resistance curve by this model supposing z = 0 in a small energy interval of width A around E = El and z = zo at other energies. A proper choice of E, (e.g. 1.54 kT,,,) and A has to be made in order to fit the experimental curves. According to this interpretation a sort of re-
*
References p . 259
246
0.J. VAN DEN BERG
[CH.IV, 0 9
arrangement scattering of the electrons would be caused by any impurity, a resonance peak occurring just near the Fermi level. No justification of this supposition was however presented. The authors further postulated certain localized electron states split by the magnetic and exchange interactions, so that the components lay just above and below the Fermi level. This phenomenological model could explain the dependence of the temperature of the maximum in the R - T curve on the concentration, the dependence of the magnetoresistance on the square of the magnetization48 and the possibility of the negative magnetoresistance. The natural interpretation is that the scattering probability of the conduction electrons depends largely on the relative orientation of the spin of e.g. the Mn-ion and the scattered electron (see below: Yosida). As Ziman20 pointed out later on, this phenomenological theory could also account for the anomalous thermoelectric power. Domenicalil70 proposed afterwards a model very similar to that of Korringa and Gerritsen, except that his “high-selectivity scattering model” made use of only the general analytical characteristics of the relaxation time z, and did not specify the details of the scattering mechanism. Hypothetical resistivity curves for gold alloys in agreement with the experimental ones could be obtained for 4 well chosen parameters related to the high-temperature resistance and the height, the width and the separation of the resonance curve.
9.2. MOLECULAR FIELD MODEL S ~ h m i t t l 7tried ~ to explain the occurrence of the maximum in the R - T curve by a model suggested by Fisherl30, assuming a sort of cooperative phenomenon of the kind proposed by Zener for the conception of ferromagnetism. If a “diluted” alloy of a transition metal in a noble metal becomes ferromagnetic, the spin degeneracy of the ground state of the impurity atom will be removed by the local field. Scattering of the conduction electrons by the impurity atoms can take place by inelastic as well as by elastic collisions (Korringa-Gerritsen considered elastic ones). The scattering cross section for elastic scattering of the conduction electrons will be different for each of the resulting levels ; the lower-lying non-degenerate states will have larger scattering cross sections. Inelastic scattering will occur as a consequence of transitions between the levels. The scattering by the Mn ions and consequently the extra resistance will depend on temperature, the filling of the levels as well as the Fermi distribution of the conduction electrons being temperature dependent. These considerations result in a maximum Referencesp . 259
CH.IV,0
9J
ANOMALIES IN DILUTE METALLIC SOLUTIONS
247
in the R-T curve and in the linear dependence of T,,, on the concentration c, but they do not account for the disappearance of the maximum in the curve for higher concentrations (c 2 1 at.%). The essential feature of Schmitt’s picture is the removal of the spin degeneracy of the ground state of the impurity ions by a local field. However Coopersmith and Brout 172 derived recently, for an interaction assumed to be of the Ising variety, the threshold concentration co for the existence of ferromagnetism in “diluted” alloys expressed in the number z of nearest neighbours in simple cubic, b.c.c. and f.c.c. lattices by the relation coz M 4, where Sat0 et al. found according to the cluster variation treatment coz > 2. This indicates for Cu-Mn alloys a threshold concentration of 16 at.% for the existence of ferromagnetism (long range ordering). Owen to Kip103 proposed later on a model with two sub-lattices A and B in such a way that short-range interactions give “ferromagnetic” (parallel) coupling A-A, B-B and “antiferromagnetic” (anti-parallel) coupling A-B. A model with both couplings interchanged should also be applicable. Schmitt and Jacobs48 used for their phenomenological considerations in principle the same model of a molecular field. For the anomaly in the electrical resistance they supposed an inelastic scattering process and for the magnetic anomalies a spin-orientation-dependent elastic one. At zero temperature and in the paramagnetic state inelastic scattering is absent. In a limited temperature range the inelastic scattering contributes to the electrical resistance. The numerical calculation by Schmitt and Jacobs resulted however in a R- T curve without a maximum and a minimum, which is only in agreement with Cu- Mn alloys of a concentration of about 1 at.% Mn. Inelastic scattering does not suffice for an explanation of the anomalies in the magnetic properties. The experimental result that A p / p K 1’ imposes certain restrictions on the type of magnetic ordering that can be present. Schmitt and Jacobs48 believed that from this experimental result they could infer: within any volume of dimensions comparable to the electron mean free path, the net magnetization in the virgin alloy is zero. As a model they used the above-mentioned one : “ferro-magnetic” domains, which are “antiferromagnetically” coupled, together with a spin dependent scattering. The authors arrive for Cu-Mn at the right sign for the magnetoresistance and at the observed dependence thereof bn the magnetization. As indicated in Section 5.3, no Knight shift was found in Cu-Mn alloys, which was expected by Owen et ~ 1 . 2 8starting from an indirect interaction via the conduction electrons between Mn ions and Cu nuclei of the form References p . 259
248
G . J. VAN DEN BERG
[CH.IV,
09
A Ses, where S is the spin of the Mn ion and s is the nuclear spin of Cu. Yosida 125 reached a better agreement with the experiments by taking into account higher-order effects of the s-d interaction. The uniform polarization of the conduction electrons, arising from the first-order exchange interaction with the Mn ions, is strongly influenced by the first-order perturbation of the wave function. This results in a concentration of the polarization of the conduction electrons in a small region around the Mn ions. Only the Cu nuclei quite near a Mn ion are subjected to an effective field originating from this polarization. As the concentration of the Mn is very small, the greater part of the Cu nuclei do not experience this effective field, which means no Knight shift and only a line broadening. This result was predicted before Yosida's explanation by Hart 123, starting from remarks made by Friede11°1p173, that one must take into account the shielding of the ionic charge of the dissolved impurity. Concerning the electrical resistivity of Cu- Mn alloys, Yosida174 started from an s-d interaction and described this in first approximation by a molecular magnetic field. The Mn ions were divided into two groups of equal numbers of ions. One group experiences a field H', the other one a field H - , which fields are the sums of the molecular fields and theexternal magnetic field H. At low temperatures an antiferromagnetic ordering of the Mn ions was assumed. Then for H = 0 the molecular fields are equal but opposite in sign. Above the Ntel temperature H f and H - are zero in first approximation. Yosida assumed a disturbing potential consisting of two parts: 1 . with a spin-dependent exchange interaction, 2. with a spin-independent interaction between the 4-s conduction electrons and the 3-d electrons localized at the Mn-ions, originating from a shielded Coulomb potential at the Mn ions. The exchange interaction is represented by an effective potential with a different sign for electrons with + and - spin. The consequence is a different scattering probability for electrons of different spins. A shift AE, of the Fermi surface occurs, which is : AE; a (W, T WI),
where W, is the total transition probability due to the spin-independent and the spin-dependent potential, I is the total magnetization of the Mn ions WZ is the cross term of both interactions. and w is constant. The term The expression for the specific resistivity is now: References p. 259
CH. IV, 4
91
ANOMALIES IN DILUTE METALLIC SOLUTIONS
249
The second term on the right, which is of great importance for the anomaly in the niagnetoresistivity, disappears only when an antiferromagnetic ordering of the Mn ions takes place in zero magnetic field. For the calculation of W , and w in eq. (11) Yosida174 represented the interaction between the conduction electrons and the Mn ions by H =CCTr(ri-R)-2CCJ(ri-Rn)(si.Sn). i
n
i
n
Here ri and R, are the coordinates of conduction electron i and Mn ion n respectively, while si and S, are the spin operators of the electron and the ions, V is the normal scattering potential and J the effective exchange integral between both particles. The electrons with + spin and - spin have the partition functions f and f'-. The Boltzmann equation is valid for each function. The collision terms split up into one term for the elastic and one for the inelastic collisions. By writing the calculated expressions for these terms in the two Boltzmann equations Yosida obtained an equation for AE; and one for AE,. With the solutions of these equations and with eq. (12) it was possible for him to calculate an expression for the specific resistivity at low temperatures. According to this result the resistivity of Cu-Mn increases monotonically from T = 0°K as the temperature rises. No minimum or maximum can be deduced. Only the magnetoresistance became the right order of magnitude and sign and appeared proportional to the square of the magnetization. The "giant" thermopower was not considered by Yosida. De Vroomen and Potters175 intended to extend Yosida's calculations in order to find a description of the anomalies in the thermopower and in the electrical resistivity. An important difference between the treatment by Yosida and that by De Vroomen-Potters is the dependence on energy of the relaxation times z+ and z- of the electrons with spin + and spin - respectively. The Boltzmann equations are reduced to two difference equations for z+ and z-. This results in the following formulae for the electrical and thermal current density : + W f?f J= C (7' + r-)-dul +
O
arl
-w
+m
W = C' J -m
where q = ( E - E,)/kT. References p. 259
V(Z+
a!" + t-)-dq ail
250
0. J. VAN
[CH.IV, 8 9
DEN BERG
From J one derives the electrical resistance. The absolute thermopower S (=17/T) is calculated by the authors from the absolute Peltier heat 17 (= WIJ). The solution of the difference equations depends on g,u,H/kT (with H = effective field). Therefore both J and the electrical resistance depend on the temperature T. This difficult calculation resulted in the same behaviour of the electrical resistance as Yosida found. In the ferromagnetic case - very large thermopower - as well as in the antiferromagnetic one - thermopower zero - and in a nearly antiferromagnetic case - thermopower smaller than in the ferromagnetic case but still abnormally large - the behaviour of the electrical resistance as a function of temperature, calculated by De Vroomen and Potters175 is analogous. The considerations concerning the thermopower by GuCnault and MacDonald176 and by Bailynsa are in many respects analogous. Here the reader is referred to the original publications. Kouve1177 proposed a ferromagnetic-antiferromagnetic model based on experiments concerning magnetic and electrical properties of Cu-Mn and Ag-Mn alloys with concentration higher than 5 and 10%Mn r e s p e c t i ~ e l y ~ ~ ~ . The model was expected to be valid to a lower concentration of about 1 at.% Mn. A fundamental assumption of this model for Cu-Mn and Ag-Mn alloys is that the sign of the magnetic interaction between neighbouring Mn atoms depends critically on the distance of separation. The RudermanKittell26 type of interaction is presumably the principal long-range coupling mechanism in the extremely diluted alloys. However, the thermoremanence results4*~lo3~131~132 suggest that even in alloys with as little as 1 at.% Mn there may be very strong short-range magnetic forces of a different origin (see 9.5). Kouvel assumes these short range forces to couple nearest-neighbour Mn atoms antiferromagnetically and those of large separation ferromagnetically (see 9.4). For fairly diluted Mn alloys the magnetic state that emerges from these considerations is that of very small ferromagnetic regions (domains) coupled “antiferromagnetically” through nearest neighbour Mn atom pairs (see the earlier proposals above). Using the Ruderman-Kittel-Yosida spin-spin coupling via the conduction electrons Marshall153 derived for an Ising model that the large contribution to the specific heat due to the addition of Mn to Cu, as published by Zimmerman et a1.142 (Section 7), is linear in temperature and independent of Mn concentration. A plausible assumption was made concerning the form of the probability distribution of the effective field for a diluted alloy (see 6.5). I
References p. 259
CH. IV, 8 91
9.3.
ANOMALIEP IN DILUTE METALLIC SOLUTIONS
CRITICISM OF THE MOLECULAR FIELD TREATMENT (SEE ALSO
25 1
DEKKER, 9.4.)
Sato, Arrott and Kikuchi 151 discussed magnetically diluted systems and considered the averaging process characteristic of the approximation adopted. “The molecular field treatment is quite inappropriate even qualitatively in the case of magnetically diluted solutions”. This statement follows from a discussion by the authors of the statistical treatment. The main conclusions are that co-operative phenomena do not occur if the atomic arrangement is random, until a finite concentration of magnetic atoms is obtained. This threshold concentration depends on the range of interaction and the number of nearest neighbours (2). Brout et al. 172 calculated for this concentration the value c o w + z by evaluating the higher-order terms in the cluster expansion. Sat0 et al. 1 5 1 used the Curie-Weiss, the cluster-variation and the average coordination number treatment and found respectively the values 0, 0.09 and 0.17 for co in Au-Fe, while Brout’s value was 0.11 and the experimental one was about 0.14. From these figures one must conclude that in the diluted alloys no long-range ordering can occur. Sat0 et al. 151 drew attention to the fact that a maximum in the susceptibility needs not necessarily imply that a long range order of spin on the one hand and the onset of antiferromagnetism in the “diluted” system on the other hand can be accompanied by almost negligible effects on the susceptibility vs temperature relation. In the low-concentration range a broad Schottky-type anomaly in the specific heat curve has been observed, which is in contrast to the appearance of a sharp I-type curve when the co-operative phenomena appear.
9.4. THE“ION
PAIR AND ISOLATED IONS” TREATMENTS
Dekke r l 7 * criticized the use of the molecular-field model because such a model should lead always to a sharp NCel-temperature. The experiments on the susceptibility, however, for small concentrations of Mn in Cu or Ag (see Section 5 ) indicate a continuous transition. Dekker considered a facecentered-cubic host lattice with solute atoms of non-vanishing spin. The interaction between nearest-neighbour solute atoms was assumed to be of “antiferromagnetic” sign and between solute atoms at larger distances from each other to be of “ferromagnetic” behaviour. The susceptibility could be described qualitatively. For the description of the electrical resistivity as a function of temperature Dekker179takes into account two categories: 1. R-T curves with a minimum only, 2. R-T curves with a minimum as well as a maximum. He References p . 259
252
G. I. VAN
DEN BERG
[CH.N,0 9
considers scattering of the conduction electrons by isolated magnetic ions and by magnetic-ion pairs. The first scattering was treated as follows : Suppose the perturbation energy operator for the electron can be written as : 25 H’(r) = 1/6(r - 0 ) - - S ( r - O ) S , . S , h2 where P denotes the position of the electron, J the exchange integral, S, and S are the spin of electron and ion respectively. The first term on the right represents the “normal” interaction between the charge of the electron and the electrical perturbation field of the solute ion. The exchange interaction is described by the second term. The resistivity due to Niisolated magnetic ions per unit of volume is
pi =
m*2k N [V 2 ne2nh3
~
+ J2s(s+ l)]
where kF is the wave number at the Fermi level. For this part of the resistivity one can easily show that a negative magnetoresistance occurs. The second scattering calculation (by pairs) proceeds as follows : Suppose the energy of the exchange interaction of both ions is E = = - (2J12/h2)S19S2,where S1and S2 are the spin operators of the ions. When the exchange integral JI2> 0, the spins in the ground state will be parallel (“ferromagnetic” interaction), when J,, c 0 an “antiferromagnetic” interaction exists. Starting from a perturbation Hamiltonian :
H’(Y)= V6(r - 0 )
25 + V6(r - R) - 2J -S *S16(~ - 0) - - S *S~S(Y - R ), A2 h2
where R denotes the position of ion 2 relative to ion 1, one arrives at the resistivity due to the exchange interaction
In this model two sorts of scattering processes are present: 1. elastic collisions ( A I = 0), 2. inelastic collisions ( A Z = f l), where I = quantum number of the resulting spin of the pair. The scattering of the electron waves by the ion pair is accompanied by interference of the scattered waves. This interference occurs in the resistivity as a function F ( k F R )(see also 9.5). A resistance minimum can occur only when J l 2 F > 0. But two cases are possible: References p . 259
CH. IV,
0 91
ANOMALIES IN DILUTE METALLIC SOLUTIONS
253
1. F < O and J12< O (antiferromagnetic pairs), 2. F > 0 and J 1 2> 0 (“ferromagnetic” pairs).
In case 1 the minimum is followed by the maximum. In case 2 the resistance increases with decreasing temperature. Qualitatively the occurrence of both types of R - T curves was made clear, though the considerations of the pair model must be considered as a first approximation. Dekker did not consider the thermoelectric power. Brailsford and Overhauser 180 have calculated the resistance from scattering by a nearest-neighbour ion pair. If the interaction is “ferromagnetic” (exchange integral J > 0, see also above), a resistance minimum should occur. One immediate consequence of the model is that the anomalous resistance change for a given temperature increment below the minimum should be proportional to the square of the concentration of the paramagnetic solute. However, this relation is expected to hold only for extreme dilution and the minimum is obtained only if the range of the effective s-d exchange potential is not too short. The experimental results do not agree with the square of the concentration for the proportionality of the slope of R-T curve below the temperature of the minimum (see Section 2). Therefore Brailsford and Overhauser considered also scattering by “single ions” and showed that the dependence of the slope of the R - T curve on the concentration becomes smaller. A weak indication of a maximum in the R-T curve was derived by the authors18 after corrections of the first calculations.
9.5. THEVIRTUAL BOUND
STATE TREATMENT
Friedel and coworkers 173.181,1*2 introduced for the study of the behaviour of transition impurities dissolved in normal metals the concept of the virtual bound levels, which phenomenon was already known in the scattering of electrons by free atoms. “These levels are a particular case of the resonance phenomena which arise when one puts together two systems with similar eigenenergies. Here, one deals with the d-states of the transitional impurities and the conductive state of the matrix. These states have indeed similar energies, for the d shells of the impurities mentioned often have a magnetic moment, indicative of a partial filling. Starting from one d state and the conductive state k, which gives rise to two states ad 5 pk, one obtains through resonance with the conductive states k’, k , etc. a region in space and in the energies, where each of the extended states presents, in the alloy, an amplitude larger on the impurity atom than in the matrix and with strong d character. Summing up the corresponding excess charges on all References p . 259
254
[CH.IV,8 9
0.J. V A N DEN BERG
the states of the continuum, one obtains a local excess of charge equal to that of the bound state with which one had started. This is what is called a virtual bound state”l16. The properties of the virtual bound states can be simply studied in two rather different models : a. a free electron model is used for the conduction electrons, while the wave functions of the alloy can be analyzed in spherical harmonics, because of the approximately spherical symmetry of the perturbation by the impurity atom ; b. Anderson183 and W o l f P 4 postulated a perturbing potential U, introduced by the impurity, which is small and repulsive. This is the case when
Fig. 41. The phase shift q-t(E)as a function of energy E of the d components for a free electron gas (1 = 2). a: bound state, 6 : virtual bound state1T4.
the base metal belongs to the same long period as the impurity and is on its right (Cu-Mn). A calculation of the width of the lorentzian for the density of states gives w M I < k I U Id > I. The main advantage of this method is the possibility of taking into account explicitly the form of the Bloch functions k of the conduction band of the matrix. This method has been applied in the case of a transitional matrix 185, Returning to model a, one can remark that the harmonic of quantum number I equal to the virtual bound state is of main interest. For the d state I = 2. The presence of the impurity atom perturbs these wave functions and induces at large distances characteristic phase shifts vt For a real bound state qr starts from the value x at the bottom E, of the conduction band (Fig. 41) and tends to zero at large energies. For a virtual one qr starts, however, from zero and its curve as a function of E joins that for the bound state if the perturbation is nearly sufficient to capture a 1 bound state. This joining takes place for a d state over a small energy width w around
.
References p . 259
CH. IV,
8 91
ANOMALIES IN DILUTE METALLIC SOLUTIONS
255
an average energy Eo, which can be taken as those of the virtual 1 bound state. For very small energies Eo one finds for a square well (ro) perturbing potential: (kr0)”-’k2 w=r1.3. ... .(2I- l)]” where k = 2(E0 - E,) when atomic units I e I = m = h = 1 are used. From this expression one can conclude: 1. w decreases when I increases. For s ( I = 0) and p (I = 1) states the concept loses physical interest because the width w in fact is large compared with the average energy Eo - E,; 2. w increases with Eo - E,. Exact computations 182,1869 187 confirm these conclusions at least qualitatively for the energies Eo - Ec of interest, of the order of the Fermi energies in ordinary metals. For transitional impurities in normal metals (I = 0, E, - Ec w 5 to 15 eV) a good approximation is
Application of the description given above to aluminium-based alloys with the transition metals from scandium to copper shows that the d shell of the impurity atom gives rise to only one virtual d state, able to contain 2(21 + 1) = 10 electrons. This state is filled progressively going through the first long period in the direction mentioned. This can be concluded from the single maximum in the atomic increase of electrical resistivity due to the long-period elements as solutes1731188. The maximum value is of the right order of magnitude when only the large phase shift v2 is considered. The thermoelectric power accress is negative in the first and positive in the second part of the seriesl87. The electronic specific heat should contain a positive contribution SC,, proportional to the impurity concentration, with a large maximum in the middle of the series. The magnetic susceptibility Sx is temperature independent; it is paramagnetic in the first half and diamagnetic in the second half of the series1s9. This is precisely what is expected in the model described above. Friedel 116 considered the various splittings of virtual d states. For the alloys mentioned above, the d-d exchange correlations play the important role. The latter can split the d” state into two dJ states with opposite spins, which would be successively filled in a transitional series. (Compare Hund’s rule for free atoms of ions.) According to Stoner’s reasoning190 for pure metals one can start from a References p . 259
256
G . J. VAN DEN BERG
[CH. IV,
09
virtual bound state with equally filled halves with opposite spins. An infinitesimal splitting produces a magnetic moment pBSpby transferring +Sp electrons from one half of the state to the other (pB= Bohr magneton). When A E represents the average difference in energy between two d electrons of parallel and antiparallel spins in an atom, the gained exchange energy 6E, = AE(6p/2)’. The spent kinetic energy 6E, = 6E. 6 ~ 1 2 ,where 6E. Sn(Em)V/2c= 6p/2 with 6n =the supplement of density of states; c = atomic concentration of impurities, V = atomic volume of the alloy. This gives 6E, = 2 ~ V - ~ d n - ’ ( E () 6 ~ / 2 ) The ~ . total energy is lowered by splitting if SE, + 6E, < 0 or Van( E ) AE>l. (14) 2c This condition, due to Blandinl82, is equivalent to that given by Anderson 183. Introducing in the inequality for 6n(E) the expression c dq -q--“-q V dE
c w/n Y wz + ( E - Eo)”
where q is the multiplicity of the 1 state, this condition is nearly equivalent to Friedel’s condition v * d E> w (15) which is a comparison of the width of the states to the maximum exchange energyp-AEfor total splitting. The maximum value of p is 5, the number of electrons or holes in the d shell. The approximate equivalence of the criteria (14) and (15) suggests that, when stable, the splitting is nearly always about total. A detailed study by Blandinls2 confirmed this equivalence. Table 2, composed by Friedel and coworkers using the conditionp.dE > w, predicts approximately for various matrices, which solutes of the first long period should present an exchange splitting and thus a permanent magnetic moment. For the calculation of w formula (13) was used and a n average exchange energy A E of about 0.8 eV was taken. The number p of electrons or holes in the d band has been estimated by assuming somewhat arbitrarily that all impurities gave one electron to the conduction band. Splitting will occur in the area of the table between the heavy lines. The results of measurements of the specific heat (Section 7) and of the electrical resistance (Section 2) regarding the occurrence of a magnetic moment are also plotted. The agreement is satisfactory and can be improved by taking into account the actual populations p deduced from the observed magnetic moments. As can be read from the table aluminium-based alloys will not show exchange splitting, that means no anomalies in the electrical resistance (Section 2), References p . 259
CH. IV,
0 91
257
ANOMALIES IN DILUTE METALLIC SOLUTIONS
thermopower (Section 3), specific heat (Section 7), or magnetic properties (Section 5), all in agreement with experimental results. For transition elements with large p values such as Cr and Mn the splitting occurs in the noble metals, Mg and Zn. Some difficulties remained concerning the evaluation of the “spin flip” term for the electrical resistivity. The broadening in energy of a virtual bound state will be accompanied by one in space. A magnetic impurity atom scatters in a different way the electrons of both spin directions, which causes different long range perturbations in the electronic density and gives rise to a long range spin polarization. In normal matrices this polarization is approximately isotropic and it oscillates with a wave length equal to half the Fermi wave length 2n [2m (E,, - E,)]-’, while the amplitude decreases as the cube of the distance. The absolute value of the electronic density is important within a range of the order of 2n(2mW)-*. Using eq. (13) one finds that this range increases when the Fermi energy decreases. The magnetic coupling I12S,*S2 between two impurities is expected to be proportional to the spin polarization produced by one impurity on the other. Thus
I,,
Ar~~.osc(2kFr,,),
where osc stands for an oscillatory function approaching a cosine for large rI2. The major contribution to the spin polarization comes from the q2 phase shift, while in first approximation the contribution of the other qi can be neglected. This amounts to neglecting the exchange polarization, the principal contribution taken into account by Yosida125, with respect to the resonance polarization. The error made in this way amounts to between 6 and 20%1*2.If the spin polarization due to an impurity is analyzed in spherical harmonics around a second impurity, the 1 = 2 harmonic gives rise to a resonance coupling and the others to an exchange coupling with its magnetic moment. But the spin polarization has a pure I = 2 symmetry only around the first impurity while its 1 = 2 component around the second impurity will be weak. Owing to the short wave length of the oscillations this is even the case for two nearest-neighbour impurity atoms. The coupling will therefore probably be mostly by exchange with the 1 2 harmonics. For such a resonance exchange hybrid coupling A is proportional to the kd exchange energy and inversely proportional to the spin of the impurities. With the coupling mentioned above one can derive a Curie temperature which is proportional to the concentration and is of the right order of magnitude for diluted Cu-Mn alloys. In Section 7 the magnetic entropy,
+
References p . 259
258
0.J. VAN DEN BERG
[CH.rv 6 9
calculated from specific-heat measurements, was discussed. This entropy of disorder of the magnetic moments may suggest that the latter are frozen in a kind of ordering, which is antiferromagnetic in its average, below a temperature TN proportional to the impurity concentration. Starting from the coupling in eq. (16) which extends over large distances and oscillates rapidly with distance, one must conclude that the effective field on a given impurity atom depends on its interaction with several other impurity atoms, which means that its magnitude and sign vary rapidly with the exact disposition of these neighbours. The distribution function for the internal fields is therefore necessarily a quasi-continuous and symmetrical curve at least as long as no external field stabilizes a given direction. This fact was also discussed by Marshall starting from the Ruderman-Kittel-Yosida interacti0n15~.From such a curve there follows at 0°K “a frozen but disordered arrangement of the magnetic moments”. Because of the symmetry of the curve for the atomic concentration of sites n(Hi) with an internal field between Hi and Hi -tdHi this arrangement should be on the average “antiferromagnetic”. At small concentrations, less than a few percent, the central line is a lorentzian with a few satellites s and s’ corresponding to pairs of near neighbours. The central line has a width 24 proportional to the concentration, and the density of sites with zero molecular field n(0) is independent of concentration (Fig. 42). The Ntel temperature TN should increase as the concentration, for kT, M pBd. Well below TN only the sites corresponding to the shaded area of Fig. 42 (,u& M kT) are thermally activated. From this there results a magnetic specific heat SC,, which should increase linearly with temperature153 and which should be independent of concentration: SC, % ~ ’ s ’ T ~ ( pB-’ o ) (see Section 7). For thermoremanent magnetization experiments, Friedel and coworkers calculated the slope of the magnetization field curve at the origin to be approximately 2n(O), which means independent of the concentration c. The
Fig. 42. Density n ( H M ) of sites with a molecular field HM in diluted alloys, such as Cu-Mn, at low ternperatures1l3. References p . 259
CH. IV].
09
ANOMALIES IN DKUTF. METALLIC SOLUTIONS
259
critical field above which saturation should set in is approximately d or proportional to c. Kouvel’s experiments 132 for large concentrations of Mn in Cu or Ag do not confirm these calculations. As Carolil69 showed, one can derive an approximate expression for the near infrared absorption by an alloy of a noble metal with a transitional solute (Au-Ni 7 at.%) starting from the virtual bound state concept. The expressions for the increase in electronic specific heat and in residual resistivity gave an opportunity to calculate the half-width of the virtual bound state (0.14 eV for Au-Ni 7 at.%). The increase of the electronic specific heat for Au-Ni became of the same order of magnitude as that for Cu-Ni, obtained from experiments. The formula for the infrared absorption of Au-Ni yielded the same value (0.137 eV) for the half width.
9.6. CONCLUSION
A fundamental approach to the problem of the chapter has not been possible up to now. Of the models presented, each accounts for some of the abnormal properties mentioned in 9.0. For several points the resonance hypothesis is suitable. The concept of the virtual bound state may be a link between the “molecular field treatment” and the “ion pair” approximation. REFERENCES 1
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A. C. Chapman and E. F. W. Seymour, Proc. Phys. SOC.(London) 72, 797 (1958). B. J. Verkin and I. M. Dmitrenko, Doklady Akad. Nauk S.S.S.R. 19, 409 (1958). lZ9J. Korringa and A. N. Gerritsen, Physica 19, 457 (1953); Commun. Leiden Suppl. No. 106. 130 J. C. Fisher, reported in Io3 and in: R. W. Schmitt and I. S. Jacobs, Can. J. Phys. 34, 1285 (1956). 131 0. S. Lutes and J. L. Schmit, Phys. Rev. 125, 433 (1962). 132 J. S. Kouvel, J. Phys. Chem. Solids 21, 57 (1961). 133 M. D. Blue, J. Phys. Chem. Solids 11, 31 (1959). 134 W. B. Teutsch and W. F. Love, Phys. Rev. 105, 487 (1957). 135 Yu. P. Gaidukov, J. Exptl. Theor. Phys. (U.S.S.R.) 34, 836 (1958); Soviet Physics J.E.T.P. 7, 577 (1958). 136 J. E. Kunzler and J. R. Klauder, Phil. Mag. 6, 1045 (1961). 137 N. E. Alekseevskii and Yu. P. Gaidukov, J. Exptl. Theor. Phys. (U.S.S.R.) 32, 1589 (1957); Soviet Physics J.E.T.P. 5, 1301 (1957). 138 J. de Nobel, Suppl. Bull. Inst. int. Froid, Annexe 1956-2, p. 97. 139 G. J. van den Berg, Proc. 7th int. Conf. low Temp. Phys., Toronto 1960 (University of Toronto Press, Toronto, Canada) 246. 140 A. J. P. Meyer and P.J. Taglang, J. Phys. Rad. 17, 457 (1956). 1 4 1 F. J. du Chatenier and J. de Nobel, Physica 28, 181 (1962); G.J. van den Berg and J. de Nobel, J. Phys. Rad. 23, 665 (1962); J. de Nobel and F. J. du Chatenier, Proc. 8th int. Congress low Temp. Phys., London 1962 (Butterworth and Co., Ltd., London, 1963) 241. 142 J. E. Zimmerman and F. E. Hoare, J. Phys. Chem. Solids 17, 52 (1960). 143 L. T. Crane and J. E. Zimmerman, Phys. Rev. 123, 113 (1961). 144 L. T. Crane, Phys. Rev. 125, 1902 (1962). 145 G. L. Guthrie, S. A. Friedberg and J. E. Goldman, Phys. Rev. 111, 45 (1959). 146 J. K. Logan, J. R. Clement and H. R. Jeffers, Phys. Rev. 105, 1435 (1957). 147 D. L. Martin, Can. J. Phys. 39, 1385 (1961). 148 D. L. Martin, Proc. Phys. SOC.(London) 78, 1489 (1961). 149 D. L. Waldorf, J. Phys. Chem. Solids 16, 90 (1960). 150 F. Bitter, A. R. Kaufmann, C. Star and S. T. Pan, Phys. Rev. 60, 134 (1941). 151 H. Sato, A. Arrott and R. Kikuchi, J. Phys. Chem. Solids 10, 19 (1959). 152 A. W. Overhauser, J. Phys. Chem. Solids 13, 71 (1960). 153 W. Marshall, Phys. Rev. 118, 1519 (1960). 154 D. L. Martin, Proc. 8th int. Congress low Temp. Phys., London 1962 (Butterworth and Co., Ltd., London 1963), 243. 155 R. Tournier, J. J. Veyssie and L. Weil, J. Phys. Rad. 23, 672 (1962). 156 R. Smoluchowski, Phys. Rev. 84, 511 (1951). 157 J. le Guillerm, R. Tournier and L. Weil, Proc. 8 t h int. Congress low Temp. Phys. London 1962 (Butterworth and Co. Ltd. London, 1963), 236; R. Tournier and L. Weil, J. Phys. Rad. 23, 522 (1962). l5~7 K,Kobayashi, J. Phys. SOC.Japan 15, 1352 (1960). 159 J. J. Becker, J. Metals 9, 59 (1957); J. Appl. Phys. 29, 317 (1958). 160 C. P. Bean and J. D. Livingstone, J. Appl. Phys. 30, 120 S (1959). 161 T. Mitui, J. Phys. Soc. Japan 10, 1023 (1955), 13, 549 (1958). lo2P. Gaunt and J. Silcox, J. Phys. SOC.Japan 17,(Suppl. B1) 665 (1962); Proc. Int. Conf. Magn. and Crystallogr. 1 (1962). 163 C. W.Garland and J. Silverman, J. Chem. Phys. 34, 781 (1961). 164 G. Seidel and P. H. Keesom, Phys. Rev. 112, 1083 (1958). lo5 J. E. Zimmerman and L. T. Crane, Phys. Rev. 126, 513 (1962). lB6 J. D. Filby and D. L. Martin, Can. J. Phys. 40, 791 (1962). lz8
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168
CHAPTER V
MAGNETIC STRUCTURES
OF HEAVY RARE-EARTH METALS BY
KEI YOSIDA
INSTITUTE FOR SOLIDSTATEPHYSICS, TOKYO,AzUU, TOKYO
UNIVERSITY OF
CONTENTS: 1 . Introduction, 265. - 2. Survey of experimental results, 268. - 3. Theoretical consideration, 275. - 4. Relation between the Fermi surface and the screw structure, 285. - 5. Summary, 292.
1. Introduction
An excellent review article of information on rare-earth metals ranging from lanthanum to lutetium obtained up to 1956 has been presented by Spedding, Legvold, Daane and Jennings l. As precisely described in that article, rareearth metals show anomalous behavior in physical properties, particularly in magnetic and electrical properties. Although the anomalies found were supposed to be connected with the magnetic ordering of the localized magnetic moments arising from the 4f electrons, it was very hard to understand what happens in these metals from experimental results on polycrystalline samples. Our understanding about rare earths has greatly been elevated by experiments on single crystals which Spedding and his collaborators have succeeded in growing. In particular, neutron diffraction experiments on single crystals of heavy elements were most powerful for clarifying the essential nature of rare-earth metals. These experiments have been performed at Oak Ridge by Koehler, Wilkinson, Wollan and Cable 2-6. In this article, we shall describe the results of magnetic and neutron diffraction experiments on single crystals and clarify the origin of the magnetic structures, showing that these structures are closely connected with the Fermi surface of the conduction electrons. The magnetic moment per atom, deduced from the temperature dependence References p . 293
265
266
[CH.V, 8 1
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of the paramagnetic susceptibility, is in good agreement with that of the trivalent ions for most cases with the exceptions of europium and ytterbium, which behave as the divalent ions in a metallic phase. Therefore, the magnetic moments of rare-earth metals are considered to arise from the electrons in the incomplete 4f shell, and to be well localized at the vicinity of the nucleus. Since rare-earth atoms have three valence electrons which become itinerant in themetallicstateandeight 5sand5pclosed-shellelectrons outsidethe4fshel1, the wave function of the 4f electrons is considered to be concentrated towards the nucleus by the strong Coulomb attraction of the effective nuclear charge. This contraction of the 4f wave function may be a main cause which makes the effect of the crystalline field less effective than the LS coupling. The good quantum number for the rare-earth ions even in the metallic state is, therefore, the total angular momentum J = L S. The magnetic moment of the ion is parallel to J and described by gJpgJ,where the Land6 factor g, is given by g, = 1 {S(S 1) J(J 1) -L(L 1))/2J(J 1) and p B represents the Bohr magneton. The heavy rare-earth metals crystallize in a hexagonal close-packed structure at room temperature and below, with the one exception of ytterbium. The crystal structure of ytterbium is face-centered cubic and it has two conduction electrons with a closed 4f shell. This fcc phase changes to a bcc phase at high temperatures. The hexagonal close-packed structure observed in the heavy series is normal and is constructed by stacking two hexagonal layers A and B, whereas the hexagonal close-packed structure seen in the light series is constructed by stacking three hexagonal layers A, B and C in the order of ABAC, the c axis being twice as large as that of the noImal structure. For samarium the construction is more complicated and the stacking sequence is ABCBCACAB.. . with a period of nine c layers. Besides this type of hexagonal close-packed structure, lanthanum and cerium have an fcc modification. Furthermore, cerium has another 'collapsed' fcc modification which appears at low temperatures. The occurrence of this modification is associated with a tranfer of a 4f electron to the conduction band. The crystal structures for the elements of the light series change to a bcc structure at high temperatures ranging from 700" C to 850" C. An exceptional case is europium, the crystal structure of which is bcc in the whole temperature range. The large atomic radius of europium metal suggests that it has two valence electrons and that the core is doubly ionized. This suggestion is consistent with the value of the magnetic moment deduced from the high temperature susceptibility. Crystal structures of rare-earth metals are summarized in Table 1.
+
References p. 293
+ + +
+ +
+
CH. V,
0 11
267
MAGNETIC S7RUCTuRES OF HEAVY RARE-EARTH METALS
TABLE 1 Crystal structures and lattice parameters of rare-earth metals at room temperature and below
structure
element sc Y
La Ce
Pr Nd
Sm Eu Gd Tb
DY Ho Er Tm Yb Lu
hcp ABAB hcp ABAB hcp ABAC fcc hcp ABAC fcc fcc' hcp ABAC hcp ABAC rhom. ABCBCACAB bcc
hcp ABAB hcp ABAB hcp ABAB hcp ABAB hcp ABAB hcp ABAB fcc hcp ABAB
lattice a
constant c
cJa
reference
3.3090 3.6474 3.770 5.302 3.68 5.1612 4.85 3.6725 3.6579 8.996 3.621 4.5820 3.6360 3.6010 3.5903 3.5773 3.5588 3.5375 5.4862 3.5031
5.2733 5.7306 12.159
1.594 1.571 1.613
7
11.92
1.62
7 7 7
8 7 8
11.8354 11.7992 a: = 23'13' 26.25
1.611 1.613 1.611
7 7
7 9
5.7826 5.6936 5.6475 5.6158 5.5874 5.5546
1.590 1.581 1.573 1.570 1.570 1.570
5.5509
1.585
7 7 7 7 7
7 7 7
The conduction band for rare-earth metals is supposed to be constructed mainly from s- and d-bands or mixtures of these bands. Such an electronic state is similar to the cases of scandium and yttrium. These two metals crystallize in a normal hexagonal close-packed structure. Therefore, apart from the two elements of europium and ytterbium, the hexagonal structure of rare earths seems to originate in the s and d characters of the conduction band. Since the ABAC structure has been found only in the light series of rare-earths, this structure may have some connection with the existence of an incomplete 4f shell. For light elements, the extension of the 4f wave function is expected to be large compared with that for heavy elements because of the smaller effectivenuclear charge and its energy level is expectedto be comparatively high. Therefore, the influence of the 4f state on the conduction band is expected to be more significant for light elements. The occurrence of a phase with an fcc structure seems reasonable because the difference in cohesive energy between fcc and hcp structures is generally small, as one can see in the cases of metallic cobalt and others. The c/a ratio is considerably smaller for the heavy series, ranging from References p. 293
268
KEI YOSIDA
[CH.V, 0 2
1.59 to 1.57, than for the light series, in which it ranges from 1.61 to 1.62 and is nearer to the ideal value of 1.63, as seen from Table 1. The difference in this ratio for two series seems to be connected with the difference in the stacking sequence. 2. Survey of Experimental Results
First we shall review the experimental results on heavy rare-earth metals. Since the experimental results on polycrystals have been described in detail in the article by Spedding, Legvold, Daane and Jenningsl, stress is laid mainly on new results on single crystal samples. Gadolinium: It is well known that gadolinium shows ferromagnetism below about 290" KIO. The saturation moment per atom extrapolated to the absolute zero of temperature is nearly equal to the value for the trivalent free ion, 7 Bohr magnetons per atom. The susceptibility above the Curie temperature obeys the Curie-Weiss law and the paramagnetic Curie temperature is reported to be equal to 302.7" K. Thus, the ferromagnetism of gadolinium is a typical one and its feature is rather in the fact that the TP law for the temperature dependence of the saturation magnetization holds very well up to about half way to the Curie temperature. This seems to result from a small anisotropy and a good localization of the magnetic moment. Russian work by Below et al.ll,l 2has recently shown that the magnetization exhibits a maximum and the coercive force shows a minimum at about 210" K in very weak fields ranging from 0.1 Oe to 1 Oe. Based on these results, Belov and Pedko12 suggested that a screw spin configuration is stable in gadolinium between 210" K and the Curie temperature in the absence of the external magnetic field. Measurements on single crystals have been made by Graham 13 and Corner et al.14. According to Graham's results, the direction of easy magnetization makes an angle to the c axis. This angle increases from about 33" at the absolute zero, reaches 90" at about 165" K, and then in the temperature interval from 225" K to 245" K decreases rapidly to zero. In this interval the anisotropy energy becomes vanishingly small. A similar behavior was also observed by Corner et al. Therefore, it is not unlikely that the anomaly found by Belov et al. in the magnetization curve is due to the decrease of the anisotropy energy. The magnetostriction and thermal expansion measurements on single crystals are reported by Bozorth and Wakiyama 15. No neutron diffraction experiments have been reported for gadolinium. Terbium: Neutron diffraction experiments have been made by Koehler, Wilkinson, Wollan and Cable on single crystal specimens of the rare-earth References p . 293
CH. V,
5 21
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
269
metals Tb, Dy, Ho, Er and Tm, which follow gadolinium. According to magnetic measurements on polycrystals 16, metallic terbium becomes antiferromagnetic below the Ntel temperature of 230" K and undergoes a transition to the ferromagnetic state at about 218" K. The effective Bohr magneton number peffand the paramagnetic Curie temperature have been determined as peff= 9.7pB and 8 =237" K by the susceptibility measurements above the Ntel temperature. The neutron diffraction measurements 6 have established that the magnetic structure in the antiferromagnetic region is a screw structure with the screw axis parallel to the c axis. The magnetic moments are parallel to the c planes. The magnetic reflections for the screw structure are given by the following Bragg condition: S = k'-k = !3 Q, where s is the scattering vector, k' and k are, respectively, the wave number vector of the scattered and incident neutron beams and !3 represents the reciprocal lattice vector. The vector Q has the direction of the screw axis and its magnitude is given by Q =(c')-la, where c' is the distance between two adjacent layers perpendicular to the screw axis and a is the angle between the magnetic moments of the atoms on these two layers. The magnetic reflections vanish when the scattering vector is parallel to the magnetic moments. Therefore, the diffraction pattern responsible for the screw structure is characterized by the appearance of a pair of satellite reflections on each side of the nuclear reflections along the screw axis. From the direction and the distance of the satellites from the normal Bragg reflections, we can presume the direction of the screw axis and the turn angle a. The turn angle in terbium is found to be temperature-dependent and decreases from 20" at 230" K to about 18" per layer at 217" K. At about 223" K a magnetic contribution to the normal lattice reflections begins to appear. The ordered moment at 4.2" K is obtained as 9.1 pBfrom the neutron diffraction data on powdered samples17. Dysprosium: A neutron diffraction pattern similar to that of terbium has also been found in dysprosium and holmium. For dysprosium 3, the satellite reflections are found in the temperature range between 179" K and 87" K. The screw axis is along the c axis and the magnetic moments are confined in the c planes. The turn angle decreases almost linearly with lowering temperature from 43.2" to 26.5" per layer, but there is found a slight departure from linearity at about 130' K. At about 87" K the intensities of the satellite reflections suddenly diminish and instead the magnetic scattering grows up References p . 293
210
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[CH.
V, 8 2
in the normal (0002) reflection in a temperature interval of about three degrees. This indicates that at this temperature dysprosium undergoes a transition of the first kind from the screw state to the ferromagnetic state. The saturation magnetization at low temperatures is estimated as 9.5 pB per atom, which is slightly less than the free ion value of 10 pB. In the range between 140" K and the ferromagnetic transition temperature, the second harmonics of the (OOOn3) satellites have been observed in addition to the primary satellites, although their intensities are very small, indicating that the screw structure is slightly modified in this temperature region. Wilkinson et al.3 have remarked that this modification is associated with the anisotropy existing in the basal plane. Magnetic measurements on single crystal samples of dysprosium have been made by Behrendt, Legvold and Speddingls. According to their results metallic dysprosium shows antiferromagnetic behavior between 179" K and 85" K and ferromagnetic behavior below 85 OK, in agreement with the results of neutron diffraction experiments described above. In this metal, the anisotropy energy is so large that ferromagnetism appears only in the c plane, but along the c axis dysprosium exhibits paramagnetic behavior even below the ferromagnetic transition temperature. The easy axis of magnetization is in the direction of <1120> and the absolute magnetic moment is deduced as 10.2 pB. The paramagnetic susceptibilities above the Ntel temperature conform to the Curie-Weiss law. The effective number of Bohr magnetons is obtained as 10.64 pB for directions both parallel and perpendicular to the c axis, and the paramagnetic Curie temperatures are, respectively, 8, = 121" K and 8, = 169 "k for these two principal directions. In the antiferromagnetic region, an external field applied in the c plane causes a ferromagnetic transition when the field strength reaches a critical value, which increases with increasing temperature. The anisotropy energy in the basal plane was not recognized above 110" K. Holmium: For the case of holmium236 a similar screw structure to that in dysprosium has been found in the temperature region between 133" K and 19" K. In this temperature range, the turn angle decreases linearly from 50" at the NCel temperature to 36" at 35" K, and this value is maintained down to about 20" K. Below 35" K, the second, the fifth and the seventh harmonics are observed. Their intensities are weak and are about one percent of those of the primary satellites. Below 19" K magnetic contributions to the normal nuclear reflections, except for (000n3)reflections, are observed, whereas for the (OOOn3)reflections the magnetic scattering is found only in the satellite reflections. This indicates References p . 293
CH. V $ 2 ]
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
271
that the perpendicular components of the ordered moments to the c axis are in a helical arrangement and the parallel components are aligned ferromagnetically. Therefore, the ordered moment is considered to rotate on a cone as one goes from one c plane to the next. The parallel and perpendicular components of the ordered moment are found to be 2.0 pB and 9.8 pB at 4.5" K, respectively. In this lowest temperature region, the turn angle is found to be 30". The abrupt decrease of the turn angle from 36" to 30" near 20" K may be related to the increase of the six-fold anisotropy in the c plane. From the study of the neutron diffraction pattern under an external field in the c plane, it is confirmed that the easy axis of magnetization is in the direction of at low temperatures. But, at high temperatures no preferred direction is found within the c plane. The magnetic measurements on single crystals of holmium made by Strandburg, Legvold and Speddinglg have confirmed that the magnetic behavior is quite consistent with the results of neutron diffraction experiments. The NCel temperature was determined to be 132" K, the effective number of Bohr magnetons was 11.2 pugand the parallel and perpendicular paramagnetic Curie temperatures were, respectively,8, = 73" K and 8,, = 88" K.
H (KILO
-OERSTEDS)
H ALONG
-z I O T O >
Fig. I . Magnetization curves (per gram) of metallic holmium for the field applied to the < 1010 > direction. Dashed lines are for decreasing field (Strandburg, Legvold and Speddingl9).
Referencesp. 293
212
KEI YOSIDA
[CH. V,
52
Below 20" K, metallic holmium shows ferromagnetic behavior along the c axis. The spontaneous magnetization along this direction is estimated as 1.7 fig in good agreement with that obtained by Koehler et aL2.The saturation moment in the c plane is obtained as 10.34 p B . Strandburg, Legvold and Spedding have observed that the magnetization curve of holmium shows an anomalous behaviour in the antiferromagnetic region which has not been observed in dysprosium. Fig. 1 shows the magnetization vs external field curves for holmium along the easy direction of obtained by Strandburg et al. at various temperatures. As seen in this figure, a rise of the magnetization corresponding to the transition from the screw state to the ferromagnetic state takes place in two or three steps at high temperatures. For the field applied to the difficult direction, namely <11ZO>, the magnetization rises in two steps. For decreasing field a hysteresis is observed in the rising part of the first step. These steps are also recognized in the magnetization vs temperature curves. Koehler et al. have recently observed the neutron patterns of a new phase corresponding to each part of the magnetization processes. Erbium : Neutron diffraction patterns for metallic erbium are quite different from those for the preceding three metals. The temperature variations of the intensities for the three representative reflections, namely (1 lZO), (1 120+)and
,Ot
I
60
3
Temperature ( O K )
Fig. 2. Temperature dependence of the ( 1 IZO), (1 120+)and (0002-) reflections of metallic erbium (Cable, Wollan, Koehler and Wilkinson*). References p . 293
CH.
V, 5 21
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
213
(0002-), obtained by Cable, Wollan, Koehler and Wilkinson4, are shown in Fig. 2. The plus and minus signs indicate the direction of displacement of the satellite positions from the reciprocal lattice points. In this figure, the normal (1 120) reflection shows only a nuclear contribution between 80" K and 20" K, but below 20" K a magnetic scattering contributes to this reflection as seen from the discontinuous rise of the intensity at 20" K. The (llZOi-) satellite reflection begins to appear at the NCel temperature of 80" K, increases almost linearly with decreasing temperature, and then the intensity of it drops down to a small and constant value below 20" K. On the other hand, the (0002-) satellite does not appear until the temperature is lowered to 52" K. The intensity of this satellite shows a discontinuous rise at 20 OK. From these results it has been concluded by Cable et al.4 that between 80"K and 52" K only the c component of the magnetic moment is modulated sinusoidally along the c axis, and that the sinusoidal modulation of the perpendicular component begins at the lower temperature of 52" K with the same period. Below 20" K, the c component is aligned ferromagnetically and a small perpendicular component still remains helically ordered. The period of this helical modulation is constant at 4.1 c. Therefore, at the lowest temperature region below 20" K the hodograph of the ordered moment forms a cone similar to that for the low temperature phase of holmium. The ordered moment along the c axis is estimated as 7.2 pB per atom, and that normal to the c axis as 4.1 pB per atom. In the high temperature phase, the period of the sinusoidal modulation is independent of temperature and equal to 3.5 c or 7 c layers. Below 52"K the period becomes temperature-dependent, increasing linearly from 3.5 c at 52" K to 4.0 c at 20" K. For this temperature phase it is impossible to distinguish experimentally whether the perpendicular component is rotated in successive c layers or whether it oscillates linearly in one direction in the c plane, being combined with the c component oscillation into a cycloidal arrangement. In this phase, weak third harmonics are observed for (1010) and (1 120) reflections, but not for (0002), indicating that the sinusoidal modulation of the c component is squared. According to magnetic measurements on single crystal samples made by Green, Legvold and Spedding20, metallic erbium is antiferromagnetic between 85" K and 19.6" K, and ferromagnetic along the c axis below 19.6" K. The saturation moment along the c axis is evaluated to be 8 pa per atom. The transition temperature from the antiferromagnetic phase to the ferromagnetic phase is shifted towards higher temperatures by a field applied along the c axis. The effective Bohr magneton number obtained from the plot of the reciprocal susceptibility vs temperature above the NCel temperature References p . 293
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[CH. V,
52
is 9.9 pB and two principal values of the paramagnetic Curie temperature are 8, = 61.7" K and 8, = 32.5 OK, respectively. An anomaly originating from the phase transition at about 52" K is not very clear in the magnetic behaviour, but near this temperature a big peak is found in the electrical resistivity along the c axis. Thulium: The magnetic structure in thulium revealed by neutron diffraction experiments 6921 is similar to that found in the high temperature phase of erbium, in the sense that there exist no ordered moments in the direction normal to the c axis. The sinusoidally modulated ordered moment develops along the c axis below the NCel temperature of 56" K. The period of this modulation is constant at 3.5 c in all the temperatures below 56" K. At 38" K the third harmonic begins to grow rather rapidly, indicating that the sinusoidal modulation of the moment becomes squared. Simultaneously, magnetic contributions in the normal (1010) and (1 120) reflections and also the second harmonics are found. This fact means that the resultant moment is induced along the c axis. Koehler et a1.2l have confirmed that the observed intensities at 4.2 "K can be accounted for on the basis of a magnetic structure in which the sinusoidal modulation of the high-temperature structure is completely squared, and in which the moments on four successive layers point upwards while those on the next three layers point downwards. No magnetic measurements on single crystals of thulium have been made, but by measurements on polycrystals Rhodes, Legvold and SpeddingZ2and Bozorth and Davis23 have reported that metallic thulium is ferromagnetic at low temperatures, having the saturation moment of about 0.5 pB per atom. This value is considered to be the polycrystalline average of 1 pB per atom along the c axis, predicted from the structure determined by Koehler et aL2I. From the paramagnetic susceptibilitiesabove the NCel temperature petf= 7.56 TABLE 2 Data obtained by magnetic measurements on heavy rare-earth metals magnetic moment peratomat0"K Tc
TN C K ) C'K)
OlB)
Gd Tb
7.12 9.25 Dy 10.19 (1 HO 10.34 11 1.7 11 Er 8 11 Tm 0.5
289 218 (1120) 85 < i o m 20
peii
(Pa)
8,
References p. 293
19.6 22
7.95 9.7 230 178.5 10.64 121 73 132 11.2 85 60
9.9 7.56
oP
(" K) P K ) C'K)
e,
61.7
169 88 32.5
8, ) ~ J ( J 1) ref. (" K)
(gJ - I
+
302.7 237 153 83
19.2 22.6 21.6 18.4
19
42.2 20
16.6 17.2
23
10 16
I8
20
CH.V, 8 31
275
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
and 8 =20" K are obtained. The results of neutron diffraction experiments and of magnetic measurements described so far are summarized in Tables 2 and 3. TABLE3 Magnetic structures of heavy rare-earth metals determined by neutron diffraction experiments
TN
TL?
Tb
T 0°K PI1 0
PL 9.1 P B T 0°K 0 PL 9.5 P B turn angle PI1
Ho
T 0°K PI1 2.oPB
PA 9.8PB turn angle
Er
T 0°K Pi1 7.2PB
PL 4.1 P B
19" K ferro 0 screw screw 30", 36" __ 20" K ferro screw
period
179"K + 43.2"
35" K
33" K 0 screw
36"
--f
oscillate oscillate 4 . l ~ , 4 . 0 -+3.5~. ~
50" 80" K
52" K
38" K
antiphase 0 3.5c
20"
0 screw 26.5"
T 0°K PI1 7.0Pcn Pl 0
--z
87"K
0 ferro
period
Tm
0 screw 18"
turn angle
DY
230" K
223" K 0 ferro
oscillate 0 3.5c
56" K
oscillate 0 3.52
3. Theoretical Consideration As described in the preceding section, heavy rare-earth metals have various types of the ordering, depending on the elements and the temperature regions. However, all the structures have a common feature in that the magnetic moments on the same c layer are parallel and that they oscillate linearly or helically along the c axis. This oscillatory nature is considered to result from the exchange interaction between localized magnetic moments, which is written as H e x = - Ci,j J(Rj - Ri) si sj 3 (1)
where J ( R j - Ri) represents the exchange integral between the two spins S, and S j situated on the i-th andj-th lattice sites. For the screw structure specified by the vector Q the components of each spin can be expressed by References p . 293
276
KEI YOSIDA
Si,= 0,
Six = S cos cli
and
Si, = S sill ai, ai= a,,
[CH.V,
83
+ Q Ri , a
assuming that the spin is rotated in the xy plane. Therefore, the energy of the spin system with the screw structure can be calculated, in the classical approximation, as
E,, = - N S 2 J ( Q ) , J(Q) =CjJ(Rj
- Ri)exp(iQ*(Rj- Ri)),
(2) (3)
where N is the total number of atoms and the Fourier component J ( Q ) is defined in the fundamental Brillouin zone. J ( Q ) is independent of Rj for the Bravais lattice and for a hexagonal close-packed structure, as Q is along the c axis, but otherwise it should be replaced by the average value over the atoms in the unit cell. From the expression (2), we see that the screw structure with the value of Q , which makes J ( Q ) maximum, is the most stable configuration of the spin for the Bravais lattices. The possibility of the screw spin structure has been considered first by Yoshimori24 and later by Kaplan25 and Villain26.J ( Q ) becomes extremal at the center and the boundaries of the Brillouin zone, but besides these it is capable of having extrema at other points of the k space, depending on the nature of the exchange interaction, as will be shown later. The expression (2) also shows that the exchange energy does not depend upon the direction of the plane of rotation of the spin. In this sense, the screw structure has a wider meaning including a cycloidal structure as a special case and a normal screw structure, in which the plane of spin rotation is normal to the screw axis, called the ‘proper’ screw. The plane of spin rotation is determined by the crystalline anisotropy energy, which is, for some cases, large enough to deform the screw structure itself. The various magnetic structures found in heavy rare-earth metals are regarded as screw structures deformed by the anisotropy energy characteristic of each atom. From this point of view, the effect of the anisotropy energy on the screw spin structure has been investigated by R. J. Elliott27, T. A. Kaplanzs and Miwa and Yosida29. A concise description of these investigations can be seen in the paper by Nagamiya 30. The anisotropy energy for rare-earth atoms in the metals arises mainly from the electrostatic interaction between the multipole moments possessed by the 4f electrons and the crystalline field produced by the surrounding charge distribution. On the basis of the point charge approximation, the crystalline potential for a hexagonal close packed structure is assumed to take the form, as given by Elliott and Stevens 31, References p. 293
CH.
V, 8 31
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
21r
V = Az(3z2 - r2) + A:(35z4 - 30r2z2+ 3r4) +
+ Az(231z6 - 315r2z4 + 105r4z2- 5r6) -t + A: (x6 - 15x4y2+ 15x2y4- y 6 ) ,
(4)
where A: are the constants determined by the distribution of charges surrounding the rare-earth ion, and the z axis is taken along the hexagonal axis. As shown by Stevens32, this potential is equivalent, in the subspace spanned by the 2 J + 1 basis functions representing the eigenfunctions for the z component of J , to
+ /?A: < r4 > [35Jt - 30J;J( J + 1) + 3J2(J + 1j2 + 255; - 6J(J + I)] + + yA2 < r6 > [23156, - 3155(5 + 1)Jt + 1O5J2(J + 1)'J; - 5J3(J + 1)3+ + 7355; - 5255(J + 1)Jt + 40J2(5 + 1)2+ 2945; - 60J(J + l)] + + y A: < r6 > 4[ ( J ~+ iJ$ + ( J -~ i ~ , , ) ,~ ] (5) where a, p and y are the constants depending on the values of S, L and J and have been calculated by Stevens32. are the average of r" with respect to the radiai part of the 4f wave function, and their values for free ions have recently been calculated by Freeman and Wats0n3~by the Hartree Fock approximation. This Hamiltonian representing the anisotropy energy can be simplified by the classical approximation as
Ha = DP, (cos 0) D =201A;J2,
+ EP4 (cos 0) + FP6 (cos 0) + G sin'% cos 649, B=8PA:J4,
F = 1 6 y A ; < r 6 > J 6 a n d (6)
G =yA: < r 6 > J 6 , where P,,(x) is the Legendre polynomial and t9 and are the polar and azimuthal angles of the vector J measured from the c axis and the a axis, respectively. It is difficult to evaluate A t , to which the conduction electrons and the 5s2p6 outer core electrons both contribute. The contribution from the deformed outer core has recently been studied by Burns 34, according to whom this contribution is less than ten percent and not important. The order of magnitude of A: may be evaluated by the contribution from the nearest neighbor ions provided with the charge of 3e, assuming that the effect of the further ions is screened out by the conduction electrons. This rough estiReferences p . 293
278
KEI YOSIDA
[CH.V, 8 3
m a t i 0 n ~ ~ 3gives 2 ~ the order of 10 cm-1 for D, E, F and G . These values are considerably large but still smaller than the exchange energy, which is considered to be the order of 100 cm-1. Table 4 shows the values of a<+> 5 2 , fi c r 4 > 5 4 and y J 6 , and the plausible signs of the anisotropy constants for the heavy rare-earth series. If the anisotropy Hamiltonian (6) is treated as a perturbation to the exchange Hamiltonian, the anisotropic part of the free energy is, to the first order, given by the thermal average of the expression (6) under the internal field produced by the exchange interaction. This is calculated as
F, = D P (cos ~ e) +FP,(COS8)
+ EP, (cos e) + -k G s i n 6 ~ c o s 6 p
,
(7)
where 8 and p are the polar and azimuthal angles of the direction of the ordered magnetic moment of the ion and is given by the following average of the Legendre function: =
SP, (cos 0)exp (u cos 0)sin 0 d 0 > f exp (u cos 0)sin 0d 0
(8)
where u is given by the product of the magnetic moment and the internal field divided by kT. Therefore, the temperature dependence of the anisotropy energy is determined by that of . This quantity is, as generally shown by Zener35, Keffer36and Van Vleck 37, proportional to the +n(n 1)-th power of the ordered moment at low temperatures and to the n-th power near the Ntel temperature. Thus, in the high temperature region, only the second order term of the anisotropy energy is dominant because the higher order terms decrease with temperature more rapidly, but at low temperatures the higher order terms also become important. The D term makes the preferred direction of the ordered moment parallel or perpendicular to the c axis, according as D is negative or positive. However, the E and F terms make the preferred direction parallel to the c axis when they are negative, but when they are positive, the preferred axis is inclined to the c axis by the angle u, which is given by cos2a=$ for the E term and cos2a = a 1 5 22/15) for the F term. For terbium and dysprosium the leading axial anisotropy energy is considered to be the D term in the whole temperature range, and the ferromagnetic transition found in these metals is caused by an increase of the G term with lowering temperature, which describes the anisotropy in the c plane. For a screw structure, the anisotropy energy in the c plane is cancelled out in its
+
+
References p . 293
CH. V,
5 31
279
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
first order, except for the special cases in which the turn angle is equal to 60" or 120°, but it stabilizes the ferromagnetic state as the magnetization is along the easy direction. Therefore, if the anisotropy energy exceeds the difference in the exchange energy between the screw state and the ferromagnetic state, which is given by P [ J ( Q ) - J ( O ) ] , a transition of the first kind is expected, assuming that J(0)is larger thanJ(3 n/c)and J($ n/c),as has first been considered by Enz38 for dysprosium. Although the exchange energy itself is much larger than the anisotropy energy as noted before, it is quite possible that its difference between the screw state and the ferromagnetic state has a comparable order of magnitude with the anisotropy energy, particularly for the case in which the period of the screw is large, as in rareearth metals. The magnetostrictive energy would also make some contribution to this transition38339. For holmium, a proper screw structure is expected to be stable in the high temperature region in which the D term is dominant. As the temperature is lowered, the E and F terms become large. In particular, it is supposed that the growth of the E term, which has a positive sign, induces a transition from the screw state to the conical structure. For erbium the F term, which is positive and comparatively large, as shown in Table 4, will make the conical structure stable at low temperatures. The D term, which is dominant at high TABLE 4 Anisotropy constants of trivalent rare-earth ions in metals
S L J gJ gJJ *
*
312
9
0.220 0.119 0.138
<
-
3 3 6
2 7
2a.P r 2 ) (cm-1) 88J4 ( r 4 > 16y.P < r e >
A8
?I2
7/2
(A)
AbO As0 $.
Tb3+
0
A2'
Gd3+
0.211 0.111 0.125
- 0.154 0.141 - 0.105
'
Ho3+
Er3+
Tm3+
2 6
312
'5J2
8
1512
1 5 6
4/3
5i4
615
71%
10
10
9
7
0.203
0.195 0.096 0.099
0.187 0.088 0.088
0.180 0.082 0.078
- 0.144 -0.055 - 0.466 - 0.207 0.325 - 0.465
0.054 0.296 0.519
0.131 0.140 - 0.328
Dy3+ 5 ~ 2
5
0.104
0.112
___-__
6
D E F G
Calculated values by Freeman and Watson and interpolated values from them.
Referencesp. 293
280
[CH. V,
KEI YOSIDA
53
temperatures, makes the preferred axis of the ordered moment parallel to the c axis for erbium and thulium. Since for this case a cycloidal structure is not compatible with this anisotropy energy, the screw structure itself will be modified to some extent. In order to clarify what happens for this case, we shall investigate in more detail the behavior of the free energy near the NCel temperature on the basis of the molecular field approximation. For rare-earth ions, the good quantum number is the total angular momentum J, and the magnetic moment is parallel to J, being expressed by g,pueJ. However, in order to simplify the notation we use S in place of J in the following. First, along the line of the Hartree approximation, we divide the spin Si into two parts; the thermal average of Si and its deviation, as follows:
si= oi + (Si - oi).
(9)
Then, the exchange Hamiltonian can be expressed as
He, = Ho
+ H,',,
(10)
Ho = c i c j J ( R j- Ri)Oj.Gi - 2 C i X j J ( R j - R i ) O j * S i ,
(11)
Hix = - CiCjJ(Rj - RJ(Sj - oj)(Si - oi).
(12)
In the usual molecular field approximation, He:, which is of the second order in the deviation, is neglected. With .this simplification for the exchange Hamiltonian the free energy of the spin system can be calculated as
F
+
= CiCjJ(Rj - R i ) b j T i
n
where Ha, represents the anisotropy energy of the i-th spin given by eq. (6). The quantity oihas been introduced as a thermal average of the i-th spin, and it should be determined by the self consistency condition. However, in the present case this condition becomes difficult to solve because gi depends on i. Therefore, we assume the i-dependence of oi in an appropriate form including a feasible number of variational parameters, and we determine these parameters so as to minimize the free energy. The simplest trial function for the present purpose will be oix= oxcos Q * K i ,
-
uiy= o,,sin Q Ri
and
oiz= o, cos Q .Ri
.
(14)
The hodograph of the ordered spin given by these relations describes an Referencesp . 293
CH. V,
5 31
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
281
ellipse whose normal is inclined by an angle CI to the c axis in the xz plane, where c1 is given by tan a =az/ax. Inserting the trial functions (14) into (13) and expanding this in a power series with respect to a,, a,,and az,we obtain the following expression for the free energy up to the fourth power of a:
F N
+ [3 ( ~ 2 , ) ~- (s;)] cz + [
(15)
where <S,"> means the thermal average of S," in the presence of only the anisotropy energy, namely,
s; = S"
d D C O S " ~exp (- H J k T ) d D exp (- H J k T )
At high temperatures, all the coefficients in eq. (15) are positive, so that the equilibrium values of a,, ayand az are given by zero. As the temperature is lowered, the coefficient of a: or that of a: becomes negative. Then, a, or a, and a,,will take a non-zero value. If the preferred direction of the spin lies in the c plane (D>0), the coefficient of a: and a,"first comes to zero at the temperature determined, in the first order of anisotropy, by
It is noticed that E and F do not take part in this expression. Below this temperature the spin system takes a proper screw structure. This case corresponds to terbium, dysprosium and holmium. On the other hand, in the case where D is negative, the coefficient of af first becomes negative. Therefore, 0, becomes non-zero, while axand a,,are still zero below the temperature determined by
In this ordered state, only the z component of the spin oscillates sinusoidally. References p . 293
282
[CH.V, 0
KEI YOSIDA
3
This case corresponds to the high temperature phase of erbium and thulium. As the temperature is lowered further, the coefficient of 0,”becomes negative. This temperature is determined by 1 - 2B+(s’ - (St)) kT,
+ 161 (-) 2J(Q> kT, -
[(St) - (FS;)~] Q: = 0 .
(19)
Below this temperature, an oscillatory ordering of the y component of the spin sets in and thus the hodograph describes an ellipse whose major axis coincides with the c axis. This state is considered to correspond to the intermediate phase of erbium. The period of the y-component oscillation is the same as that for the z-component oscillation. If we assume a different period for the y-component oscillation, the last term of eq. (19) is multiplied by a factor of two. This means that the state in which the z and y components oscillate with the same period, is stable. For the x component the situation is somewhat different. If the oscillation of the x component appears, the ellipse will begin to tilt. However, this state is not favorable at all to the D term of the anisotropy, as well as the exchange energy. Therefore, the oscillation of the x component is not expected to appear except for special values of E and F. Eq. (15) for the free energy includes only J(Q). Therefore, the equilibrium value of Q is given by the value for which J ( Q ) takes a maximum, and it is independent of the temperature so far as the trial function (18) is assumed. However, the assumed form for ot does not satisfy the self consistency condition. This means that higher harmonics appear superposed on the fundamental tone. For the case in which only the z component is oscillatory, higher harmonics with odd multiples of the fundamental frequency are superposed. The amplitude of the lowest higher harmonic can be calculated in a similar way, or from the self consistency condition by putting oiZ= a,cosQ.Ri i a’~os3Q.R~.
(20)
The result is obtained for positive 4343, as,
It is seen from this result that this component tends to make the sinusoidal modulation squared because of its negative sign. This higher harmonic has an effect of changing the period, as is easily seen from the fact that in the References p . 293
CH. V, 9 31
MAQNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
283
expression of the free energy a term including J(3Q) appears. But this change in the period is very small. Actually, the change of the period is not observed in the high temperature phase of erbium nor in the whole temperature range of thulium, in which only the z component of the spin is ordered. The change of the period with temperature is observed for the proper screw structure of terbium, dysprosium and holmium, and for the intermediate phase of erbium. Elliott 27 has introduced the quadrupole-quadrupole interaction in order to explain the temperature dependence of the pitch of the screw. The quadrupole-quadrupole interaction between nearest neighbor atoms is shown to contribute the following additional term to the free energy (15) 29: 400
(1 + ~ c o s ~ Q c x’ )
e2 A = a2 < r2 )2 -IJ
R5
(n,),
where a is the constant which was defined in eq. ( 5 ) , R the distance between neigboring atoms on two adjacent c layers, 123 the cosine of the angle between the c axis and the direction joining these two nearest-neighbor atoms, and c’ a half of the lattice parameter c. For the case of D > 0, namely, ox= by= 0 and oz=0, Q has the following temperature dependence :
where Qodenotes the value of Q which makes J(Q) maximum. This result shows that the turn angle per layer decreases linearly with lowering temperature, because A is negative and sin 2Q0c’ is positive. This temperature dependence of the turn angle qualitatively agrees with the experimental results. However, the estimated value for the right-hand side of eq. (25) turns out to be too small in its order of magnitude when compared with the experimental results in dysprosium and holmium. For the rare-earth metals, the pseudo-quadrupole interaction, or more generally, the indirect interaction having a biquadratic form with respect to two interacting spins, is expected. Such higher order Referencesp . 293
284
[CH.V,
KEI YOSIDA
53
interactions are likely to make the pitch of the screw temperature-dependent. The other point which is to be mentioned about the change of the pitch with temperature is that the quadrupole-quadrupole interaction should also change the period for the sinusoidal modulation of the z component of the spin, as seen from eq. (22). In deriving (22), we have treated the spins classically and assumed that, for the case of the longitudinal oscillation of the ordered spin, the transverse component is averaged out by the thermal motion. However, for the temperature region in which the axial anisotropy energy is effective, the Ising model may be a more reasonable approximation. In such a case the biquadratic interaction is ineffective for changing the period with temperature. For thulium, in which all the parts of the anisotropy energies make the spin moment parallel to the c axis, the Ising model will be a good approximation in the whole temperature range. Foi this case we can understand consistently the reason why the period is independent of temperature, on the basis of the idea of the biquadratic interaction. Finally, we mention the effect of the anisotropy energy and the external field in the c plane on the proper screw structure. For simplicity, we confine our consideration to the special case in which the axial anisotropy is so large that the spin moment is restricted in the c plane. For this two dimensional case, the energy is expressed at the absolute zero of temperature as
Minimizing this equation with respect to pi, we obtain the equation determining pi as - 2S2 C jJ(Rj - Ri)sin ( p j - qi) - 6G sin 6pi = 0. (27) In the absence of the anisotropy energy, pi is given by pp = Q.Ri
+
CI,
where u is an arbitrary phase angle. A solution of eq. (27) can be obtained in a power series of G as follows: qi = qy
+ 6sin6py + ...
(28)
and 6 is calculated as
Since exp (ipi) is expressed with the use of (28) as exp (iqi) = exp (ipp) + 4s (exp (7ipp) - exp (- Siyly)) References p . 293
+ ...,
(30)
CH.
v, 0 41
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
285
the deformation of the screw structure, due to the six-fold anisotropy energy, gives rise to the seventh and fifth order satellites in neutron diffraction patterns which have been observed in holmium by Koehler et a1.6. However, the second order satellites also observed in holmium can not be explained. For the external field the Zeeman energy -gpBSH cos y i comes in place of G cos 6pi in eq. (26). Accordingly, the deformation of the screw due to a small external field gives the constant polarization and the second harmonic. The deformation of the screw by strong magnetic fields in the c plane has been studied by Herpin and MCriel 40, Enz 38 and more generally by Nagamiya et a l . 3 0 1 4 1 . According to their results, a transition from the modified screw structure described above to a new phase occurs, in which the spins oscillate back and forth in the c plane around the field direction and then this fan-like structure changes to the ferromagnetic state as the field is further increased. This behavior accounts for the magnetization curve obtained in the antiferromagnetic region of dysprosium. However, in order to explain the magnetization curve with two or three steps obtained in holmium, the present simple treatment should be improved by taking into account the effects of temperature and anisotropy energy, including the D,E and G terms. 4. Relation between the Fermi Surface and the Screw Structure
In the preceding section, it has been shown that the various types of magnetic structure found in heavy rare-earth metals can be regarded as the screw structure, deformed by the comparatively large anisotropy energy characteristic of each trivalent rare-earth ion. The oscillatory character of the magnetic structure is due to the exchange interaction, for which the Fourier component of the exchange integral J ( Q ) has its maximum at a non-zero value of Q . It would be reasonable to consider that the main part of the exchange interaction in rare-earths is produced by the exchange coupling of the conduction electrons with the localized spins, because the overlap between the 4f wave function of two neigboring atoms is expected to be small. Therefore, the oscillatory character of the exchange interaction will have some connection with the conduction band structure. The exchange interaction between the conduction electrons and the 4f spins is assumed as the following conventional form42:
where si and ri and Snand Rnrepresent the positions and the spins of the conduction electron and the rare-earth ion, respectively. This expression can be written in a more fundamental way as References p . 293
286
[CH.V, 5 4
KEI YOSIDA
Hex
=
-N-lxqxk,ng(q)e
+ a;+qfak&sn-
- iqR, [(a;+qtakf
+ a;+q&akfSn+]
- a;+qJ,ak+)Snz +
(32)
Y
where @ * k t and a k t are the Fermion operators for the conduction electron with the wave vector k and the up spin, and S,,, is given by S,,, = S,,, & isny. Generally, the coupling constant of the s-f exchange interaction g (4) will depend on the vectors k and k + q , but the q-dependence is the most important in the present case. By using the adiabatic approximation and by treating eq. (32) as a perturbation to the kinetic energy of the free electrons, we obtain the Ruderman-Kittel interaction43 between the localized spins, and then the Fourier transform of the exchange integral J ( Q ) can be calculated for this indirect exchange. However, in order to look at the Ruderman-Kittel interaction from another angle, we first put in eq. (32)
S,,, = SCOSQ-R,,, S,,, = SsinQeR,, and
S,,, = 0 ,
(33)
assuming a complete helical ordering for the localized spins. Then, eq. (32) can be expressed as Hex
=
-S
+
~ K ~ k g ( ( K QI){F(-K)u;+Q+K$ukt
+
(34)
+F(K)a;tak+Q+K&>,
1 F ( K ) = -Ciexp(if(.R,), N"
(35)
where K denotes the reciprocal lattice vector, F ( K ) the structure amplitude, and Nu the number of the atoms in the unit cell. The summation over i is taken over these atoms. This equation expresses the periodic potential with the wave vector Q acting on the conduction electrons, and each summand specified by K gives rise to an energy gap across the plane + ( K + Q ) in the k space, where - and signs are taken for up and down spins, respectively. The magnitude of this energy gap is given by 2Sg(l K + Ql) IF(K)I. If we treat the conduction electrons as free, the change in the electronic energy caused by the formation of these new Brillouin zone boundaries can be calculated to the second order in g(1 K + Ql) by the usual method of perturbation as
+
Referencesp . 293
CH.v, I 41
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
287
where n k is the occupation number of the state with k, and Ekis its kinetic energy in the absence of the s-f exchange interaction. E, is the Fermi energy, N, the total number of the conduction electrons and kf the Fermi wave vector. The functionf(x) is defined by
and it is shown in Fig. 3. Eq. (36) is, of course, equal to the sum of the interaction energy between the
- x
Fig. 3.
Plot off(x).
4f spins interacting with the Ruderman-Kittel interaction 429 43 and their self energies. The self energy is calculated as 3 N, dEzs = - N S 2 - - - N N - 1 C q g ( q ) 2 f 8 NE,
Therefore, J ( Q ) defined by eq. (3) is obtained for the Ruderman-Kittel interaction as
Since it is very difficult to proceed to the higher order calculation particularly at non-vanishing temperatures, we shall discuss the ground state energy for the screw structure on the basis of eq. (39). g(q) will drop to a small value when q becomes larger than the reciprocal of the 4f radius, but if we assume that g (q) is Referencesp . 293
288
KEI YOSIDA
[CH. V,
84
constant, J ( Q ) can be expressed by the sum in the real space as
where the sum over j is taken over all the lattice points in the real space except the origin, and G(x) is defined by G(x) =
xcosx - sinx x4
The change of J ( Q ) with Q along the c axis has been calculated by Woll and Nettel 44, Yosida and Watabe 45, de Gennes and Kaplan and Lyons 46 for a hcp lattice with three valence electrons per atom. Their results show that J ( Q ) actually has its maximum at the non-zero value of Q corresponding to a turn angle equal to 50"- 60". The ambiguity in the numerical value originates from the difference in the accuracy of calculation. In order to gain an insight into the behavior of J(Q), it would be more convenient to use eq. (39), i.e., the expression in the k space. If we assume that g ( q ) is constant, J ( Q ) - J ( 0 ) is proportional to
The behavior of I ( Q ) is determined by the reciprocal lattice structure, the diameter of the Fermi sphere and the functional form off(x). As shown in Fig. 3 , f ( x ) is monotonically decreasing. But its second derivative is negative for x < 1 and positive for x> 1, and the first derivative becomes minus infinity at x = 1. Therefore, the contributions to I ( Q ) from the lattice points TABLE 5
Values of f ( x ) , F ( K ) and number of equivalent points for reciprocal lattice sites of hcp with C / Q = 1.57. kr = 1.6152n/a corresponds to three valence electrons per atom. K is expressed by K = (21r/a). (nz,nyl/$, a nJc)
0 0.7149 0.7887 0.8164 1.0645 1.2382 1.382 1.4298
References p . 293
2 1.6135 1.5118
1.4682 0.7832 0.5177 0.3973 0.3614
1 (1 i id31 / 4 1 (3 i i d 3 ) / 4 (1 f i d 3 ) / 4 1 id3)/4 (3 (1 i i d 3 ) / 4
+
1 6 2 12 12 6 12 6
CH. v, 6
41
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
289
with K<2kf become considerably different from those from the lattice points with K > 2kf and the major parts of the change in I ( Q ) come from the contribution of the lattice points for which Kis near 2kf.Table 5 shows the values off(+K/k,), F ( K ) and the number of equivalent points for the reciprocal lattice points of the hexagonal close-packed structure with the c/a ratio of 1.57, assuming three valence electrons per atom. As seen from this table, for the first four sets of the reciprocal lattice points, namely, (000), (110), (002) and (1 1l), x is less than unity and the Brillouin zone boundaries produced by these lattice points cut the Fermi surface. The contributions to I ( Q ) from these sets of reciprocal lattice points decrease with increasing Q , as expected from the functional form off (x) described above. However, the contributions from larger values of K increase with Q , except for the sets for which K, is small compared with K, and K,,, such as (200) and (220).Thus, the increase in J ( Q ) with Q is caused by the contributions from the sets of K larger than 2 kf.In particular, the contribution from the (112) set of points is largest because +K/kf is close to unity, and the derivative of J ( Q ) becomes plus infinity at the value +Q/k,=0.089, satisfying the condition for the Kohn anomaly 47, namely,
I KI
12
-QJ
=2 k ,
(43)
reflecting the behavior off(x) at x = 1. For this value of Q , the plane of the energy discontinuity just touches the Fermi surface. For larger values of Q the increasing tendency of I(Q) is cancelled by the decreasing tendency of the contributions from the inner four sets of K, and a maximum appears at a slightly larger value than that determined by (43). The precise value for maximum J ( Q ) depends on how far the summation is taken over K, namely, on the effective range of g (q), since the convergence of this summation is not
J
(0)-
References p. 293
290
[CH. V, 0
KEI YOSIDA
4
good. Taking the summation up to K=(117), we obtain +elkf -0.11, corresponding to a turn angle of about 50" K. The Q-dependence of J ( Q ) is schematically shown in Fig. 4. As the effective range of g(q) becomes smaller and, therefore, the upper limit of Kfor the summation in I ( Q )becomes smaller, the value of Q denoted by Q, which makes J ( Q ) maximum will get smaller, since the contributions from the larger values of K tend to increase with Q. I n Fig. 5 , the turn angles of heavy rare-earth metals at the NCel temperature and low temperatures are plotted against the atomic number. In this figure, we can see that the Q,value decreases as the atomic number decreases. A reason for this tendency
02-
0,c2lT
01-
Gd
Tb
Dy
Ho
Er
Trn
Fig. 5. Values of Qo for heavy rare-earth metals. Solid circles and crosses indicate the values at the Nee1 temperature and the constant values at low temperatures, respectively.
may be attributed to, besides the difference in band structures, the fact that with decreasing effective nuclear charge the 4f radius increases relative to the lattice parameter and the range of g(q) becomes shorter. The change in the c/a ratio will not give such a large change in Q,. According to the experimental results, the turn angle, which is proportional to Q, decreases linearly with lowering temperature and reaches a constant value at low temperatures, except for Tm, which has a temperature-independent Q,. If we attribute the cause for this temperature variation of Q, to the biquadratic interaction, the constant value of Q, at low temperatures can be considered to correspond to the value determined by eq. (43), namely, the value for the Kohn anomaly, since the biquadratic interaction can not make the value of Q, smaller than this value where J @ ) has an infinite derivative. If this interpretation is correct, we can deduce the radius of the Fermi sphere from the constant value of Qo at low References p . 293
CH. V,
5 41
29 1
MAGNETIC STRUCTURES OF HEAVY RARE-EARTH METALS
TABLE 6 kfo is the Fermi wave vector for free electrons
Radius of Fermi sphere.
kriizikio
Gd
Tb
DY
Ho
Er
Tm
1.045
1.034
1.021
1.006
0.996
0.984
temperatures. Table 6 shows the radius of the Fermi sphere in the direction of K1lz deduced in this way. In this table the value for gadolinium is evaluated by extrapolating the values for heavier metals. It seems that for gadolinium a simple ferromagnetic state is stable. Therefore, it is supposed that for this metal J ( Q ) has its maximum also at Q = 0, and this value is larger than the other maximum near the point of the Kohn anomaly. An increase of the overlap between the 4f wave functions with decreasing nuclear charge may make the direct exchange interaction appreciable, especially for gadolinium, terbium and also for europium. If the direct interaction is ferromagnetic, this also has the tendency of decreasing the turn angle. The neutron diffraction experiments on Tb-Y alloys made by Koehler et aZ.48 are quite suggestive. The change in the turn angle with concentration of Y is shown in Fig. 6 . As this figure shows, the turn angle increases and its temperature variation becomes smaller with the concentration of yttrium. For high concentrations of yttrium the turn angle becomes temperatureindependent. The disappearance of the temperature dependence of the turn I
I
0
25
I
I
I
I
50
75
100
Concentration (at % Tb in Y )
Fig. 6. Interplaner turn angle of Tb - Y alloys. Open and solid circles indicate the values at the Ntel temperature and the values at low temperatures,respectively (Koehler, Child, Wollan and Cable4*).
References p . 293
292
KEI YOSIDA
[CH.V, 0 5
angle means that the biquadratic interaction is short-ranged. The increase of the turn angle with yttrium may be due either to the fact that the lattice parameter of metallic yttrium is slightly larger than that of terbium or to the fact that the ferromagnetic direct interaction, which is short-ranged, makes the turn angle smaller in terbium. Experimentally, the temperature range in which the screw structure is stable is made broader, and for concentrations higher than 25 at % yttrium, the ferromagnetic region disappears. This will be consistent with a rapid increase in J(Q,)-J(0). The value of the turn angle for low concentrations of Tb tends to the value of about 48", which is slightly smaller than the value for erbium and thulium. This shows that the Fermi surface of yttrium is quite similar to that of heavy rare-earth metals although the valence electrons of yttrium are in the 4d- and 5s-states. The screw arrangement of the ordered moment gives rise to the energy gaps across the planes of :(KfQ) in the k-space, as mentioned before. These new energy gaps cause an increase of the effective mass or a decrease of the effective number of the conduction electrons. Therefore, the electrical resistivity will increase with a growth of the screw-type ordering. On this basis, the anomalous behavior of the electrical resistivity of rare-earth metals has been discussed by A r r ~ t and t ~ Mackintoshso, ~ and detailed calculations have been made by Miwas1 and also by Elliott and Wedgwood55. 5. Summary
It has been shown that the magnetic structures found in the heavy rare-earth metals are stabilized by the exchange interaction, which by itself makes a simple screw structure stable, and by the crystalline anisotropy energy. The dominant anisotropy energy is the one-ion anisotropy energy possessed by each rare-earth spin, and is expressible in terms of the spherical harmonics up to the sixth order. The exchange interaction arises mainly from the indirect interaction produced by the exchange interaction between the conduction electrons and the localized spins, though the direct exchange may have some minor effects on the magnetic properties for terbium and gadolinium. The indirect interaction plays a decisive role for the screw structure of the ordered spins. In particular, the occurrence of the screw structure is connected with the fact that some of the Brillouin zone boundaries are situated just outside the Fermi surface. This has been shown for a hcp lattice with three valence electrons per atom on the basis of the free electron model. Since the valence electrons in rare earths are in the 5d- and 6s-states, the free electron picture should be modified to some extent. However, the References p. 293
CH. V]
MAONETIC STRUCTURES OF HEAVY RARE-EARTH METALS
293
essential mechanism for the screw structure is supposed to be the same. In concluding this article, we add a mention of the light series of rare-earth metals. The crystal structure of light metals is ABAC-hexagonal, and for this structure the structure amplitude IF(K)I for K112 is only a half of the value for ABAB. On the other hand, the 4f wave function is more extended and its energy level is expected to be higher. This makes the direct exchange larger and the range of g(q) smaller. Thus, the maximum of J ( Q ) a t Qo may be smeared out. A large 4f orbit also makes the anisotropy energy larger and the quadrupole-quadrupole interaction more important, and further increases the degree of the s-f mixing, which is expected to be small for the heavy series, as discussed by Roche1-5~. For europium, which is the last element of the light series, the s-f mixing and the effect of the direct exchange might be appreciable since its 4f radius is particularly large because of doubly ionized cores. According to neutron diffraction experimentss3, metallic europium has a screw structure below 87" K. The screw axis is along the <100> direction and the interplaner turn angle is equal to 50". For this case, the free electron picture does not work successfully for explaining this screw structure. Therefore, the Fermi surface of europium seems to be considerably deformed from a sphere. The deformation of the Fermi surface and the anomalous magnetic behavior at low temperatures found by Bozorth and Van Vleck54 might also be attributed to the large 4f radius and high 4f energy levei.
Acknowledgments I should like to thank Dr. Koehler for his kind advice in the description of the experimental results. Notes added in proof:
1. The magnetic and electrical measurements on single crystals of metallic terbium have been reported by D. E. Hegland, S. kgvold and F. H. Spedding, Phys. Rev. 131,158 (1963). 2. The spin structures of heavy rare-earth metals have been discussed by A. W. Overhauser on his viewpoint of the spin density wave, J. Appl. Phys. 34, 1019 (1963). REFERENCES
2 3
F. H. Spedding, S. Legvold, A. H. Daane and L. D. Jennings, Progress in low temperature physics, Ed. C. J. Gorter, Vol. 11, Chap. 12 (North-Holland Publishing Co., Amsterdam, 1957). W. C. Koehler, J. Appl. Phys. 32, (1961) 20s. M. K. Wilkinson, W. C. Koehler, E. 0. Wollan and J. W. Cable, J. Appl. Phys. 32, 48s (1961).
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J. W. Cable, E. 0. Wollan, W. C. Koehler and M. K. Wilkinson, J. Appl. Phys. 32, 49s (1961). 5 M. K. Wilkinson, H. R. Child, W. C. Koehler, J. W. Cable and E. 0. Wollan, J. Phys. SOC.Japan 17,Suppl. E I I I , 27 (1962). 6 W. C. Koehler, J. W. Cable, W. 0. Wollan and M. K. Wilkinson, J. Phys. SOC.Japan 17,SUPPI.B-111, 32 (1962). 7 F. H. Spedding, A. H. Daane and K. W. Hermann, Acta Cryst. 9,559 (1956). 8 C. J. McHargue, H. L. Yakel Jr. and L. K. Jetter, Acta Cryst. 10,832 (1957). 9 F. H. Spedding, J. J. Hanak and A. H. Daane, Trans. AIME 212, 379 (1958). 1" J. F. Elliott, S. Legvold and F. H. Spedding, Phys. Rev. 91, 28 (1953). 11 K. P. Belov, R. 2. Levitin, S. A. Nikitin and A. V. Pedko,Zhur. Eksp. Theoret. Fiz. 40,1562 (1961). 12 K.P. Belov and A. V. Pedko, Zhur. Eksp. Theoret. Fiz. 42,87 (1962). 13 C. D. Graham Jr., J. Phys. SOC. Japan 17, 1310 (1962). 14 W. D. Comer, W. C. Roe and K. N. R. Taylor, Proc. Phys. SOC.(London) 80, 927 (1962). 15 R. M.Bozorth and T. Wakiyama, J. Phys. SOC.Japan 17, 1669, 1670 (1962). 16 W. C. Thoburn, S . Legvold and F. H. Spedding, Phys. Rev. 112, 56 (1958). 1 7 W. C. Koehler, E. 0. Wollan, M. K. Wilkinson and J. W. Cable, Rare-earth research developments conference, Lake Arrowhead, California (1960). 18 D.R. Behrendt, S. Legvold and F. H. Spedding, Phys. Rev. 109,1544 (1958). 19 D. L. Strandburg, S. Legvold and F. H. Spedding, Phys. Rev. 127,2046 (1962). 20 R. W. Green, S. Legvold and F. H. Spedding, Phys. Rev. 122,827 (1961). 2 1 W. C. Koehler, J. W. Cable, E. 0. Wollan and M. K. Wilkinson, J. Appl. Phys. 33, 1124 (1962); Phys. Rev.126,1672 (1962) 22 B. L. Rhodes, S. Legvold and F. H. Spedding, Phys. Rev. 109, 1547 (1958). 23 D. D. Davis and R. M. Bozorth, Phys. Rev. 118, 1543 (1960). 24 A. Yoshimori, J. Phys. SOC.Japan 14, 807 (1959) and T. Nagamiya, J. phys. radium 20, 70 (1969). 25 T. A. Kaplan, Phys. Rev. 116,888 (1959). 26 J. Villain, J. Phys. Chem. Solids 11, 303 (1959). 27 R. J. Elliott, Phys. Rev. 124,346 (1961), J. Phys. SOC.Japan 17,Suppl. B I , 1 (1962). 28 T. A. Kaplan, Phys. Rev. 124, 329 (1961), J. Phys. SOC.Japan 17,Suppl. B I , 3 (1962). 28 H. Miwa and K. Yosida, Progr. Theor. Phys. (Kyoto) 26, 693 (1961), J. Phys. SOC. Japan 17,Suppl. B-I, 5 (1962); H. Miwa, thesis, University of Tokyo (unpublished). 30 T. Nagamiya, J. Appl. Phys. 33, 1029 (1962). 31 R. J. Elliott and K. W. H. Stevens, Proc. Roy. SOC.A 215,437 (1952). 3 2 K. W. H. Stevens, Proc. Phys. SOC.(London) A 65, 209 (1952). 33 A. J. Freeman and R. E. Watson, Phys. Rev. 127,2058 (1962). 34 G. Burns, Phys. Rev. 128,2121 (1962). 35 C. Zener, Phys. Rev. 96, 1335 (1954). 36 F. Keffer, Phys. Rev. 100, 1692 (1955). 37 J. H. Van Vleck, J. phys. radium 20, 124 (1959). 38 U. Enz, J. Appl. Phys. 32,22s (1961), Physica 26, 698 (1960). 3 9 J. Smit, J. Phys. SOC.Japan 17,9 (1962). 40 A. Herpin and P. Meriel, C. R. Acad. Sci. 250, 1450 (1960). 41 T. Nagamiya, K. Nagata and Y.Kitano, Progr. Theor. Phys. (Kyoto) 27, 1253 (1962), J. Phys. SOC. Japan 17,Suppl. B-1, 10 (1962). 42 T. Kasuya, Progr. Theor. Phys. (Kyoto) 16, 45 (1956); K.Yosida, Phys. Rev. 106, 893 (1957). 43 M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954). 44 E. J. Woll Jr. and S. J. Nettel, Phys. Rev. 123,796 (1961). 4
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K. Yosida and A. Watabe, Progr. Theor. Phys. (Kyoto) 28, 361 (1962). P. G. de Gennes, C. R. Acad. Sci. 247,1836 (1958), also to be published; T. A. Kaplan and D. H. Lyons, Phys. Rev. 129,2072 (1963). 47 W. Kohn, Phys. Rev. Letters 2, 393 (1959); E. J. Woll Jr. and W. Kohn, Phys. Rev. 126, 1693 (1962). 48 W. C. Koehler, H. R. Child, E. 0. Wollan and J. W. Cable, J. Appl. Phys.34,1335 (1963). 49 A. Arrott, ‘Antiferromagnetism in Metals and Alloys’ in Magnetism, Ed. G. Rado and H. Suhl (Academic Press Inc., New York, to be published). 50 A. R. Mackintosh, Phys. Rev. Letters 9, 90 (1962). 5 1 H. Miwa, Progr. Theor. Phys. (Kyoto) 29, 477 (1963); also thesis, University of Tokyo (unpublished). 52 Y.-A. Rocher, Advances in Physics 11,232 (1962). 53 C. E. Olsen, N. G. Nereson and G. P. Arnold, J. Appl. Phys. 33, 1135 (1962). s4 R. M. Bozorth and J. H. Van Vleck, Phys. Rev. 118, 1493 (1960). 55 R. J. Elliott and F. A. Wedgwood, Proc. Phys. SOC.(London) 81,846 (1963). 45
46
C H A P T E R VI
MAGNETIC TRANSITIONS BY
C. DOMB KING’SCOLLEGE, UNIVERSITY OF LONDON AND
A. R. MIEDEMA
KAMERLINGH ONNESLABORATORY, LEIDEN
CONTENTS: 1. Introduction, 296. - 2. The Ising model, 299. - 3. The Heisenberg model, 304. - 4. Remarks on the analysis of experimental data, 307. - 5. Experimental data, 310. - 6. Comparison between experiment and theory, 333. - 7. Conclusions, 339.
1. Introduction Transition in solids involving discontinuous behaviour of specific heats and allied thermodynamic functions have for many years claimed the attention of experimental and theoretical research workers. First order transitions could be understood in terms of the standard thermodynamic picture of independent phases in equilibrium ;but particular interest has been attached to higher order transitions which did not involve a finite change of entropy at the transition point. The earliest example of such a transition was the Curie point of a ferromagnet with its associated specific heat anomaly, disappearance of spontaneous magnetization, and infinite magnetic susceptibility. A second example was provided by order-disorder transitions in alloys, and, as experimental work on the physics of the solid state progressed, more and more such anomalies were discovered, arising from antiferromagnetism, rotation of molecules in solids and a number of other causes. Development of low temperature techniques in recent years has led to a great increase in experimental data available on magnetic transitions. In fact any solid with magnetic interactions between its constituent units will lead References p . 340
296
CH. VI, 5 11
MAGNETIC TRANSITIONS
297
to a transition point at a temperature of order Elk, E being the magnitude of the interaction; for most magnetic salts these temperatures are of the order of degrees or fractions of a degree Kelvin. The first theoretical attempt to account for the properties of a ferromagnet was made many years ago by Pierre Weiss 1, who postulated the existence of a large “internal field”. With the aid of this hypothesis Weiss was able to reproduce the most important physical features of a ferromagnet, the existence of a Curie temperature T,, of a spontaneous magnetization below the Curie temperature, and of a magnetic susceptibility proportional to 1/(T - T,) above the Curie temperature. A similar treatment for antiferromagnets was given later by Landau2 and NCe13. An extensive review of molecular field treatments of antiferromagnetism can be found in ref. 3’. However, the origin of the “internal field” was not discussed in detail, and thus a statistical formulation in terms of atomic interactions was not possible. In 1925 king4 attempted such a formulation, although the form of interaction which he took was somewhat empirical; in 1928 Heisenberg5 used a quantum mechanical treatment of atomic forces, and suggested an interaction with a more rigorous theoretical foundation. The king and Heisenberg models will be considered in detail in Sections 2, 3 respectively. Using simple mathematical approximations the results of the Weiss theory can be re-derived for either of these models. But although qualitative agreement with experiment is thereby provided, further investigation shows that the mathematical approximations are inadequate as a basis for detailed comparison with experiment. In connection with the analogous problem of order-disorder transitions in alloys Bragg and Williams introduced the fundamental concept of long range order which is essential to a clear understanding of the nature of most higher order transitions ; they derived formulae essentially equivalent to those of Weiss. Bethel showed how to introduce a parameter to take account of short range order, and hence laid the foundation of an improved approximation for the Ising model. However, the complexity of the statistical problem, and the inadequacy of the approximations which had been introduced were clearly demonstrated by the exact solution by Onsager8 in 1944 of the king model for the two dimensional quadratic lattice. This exact solution depends on an infinity of order parameters, whereas approximations take account of only one or two of them. The form of the specific heat curve in Onsager’s solution differs substantially from those obtained by the approximations, but differs equally markedly from experimentally observed specific heat curves. PerReferences p . 340
298
C. DOMB A N D A. R. MIEDEMA
[CH. VI,
81
haps the most striking difference is the large “tail” of the Onsager curve, which shows that a substantial fraction of the total entropy change of the system takes place in the temperature region above the Curie point. It is well appreciated that the interactions in solids which give rise to specific heat anomalies are usually quite complex, and the Ising and Heisenberg models are only put forward as a preliminary first step. But the validity of the model can only be properly assessed if reliable theoretical information is available regarding its properties. Thus we wish to know if the differences between the Onsager specific heat curve and experimental results are due to the two dimensional nature of the Onsager solution, or to the inadequacy of the Ising model as a representation of the interactions. Unfortunately the methods which were introduced by Onsager and subsequent authors for the exact solution of the Ising model for two dimensional lattices fail completely in three dimensions, and virtually no progress has been made with the exact approach to this problem. Theoretical research has thus been confined largely to improve closed form approximations, and to the derivation of exact series expansions at high and low temperatures. But particular techniques have been developed in the past few years which enable such series expansions to be extended considerably97lo, so that predictions of the asymptotic form of the coefficients (which is closely related to critical properties of the model) can be made with considerable confidence. When the coefficients in the series expansions are not uniform in sign some form of transformation equivalent to analytic continuation must be used before critical properties can be calculated. The introduction by Baker11 in 1961 of the Pade approximant has been very fruitful in this connection and has substantially added to our knowledge of the mathematical behaviour of thermodynamic functions in the critical region. Progress with closed form approximations has been less spectacular, but has in general provided confirmation of the predictions of series expansions. In regard to the Heisenberg model theoretical developments have proceeded more slowly. Series expansions are more difficult to derive and calculations available so far provide fewer terms. The spin wave picture introduced by Bloch12 in 1930, has been particularly helpful at low temperatures and by using in addition the calculations of spin wave interactions made by Dysonl3, thermodynamic properties can be predicted in this temperature range. However, the formulae cease to be valid well before the critical temperature is reached. Nevertheless the authors feel that sufficient theoretical information is now available regarding the properties of the Ising and Heisenberg models to References p . 340
CH. VI, 0
21
299
MAGNETIC TRANSITIONS
warrant a serious comparison with experimental results, and it is with this aim that the present article has been written. They do not claim to deal exhaustively with all experimental data available on magnetic transition points. But they hope to focus interest on a number of substances for which a fair measure of agreement between theory and experiment has been reached; and to draw attention to further experimental data and theoretical developments which are needed in this field. 2. The Ising Model
The lsing model assumes a semi classical type of interaction between magnetic spins in a crystal lattice. An atom or ion of spin s has (2s 1) possible orientations relative to a magnetic field, and the interaction between the spins is taken as proportional to sciscj,the product of the components of the spins. We may thus write for the Hamiltonian in a magnetic field H
+
<
the summation being taken over all nearest neighbour pairs i , j in the lattice. The constant factors in (1) have been chosen so that as s varies the maximum interaction between neighbouring spins remains equal to J' and the maximum magnetic moment remains equal to m ; this is useful for comparing the thermodynamic and magnetic properties of the model for different values of s and particularly for considering the limiting case s+co. The Ising model thus corresponds to extreme magnetic anisotropy since there is no interaction between x and y components of spin. When s = 3 only two orientations of spin are possible, parallel or antiparallel to the external magnetic field. This is the original model considered by Ising, and a number of simplifying features arise. Most theoretical discussions (including the exact two dimensional solutions referred to in Section 1) correspond to this case. But it has also been possible using series expansions to assess the effect of change of s. In this section we shall endeavour to provide a general summary of the critical properties of the Ising model. We shall only quote the results, but reference will be given to original papers from which a detailed justification can be obtained.
2.1. GENERAL REMARKS. THERMODYNAMIC PROPERTIES The thermodynamic and magnetic properties of one dimensional models References p , 340
300
C. DOMB A N D A. R. MIEDEMA
[CH. VI,
T A B L E1 Critical values for king model(s Lattice structure
4
Linear chain Honeycomb Simple quadratic Triangular Diamond Simple cubic Body centered cubic Face centered cubic Weiss approximation
2 3 4 6 4 6 8 12 a,
= 3)
S, - S c
Ec
EQ
k Tc
Sc -
4 J’
k
k
k Tc
EC -k Tc
0 0.506 0.567 0.607 0.676 0.752 0.794 0.816 1.000
0 0.265 0.306 0.330 0.511 0.560 0.586 0.591 0.693
0.693 0.428 0.387 0.363 0.182 0.133 0.107 0.102 0.000
0 0.227 0.258 0.275 0.418 0.447 0.460 0.463 0.500
cc 0.761 0.623 0.549 0.322 0.218 0.169 0.150 0.000
~
02
-
~~~
do not show discontinuities except at T = 0. Two and three dimensional lattice models have Curie points with characteristic discontinuities, and it is usually convenient to regard a one dimensional model as having a Curie point at T = 0. The dimensionality of a lattice model rather than its detailed lattice structure is the prime factor in determining the behaviour of the modelg. Thus the two dimensional triangular lattice (coordination number q = 6 ) is much closer in behaviour to the simple quadratic (S.Q.) lattice ( q = 4) than to the simple cubic (S.C.) lattice (q = 6). Similarly the loosely coordinated diamond lattice (q = 4)14 is nearer to the close packed cubic lattice (q = 12) than to the simple quadratic lattice (q = 4). The Weiss “internal field” approximation represents an asymptotic limit as q +00. Increasing q, or equivalently increasing the range of interaction, will make the properties of the model approach those of the Weiss approximation. The above features are well illustrated by reference to Table 1 in which critical properties of the Ising model of spin 3 are listed for a number of lattices. In order to assess the difference between the specific heat curves for the various lattices it is convenient to take T, as unit of temperature, and consider the magnetic part of the specific heat C , as a function of t( = TIT,). Then 1
0 m
References p. 340
CH. VI,
8 21
MAGNETIC TRANSITIONS
301
so that a tabulation of S, and S , - S, for various lattices enables us to compare the magnitude of the specific heat curves below and above the Curie temperature. The sum of the two terms, S,, is the same for all lattices and is equal to kln2(= 0.693k). These quantities are also particularly useful for comparison with experiment, and hence for testing the validity of a given model, since they do not depend on the magnitude of the interaction energy J’. It is useful similarly to tabulate (E, - Eo)/kT,
= 0 m
which represent directly the areas under the specific heat curve below and above the Curie point. The sum of these two terms, - Eo/kT,, is no longer constant, but decreases from co to a limiting value of 4 as q+m. It will be seen that there is a major difference between the results for two and three dimensional lattices. The “tail” of the specific heat curve is much smaller for the latter, and we can state definitely that one of the major differences between the Onsager curve and experimental observation is due to the two dimensional character of the Onsager calculation. The figures quoted in Table 1 are exact for two dimensional lattices and estimates of S,, E, should not be in error by more than 1% for three dimensional lattices. The error in the critical temperature is very much smaller. Calculations for different values of s have been confined to the face centered cubic (F.C.C.) lattice as a suitable representative of a three dimensional model15; the high temperature series expansions derived converge smoothly and the accuracy of the estimates should be comparable with those of spin 3. The most immediate effect of increasing s is to increase the total entropy change of the system from T = 0 to T = co, which is given by kln(2s + 1). But it will be seen from Table 2 that nearly all of this increase in entropy takes place in the region below the Curie temperature; even when s goes to infinity the increase in entropy above the Curie temperature is not more than 30%. Critical constants are given in Table 2 for s =$, 1, 2, co.Values for intermediate s can readily be found by interpolation in 11s. The specific heat singularity for two dimensional lattices with s = 3 is References p . 340
302
[CH. VI,
C. DOME AND A. R. MIEDEMA
52
TABLE2 Critical values for Ising model with general s (F.C.C. lattice) S
3~kTc qJ‘(s 1)
sc
S, - S c
EC - EQ
k
k
kTc
Ec -k Tc
0.816 0.851 0.864 0.874
0.591 0.983 1.486 co
0.102 0.116 0.123 0.131
0.463 0.721 0.990 1.541
0.150 0.160 0.167 0.175
3 1
2 03
-
+
of the form Alnl T - T, I, so that there is a logarithmic infinity on both sides of the Curie point. For three dimensional lattices it is fairly clear that the specific heat remains infinite at the Curie point, but its detailed form has not yet been properly established. There is a good deal of evidence to suggest that it has a logarithmic infinity below the Curie temperature1411~; on the high temperature sidela. 17 it is either logarithmic with a substantially higher value of A , or of the form (1 - T,/T)-”” with n about 5. There is not much detailed information about the effect of change of s on the specific heat singularity. Qualitatively we can say that an increase in s sharpens the high temperature part of the curve, but the precise magnitude cannot be assessed with the present number of terms available.
2.2. MAGNETIC PROPERTIES Although the inverse of the magnetic susceptibility of a ferromagnet in zero field varies linearly with temperature at sufficiently high temperatures, it curves appreciably on approaching the Curie point. For a two dimensional model (s = 4) the behaviour very near the Curie point is given by xo
N
B(1
- Tc/T)-”4.
(4)
This result was first suggested on the basis of an analysis of high temperature series expansions 18 and subsequently received more rigorous theoretical justificationl9. For a three dimensional model the corresponding estimate is xo
N
C(1
- Tc/T)-5’4.
(5)
Both formulae (4) and ( 5 ) remain valid when the spin differs from 3; in fact change of spin has only a small effect on the high temperature susceptibility, the magnitude being comparable with that due to change of lattice structure in a given dimension. At low temperatures the spontaneous magnetization deviates from unity References p . 340
CH. VI,
5 21
MAGNETIC TRANSITIONS
303
by terms of the form exp ( -a/kT). Near the Curie temperature (for s =)I! it goes to zero as M, N D(1 - T/TCJ1’8 (6) for a two dimensional model7 and as
M0 N E(l
- T/Tc)5’16
(7)
for a three dimensional model14.20. The first suggestion for the index in but subsequent investigation seems to indicate that is more (7) was likely to be correct. There appear to be no investigations of the behaviour of spontaneous magnetization for s different from 3. In the absence of a magnetic field, the thermodynamic properties of an antiferrornagnet and a ferromagnet are identical for “ordering lattices” (is. similar to the S.Q. or S.C. which can be decomposed into two equivalent sub-lattices). The initial magnetic susceptibility of an antiferromagnet for two-dimensional lattices has the form
near the Ntel point. This is a function having a maximum at a temperature
T, above the NCel temperature TNand a vertical tangent at the NCel point. The ratio T,/T, is approximately 1.537 for the S.Q. lattice and 1.688 for the honeycomb lattice. For three dimensional lattices23 the formula (8) needs only slight modifica-
where H - and H, indicate different values of this constant below and above the NBel temperature. The general form of the curve is the same as before, except that T,/T, is now closer to unity, typical values being 1.098 for the S.C. lattice and 1.065 for the body centered cubic (B.C.C.) lattice. Non ordering lattices (like the triangular lattice in two dimensions and the F.C.C. in three dimensions) have the particular feature that thermodynamic and magnetic quantities are continuous at all temperatures. For the triangular lattice an exact solution exists showing that the model has a finite entropyg at T=O; since this contradicts the third law of thermodynamics there must be other physical factors ignored by the model (e.g. small interactions of a different type) which would become significant a t the lowest temperatures Z4. The magnetic susceptibility has been accurately estimated over the whole temperature range25. For the F.C.C. lattice, on the other hand, Danielian has shown26 that there is no longer a finite entropy at T= 0, and hence no contradiction to the third law. References p. 340
304
C.DOMB AND A. R. MIEDEMA
[CH. VI,
53
For antiferromagnets with spin different from f no detailed calculations have been undertaken, but certain general arguments 2’ indicate that the magnetic behaviour in the critical region is qualitatively similar to that with s = 3.
3. The Heisenberg Model Using a valence bond type of approximation it can be shown that the energy of interaction of two atoms i, j with spins si, s j contains a term equal to - 2Jsi*sj,where J i s usually referred to as the exchange integral. The Hamiltonian for a ferromagnet suggested by Heisenberg5 is of the form
To compare models with different spin and to allow s to tend to infinity, it is convenient to rewrite (10) in the same form as (l),
i
where J’= 2Js2 and m = gps. The Heisenberg model corresponds to magnetic isotropy. It has been suggested that a model intermediate between Ising and Heisenberg with Haniiltonian
would conveniently take account of general anisotropy, but the properties have not so far been calculated in any detail.
3.1. GENERAL REMARKS. THERMODYNAMIC PROPERTIES Calculations for the Heisenberg model are much more difficult to carry out than for the Ising model. The only exact results available are for certain properties of a one dmensional chain; all estimates of critical behaviour depend on high temperature series expansions. No discontinuities arise in one or two dimensional systems (Curie temperature T, = 0) and we must proceed to three dimensions to derive standard properties of a ferromagnet. Curie temperatures are therefore lower, and specific heat “tails” larger than those for the Ising model with corresponding parameters. References p . 340
CH. VI,
8 31
305
MAGNETIC TRANSITIONS
The F.C.C. lattice is again chosen as a representative three dimensional lattice15 since it provides smoothest convergence for a given number of terms. The estimates are less reliable than for the Ising model; they are best for s+o3 and decrease in reliability as s decreases to 4.However, it is still considered that errors in critical constants should not exceed a few percent, and hence that a basis is provided for detailed comparison with experiment. T A B L E3 Critical values for Heisenberg model with general s (F.C.C. lattice) 3skTc S
qJ’(s
+1 2 CO
+ 1)
0.679 0.747 0.774 0.798
SC k
0.428 0.810 1.305 CO
S,
k
Ec - EO ~kTc
_ -Ec
0.265 0.289 0.304 0.322
0.291 0.555 0.833 1.406
0.439 0.449 0.459 0.474
-
Sc
kTc
From the results given in Table 3 it will be seen that the tail of the specific heat curve is nearly three times as large as for the Ising model. Only qualitative information is available regarding the form of the specific heat curve near the singularity; on the high temperature side it is less steep than the Ising curve with the same s, the Heisenberg curve for s = co being comparable with the king curve for s = 3.
3.2. MAGNETIC PROPERTIES As we might expect the curvature of the high temperature inverse susceptibility near the Curie point for the Heisenberg model is greater than for the Ising model and is given approximately by
xo z K ( l - T,/T)-4’3. This result has been established from high temperature series expansions by two independent methods, analysis of the magnitude of the coefficients15 and use of the Pade approximant 28. It seems to be valid for all s, although additional terms are desirable for confirmation when s = 3. The low temperature behaviour is very much more complicated than for the king model. Blochl2 first introduced the concept of a spin wave, and showed that at sufficiently low temperatures the spontaneous magnetization differs from unity by a term of order T - % .Dysonl3 first took proper acReferences p . 340
306
C . DOME AND A.
[CH.VI, 0
R. MIEDEMA
3
count of the interaction between spin waves, and hence derived the extended formula M/Mo = 1 - CIlT3 - UzT’ - C13T3 - /?1T4 O ( T e ) , (14)
+
where al,u2, a3, PI can readily be calculated for various lattices. But the formula (14) does not extend to the neighbourhood of the Curie point and we have no precise information about how the magnetization tends to zero. On general grounds we might expect a formula of the type b(1 - T/T,)” where n is less than &, the value for the Ising model. For the Heisenberg model there is no longer symmetry between a ferromagnet and an antiferrornagnet, except in the limit s+ 00. Even the ground state of an antiferromagnet is quite complicated, and an exact calculation exists only for a linear chain (s = +). BetheZ9showed that the lowest energy of a linear chain is E , = - NJ(21n2 - 0.5) =
-+NJ
x 1.7726,
(15)
i.e. 1.7726 times lower than - 2NJ(+)’, the corresponding value for a ferromagnet (some errors in Bethe’s work were corrected by Hulthen30). More generally the ground state energy for any lattice can be written31
+
E , = - 4Nq-2Js2(1 Y / ~ s ) ,
(16)
where y must lie between the limits
O
Mf = 4NgP(s - CI),
(17)
where approximations to CI are 0.197 for an S.Q. lattice, 0.078 for an S.C. lattice and 0.075 for a B.C.C. lattice. At low temperatures spin wave theory predicts that in the absence of anisotropy the excitation spectrum should follow a Debye pattern, and hence for cubic lattices the magnetic specific heat should be proportional to T 3 .As the temperature tends to zero also the perpendicular susceptibility should approach a limiting value
xI = NgZP2/4qJ. Little definite information is available regarding the thermodynamic behaviour of an antiferromagnet near the transition point. The transition temperature will in general be higher than for a ferromagnet3l”. In the limit References p. 340
CH. VI,
8 41
MAGNETIC TRANSITIONS
307
s+co the thermodynamic properties in the absence of a field are identical
with those of a ferromagnet; hence for large s we can use calculations for a ferromagnet to provide a reasonable approximation. Regarding the magnetic susceptibility in zero field Fisher %'* 28 has given general arguments to show that the behaviour is similar in form to that for theIsing model (maximum in xl, above the NCel temperature, vertical tangent at the transition point). In regard to the perpendicular susceptibility in cases of strong anisotropy Fisher has shown that the behaviour near the transition temperature is very similar to that of the parallel susceptibility; xL tends to a finite limit as T-t 0 and exact solutions can be given for the honeycomb and S.Q. lattices.
4. Remarks on the Analysis of Experimental Data Widely different experimental techniques have been used in investigating magnetically ordered materials. Apart from the usual magnetic and thermal measurements, conclusive information on the magnetic structure is obtained for many substances from neutron diffraction, nuclear resonance or electronic resonance experiments. The experimental data, given in the next section, will concentrate on those magnetic materials for which a reasonable insight into the kind of magnetic ordering and the range and type of magnetic interactions has been obtained, so that the results (mainly from thermal data) can be compared with the theoretical predictions given previously. 4. I. THERMAL DATA
Generally three terms contribute to the specific heat of a magnetic substance : a lattice term, an electronic term and the magnetic specific heat. Thus the final term is obtained by subtracting the contributions arising from lattice waves and electrons and this may reduce the accuracy considerably. Starting with the magnetic metals, the electronic terms may be written in the free electron model as
c, = C" = yT(1 - ~ n 2 ( ( T / T o ) 2 ) ,
(19)
where y = n2nk/2T,, Tois the degeneracy temperature and tz is the number of effective electrons. The contribution arising from the lattice waves can be represented by (see for instance 32) :
c, = (1 + a C T ) f p ) ¶ References p . 340
308
C.DOMB AND A. R,MIEDEMA
[CH.M, 5 4
where f (0, J T ) represents the usual Debye function, CI is the coefficient of volume expansion and G the Gruneisen constant. The constants y and 8 D can be obtained by matching the theoretical specific heat in a temperature region where the magnetic contribution is negligible (see Hofmann et a1.32).In fact y is obtained by extrapolating CT-' to zero temperature while 8, is fitted at temperatures much below the temperature of the magnetic transition for Fe, Ni and Gd and at temperatures far above Tc for a lot of rare earth metals. Most antiferromagnets are favorable in having no electronic specific heat; on the other hand the lattice specific heat may no longer be describable by one simple Debye function. As shown by Hofmann et a1.32 the specific heat of a large number of binary salts can be described by two Debye parameters, one Debye temperature for each type of atom in the binary compound. It1 other words the specific heat of a binary compound R,X, can be represented by the relationship :
where OR and Ox refer to the Debye temperatures associated with R and X, respectively. The values of these two parameters can be found by matching the experimental specific heat or the entropy to the theoretical values at temperatures widely different from the critical temperature. When matching the entropy at T >> T, the magnetic contribution is defined to be Rln(2s + I). As shown by Hofmann et ul. the two Debye constant procedure gives an excellent description (accuracy within one percent) of the specific heat of some diamagnetic salts (e.g. ZnF,). However, it cannot be used for salts with more than two atoms in the molecule or for those binary salts which have a layer type structure, e.g. MnCl,. Another procedure for subtracting lattice specific heat was discussed by Stout and Catalan0 33, when analysing their data on the antiferromagnetic halides. They derive the lattice contribution for a magnetic salt from the specific heat versus temperature curve of an isomorphous diamagnetic salt by changing the temperature scale by a factor which differs only slightly from 1. This scale factor can be found by matching the specific heat or the entropy at relatively high temperatures. If the principle of corresponding states were exactly followed one would expect that at higher temperatures the scale factors as derived from the entropy or the specific heat would be equal to one another and temperature independent. In practical cases this is not found to be the case. When applying the corresponding states References p. 340
CH. VI,
8 41
MAGNETIC TRANSITIONS
309
principle, one has a choice of using the specific heat or the entropy curve and moreover one may have to accept a scale factor which varies slowly with temperature. When evaluating the thermal data given in the next section the different methods for subtracting the lattice specific heat were compared. In general the results were only slightly different. For compounds with transition temperatures above 20°K we estimated the accuracy to be 25 percent for the fraction of the entropy removed above T,, 20 percent in the fraction of the energy gained above T,,and to be 15 percent for the total energy. The uncertainty arises mainly from the extrapolation necessary on the high temperature side, where the specific heat tail may give a large contribution, especially to the energy. For salts with lower transition temperatures the accuracy may be considerably better. 4.2. THEEXCHANGE CONSTANT
In discussing the properties of ferro- or antiferrornagnets one of the problems is the determination of the exchange constant J. When the magnetic interactions are predominantly of the exchange type, and isotropic, and occur among nearest magnetic neighbours only, J can be derived from the following observed quantities: (1) The Curie-Weiss constant B, which is related to J by the formula
6’=
+ l)qJ/3k,
~S(S
(22)
where q stands for the number of interacting neighbours, s is the spin and k Boltzmann’s constant. (2) The magnetic specific heat at temperatures much higher than T,, which is proportional to T - ’ . The constant CMT2(per gram ion) is related to J by: C,T2/R = 3s2(s + 1)’qJ2/3k2.
(3) The total energy per gram ion gained in the ordered state at T=O which is given by EIR = s2qJ/k.
(24)
Formula (24) may be expected to be applicable for ferromagnets while for (Heisenberg) antiferromagnets the energy gain may be somewhat larger. According to Anderson34 this energy gain may for an ordered antiferromagnet exceed that calculated from formula (24) by a factor which equals 1 y/qs, as discussed in Section 3.2. Since 0 < y < 1 this
+
References p . 340
310
C. DOMB AND A.
R. UIEDEMA
[CH.VI, 5
factor in many practical cases does not differ much from 1. In the next section we have used y=O.5 for those structures for which the exact value of y is not available, when evaluating J from the energy for antiferromagnets. (4) The temperature dependence of specific heat or spontaneous magnetization in the ordered state. For a ferromagnet spin wave theory predicts (Section 3.2) that at temperatures much below T,both the specific heat and the deviation of the magnetization from saturation should be proportional to T*.The value of the proportionality constant has been used in practice to determine J. (5) The zero field susceptibility of an unaxial antiferromagnetic crystal, as measured perpendicular to the alignment axis. According to both molecular field and spin wave theory xL is given by:
xL = NgZP2/4qJ= C/20,
(25)
where C is the Curie constant. The values of J derived from different sources may agree if the assumption of interactions among one kind of magnetic neighbours only is justified. If no agreement exists, the formula for 8, CMTz, E and xL can easily be modified, so that two or more exchange constants are taken into account.
5. Experimental Data 5.1. FERROMAGNETS 5.1.1. Introduction
The experimental data available on ferromagnetism have been extended considerably during the last few years, especially because of the discovery of a number of ferromagnetic insulators. The most promising examples are EuO (T, = 77"K35), CrBr, (T,= 37"K36), GdCI, (T, = 2.20"K37),twoisomorphous copper salts CuK,CI4.2H,O and Cu(NH4),C14.2H,0( T, = 0.88, 0.70°K38) and Dy(CzH5S04)3.9H,0 (T, = 0.13"K39). For the latter four salts a fairly complete set of experimental data has been obtained, which includes the spontaneous magnetization, the zero field susceptibility and the temperature dependence of the specific heat. The magnetic behaviour of the latter four ferromagnets and also of a number of cobalt tutton salts, which will be discussed further on, deviates from that of the usual high temperature ferromagnets in the absence of hysteresis and remanence. Measurements of different specimens of the comReferences p . 340
CH. VI,
0 51
311
MAGNETIC TRANSITIONS
pounds show that in all these crystals the ballistically measured apparent susceptibility ' becomes independent of temperature for T below T, and for the easy axis, and is equal to the value of 1/N per cm3, where N is the demagnetizing factor. This is to be expected for a ferromagiietic substance magnetized in domains, in which the magnetization will be such that within a specimen the external and demagnetizing fields cancel. In the ordered state the susceptibility decreases with decreasing temperature if measured with alternating fields ; relaxation effects occur with relaxation times strongly increasing with decreasing temperature40 being near T, of the order of lo-" sec in CuK2C14.2H20and Cu(NH4)C1,*2H20and of 10-2-10-3 sec in the other compounds mentioned. The differences may be related to the dimensions of the domains. For the two copper salts the anisotropy energy is of the order of 1Ov2kTCfor which a simple calculation based on considerations by Kittel41predicts that the domain thicknesses are 10-2-10-3 cm (and the number of spins in a domain wall M 30), while for Dy(C2H5SO4),*9H20, GdC1, and the cobalt tutton salts the anisotropy energy is of the order of kT,,for which the domain thicknesses are about cm. 5.1.2. Zero Field Susceptibility The apparent zero field susceptibility for a ferromagnet in the paramagnetic region just above T, increases steeply with decreasing temperature for T > T,,
I
0021w~_OD4
0.1
,
02
a4
Fig. 1. The apparent zero field susceptibility of two ferromagneticcopper salts at temperatures above T,.It is shown that the susceptibility of an infinitely long sample can be described by a power law of the form x/C = A(1 - Tc/T)-n.Both x and 1 - Tc/T are plotted on a logarithmic scale, 0 CuK2C14 * 2Ha0 A Cu(NH4)z c14 * 2Hz0.
* In the following susceptibility is used for the magnetic moment divided by the external field; if not otherwise stated the graphs give the susceptibility for a spherical sample. References p . 340
312
C.DOMB AND A. R. MIEDEMA
[CH. VI,
05
but as a consequence of the mentioned upper limit of N-' does not increase to infinity. In order to make a comparison with theory, which predicts that x is proportional to ( 1 - T,/T)-", the experimental values have been corrected to those for an infinitely long cylinder for which N = 0. The results for CuK,C14.2H,0 and Cu(NH4),C1,.2H,0 are plotted in Fig. 1. Both x and 1 - TJT are plotted on a logarithmic scale. It may be seen that a power law of the desired form is quite accurately followed with an exponent 12 = 1.36 and 1.37 for CuK and CuNH,, respectively. The proportionality constant, A , equals in units of the Curie constant 1.3 deg-l and 2.0 deg-' for the two salts. The susceptibility of GdCl, is also quite well represented by the power law with n = 1.33 and A = 0.040 e.m.u./cm3. The zero field x of Dy(C2HSS0,),*9H20 is not described by a formula of the desired form. 5.1.3. Thermal Properties Results derived from the temperature dependence of the magnetic specific heat, C,,, are collected in Table 4. The exchange constants (if possible average values from different sources) are also given. The entries in the table are (1) the critical temperature T,, (2) the total magnetic energy E gained by the magnetic ordering which is obtained by integrating the C , versus T curve, (3) the ratio, (Em- E,)/kT,, of the fraction of the energy gained at temperatures above T, and kT,, (4) the magnetic entropy removed in short range ordering processes, (5) the spin value s, (6) the exchange constant J, and (7) the value of 3kTC/2qJs(s+I), which should be equal to 1 in the molecular field approximation. Fe: The characteristics for iron have been derived from the curve given by Hofmann, Paskin, Tauer and Weiss32. The electronic and lattice contributions are given by y = 50 x lop4J/mole deg' and 8, = 432°K. The energy given in Table 4 is considerably higher than that given by the authors themselves, owing to a difference in the extrapolation of C , on the high temperature side. The exchange constant is derived from measurements of the spontaneous magnetization as a function of temperature much below T,(Fall0t4~J/k= 205°K) and from E(J/k = 170°K). The structure of iron is B.C.C. and thus q = 8. Ni: The specific heat curve for nickel has been derived by Hofmann e.a. from data of NCel43 and of Sykes and Wilkinson44. The best values of y and are 52 x J/mole deg' and 390"K, respectively. The effective spin of nickel, as determined from the saturation magnetization is 0.3 which in the Van Vleck model can be considered as the nickel ions being for sixty percent of the time in a magnetic 3d9 state (s = +) and 40 percent in the References p . 340
n
P
,$ M
ul
Y
T A B L E4 Thermal properties of some fcrromagnets Substance FC-
1043
1 1 x 103
0.32
1
190
0.53
Ni
630
23 x lo2
0.21
)for60 %
260
0.68
Gd
292
36 x lo2
0.36
-
CrBrs
37
I
2:
2
R
rl
rd ...
.. .
3 2
5.4; 0.88
0.80
0.302
0.14
0.221
0.16
GdC13
2.20
CuKzCla. 2Hz0
0.88
5.5
0.40
0.22
1 2
CU(NH4)z Cia. 2Hz0
0.70
3.9
0.36
0.22
2
27
0.29
1
I
314
C. DOMB AND A. R. MIEDEMA
[CH.VI, 4 5
3d" state (s=O). Thus in order to get the magnetic energy for a mole of magnetic nickel one must divide E by 0.6. The exchange integral can be derived from this energy by using the molecular field model and taking the effective number of interacting neighbours 0.6 q, where q = 12 for nickel. The result is J/k = 260°K which compares favourably with the J values obtained from low temperature magnetization data of J/k = 230"K42, J/k = 290"K 45 and J/k = 250"K46.The latter value was measured for nickel films by Nos6 using spin wave resonance techniques, who found J to be independent of temperature. CrBr, : The susceptibility of CrBr, follows, according to Tsubokawara 36 a Curie-Weiss law with 19 = + 47°K. The crystal becomes ferromagnetic at 37°K. The crystal structure is such that the magnetic atoms are arranged in layers. Each magnetic atom has 3 neighbours in the same layer and 2 in the neighbour layers. The nearest magnetic neighbours within a layer are strongly coupled by superexchange through a single intervening atom, whereas exchange between spins in adjacent layers is weaker because 2 intervening atoms separate the spins. The two exchange constants have been measured by N.M.R. techniques, using the Cr 53 nuclei (Gossard, Jaccarino and Remeika 47); J1 = 5.4k and J2 = 0.88kYwhich values are in excellent agreement with the Weiss constant of 47°K. In calculating kTc/qJ for CrBr, we used for qJ the sum 35, 23,. GdCl,: The ferromagnetic ordering in GdCI,, reported by Wolf, Leask, Mangum and Wyatt37148 is caused by both exchange and dipolar interactions. The dipolar interactions cause a strong preference for alignment along the crystallographic c axis of this simple hexagonal crystal. No representative value of J can be derived from the data and thus no value of kTJqJ. If one nevertheless, wishes to compare the strength of the magnetic interaction with the transition temperature, a useful quantity may be 3sRTC/2E(s+ 1) which for a pure exchange ferromagnet equals 3kTc/2qJs(s+ 1). Its value for GdC1, is 0.79. CuK,C14.2H,0 and Cu(NH4),CI,*2H,0. The two copper compounds38 apparently are examples of a simple Heisenberg ferromagnet with dominating nearest neighbour interactions. The crystal structure is tetragonal and deviates only slightly from B.C.C. (colao= 1.05). The crystals reach the ferromagnetic value of N-' for the apparent susceptibility in both a and c axis (experimental accuracy w 10%). The exchange constants have been obtained from (1) the Curie Weiss constant 8, which is practically isotropic, (2) the specific heat constant C M T Zat T >> T,, (3) the values for E and (4) the specific heat below T, and spin wave theory. The values obtained for the
+
References p . 340
CH. VI, 8 51
315
MAGNETIC TRANSITIONS 300 J molt OK
100
30
10
03
01 CMt
003
T_
01
02
03
04
06
08
1 OK
Fig. 2. Comparison of the specific heat versus temperature curves of two ferromagnetic copper salt with spin wave theory. The solid lines are calculated with Dyson's formula: CM= Do(kT/J)f. Dl(kT/J)* Dz(kT/J)* D3(kT/J)4... by fitting the parameter J. For the constants D f the numerical values for a B.C.C. lattice have been used (DO= 5.68 x D1 = 1.56 x DZ= 6.45 x 0 3 = 3.70 x The dashed line shows for the K salt the contribution of the Bloch Tf.term alone.
+
+
+
Fig. 3. Specific heat anomalies of three ferrornagnets. The curves for nickel (s = f), CuKK14 * 2HzO(s = 3) and GdCla(s = g) are plotted versus T/Tc.On the high temperature side the curves clearly show the increase in steepness with increasing'spin value,
A nickel corrected for 60 % spins References p . 340
-C~KzC14.2H20, - - -GdC13.
Cu(NH4)z Clr. 2Hz0
316
C. DOMB A N D A . R. MIEDEMA
[CH. VI,
55
K salt are Jlk = 0.30, 0.295, 0.33 and 0.282"K from the 4 sources, respectively, and J/k = 0.24, 0.21, 0.24 and 0.222 for the NH, salt. An illustration of the fit to spin wave theory is given by Fig. 2. The solid line is calculated from a formula given by Dyson13 fitting the only parameter J . For the K salt the contribution of the T* term alone is also shown (dashed line in Fig. 2) and it will be clear that the influence of the higher order terms is quite important for the specific heat. In Fig. 3 it may be seen that the specific heat tail at T > T, is steeper for larger spin values. The solid curve gives the average values for CuKzCI,-2H,0 and Cu(NH4),C1,*2H,O. The dashed curve gives C , for GdC1, plotted on a reduced temperature scale. One may say that in the latter salt the specific heat rapidly rises from a practically T-' dependence to rather high values, whereas C, rises more gradually for spin 3. Fig. 3 shows also some points for nickel (corrected for the 60 percent spins). The points fit nicely to the curve for the copper salts which may be an indication that the magnetic interaction in Ni and in the copper compounds (superexchange through two intervening atoms) is described by the same formula. (The coordination numbers are not very different, q =8 for the copper salts and effectively 7.2 for nickel.) 5.1.4. Dysprosium Ethyl Sulphate
The data obtained on this salt by Cooke, Edmonds, Finn and Wolf39 have not been included in Table 4. The magnetic properties are rather special in that the magnetic coupling between the dysprosium ions is of the dipolar kind. The peculiar crystal structure causes furthermore a dominating magnetic coupling with the two nearest magnetic neighbours, such that the crystal can be considered as consisting of a bundle of fairly isolated linear chains. The chain structure reduces the critical temperature and a large fraction of the magnetic entropy (about 5) is removed in short range ordering processes. Calculations based on the Ising model (justified by the complete anisotropy of the g values) were given in ref.30. Taking into account also the interactions with next near neighbours on a chain agreement between the calculated and the experimental x versus S and S versus T relations was obtained. 5.1.5. Influence of a Magnetic Field
The approach of the magnetization to saturation as a function of H is similar for many pure ferromagnets. Below the Curie temperature the magnetization, M , at first rises at a constant rate independent of the particular References p . 340
CH.VI, 5 51
317
MAGNETIC TRANSITIONS
value of the temperature, corresponding to an apparent susceptibility 1/N. This continues to a certain value of the field above which the magnetization rapidly approaches a constant value. These values of M , which are reached in rather small fields, may be considered as the spontaneous magnetization,
I
0-
I
05
I
15
I 0
I
20
Fig. 4. The influence of a relatively weak magnetic field on the specific heat of a Heisenberg ferromagnet (s = 3). The curves shown have been obtained for CuKzClr. 2Hz0 in which salt the exchange energy corresponds to about lo4 Oe. The zero field curve is the average one obtained for CuKK14. 2H20 and Cu(NH& Cl4- 2H20, H=O 0 CuKzC14.2HzO - - - - H = 185Oe A Cu(NH4)z Cia. 2Hz0 H = 970 Oe. ~
and can be used as a test of the validity of spin-wave theory. The T* law is usually found at temperatures far below T, and the corresponding values of the exchange parameter can be compared with J values from other sources. The influence of a magnetic field on the specific heat has been studied for GdCl, and CuK2C1,.2H20. For GdC1, it was found that fields, corresponding in energy to kT,, change the character of the specific heat curve so that the lambda type anomaly is replaced by a rather flat and much lower maximum. The measured CH versus T curves were surprisingly well described by calculations based on the molecular field model. For CuK2C1,.2H20 the C, uersus T curves were studied in relatively much smaller fields. Fig. 4 shows that a field of 185 Oe (in energy comparable with kT,) already influences the anomaly considerably, and this would References p. 340
318
C. DOMB AND A. R. MIEDEMA
[CH.VI, 5 5
be in disagreement with a molecular field model. With increasing field strength the temperature of the specific heat maximum shifts at first to lower temperatures, reaching a minimum for H w 2 x lo2 Oe.
5.2. THERAREEARTHMETALS The experimental data concerning the rare earth metals can be summarized as follows (see for instance Spedding, Legvold, Daane and Jennings49). The susceptibility of Ho5O*5I,Dy52 and Er53 follows a Curie-Weiss law with a positive Curie-Weiss constant at relatively high temperatures. At a certain temperature there is a maximum in the x versus T curve. Above this maximum x is independent of the field; below the maximum x becomes field dependent in such a way that (ajy/aH), is positive. At lower temperatures x goes through a minimum and increases rapidly. The maximum in x is accompanied by a pronounced anomaly in the C, versus T curve as shown for dysprosium in Fig. 5. At lower temperatures CM shows another lambda type anomaly at a temperature where the metal becomes ferromagnetic. For terbium54 the second anomaly is not found but the metal is ferromagnetic at very low temperatures and apparently antiferromagnetic near 220°K. The absence of an order-order anomaly may be due to the smallness of the difference in energy between the a.f. and f. states, which agrees with the fact that the antiferromagnetism near 220°K is destroyed by a very small magnetic field. Samarium55956 shows two maxima(T = 105 and 13°K) in its CM versus
Fig. 5. The magnetic specific heat of two rare earth metals. The curve for dysprosium shows two sharp maxima, that for terbium only one, whereas both metals undergo two magnetic transitions. References p. 340
CH.VI,
8 51
319
MAGNETIC TRANSITXONS
TABLE 5 Thermal properties of some rare earth metals Element Sm
Gd Tb
State OHslz SS,p
7F0
DY
'Hi618
Ho
5 1 ~
Er
41i5/z
E (Jimole)
Em-Ec RTc
9.5 X 10' 36 x 10' 31 X 10' 25 X lo2 15 X 10' 10 X 10'
0.26
-
0.40 0.32 0.22 0.33
Sm
-
SC
Twit
k
(" K)
0.14 0.36 0.26 0.20 0.13 0.18
106; 13 292 228 178; 85 132; 19 85; 20
T curve but there is no sign of ferromagnetism; this suggests that the 13°K anomaly corresponds to a change from one type of a.f. ordering to another one. Gadolinium57 shows a single transition from the paramagnetic to the ferromagnetic state at T,= 292°K and the specific heat curve is similar to that of terbium, shown in Fig. 5. Both curves show a rather large specific heat at temperatures near 0.3 T,(a maximum in C,/T versus T). The characteristics of the specific heat curves of the rare earth metals are collected in Table 5. The total energy and the fraction of the energy and entropy removed above the highest transition temperature are given. For all the metals the total entropy fits nicely to Rln(2J' + l), where J' is the total magnetic quantum number of the magnetic ion. The specific heat data have been analysed using the same electronic specific heat for all rare earths ( y = 1.0 x J/mole deg2), while the lattice specific heat measured for the diamagnetic lanthanum has been used, multiplying the temperature by a scale factor such that the correct value for the specific heat at room temperature is obtained. The lattice contribution increases from Sm to Er. The specific heat data on thulium58 have not been included in the table, since there are weak maxima in the C, versus T curve far above the transition temperature. 5.3.
COMBINED
FERROAND ANTIFERROMAGNETISM
Near 1°K single crystals of the cobalt tutton ~alts5~940 show a magnetic behaviour which is extremely anisotropic. The crystals become apparently antiferromagnetic according to the susceptibility data obtained in the a-c plane, but ferromagnetic according to measurements in the direction perpendicular to this plane (K3). For CoK2(S04), .6H20 the x versus T relation References p. 340
320
C. DOMB A N D A. R. MIEDEMA
[CH. VI,
05
is shown in Fig. 6 . At the critical temperature 3: starts decreasing for the
Kidirection, becomes nearly independent of temperature for the K2 direction (Kiand K2 are mutually perpendicular directions in the a-c plane) and reaches the ferromagnetic value of N - for the K3 direction. The spontaneous magnetization is about 50 percent of the saturation moment. In the tutton salts there are two lattice positions for the magnetic ions, which are different in relation to the axis of the axially symmetrical crystal field. Both the magnetic exchange and dipolar interactions favour a parallel orientation for the magnetic moments occupying equivalent lattice sites and antiparallel orientation for those on different lattice positions 60-62. Thus an ordering into two sub-lattices is to be expected below T,. However, because of a strong preference for the crystal field axes arising from the anisotropy of the exchange and hyperfine structure interactions, the magnetization vectors of the two sub-lattices are not simply antiparallel but make a large angle with each other (46" for CoK to 82" for CoCs), as shown in Fig. 7. It may be seen that there is a net magnetization in the K3 direction which accounts for the observed ferromagnetism. The anisotropy of the exchange interactions can be understood from the following. The ground state of a free cobalt ion is, according t o Hund's rule, 4F,. In a tutton salt this state is split by the electrical crystal field in such a way that at very low temperatures only one doublet is populated, which can be characterized by a fictitious spin s = and strongly anisotropic but axially symmetrical g-values. Both angular momentum and spin moments contribute to the fictitious spin s'; as shown by Abragam and Pryce63 a separate set of g values (g\l,g:, gsl = 4.7, gS, = 2.5) can be associated with each of them. For any oiientation of s' in space the direction and magnitude of the intrinsic spin are found by multiplying the component of s' parallel to the crystal field axis by gsl and the component perpendicular to this axis by g;. As a consequence the exchange interaction between two cobalt ions becomes strongly anisotropic, since :
+
= i>k
i>k
-2
c
J'sfsJ.
(26)
i>k
The ordering in the cobalt tutton salts is neither antiferromagnetic nor ferromagnetic, but somewhere in between. It seems reasonable, however, that because of the predominating anisotropy the cobalt salts are examples of the three dimensional Ising model, which is symmetrical in regard to thermal properties for f. and a.f. ordering. References p . 340
CH. VI,
5 51
321
MAGNETIC TRANSITIONS
3
Fig. 6. The apparent zero field susceptibilities of CoKz(S04)z. 6H2O near the critical temperature. K1, Kz and K3 are three mutually perpendicular directions. The magnetic behaviour is apparently a.f. in the KI, but f. in the &-direction, A KI direction, 0 Kz direction, E l K3 direction.
K,
dirhction
I
'
- - - - - ' - - K- +direction --------I
(0)
Ib)
Fig. 7. Positions of the magnetic sublattices in magnetically ordered cobalt tutton salts. TI and TII represent the preferential axes for the two cobalt ions in the unit cell. There are two possibilities for the ordered structure (a, b) which show a net magnetization in the K3 direction. The angle between TI and TIIis about 80°,the angle between MI and MII varies from 46" for CoK to 82" for CoCs tutton salt. References p . 340
322
[CH.VI, 8 5
C. D O N A N D A. R. MIEDEUA
TABLE6 Thermal properties of four cobalt tutton salts, which are possibly representative for the three dimensional king model
sco-
Tutton salt
T,('K)
___
k
(CT2/R)er x 104
(CT2/R)dlp x 104
(CT2/R)h.f.s. x 104
CoK CoNH4 CoRb COCS
0.134 0.084 0.092 0.072
0.105 0.13 0.09 0.14
72 10 17 3
23
27 18
19 18
24
sc
1s
23
The dipolar and exchange contributions to the specific heat in the paramagnetic state are not simply additive; the small cross-term is included in (CT2/&=. The h.f.s. interactions are strongly anisotropic. The h.f.s. constant is about 10 times larger for alignment parallel to the crystal field axes than perpendicular to this axis.
From Table 6 it may be seen that the fraction of the magnetic entropy removed above T, is practically equal for the four salts. In calculating this fraction a correction for the entropy (at T,)removed from the nuclear spin system was applied to the experimental data. In order to show the relative magnitudes of the exchange, dipolar and hyperfine structure interactions, their contributions to the specific heat in the paramagnetic state are also given in Table 6 . For CoNH,- and CoK-tutton salt the temperature dependence of x in the ferromagnetic direction has been analyzed in the region just above T,. From data by Garrett59 we found for CoNH4 that in the formula x = =A(1 - T,/T)-" a value of n = 1.27 (accuracy w O.lO), while for CoK according to recent Leiden measurements62 n = 1.22 f 0.03. 5.4. ANTIFERROMAGNETISM
For about 25 antiferromagnetic salts both magnetic and thermal properties have been investigated. The transition from the paramagnetic to the antiferromagnetic state occurs at a critical temperature TNand is indicated by a lambda type anomaly in the specific heat versus temperature curve. Near T N the susceptibility shows a maximum as a function of temperature and it becomes strongly anisotropic below TN. In the simple case of a two-sublattice uniaxial antiferrornagnet x decreases strongly with decreasing T for the direction parallel to the axis of alignment of the sub-lattice magnetization (x,,) and it is fairly independent of temperature perpendicular to this axis (xL), as illustrated by Fig. 8 for MnCI2.4H,0. The thermal data of some salts, which can be considered as simple antiferromagnets are collected in Table 7. By a simple antiferromagnet we mean References p . 340
8
TABLE 7 Thermal data on some “simple” antiferromagnets
k
% ;i:
F! c P
TN
Salt
s
(OK)
5
MnFz
2
MnClz. 4H20
5 2
MnBrz . 4Hz0
2
FeFz
2
NiFz NiClz. 6HzO COFZ
61
(Sm
-
Sc)
k 0.26
Em - Ec
E (J/mole) 780
$
RT, 0.46
0.72
- 2.0
8
Stout e.a.33
+
J/k
references specific heat
3kTc 2qJs(s 1)
q
(OK)
1.62
0.25
19.6
0.46
0.79
- 0.058
6
Friedberge.a.65
2.14
0.34
29
0.56
0.69
-
0.088
6
Kapadnis e.a.66
18.3
0.21
930
0.41
0.66
- 3.7
8
Stout e.a.33
1
73.5
0.32
755
0.62
0.63
- 11
8
Stout e.a.33
1
5.3
0.44
62
0.92
0.56
- 1.2
6
Robinson e.a.‘3
1 2
31.1
0.14
5
Stout e.a.33
co1 v1 Y
E
4
#
x
TA BLE8 Lattice parameters in the antiferromagnetic fluorides Salt
ao
co = rn.,,.
rn.n.n.
MnFa
4.87
3.31
3.81
FeF2
4.69
3.31
3.10
CoFz
4.69
3.18
3.68
NiFz
4.65
3.08
3.62
znF2
4.70
3.13
4.68
Jn.n.n.lk from 6 (OK) - 2.2 -
3.7
- 10
Jn.n.n./k
Jn.n.n./k
from C M T ~ - 1.9
from E - 1.9
3.8
- 3.6
-
- 12
Jn.n.n./k
from X I - 1.9 -
3.5
-11 W
2 !
324
C. DOMB A N D A. R. MIEDEMA
[CH.
VI, 5 5
Fig. 8. The zero field susceptibility of a single crystal of MnClz 4&0 (after Van den B r ~ e k @ whose ~), properties may be representative of those of an Heisenberg antiferromagnet with s = 4, 0 c - axis A b - axis.
that (1) a two sub-lattice ordering is possible, (2) the magnetic interactions among one type of magnetic neighbours predominate and (3) a given magnetic ion and its neighbours build a three dimensional lattice. Four of the compounds mentioned in Table 7 are iron group fluorides. Their crystal structure is body centered tetragonal with an ao/co ratio of about 1.5. The magnetic ions have two nearest magnetic neighbours (interaction Jn.J at a distance of % 3.2 A and 8 next near neighbours (Jn.n.n,) at w 3.6 A as may be seen in Table 8. Neutron diffraction experiments have shown that in the ordered state next near magnetic neighbours become oriented antiparallel and nearest neighbours parallel, which indicates that owing to the presence of the fluorine ions Jn.n,n. is at least as important as J,.,.. MnF, : From E.S.R. experiments Brown, Coles, Owen and Stevenson68 derived both Jn.n.n. and Jn.n.for manganese ions incorporated in a ZnF, crystal. They obtained .Tn.n.n. = - 4k and Jn.n, = + 0.4k. Comparing the distances separating the positive ions in ZnF, and MnF,, one may expect and Jn.n.to be smaller in MnF,. It turns out that agreement both Jn.n.n. between calculated and experimental values can be obtained for the CurieWeiss constant, the specific heat constant, the total energy and xL at zero temperature", assuming Jnan, to be zero and Jn.n,n, = 2.0 k, as shown in Table 8. If a positive exchange constant Jn.n.were present, the value for Jn.n.n. derived from 0 would be too low, the ones derived from CMTzand E References p . 340
CH. VI,
5 51
MAGNETIC TRANSITIONS
325
would be too high and the one derived from xL would be correct. MnF, can be considered as an example of a Heisenberg antiferromagnet since the susceptibility at not too low temperatures is completely isotropic. FeF, and NiF, : The values of Jn.n.n. derived from different sources under the assumption Jn.n. = 0 agree well for both salts. For NiF, 1x0 data on xL are available. The susceptibility at high temperatures is practically isotropic for NiF, and shows an anisotropy of about 25 percent for FeF,70. CoF,: For the cobalt salt no value for the exchange parameter can be derived. The exchange must be expected to be strongly anisotropic (M 80 percent) since both orbital and spin angular momentum contribute to theeffective spin (Section 5.3). CoF, may be not very different from the Ising model. NiC1,.6H,O: In the monoclinic crystal NiCI,.6H,O the atoms are arranged in layers perpendicular to the c axis. Each magnetic atom has two nearest magnetic neighbours in different layers and four next near neighbours in the same layer. It happens that both Jn.n, and Jn,n.n. have a negative sign and are nairly equal as derived from the experimental values for the quantities mentioned above. The susceptibility 7l is isotropic at high temperatures. MnCl,.4Hz0 and MnBr, *4H,O: The crystal structure is monoclinic but the position of the magnetic ions are not known. It turns out that the four experimentally available quantities 72-74 can be fitted by one exchange constant if the coordination number q equals 6, which suggests that the arrangement of the magnetic atoms is similar to that in NiC1,.6H,O. The high temperature susceptibility is isotropic. The temperature, T,, at which xII shows its maximum is for the antiferromagnets of Table 7 (for which x has been measured) slightly but definitely higher than T,, the temperature of the specific heat anomaly. The ratio TJT, equals 1.05, 1.05 and 1.03 for MnF,, MnCI, . 4 H z 0 and MnBr, *4H,O, respectively, 1.05 for FeF, and about 1.15 for NiC1,-6H20. (Because of the presence of higher order levels at energies of m look, no reliable data on TJTC can be given for CoF,.) In some cases a weak maximum for x1 has also been observed, the temperature of this maximum being less different from T,.
+
5.5. LAYERTYPEANTIFERROMAGNETS Well known examples of layer type antiferromagnets are some anhydrous iron group chlorides. In these salts the metal ions are arranged in planes, forming triangular networks, each layer of metal ions being separated from an adjacent layer by two layers of chlorine atoms. The shortest distance between metal atoms of different layers is about 5.8 A, while the nearest Rejerences p . 340
326
C. DOMB AND A. R. MIEDEMA
[a. w, § 5
neighbour separation is about 3.5 A. The exchange interactions have the ferromagnetic sign for nearest neighbours in a layer (J1> 0)while the interaction between layers (5,) leads to an antiferromagnetic ordering. In FeC1,75, NiC1,76 and CoC1, I 5, 1 is much smaller than IJ1I, as is demonstrated by the fact that the a.f. ordering is easily destroyed by a magnetic field77. Again from 8,C M T z E , and xI (estimated) the two exchange constants can be determined. For FeCl, the results are J1 = 2.2k, J2 = - 0.8k while in NiC1, = Ilk and 5, = - 3k (zl= 6, z2 = 2). For the strongly anisotropic case of CoC1, no exchange constants have been derived. It may be seen in Table 9 that the amount of entropy removed in short range order processes for these layer type salts is considerably larger than for the corresponding fluorides. We have not succeeded in deriving exchange constants for the other compounds of Table 9. The hydrated crystals CoCl,-6H,O, CoBr,.6H20 and NiBr, - 6 H z 0 are presumably examples of layer type antiferromagnets in which both the interaction between layers and that within a layer favours antiparallel orientation (as in NiCl, -6HzO). In CuCI, .2H20 and MnCI, the strongest (negative) interaction seems to exist with two nearest neighbours. As discussed by Marshallas (for CuCl,.2H,O) the crystals may be looked upon as consisting of a number of chains, which are coupled by a relatively small positive interaction (exchange for CuCI, -2H,O, dipolar interaction for MnCI,). As a result the crystals are neither examples of simple antiferrornagnets nor of linear chains, since the interactions between chains are no orders of magnitude smaller than the interactions within the chains. According to Spence and Murtys7 the magnetic ions are arranged in pairs in LiCuC1,.2Hz0. For such a pair of copper ions the triplet state corresponding to parallel orientation seems to be energetically the most favourable. The transition to the a.f. state then corresponds to an ordering of the total spin 1 of the copper pairs. 5.6. ANTIFERROMAGNETISM IN SPECIAL LATTICES Even when the exchange interactions occur only among nearest neighbours, complications may arise. In a face-centered cubic lattice each magnetic ion has 12 nearest magnetic neighbours which are arranged in such a way that two nearest neighbours of a given magnetic ion are also nearest neighbours to each other and thus no ordering with all interacting spins antiparallel is possible. It turns out (Van Vleck, Carter e.a.aa) that there are a number of possibilities for the ordered state of a F.C.C. antiferromagnet in which References p . 340
p
TABLE9 Thermal properties of a number of antiferromagnetic salts.
P
Salt
S
11 II Tc
(OK)
Q Lu
0
FeCh
2
NiClz
1
i
23.5
52
1
1 2 5
NiBrz. 6HzO
1
1
CoClz. 6HzO
1 2
coc12
2
1
2
CoBrz 6 H z O
1 I 1 I
3.1
4.3 4.4
i
1 I 1
(S, - Sc)
k 0.64
0.46 0.28 0.65 0.36 0.35
. _
1 2
CuClz. 2H20 MnClz
______ 5
2
-
4.3 1.96
0.35
1
0.67
II i
1
1 ~
~
E (J/mole) 4.0 x lo2 6.0
102
1.8 x lo2
77 20
i
1 1 I
23
E m - Ec 1.33 0.92 0.61
1.11 0.81 0.65
1
I ~
II
30
I
i1 1
1 1
I 1
$
rm
CMT~ T S Tc (J0Kjmole) 6.3 x lo3 2.0
104
references specific heat
1 Trapeznikow e.a.78 '
Busey e.a.79
2.9 x lo3
Chisholm e.a.80
2.1 x 102
Spencee.a.81 Robinson e.a.87
34
Forstat e.a.82 46
Friedberg e.a.S3 1.21
I
Murrays4 49
LiCuCls. 2H20 The compounds under A and B are layer type antiferromagnets with positive exchange interaction within a layer for A and negative interaction within layers for B. The group C salts are examples of linear chain salts where the interactions between chains are relatively not very small.
lQ
4
328
C. DOMB A N D A. R. MIEDEMA
[CH. M, 0
5
there are two energetically favourable oriented pairs against one unfavourable pair. Which of the possible superlattice structures is preferred depends on the next near neighbour interactions, weak though they may be. Examples of F.C.C. antiferromagnets with predominating n.n. exchange interactions are available in the ZnS type of MnSg8-90, with spin +, two chloroiridates 91 with spin 3 and two nickel hexamine halogenides 92 with spin 1. The x versus T curves of the salts are shown in Fig. 9. In order to obtain comparable scales the temperature is plotted as T/8 and the susceptibility as &/C. An important feature of the curves is that they do not correspond to a Curie-Weiss law in a large temperature range above T,. The critical temperatures as derived from susceptibility (MnS, (NH,), IrCI,, K, IrCI,) or from specific heat data (Ni(NH3),Br2, Ni(NH3), JZg3) differ only slightly from T,=8/10 the extreme values being T, =8/9.3 and 8/11.5. The specific heat versus temperature curve for Ni(NH3),Br2 is given by Fig. 10. A correction has been made for the contribution of the ammonia molecules to the specific heat, which apparently becomes very large below 0.2"K. At the transition temperature about 60 percent of the entropy has already been removed in short range ordering processes. The total energy (which, because of a rather uncertain high-temperature extrapolation, is not too accurately known) equals 17 J/mole. According to simple molecular field theory, this energy should be equal to +NqJs2, since only one third of
I
Fig. 9. Susceptibilitiesof some F.C.C. antiferromagnets near their critical temperatures, MnS after Carter and Stevens88 (s = 5, 8 = 93" K) 0 0 KdrCl6 after Cooke e.aegl (S = + , O = 20"K) (NH4)Z IrCh after Cooke e.a. (S = +,B = 32" K) A v Ni(NH& Brz after Palma e.aSg2 ( s = l , O = 7'K). References p. 340
.
CH. VI,
5 51
329
MAGNETIC TRANSITIONS
I 0
Tie
I
I
02
0.4
(
Fig. 10. The magnetic specific heat of an F.C.C. antiferromagnet with s = I . The curve is measured for Ni(NH3)sBrz and has been taken from data of Ukei and Kanda94 ( T > lo K) and Van Kempen e.a.gs ( T < 1" K). The temperature is plotted in units 8, the Curie-Weiss constant, which equals 7" K.
the neighbours of a given spin are effective for the ordering. This leads to RBIE = -4 for spin 1, which can be compared with the experimental result RBiE = -3.5. A considerable lowering of the transition temperature, compared to that of a corresponding simple antiferromagnet, is found in crystals where the magnetic atoms are arranged in magnetically isolated linear chains. A clear cut example is Cu(NH,),SO, .H,O, whose crystal structure, according to X-ray investigations 95, is such that the copper ions form chains parallel to the c-axis linked as follows:
- C U 2 + - O H , - cu2+ - OH2 - c u 2 + while the linkage between two Cu2+ ions on neighbouring chains is the following: - Cu2+ - NH, - SO, - NH, - Cuz+ - . Both x and C, show broad maxima as function of T i n the liquid helium temperature range Long range magnetic ordering is reached at a much lower temperature (T,= 0.37"K) as indicated by a small lambda type anomaly in the C , versus T curve 98. Experimental data obtained on CuSO,. 5H,O and CuSeO,. 5 H 2 099 indicate the presence of two practically independent and equally populated 96997.
References p . 340
330
[CH.W,5 5
C. DOMB A N D A. R. MIEDEMA
systems of copper ions. One system behaves very much like the copper ions in Cu(NH,),SO,~H,O and looses its entropy near 1°K; the other copper ions remain paramagnetic to below 0.1"K. The two crystals become antiferromagnetic at T = 0.029"K and 0.046"K, respectively. J
mol.OK 3.0
u)
1.0
54 . 0
1.0
2.0
Fig. 11. Specific heat versus temperature curves for two linear chain crystals with s The dashed line represents calculations for a closed ring of 10 atoms. 0 Cu(NH3)4 so4. HzO data from Fritz and PinchQ7,Ukei and Haseda e.a. 98, A Cu(SO4) * 5Hz0 data from Duyckaerts100, Geballe and Giauque'O1 and ref.eg, _ - _ - Calculated by GriffithsIo2.
=
3.
Fig. 11 shows the specific heat curves for Cu(NH,),SO,.H,O and CuSO4.5H20; for the latter salt a small contribution from the copper ions still paramagnetic below 0.1"K has been subtracted. The temperature is plotted as kT/J. The values of J are obtained from t9 and CMT2at high temperatures*, keeping in mind that for CuSO,. 5Hz0only half the number of ions contribute. The experimental data coincide nicely for the two copper salts, the specific heat showing its maximum value of 2.99 J/mole at kT/J =0.95. One may see that the lambda type anomaly occurring near kT/J =0.1 in Cu(NH,),SO, H20has practically no influence on the characteristics of
-
* The 0 values reported for Cu(NH3)&04 * HzO, all measured in the same temperature range, differ somewhat for different authors. For the powdered salt Kido and Watanabeln3 report 6' = - 4"K, Fritz and Pinch87 found 6' = - 5.3'K, while according to the analysis by Eisenstein1O4their value should be-3.3'K. Watanabe and Hasedage report for thesingle crystal an isotropic 6' of about - 1°K but their Curie-constant is considerably lower than that calculated from the g-values observed in E.S.R. experiments. Using the higher Curieconstant the 6' for the singlecrystal will be about-3°K. The valueforJfromJ/k =-3.34"K used in figures 11 and 12 is derived from the specificheat constant and seems to correspond to the average value for 0. References p , 340
CH. VI, 9 51
331
MAGNETIC TRANSITIONS
the curve which hence may be considered to be representative of the specific heat of an isolated (Heisenberg) chain. The susceptibility of Cu(NH,),SO,*H,O is plotted in Fig. 12. The curve shows a broad maximum at about kT/J'= 1. It is evident that for any finite antiferromagnetic chain, which contains an odd number of spins, x has to increase with increasing T at sufficiently
I
f
0
kT/J
ID
2.0
3 . 0
4.0
Fig. 12. Comparison of the apparent zero field susceptibility of two linear chain crystals with calculations for a closed ring of 10 atoms, 0 Cu(NH& so4 * HzO after Haseda e.a.O8908,
A
Cu-dihydroxy-para-quinone(rather short chain) after Haseda e.a.Io5, - - - - calculated for a closed ring of 10 atoms (after Griffiths).
low temperatures. Such an end effect has been observed for the organic compound Cu-dihydroxi-para-quinone, where the copper ions are arranged in chains which are rather short because of the small size of the crystallites. Fig. 12 shows that there is no difference between a short chain and a long chain at relatively high temperatures (kT/J > 1.3), but a large difference at lower temperatures. The rise of the susceptibility of Cu-dihydroxi-paraquinone corresponds to an estimated average length of the chains of about 15 magnetic atoms. An example of a linear chain crystal with anisotropic intra-chain interaction is K,Fe(CN), which according to data of Duffy e.a. 93 becomes antiferromagnetic at T = 0.129"K. By observing the paramagnetic resonance spectrum of Fe-Fe pairs in dilute crystals Ohtsuka106 found that the jntrachain exchange interaction constant J was 25 percent anisotropic and about ten times larger than J' between chains. The specific heat curves show a References p . 340
332
[CH. VI,
C . DOMB A N D A. R. MIEDEMA
Fig. 13. The magnetic specific heat of CuClz (s = 4)and CrClz (s Chisholm107.
= 2)
55
after Stout and
pronounced A-type anomaly, while above T, C , decreases slowly without having a maximum. The large differences between K,Fe(CN)6 on the one hand and the copper salts mentioned above on the other hand suggests that, if some anisotropy is present, long range magnetic ordering may take place at a much higher temperature for a given ratio of J’/J. Other antiferromagnets for which the C , versus T curve shows not only a lambda type anomaly, but also a much broader and low maximum above T, are CuCI, and CrC1,. The compounds (Fig. 13) were investigated by Chisholm and Stout 1°7 who found that for CrCI, 73 percent of the magnetic entropy is removed above T, and about 80 percent for CuC1,.
(0)
Ibl
Fig. 14. Possible antiferromagnetic structures for CrClz and CuC12. The structure (a) is favourable if second near neighbour interactions are strongest (phase differences being given by and -) while in the structure (b), observed for CrClz by Cable e.a.lo8, first and third near neighbours are antiparallel, bo = 5.98 A, co = 3.48 A. ao = 6.64 A,
+
References p . 340
CH. VI, 0
61
MAGNETIC TRANSITIONS
333
The crystal structure of the compounds differs slightly from body centered tetragonal; the length of the c axis is much shorter than that of the a and b axes. Each magnetic ion has two nearest neighbours (a,a,, b,b, in Fig. 14b) 8 next near neighbours (alb,, a,b2, alcl) and 4 third near neighbours (b,cl). Chisholm and Stout suggest that the interactions among nearest neighbours predominate and this suggestion seemed to be confirmed by observations of the antiferromagnetic structure of CrCI, by Cable, Wilkinson and Wollan108 (Fig. 14b). However, the large values observed for the Curie-Weiss constant make a linear chain structure very unlikely. Combining the obtained values for B and CMT2one arrives at a number of interacting magnetic neighbours larger than 10 in both salts (CuCl,: B = 93"K1O9,CMT2m 1.3 J"K/mole, E -3.3 x l o 2 J/mole; CrC12: 8 = 128"K11°, CMT2m 1.5 J"K/mole, E w 5.4 x lo2 J/mole). Hence it also seems possible that the interactions among magnetic neighbours up to the third are of comparable magnitude. The a.f. structure attained may be similar to that in MnF, (Fig. 14a) if second n.n. interactions are strongest, whereas the structure found by Cable e.a. corresponds to stronger first and third n.n. interactions. In either case a considerable lowering of T, may arise from the fact that not all interacting spins can be oriented antiparallel below T, (observed e/T, values are 8 and 4 for CrC1, and CuCI,, respectively). The susceptibility of both CrC1, and CuC1, shows a maximum around TIT, =2.5. The observed maximum values are Ox/C = 0.45 for CuCI, and Ox/C =0.7 for CrCI,. 6. Comparison between Experiment and Theory
It is the purpose of the present section to analyse some of the experimental results outlined in the previous section to find how well they compare with the theoretical estimates for simple king and Heisenberg models (i.e. with nearest neighbour interactions only) given in Sections 2 and 3. By this means we can focus attention on substances for which the simple model is a reasonable approximation, and we can suggest improvements in the model which might lead to better agreement with experiment. 6.1. FERROMAGNETS Of the salts discussed in Section 5.1 those for which the Heisenberg model seems most appropriate are CuK2C1,-2H20 and CU(NH,)~C~,* 2H20. The magnetic susceptibility immediately above the Curie temperature can be represented by a relation x = A(1 - T,/T)-", (27) References p. 340
334
C. DOME AND A. R. MIEDEMA
[a. w, 6 6
where n = 1.36, 1.37 respectively; this is in reasonable agreement with the theoretical formula (13). For an F.C.C. lattice with s = + theoretical estimates of critical parameters are given in Section 3.1 ; comparison with Table 4 shows that the agreement for the above two salts is satisfactory as a first approximation. If we wish to proceed to a finer analysis we should compare with theoretical estimates for a ,B.C.C. lattice, which is a good approximation to the lattice structure of the two salts. Although such estimates are not available, analogy with theIsing model leads us to think that they will differ by only a few percent from those of the F.C.C. lattice, kTJq.7, Sc/k and (Ec- Eo)/kTc all being slightly smaller for the B.C.C. lattice. Recent calculations by Domb, Sykes and Wood111 for the Heisenberg model show that 3kTc/2qJs(s+ 1 ) is about 6% smaller for the B.C.C. than for the F.C.C. We thus find that the experimental value of kTC/qJ(Table 4) is about 20% higher than the theoretical calculation (Table 3) and that of ( S , -Sc)/k is comparably lower. Both of these effects would be taken into account by the introduction of a small admixture of second and more distant neighbour ferromagnetic interaction. The slight deviations in the values of the exchange constant J obtained by different methods (Section 5.1.3) would also arise naturally from such additional interactions, since each method yields a particular type of average of J as a function of distance. For GdC1, we should expect the dipolar interactions to cause some complications, but we may observe that the exponent n in (27) is in excellent agreement with a simple Heisenberg model (13), and the critical data illustrate nicely the increase in kT,/qJs(s + 1) and ( S , -Sc)/k as the spins increase from to 3. A corresponding increase in sharpness of specific heat above the Curie point (Section 3.1) is illustrated by the experimental curves in Fig. 4. Because of the smallness of the fraction of the magnetic entropy removed above T,(S, - S, = 0.13k In 8) it is not surprising that the thermal properties of GdCl, are quite well described by a molecular field approximation, as was reported by Leask, Wolf and Wyatt 48. From Fig. 2 it will be seen that there is excellent agreement at low temperatures between experimental and calculated values of C, for the two copper chlorides. Whilst this is encouraging, the significance of the good fit, even at temperatures near T,, should not be overestimated, since a relatively small interaction between next near neighbours can cause considerably changes in the higher order terms in Dyson’s f o r m ~ l a ~ ~ J ~ ~ . When we come to discuss the results for ferromagnetic metals there are
+
References p. 340
CH. VI, 8 61
MAGNETIC TRANSITIONS
335
difficulties both on the experimental and theoretical side. The magnetic contribution represents only a small fraction of the total specific heat, and the result of the subtraction process discussed in Section 4. I , can only be regarded as a crude approximation to the magnetic specific heat. Also we should not expect substances like Fe and Ni t o be adequately represented by a Heisenberg model, and we do not therefore attempt a detailed comparison with theory. It is interesting to note empirically that for Fe the value of 3kTc/2qJs(s 1) is appreciably Zower than the theoretical estimate for the Heisenberg model and ( S , -Sc)/k is somewhat higher; this is the behaviour we should expect if second neighbour interactions were opposite in sign to nearest neighbour interactions i.e. antiferromagnetic. Deviations from T‘ dependence of the spontaneous magnetization for Fe and Ni are larger than expected from Dyson’s theory; this may be due to the influence of the conduction electrons and to distant neighbour interactions 112. For Gd it has been suggested that the Heisenberg model should provide a suitable approximation, but a closer investigation reveals a number of difficulties. Values of J derived from the Curie-Weiss law, and spin wave theory at low temperatures differ appreciably. The problem has been discussed by Goodings113 and the suggestion of an ‘‘optical’ytype spin wave has been considered in detail. Although this seems to be a step in the right direction a substantial discrepancy remains which might possibly be explained by taking more distant neighbour interactions into account. Accurate susceptibility data immediately above the Curie temperature would be particularly useful as a test of the validity of the Heisenberg model; preliminary measurements at Leiden indicate an exponent in (27) much lower than the theoretical prediction of 1.33, and the concept of Gd as a simple Heisenberg substance may have to be abandoned. Not much can be said on the other rare earth metals where the magnetic ordering is rather complicated. From Table 5 one may notice that the values of ( S , -Sc)/k are for all the metals in between the limits for the Heisenberg and the Ising model, while furthermore it is reassuring that ( S , -Sc)/k is smaller for those metals with the larger orbital angular momentum (which may provide a crude measure of anisotropy).
+
6.2. COBALT TUTTONSALTS In Section 5.3 it was suggested that the cobalt tutton salts might reasonably be considered as Ising models of spin 4.It will be noted that the values of ( S , - S&k in Table 6 are smaller than the corresponding values in Table 4 References p . 340
336
C.DOMB AND A. R. hUEDEMA
[a. M,B 6
by a factor of two or more, and this is a characteristic theoretical prediction of the difference between the Ising and Heisenberg models. In fact estimates of (S, - Sc)/k for the Ising model (s = 4) for F.C.C., B.C.C. and S.C. lattices are 0.102, 0.104, 0.133 respectively, and are very comparable with the values in Table 6. It is also interesting to observe that the exponent in (27) for the ferromagnetic direction, as measured for CoK,(SO,), .6H20, is smaller than that of the Heisenberg model, and this again is in accordance with theoretical predictions, But a more detailed comparison of theory and experiment can only be madeif specific calculations are undertaken to fit each particular salt. The results given in Table 6 show a strong similarity between CoK, (SO,), .6Hz0 where the magnetic interactions are mainly of the exchange type and the corresponding CoNH and CoCs-salts, where dipolar interactions are dominant. This may suggest, together with the good fit with theory found for ferromagnetic GdC13, where dipolar interactions are also important, that the thermal properties are not very sensitive to the actual type of interaction.
-
6.3. ANTIFERROMAGNETS For the Ising model the thermal properties of an antiferromagnet are identical with those of a ferromagnet, provided the lattice is “even” in structure (e.g. S.C., B.C.C.). For the Heisenberg model the above statement is only true for s = co ;for finite s the a.f. critical temperature is higher3’” than the f., and one would expect the difference to become more marked as s decreases. Detailed calculations are not available for the magnitude of the difference between the thermal properties of a ferro- and antiferromagnet. Estimates of the difference between the energies of the lowest states (Section 3.2) can be used as a preliminary guide to the agreement to be expected between experiment and ferromagnetic theory. From the discussion in Section 5.4 MnFz can be regarded as an example of a Heisenberg antiferromagnet, and since the spin s = 3 the thermal properties should not differ markedly from those of a ferromagnet. In fact reference to Table 7 shows satisfactory agreement with the theoretical calculations of Section 3. For NiF, magnetic isotropy is retained and hence the Heisenberg model is still appropriate; the spin drops to s = 1, and the decrease in 3kTc/2q.7s(s 1)and correspondingincrease in (S, - Sc)/kand (Em- Ec)/kTc need further investigation. For FeF, there is increased anisotropy, and hence a move in the direction of the Ising model; for CoF, the anisotropy is much more marked, and the value of 0.14 for (S, - Sc)/k is well within the Ising
+
References p . 340
a. w, § 61
MAGNETIC TRANSITIONS
337
range. NiC1,.6Hz0 should be comparable with NiF,, and the decrease in kT,/qJ and increase in (S, - S,)/k observed experimentally may be due to looser coordination (q = 6). The properties of MnClz*4H,0 are very close to those of MnF,; MnBr, .4Hz0 deviates somewhat from both of these and a more detailed investigation of the interactions would be needed to account for the differences. Susceptibility maxima at a temperature above Tc are a theoretical prediction both for the Ising and Heisenberg models (Sections 2.2 and 3.2). The difference between the temperature of the maximum and X, decreases with increase in coordination of the lattice. Its magnitude is 5-10% for 3 dimensional lattices for the Ising model, but no detailed calculations are available for the Heisenberg model. The experimental figures given at the end of Section 5.4 are comparable in magnitude with estimates based on the Ising model. To sum up we can say that agreement between experiment and theory for the above salts is quite satisfactory, and more detailed calculations on the properties of the Heisenberg model of an antiferromagnet are needed before a more refined comparison can be made. No comparison with theory can be made for the antiferromagnets listed in Table 9. Quite generally one may say that the values of ( S , - S,)/k are higher than the values for the corresponding simple antiferromagnets and may be due to quite low values of q, the number of nearest neighbours.
6.3.1. Linear Chain Antiferrornagnets We have noted in Section 5.6 that the two copper salts Cu(NH,),SO,.H,O and CuSO, * 5Hz0 may be considered approximately as isolated Heisenberg chains. We may therefore compare experimental data on these substances with linear chain calculations. It is interesting to note that the specific heat curve in Fig. 1 1 goes linearly to zero on the low temperature side, being described by C,kT/J = 3.9 J/mole" K. The important theoretical prediction, due to Bethe29 and Hulthen30, that the ground state energy of a Heisenberg linear chain should exceed that correspondingto the molecular field model ( E = 2NJS2)by a factor 1.77 can be checked. The curve of Fig. 1 1 corresponds to a value of 1.75 for this factor. The dashed line drawn in Fig. 1 1 represents calculations by Griffiths102 for a closed ring of 10 magnetic atoms. This line coincides with the experimental curve for kT/J > 1 but it goes more steeply to zero at kT/J < 0.5. Fig. 12 shows the susceptibility of Cu(NH,),SO,.H,O plotted as Jx/Ck versus kT/J, and this is again compared with the curve calculated for a References p. 340
338
C. DOME AND A. R. MIEDEMA
[a.v1,g6
closed ring of 10 magnetic atoms. There is satisfactory agreement between the two curves above the maximum, but the calculated curve for 10 atoms turns down at a higher temperature than the experimental one; this is undoubtedly due to the finite length of the chain. Measurements of the perpendicular susceptibility in the ordered state of C U ( N H , ) ~ S O ~ - Hdo ~ Onot agree with the results of elementary theory. For s = 3 both molecular field and spin wave approximations predict that xL should be given by JXJCk = 0.5; the observed value is JxJCk = 0.22. However the latter value fits remarkably well with the value expected from statistical theory (Hulthenso) for the susceptibility of the "unordered" linear chain at T = 0; Hulthen obtained JxJCk = 0.23. 6.3.2. Spin Wave Theoryfor Antiferrornagnets For three dimensional antiferromagnetic lattices, spin wave theory predicts, that in the absence of anisotropy, the magnetic specific heat beginning at very low temperatures should vary according to a cubic law. This prediction is not easily verified since it must be expected to hold at temperatures far below T,,at which temperatures C, becomes very small and can hardly be distinguished from the specific heat of the lattice waves. Furthermore if some anisotropy exists, the cubic law should only be followed at temperatures in energy comparable with the anisotropy energy; at lower temperatures the dependence should become more rapid than any power of T. This prediction has been verified for MnF2 by Catalan0 and Phillips114, who found the spin wave contribution to be quantitatively in agreement with that calculated from magnetic data. Quantitative agreement between theory and experiment is also obtained for MnCO, by Borovik-Roman~v~~~. In MnCOj the spins lie in the plane perpendicular to the principal axis of the crystal and in this plane there is practically no anisotropy. As a result the spin wave spectrum divides into two branches, for one of which there is a significant gap in the energy spectrum, while for the other branch there are practically no gaps. According to Borovik-Romanov the magnetic specific heat of MnCO, shows a T 3 dependence at temperatures below 4"K , a steeper rise above this temperature tending to another T 3 law with a doubled coefficient above 6"K(T, = 32.4"K). For many other antiferromagnets the predicted T 3 law for the specific heat and the corresponding T 2 dependence for the spontaneous magnetization have not been found, but this may be ascribed to the fact that the temperatures at which the experimental data were taken, are not low enough compared to TN. References p. 340
Because of the zero point vibrations in the spin wave spectrum the degree of alignment per magnetic ion for the ground state of an ordered antiferromagnet is expected to be smaller than the classical value, and at T = 0 is given by : (s) = s
-a,
where the value of u depends on the type of lattice (see Section 3.2). Experiments of Montgomery, Teany and Walshlle on KMnF, suggest that this prediction is not realized. By comparing the hyperfine structure constant obtained from electron spin resonance experiments with that derived from the nuclear contribution to the specific heat in the a.f. state they obtained <s)/s = 0.998 f 0.015, where a reduction of 3 percent is predicted. A similar result is obtained for MnF,, combining E.S.R. data of Clogston et aZ.117 with specific heat data of Cooke and Edmondslla i.e. (s)/s = 1.01 f 0.02. 7. Conclusions
We may first express moderate satisfaction at the agreement achieved between theory and experiment. Substances have been identified for which either the Ising or the Heisenberg model can be regarded as a reasonable first approximation, and available experimental data on the thermal and magnetic properties of these substances are in fair agreement with theoretical predictions. Copper salts have provided good examples of Heisenberg models of spin and investigations of new copper salts may be fruitful. Favourable representation of the king model has been obtained from cobalt and rare earth salts at low temperatures. Further experimental data on ferromagnets would be desirable but this is not easy to achieve since the number of ferromagnets seems fairly limited. This is a consequence of the fact that a substance in which both f. and a.f. interactions occur, still has the possibility of becoming ordered antiferromagnetically, in such a way that a ferromagnetic coupling exists among ions belonging to the same sub-lattice. More complete data on recently discovered ferromagnets such as EuO, EuS, EuSello (Heisenberg model s = 4)and Nd ethyl sulphate (Ising model s = +) should be forthcoming shortly. On the theoretical side a serious gap in our knowledge at present is the behaviour of the Heisenberg model of an antiferromagnet for low spin values. An even more pressing requirement is a proper understanding of the nature of the super-exchange interaction. The number of “simple antiferromagnets” for which comparison with present theory can be made is remarkably small and it is hard to predict in which salts simple antiferromag-
+
Rtfirences p . 340
340
C. DOMB A N D A. R. MIEDEMA
[av, .8 7
netism could be expected. Considerable progress might be made if a better understanding of the underlying mechanism were available. Experimental results on the thermal properties of simple antiferromagnetsin fixed external magnetic fields of varying strength, like those reported for MnCI2.4H20 by Voorhoeve and Dokoupil lZo, might also furnish useful additional information. Antiferromagnetism in “non ordered” lattices such as the F.C.C. presents a special theoretical challenge. Experimental results seem to agree that O/T,M 10 for a variety of spins. Information on the speciiic heat of an F.C.C. antiferromagnet is available for s = 1[Ni(NH,),Br,]; corresponding measurements of the chloroiridates (NH,), IrCl,, K2 IrC1, would provide data for s = Incidentally linear chain crystals can readily be distinguished from other crystals in which T, is reduced considerably (e.g. F.C.C. and structures like CrC12) from the ratio of the temperature at which the broad maximum in specific heat occurs and the Curie-Weiss constant O ; this ratio is about 1 for a linear chain. More precise information on the effect of distant neighbour interactions could help in the arrangement of a more detailed comparison between theory and experiment. Calplations could also be undertaken for specific interactions with a given degree of anisotropy if this was known independently from resonance experiments. Finally we may observe that important information on spin waves in magnetic crystals might be forthcoming from thermal conductivity measurements. If the heat transfer is determined by the magnetic system, as may be possible at temperatures near 1”K, data on the interactions between spin waves can be obtained.
+.
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117,1222 (1960).
'
A. H.Cooke and D. T. Edmonds, Proc. Phys. Soc. (London) 71, 517 (1958). llg S. van Houten, Phys. Letters 2,215 (1962). 1*0 W. H. M.Voorhoeve and Z. Dokoupil, Communications (Leiden) 328b; Physica 27, 118
777 (1961).
CHAPTER V I I
THE RARE EARTH GARNETS BY
L. m E L , R. PAUTHENET
AND
B. DREYFUS
UNIVERSITY OF GRENOBLE, FRANCE
CONTENTS: 1. Introduction, 344. -2. The magneticproperties of the rare earth garnets, 346. - 3. The levels of rare earth ions in the garnets, 369.
1. Introduction (by L. N ~ L ) The rare earth garnets, of general formula Fe,M,O,, where M is a trivalent rare earth ion, form an important class of magnetic compounds whose properties can be remarkably simply and quite accurately interpreted by a theory of three ferrimagnetic sub-Iattices. Theoretically, their study is of interest because of the large number of atomic substitutions that can be realised in the lattice and one hopes that t h i s may lead to a solution of the problem of magnetic interactions. Their practical interest, especially at high frequencies, is considerable, as they are excellent electrical insulators, preparable as monocrystals and characterised by very narrow resonance lines. The story of the rare earth garnets started in 1950, at Strasbourg. Forestier and Guiot-Guillain showed1 that, on heating an equi-molecular mixedprecipitate of Fe203 and M,O, where M is a rare earth such as Nd, Er, Y, Sm, Pr, La, they obtained strongly ferromagnetic products which they identified as Fe MO, and whose Curie points ranged from 520 to 740°K. In 1951, Guiot-Guillaina showed that the ferrites of praesodymium and lanthanum had a perovskite-type structure. Continuing their studies in 1952 and 1953, Forestier and Guiot-Guillain3 brought to light a curious fact: with M=Yb, Tm, Y, Gd and Sm, the aforementioned products had two Curie-points about a hundred degrees apart. These points varied regularly from one metal to the next as a function of the atomic radius of M. On passing through the upper Curie-point one obtained a strong thermoremanent magnetisation. These results attracted the attention of the “Laboratoire de Magn6tisme” References p . 382
344
CH.VII, 5 11
THE RARE EARTH GARNETS
345
at Grenoble: Pauthenet and Blum4prepared a gadoliniumferrite and showed in 1954 that, besides the two Curie-points Oi = 570°K and O2 = 678°K already mentioned, there was a third temperature O3 = 306°K at which the spontaneous magnetisation was zero. It certainly seemed that this was a compensation temperature, in the sense of Nbel's theory of ferrimagnetism ; that is, a temperature where the magnetisation changes sign. At the same time, N6el5 proposed a likely explanation of these facts on the supposition that the Fe3+ ions in the lattice formed a ferrimagnetic ensemble A having a spontaneous magnetisation opposite in direction to that of the sublattice B of M3+ ions. The molecular coupling field was sufficiently weak such that for temperatures above that of liquid air the magnetisation of the B ions was proportional to the field. In particular, NBel suggested that the point O3 corresponded to an exact compensation of the magnetisations of the ferrimagnetic A ions and B ions. A few days latere the existence of such compensation points was confirmed for the ferrites of dysprosium and erbium, at 246" and 70°K respectively; at these temperatures the remanent magnetisation changed sign. In spite of this success the supposed ferrimagnetic distribution of the Fe3iions was inconsistent with the recently confirmed perovskite structure of the compound FeMO,. At the request of L. NCel, Remeika at the Bell Telephone Laboratory prepared monocrystals of FeMO, which, after study by S. Geller7, were sent to Grenoble. Starting from the idea that the observed phenomena were due to a new type of compound, Bertaut and Forrat showed in January 19568 that one was dealing with a cubic compound Fe,M3Ol2, space-group 0 : ' l a 3d, with eight molecules per unit cell - an entirely unexpected discovery. The ferrimagnetic distribution of the Fe3+ ions was thus 24 ions on the tetrahedral sites 24d, oppositely magnetised to the remaining 16 Fe3+ ions on the octahedral sites 16a. In the following months, many experiments quantitatively verified the accuracy of this model. In particular, Pauthenet, after his work on pure Fe GdO, and Fe3GdsO12, showed9 that gadolinium ferrite (the object of his first investigations4) was in fact a mixture of 0.26 g of garnet, 0.64 g of perovskite and 0.1 g of Gd203; the points O1 and O2 were respectively the Curie-points of the garnet and perovskite. He soon completed his results with a study of the complete series of rare earth garnetslo, while, with AlBonard en Barbier he studied11 the properties of the isolated sublattice of Fe3+ ions in yttrium ferrite. All these results were communicated to the Congress of Physics at Moscow1a (25-31 May, 1956). References p. 382
346
m.m, F, 2
L. NhL, R. PAWHENET AND B. DREYHJS
The same research teams also did the first work on substitutions13, the accurate determination of the parameters14 and, with Herpin and Meriel15, structure determination by neutron diffraction. Following this, the Bell Telephone Labs. conducted a beautiful series of experiments on substitutions in the garnets, originating in the work of S. Gellerls. The two following articles are devoted to the garnets, the first by R. Pauthenet on their magnetic properties, and the second by B. Dreyfus on their optical and thermal properties, especially at low temperatures. In view of the numerous articles on the garnets since their discovery, it has been necessary to make a limited choice of the subject treated.
2. The Magnetic Properties of the Rare Eartb Garnets (by R. PAUTHENET) The aim of this section is to describe only the more important results obtained from the study of the magnetic properties of the garnet-type ferrites in a static field; in particular, those results useful for the understanding of many interesting problems that these compounds pose. 2.1.
PREPARATION OF THE
EARTH
GARNET-%
Fl3UUIW
There are two methods used for the preparation of the garnet-type ferrites in a polycrystalline form. The first consists of a solid-state diffusion of an intimate compressed mixture of oxides, obtained by heating for eight hours between 1340 and 1380°C. The second consists of the thermal decomposition of a homogenous co-precipitate of the nitrates at 400" C followed by baking at 1200°C for several hours8. The fkst method is quicker but the products are a little less pure and homogenous than those obtained by the second. This latter can be improved in detail to obtain very pure compounds17. The garnet-type ferrites exist for yttrium and all the rare earths except lanthanum, cerium, praesodymium and neodymium; preparation with prometheum, which is radioactive, has not been attempted; the ionic radius of the rare earth must be less than 1.13 A (Table 1). It became quickly evident that work with monocrystals was of great interest. Nielsen and Dearborn18 perfected a method of preparation using PbO as a solvent; study of the phase diagram F e z 0 3 - M z 0 3 - P b 0 showed that a satisfactory initial composition was MFe203- 3.5 Mz03 52.5 Pb 0. The method consists in heating the above mixture in a platinum crucible for five hours, cooling at less than 1" C per hour to about 900" C and finally quenching it. One can thus obtain monocrystals of garnet-type ferrite of about 1 cm in size mixed with crystals of magnetoplombite, rare References p . 382
-
,;i
f:
2
h)
P
Y
TABLE 1
z
Y
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
L
0
0
3
5
6
6
5
3
0
3
5
6
6
5
3
0
S
0
0
-
1 2
-2
3 2
-4
5 2
6 2
-7 2
-6
-5
4 -
3 2
2 2
1 2
0
J
0
0
5 2
15 2
8
-" 2
1.07
1.05
Ma+
ionic radius of Goldscbmidt (A)
2
4
-
9 2
2
4
-
5 2
0
-
2
7 6 2
2
-
2
6
5 7 0
2
b u4
1.06
1.22
1.18
1.16
1.15
1.13
1.13
1.11
1.09
1.04
1.04
1.00
0.99. X
a f 0.004A
12.37s
12.524 12.516 12.465 12.458 12.409 12.381 12.360 12.321 12.291 12.271
JOPB
9.44 f0.55
0 8.32 9.32 5.15 30.3 31.4 32.5 27.5 23.1 2.0 f0.3 f0.3 f 0.3 f 0.3 f 0.3 f 0.3 f 0.3 f 0.1 f 0.3 f0.05 564
568
563
567
ecoK f2
290
246
220
136
e, OK
-24
-8
-32
-6
eKDK f2
B
,
560
578
566
556
549 4<ec
84 ec<20
-8
548
549
I
348
L.N h , R. PAUTHENKT AND B. DREYFUS
[CH.W,8 2
earth orthoferrites and iron oxide. The crucible used is quickly corroded and Nielsenln has suggested the use of Pb F, as a solvent to prevent this; fusion is then lowered to about 1250°C.
STRUCTURE 2.2. CRYSTAL The crystallographic study of the rare earth garnet-type ferrites has been done by Bertaut and Forrat 8914 and taken up again by Geller and Gilleo20*21.
Fig. 1. Positions of the magnetic ions in the garnet structure.
These ferrites are cubic in structure and are isomorphous with the garnet Ca3 Fe (Si 04)3(Fig. 1). There are eight units of Fe,M301, in the unit cell. The oxygen ions define three types of sites at which the magnetic ions are found; 16 Fe3+ ions occupy the 16a positions which are at the centre of an octahedral arrangement of oxygen ions (Fig. 2a); 24 Fe3+ ions occupy the positions 24d at the centre of a tetrahedron (Fig. 2b); hally the 24 M3+ ions occupy the positions 24c inside a polyhedron with eight vertices and twelve edges, which can be appropriately represented as an irregular hexahedron obtained on distorting a cube by torsion about a tetrad axis followed by a slight distortion of the faces (Fig. 2c). The totality of the a-sites References p . 382
CH. W,9 21
349
THE RARE EARTH GARNETS
defines the sub-lattice (a); similarily we have the sub-lattices (d) and (c). A Fe3+ ion in a 16a position will be written Fe3+ (a); its position in the unit cell is (O,O,+); the Fe3+ (d) ions are at (O,&,+)and the M3+ (c) ions at (a,+,$) and (O,+,+); the 96h oxygen sites are defined by x = - 0.027,, y = 0.057,, z = 0.149, and are given by t-xyz;
xjj(3-4;
(b- + v)(t- x>ca + 4
(+-x)yZ; ;
qt.-yY)z;
(4--y)(4--x)(4-z);
(a - Y ) ( $ + X)(S + 4 ; (8 + Y>(++ XI(& - 4.
The crystal parameter a (Table 1) decreases regularly with the ionic radius of the rare earth, in agreement with the rule for lanthanide contraction, from 12.52, A for 5Fe,03. 3Sm,03 to 12.277 for 5Fez0,. 3Lu,O3. To discuss the magnitude and sign of the magnetic interactions it is useful to know the inter-ionic distances between nearest neighbours and the bond angles between neighbouring magnetic ions and the intermediate oxygen ion; these are given in Table 2; yttrium ferrite furnishes a typical order of magnitude for the distances.
2.3. MAGNETOSTATIC PROPERTIES 2.3.1. Experimental Results Experiment shows that for temperatures less than about 550°K and for fields between 5000 and 20000 Oe, the magnetisation J of the ferrites of yttrium, samarium, europium and lutetium varies with the field according to the law of approach to saturation J = J , (1 - a/H) ; from this we can find the spontaneous magnetisation J, by extrapolation. For the ferrites of gadolinium, terbium, dysprosium, holmium and erbium (a)
Referencesp . 382
(b)
(4
the variation of magnetisation with field can be represented by the linear expression J = Js xH,
+
except near liquid helium temperatures; the same expressionalso describesthe behaviour of thulium and ytterbium femtes except at very low temperatures. For each ferrite, Fig. 3 shows the temperature variation of the spontaneous magnetism expressed in Bohr magnetons (pB) relative to one grammemolecule 5FeZO3. 3M,03. For the ferrites of gadolinium, terbium, dysprosium, holmium and erbium, the magnetisation is large at low temperatures, rapidly decreases as the temperature increases, becomes zero at a temperature 8, (Table I), reappears, and then finally disappears at a Curie temperature 8, which is in the region of 560°K (Table 1) for these ferrites. These experimental results do not allow us to differentiate between a law in T* or TZfor the approach to absolute zero; in either case one finds the magnetisation at absolute zero, Jo, by extrapolation (Table 1). For thulium ferrite, experiment allows us to determine the temperature 8, as being between 4.2 and 20.4"K; for ytterbium.fenite the spontaneous magnetisation is small near absolute zero, 8, being 7.7"KZa; it then increases with the
Fig. 3. Variations of the spontaneous magnetisation versus temperature for the magnetic garnets. References p . 382
~~
Neighbowing ioq
A
B
g
Site
Nature
Fe3+ (a)
Fe3+(a)
Number Distance
n
3
Fe3+(d)
M3+ (c)
dA
8
3 5
YIG
Bond
5.35
Fe3+ (a) - 0 2 - - Few (d)
- 02-
Angle 126.6
Fe3+(d)
6
“s
3.46
Fe3+ (a)
- M3+ (c)
102.8
(1)
M3+ (c)
2
aii
3.46
Fe3+ (a) - 02-- M3+ (c)
104.7
(2)
02-
6
2.00
Fe3+ (d) - 02-- M3+ (c)
122.2
(1)
Fe3+(a)
4
3.46
Fe3+ (d) - 02-- M3+ (c)
92.2
(2)
Fea+ (d)
4
3.79
M3+ (c) - 0’- - M3+ (c)
104.7
MS+ (c)
2
3.09
Fes+ (a) - 0 2 -
- Fe3+(a)
147.2
M3+ (c)
4
3.79
Fe3+(d) - 02-- Fe3+(d)
02-
4
as
Fe3+(a)
4
a-
5 8
3.46
I
3.09
(1) distance Y3+ - 0 2 -
= 2.43
8,
4 6 a8
3.79
(2) distance Y3+ - 02-= 2.37
A
a -6
3.79
(3) for 4 neighbouring ions 0 2 -
5
5 6 1 6
1.88
Fe3+(d)
2
FeS+(d)
4
M3+ (c)
4
02-
4
2.37
02-
4
2.43
a-
86.6; 78.8; 74.7; 74.6
8
after Bertaut and Forrat**l7 Geller and Gdeo18$lo
(3)
352
L.N ~ L R. , PAUTHENET AND B. DREYFUS
[a. w,§ 2
temperature, passes through a maximum decreases and becomes zero at 0, = 548°K. For the ferrites of yttrium, samarium, europium and lutetium, the variations (Js, T) are regular and similar to those of a normal ferromagnetic; the Curie points are about 560°K. Fig. 4 shows the variation with temperature of the inverse of the susceptibility, x, for one gramme-molecule of several compounds; also shown
45
40
5
-4 0
Fig. 4. Variations of the reciprocal superimposed susceptibility versus temperature, below Curie point, for some magnetic garnets.
are the Curie-Weiss lines defined by the theoretical value of the Curie constant C, as deduced from the ground state of the free rare.earth ion. Neglecting for the moment the region of saturation at very low temperatures and the region near the Curie point, we can see from the two curves that the susceptibility is very near that of the free rare earthion. The order of magnitude of the paramagneticCurie point, Op, is defined by extrapolation of the Curie-Weiss lines (Table 1); they are negative and near absolute zero; they give us, thus, an indication of the strength of magnetic interactions between the rare earth ions, namely that they are weak and negative. These ferrites are paramagnetic above 570°K and their susceptibilities have been measured up to about 1500"KZS. Note for the moment that the curves (l/xm,T) (Fig. 5 ) are strongly concave towards the temperature axis, even more so than those observed for the spinel-type ferrites24.25; their References p. 382
THE RARE EARTH GARNETS
353
Fig. 5. Variation of the reciprocal susceptibility versus temperature, above Curie point, for Gd IG.
analysis yields important information on the magnetic interactions, which we shall treat in the following paragraph. 2.4. MAGNETIC INTERACTIONS We shall firstly make some remarks particular to the magnetic behaviour of these ferrites. Consider the ferrites of yttrium and of gadolinium; take the sum of the absolute values of the moments of the magnetic ions in the molecule at absolute zero. Knowing that the moment of a Fe3+ ion is 5pgr that of Y3+ is zero and that of Gd3+ is 7pB, we find the sums of 5OpB for 5 Fe203*3Y,03 and 92pBfor 5Fe20, 3Gdz0,; that is, much higher than the respective measured values of 9 . 4 4 ~ and ~ 30.3pB. Let us add that the temperature variation of the spontaneous magnetisation (Fig. 3) is quite different from that of iron, say; certain of these curves show a zero spontaneous magnetisation at a temperature below the Curie point which then reappears - just like that observed for the spinel-type Fe2.5 -xCr,04 26. ferrites of formula LiOa5 Also the variation of the inverse of the paramagnetic susceptibility with temperature above the Curie point is concave towards the temperature axis (Fig. 5). These remarks show us a striking analogy between the observed behaviour of these ferrites and that observed for the spinel-type ferrites 25-27; the interpretation of these results must thus be sought within the framework of a ferrimagnetic model 27. References p . 382
L.&,
354
R. PA-
[a. w, B 2
A N D B. DREYFUS
To account for the observed magnetisation at absolute zero we are led to adopt the following model: There are strong negative interactions between sub-lattices (a) and (d) which orient the Fe3+ ions on the octahedral sites in an antiparallel direction to the Fe3+ ions on the tetrahedral sites; the interaction between the ions on the sub-lattices (c) and (d) are negative but weak and the M" ions on sites (c) are magnetised in an antiparallel direction to that of the resultant moment of the Fe3+ ions. The values of the magnetisations at absolute zero for yttrium and gadolinium ferrites are, according to this scheme, lopB and 32pB respectively; these are in good agreement with the observed values. Let us now see if our model is compatible with the known facts on magnetic interactions in the ferrites. Being insulators, the magnetic interactions cannot occur via the conduction electrons in the garnet-type ferrites; the magnetic ions are separated from each other by the large oxygen ions and this separation is too large to give rise to an appreciable direct exchange28; however this situation can give rise to important superexchangeinteractions29 in certain cases. The theory showsSO that these indirect exchange interactions decrease with the separation of the magnetic ions and increase as the angle formed by triplet M3+ - 02- M3+ tends to 180". In the garnet structure the bond angle Fe3+(a)- 0'- - Fe3+(d)is 126'6' (Table 2) and the distance Fe3+ - 02-about 2A; this arrangement is very favorable for strong negative interactions between the sub-lattices a and d. Similarily the bond angle M3+(c)-0'- - Fe3+(d) is 122O2'; the distance M3+(c)-02- is 2.43A, greater than the previous case; one can thus suppose that the M3+(c) - Fe3+(d) interactions are negative and weaker than Fe3+(a) Fe3 (d) interactionspreviously considered. The bond angles Fe3 (a) - O2- M3+(c)are 102"8' and 104"7',both near 90"; consequently these interactions are weak. Continuing this line of reasoning one can expect that the interactions between the iron ions on sub-lattice (a) are strong, the bond angle Fe3+(a) - 0'- - Fe3+(a) being 147'2'; on the other hand the interaction between the Fe3+(d) ions should be weak as the bond angle lies between 74'6' and 86'6'.
-
+
+
2.5. INTERPRETATION OF EXPERIMENTAL RESULTS
A. Magnetisation at absolute zero Let us generalise this ferrirnagnetic model for the rare earth ferrites, excepting samarium and europium. The Fe3+ ion is in an S-state with a well defined moment at absolute zero of 5,uB; for the series of rare earth Rejkrences p . 382
cH.VnY821
THE RARB EARTH GARNETS
355
ions from terbium to ytterbium inclusive, the moment at absolute zero is ( L + 2s) pBfor the free ion under the hypothesis of Russel-Saunders coupling; L and S are the orbital and spin quantum numbers respectively (Table 1). Thus the moment of 5Fe203 3M203at absolute zero must be
-
J o = [6(L
+ 2s) - lo] ~ l g .
The agreement of the values calculated from the expression (Fig. 6) and those observed is satisfactory only for yttrium, gadolinium and lutetium ferrites; for the other femtes the rare earth moment must be less than the theoretical value ( L + 2 S ) p B . We already know that in the first series of transition
Fig. 6. The absolute saturation of the magnetic garnets.
elements the orbital moment is largely “quenched” ; the magnetic moment is due mainly to the spin; also, that this is due to the asymmetry of the crystalline field acting on the magnetic ion. While this may not be exactly the same case for the rare earths in the garnet-type ferrites one can well conceive a similar “quenching”. The rare earth ion is surrounded by eight oxygen ions at the vertices of an irregular hexahedron, which produce a strongly asymmetric crystalline field acting on the ion. This field raises the degeneracy of the energy levels; in particular, the moment of the ground level is no longer that of the free ion. We also note that the separations of these split levels from the ground level can be in the order of the energy corresponding to the measuring temperature; thus there is a contribution to the magnetisation from these sub-levels which is temperature dependent. The various theoretical investigations91-88 of this subject confirm our viewRefirencesp. 382
356
L. NJkL, R. PAUTHENET A N D B. DREYFUS
[am ., 9 2
point. This ferrimagnetic model has also been confirmed by the neutron diffraction of yttrium ferrite at room temperature's. 34.
B. Yttrium ferrite. Magnetic interactions between iron ions We know that the Curie points of these ferrites lie in the narrow band 548 to 578"K,whatever the rare earth ion (Table 1). From this we conclude that the Curie points are independent of the interactions between the rare earth ions, or, otherwise stated, these interactions are weak; it follows that the Curie points are determined by the interactions between the iron ions which are strong and of the same strength in the various ferrites. As the temperature increases, the spontaneous magnetisation of the rare earth ions decreases more rapidly than that of the iron ions, due the weak coupling between the former; the resultant spontaneous magnetisation, equal to the difference between these two contributions, has the sign of that of the M ions at low temperatures, is zero at the compensation temperature 8, (where the magnetisations of the M3+ and Fe3+ ions are equal and opposite) and then changes sign above 8,. We illustrate this behaviour, noting that the part of the curve (Js,T),which is symmetric in the abcissae above O,, is the extension of the curve drawn below 0, (Fig. 3). The compensation temperature is found experimentally by following the thermal variation of the remanent magnetisation of a specimen in a strong field at low temperaturesby means of a ballistic galvanometer; as the temperature rises the galvanometer deflection falls, is zero at 0, and then changes sign for higher temperatures. In an external field the variation of the magnetisation of the iron ions in this ferrite is very weak, which fact is also borne out by measurement on yttrium and lutetium ferrites, However, for temperatures above 50"K,the magnetisation of the M ions is far from saturation, and the resultant variation of the magnetisation of the specimen is that of the M ions. This variation is superimposed on the magnetisation in zero external field, and is of a paramagnetic nature. These thermal variations have been quantitatively interpreted by an extension of the molecular field approximation successfully developed by Neela7 to explain the magnetic properties of the spinel-type ferrites of two magneticsub-lattices; theextensionisto three magnetic sub-latticesin this case. The relative proportions of magnetic ions on the sites a, d and c are respectively L = & , p = 3 , v = 3 . The molecular field approximation reduces to the assdption that the action of the nearest neighbours on a Fe3+ (a) ion is equivalent to a magnetic field ha which is the sum of the three fields ha,, ha,, ha,; the first ha,,, is due to the Fe3+(a) ions; it is proportional to Referencesp . 382
the magnetic moment of these ions, Jayand to their number, 1; the field had represents the action of the neighbours Fe3+(d); it is proportional to their magnetic moment Jd and their number p ; laacand the molecular fields kd for the Fe3+(d) ion and k, for the M3+(c) ion are defined in a similar way. These three molecular fields are written
where the molecular field coefficients, nij, are proportional to the exchange integrals. At temperature T, in external field (H), the partial magnetisations Jay Jd and J, are the solutions of the three simultaneous equations
Ji = JoiBJt{&(hi
+ If)} ;
i = a,d, c.
Here .Ioi is the moment at absolute zero of the ion on the site i, and B,, is the Brillouin function of the ion i where Ji is its quantum number. The spontaneous magnetisations J,,, Jds and J,, are the solutions of the same equations with H = 0; the resultant spontaneous magnetisation is
J, = 1J,,
+ pJdS+ Y J,, .
At high temperatures the Brillouin function can be replaced by the first term of its series development and we have,
where C,is the Curie constant of the ion i. When the rare earth ion is non-magnetic, as is the case with yttrium and lutetium ferrites, the problem reduces to that of a substance with two sublattices. We know that, theoretically, the thermal variation of the inverse of the paramagnetic susceptibility above the Curie point is a hyperbola24, 1 T 1 -=-+--xm
c
xo
CT
T-6'
where C is the Curie constant for all the magnetic ions in the ferrite and l/xo, CT and 8 are functions of the molecular field coefficients, n,,. Referencesp . 382
358
L. NhL, R. PAUTHENETAND B. DREymrs
[aw, .62
2.6. TEMPERATURE VARIATION OF THE INVSRSB OF THE PARAMAGNETIC SUSCEPTIBILITY ABOVE THE CURIE POINT AND THE SPONTANEOUS MAGNKITSATION OF YTTRIUM FERRITE
We have applied the above treatment to the case of yttrium ferrite. From the experimental curve (Fig. 7) we have been able to fit a hyperbola of the above type which is a good approximation for the values C = 50, l/xo = =30.5, t~ = 990 and 8 = 570. Before going further, it is necessary to note that the experimental Curie constant, C, is greater than the theoretical value
Fig. 7. Experimental and theoretical curves in the ferrimagnetic and paramagnetic range of temperature of YIG.
C' =43.77 for the ten Fe3+ ions. The same result had already been found for
the Curie constants of the spinel-type ferrites. Nee136 explained this as being due to the effect of thermal expansion on the magnetic interactions. On the hypothesis that the molecular field coefficient is a linear function of the temperature, 4 1 = no&
+ YT),
we have the following relation between C and C':
-1 --_1 +-.7
c
C'
xo
From this we deduce that y = - 1.3 x for yttrium ferrite, which agrees well with the values observed for the spinel-type ferrites; knowing l/xo, y and 8 this correction gives the following values of the molecular field coefficients: nOaa= - 352, noad = - 742 and nod,, = - 210. We note that these interactions are negative and that the strongest is between the magnetic ions of the sub-lattices(a) and (d), as expected. At References p . 382
fx. w,8 21
THE RARE E A R m GARNETS
359
very low temperatures the molecular field acting on a Fe3+(a) ion is about 5.3 x lo6 Oe and that on a Fe3+(d)ion 3 x lo6 Oe. Knowing the coe5cients n,,, we can now solve the set of equations (1) for a given temperature and H = 0. We thus find the theoretical thermal variations of the spontaneous magnetisations of the iron ions on each sublattice, (J,,, T) and (&,, T);this also gives us the thermal variation of the total spontaneous magnetisation (Js,T).Comparison with the experimental curve (Fig. 7.) shows satisfactory agreement with this theory. Robert 36 has recently obtained some very interesting experimental results for the thermal variation of (J,,, T)and (J&, T)in yttrium ferrite. Starting from Solomon's experimental results37 from the Mossbauer effect in this ferrite, he measured the nuclear magnetic resonance frequencies of the Fe57 ions on the a and d sites between 1.7 and 400°K.Suppose that the hyperfine structure is given by a term A S*Z,I being the nuclear spin and A a constant independent of the temperature; because of the strong exchange coupling between the electron spins, each nucleus experiences an interaction dependent only on the mean value of the spin, ( S ) , of the ion in the corresponding sub-lattice. The nuclear magnetic resonance frequency is thus A ( S ) and its variation measured against temperature is a sensitive measure of the mean value of the spin and hence of the spontaneous magnetisation at the site considered. Considering the approximations made, the agreement between these two types of curves, while not perfect, is satisfactory (Fig. 8).
Fig. 8. Theoretical variations of the spontaneous magnetisations of the magnetic ions on the a, d and c sites in the case of Gd IG. References p . 382
360
[a. vn, 0 2
L. Nh., R. PAUTHKNET AND B. DReypuS
C. Garnet-type ferrites with magnetic rare earth ions. Magnetic interactions between iron and rare earth ions Since the Curie points of these ferrites are of the same order of magnitude as that of yttrium ferrite, we shall suppose that the interaction between iron ions is the same in these ferrites as those we have just defined for the yttrium ferrite. We shall take into account the magnetic interactions between iron and rare earth ions by deking a mean molecular field coefficient, n, such that its product by the sum of magnetisations of the iron ions is equal to resultant of the molecular fields R,, and hcd, due to the ions Fe3+(a) and Fe3+(d) respectively; n thus satisfies the equation n(nJas
+ pJds> = ncaaJar + %dpJds
*
At the compensation temperature equation 3 of the set (1) becomes
from which, using the above results, we find 1
n=--
1
for T = 8,.
vxc
This value of n is negative and can be directly deduced from experiment, as its absolute value represents the inverse of the paramagnetic susceptibiIity superimposed on the ferrimagnetism of the iron ions at the compensation temperature 8,. To relate n to the exchange integral which it represents, we note that exchange coupling acts only by means of the spin angular momentum; we are thus led to represent the exchange integral by a coefficient n' related to n by:
the variations of n' from one rare earth to another should be due only to the variation in distance between the ions, all other parameters being constant (Table 3). T A B L E3
M n'
1
I
References p . 382
Gd
Tb
DY
Ho
- 107
- 86
- 93
- 67
Er
Tm
- 66
- 58
f2H. w, 8 21
THE RARE EARTH GARNETS
361
-
Wolf and Van Vleck38 have found n' = 103 for europium ferrite by an entirely different method; this is in good agreement with the above values. However, with the exception of europium and gadolinium ferrites, we should not attribute more than an order of magnitude importance to these values; for the other ferrites, whose compensation temperatures decrease as the atomic number of the rare earth increases (Table 1) we must make corrections to take into account the saturation of the rare earth ion and the effect of the crystalline field on its moment. For example: the molecular field acting on a Gd3+ ion in gadolinium ferrite at 100°K and due to the iron ions is about 3.7 x lo5 Oe; that is, about one tenth of the field acting on the ion irons. 2.7. MAGNETIC INTERACTIONS BETWEEN RARE EARTHIONS We have already indicated that the experimental values of the paramagnetic Curie points, for the superimposed paramagnetic susceptibility below the Curie point, are not accurate; they are impossible to measure in the case of thulium and ytterbium ferrites; from the relation 8, = nccvC,, we find the molecular field coefficient, nco which characterises the interactions between the rare earth ions ; consequently these coefficients are not well determined. Fortunately, their order of magnitude is small and we can neglect them in certain problems. Taking gadolinium ferrite at 100°K as an example, the rare earth interaction is equivalent to a field of about 50000 Oe; this is about one seventh of the field due to the iron ions acting on the rare earth ion, and about one hundreth of the interaction between iron ions.
2.8. THERMAL VARIATION OF THE SPONTANEOUS MAGNETIZATION AND OF THE INVERSE OF THE PARAMAGNETIC SUSCEPTIBILITY IN THE RAREEARTH FERRITES
We suppose that the thermal variations of the spontaneous magnetisations (J,,, 2') and (Jds,T)of the iron ions on sites a and d are the same as those established in examining yttrium ferrite (Fig. 8). From the known values of n and n,,, we deduce the spontaneous magnetisation of the rare earth ion (J,,,T ) ,as the solution of the equation
using our previous results. Because of the weak coupling between rare earth ions, the variation found as solution of this equation is quite different from that of the iron ions and is in agreement with our previous hypotheses. References p . 382
362
L. NhL, R. PAUTHENET AND B. DREYFUS
[a. vn, P 2
Knowing the thermal variations of the spontaneous magnetisation of each sub-lattice we can thus deduce the total resultant spontaneous magnetisation (Js,T). The inverse of the superimposed susceptibility is given by
where B;,(z) is the derivative of the Brillouin function with respect to z for the ion M at the value corresponding to the spontaneous magnetisation, J,,, and the relevant temperature. The inverse of the paramagnetic susceptibility above the Curie point is theoretically a third degree equation,
I T 1
aT+a
Generally, for a substance with p sub-lattices, such a curve is of degree p . Albonard23 has developed these calculations and has attempted to determine all the coefficients nii, particularly those for the rare ion; this work is interesting, but it does not bring to light any more fundamental information on the interactions than that already developed.
2.9. THERAREEARTHG A L L A ~ S We have been able to aErm that the rare earth ion plays an important role in the magnetic properties of the garnet-type ferrites. The form of the thermal variation of the spontaneous magnetisation is determined by the rare earth ion and we have also seen that the moment of this ion at low temperatures is less than that of the free ion. In order to fur the magnetic behaviour of this ion it is interesting to study its properties in a garnet-type structure where the only magnetic ions are those of the rare earths themselves; this is realised by substituting a non-magnetic ion such as gallium, aluminium, indium, scandium or chromium for the ferric ion in the ferrite. This substitution is most complete for gallium, with which it has been possible to prepare gallates with yttrium and all the rare earths from praesodymium to lutetium except prometheum. The crystal parameters vary between 12.25 and 12.57A, which is of the same order of magnitude as in the ferrites; we may also assume that the distances between rare earth ions and thus the dimensions of the polyhedron of oxygen ions surrounding the rare earth ion, are the same in the ferrites and the gallates; thus, the exchange interactions between rare earth ions are very weak. The very accurate measurements by Wolf et al. 38 giving ep= - 0.048 f 0.01“K for 5Ga203 3Yb203
-
References p. 382
confirm our point of view. More particularily, we shall examine the magnetic behaviour of ytterbium gallate at low temperatures. Starting from the variation of ( J , H ) measured at temperatures 2.35, 4.2 and 20.4”K we have plottedJ/Joc as a function of H/T39, where Joc is the moment of Yb3+ at absolute zero in its ground state 3. The curve thus obtained is unique (Fig. 9) and differs considerably from that of the Brillouin function for 3.
0
-morn
n/r
Fig. 9. Variations of the relative magnetisations versus H/Tfor Gd Ga G and Yb Ga G and the corresponding Brillouin function.
On the other hand, the same curve plotted for gadolinium gallate, relative to the value of the moment of Gd3+ in an molecular field, shows that the moment of this ion is given by the Brillouin function even for large values of HIT.The difference in the magnetic behaviour of these two ions illustrates the “quenching” of magnetic moments in rare earth ions due to the crystalline field caused by the neighbouring oxygen ions. The ground state of Gd3+ is ,% and this is not split by the crystalline field; on the other hand the Yb3+ ion has ’F, for its ground state; assuming a cubic crystalline field, this eight-fold degenerate level is split into two doublets, r6 and r7 and a quadruplet, r8 (Fig. 10); under the real orthorhombic symmetry, the quadruplet is further split into two doublets. The ground level is r7 and it is little altered by the orthorhombic symmetry; under these conditions of cubic symmetry the theoretical calculation of the magnetic properties of Yb3+ are more easily effectedal. The magnetic moment is determined by the matrix elements for the states r7 which give a term in HIT, and by References p . 382
364
L.N k . , R. PAUTHENET AND B . DREYFUS
[aw, . 82
the non diagonal elements between r7 and r8 which give a constant paramagnetic term. The latter may be obtained from the paramagnetic susceptibiiity40 curve and one then can deduce the energy-gap between r 7 and r8;this has the amazingly large value of 570 cm-', which, however, has been confirmed independently by optical absorption measurements41. It follows that the constant paramagnetism is small; it is negligible at low temperatures compared to the term in HIT which can be exactly calculated, and
Fig. 10. The energy levels of the Yb3+ion.
its theoretical variation is in good agreement with the experimental curve (Fig. 9). The departure from cubic symmetry at each site c gives rise to an anisotropic Land6 g-factor, which has been experimentally observed both by paramagneticresonance42and by anisotropy measurementsin a static field43; it has been explained by a treatment of the actual orthorhombic symmetry40. A second feature particular to the rare earth ion is its paramagnetic behaviour at high temperatures. As the exchange interactions are negligible, it is interesting to compare the product xT,of the paramagnetic susceptibility with the temperature, with the Curie constant C of the ground level (Fig. 11). Agreement is perfect in the case of gadolinium gallate; for erbium and ytterbium XT tends assymptotically to C while for praesodymium, neodymium, samarium and europium gallates XTincreases continuously with the temperature; finally, for the gallates of terbium, dysprosium and holnlium, xT starts by increasing with the temperature, tends towards C and then decreases for temperatures above about 900°K. These results typify a general type of paramagnetism in the rare earths that Van Vleck44 developed to interpret the susceptibilities of Sm3+ and Eu3 . For these latter at ambient temperatures the multiplet splitting is in the order of the thermal agitation energy, and these higher multiplet +
References p . 382
levels contribute progressively to the paramagnetism; for the other rare earths, the multiplet splitting is much larger, and the magnetic contribution due to the higher levels is negligible at ambient temperatures; at higher temperatures of about 1500"K, where measurements have been made, this contribution becomes appreciable. Using Van Vleck's results for the paramagnetic susceptibility x, we have,
WJis the energy of the state with quantum number J and aJ is the constant paramagnetism term; it is the sum of the squares of the non-diagonal matrix elements. The expression may be evaluated either from experimental values for the energy levels45 or by direct calculation of these levels44; it is then necessary to adjust the screening constant 0. Satisfactory agreement is obtained for a = 34, which is quite compatible with that used by Van Vleck in his study of Sm3+ and Eu3+. 2.10. EUROPIUM AND SAMARIUM FERRITE~
As we know the magnetic behaviour of the ions Sm3+ and Eu3+, we have
Fig. 11. The paramagnetism of the gallium garnets and the theoretical curves from Van Vleck's theory.
References p . 382
366
L.NhL, R. PAWTHENET AND B. DREYPUS
[a.w,§2
left until now a discussion of their femtes. We shall do so in light of what we have just seen for their gallates. The ferrimagnetic behaviour of these ferrites below their Curie points has been treated by Wolf and Van Vleck88 and by Van Vleck46;it may be summarised as follows. In the approximation of cubic symmetry the effect of the crystalline field on the ground state, 'FO, of Eu3+ may be ignored, as this is a state with J = 0; also, the first excited level with J = 1 is not split by a cubic field. The interactions between the rare earth ions may also be neglected as compared to the interactions between the iron ions which may be represented by a molecular field H, = n'(Ws, + p&). In zero external field, the rare earth ion is magnetised by the molecular field alone; the calculation of the resulting spontaneous magnetisation is of the type developed by Van Vleck44 to find the magnetisation of the free ion; one must consider both diagonal and non-diagonal terms for the different levels of the multiplet. Here, the molecular field plays the part of the external field but, one must remember in calculating the terms, that this molecular field acts only on the spins. Wolf and Van Vleck have done these calculations for the first three levels J = 0 , l and 2; they adjusted n' such that there was agreement with the experimental value of J, at absolute zero; their value of n' = 103 is in agreement with ours of n' = 107 which was determined from gadolinium ferrite by quite different method. Knowing the thermal variation of the spontaneous magnetisation of the iron ions (taken as that of yttrium ferrite), they were able to calculate the thermal variation of the spontaneous magnetisation of both the Eu3+ ion and that of its corresponding ferrite. This latter variation is in perfect agreement with the experimental curve. This same treatment has not been able to account for the magnetic properties of samarium ferrite in a quantitative manner46; it presents, however, the interest that at a certain temperature the moment af Sm3+ in the exchange field must change sign. There is an important contribution to the susceptibility from the non-diagonal elements between levels 4 and 4;this represents a constant moment. The diagonal terms vary as 1/T and are of opposite sign to the non-diagonal ones; thus, the sum of the two changes sign at a certain temperature. This effect has not yet been studied in samarium ferrite but it has been observed in the alloy Sm Ala (ref. 47. It could explain the low value of the moment of Sm3 . +
2.1 1. SUBSTITUTIONS IN THE RARE EARTHF ERRIm The problem of the substitution of the various ions (magnetic or not) in the rare earth compounds is interesting in that it gives us information on Referencesp . 382
CH.W, 21
THE RARE EARTH OARNETS
367
magnetic interactions, the properties of a particular ion and, like for the substitution by zinc in the spinel-type ferrites, it can have a practical interest. The substitution of the ferric ions was the first to be thought 0f48-6~; it is complete with the ions A13+ and Ga3+ and is partially complete with Cr3+, Si3+ and In3+.As the A13+and Gd3+ ions are smaller than the iron ones, they preferentially occupy the smallest crystallographic positions; that is the tetrahedral sites, 24 d. The Si3+ and In3+ ions are bigger than those of iron, and they preferentially occupy the octahedral sites 16a. A Cr3+ ion, which is much smaller than an iron one, preferentially occupies an octahedral site; this is probably as a result of its electronic codguration d2 sp3 which is favorable to the formation of an octahedral bond. Next we may consider substitution of the rare earth ion. We have seen that the garnet-type ferrites exist for the rare earths between samarium and lutetium; evidently one can effect a substitution by any rare earth in this group. However the main interest is that of substitution by the lightest rare earths, La, Ce, PryNd, which do not give simple ferrites as their ionic radii are too large. However one can have partial substitution in compounds of the type x M z 0 3with M = La3*, Pr3+,Nd3+ 51-63; the limit5Fez03 (3 - x)YZO3* ing values of x are 2 for Nd3+, 1for Pr3+ and 0.5 for La3+. Cerium cannot be introduced into this structure. From what has been said on the ferrimagnetic model of the garnet-type ferrites, the substitution of a magnetic ion (e.g. neodymium) for Y3+ should decrease the resulting moment, because the rare earth moments are antiparallel to those of the iron ions; in fact experiment shows that the resultant moment of the ferrite increases with the amount of neodymium. This unexpected result has been explained by W0lf54. In the molecular field model we wrote the exchange energy as He,
=
- n'SFc
* STRI.4 Y
where SFe is the resultant spins of the iron ions on the a and d sites and S,, is the rare earth spin. As the coefficient n' is negative, these two spins are antiparallel in equilibrium. For the second rare earth sequence (gadolinium to ytterbium) the magnetic moment of the rare earth is parallel to the spin direction (J=L S); consequently the rare earth moment is in the opposite direction to that of the resultant moment of the iron ions on the a and d sites, and the moment of the ferrite is the difference between these two moments. For the first sequence (cerium to europium) the magnetic moment of the rare earth is in the opposite direction to its spin (J = L - S ) and hence the rare earth moment is to be added to the resultant moment of the iron ions.
+
References p . 382
368
L. N ~ L R. , PAUTHENET AND B. DREYFUS
[CH.W,4 2
2.12. OBSERVATION OF ELEMENTARY DOMAINS Slices of the rare earth ferrites (type garnet and gallate), of thickness about 20-50 microns, are transparent to visible light; they can form the basis of transmission experiments which are important from both a fundamental and applied viewpoint. Dillon55 was the first to observe wide transmission and absorption bands in the visible spectrum for yttrium ferrite. He observed a Faraday rotation of a plane polarised beam propagated parallel
Fig. 12, Magnetic domains in YIG (from Dillonso).
to the direction of spontaneous magnetisation, the birefringence of a plane polarised beam propagated in a direction perpendicular to the spontaneous magnetisation and the dichroism for circularily polarised light. An important practical application of this optical Faraday effect has been the direct observation of the elementary domains55-57. If a slice of a monocrystal is examined under a polarising microscope one sees an image generally ressembhg Fig. 12. The regions of different contrasts represent the domains magnetised in different directions; the maximum contrast is that for two domains at 180" and whose magnetisations are parallel and antiparallel to the optic axis. By adjusting the analyser with respect to the polariser, one can vary the contrast; when polariser and analyser are parallel there is no contrast between the domains themselves, but the Bloch walls show up as black lines. One can also follow the movements of these Bloch walls in an external field and see their disappearance at the Curie point. We point out that the observed domains are those of the whole of the References p . 382
CH. VII, 8 31
THE RARE EARTH GARNJXS
369
sample, and not just those of its surface as is the case for the Bitter powder technique. However, the specimens must be very thin and the domain structure depends on the surface state - unless all contrast has disappeared. This is difficult to realise in practice as one can use neither electrolytic polishing nor chemical etching; but with careful mechanical polishing excellent patterns may be observed.
3. The Levels of Rare Earth Ions in the Garnets (by B. DREYFUS) As we have seen in the first part of this article, the central problem in the study of the garnets is the determination of the magnetic properties of the rare earth ions. They are acted on by a non-cubic crystal field and an exchange field due to the Fe+3ions. Since the orientation of the crystal field, with respect to the cubic axes of the crystal, varies from one ion in the unit cell to the next (for a given general direction of the magnetisation there are six different sites), and further, since the splittings due to the crystalline and exchange fields are not always much larger than the thermal agitation energy, the resulting situation is extremely complicated and this has attracted the notice of many research workers. In the following paragraphs we shall examine some of these investigations devoted principally to the low temperature region and the determination of the rare earth levels; we refer the reader to the original articles where he will find a much more complete set of references. In particular, it has not seemed possible to consider the very many investigations devoted to paramagnetic and ferrimagnetic resonance, the Mossbauer effect, nuclear resonance, etc. 3.1. CRYSTALLINE FIELDS We know that the internal coupling in the 4f shell leads to a term with a given L and S (following Hund's rule, generally). The spin-orbit coupling splits the term into corresponding multiplets labelled with the quantum number J ( J = L S). The elements to the right of gadolinium in the periodic table have J = L S, whilst those to the left have J = L - s;gadolinium itself has an orbital moment L=O. This description corresponds to the free ion, but the rare earth ion in a crystal is subject to the perturbation of the crystalline field which is due to its environment; this field partially raises the degeneracy of the ground multiplet. The crystalline field is characterised by its symmetry and its strength. These questions have been treated by Bethe in his classic paper of 1929 where he shows how the theory of groups can simplify the problem by taking fullest advantage of the symmetry of the crystal.
+
References p. 382
+
370
L.!&EL,
R. PAUTHENET AND
8. DRWFUS
[m.w,I 3
In the garnets the determination of the energy and the properties of each sub-level (gyromagnetic ratio and splitting due to an external or exchange field) is indispensable for the understanding of the many macroscopic properties: susceptibility, spontaneous magnetisation, anisotropy, optical absorption spectra, etc. The crystalline field acting on the 4f electrons can be developed in terms of spherical harmonics, Y;",with 1 = 2,4 or 6. We need only consider even terms, and terms higher than sixth order are zero. Various degrees of approximation have been used; firstly, that of a field of cubic symmetry in which only spherical harmonics of fourth and sixth order appear. White and Andelin32 have treated a cubic potential of fourth order together with a magnetic field in direction [l,O,O] or [1,1,1]. The results obtained from a computer (the order of the matrices, W + 1, can be as high as 17) give the energies as a function of the ratio of the strength of the magnetic field to the strength of the crystalline field. One has also taken into account sixth order terms to find the energies and wave functions when considering a crystalline field of cubic [email protected] refined calculations have tried to take into account the actual, non-cubic, symmetry60. Ayant and Thomas61 have used a distorted cube as their model, as well as a more rigorous one40 where they have replaced the true position of the oxygen atoms by point charges (electrostatic model). Finally, Dillon and Walker62 have performed certain more complete calculations regarding the orientation of the field. 3.2. SPECIFIC HEATS The thermal excitation of the various energy levels gives rise to a contribution to the specific heat. We should note however that tbe rare earth ions in the crystal are not isolated, but are magnetically coupled to the Fe3+ ions. We then have collective oscillations (spin-waves or magnons), and one can speak of the energies of individual ions only with care. Many measurements have been made on powdered garnets63-65. The most complete are those of Hams and Meyere6 in the range 1.3"K to 20°K. The speciflc heat may be considered as the sum of three terms
C = Chtt + C m n p + Cnucl; C,,, is the lattice contribution which may be characterised by a Debye temperature, 8. Assuming similar structures, we can suppose that the force
constants are the same for all the garnets to a fist approximation, and thus 8ocM-* where M is the molecular mass. We probably do not make a References p . 382
ca.wO31
THE RARE EARTH 0ARm
371
large error in taking 8 as that of Lu I G which has been very accurately determined as 458°K.The upper limit is that of YIG at 8 = 572'K. The nuclear contribution, Cnucl,is due to the hyperfine coupling With the electronic moments, and gives rise to a Schottky anomaly; only its "tail", varying as T-', is observable in the temperature range studied. The contribution of the Fe67 can be estimated from the Mossbauer effect or by nuclear resonance; it is too small to be measured. However the contribution from the rare earth nuclei may be appreciable:
Here Cnuclis for one mole of active nuclei (correcting for isotopic abundance), I is the nuclear spin and A the energy separation between successive nuclear levels in the hyperfine field; the contribution due to quadrupolar couplingofthe nucleus withthe gradient of the electricfield has been neglected. In the absence of measurements below 1.3"K, it has been possible to determine A only for Tb I G ( A = 0.10 cm-') and Ho I G (A = 0.181 cm-'), for which this term is the most important. These values are compatible with those determined by electronic paramagnetic resonance in the other rare earth salts. d may be calculated if one knows the exact electronic wave function for the ground level. One must take into account the inequivalence of the different sites. In particular, Cnuclof a monocrystal depends on the orientation of the spontaneous magnetisation with respect to the axes. It seems that such a calculation has not been performed. Finally, the last term, Cmag,is the most interesting. In YIG and Lu I G only the Fe3+ ions are magnetic, the anisotropy is weak, and the theory of spin waves gives a specific heat varying as T*.
(k. 5)3
Cmag= 0.235
JIMd",
where D is the coefficient for the acoustic branch of the spin-waves of frequency o and wave-vector k hw = Dk2a2 (a is the lattice parameter). D can be calculated from the exchange integrals
and is proportional to the sum
Douglass67 has determined the spin-wave spectrum of YIG at the centre Refirems p. 382
372
L.N&L, R. PAUTHENET AND B. DREYNS
[aMI, .B3
and corners of the Brillouin zone. His calculations show that it is only the acoustic branch of the spin-waves that is excited in an appreciable manner at 20°K. Although the variation in T*is verified experimentally, the values of D measured on several samples of YIG vary from 16 to 27 cm-'. For L u I G the only value is 27.1 cm-'. These values do not agree perfectly with the exchange integrals calculated from magnetic measurements or by microwave instability es-69. In the case of Gd I G the spin-wave frequencies have been calculated for the centre of the zone70. This shows that a certain number of optical branches have frequencies low enough to be excited at 20°K (there are 11 modes with frequencies between 27°K and 46°K). Tinkham71 has shown that, due to the slight differences between the gyromagnetic ratios of the ions on different sites, the frequencies of the optical branch are practically constant for most of the zone; these collective excitations have a groupvelocity which, if not zero is very small, and are well represented by individual excitations having the same energy as the separations of the sublevels of the rare earth ions. This enables us to justify the simple model adopted by Harris and Meyer; they consider the sub-lattices (a) (d) and (c) as constituting a continuum in which spin-wave theory is valid; this gives a term in T + ;they then add the contribution due to the rare earth ions, treated as systems having individual energies, and under the influence of the molecular field due to (a) and (d). The separation of adjacent levels taken from their results is AE = 28.6 cm-', in good agreement with Pauthenet's magnetic measurements (AE = 24.9 cm-l). Detailed calculations of the magnon spectrum for the whole of the zone are now under way72. The orbital moment is non-zero in the other garnets and we must accurately consider the action of the crystalline field. The example of Yb I G is the simplest; we must distinguish: a. The contribution from the acoustic branch, varying as T*. This is now smaller (than that of Y I G) due to the large anisotropy which converts this term into an exponential. We should also add that D itself is increased because of the smaller degeneracy of the ground levels. The resultant specific heat is negligible. b. The optical modes of the individual ions in the molecular field. These separate into two groups due to the inequivalence of the sites; when the spontaneous magnetisation is in the [l 111 direction the mean value of the corresponding splitting is 25 cm-', which is in excellent agreement with the optical measurements of Tinkham and Sievers72.74 who find two rays at 23.4 cm-' and26.4cm-'. References p . 382
CH.VII,
8 31
THE RARE EARTH GARNETS
373
c. A mode corresponding to the exchange resonance between the YbJt and Fe3+ ions which is accidentally situated at low frequencies. Calorimetric measurements give this mode a mean energy of E = 17 cm-I while, for k = 0, optical measurements give E = 14 cm-l. We shall not discuss the other garnets (Ho, Tb, Er, Dy, Sm). Their general behaviour is well understood, but there are still many details to be explained.
3.3. THERMALCONDUCTION We cannot leave the subject of spin-waves without drawing attention to the fact that their contribution to thermal conduction has recentIy76~76 been shown. The measurements were on Y I G between 0.4"K and 20°K. This substance was chosen because of its weak anisotropy and, hence, large magnon specific heat. The conductivity is given by the classical formula
u is the group velocity dwldk, which is proportional to k for the dispersion relation h w = Dk2 and 1 is the mean free path. Douglass has taken advantage of the influence of an external field, since the dispersion relation, hm = g@H -iDk2, leads to a reduction of the specific heat, C ; he thus separates the magnon and phonon contributions. He used a field of about 20 kOe thus causing a gap of about 2°K in the spectrum. Apart from some complications due to the existence of domains, the magnon conductivity in zero field at 0.5"K has been estimated as 70% of the total conductivity. The magnon conduction follows a law in T 2 , which agrees with the theory77.78 for a free path independent of k. This free path is of the order of 2 x cm and is probably determined by paramagneticimpurity scattering which is very effective for long wave lengths.
3.4. SPECTROSCOPIC INVESTIGATIONS The interaction of electromagnetic radiation with a large number of garnets has been studied both for powders and monocrystals, and principally by absorption. The lattice vibrations in the garnets make them strongly absorbent in a spectral region from about 100 cm-' (loop) to about 800 cm-' (12,5p)70. We note in particular that the vibrations of the SiO, tetrahedron can be detected; these are formed by the presence of Si4+ ions as impurities and play an important role in many properties (particularily anisotropy), due to the Fe2+ ions which appear to keep the crystal electrically neutral. We recall also the work on the optical absorption of the Fe3+ ions, whose References p . 382
374
L.NfreL, R. PAUTHENET A N D B. DREYFUS
[CH.W,8 3
spectrum starts at about 10000 cm-' and has been studied up to about 35000 cm-'80#81. Two transparent regions are left; the first is situated in the near infra-red and visible region where transitions between the ground and excited multiplets may be observed; the second is in the very far infrared (between 10 cm-' (lOOOp) and 100 cm-' (loop)) where the principal transitions are those within the ground multiplet. We recall that in absorption measurements the line intensity is proportional to the population of the lower level. At very low temperatures the only transitions are those from the lowest sub-level; but at temperatures above that of liquid hydrogen (kT- 15 cm-') we have the addition of a spectrum due to transitions from one of the excited sub-levels of the ground multiplet.
3.5. THE VISIBLE AND
NEAR INFRA-RED SPECTRUM
Due to the inequivalence of the sites there are in general at least two spectra present, and more than two if the spontaneous magnetisation is arbitrarily inclined to the crystal axes. It seemed that one could get over the further complication of the exchange field splitting by using isomorphous samples such as gallates, aluminates, etc. but it appears that the crystalline field itself is quite sensitive to these substitutions, and that one should regard electrostatic models with prudence. The simplest case is that of Yb I G, studied by Wickersheim and White82g 8s in the near infra-red. Yb3+ has two multiplets, J = J and J =3 (Fig. 13). The observed tranSPIN-ORBIT COUPLING
CUBIC
FIELD
RHOMBIC FIELD
fg. 13. Energy level scheme for Yb+++in YGaG.
References p. 382
CH. W,8 31
375
THE RARE EARTH OARNETS
sitions are from the ground sub-level to the various excited levels of the J = 3 multiplet. The molecular field (whose effect is not shown in Fig. 13) splits each sub-level into two, and this splitting is different for the different sites. So that, at not too low temperatures (T = 77"K),there are four absorption lines centred on the 10 316 cm-I transition. In zero external field, the spontaneous magnetisation is along [1111, there are two types of sites and hence eight lines. A study of the shift of these lines as a function of the angular variation of M, allows one to deduce the Zeeman splitting. For example: the ground level for M, parallel to [ l l l ] is split into two levels with separations 22.1 cm-' and 25.3 cm-' for the two sites (in good agreement with specific heat measurements and optical measurements in the far infra-red). The splitting as a function of angle can be written, using an effective field, as H =- p*H,.,, where p is the magnetic moment due to the spin (as we are dealing with an exchange field) of the level considered.
S'being an effective spin of 3 and g' the gyromagnetic tensor which can be found by electronicresonance (herethe determinationwas on Yb Ga G)48384. The effective field is produced by the exchange interaction with the Fe3+ ions, Thus, even when the exchange interaction is isotropic (&=constant), it is natural that the splitting depends on the angle, becausef the anisotropic g-factor. Now, optical measurements show that the exchange field itself is anisotropic, being a tensor with the same principal axes as g. By way of example we give below the values of Herrand g (for Yb GaG) in the t h e principal local directions of the rare earth ion. T A B L E4
Hexcb =
gJ = 8/7
gJ
2(gJ - 1) for Yb+++(2F710)
References p . 382
87.200 2.85
153.000 3.60
169.000 3.78
349.000
611.000
678.000
376
L. N h & R. PAUTHENET AND B. DREYFUS
[CH.W,5 3
Starting from a detailed knowledge of the energy levels one can calculate macroscopic properties such as the spontaneous magnetisation of a monocrystal as a function of angle, its anisotropy etc. Such calculations have been performed 22 and show good agreement with experiment. Measurements have also been made on othe1 garnets. The same Yb3+ ion has been studied diluted in Y GaGaO, which has the advantage of avoiding the complications caused by the molecular field; the splittings of the 4 and 3 multiplets were found and these agreed with susceptibility measurements31 which give about 500 cm-' for the separation of the first excited level of the ground multiplet. The spectrum contains additional lines whose intensity is temperature dependent, thus bringing the lattice vibrations into evidence. In the case of Er I G85 the splittings and transitions of the ground multiplet J = $6 and the excited multiplets and Q, have been measured; the splitting of the ground level, as deduced from the hypothesis of an electrostatic cubic field, is in satisfactory agreement with the susceptibility of Er Ga G86. However, this simplified model does not seem able to completely explain the complexity of the spectrum.
3.6. THEFAR INFRA-RED SPECTRUM From loop the garnets become transparent again and it is possible to study transitions within the ground multiplet. This has been done by Tinkham and Sievers73~74between 10 cm-' and 100 cm-l. The 10 cm-I limit is not very important, since, if there were levels with lower energies, they would influence the specific heat. The spectra show several types of lines, amongst which one, at about 80 cm-' and common to all the measured garnets, varies slowly as a function of molecular mass and has been attributed to lattice vibrations. One next observes absorptions corresponding to the energies of the individual ions, This result, at first sight natural isn't completely so, since the electromagnetic waves have a large wavelength (k= 0) and interact with the crystal as a whole. In reality, then, it seems that these absorptions are concerned with collective oscillations (spin-waves), particularily their optical branches. However, if all the sites (c) are identical, the optical modes for k = 0 are magnetically compensated70 and no coupling with an electromagnetic wave is possible. Tinkham 7 1 has shown that if the gyromagnetic ratios of the (c) ions are not quite the same, then the above selection rule is modified and absorptions corresponding to the optical branch become observable. In the case where the Fe3+ ions are situated on the same sub-lattice 1 Referencesp. 382
CH. W,8 31
377
THE RARE EARTH GARNETS
and the identical rare earth ions on sub-lattice 2, there are two possible modes of resonance
- M z H , 0,= - I[y,M, - ylM,] MllYl - MZIY, These modes are for k =O in an external field, H, and neglecting anisotropy. 1is the molecular field coefficient between 1 and 2. The first mode is that of normal ferromagnetic resonance and is situated in the micro-wave region, except when we have to take into account a large anisotropic energy. The second is called exchange resonance87 and is normally found in the infrared; its intensity is proportional to (yl - y 2 ) 2 . One thus has two modes, neither of which correspond to individual transitions. The spectrum as a function of k, is shown in the upper half of Fig. 14. In the case of two slightly different sites (2), Tinkham has shown that the optical branch spectrum may be represented by considering half the preceding Brillouin-zone, that is, by folding back the dispersion curve (lower half of Fig. 14). For k = 0 one now sees another mode with w2 = YAM, appear; this is in addition to the exchange-mode, and corresponds to the individual precession of ions (2) in the exchange field created by the ions (1). This mode has an intensity proportional to (dy,)' where 6y2 is the difference between the 0 0
=
o~
0.2
0.4
0.6
0.8
1
I
I
I
1.0
ka/r
Fig.14. Spin wave spectra of two and three sublattice models of rare earth iron garnets. Since the branch near the unperturbed iron spin-wave branch Sa, is essentially the same in both cases, it is drawn only once. The two sublattice-case, shown above, gives one optical branch, and the three sublattice case gives two, as shown in the lower part of the figure. Note the tenfold difference in scale between the positive and negative frequency branches. References p . 382
378
L. N I h , R. PALJTHENETA N D B. DlzBypuS
[CH.W,0 3
gyromagnetic ratios of the two different (2) sites. We note in passing that for k =O the dispersion of the branch o2is weak, which fact has been used to justify the Harris and Meyer model66 for the specific heat (see above). In as much as AM, varies little between 0°K and 70"K,the individual transitions are relatively insensitive to temperature variations; this has allowed their identification in the spectra. In the case of Gd I G no transition has been observed, apart from that due to the lattice at 80 cm-'; this is explained by the fact that the Gd3+ ions are isotropic and, after what we have just said, the intensities of the individual transitions are thus zero. From magnetic and specific heat measurements, the exchange resonance ought to be at 28 cm-'. Now it accidentally happens that the gyromagnetic ratios of the Fe3+ and Gd3+ ions are both equal to 2. The intensity of the corresponding transition is then zero; indeed, such a transition has not been observed. Yb I G, already extensively studied in the visible and near infra-red, has been the object of an intensive investigation. One again finds the individual transitions6OV83 at 23.1 cm-' and 26.7 cm-' for the two types of sites, and the magnetisation is parallel to [1111. The anisotropic gyromagnetic ratios and the exchange field have been found by application of an external field, variable in strength and direction. Although less precise than those of Wickersheim, these measurements agree with his and confirm the existence of an anisotropic exchange field. Other, more complex, spectra have been observed (Sm I G, Ho I G, Er I G); in the case of Er I G and Sm I G it seems that the temperature variations can be explained only by considering the usually neglected interactions between rare earth ions. Finally, the rest of the garnets shows transitions which are temperature dependent (for Yb I G, they vary from 14 cm-' at 0°K to 2Ocm-' at 70°K);these transitions are attributed to the exchange resonance, o,= = - [y2M, - ylM2] which is temperature dependent, through the rare earth magnetisation, M2. In Sm I G a second temperature dependent transition has been identified as a ferromagnetic resonance; this latter is in the far infra-red due to a strong magnetocrystallineanisotropy, which acts as a strong temperature dependent effective field.
3.7. h Z A S T I C SCArrwINa OF NEUTRONS The determination of the spin-wave spectrum, w(k), is possible by the inelastic scattering of neutrons. The analysis is made from the diffracted Rejerences p . 382
CH. Vn, 8 31
THE RARE EARTH GARNETS
379
neutrons, taking account of the conservation of energy and momentum. The dispersion of the optical branches is weak and the energy is practically that of the rare earth ion in the molecular field'l. The energies of the Yb3+ ion have been founds8 at 80°K and 300°K in the case of Yb I G. The energies transferred are respectively 3.0 meV and 2.4 meV, in agreement with magnetic, optical and thermal measurements. An energy transfer of 0.07 eV (610 cm-') has also been noted; this corresponds to the alreadynoted transition within the ground multiplet. 3.8. GIANTANISOTROPY Though it is not possible to treat the numerous investigations on resonance in various garnets, the position and width of lines, relaxation times etc., we shall make an exception in the case of Dillon's et uLB2~*9-91. Their particularly spectacular results shed light on the energies of the rare earths as a function of the orientation of the applied field. These authors studied the behaviour of a monocrystal of Y I G containing a rare earth, such as Tb, as an impurity in concentration 0.01% to 0.19%; the ferromagnetic resonance was investigated at liquid helium temperatures. The resonance
Fig. 15. Hres in (110) at 1.5OK in YIG (0.2%Tb). The frequency was 22 989 Mc/sec. The peaks as seen in this plane are designated by Roman numerals. The dashed curve in Hres for the purest YIG with which we have worked.
Rqfwencesp. 382
380
tm. w,8 3
L.NJkL, R. PAUTHENET AND B. DREYFUS
is detected, as a function of the orientation of the applied external field relative to the crystal axes. In spite of the small amount of impurities, the resonance field at certain angles varies very sharply (Fig. 15). These peaks have been studied as functions of numerous physical parameters: concenZOQ
IW 160 140
120
loo 80
so 40
20 $ W
0
-20 -40
-
60
-80 -100
-120 -140
-160
-I w -200
-
220
-140
0.
lo. 20’ ?,o.
40.
50’
60.
70.
10.
SO.
100-
Fig. 16. Energy levels of one site of Tb+++in YIG. The spontaneous iron magnetization is in (1 10) plane.
tration of Tb, temperature, resonance frequency, etc. The theory which has been givensa,9% 93 shows the fundamental importance of “near-crossing” of levels under the combined influence of crystalline and exchange fields, for certain orientations of the latter. At these near-crossing points, where one level becomes very close to that immediately above it, there is a large curvature of the level. We can describe this by saying that there is both a large magnetic susceptibility and a large anisotropy of the particular level. These two descriptions are in some sense equivalent, and one can study the resonance equations starting from one or the other. If one uses the anisotropy description, the ferromagnetic resonance 9 2 ~ formula 9 ~ becomes
References p . 382
CH. w, 5 31
,381
THE RARE EARTH GARNETS
w is the frequency in the resonance field H,,,, and Haand H, are the effective anisotropy fields for the two principal directions u, p. H, is related to the curvature of the level by 1 a2E Ha=---. M 89,”
The appearance of a sharp variation in Ha,@,thus explains the observed peaks in H,,,. Dillon and Walker62 have tried to explain their results quantitatively by introducing an isotropic exchange field, and a 3-parameter crystalline field for the spherical harmonics of order 2, 4 and 6 ; each harmonic was chosen to satisfy the electrostatic point-charge model compatible with the crystal structure (Fig. 16). General agreement is good, even quantitatively; unhappily, the results are insensitive to large variations of the four parameters introduced, and it does not seem possible to find the crystalline field easily and accurately in this way. OF THE “RECONSTRUCTION” 3.9. AN EXAMPLE
OF A
GARNET
All the parameters obtained on the energy levels, as a function of the orientation of the spontaneous magnetisation, enable us to envisage the calculation of all the microscopic properties starting from these primary parameters. This has been done by Ndel and Pauthenet for the simple garnets YIG and Gd I G, and has been attempted 22 for the simplest of the remaining garnets, Yb I G . Starting from Wickersheim’s results 83, Henderson and White22 calculated the partition function for Yb I G. They supposed that the magnetisation of the iron ions was temperature independent, which is fully justified below 80°K.For M, in the plane 110, there are only four inequivalent rare earth sites, and optical measurements give the splittings of these four sites. The magnetic partition function was taken as the sum of the partition functions of the individual ions (which we have seen is justified for the optical branches of the spin-wave spectrum). The calculations were performed on an electronic computer, which then enabled an a priori calculation of the free energy as a function of temperature and orientation. One thus sees that the direction of easy magnetisation is [1111. One also deduces the values and thermal variation of the anisotropy constants, & and K2.These values agree with the static measurements of the couple necessary to maintain the magnetisation in a given direction. The saturation magnetisation of a monocrystal varies with the angle, and the magnetic compensation point, found as 7.7”K, agrees with Pauthenet’s References p . 382
382
L. N ~ L R. , PAUTHENET AND B. DREYFUS
[m. w
measurements. The specific heat and the adiabatic cooling of a monocrystal by rotation in an external field were also calculated. REFERENCES H. Forestier and G. Guiot-Guillain, C. R. Paris 230,1844 (1950). G. Guiot-Guillain, C. R. Paris 231, 1832 (1951). 3 H. Forestier and G. Guiot-GuilIain, C. R. Paris 235, 48 (1952); G. Guiot-Guillain, C.R.Paris 237, 1654 (1953). 4 R. Pauthenet and P. Blum, C. R.Paris 239, 33 (1954). 5 L. Nkl, C.R. Paris 239,8 (1954). 6 G. Guiot-Guillain, R. Pauthenet and H. Forestier, C.R.Paris 239, 155 (1954). 7 S. Geller, Phys. Rev. 99, 1641 (1955). F. Bertaut and F. Forrat, C.R. Paris 242,382 (1956). 9 R. Pauthenet, C.R. Paris 242,1859 (1956). lo R. Pauthenet, C.R. Paris 243,1499 (1956). R.Aleonard, J. C. Barbier and R. Pauthenet, C.R. Paris 242,2531 (1956). 12 L. NM, F. Bertaut, R. Pauthenet and F. Forrat, Comptes Rendus de l'Acad6mie des Sciences d'U.R.S.S. 21,6 (1957). 13 F. Bertaut and F. Forrat, C.R. Paris 243,1219 (1956). F. Bertaut and F. Forrat, C.R.Paris 244,96 (1956). 15 F. Bertaut, F. Forrat, A. Herpin and P. Meriel, C.R. Paris 243,898 (1956). 1 6 S. Geller and M. A. Gilleo, Acta Cryst. 10,243 and 787 (1957). 1' W.P. W olf and G. P. Rodrigue, J. Appl. Phys. 29,105 (1958). l8 J. W.Nielsen and E. F. Dearborn,J. Phys. Chem. Solids 5,202 (1958). J. W. Nielsen, J. Appl. Phys. 31, 51 S (1960). 20 S. Geller and M. A. Gilleo, J. Phys. Chem. Solids 3,30 (1957). 21 M. A. Gilleo and S. Geller, Phys. Rev. 110,73 (1958). 28 J. W. Henderson and R. L. White, Phys. Rev. 121, 1627 (1961). 23 R. Aleonard, J. Phys. Chem. Solids 15,167 (1960). 24 A. Serres, Ann. Phys. (1932). 26 M. Fallot and P. Maroni, J. Phys. Rad. 12,256 (1951). a@ E. W. Gorter, Phillips Research Repts 9,403 (1954). 27 L. Nkl, Ann. Phys. 3, 137 (1948). *8 L. Nbl, Ann. Phys. 5,232 (1936). 28 H. A. Kramers, Physica 1 , 182 (1934). 80 P. W.Anderson, Phys. Rev. 79,350 (1950). 81 Y. Ayant and J. Thomas, C.R. Paris 248,387 (1959). 88 R. L. White and J. P. Andelin, Phys. Rev. 115, 1435 (1960). 8s W. P. W olf,Proc. Phys. SOC.74,665 (1959). 34 E, Prince, Phys. Rev. 102, 675 (1956). 35 L. Nkl, J. Phys. Rad. 12, 259 (1951). 86 C. Robert, C.R.Paris 251,2685 (1960). 87 I. Solomon, C.R. Paris 251,2675 (1960). 88 W.P. Wolf and J. H.Van Vleck, Phys. Rev. 118, 1492 (1960). 39 R. Pauthenet, J. Phys. Rad. 24 388 (1959). 40 Y. Ayant and J. Thomas, C.R. Paris 250,2688 (1960). 41 R. Pappalardo and D, L. Wood, J. Chem. Phys. 33,1734 (1960). 48 D. Boakes, G. Garton, D. Ryan and W.P. W olf,Proc.Phys. SOC.74, 663 (1959). 43 K.F. Pearson and R. W.Teale, Proc. Phys. SOC. 76,388 (1960). 44 J. H. Van Vleck, Electric and Magnetic Susceptibilities (Oxford, 1932). 2
a. w] 46
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THE RARE EARTH OARNETS
383
G. H. Dieke and L. A. Hall, J. Chem. Phys. 27,465 (1957). J. H. Van Vleck, J. Phys. Soc. Japan 17, 352 S (1961). V. Jaccarino, B. T. Matthias, M. Peter, H. Suhl and J. H. Wernick, Phys. Rev. Letters
5, 251 (1960). M. A. Gilleo and S. Geller, J. Appl. Phys. 29, 380 (1958). 4e R. Pauthenet, J. Appl. Phys. 29, 253 (1958). G. Villers, R. Pauthenet and J. Loriers, J. Phys. Rad. 20, 382 (1959). G. Goldring, M.Schieber and Z.Vajer, J. Appl. Phys. 31,205 (1960). A. Aharoni and M. Schieber, J. Phys. Chem. Solids 19,304 (1961). S. Geller, H. J. Williams and R. C. Sherwood, Phys. Rev. 123, 1692 (1961). 64 W.P. Wolf, J. Appl. Phys. 32,742 (1961). 65 J. F.Dillon, J. Appl. Phys. 29, 5839 (1958). " J. F.Dillon, J. Appl. Phys. 29, 1286 (1958). J. F.Dillon, J. Phys. Rad. 20,374 (1959). 68 Y.Ayant and J. Rosset, Ann. Inst. Fourier, Grenoble 10,459 (1960). " R. Pappalardo, J. Chem. Phys. 34,1380 (1961). 'O R.Pappalardo, J. Chem. Phys. 31, 1050 (1959). 61 Y.Ayant and J. Thomas, C.R. Paris 248,1955 (1959). 6a J. F.Dillon Jr. and L. R. Walker, Phys. Rev. 124, 1401 (1961). 6s D.T. Edmonds and R. G. Pekisen, Phys. Rev. Letters 2, 499 (1959). 64 J. E. Kunzler, L. R. Walker and J. K.Galt, Phys. Rev. 119, 1609 (1960). 66 S. S. Shinozaki, Phys. Rev. 122,388 (1961). 66 A. B. Harris and H. Meyer, Phys. Rev. 127,101 (1962). 87 R.L.Douglass, Phys. Rev. 120, 1612 (1960). e.8 R. C. Le Craw and R. L. Walker, J. Appl. Phys. 32, 167 S (1959). 139 E. H. Turner, Phys. Rev. Letters 5, 100 (1960). 70 B. Dreyfus, J. Phys. Chem. Solids 23,287 (1962). 7 1 M. Tinkham, Phys. Rev. 124,311 (1961). 48
''
A. B. Harrris, private communication. A. J. Sieven and M. Tinkham, Phys. Rev. 124, 321 (1961). 74 A. J. Sievers and M. Tinkham, Phys. Rev. 129, 1995 (1963). 78 B. Liithi, J. Phys. Chem. Solids 23,35 (1962). 76 R. L. Douglass, Phys. Rev. 129, 1132 (1963). 77 H. Sato, Progr. Theoret. Phys. (Kyoto) 13, 119 (1955). 78 V. G. Bar'yakhtar and G. I. Urushadze, Soviet Phys. JETP 7,875 (1958). 79 K. A. Wickenheim, Lefever and Hanking, J. Chem. Phys. 32,271 (1960). *O J. F. Dillon, J. Appl. Phys. 29, 539 (1958). 81 A. M.Clogston, J. Appl. Phys. 31, 198 S (1960). 88 K.A. Wickenheirn and R. L. White, Pbys. Rev. Letters 4, 123 (1960). 88 K.A. Wickersheim, Phys. Rev. 122,1376 (1961). 84 J. W. Carson and R. L. White, J. Appl. Phys. 31, 535 (1960). 85 B. Dreyfus, J. Verdone-Thuilier and M. Veyssie-Counillon, C.R. Paris 252,1928 (1961) and 256,646 (1963). 86 J. Thomas,private communication. 87 J. Kaplan and C. Kittel, J. Chem. Phys. 21, 760 (1953). 88 H. Watanabe and B. N. Brockhouse, Phys. Rev. 128,67 (1962). 88 J. F. Dillon Jr., Phys. Rev. 111, 1476 (1958). 90 J. F. Dillon Jr. and J. W.Nielsen, Phys. Rev. 120, 105 (1960). J. F. Dillon Jr., Phys. Rev. 127, 1495 (1962). 1x3 C.Kittel, Phys. Rev. Letters 3, 169 (1959). 88 C. Kittel, Phys. Rev. 117, 681 (1960). '2
78
CHAPTER VIII
DYNAMIC POLARIZATION OF NUCLEAR TARGETS BY
A. ABRAGAM
AND
M. BORGHINI
CENTRE D’ETUDES NUCL~AIRES DE SACLAY, FRANCE
CONTENTS: Introduction, 384. - 1. Dynamic polarization at low temperatures, 385. 2. Spin temperature theories of dynamic polarization, 400.- 3. Experimental arrangements and results, 415. - 4. Future developments, 440.
Introduction In a recent review of the problem of polarized targets1 the following statements could be found about dynamic polarization methods: “These methods represent the most promising methods of polarizing protons for investigating the important problem of nuclear forces by p-p and n-p scattering. However in view of the stage of development of those methods it can hardly be said that they have been established as practical methods for nuclear reactions”. During the last three years the situation has changed dramatically. Important experiments on p-p scattering2 and nf -p scattering3 have been performed recently with dynamically polarized targets and many more are planned for the near future. Although the subject is still progressing at a fast rate, theoretical and experimental results gathered so far, warrant a survey of its present state. While the basic principles of dynamic polarization can be stated in simple terms, a real understanding of the processes that take place between the various systems participating to the phenomenon, namely nuclear spins, electronic spins, lattice vibrations, helium bath, microwave and radiofrequency fields, requires a knowledge of some aspects of electron and nuclear resonance, such as spin lattice relaxation, spin-spin dynamics, spin temperature in the rotating frame, etc., unfamiliar not only to nuclear physicists but also to many solid state physicists. References p . 446
3 84
cH;MII,8 11
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
385
We thought it worthwhile to give a rather full discussion of these points, insofar as they are relevant to the problem of dynamic polarization, even though parts of this discussion which have not been published before, may have a somewhat tentative character. This survey is devoted mainly to the only method that has so far yielded practical results in the production of polarized proton targets, namely the so-called “solid effect”. Some other methods are considered briefly in the first and last chapter. After an introductory section, Section 2 contains a detailed discussion of the theory of the “solid effect” while Section 3 gives a description of experimental methods and results, some of them again published for the first time. In conclusion some future directions of development are considered.
1. Dynamic Polarization at Low Temperatures 1.1. ELECTRONIC AND NUCLEAR PARAMAGNETISM : GENERALITIES 1.1.1. Polarization We define the polarization of a system of spins Z along an axis Oz as
where ( ) is a quantum mechanical ensemble average. The privileged direction Oz is chosen according to the particular conditions of the problem : external magnetic field, symmetry axis of an electrostatic crystal field, direction of propagation of an atomic beam, etc. In the simple case of non-interacting spins 3, with magnetic moments ,u in equilibrium with a thermal bath at a temperature T, the polarization in an applied magnetic field H is given by P = tanh(pH/kT)
where k is the Boltzmann constant (k z 1.36 x cgs). This formula is a special case of the Brillouin formula for arbitrary spin value I: p = -21 1coth - coth 2 PHFT. 21
+
rGg) &
1.1.2. Electronic Paramagnetism in Solids 4 In solids, atoms experience interatomic forces which insure the cohesion of the solid and are responsible for its particular structure. The effects of these References p . 446
386
A. ABRAOAM AND
M.BORaHINI
[CH.MIL, 8 1
forces can often be described as those of a local static electric field that acts on each atom (or ion) and will affect its energy levels. On the other hand, these forces allow elastic vibrations of atoms around their mean positions. In crystalline solids, these vibrations are made of quasi-harmonic modes, whose excitation is described by the number of corresponding quasi-particles, or phonons, which exist in the solid. The thermal excitation by a bath at a temperature Tleads to a mean number of phonons per mode of frequency v, given by Planck's formula:
n(hv) = (ekvIkT - I)-'. At low temperatures, the interactions between these different oscillators are very weak: at 1"K, the mean free path of a phonon may be of the order of one centimeter, and the phonons are mainly emitted or absorbed on the surface of the crystal, provided that there are not too many imperfections such as dislocations or impurities inside the crystal. The interaction between these vibration modes and the spins will produce energy exchanges between the spin system and the lattice, which is made of all the degrees of freedom of the solid other than those of spins.
A solid will exhibit electron paramagnetism if it contains electron spins that are not all paired off in closed shells or in valence bonds. A fist example of paramagnetism is that of conduction electrons in metals or semi-conductors: their orbital angular momentum is usually (although not always) quenched and their magnetic moment is given by: P=-gBS
where g S 2 and B = I e I tt/2mc w 0.927 x cgs is a Bohr magneton. Some molecules, or free radicals, have in their ground state, one unpaired electron; some of them are stable in a concentrated solid state, such as DPPH ((C,H,) ZN - NC6(N0,),H,): the paramagnetic electron is again in a state where g = 2, the orbital momentum being quenched. Irradiated solids often exhibit electronic paramagnetism, whose featuresdepend on the nature of the irradiation. Generally g is in the neighbourhood of 2. Another example, which will mainly be considered now, is that of atoms of transition elements in non-conducting solids: iron group, rare earths, palladium group, actinides. In these atoms, some internal electronic shell is incompletely filled, and exhibits electronic paramagnetism. The electrons of the external shells are engaged in chemical bonding and are non magnetic. References p . 446
cH.=,011
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
387
We shall take as an example the case of cerium ion Ce+++ which has one 4f electron outside of closed shells. In a free ion, the strong spin-orbit coupling U s lifts partially the (2s + 1) (21 + 1) = 2 x 7 = 14 fold degeneracy of a 4f electron into two multiplets j = 5 and j = 3. In a crystal the local crystalline field, rather smaller than the spin-orbit coupling, lifts partially, and according to its symmetry, the degeneracy of each of these multiplets. For example, in a double nitrate of cerium and magnesium CMN, the crystal field has trigonal symmetry at the site of the cerium, and splits the ground multipletj = 3 into three so-called Kramers doublets : such doublets occur in atoms with an odd number of electrons; their degeneracy cannot be lifted by an electrostatic field, however low its symmetry, but only by a magnetic field. The two states I a ) and I B into which this field splits the doublet are called Kramers conjugate. In CMN the spacings between these three doublets are of the order of 30°K or more. At helium temperature, only the lowest doublet is populated. The static paramagnetism can then be described as that of a fictitious spin, the “effective spin”, S = $, whose magnetic moment differs from that of the free electron, and turns out to be strongly anisotropic p= -gp.s (1)
>
where g is a second rank tensor. In CMN, 2 being the direction of the crystal field axis, gllz = 0.02365, g1z 1.83’. We shall come back jn more detail to the case of cerium, and we shall also consider the case of neodymium, which is important to obtain high proton polarizations. In the following general considerations, we shall use the symbol S to represent the real or the effective electronic spin as the case may be, with a magnetic moment given by the relation (1) where g is a tensor. Order of magnitude of the electronic polarization. In easily obtainable fields, at liquid helium temperatures, electronic polarizations are fairly high; for instance, with g = 2, H = 24000 Oe, T = 1”K, P, = 92%. Fig. 1 shows the polarization for S = 4 as a function of gH/2T. 1.1.3. Dynamic Interaction between Electronic Spins and the Lattice.
Spin-Lattice Relaxation
The modulation of the crystal field by the vibrations of the solid produces transitions in which the spin system and the lattice exchange energy; if we consider for instance spins S = 3 in a lattice at a temperature T,in a magnetic References p . 446
388
A.ABRAGAM AND hi. BORGHINI
too
-
60
-
60
-
40
-
20
I
% [kQ/*K)
I
io 20 Fig. 1. Polarization of an electronic spin 4 as a function of g-value, magnetic field and temperature. 0
+
field H, along Oz, with eigenstates S, = 1 ) and I - >, the exchanged energy is gaH. It is well known from statistical mechanics that the probabilities of the two inverse transitions W- -,+ and W ++ - are unequal and that their ratio is given by W - + + / W + + - = exp(- g j H / k T ) . (2)
This leads to the Boltzmann ratio between the populations of the states
1
+ )and1 - ) :
N + / N - = exp (- g j H / k T )
and to the polarization P, = tanh (- g P H / 2 k T ) .
If, at a given time, the spin system is not in equilibrium with the lattice, its polarization will tend toward its equilibrium value with a relaxation time constant T, given by l/Te = W+*- W-++.
+
=-
For spins S 3, the return to the equilibrium is not always describable by an exponential decay with a single relaxation time, but we shall mainly deal with spins and disregard this complication. Among the various spin-lattice relaxation mechanisms that exist in solids we shall single out two processes that are in particular responsible for electron-spin relaxation at helium temperature in some hydrated rare-earth salts with Kramers degeneracy, particularly suitablefor dynamic polarization. The first is the so-called direct process7 where the energy required for the spin flip from a Kramers state to its conjugate is conserved by emission or absorption of a phonon of the same energy hoe = gbH. The probabilities W++ - and W- + + for such a flip are proportional respectively to (n 1)
+
+
Referencesp . 446
=vm,§ll
389
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
and n, where It is the number of phonons of frequency o,present per vibration mode in the crystal and the relaxation rate l/Td = ( W+-.- W- -,+) is proportional to (2 n 1). If the phonons are in thermal equilibrium at a temperature T :2n 1 = coth (ho,/2 kT) z 2 kT/fiw,if kT >> ha,. This describes the temperature dependence of the relaxation rate. More precisely it can be shown that
+
+
+
This temperature dependence of l/Tdis sometimes masked by a phenomenon called the phonon bottle-necka: the beat capacity of the phonon system being finite its ability to relax the spins depends on the thermal contact between the phonons and the helium bath. The time constant z for a phonon to be cooled by the bath can be crudely estimated to be of the order of l/u where I is the linear dimension of the crystal and u the velocity of sound. There will be a phonon bottle-neck if the rate of flow of energy from the spins to the phonons namely Espins/Td is much larger than the rate of energy flow phonons-bath: Eph/Z. In that case the observed spin relaxation rate will be 1/Th %3 (l/z) (Eph/~sp,ns). Since Espins is proportional to l/Tand Eph to T the temperature dependence of the observed relaxation rate should be quadratic rather than linear as in the true direct process. The phonon bottle-neck has actually been observed in various salts9JO. It can be shown that the dimensionless parameter CT = (Espins/Te)/(Eph/r), which determines the occurrence of the bottle-neck, is proportional to the square of the Larmor frequency o,and it is clearly proportional to the concentration of electron spins. The larger the electron spin frequency and concentration the more severe the bottle-neck. The second relaxation process of importance in the rare earth salts with Kramers degeneracy is the so-called Orbach process11: it involves an intermediate real transition from a state la) of the ground Kramers doublet to a state Ic) of an excited doublet with absorption of a phonon of energy k6, equal to that of the excited doublet, followed by a fast decay from Ic) to 15) with emission of a second phonon; this process is only effective if phonons of the energy k0, exist in the crystal, that is if 0, < 6,, Debye temperature of the crystal. The number of phonons of energy k6, present in the crystal is proportional to exp (- 6,/T), which is thus the temperature variation of the relaxation rate for this process. Since the phonons responsible for the Orbach process are far more energetic and correspond to a far greater density of modes than those of the direct process, the parameter B defined previously will be very small and no phonon bottle-neck should occur in general. References p . 446
390
A.ABRAOAMAND M. WlRoSw
[CH.Vm,8 1
1.1.4. Interaction between Electronic Spins and Radiofrequency Fields. Resonance. “Saturation” The interaction El.gB.S between a spin S and a magnetic field HI oscillating at a frequency a,may produce transitions between two spin levels li) and l j ) , if o is near the characteristic frequency a,given by hw, = E*
-Ej
where E1 and Ej are the energies of the levels, and if the matrix element (ilH, -g/3*S/j)is non zero. For instance, if a spin 3 experiences a constant magnetic field H along 0 2 , and if HIis directed along Ox 10 2 , ha, = gPH and the transition probability by unit of time induced by the field Hi, is given by
wo = +ng2/?2h-2H: x f(0- 0,) where 2iY, is the amplitude of the oscillating field, andfis a form function, normalized to: +W J f(o-W,)d(w-o,)=l -m
which depends on the Width of the levels. A distinctive property of the transitions induced by an r.f. field, as contrasted with those induced by the lattice, is that the probabilities for the two inverse transitions are equal, for example
WO(++-)/K3(-++) =1 because of the high number of photons which are present in the r.f. field. Thus, a magnetic field at a resonant frequency tends to equaIize the populations of the spin levels between which it induces transitions. It will equalize them if its frequency is exactly equal to the resonance frequency of the spins, and if the transition probability is much greater than the corresponding probability induced by the coupling with the lattice: w,T,>>1. The resonance is then said to be saturated. 1.1.5. Nuclear Paramagnetism. Magnetic Moment and Spin, Larmor Frequency We consider nuclei for which the spin I is not zero. The magnetic moment p of a nucleus is aligned With its spin I, and its gyromagneticratio yn is defined by : Referencesp . 446
CH.Vm,0 11
391
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
p = y,hZ. The nuclear moments are smaller than the electronic ones by a few orders of magnitude: for instance, the ratio of the proton to the free electron moment is Ga. The natural thermal polarization of nuclei is thus much lower than the electronic one in a given magnetic field, at a given temperature; for instance, for protons at 1"K, in 24000 Oe, P, = 0.24%. The interaction with an applied magnetic field H is described by the Zeeman hamiltonian &'z = - ynhI.H. The motion of a spin in such a field is a uniform angular precession around the field at a rate on= 2nvn = - y,H, called its Larmor frequency. We define an anticlockwise precession as positive. With this sign convention, a,,is positive for a negative magnetic moment. As for electronic spins, an r.f. field HI, normal to a main constant field H, of frequency o equal to the nuclear spin Larmor frequency, can induce transitions between spin states, with a probability W , proportional to H : ; if the field has a very small amplitude, it does not change the populations of the levels. The observed resonance signal then provides a relative measurement of the nuclear polarization. I. 1.6. Nuclear Spin-Lattice Relaxation The most effective nuclear relaxation mechanism in solidsat low temperatures is the interaction of the nuclear spins with fast relaxing electronic spins. The magnetic interaction between a nuclear moment Z and an electron of spin s and orbital moment I is given by an operator expression:
1
1 s r(s*r) + -3 - .- + 3r r3 r2
and, by taking the expectation value of d over the electronic wave-function, one obtains a hyperfine interaction of the general form
where d is a second rank tensor. The effects of an interaction such as (3) depend on the substance of interest ; we shall consider successively the cases References p . 446
392
A. ABRAGAM AND M.BORGHfNl
[am, .81
of nuclei in metals, of nuclei belonging to paramagnetic atoms, and of nuclei of diamagnetic atoms in solids containing a few paramagnetic centers; to each case, there corresponds a different type of relaxation and a different method of polarization: a method proposed by Overhauserlz, a method proposed by Jeffriesls and a method proposed by Abragam and Proctor14, the “solid effect”. The hyperfine interaction X h f s has, first, static effects, such as changes in the nuclear eigenstates and energy levels; it has also dynamic effects which provide a powerful relaxation mechanism for the nuclei. This relaxation can occur in two ways : the modulation of the tensor d by the internal motions of the lattice can produce energy exchange between the nuclear spin system and the lattice : this process is sometimes referred to as relaxation of the first type15; the static hyperfine coupling %hfs can also mix the spin eigenstates of the electron-rtucleussystem in such a way that the relaxation hamiltonian of the electrons will induce otherwise forbidden transitions in which energy is exchanged with the lattice and where the nuclear state is also changed; this corresponds to a second type of relaxation. The relaxation of nuclei in metals (and in liquids) is of the first type; it is of the second type in diamagnetic solids at low temperatures; the two types may be simultaneously effective in some intermediate cases. We single out for discussion the relaxation by paramagnetic impurities in solids at low temperatures, of paramount importance for the problem of dynamic polarization. In the case of two atoms at a distance r from each other, the hyperfine coupling between the electron spin of the first atom and the nuclear spin of the other is mainly a relatively long range dipolar interaction
where S is the effective spin of the atom, g its gyromagnetic tensor. If Oz is the direction of the applied magnetic field H,the terms in I,S, in this interaction can change the nuclear resonance frequency, but this effect is negligible in high fields; the dipolar interaction is then a perturbation and its other terms which do not commute with the Zeeman interactions, cause a slight change in the eigenstates by mixing the unperturbed eigenstates 1 ), I - ), I - ) and I - - ) (we take Z = S = 4); the most effective of these terms are those which mix states with neighbouringenergies, i.e. I+S, and Z-Sz: the new eigenstates are shown in Fig. 4, in the case of a negative nuclear moment. This is a justscation for the neglect of the scalar
++
+
References p . 446
+
CH.VIII, 8
11
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
393
electron-nucleus coupling which can only mix the states I + - ) and I - + ) separated by the electron frequency o,>> I wnI . Because of this mixing, the electronic relaxation interaction, represented by a hamiltonian of the form orH,(t).S,where H,(t) is a “random” local field responsible for the electron relaxation, can induce transitions between the states la) and Id) and the states Ib) and Ic), with a probability smaller by a factor 82 than that of the electronic relaxation transitions [a)- Ic) and Ib) t-, Id), with E E (gP/rS)/H if, for instance, g is isotropic. More precisely, the dipolar coupling between an electronic spin S of relaxation time T, and a nuclear spin Zsituated at a distance r from it, S Zmaking an angle 8 with the applied field H, produces a relaxation probability for the nuclear spin given by c g2P2 1 w(r,e)= r6 - = ~ - - - s ~ ~ ~ B c o s ~ o s I()-s.+ r6 H 2T, This probability falls off very rapidly with the distance r and if the nuclear spins did not interact with each other, paramagnetic impurities would be a poor relaxing agent. 1.1.7. Spin Dixusion The interaction between two nuclear moments at a distance r from each other, is given by the dipole-dipole hamiltonian
The static effect of this interaction for a system of many nuclear spins is to broaden the energy levels, but this effect is of little importance in the problem of relaxation and dynamic polarization. In this respect, a much more important dynamic consequence of the dipolar interaction is the process of spin difisionl6 which is necessary to relax or polarize high densities of nuclei in insulating solids : the terms Z, ZL and Z- Z; of this interaction can induce transitions between the eigenstates of the two spins, which can be looked at as simultaneous flips, in opposite direction, of the two spins, and which conserve energy if the two spins have the same Larmor frequency ;the lattice is not involved in the process, which is thus temperature independent, and which is much more rapid than the spin-lattice relaxation: typically, the probability W of a transition for two neighbouring protons at a distance a w 3 A is of the order of 104 sec-’. Successive simultaneous flips of neighReferences p . 446
394
A. ABRAGAM AND
M.BORQHTM
[CH.VIlI, 4 1
bouring spins provide a mechanism of diffusion for the spin perturbations which tends to maintain the internal thermodynamical equilibrium of the spin system. It is easily shown that the nuclear polarization p(r) considered as a continuous function of the position r of the nucleus obeys a diffusion equation of the form :
where the diffusion coefficient D % Was is of the order of in typical cases. Equation (7) is valid ifp does not vary appreciably over the distance between two neighbouring spins and if the anisotropy of the dipolar coupling between spins is neglected. In the presence of paramagnetic impurities at positions re equation (7) has to be replaced by
-- - D V 2 p ( r )- C ZeJr, - rJ- 6 [ p ( r ) - pol at
where p o is the thermal equilibrium value of the nuclear polarization and C = gab2 S(S + l ) / H T , corresponds to an angular average of (5). It can be shown17118 that the mean value jj of the polarization, taken over the entire sample, obeys a much simpler equation:
6
N, is the concentration of paramagnetic impurities and b a characteristic length which depends on C and D in a way concerned with the details of the spin diffusion in the immediate neighbourhood of the paramagnetic impurities. If D remains constant down to the nuclei nearest neighbours of paramagnetic impurities it can be shownl7JB that b % CfD-' and T,, T$. If because of local electronic fields the nuclei in the neighbourhood of the paramagnetic impurities have different Larmor frequencies thus quenching the spin diffusion, the theory becomes more involved and a different dependence T,, = f(T,)is expectedle*20.In some cases an empirical relationship T, Ttwith p = has been observedle921. Care should be taken to compare T,, with the true electron relaxation time T,,which is not observed in the presence of phonon bottle-neck.
-
-
References p ; 446
+
CH.Vm,8 11
1.2.
DYNAMIC POLARIZATION OF NUCLEAR TARGFXS
395
DYNAMIC POLARIZATION : GENERALITIES
1.2.1. Metals. Overhauser Efect The main part of the hyperfine interaction in metals between a nuclear spin Z and the spin S of the conduction electrons is the scalar interaction *~s,meta~
= Ak1.S.
This interaction contains terms Z+S- and 1-S+ which produce simultaneous Aips of the electron and nuclear spins, the energy A(we - w,) involved in such a fip being provided or taken up by the “lattice”, which in the present case is the translational energy of the electrons. This process is a mechanism of nuclear relaxation of the first type. Assume for simplicity Z = 3 and let us call W(+ -) -,( - +) the transition probability for a double flip, S, going from + to - and I, from - to + ; the energy exchanged with the lattice is A(we - 0,) where we and w, are the Larmor frequencies of the two spins, and according to formula (2)
If N* and n* are the populations of the spins S and the spins Z, one has, under steady state conditions:
N + n - W(+-)+(-+) = N-n+ %-+)+(+-).
(11)
Independently of the electron-nuclear coupling, electrons have much more powerful relaxation mechanisms of their own that insure the condition
N+IN- = exp-
- hue kT
and according to (10) and (11) n + / n - is just equal to the natural Boltzmann factor exp (- tuUn/kT).If on the other hand an intense r.f. field saturates the pure electronic transition AS, = & 1, it imposes the condition:
N+/N- = 1 and, by (10) and (11):
n + / n - = exp
fi(%
- W”) kT
Awe
m exp-
kT
which is the Overhauser result 12. In the foregoingwe have tacitly assumed that the electrons obeyed Boltzmann rather than Fermi statistics. The argument although slightly more complicated is easily extended to the latter case. References p . 446
396
A. ABRAGAM AND
[CH.Vm, 8 1
M. BORGHINI
The process is often visualized by considering a typical pair I, S: the the eigenstates are shown on the Fig. 2, with their energy separations, the nucleus being taken with a negative moment, The hyperhe coupling, modulated by the motion of the electrons, induces the relaxation transition be-
Fig. 2. Energy levels of an electronic spin and a nuclear spin Z = 1. in a metal. A symbol such as I > represents a state where SE= + , I z = f.
+> -
I-
++ +
& I
+
>
tween the states I + - and I - + >; if one saturates the electronic transition at frequency we,the populations of the levels connected by the r.f. field become equal, while the relaxation transition imposes the Boltzmann ratio exp { A (ae- o,)/kT}between the states [ - ) and I - ) :the populations of the four states [ ), 1 ), I - } and I - - ) are thus in the ratio 1 : 1 : exp { - A (0,- o,)/kT}: exp {- A (ae- w,)/kT} and the nuclear polarization is given by
++
+
+
+
+
The Overhauser effect, originally proposed for nuclei in metals, where the nuclear relaxation is of the first type, has been extended to other situations such as liquids containing paramagnetic ions or free radicals in solutionls. It is the relative Brownian motion of these ions or free radicals with respect to the nuclear spins, that is responsible for the “first type” character of the nuclear relaxation. The Overhauser effect has also been observed in solids containing paramagnetic free radicals strongly coupled by exchange 22. There, it is the “motion” of the orientation of the electron spin, caused by exchange, rather than the motion of the atom bearing that spin, that is responsible for a nuclear relaxation of the first type. The Overhauser effect was first demonstrated in metallic lithium, in a field H = 30.3 Oe, at a temperature T = 70°C23. References p. 446
CH.VIII,
g 11
397
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
1.2.2. Nucleus belonging to a Paramagnetic Atom or Ion The spin S appearing in the formula (3) for the hyperfine coupling is the fictitious effective spin as defined earlier. The hyperfine interaction may have the full generality given by a tensor of second rank, but may also take simpler forms such as the scalar form
AhZ-S or, as is often the case, when the crystal field has an axial symmetry of axis Oz, the form
AhZ,S,
+ + B h ( I + S - + Z-S+).
I a>-
1b>= k
I t ->
lo= 9p-
Id>->
q l - + >
Fig. 3. Energy levels of the electronic spin S = 3 and the nuclear spin Z = 3 of a paramagnetic atom or ion, coupled by an isotropic interaction h A I - S in a strong magnetic field H. The admixture coefficients are q = hA/Z&H<< 1 ; p = ( 1 -qZ)’w 1.
In any case, at least in the high magnetic fields required to produce significant orientations, this interaction is smaller than the Zeeman energy of the electron spin; nevertheless it is usually much larger than the Zeeman energy of the nucleus. The static effect of this hyperfine interaction is to change the energy levels of the system: let us take, for example, the case of an isotropic hyperfine interaction fi A Z-S:fi A is supposed to be much smaller than gPH, where H i s an applied magnetic field; the eigenstates and energy levels are given in the Fig. 3 ; the allowed transition frequencies are thus gPH + Ah and QPH - Ah, Besides this change in the energy levels, the hyperfine interaction produces also a change in the eigenstates of the system so that some of them are not eigenstates of either I, or S, but a mixture of both. A dynamic consequence of this mixing of states is that, besides the allowed electronic transitions [a) Ic) and [b) et Id), the relaxation hamiltonian of the electronic spin, which can be written in the form OL H&).Scan induce the “forbidden” tran-
+
-
Referencesp . 446
+
sition Ib) t,Ic) which changes the expectation value of the nuclear spin, causing a nuclear relaxation process of the second type. If one saturates now one of the allowed electronic transitions, for instance la) * Ic) with an intense r.f.field Hinormal to H, this relaxation acts between Ib) and Ic), whereas the pure electronic relaxation acts between Jb) and Id), and the populations of the levels la), Ib), Ic), Id) are seen to be in the ratio 1 : 1: exp (- gPH/kT): 1l6, and the nuclear polarization is given by P, =
1 3
- exp (- gBH/kT)
+ exp(-
gBH/kT)
with a maximum value of 3. If one saturates, with two r.f.fields for example, the two allowed electronic transitions, one can see in the same way that the nuclear polarization is given by P, = tanh (gpH/2kT) with a maximum value of + 1. An other type of dynamic polarization is possible13 in the same system: as a consequence of the mixing of states due to hyperfke coupling, an r.f. field HI,paraZZeZ to H, can saturate the “forbidden” transition Jb)c+ Ic), which corresponds to a simultaneous flip of the electronic and nuclear spins; the electronic relaxation acting between the states la) and Ic), and the states Jb) and Id), will insure that the populations of the levels la), Ib), Ic), Id) are in the ratio exp (- gpH/kT) : 1 : 1 :exp (gj?H/kT).This corresponds to a nuclear polarization P, = - tanh(gBH/2kT) with an extremal value of - 1. More complicate situations can arise when the nuclear spin is greater than +, or when the hyperfine interaction is not isotropic, but in each case the saturation of allowed and/or “forbidden” transitions can produce nuclear polarizations of the order of the electronic one, as well as nuclear alignmen t 24. The main difference between the two kinds of orientation lies in the dynamics of the effect: when allowed transitions are saturated, the electronic Boltzmann factor is transferred to the nuclei by a forbidden relaxation transition, and the time constant for this process, which is the nuclear relaxation time, is long. On the other hand, when forbidden transitions are saturated, the rate for the increase of the nuclear orientation is the transition probability induced by the r.f. field, which can be very high if sufficient power is available, and can thus be made very fast. Referencesp . 446
CH.VILI.8
11
399
DYNAMIC POLARIZATION OF NUCLEAR TAROETS
Dynamic polarization inside paramagnetic ions has been demonstrated first*&on cobalt-60 in La,Mg3(N03)iZ, 24 D,O at 1.5”Kand used to study pray anisotropy. 1.2.3. Polarization of a Nucleus near a Paramagnetic Atom. “Solid Efect”
As shown in Fig. 4 there are now two relaxation transitions which flip simultaneously both spins, the equation for the populations, instead of being given by (1 1) is n+“+q+++--)+N-w(-+++-,]= = n - [N+ W(+-+-+)N- q---.+ +J.
+
(12)
The relaxation transitions 1 + + ) S [ - - ) have now the same strength as I + - ) e I - + and can be written in the form
>
W(+++--) = w‘exp{ h(w,
+ on)/2kT}
q--+++) = w‘exp{-h(o, + wn)/2kT} q+-+-+) = w‘exp( h ( o , - on)/2kT} W,= w ’ exp { - h (0, - o n ) / 2 k T }. + + + -)
Without any r.f.field, N + / N - = exp (- FioJkT)and n+/n- = exp (- ho,/kT) as it should. If an r.f.field “saturates” the allowed electronic transitions, we get N + / N - = I, and one can see that again n+/n- = exp (- hwn/kT):there is no nuclear polarization by Overhausereffect”.This is to be contrasted with the preceding case of a nucleus inside a paramagnetic atom, where the hyperfine interaction was scalar, and could allow the relaxation transition I - sl: I - + ) only. Here, the dipolar coupling allows equally the relaxation transitions I + - ) a 1 - + ) and I + + ) I - - ),which have opposite effects on the nuclear polarization.
+ >
lo>.pJ*+)-d+->
Fig. 4. Energy levels of an electronic spin S = coupled to a nuclear spin 1 = +, at a distance r, by a dipole-dipole interaction,in a strong magnetic field H. The admixture coefficients are q = gb/(ra H ) << 1 ; p = (1 - q y w 1.
+
Ic>= pO+ql--> Id
Referencesp . 446
>
pl-->-q
I+ +>
400
A.ABRAGAM A N D M. BORGHINI
[CH.Vm,8 2
If an r.f.field “saturates” one of the forbidden transitions, for instance [ + ) at frequency 0,- on,the rate of flips due to the adverse transition I ) [ - - ) may be neglected, and as now since this is a transition induced mainly by an W,+ - +) = W,- + + + applied r.f.field, equation (12) gives
I+-)
-
++
n + / n - = N + / N - = exp(- ho,/kT). The nuclear polarization is highly increased and equals the electronic one : this is dynamical polarization by “solid effect”l4. ) [ - - ) transition which is “saturated” by the If it is the [ r.f.field, at frequency 0,+ on, one finds
++
n + / n - = N - / N + = exp(hw,/kT); the nuclear polarization is again increased in absolute value, but has a reversed sign with respect to its natural value. This is the principle of dynamic polarization by dipolar coupling between nuclear and electronic spins ; in practical cases, many complications arise which we shall consider in the following sections. We must mention now that the process that we have just shown to be effective for a pair of neighbouring spins I and S, can be used to polarize a great density of the same nuclei of diamagnetic atoms: this possibility is due to the spin diffusion between nuclear spins, as mentioned earlier. The solid effect has been demonstrated first at room temperature in LiF, where the spins of F19 played the role of electronic spins S, as they have a greater moment and a shorter relaxation time than the spins of LiS, which were taken as spins I. On the application of an intense r.f.field at a frequency o = w ( F 9 & o(Lie), an increase in LiS polarization by a factor & y(FlD)/ y(Li6) = k 6.5 was observed as expected14. The solid effect was first observed with electronic spins in the course of a search for an Overhauser effect at room temperature in charcoal containing free radicals and adsorbed benzene, but the cause of the positive and negative polarizations obtained on each side of the electronic resonance frequency was not recognized26. 2. Spin Temperature Theories of Dynamic Polarization
In the last section, we showed how the saturation of transitions involving simultaneous flips of electronic and nuclear spins by means of strong resonant r.f. fields, could lead to an enhancement of the nuclear polarization. In order to derive these results, we used rate equations for the populations References p . 446
CH. WII, 8 21
DYNAMIC POLARIZATION OF NUCLEAR TARGET3
401
of the various Zeeman levels of the electron and nuclear spins, following the classical treatment of Bloembergen, Purcell and Pound27 (BPP for brevity). The rate of a transition between two Zeeman levels was taken to be proportional to H:f(w), HI being the amplitude of the driving r.f. field, andf(o) a shape function describing the broadening of the Zeeman levels by dipolar spin-spin couplings or any other broadening agent. BBP had recognized, and Portis has further emphasized26 the necessity of distinguishing between so-called inhomogeneous broadening where f(o)represents a distribution of Larmor frequencies among non-interacting spins, and homogeneous broadening caused by interaction between like spins. In the latter case, it has been thought for a long time that the rate equations method of BPP was at least qualitatively correct, until Redfield demonstrated that it could lead to completely wrong results and indicated the correct approach to the problem of saturation of interacting spins in a solid29. Since the “solid effect” involves the saturation of certain spin transitions by strong r.f. fields in solids, it was normal to extend Redfield’s method to that problem, as first suggested by Solomon30. Redfield’s original theory is only valid for very strong r.f. fields. It was extended to the case of r.f. fields of arbitrary strength by Provotorov31 and generalized by one of us (M.B.) to cover the problem of dynamic polarization32. We shall refer to it in the following as the RSPB theory. The search for suitable nuclear target materials and high nuclear polarizations has so far precluded careful, systematic investigations of the phenomena of dynamic polarization under conditions where experimental results could be confronted in detail with the predictions of the RSPB theory. There is little doubt however that the RSPB approach to the problem of dynamic polarization is essentially sound, and despite its apparent complication, and the present lack of experimental confirmations, a description of this theory seems warranted. We devote to it the present chapter. The reader who is not particularly interested in spin dynamics can skip it. 2.1. LIMITOF VERY STRONG r.f. FIELDS. HOMOGENEOUS SPIN SYSTEMS
Since, despite its great success, Redfield’s original theory is still unfamiliar to non-specialists, we begin by recalling its main features. Further details can be found in29 and 33. 2.1.1. One Spin Species
Consider a system of spins of a single species S, in a field H along 0 2 ,with a Larmorfrequencyws = ysH, and spin-spininteractions SSs, submitted to
-
References p . 446
402
[CH.MI,8 2
A.ALiRAGAMAND M.BOROHINI
an r.f. field Ifls along Ox, normal toH, offrequencyo. The total Hamiltonian
~=ws~js~+2w,Z;S,Icostot+~~, where wl = - ysH,, is time dependent and cannot be used directly for a thermodynamical description of the spin system. If one performs the canonical transformation USU- to a rotating frame of reference, of frequency w, by means of a unitary operator U = exp( - i oS,t), where S, = c j S i , the system is described by the effective static Hamiltonian
where i@& represents the secular part of i@=,which commutes with U,and 5Y* is the effective Zeeman Hamiltonian in the rotating frame. In most cases los- 01 >> lull and &Y* m (os- m) S,. Redfield assumes that the statistical behaviour of the spin system in the rotating frame is describable by assigning to it a spin temperature Ts. With the assumption of a spin temperature in the rotating frame, the spin density matrix +t in that frame, has the form
u* = exp (- fi&/kTS)/Tr exp (- h&/kT,) or, in the high temperature approximation, the form
d w 1 - hS*/kTs= 1 - /&H* with
= h/kT,.
(13)
It follows from (13) that the expectation value of the effective Hamiltonian (S*)= Trace (0*&7*> is given by:
+
=
< X * )= <9*) (S&>B S [(as -
+ a:]
where wt = Trace (&7&)2/Trace (S:) and that: I
We have neglected w: compared to wi and (as- w)2, The coupling of the spins with the lattice affects differently the expectation values of (Z*) and
where I ,is the thermal equilibrium value of the Zeeman energy in the laboratory frame in the absence of the r.f. field and: References p . 446
~ . ~ 0 2 1
DYNAMIC POLAREATION OF NUCLEAR TARQETS
403
Tiis the usual spin-lattice relaxation time and &at), stands for the rate of change induced by the coupling with the lattice; a is a number different from unity. For a purely dipolar spin-spin coupling it is equal to 2 if there is no correIation between the relaxation of two neighbouring spins and to 3 for complete correlation. The latter is the more likely for long wavelength phonons responsible for the direct process, the former for the short wavelength phonons of the Orbach process. For argument’s sake we shall take a = 2. By writing that under steady state conditions
the value and making use of (14) we find for the inverse spin temperature /?,
where pL = h/kTLis the inverse lattice temperature. The “local field” HL defined by yi HZ = mi, is related - to the second moment of the unsaturated resonance line by HE = 4AH2. 2.1.2. Two Spin Species If the solid contains two spin species S and I, with respective Larmor and if a strong r.f. field is applied at a frequency w frequencies 0,and oI, near w,, the same canonical transformation U as before leads to an effective time independent Hamiltonian &?* given by:
where #; is the secular part of the S, I interaction. If S and I are both nuclear spin species and if w is in the neighbourhood of %, usually l q l 10~ , - wI and the assumption of a single spin temperature for the whole Hamiltonian X * is not valid, the Zeeman Hamiltonian y I, remaining at the lattice temperature. However in the dynamic polarization process by the solid effect, Iws - 01 and loll are either equal or at least of the same order References p . 446
404
A. ABRAOAM AND
[CH.WI,5 2
M.BORC3HINI
of magnitude and the assumption of a single spin temperature for the whole Hamiltonian 2*may be acceptable. Then the same kind of argument which led to (15) for a single spin species, gives the expression (17) for the common inverse spin temperature of the two species system
4%- 0 )
_ --
BS
BL
(us- 0)'
+ 202 +
NI Tf I ( I - -j Ns TIS ( S
+ 1) ' + l)@'
(17)
where Ns and NI are the number of spins in the sample, Tf and Ti the spinlattice relaxation times, and OIL'is given approximately by 20: = [2 Trace (&!s)z
+ Trace (Z;)']/Trace
S:
.
(18)
In (18) we have omitted terms of the order of (Ts/Tj)Trace (3?i)/Trace(2f'&)2 which are negligible except at unusually low electronic spin concentrations, and neglected Ts/T: with respect to unity; OIL is then related to thecontribution from the spins I to the second moment of the unsaturated resonance line, by 7 7 0;;" = 3 AHss + 4 AHls. Very often
ms
is also small compared to unity; formula (17) then reads
We shall make this approximation henceforth. 7>> A 2 If the concentration of electron spins is not too low, AHss H, the electron line is homogeneously broadened and % I 2 x wz. To appreciate the significance of the formula (17) which gives the spin temperature of the effective Hamiltonian .#* in formula [16), we must realize that the nuclear Zeeman Hamiltonian w, I, in that formula is the same as in the laboratory frame. In other words (17) expresses the cooling of the nuclear spins in the laboratory frame and gives directly the enhancement of the nuclear polarization. Referencesp . 446
CH. WI,8 21
2.2.
405
DYNAMIC P0LARV;ATION OF NUCLEAR TARGETS
ARBITRARY r.f. FIELDSTRENGTHS. HOMOGENEOUS SPINSYSTEMS
2.2.1. One Spin Species
Even for the case of one spin species, the foregoing theory based on the assumption of a unique spin temperature for the whole effective Hamiltonian is valid only asymptotically for very strong r.f. fields. While, because of the strong spin-spin couplings that exist in solids it is legitimate to assign to a* and to A?',& separately, definite spin temperatures, we must remember that these two Hamiltonians have different spin-lattice relaxation processes which lead to different values for these temperatures. On the other hand the r.f. Hamiltoniaa o,S, which commutes with neither S?* nor 2'; tends to equalize these temperatures and the net result is a compromisebetween the conflicting effects of r.f.field and spin-lattice relaxation, as shown by Provotorovsl. In his theory the same canonical transformation U is performed, but the density matrix in the rotating frame is taken of the form o*
-
exp(
- E Z T ~- y z & ) w 1 - ct S;- y%is
,
that is, with different temperatures for the effective Zeeman energy S*and the spin-spin interaction energy X$,. The exchange between these two reservoirs, which is produced by the r.f. field, is shown to be described by
- o)is a transition probability per unit time, proportional to where Wo(os Hf,which can be shown to coincide with the usual transition probability used in BPP rate equations. The general form of these equations can be predicted from simple minded arguments: the first equation expresses the fact that the r.f. field tends to equalize the inverse temperatures 01 and y of 3Y* and X& whilst the second is a consequence of the conservation of the total spin energy ( Z ' )= % .(OS - o)2 yw;
+
+
These equations have been derived by Provotorov from first principles using quantum mechanical calculations. By writing
References p . 446
406
A. ABRAGAMAND M. BORGHINI
[aa, . g2
the steady state temperatures are then given by
and by where A = w, - o. These formulae reduce, as they should, to :
where /Is is given by formula (15), in the case where WoTl >> 1. 2.2.2. Solid Effect Theory We consider now two spin species S and I, which represent electronic and nuclear spins, with Larmor frequencies w eand on, and with we >> on.The total Hamiltonian in the presence of an r.f. field of frequency o = a,,reads
We shall suppose that the electronic line is homogeneously broadened and at first that there is no spin diffusion; different nuclei may have different dipolar interactions with the electronic spins, and consequently, different Similarly they will have different flip probabilities per relaxation times T,!. unit of time W''induced by an r.f. field applied at a frequency o near 0,. We shall see however that the so-called "saturation parameter" s' = W"Td wilI be the same for ali the nuclei; anticipating the fact that the r.f, power and the nuclear relaxation enter the formula giving the nuclear polarization through this parameter only, we shall not specify the particular value of T: nor W",and write T,and W' as if they were unique. This procedure is valid only to establish the steady state value of the polarization; transient behaviours are of course different for different nuclear spins. After the transformation to the rotating frame of reference expressed by the operator U,the effective Hamiltonian is:
where again A = o, References p . 446
- w,and where #',&
and #: are the secular parts of
CH.vJ.u, 0 21
DYNAMIC F’OLARIZA~ONOF NUCLEAR TARGETS
407
the spin-spin interactions. Generalizing further Provotorov’s theory we assign to the nuclear Zeeman Harniltonian an inverse temperature 8, distinct from those, a and y, of 9’; and #&. The form of the density matrix is then taken to be d - e x p [ - a d Z j S . i - B o , ~ ~ I I t - yM&] where contributions from Hu and Hi are omitted. While the contribution to 17*from &‘,I is easily seen to be negligible, the best justification for dropping &:‘ is that it simplifies matters considerably without affecting appreciably the final result. As before, the theory has only been worked out for the high temperature approximation which is unfortunate for a description of low temperature dynamic polarization, but will give a physical insight into the situation. In this approximation, the density matrix can be written:
where a, and y are the inverse temperatures to be determined. As before, in order to obtain the steady state solution, one writes for the total rate of change of the various inverse temperatures involved :
a@),
are the relaxation variations given by
(21.1)
(21.2)
(21.3) 3/,
= h/kT,, inverse lattice temperature, is a very small quantity. slat),, stands for the rates of change due to the r.f. field, given by
(22.1) Referencesp . 446
408
(22.3)
Wo(A)is the same transition probability as in Provotorov’s theory and describes the direct action of the r.f.field on the electronicsystem; W’(d - 0,) and W’(A 0,) represent the probabilities for the two “forbidden” transitions giving rise to solid effect. We have neglected the indirect action of the dynamica1polarization on the electronic inverse temperatures 01 and y by omitting terms proportional to in the first and third equations (22). It can be shown that the effects so neglected are of the order of (N,/Ns)(Te/Tn) much smaller than unity in most practical cases. We shall not write out the complete and rather cumbersome formula valid in the general case. As before, the general form of the equations (22) can be deduced from energy balance considerations: when a photon of energy hm is emitted or absorbed, with a simultaneous change hme & ha, in the Zeeman energies of the electronic and nuclear spin system, the energy difference h(me - w k 0,) = h(A i-w,,), has to be taken up or given off by the spin-spin energy. As in the simpler case of equations (19), a derivation of (22) from first principles is possible. It can be shown in particular that the “forbidden” transition probabilities W‘ are proportional to H fand, for a given spin Z, to
+
where r is its distance to its neighbouring electronic spin, and 8 is the angle between the applied field H and the spatial vector IS. In the absence of nuclear spin diffusion the relaxation time of this spin Z is given by 1/T, = &2/Te. One then sees that W’T, is independent of c2, and does not depend on the particular position of the nuclear spin. The exact expression of the function W’(m) is rather involved. It exhibits a bell-shaped behaviour with a maximum at m = 0 and a width of the order of m,. The general steady state solution for the nuclear spin inverse temperature, which gives, at the same time, the enhancement of the nuclear polarization, deduced from (22), reads : References p. 446
CH.Vm,8 21
409
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
-
where W - = W’(d w,) and W + = W‘(d + w,) are the transition probabilities for the two opposite solid effects. An inspection of (23) shows that two effects corresponding to the two terms in the numerator are able to cool the nuclear spins and consequently to enhance their polarization: the effect of the first term can be interpreted as a pure “solid effect”, to which the spin-spin interaction temperature does not participate, and which tends to polarize the nuclei with a maximum enhancement of wela,, = y e / y n ; the second term produces a direct cooling of the electronic spins, with participation of their spin-spin interaction temperature. Through the same “forbidden” transitions as those of “solid effect”, it can cool the nuclear spins, to a cooling, or polarization enhancement, given by w,(w, - o ) / [ ( w e - o ) z + 2 021 with a maximum value of we/2J2wL reached for A = wL,/2. Depending on the respective values of w, and of 2 /, 2 w,, the first or the second of these maximum enhancements will be greater: if 0, < 2 oL,“solid effect” will be more effective, and the maximum polarization will be obtained for the r.f. frequencies we w, and we - w,; if w, > 2 4 2 wL the second effect may, at least in principle, give polarizations higher than those of “solid effect”, for r.f. frequencies equal to we 4 2 wL and we - J2 wL, which are situated well inside the electronic resonance line. If w, >> 2J2 wL,one of the limitations for these large enhancements will be the small value of W - and W f which are maximum for we - w = f 0,. The variation of the enhancement of the polarization as a function of r.f. power is given, for some typical values of the parameters in fig. 5. We consider now some limiting situations, in order to compare the results to usual theories: a) if the electronic resonance line is very narrow so that its width is much smaller than the nuclear resonance frequency, and if w R we - a,,W,and W +are negligible, and
42
+
+
which is the usual result obtained from rate equations for the populations of Zeeman levels, which turns out to be valid in this case because the spinspin interactions HSsdo not participate to the process. Let us take a typical References p . 446
410
A. ABRAGAM AND
[CH.Mn, 8 2
M. BORGHINI
60
R.F.
power (~rbitrary unib)
Fig. 5. Example of the theoretical dependence of the enhancement E of the polarization of a nuclear spin Z = 4 on the r.f.field power P, according to formula (23). Numerical constants are relyn = 600;H = 3600 Oe; HI, = 25 Oe; the electronic resonance line has a Lorentz shape, with a half-width at half intensity equal to HL.The curves are drawn for different values of the parameter A' defined by gPA' = A(we -w) : 5; 20;35;50;100 Oe.
example of electronic spins with g = 2, in a magnetic field of 15 kOe, thus with a Larmor frequency of 40 GHz and with a resonance line width of 10 Oe. The probability W, induced by a ref.field Hi of 0.1 Oe, obtained for instance in a rectangular resonance cavity of 0.1 cma, with a @value of lo00 and a r.f. power of 1 mW, is 2 x 104 sec-'. For nuclei distant of 5A from these electronic spins, the mixing coefficient e is R and the probability W' for the forbidden transition, W' = e2Wo, is 2 sec-' to be compared with nuclear relaxation times comprised between about 0.1 and 10 sec typically. On the other hand if o m at,W - and W' are negligible and, though the electronic spins are highly cooled by the r.f. field, the nuclear spins stay at the lattice temperature and are not polarized, the thermal contact between them and the electrons not being established. If the electronic resonance line is broad with respect to the nuclear frequency, and in the case of low r.f. power, one finds from (23) that 0, _B ---
BL
w-T,- W+T,
@"1+ W-T,+ W + T , +
A2 + 20;
WO Te
20:: The two types of solid effect transitions have opposite effects; the situation References p . 446
CH.MI,0 21
411
DYNAMIC POLARIZATION OF NUCLEAR TARQETS
is said to be "differential" for the form of the nuclear polarization as a function of the r.f. frequency is approximately that of the derivative of the electronic line. Finally, if the r.f. power is very high, /3 is given by
B/B,
=
(we
- @)/{(me -
+2 4 )
and one sees that there is no more a differential effect, the form of /3 as a function of r.f. frequency o being always that of a dispersion curve. This formula is the same as formula (17'). As already stated at the beginning of this section this theory has not received the test of precise experimental confirmations but it gives correct orders of magnitudes in practical cases.
2.3. HOMOGENEOUS SPIN EFFECT THEORY
SYSTEMS WITH
NUCLEAR SPIN DIFFUSION.
SOLID
We consider now dynamic polarization by electronic spins with a homogeneous resonance line, in the case where there is spin diffusion inside the nuclear spin system. The density matrix in the rotating frame can be written as:
and /3(rn) obeys the diffusion equation (24) obtained by combining equations (8) and (22.2):
+
+
where T(A - wn)/r6 = W ' ( A - on)and r(d wn)/r6 = W ' ( A on) are the forbidden transition probabilities induced by the r.f. field. They verify approximately the relations
r(A- o,)/C = W, ( A - 0,) T,and r ( A + wn)/C = W, (A + on)T,, where W, is the allowed transition probability induced by the ref. field (actually, the form functions contained in W' and in W, are not quite the same as a detailed calculation can show, but we disregard this complication). Referencea p , 446
412
A. ABRAGAM AND M. BORGHINI
[CH. WI,
82
With this approximation the diffusion equation can be rewritten as:
r--r+ . u 0,c + r- + r+ the rate equation for p has then the same form (25) A
Pst = -
C BL c +r - +r+
;
as equation (9) in the
absence of the r.f. field :
-dtdp
=
- 4aNebD(p - &).
(25)
Using the definitions above of r-,r+and C the reader will be able to ascertain that fist is identical with the earlier expression (23) in which one replaces now W-T, by Wo(A - w,) T, and W'T, by W,(A w,)Te which were respectively equivalent in the case of no spin diffusion. Although the local behaviour of the polarization is seen to be modified by spin diffusion, the dependence of its average value in steady state on r.f. power and frequency is the same as without spin diffusion; this is obviously due to the fact that for each nuclear spin, the relaxation and dynamic polarization probabilities due to its coupling with the electronic spins are strictly proportional to each other; this is no longer true if different electronic spins S contribute differently to relaxation and dynamic polarization, as we shall see now.
+
2.4.
RELAXATION AND POLARIZATION BY UNLIKE
ELECTRONIC SPINS IN THE
CASEOF NUCLEAR SPINDIFFUSION (LEAKAGE) We consider the case of two electronic spin species S1 and S, which tend to relax or polarize the nuclei towards different inverse temperatures PIand 8,.
This will be the case if the homogeneous spins S, contribute to both nuclear polarization and nuclear relaxation whereas the spins Sz are impurities present in the sample that contribute to nuclear relaxation but, because of their widely different Larmor frequency, not to dynamic polarization; the diffusion equation reads
a -P at
= DV'
P - r1zi Irl-
References p . 416
rnI-6 (S - 81) - r z ~2
Ir, - rnI-6 (B - P J
CH.Mn, 8 21
413
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
and the average value equation18
fl
of the nuclear inverse temperature obeys the
where bl and b, are functions of the respective Fl and I’,depending on the details of the spin diffusion; the steady state enhancement of the nuclear polarization under solid effect is thus given by
where Po is the inverse temperature that would be given to the nuclei if the spins S , did not exist in the sample. If the spins S1 are used to polarize dynamically the nuclei under the action of an r.f. field Hl of frequency w and if the spins S, are relaxing impurities, we may write rather generally, as may be seen from Section 1: b, (~~+r++r-)”thc~W ~ ~i(/C~O +, C, rO ,+ w;,r) =~ wo(~-w,) E W,, where T, is the relaxation time of spins S1 and Wo(w)is the probability of the allowed transition of spins S1 induced by the field Hi;if l/Tn,l and l/Tn,zare the respective contributions of spins S , and of spins S , to the total nuclear relaxation rate l/T,, formulae (27) and (23) give for the enhancement of the nuclear polarization, with Wo = Wo(w)
-
-
-
for instance, if the electronic line width is much smaller than the nuclear frequency, and if A = w,, we shall have W W: 0 and for p = this formula becomes we WO-Te -=(28) L’ w n i + w ; T e +T~ n,~ 2l + W ; T e ’ N
a,
B
(28) can be used to investigate the effect of spurious electronic impurities on nuclear polarization. Formula (26) can be generalized to the case of an arbitrary number of References p . 446
414
A. ABRAGAM AND
M. BOROAW
[am .,0 2
electronic spin species and at least in principle applied to the problem of polarization by electronic spins having an inhomogeneous resonance line in the presence of nuclear spin diffusion. 2.5. INHOMOGENEOUS ELECTRONIC SPINSYSTEMS
We have supposed in the preceding that the different spins S had the same Larmor frequency, and that the width of their resonance was due to their dipolar interactions &’=. Situations can arise where these conditions are not fulfilled, for instance if the Larmor frequencies have a distribution due to lattice defects or if the line width is caused mainly by the interactions of the spins S with their neighbouring nuclei. Usually no distinction is made between these two broadening mechanisms, described as inhomogeneous broadening. While the behaviour of the electronic spins may well be insensitive to the precise origin of this inhomogeneity, the dynamic polarization of the nuclei is likely to be affected if these very nuclei are responsible for the electronic inhomogeneous broadening. The theory of solid effect has not been worked out so far for either case and we shall consider very briefly the simpler case of a distribution of Larmor frequencies. In this case the electronic spin system may be visualized as a collection of spin packets having different frequencies. The homogeneous width of a single spin packet due to the electronic spin-spin couplings is smaller than if the spinshad all the same frequency, for the distribution of Larmor frequencies impedes the flip-flops among the spins. The different packets may behave independently of each other, or, on the contrary may be connected by the so-called “cross-relaxation” or “spectral diffusion” produced by the dipolar spin-spin couplings34- 36. We consider first independent spin packets. On the application of an r.f. field H , , the effective Zeeman energies of the different spin packets S’ take different inverse temperatures d in the rotating frame; in particular, the two packets S( +) and S( -) with Larmor frequencies we( + ) and we(-) such thatA(+)= a,(+)-o = - o , a n d d ( - ) = w o , ( - ) - w = w,,takeopposite inverse temperatures a( +) = - u( -). These two packets will be the only ones to contribute to the dynamic polarization of the nuclei if the width of a packet is much smaller than the nuclear frequency w,. Their contribution to nuclear polarization will subtract from each other. If 0, is much smaller than the width of the’envelopeof the electron spin packets, the dependence of the nuclear polarization on the frequency will be proportional to the derivative of that envelope and have roughly the shape of a dispersion curve. In the past, the observation of such shapes for dynamic polarization curves has References p . 446
5 31
CH. Wr,
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
415
been taken as evidence of the extreme inhomogeneous behaviour of the electron line. One should beware of such conclusions for, as we saw before in the extreme opposite case of completely homogeneous line, the dynamic polarization curve still has a similar dispersion shape. If, as is often the case, the spin packets are not independent and the crossrelaxation causes perturbations affecting one spin packet to diffuse through the entire curve, the distribution of spin temperatures among the various spin packets under the action of an r.f. field becomes complicated. To show qualitatively how spectral diffusion could modify the previous results on dynamic polarization, we shall consider a limiting case: we suppose that the spin packets are narrow with respect to the overall frequency distribution and that spectral diffusion is sufficiently strong to equalize the temperatures of the Werent spin packets; the common inverse temperature in the rotating frame can then be shown to be
where a,is the mean frequency of the distribution and 2 its second moment with respect to we.The two groups of spin packets S ( +) and S ( -) will now add rather than subtract their contributions to nuclear dynamic polarization; however, the frequency dependence of dynamic polarization will still have a dispersion shape as follows from (29).
3. Experimental Arrangements and Results We shall describe now some experimental studies of dynamic polarization at low temperatures under optimum laboratory conditions and their extension to nuclear scattering experiments. Two such experiments, with polarized proton targets, have been performed until now: scattering of polarized 20 MeV protons by a thin target of lanthanum magnesium nitrate (LMN)2; scattering of 246 MeV z' by a large target of one cubic inch of LMNa. It turns out that the experimental arrangement used to polarize large targets is very similar to laboratory apparatus; on the other hand, the polarization of thin targets for low energy scattering presents special difficulties. We shall therefore adopt the followingplan: 1- Experimentalresults; 2 - Laboratory apparatus and large targets; 3 - Thin targets.
3.1. EXPERIMENTAL RESULTS We review here the experiments that have led to nuclear polarizations useful, or potentially useful, for polarized targets; in this respect, Overhauser and Referencesp . 446
416
A. ABRAGAhf AND
M. BORGKINI
[CH.Vm,8 3
Jeffries methods are not so interesting as “solid effect” and they will be omitted from this review; although the thickness of metallic targets to be polarized by Overhauser effect is not in principle limited by the penetration depth of the microwave magnetic field97 to values of the order of 1 micron or less at the low temperatures and high frequencies necessary for convenient polarizations, practical difficulties are such that no polarization useful for nuclear scattering has been yet obtained in metals (see however Section 4). Jeffries’ method applies to nuclei inside paramagnetic ions, which must be present in the sample at a rather low concentration, in order to avoid a broadening of the electronic resonance lines due to spin-spin interactions, which would prevent the necessary separation between the different allowed and “forbidden” transitions. The “solid effect” can be used to polarize relatively high densities of nuclei, and we shall restrict ourselves to such cases, omitting for example, the polarization of the low abundance (4.8 %) silicon isotope with non-zero spin, Si29, in silicon crystals slightly doped with phosphorus38. It will be seen that, in most cases, besides the polarized nuclei of interest, the target will contain other nuclei; in nuclear experiments, the events produced by the beam on these “parasitic” nuclei may sometimes be eliminated by coincident countings of two outgoing particles, as for instance in p-p scattering at 20 MeV, or by a kinematical reconstruction of the event, after measuring the impulsion momentum of one outgoing particle. 3.1 .l. Dynamic Polarization of Protons 1. Protons in plastics. Dynamic polarization of protons at helium temperatures has been performed in lucite where a variable amount of the free radical DPPH was dissolved through an intermediate dissolution in benzene or chloroform, in a magnetic field of about 12 kOe, with an r.f. frequency in the 8 mm wavelength range, around 35 GHz. Early experiments had given polarizations of 1.5% at 4.2”K3QY then of 4.5% at 1.5OK40 for concentrations of 10% in weight of the free radical. Further investigations on doped or irradiated plastics, performed at 9 GHz41 or at 35 G H z ~have ~ , so far never given reproducible enhancements of the nuclear polarization higher than 60, which would represent polarizations of some 10% in practical limit conditions: o,= 72 GHz, T = l0K, if the same enhancements were obtained under those conditions. For instance, at 9 GHz, 1.2”K, the maximum published enhancement, in polyethylene irradiated with fast neutrons, was of 4041. Enhancement of 20 at 0.5”K has been obtained43also in a sample of polyethyleneirradiated with fast neutrons which gave enhancements of 30 at 1.6”K. The inhomogeneous broadening Referencesp. 446
CH.vm, 8 31
417
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
of the electron resonance line ( x 140 Oe) may explain these low enhancements, but in plastics doped with free radicals, the lines are narrower (NN 10 to 20 Oe) and the low enhancements are not understood. 2. Protons in lanthanum magnesium nitrate. Dynamic polarization by solid effect in lanthanum magnesium nitrate (LMN) has been extremely fruitful, and has led to increasing proton polarizations: 1.5% at 9 GHz, 1.2OK44; 3% at 9 GHz, 1.7OK21; 19% at 36 GHz, 1.5OK40, in LMN doped with various amounts of cerium; then 51% at 50 GHz, 1.350K45; 70% at 72 GHz, l.2OK46, in LMN doped with 1% neodymium enriched to 98.5% in even isotopes. We begin by giving some details on this interesting substance.
LMN. Crystal structure; lattice parameters. LMN = Laz Mg, 24 H20, is used for dynamical polarization as single crystals. Powder X-ray diffraction studies47 indicate a rhombohedra1 unit cell with sides 13.1 A and interaxial angle 49". There are two equivalent sites for the divalent ions, each surrounded by water molecules. There is one site for the trivalent ion; the nearest hydrogen atoms are 4.9 A away48, which agrees with a rough magnetic resonance estimate4e(4.5 A). The density is50 d = 1.99 g/cmst. The average spacing is 8.5 A between La ions and 3.0 A between protons, but they are paired in water molecules in which their distance is about 1.7 A; the average spacing between water molecules is 3.6 A. The Debye temperature is BD = 60OK51, the velocity of sound is generally taken as 2.5 x 105 cmf sec.52153.The single crystals grow in flat hexagonal plates.
'
Electronic paramagnetism. Resonance lines. Ce' '' and Nd' ' have respectively one and three 4f electrons; the ground multiplets are respectively J = $ and J = The crystalline field has trigonal symmetry; its axis Z is normal to the plane of the crystal and it splits each ground multiplet into several Kramers doublets. a. Cerium. The spin-Hamiltonian for the ground doublet is
8.
where 2 is the crystalline field axis, and
t The density of LMN as calculated from the dimensions of the unit cell is 2.073 g/cm3; preliminary messurements at room temperature give 2.075 f0.002g/crn3V.W. P.Brogden, private communication]. References p. 446
418
A. ALtRAGAM A N D M. BOROHINI
g II = 0.0236 (ref. 5) g, = 1.826 f 0.001 (ref.54)
[CH.VIU,
33
.
The nearest excited doublet has an energy higher by an amount equivalent to 34OK65. The resonance line width depends on the concentration C of cerium ions, on the resonance frequency, and, to a lesser extent, on the temperature: with H 12,at 1.7"K: at 9 GHza1: C Hpp
at 36 GHz:
(%)
(Oe)
C (A) Hpp(0d
5 19 5
20
1 7 1 12
0.2 5
0.05 5
0.5
12
where Hppis the peak-to-peak width of the derivative of the paramagnetic absorption line. b. Neodymium. The spin-Hamiltonian for the ground doublet is
811 = 0.362 f 0.010
g1 = 2.702 f 0.006.
The nearest excited doublet has an energy higher by an amount equivalent to 47.6"K57~68, The resonance line-width has been found to depend on concentration C, resonance frequency w, and on the angle 0 between the applied magnetic field H and the crystalline field axis 2: at 35 GHz, with H I Z, and at 1.5OK69:
for a concentration of 1%, with H 1Z, for w = 9 GHz, Hpp= 3.2 Oe45; for w = 35 GHz, Ifpp = 4.5 Oe. The variation of the width Hppwith the angle 8, measured at 4.2"K, 9 GHz, in a crystal with 1% Nd is given in Fig. 6 6 0 . Referencesp . 446
~ . ~ , 0 3 1
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
' 0-
419
Fig. 6. Line-width of 1 % enriched Nd+++ in LMN,at 4.2"K, 9GHz vs. angle 0 between crystalline field direction 2 and magnetic field H.
Hyperfine interaction: Natural neodymium contains about 20% of odd isotopes; dynamic polarization is usually performed with enriched neodymium, containing 98.5% or more of spinless isotopes. Electronic relaxation time. Measurements of the recovery time of the electronic absorption signal after its saturation by an intense r.f. field give the results presented in Figs. 7, 861 and 910, Cerium: The direct process (see Sect. 1) is always masked by phonon bottle-neck for usual dimensions (a few millimeters or more), or at higher temperatures, by Orbach process; it can nevertheless be estimated, and the following empirical relations have been established10te1 for the relaxation 1/T1,of & the Orbach process, and of the direct process rates l/Z'l,o, l/Tl,d, without and with phonon bottle-neck, respectively :
l/Tl,o = 2.7 x lo9 e-34tT l/Tl,d = 25 x low3H 5 / P e , 1/T1,& =a X H2/lCP:, where H is the applied field in kOe, I(cm) the crystal thickness, P, the electronic polarization, and 01 a numerical coefficient, which was found to be equal to 5 for a concentration C = 0.2%,and to 20 for C = 2%. Neodymium: The direct process, which is about ten times slower than in the case of cerium, is not masked in relatively low fields (ex : 2.48 kOe and 2.56 kOe, Fig. 8) by the phonon bottle-neck which occurs however in higher fields (ex : 9.1 kOe, Fig. 9). References p , 446
420
A. ABRAGAM AND
[CH.WI, 5
M.BORGHINI
, , , , , I
Nd
S(‘K-1)
in
I
, , , , , , I , ,
1
LailNg,(NO,)lz.ZIH,O
) , , , , , I
,
rlH
Fig. 8. Relaxationtime Teof Nd+++in LMN vs. temperature in 2.48 and 2.56 kOe ‘I1.
Fig. 7. Relaxation time Teof Ce+++in LMN vs. temperatureel. Some data are quoted from refs.l0.50.55,117.
Fig. 9. Relaxation time Te of Nd+++ in LMN vs. temperature, at 9.1 kOelo.
Referencesp . 446
7
3
a. m,B 31
421
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
Empirical relations have been obtained1036 :
l/7'l,o % 5 x lo9 e-47.6JT' I / T ~= , ~2.5 x 10-3 H ~ I P , , l/Tl,b = 15 x H2/ZCP:. Hydrogen nuclei.
a. Relaxation time. LMN with cerium. In a field of 13 kOe62, the relaxation time T,, of the protons obeys the empiricalrelation T,, = 80 C-l T-' between 1.5 and 2.1°K, for C e"' concentrations between 0.5 and 5% (C is measured in %). At 3 kOe21, T,, = 4000 C-' T-7in the same temperature range, for concentration C between 0.1 and 1%. At lower concentrations, T,,tends towards a constant value, probably due to some impurities; at higher concentrations, T,,becomes proportional to C2, which is not understood. In this temperature range, and in a field of 3 kOe, the electron relaxation is due to the Orbach process and the exponential variation of the electronic relaxation time T, can be approximated empirically by T, T-14. Since for the same field and temperatures T,, T - 7 , for all concentrations between 0.05 and 5%, one sees that spin diffusion is responsible for the nuclear relaxation, even at relatively high concentrations, and that the relation between T,,and T, is T,, T? probably due to the quenching of spin diffusion in the immediate neighbourhood of the paramagnetic ions. LMN with neodymium. With C = 1% (this concentration is the concentration in the water solution used to prepare the crystals; as neodymiuni enters the crystals more reluctantly than lanthanum, the concentration in the crystal may be quite different), in a field H of 20 kOe, and in 100 mm3 crystals45. T ("K) 2.1 1.35 T,,(sec) 720 1200. b. Dynamic polarization. LMN with cerium. 1. At 9 GHz, with H 12 = 3.5 kOe, T = 1.7"K, the maximum enhancements21 of the nuclear polarization are 50 with a concentration C = 5%, and 160 with C = 1%. The variation of the enhancement E as a function of the microwave power P is given in Fig. 10, curve 1, for a crystal with C = 5%. 2. At 35 GHz, with H 12 = 13.5 kOe, and for concentrations C = 0.3, 0.5 and 1%, the maximum enhancements were of 230 at 2.1"K and 200 at N
N
-
References p . 446
422
A.ABRACiAMAND M.BOROIWI
P Power
Fig. 10. Experimental enhancement E of the proton polarization YS. r.f.field power (power unit is arbitrary for each curve): (1) LMN,5 % Ce, at 1.7'K, 9 GHz; (2) LMN, 0.5 % Ce, at 1.5"K, 35 GHz; (3) Solid DI, at 4.2"K, 25 GHz; (4) Irradiated polyethylene, at 77"K, 9 GHz.
lS"K62. The variation of the enhancement as a function of the microwave power P is shown on curve 2 of Fig. 10, for a crystal with C = 0.5%. For comparison, one correspondingcurve (3) for polarization in solid deuterium63 and one curve (4) for one irradiated plastic64, are shown. 3. At He3 temperatures, down to 0.55 f 0.05"K,with we = 9 GHz, C = 0.8%, enhancements of 129 f 10 and - (118 & 10) have been obtained43, in a 6 x 6 x 2 mm crystal, which correspond to proton polarizations of 8%; the difference between the values of the applied magnetic H corresponding to maximum positive and negative enhancements, is 21 f 2 Oe, whereas the difference between the fields corresponding to the extrema in the electronic resonance signals is 16 & 1 Oe. The maximum enhancements are obtained for at^ r.f. power of 1 m W in a cavity with a Q-value of 1OOO. and is equal to The proton relaxation time is proportional to T-le6' * *'", 920 f 80 sec for T = 0.32"K. LMN with neodymium. Few results are published 45 :for crystals of LMN, 1% Nd enriched to 98.5% in even isotopes: (GHz) 35 50 50
H(kOe) 17.7 20.3 20.1
TTK) 2.07 2.10 1.35
E, 300
400 340
P,(% 26 39 51.
With a frequency of 70 GHz, in the 4 mm wavelength range, polarizations of about 70% have been obtained46; at these high polarizations, the shape of the nuclear resonance line changes, and the comparison between the References p . 446
a. VJn,8 31
DYNAMIC POLARIZAnoN OF NUCLBAR TARGETS
423
natural and enhanced polarizations must be done by measuring the total area under the absorption resonance curve, rather than by comparing the height of the signal observed, with and without dynamic polarization. Polarization time constant. LMN with cerium: at 9 GHz as well as at 35 GHz, the polarization time constant has been found to be shorter than the nuclear relaxation time; for instance, at 35 GHz the following empirical relation has been found62 between the polarization time r,,, the relaxation time T,, and the ratio between the dynamic nuclear polarization pn and the natural electronic polarization P,": 1 - z,,/T, m p,,/P,". LMN with neodymium: Few results are published: with relaxation times of ten to twenty minutes, it is stated46 that, for the maximum enhancements, a steady state is reached in about five minutes, in LMN doped with 1% neodymium.
3. Solid hydrogen. The molecule of orthohydrogen, with total nuclear spin Z = I, + Z2 = 1, is, at low temperatures, in a state of rotation J = 1, because of the Pauli principle. The natural relaxation time of the proton spins, which is due to their coupling with the molecular rotation, is relatively short, of the order of 1 second near l0K65187.Paramagnetic hydrogen atoms, created by y-irradiation, have a relaxation time of the order of 0.1 sec 68, and have thus a negligible effect on the nuclear relaxation, in view of the fact that atomic hydrogen cannot be produced in high concentrations in molecular hydrogen; maximum concentrations69 are of 7 x 10l6 atoms/cmS. Dynamic polarization by solid effect has Iittle chance to succeed in these conditions, and has not yet been tried. 3.1.2. Dynamic Polarization of Deuterons
Dynamic polarization of deuteronshas been tried in solid deuterium 0nly7~971, although one can hope that significant enhancements of their natural polarization could be obtained in crystals of LMN grown with heavy water.
Solid deuterium. In the ground state, only molecules with total nuclear spin I = I, + I, equal to 0 and 2 are present, in the state of rotation J = 0. If their number is given by the statistical weights, the maximum polarization which can be achieved is of $ = 83.3%. Present results, though encouraging, have only reached a value of I.l%as: the electronic spins are those of deuterium atoms: the samples are grown by passing gaseous deuterium through an electrodeless r.f. dischaTge in a "dry-filmed" glass tube, and condensing it into a microwave cavity whose temperature is maintained at 4.2"K by a Referencesp . 446
424
A. ABRAGAM AND
M. BORGHINI
I-.
m,§ 3
bath of liquid helium. The density of atoms was approximately 5 x 1016/cm3, and it is hoped to increase it by one or two orders of magnitude. Because of the hyperfine interaction in the atoms, the electronic resonance line has a triplet structure; the lines have a width of some 2.5 Oe. Polarization in the neighbourhood of the central line leads to higher enhancements than in the neighbourhood of the lateral lines by a factor of the order of five. Maximum enhancements were of 160 at 4.2”K, in a field of 8.5 kOe, and of 60 at 12°K. The variation of the enhancement as a function of the 24 GHz microwave power is given in Fig. 10 curve 3. Maximum power is of 0.5 W in a cavity which has a Q-value of about 5000 and a volume of about 5 cm3. 3.1.3. Dynamic Polarization of Lithium and Flwrine Nuclei Dynamic polarization of Li 7 and Fl9 has been observed simultaneously in lithium fluoride irradiated with X-rays 44 ;F 19 has been polarized also in LiF irradiated with thermal neutrons40 and in calcium fluoride CaF, doped with cerium 72. Lithium fluoride. X-ray irradiation produces paramagnetic colour-centers, with a resonance line-width at half intensity of about 100 Oe, due to the inhomogeneous broadening caused by the nuclei around the electronic centers. At 1.6”K, with we = 9.4 GHz, equal enhancements of 17 were observed for the Li and the F 19 resonance signals : this was explained by the so-called “differential effect” 44; as nuclear resonance frequencies are much smaller than the electronic resonance line-width, electronic spin packets for which cue - w = w, and we- = - w, where w is the microwave frequency, give opposite contributions to the nuclear dynamic polarization which partially cancel each other; the net enhancement is thus proportional to (we/w,) w, df(w)/d w, wheref(o) is the line shape of the inhomogeneously broadened electron line; the interpretation in which all the electronic spins have the same temperature, in a suitable reference frame, which is then transmitted to the nuclei, leads also to equal enhancements for the polarization of all the nuclear species. With we = 36 GHz, enhancements by a factor 60 of the Li7 resonance were measured, which correspond at 1.2”K to a polarization of 3.9%; an equal enhancement was postulated for the Fl9 signal corresponding to a polarization of 5.5%44. In neutron-irradiated lithium fluoride, enhancements reaching 75 were measured in fields of 13 kOe, at 1.5”K,corresponding again to polarizations of the order of 5.5%40. Referencesp . 446
CH.Vm,
8 31
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
425
CalciumJEuoride.Single crystals of CaF, doped with about 0.005% of cerium, ions replacing Ca+ ions, show three magnetically different sites, the Ce’ with crystal fields of trigonal symmetry whose axes are directed along the three cube edges of the crystalline structure. Each ion may be described by an effective spin 4 with a spin-Hamiltonian: +
+
+
Af= g p z s z+ 81 (KS,+ H,S,) where gll = 3.038 & 0.003 and gL = 1.396 -t 0.002 and where x , y , z are the three crystalline directions 73374. The electronic resonance line width depends on the orientation of the applied magnetic field with respect to the crystal principal directions: with H in the [lo01 direction, two third of the Ce ions have the same resonance frequency, and the measured line-width is about 5 Oe; as the sum of the dipolar fields produced by the eight neighbouring fluorine nuclei is zero because of the symmetry, this width may originate in a small contact interaction between the electronic spin and these nuclei, or in a small tetragonal distortion of the environment due to the charge compensation. Dynamic polarization experiments72 were done in a field of 4.5 kOe, with a microwave source of 2 watts at about 9 GHz. Maximum enhancements of 20
- 20
I
F”
~regueny
Fig, 11. Enhancement E of the FIB polarization in CaF2, 0.005 % Ce+++,as a function of the difference between the frequency me of the centre of the electronic resonance line and the frequency w of the r.f. saturating field; w is held constant and His varied; Hvalues are indicated by the F19 resonance frequency72.
References p . 446
426
[CH.Vm,4 3
A. ABRAGAM AND M. EORGHINI
70 were obtained at 20"K, and found to decreaserapidlybetween 14and 12°K. An interesting feature of this sample is that, because of the narrow linewidth, three satellite-lines of decreasing intensities on each side of the main electronic line, can be observed; they correspond respectively to simultaneous flips of the electronic spin and of one, two or three neighbouring fluorine nuclei; to each of these satellite line frequencies, there corresponds a peak in the dynamic enhancement of the fluorine resonance signal (Fig. 11); the ratio of the maximum enhancements corresponding to each of these processes has been derived from simple detailed balance arguments, as those of Section 1: suppose that v nuclei flip in the same direction when an electron flips : if the microwave field equalizes the transition probabilities
w+ + + .. ,(
the equation
+)+ -,(
- - ..-)and W-d- - ..-I+
+,(
+ + ..+)
N + n + n + . . - n +W+,(++..+)+-,(--..-) = = N-n-n-...n- W-,(--..-)++,(++..+)
leads to
n + / n - = exp(hw,/vkT).
As, in the experiments under consideration, the polarizations are small, one expects that the maximum enhancements for the transitions involving one, two or three nuclei, are in the ratio 1 : 3 : 3. Experiment gives ratios of the order of 1 : : at 20°K and 1 :3 : at 12"K, although the satellite intensities as observed in the ESR spectrum are rather well accounted for by the theory. The interpretation of polarization experiments is difficult in view of the fact that the nuclear signal is that of the fluorine nuclei which are distant from the cerium ions and are polarized by spin-diffusion; the first fluorine neighbours of each electronic spin experience a field of about 100 Oe due to their dipolar coupling with this spin, and their resonance is not observed and it is likely that they do not participate to spin-diffusion with the other nuclei either. There is thus no simple relation between the satellite intensities and the enhancements produced on each of these satellites.
+&
&
3.2. LABORATORY APPARATUS AND LARGE TARGETS The sample to be polarized must be cooled at a low temperature T, must experience a constant and uniform magnetic field H a n d an r.f. field H J - H of frequency w near the electronic Larmor frequency we which induces the polarizing transitions, and an r.f. field H i of frequency w' near the nuclear Larmor frequency on, which provides the nuclear magnetic resonance signal used to measure the enhancement of the nuclear polarization due to the References p . 446
CH.m,0 31
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
421
dynamic effect. T is in the liquid helium temperature range; generally H has a value comprised between 3 kOe and 25 kOe, w evaries between 9 GHz and 72 GHz and w, between a few MHz and 100 MHz, roughly.
3.2.1. Cryogenics Dynamic polarization is interesting for nuclear targets when performed at liquid helium temperatures, and if possible, in liquid helium. Almost all experiments have been performed at liquid He 4 temperatures, between 4.2"K under atmospheric pressure, and 1.2"Kunder reduced pressure :the cryostats are thus quite standard: metal as well as glass Dewars, are used, with a radiation shield cooled to 77°K by a bath of liquid nitrogen. The sample contained in a microwave resonator is immerged in the helium bath. The lowest temperature which can be obtained in a particular case depends on the power dissipated in the cryostat and on the pump used to reduce the pressure on the bath. Fig. 12 shows the pT curve for He 4. The dissipation of liquid He4 is of the order of 1.4 litre/hour.watt at 4.2"K, and of 1.1 litre/ hour-watt between 3 and 1°K; the volume of liquid He4 which must be evaporated in a standard cryostat to cool a given He4 bath from 4.2"K to
Fig. 12. Pressure-temperaturedependence of liquid He4.
References p . 446
Fig. 13. Pressure-temperaturedependence of liquid Hes.
428
A. ABRAGAM AN0 M. BORGHINI
[CH. WII,
83
its final value T is about 45% of the initial volume for temperatures T below 1.6OK75. A few dynamic polarization experiments have been done at liquid He3 temperatures43; we give in Fig. 13 the aspect of thep, T relation76 for He3; liquid He3 is generally used in closed-loop refrigerators and its circulation is about 3 litre/hour.watt below 2.5"K. Many details about He3 cryostat construction may be found in 77.
3.2.2, Microwaves and Electronic Resonance The microwave circuits used in dynamic polarization research are quite standard and may be very simple: in principle, to saturate the spin transitions, they may consist in a continuous power source, a waveguide or a coaxial line, and a resonator containing the sample; in fact it is always convenient to be able to observe the level of the microwave oscillation, the frequency response of the resonator, the electronic resonance signal ; the simplest direct detection system is generally used, as shown on Fig. 14, where the device ( T ) feeds into the resonator part of the incident power coming from the source and into the crystal detector part of the power reflected by the resonator and the sample; this device may be a magic tee, where half of the incident power is lost in a matched load and half of the signal power is lost in the incident line, or a directional coupler of given attenuation A where the fraction 1/A of the incident power is lost in a matched load and (1 - 1/A) of the signal power is lost in the incident line, MAGIC TEE e- CIRCULATOR
SOURCE
.r.h
.......
T
ia-
LOCK-lH
CRYSTAL
FREQUENCY
STABILIZER
Ft RECORDER
MODULATOR
REFEREPICE PHASE TUtiIH(6
Fig. 14. Block-diagram of a simple microwave apparatus used in dynamic polarization. References p . 446
CH.WI,B 31
DYNAMIC POLARIZATTON OF NUCLEAR TARGETS
429
or a three-port ferrite circulator, where almost the total incident power Is directed towards the resonator, and practically the whole signal power directed towards the detector. If the source is electricallytunable, a frequency stabilization is often added to the circuit and may be one of the various types described in the text-books 78179. With a typical electronic line-width of 20 MHz, with a resonance frequency of 50 GHz for example, stabilities of the order of a few are required.
Sources. Convenient sources are continuous and are tunable, either mechanically (m.t.) or electrically (e.t.); low powers of a few milliwatts are sufficient to polarize small samples, smaller than 100 mm3 for example; high powers, of one watt or more are needed to polarize large samples, of several cms. In X-band, around 9 GHz, backward-wave oscillators, klystrons and magnetrons give any power up to 200 W; in Q-band, around 35 GHz, reflex klystrons (m.t. + e.t.) give powers up to 500 mW, floating drift-tube klystrons (m.t.) reach 15 W; in E-band, around 72 GHz, reflex klystrons (m.t. + e.t.) deliver up to 200 mW, floating drift-tube klystrons (m.t.) 1 W, and carcinotrons (e.t.) reach 15 W. Waveguides. In the mm-wavelength range, and if high powers are needed, the dissipation in waveguides may have to be considered: the best standard 8-mm waveguides (silver-lined copper) have measured attenuations of about 0.7 dB/m; the best standard 4-mmwaveguides (“standard silver”) measured attenuations of more than 3 dB/msO;oversized waveguides have fortunately smaller losses and may be used if a particular mode of propagation is not imperative; experimental helix-waveguides, strip-lines and H-waveguides have also relatively low attenuations, but their insertion losses may be still high (6 dB for example, with H-waveguides)81. Microwave resonators. Resonant cavities are generally used, although helices are employed at Oxfordsz; rectangular cavities in the TE,,, mode or cylindrical cavities in the TE,,, or TE,,, modes, for small samples with dimensions of the order of the wavelength located in a region of maximum magnetic field ;rectangular or cylindrical cavities of large volumes, resonating in many degenerate modes, are used with large samples45**3,which may fill completely the cavity. Helices have small sizes and are difficult to use at mm-wavelength frequencies; they have the advantage of being relatively broad-band and they permit easily to apply the nuclear r.f. field 23; to the sample. Reference8 p . 446
430
A. ABRAGAMAND
M. EORQHINI
[=.wn,93
LOCK -IN
RECEIVER
_I
RECORDER
I
A. F. MODULATION
REFERENCE PHASE
TUNINS
P i . 15. Block-diagram of a simple nuclear resonance detection, using a Q-meter.
3.2.3. High Frequencies and Nuclear Resonance Standard detection apparatus are used to observe and to measure the nuclear resonance signals84: Q-meter, bridge, crossed-coils arrangement85, Pound-autodyne88, Robinson-autodyne87. The simplest one is the Q-meter (Fig. 15); the sample is located inside an inductance coil L, tuned to the desired frequency by a variable condenser C. This tuned circuit is connected in parallel between a h.f. generator and a low-noise receiver. Any change in the real part of the complex nuclear susceptibility of the sample modifies the signal reaching the receiver and is detected; by modulating the main field H or the h.f. frequency by an amount greater than its width, the resonance line may be displayed on an oscilloscope. For better measurements, the sensitivity is generally increased by a phase-sensitive or “lock-in” detection: the nuclear susceptibility is slightly modulated at a low frequency either by adding to the main field H a n alternating field H’ smaller than or of the order of the line width or by frequency modulating the h.f. generator; after h.f. detection, the low-frequency signal is detected in a phase-coherent detector where a reduction of the band-width by large time constants improves the signalto-noise ratio, and is then displayed on a pen-recorder. The entire resonance line is describedby a slow scanning of the field Hor of the frequency0 ’ .The recorded signalis proportional to the derivativeof the absorption resonance line. With the low quality of the coils located in microwave devices the frequency response of a Q-meter is rather flat, and frequency scanning can be used without distortion of the line shape. This is not the case with the sharper References p . 446
a. m s 4 31
431
DYNAWC POLARIZATION OF NUCLEAR TARGETS
frequency response of a bridge. The Pound-autodyne is non-linear for the high polarization signals, and cannot be used. The Robinson-autodyne is linear, and can be used also with frequency scanning and/or modulation. Crossed-coils arrangement is mechanically more complicated and is used only exceptionally in this fie1d 88. Nuclear resonance coil. Nuclear resonance coils are often made of a few turns of thin wire, wound around the sample or in its immediate vicinity, in a position inside the cavity where they are not or only slightly coupled to
Fig. 16. Example of arrangement of a sample and a nudear resonance coil inside a rectangular cavityee, used at Harwell. (1) Dielectric coupling slug. (2) Electroformed copper cavity; resonant frequency in the TEola mode is 35.2 GHz. (3) Coil former and crystal holder. (4) Nuclear resonance coil. (5) LMN crystal. (6) I microwave choke.
+
the microwave field; then the Q-value of the cavity is practically the same as without the presence of the coil. Fig. 16 shows such an arrangement88.
3.2.4. Magnetic Fields Until now, the main magnetic field H has been produced by standard laboratory electromagnets;cryogenic magnets and superconducting coils have not beenusedin thisarea of research.The requirements on the magnetic field Hare: a) an inhomogeneity smaller than the electronic line width in the volume References p . 446
432
A. ABRAGAM AND M. BOROHlNI
[a. =,5 3
of the sample that is, for instance, for a field of 20 kOe, a line width of 5 Oe, an inhomogeneity of about lo-'; b) a constancy in time of the same order, 10- ',to be held over long periods. 3.2.5. Large Target of Polarized Protons at Berkeley 8 3 ~ Q o
A target of LMN, 1% enriched Nd (1.5% of odd isotopes), about 2.5 x 2.5 x 2.5 cm large, with a mean polarization of 22%, has been used for the elastic scattering of 246 MeV positive pions. The target was made of 4 large single crystals with crystalline axis normal to the magnetic field ;the frequency of irradiation was in the 8 mm-wavelength range; a floating drift-tube klystron of 15 W of power fed through a horn a rectangular high-mode cavity of about 30 cm3 (Fig. 17); about one watt was dissipated in the cavity; the microwave circuit contained a directional-coupler and a direct crystal detection, in order to look at the electronic resonance. For a given u.h.f. frequency, the magnetic field was adjusted for the best nuclear polarizations. A large vertical cryostat capacity of 20 1helium, with a liquid nitrogen reservoir, a mechanical pump of 700 m3/h, were used to cool the sample at about 1.2"K; liquid helium dissipation was between 50 and 100 1 a day. The lower
Fig. 17. Example of a large many-modes resonant cavity8s, used at Berkeley to polarize 4 L M N crystals, of total volume 2.5 x 2.5 x 2.5 cms. (1) Coaxial line. (2) Nuclear resonance coil. (3) Septum, used to guide the h. f. magnetic field lines. Two crystals are located on each side of this septum. References p . 446
CH. vm,8 31
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
43 3
portion of the dewar was made of thin mylar to allow for the particles arrival and departure. The magnetic field of 9 kOe was produced by a specially-built magnet with a large gap, and widely separated coils. The nuclear polarization was measured by comparing the resonance absorption signal with and without dynamic polarization; a Q-meter with frequency scanning and field modulation was used, which continuously passed over the nuclear resonance line. The coil was made of teflon insulated wire whose shape is depicted in Fig. 17 and was designed to provide a uniform signal from the whole target: the average polarization of the target was estimated to be known with a relative accuracy of f 15% (22 f 3%). The particles detection consisted in four a-counters and two p-counters ; the outgoing a+ and p were measured in coincidence when the kinematics allowed it; a telescopewith copper foiIs was used to measure the a+-momentumwhen the p was not observable. The experiment gave the polarization of the a+-p reaction as a function of the scattering angle, but the final results are not yet available. The same target where the 8-mm klystron is replaced by a 10-watts carcinotron at 72 GHz, has given polarization of at least 50%, and is intended to be used for p-p scattering between 2 and 6 GeVQO. 3.3.
THINTARGETS
3.3.1. Thin Target of Polarized Protons at Saclay
A thin target of polarized protons has been used in Saclay for the scattering of a beam of polarized protons of 20 MeV2. The experiment was done in the following way (Fig. 18): A. General description 1. Polarized proton beam. A beam of a-particles of 44 MeV produced by a cyclotron could strike a polyethylene target of 0.1 mm, knocking out protons. Protons emitted at an angle of 24" with respect to the incoming a-particles are almost completely polarized along a direction perpendicular to the a-p scattering plane91.92. In the geometry of the experiment, this plane was horizontal and the proton beam polarization vertical and directed toward the ground with a value Pb = 98 f 2%. The intensity of the beam was 1.5 x 106/cm2/sec and the energy 20 MeV with a spread of 1.4 MeV. This beam was focussed unto the centre of an electro-magnet where was located the polarized target. 2. Polarized target. The target was a single crystal of LMN containing 0.3% of cerium; the protons of its water of hydration were polarized dynamically in a vertical direction by solid effect: pt = f (20 f I>%. The References p . 446
434
A.ABRAQAMAND M. BORQHIN
ta.vII1,83
crystal located in a microwave cavity resonating at 35 GHz, was cooled, in vacuum, at a temperature near 1.6"K, in the vertical magnetic field H of the electro-magnet, with H = 13 kOe. 3. Particles detection. After the scattering, the two outgoing protons were counted in coincidence in two large angle CsI crystals located in the horizontal plane, at 45" on each side of the axis of the incoming proton beam. Two coincident particles were counted only if the energy of each of them
Fig. 18. General scheme of the 20 MeV p-p scattering done at Saclay 2,
was above 1.5 MeV and if their s u m was above 10 MeV, thus discriminating against spurious scattering events on nuclei of the target other than protons. The measurements were done in succession with upward and downward polarizations during periods comprised between half an hour and one hour, 4. Results of the measurements. The quantity measured was the spin-spin correlation coefficient CnnB3.In this experiment, where the polarizations Pb of the beam andp, of the target were parallel to each other and normal to the beam direction, the cross section of the p-p scatteringwas given by tspo1. = C T , , ~(1 ~ . pt'pb'Cn,). The result of the measurements was: C,, = - 0.91 f 0.05; this figure is possibly subject to a recalibrationto be described later on. This experiment has certain special features which warrant a more detailed description. These features are briefly outlined below, a) The fact, that this is the fist experiment of nuclear scattering performed on a polarized proton target and that the details of the experiment are available to the authors of this review at first hand, is not perhaps a very cogent reason for indulging in a detailed description.
+
References p . 446
m.vn2(%31
DYNAMIC POLARWATION OF NUCLEAR TARGETS
435
b) A more important consideration is the fundamental character of the measurement of the C,,,coefficient which provides direct and unambiguous information on the nature of the nuclear forces94. It turns out that this information is both more fundamental and more difficult to obtain by other means for Iow energy p-p scattering than for higher energies. c) It is precisely the low energy and short range of the incoming protons which make the experiment difficult in many respects. The target must be very thin (0.12 mm), the cavity have very thin walls or at least very thin windows, and no liquid helium is allowed to be found on the path of incoming or outgoing beams. This is to be contrasted with a normal laboratory dynamic polarization experiment performed in a standard cavity cooled inside a standard helium dewar. The difficulties in producing and measuring dynamic nuclear polarizations under these conditions were such that although proton polarizations of 20% were observed at Saclay as early as 1960, the actual scattering experiment could not be done until August 1962.
B. Target. Polarization measurement The target was located in the 45 mm gap of a Varian electromagnet, mounted with tapered pole caps, which produced a field H of about 13 kOe, stabilized to lo-’ over long periods. A chamber was adjusted between the pole caps and had apertures for the beam entry, three detectors, one of them in the
0 0 Fig. 19. Resonant cavity for 20 MeV p-p scattering. (I) Helium incoming flow. (2) Helium pumping line. (3) Microwave resonating volume. (4) LMN crystal. (5) Nuclear resonance coil. (6) Coaxial line. (7) Microwave guide. (8) Coupling iris. (9) Cshaped body of the cavity. References p . 446
436
A. ABRAGAM AND
M. BORGHINI
[CH. VIII,
83
axis of the beam, and the cryogenic apparatus of the target. The target was a plane single crystal of LMN of 4.4 x 3.0 x 0.12 mm, whose crystalline axis was normal to the plane; the field H was parallel to the plane of the crystal. The target was located in a rectangular copper cavity resonating at 35 GHz in the fundamental mode TE,,, (Fig. 19); the dimensions of the cavity were 7.12 x 3.56 x 5.40 mm. The incoming and outgoing beams passed through thin windows in the 7.12 x 3.56 walls; the first window was a foil of 3p of copper; the second was made of l p of copper 2% diffused tin, on the outside of the cavity, and of 0 . 5 ~of pure copper in the inside; these two foils were mounted on the body of the cavity by two flanges and 0 = 1 mm brass screws; the body of the cavity was made of C-shaped bulk copper, closed by a platinum iris of 3.56 x 5.40 x 0.3 mm with a coupling hole of o = 1.7 mm, situated in front of a monel waveguide; the waveguide had a tapered transition from the standard section 3.56 x 7.12 mm to the iris section. It was connected to the microwave source (40 mW reflex klystron) through a magic tee. The klystron frequency was stabilized on a reference high-temperature-compensatedcavity, by means of a small 450 kHz frequency modulation and a phase-coherent detection of a signal provided by this cavity. The two main arms of the C-shaped copper each had a ledge of 0.3 mm depth, 0.5 mm width, in order to receive the crystal which was glued unto them, parallel to the exit window, without touching it. The glue which was of the freon type linked the crystal mechanically and thermally to the C-shaped body of the cavity. This body was cooled to approximately 1.6"K by liquid He4 flowing inside one of the two arms of the C; the cryogenics are described in the next section. An inductance coil, made of a single rectangular turn of copper wire, of = 0.1 mm, was placed inside the resonant cavity, parallel to the plane of the crystal, just behind it. It was connected through a coaxial line ending in the other arm of the body of the cavity to a Q-meter circuit. This coil could be used to observe the resonance signal of the protons of the crystal, or to destroy by saturation their polarization: as the signal-to-noise ratio of the natural signal was bad (between 3 and 5), the usual comparison between natural and enhanced signals could not be used to measure accurately the polarization; therefore an alternative method, described below, was used to measure directly the enhanced polarization. The Sn impurity in the copper foil was intended to increase the resistivity of the foil and thus allow the magnetic field of the coil to extend through it nearly freely. The maximum polarizations were obtained with a microwave power P in the cavity of about 3 mW, corresponding to a maximum field HI of 150
+
References p . 446
CH. MI,fi 31
DYNAMIC POLARIzAnoN OF NUCLEAR TARGETS
437
mOe. If this power was steadilyincreased, the resonant absorption by the electronic spins,even at the distance w,from the centre of the electronic line, produced for a certain level P, an abrupt change in the temperature of the sample rising tovalues suchthat no dynamicpolarization could be obtained anymore; upon reduction of the applied power, the sample temperature showed an hys- ' tereticbehaviour, and a sudden drop in temperature occurred at a level Pz
Fig. 20. Typical signal of the Lorentz field measuremente5. (a) The protons are being polarized then the magnetic field U is brought to the electronic resonance line centre value. (b) The proton polarization is reduced to zero by saturation with an intense h.f. field at the proton Larmor frequency. (c) The field H is changed by some 500 mOe.
one measured the internal Lorentz field HL of the polarized protons by observing the shift that it produced on the cerium resonance line whose signal-to-noise ratio is much greater (see Fig. 20). At the end of a period of dynamic polarization in a field of 13 290 k 21 Oe, the field H was suddenly moved back to the resonance value of 13 290 Oe, after a sharp reduction by 20 dB of the microwave level. This reduction prevented a heating of the crystal which would have shortened the nuclear relaxation time too much; in the centre of the electronic resonance line, the slope of the recorded signal (which is the derivative of electronic absorption) is largest and the sensitivity is maximum. The proton magnetization shifted then the electronic resonance line by an amount equal to HL.This shift decreased toward zero with the proton relaxation time constant T,,and was recorded; one could also destroy quickly this shift by applying an intense r.f. field Hi at the nuclear resonance frequency. The sign and magnitude of the change that this destruction produced in the electronic signal was then compared to a change in the signal due to a known variation of the external magnetic field H. The signal being a linear function of field over a region large compared to HL, HL could be References p . 446
measured in that way. The Lorentz field HL is related to the proton magnetization by HL = kM;in LMN for 100% polarization, M = 0.532 Oe (for a density d = 1.99)'. The numerical constant k has been taken equal to 4 n to obtain the value of C,, quoted above. As the sample was a thin plate, this would be correct if the proton environment of the cerium ions had cubic symmetry; a recent X-ray analysis48 of the actual structure and a calculation06 of the static field created on a cerium ion by its 561 nearest neighbouring protons, supposed to be all equally polarized, would lead to increase this value of k by 10% (if the proton polarization is in the plane of the crystal), and thus increase the value of C,, by the same amount. But, as local inhomogeneities of the nuclear polarization, especially in the immediate neighbourhood of the cerium ions, can arise from the dynamics of the polarization and of spin diffusion, the correct value of k to be used must be calibrated in a separate dynamic polarization experiment, in similar conditions but without the geometry difficulties of the target. Shifts of the electronic resonance line of the order of 0.5 Oe were measured, varying from one period of polarization to another, but constant within 1% during each period. Such shifts correspond to polarizations of about 20%, if k = 4 n; a typical signal is shown in Fig. 20; the total trace corresponds to 20 sec; during this time, the stability of the klystron frequency and of the magnetic field was better than A relative uncertainty of 3% on the polarization was assumed to take into account the transient behaviour of the polarization during the measurement.
C. Cryogenics97 The horizontal access to the centre of the magnet, its exiguity (0 = 43 mm), the necessity of cooling the target crystal in the vacuum, with less than 5 mg/cms of matter on the beam trajectories, has led to the construction of a special type of cryostat (Fig. 21): no liquid nitrogen is used; the total length between room temperature and the crystal temperature is of 30 cm; liquid helium flows continuously from a standard 10 or 25 litres container, through a vacuum isolated transfer line, into the cryostat. After a separator where the vapour produced in the transfer line is pumped away through a spiral tube which cools, down to 40°K, a copper radiation shield (with two 0.5,~ aluminium windows on the beam trajectories), the liquid is filtered against solid air dust. It is then precooled down to 1.8"K in a heat exchanger by the
t
See note p. 417.
References p . 446
a. m,El 31
DYNAMIC POLARIZATION OF NUCLEAR TARQETS
439
--c
Fig. 21. Cryostat for the 20 MeV p-p scatterings'. a (1) LMN crystal. (2) C-shaped body of the resonant cavity. (3) Waveguide. (4) Coaxial
.
line. (5) Liquid helium bath at 1.6"K. (6) Pumping line. (7) Radiation shield. (8) Magnet yoke. (9) Flange. For clarity, the cavity, on the right of (a-b), has been rotated by 90" around the axis of the apparatus. b. (1) Helium transfer line. (2) Phase separator. (3) Filter. (4) Heat exchanger. (5) Expansion needle valve seat. (6)Copper tube Oint. = 0.4 mm. (7) Spiral tube, used to cool the radiation shield (A.7).
helium vapors coming from the bath, then cooled to a lower temperature T by expansion through a tungsten needle valve (0 = 1 mm, 3 0 = 0.5"). Finally it arrives, through a copper tube (0 = 0.4 mm) in a blind cylindrical hole (L= 20 mm,@ = 4mm) inside one of the arms of the Gshaped body of the cavity; the cavity cools the crystal at the edges through the glue used to iix it ;helium is then pumped back by a 25 m 3/h mechanical pump through a stainless steal pumping line of increasing diameter which envelops all the preceding devices, including part of the transfer line. A stainless steel flange, with brass screws and a 0.1 mm thick teflon O-ring, is mounted on the narrower part of the pumping line which connectsit to the cavity, and allows an easy replacement of the cavity. The overall dissipation of the cryostat is of 0.5 l/hof liquid helium, 0.1 1of which being used to cool the cavity. In this experiment the temperature T of the liquid helium bath inside the cavity wall was of 1.6"K, and did not fluctuate nor shift by more than 0.01"K during periods of hours. Since then temperatures of 1.25"K have been obtained in this cryostat 98. References p. 446
440
A. ABRAGAM A N D
M. BORGHIM
[CH.Wr, 8 4
3.3.2. Thin Target of PoZarized Protons at HarweZP8 A target of polarized protons is under construction at Harwell, which will be used to measure the spin correlation coefficient C,, in p-p scattering at 142 MeV, with the synchrocyclotron, at 60" and 90" (centre-of-mass). The target will be a single crystal of LMN, 1% enriched Nd, of about 3.5 x 7.1 x 0.5mm; it will be cooled in the vacuum and glued on the walls of a microwave cavity resonating at 35 GHz (Fig. 16). The entrance and exit walls of the cavity will be of about 25,u thick. This apparatus has been tried in laboratory conditions, with liquid helium inside the cavity and around the crystal, and, with a microwave power of 4 mW, enhancements of the nuclear resonance signal by a factor of 460 were obtained at 1.37"K, corresponding to 31% polarization. In the arrangement which will be used for the nuclear scattering experiment, because of an expected poor signal-to-noise ratio of the unenhanced nuclear signal, the polarization will be measured by observing only the enhanced signal, calibrating it with a beam of unpolarized protons, using published figures for p-p polarizations. 4. Future Developments
In a subject moving so fast it would be rash to attempt to predict future developments, However a few experiments that are known to the authors of this review to be in an active state of preparation (among probably many others) can be mentioned. At Berkeley p p scattering of high energy (several GeV) protons on a polarized target is being currently studied. At Harwell a similar experiment is being planned in the 150 MeV range (as already mentioned earlier). At Saclay the low energy p p scattering is being repeated with improved accuracy and a more versatile geometry. Scattering of polarized deuterons, for which a very efficient source is now availableDQJ00,on polarized protons is also being considered there. At CERN an important experiment involving the reaction a+ p + Z+ + K + intended to measure the Z/K parity is being prepared using a polarized proton target manufactured at Saclay. The measurement of the parity of the 5- particle should follow.
+
4.1. NUCLEIOTHER THAN PROTONS With respect to nuclear experiments with targets using other nuclei than References p . 446
a. m,o 41
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
441
protons (or deuterons) it is clearly up to nuclear physicists to decide whether a specific experiment with a polarized target of a given nuclear species is worthwhile attempting. It would be futile to attempt polarizing nuclei in the hope that some day some nuclear physicist might be tempted. The example of low energy p-p scattering has shown abundantly how much of the target construction depends on the specific nuclear physics experiment. It may be worth pointing out that He3 nuclei in the gaseous phase and therefore at a very low density have been polarized by charge exchange with metastable helium atoms 101, themselves polarized by optical pumping lo2 and that this orientation has been demonstrated by a nuclear scatteringlO3. Among nuclei that could be made available for polarized targets we have already mentioned Fl9 for which polarizations of 5% were claimed and higher ones could probably be achieved by solid effect. Li 7 and Naz3 could be polarized by the Overhauser effect lo4*l O5 in the metal. In Li 7 in particular, polarizations of the order of 70% were obtained at 4 mm and 1.5OK106. These polarizations cannot be used directly for nuclear scattering, for the small metallic particles used in the experiments were formed inside an environment of respectively, sodium hydride NaH and lithium fluoride LiF. The proportion of polarized Li7 or Fl9 in the metal is thus small compared to unpolarized nuclei of surrounding ionic lithium or sodium and the overall nuclear polarization is small. Bulk samples or thin films of sodium and lithium have not been so far prepared with a purity sufficient to exhibit very narrow electron lines that would permit large Overhauser polarizations. An effort in that direction could no doubt be made if sufficient interest was displayed by nuclear physicists. 4.2. TARGET SIZE AND
NON-RESONANT
DYNAMIC METHODS
It has sometimes been said that large targets are incompatible with large polarizations : the latter, it was argued, imply high microwave frequencies, short wavelengths and therefore small microwave cavities that could not house large samples. The fallacy of this statement was first demonstrated by Schmugge and Jeffries45, who, by using multimode cavities, were able to polarize samples with linear dimensions much larger than the microwave wavelength and thus paved the way to polarized targets for high energy physics. It is still true however that the need for a complicated (and expensive) microwave apparatus involving the dissipation of large quantities of liquid helium is a liability and it was natural to look for dynamic methods that would either require no microwaves107J08 at all, or perhaps a pumping References p . 446
442
A. ABRAGAM AND M.BOROHIM
[CH.VIII,
84
frequency much lower than the one used in the conventional solid effectlo8. The common principle of these methods is the following: Consider a system made of three parts: a thermal bath (the lattice), a cooling fluid (the electron spins) and the system to be cooled (the nuclei). The cooling cycle is as follows. The fluid is brought to the temperature of the bath and then disconnected from it. Cooling to a much lower temperature by an adiabatic change in the external parameters of the system then takes place after which the fluid is connected to the system to be cooled, warming irreversibly in the process. It is then disconnected from the system to be cooled, the external parameters are brought back to their initial values, contact with the thermal bath is reestablished, and the cycle starts again. Assume at first that after the electrons (and the nuclei) have been brought to an equilibrium temperature Toin a high field Ho further contact with the lattice can be disregarded, either because the heat capacity of the lattice is negligible or because the electron spin lattice relaxation time T, is sufficiently long (the much longer nuclear relaxation time T,will be assumed infinitefor simplicity). Because of their widely different Larmor frequencies cot and co: in the field Ho, electrons and nuclei are not on “speaking terms” and any exchange of energy between the two systems will be extremely slow. If by some means, to be discussed presently, the electron frequency me can be reduced adiabaticalIy in a h i t e field H*to a much smaller value co: such that the difference co: - co: is comparable with the electron line-width wL(which can be either larger or smaller than 03, two things will happen. Firstly, the electron temperature will be reduced to a much lower value T k: To WJW; or T % To w&$ if coi > wL. Secondly, electrons and nuclei will now be “on speaking terms” and in a time z, which we shall call the mixing time, will come to a common temperature, which will depend on the relative values of their heat capacities in the ‘‘mixing” field He, One of the ways in which the electron frequency w ecan be reduced from a large value 0: to a small value co‘, in a finite field H*, is the following107Jo8 (other schemes are described in ref.108). If for the paramagnetic spins considered the gyromagnetic ratio g becomes very small for a certain orientation of the crystal with respect to the magnetic field, a rotation of the crystal into that orientation will achieve our purpose. Protons have actually been cooled in this manner in a crystal of ceriummagnesium nitrate 49. This method so far is still incomplete because when the thermal mixing occurs the nuclear heat capacity is very much larger than that References p . 446
m,8 41
DYNAMIC FOLARIZAnoN OF NUCLEAR TARGETS
443
of the few electron-spins and the amount of cooling achieved is small. It is possible in principle to improve on this "one-shot" cooling by noticing that it is precisely when the electrons are at their coldest that they come into thermal contact with the nuclei. Altering the orientation of the crystal back to its original position before the mixing will uncouple the nuclei from the electrons. One can then rely on the electron spin-lattice relaxation mechanism to restore the now warm electrons to their original temperature To while the nuclei remain cold, and repeat the mixing procedure again and again. However, since the mixing time is not necessarily short compared with the electronic 2". the method would be clearly greatly improved if T, could be made long while the electrons are cold and coupled to the nuclei, and short when they are warm and being cooled by the lattice. This can actually be achieved in the present schemelog. For lattice temperatures of the order of 1°K in most crystals the electronic relaxation mechanism will be the direct process (absorption or emission of a single phonon). For an electronic Kramers doublet the relaxation rate will then vary as7
where 8 defines the orientation of the field with respect to the crystal axes and # (0) is a function of the orientation which will depend on the detailed structure of the excited levels of the paramagnetic ion (or radical) concerned. Consider to be specific, cerium-magnesium nitrate. For 6 = 0 i.e. when ' the field is parallel to the crystalline Z-axis,
d o ) = Q,,
0
andT, '(0) is several orders of magnitude smallerthan, say, T ; '(3 n)(especially as it can be shown that in that particular case #(@ also becomes very small when 8 = 0). If the mixing time z turns out to be less than or about equal to T,(O),a continuous rotation of the crystal in an applied field Ho could possibly lead to a steady state nuclear spin temperature of the order of
where To is the lattice temperature. Instead of cerium double nitrate, dysprosium ethyl sulphate, where g, = 0110, could also be tried. The advantage of this scheme over current methods of dynamicpolarization is that no microwave is needed and that the large applied field Ho needs not be homogeneous.
Re..rences p . 446
444
A. ABRAGAM AND
M. BORGHINI
[CH.Vm,8 4
This polarization scheme was first tested by rotating in liquid helium an LMN crystal doped with 5% cerium111. In a field of 5 kOe an increase of the proton polarization by a factor FZ 15 was observed. Similar results were obtained112 using a slightly different scheme: the crystal was fixed with the Z-axis parallel to the field H* of the magnet. A second field H i at right angle to H’ could be put on and off by means of a pulsed coil at a controlled rate. In effect the field rather than the crystal was rotated. In fields of 2 kOe proton polarization enhancements of the order of 40 were observed. In either experiment the enhancements were found to decrease when the mixing field H* was increased, an unfortunate circumstance for the obtention of large polarizations. The explanation must be sought in the fact that gI1 of C e + + +in LMN, although much smaller than that of a free electron, is still about 8 times greater than that of the proton and the electron proton spin-spin mixing is incomplete. This drawback can be overcome in the pulsed field method at least in principle by giving to the pulsed field H i a value much larger than H*. It may also be possible to improve the mixing and increase the theoretical enhancement as well by applying, in the mixing field, bursts of r.f. at a frequency w =g ,,@H*>>wn (scheme 3 of ref. 108). In ruby, enhancements of the order of 30 in the polarization of A l 2 9 have been observed when a single crystal was rotated in liquid helium113. The principle is similar to the one outlined above and discussed in more detail in ref.lO8. During the rotation, crossings between electronic energy levels of paramagnetic Cr corresponded to very low electron spin temperatures, cooling Al 29 nuclei by electron-nuclear spin-spin coupling. A very different method, more related to the Overhauser effect than to the solid state effect is that of “hot electrons”. By injecting fast electrons into a crystal of Ins b 114 it proved possible to increase the nuclear polarikation of indium and antimony by factors reaching one hundred. The gist of the principle is that in the Overhauser effect it is essential to create a difference between the temperature of the electron spins and the “lattice” represented by the kinetic energy of the electrons. In the conventional Overhauser effect this is achieved by heating the spins of the electrons above the temperature of their kinetic energy with a microwave field. As this experiment has demonstrated, the reverse is also effective. A last method of dynamic polarization 115 which deserves to be mentioned at least for esthetical reasons is the thermal mixing in zero applied field between two spin systems S and I which have the following features : the system S has a short spin lattice relaxation and a large zero-field splitting ho,. By means of an r.f. field of frequency w w w, this system can be brought into +
References p . 446
+
+
ax.m,§ 41
DYNAMIC POLAWATION OF NUCLEAR TARGETS
445
a “rotating” (or rather oscillating) frame to a very low temperature, cooling the spins I in the laboratory frame to a comparable temperature by means of the I - S coupling. If a small field H sufficient to uncouple the spins I and
S is then put on adiabatically, the spins I are highly polarized. The most remarkable feature of this method which sets it apart from other types of dynamic polarization is that the polarization reached by the spins I can be shown to be much larger than that corresponding to the Boltzmann factor exp (hw,/kT).The theoretical derivation and the experimental verification of these results at liquid nitrogen temperature with a frequencyuJ2n = 35 MHz can be found in refel’s. Preliminary results have been observed at helium temperature and a frequency of 300 MH~115‘. Whether any of these methods will actually prove competitive with the “solid effect” remains to be seen. 4.3. TARGET MATERIALS The main defect of LMN as a polarized proton target material is the presence of a large number of nuclei other than protons. Whereas at low energy, spurious scattering by these nuclei is easily discriminated against using coincidence techniques, this becomes increasingly difficult as the energy of the beam goes up, requiring a careful kinematic analysis of each event. In particular the study of resonances in high energy physics, where three body reactions occur, seems hardly feasible at present with this material and it would be highly desirable to find others with a greater proton content. Polarizing solid hydrogen seems a formidable prospect for reasons given in Section 2; HD which is not, at least in principle, subject to the same difficulties since the ground state of the HD molecule has no orbital rotation,has not yielded any significant results so far ;plastics have proved unrewarding. Perhaps when a better understanding of the details of the solid effect process, acquired through careful laboratory experiments, will be reached, larger polarizations in these proton rich substances will become possible. Another likely candidate which does not seem to have been tried so far is LiH containing F centers whose inhomogeneous electroliic line width can be narrowed appreciably ll6 by using isotopically pure Lia. In these directions anyway seems to lie the road to progress. The authors wish to thank their many colleagues who contributed to this article by discussion, letter or by kindly making available unpublished data. They are specially grateful to Professor C. D. Jeffries who has kept them informed of his beautiful work well ahead of publication. References p . 446
446
A. ABRAGAM A N D
M.EQRGHINI
[aM .n
REFERENCES GENERAL REFERENCES ON NUCLEAR ORIENTATION E. Ambler, Progr. in Cryogenics, Vol. 2, ed. by K. Mendelsohn (Heywood and Co., London, 1960) p. 1. M. J. Steedand and H. A. Tolhoek, Progr. in Low Temp. Phys. Vol. 2, ed. by C. J. Gorter (North-HoIl. Publ. Co., Amsterdam, 1957) p. 292. W. J. Huiskamp and H. A. Tolhoek, Progr. in Low Temp. Phys. Vol. 3, ed. by C. J. Gorter (North-Holl. Publ. CO.,Amstadam, 1961) p. 333. C. D. Jeffries, Progr. in Cryogenics Vol. 3, ed. by K. Mendelsohn (Heywood and Co., London, 1961) p. 129. L. D. Roberts and J. W.T. Dabbs, Ann. Rev. Nuclear Sci. 11 175, (1961). C. D. Jeffries, Dynamic Nuclear Orientation (Interscience, 1963).
TEXT REFERENCES
J. M. Daniels and J. Goldemberg, Repts. on Progr. in Phys. (1962) p. 1. A. Abragam, M. Borghini, P. Catillon, 3. Coustham, P. Roubeau and J. Thirion, Phys. Letters 2, 310 (1962). 8 C. Schdtz, G. Shapiro, W. Troka, L. van Rossum, J. Arens, F.Eetz, 0. Chamberlain, H. Dost, B. Dieterle and C. D. Jeffries, Bull. Am. Phys. Soc. 8, 325 (1963); 0. Chamberlain, C. D. Jeffries, C. H. Schdtz, G.Shapiro and L.van Rossum, Phys. Letters 4,293 (1963). 4 General reviews on paramagnetic resonance theories and experimental results have been published in Repts. on Progr. in Phys.: B. Bleaney, K. W. H. Stevens, 16, 108 (1953); K. D. Bowen and J. Owen, 18, 304 (1955); J. W. Orton, 22, 204 (1959). See also J. H. van Vleck, Theory of Electric and Magnetic Susceptibilities (Clarendon Press, Oxford, 1932). 6 C. D. Jeffries, private communication. 6 A. H. Cooke, H. J. Duffus and W. P. Wolf, Phil. Mag. 44,623 (1953). 7 J. H. van VIeck, Phys. Rev. 57,426 (1940). 8 J. H. van Vleck, Phys. Rev. 59, 724 (1941). 9 B. W. Faughnan and M.W. P. Strandberg, J. Phys. Chem. Solids 19, 155 (1961). 10 P. L. Scott and C. D. Jeffries, Phys. Rev. 127,32 (1962). l1 R. Orbach, Proc. Roy. Soc.A m , 456 (1961). 1% A. W. Overhauser, Phys. Rev. 92,411 (1953). 18 C. D. Jeffries, Phys. Rev. 106, 164(1957). 14 A. Abragam and W. G. Proctor, Compt. Rend. 246,2253 (1958). l6 A. Abragam, Pbys. Rev. 98, 1729 (1955). 16 N. Bloembergen, Physica 15,386 (1949). 3 7 G. R. Kutsischvili, Publ. Georgian Inst. Sci. 4,3 (1956). 18 P. G. de Gennes, J. Phys. Chem. Solids 3,345 (1958). 19 W. E. Blumberg, Phys. Rev. 119,79 (1960). 20 G. R. Kutsischvili, J. Exptl. Theoret. Phys. (LJSSR) 42, 1312 (1961); Soviet Physics JETP 15,909 (1962). 21 0.Leifson and C.D. Jeffries, Phys. Rev. 122, 1781 (1961). 22 H. G. Beljers, L. van der Kint and J. S. van Wieringen, Phys. Rev. M, 1683 (1954); A. Abragam, A. Landesman and J. M. Winter, Compt. Rend. 246, 1849 (1958); R. H. Webb, Phys. Rev. Letters 6, 611 (1961). 1 2
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48
49
448
A. ABRAGAMAND M. BORGHINI
[CH.Vm
L. H. Piette, R. C. Rempel, H. E. Weaver and J. M. Flournoy, J. Chem Phys 30, 1623 (1959). 7 0 G. A. Rebka Jr. and M. Wayne, Bull. Am. Phys. Soc. 7,538 (1962). 1' M. Sharnoff, J. T. Sanderson and R. V. Pound, Bull. Am. Phys. SOC.7, 538 (1962). 72 Susan Read, Thesis Oxford (1962). 3 ' J. M. Baker, W. Hayes and D. A. Jones, Proc. Phys. SOC.73, 942 (1959). 74 J. M. Baker, W. Hayes and M. C. M. O'Brien, Proc. Roy. SOC.A 254,273 (1960). 75 See H. van Dijk and M. Durieux, Progr. in Low Temp. Phys. Vol. 2, ed. by C. J. Gorter (North-Holl. Publ. Co., Amsterdam, 1957) p. 431. 76 See E. R. Grilly and E. F. Hammel, Progr. in Low Temp. Phys. Vol. 3, ed. by C. J. Gorter (North-Holl. Publ. Co., Amsterdam, 1961) p. 113. 77 See K. W. Taconis, Progr. in Low Temp. Phys. Vol. 3, ed. by C. J. Gorter (NorthHoll. Publ. Co., Amsterdam, 1961) p. 153. 78 R. V. Pound, M. I. T. Radiation Laboratory Series 11(McGraw Hill, 1947) p. 58. 79 A. F. Harvey, Microwave Engineering (Academic Press, 1963). 80 F. A. Benson and D. H. Steven, Proc. I.E.E. 110, 1008 (1963). 81 See Proc. Symp. on Millimeter Waves (Polytechnic Press, Brooklyn, 1959). 82 M. A. H. Mac Causland, Thesis Oxford (1959). 8s 0. Chamberlain, C. D. Jeffries, C. Schultz and G. Shapiro, Bull. Am. Phys. SOC. 8, 38 (1963). *4 See E. R. Andrew, Nuclear Magnetic Resonance (Cambridge University Press, 1955). 85 F. Bloch, W. W. Hansen and M. E. Packard, Phys. Rev. 69,127 (1946); 70,474 (1946). 86 R. V. Pound, Phys. Rev. 72, 527 (1947); Progr. Nucl. Phys. 2, 21 (1952). 87 F. N. H. Robinson, J. Sci. Instr. 36,481 (1959). 88 J. Combrisson and I. Solomon, J. Phys. Radium 20, 683 (1959). 88 T. W. P. Brogden, private communication. 90 C. H. Schultz, private communication. 91 G. C. Phillips and P. D. Miller, Compt. Rend. Congr. Int. Phys. Nucl. (Dunod, Paris, 1959) p. 522. 9 2 K. W. BrockmanJr., Compt. Rend. Congr. Int. Phys. Nucl. (Dunod, Paris, 1959) p. 350. 93 L. Wolfenstein, Ann. Rev. Nucl. Sci. 6, 43 (1956). 94 J. Raynal, Nucl. Phys. 28, 220 (1961), J. Phys. Radium 22, 560 (1961). 85 A. Abragam, M. Borghini and M. Chapellier, Compt. Rend. 255, 1343 (1962). 95 N. Ford, private communication from C. D. Jeffries. 97 P. Roubeau, to be published. 88 P. Roubeau, private communication. 99 R. Beurtey, R. Chaminade, A. Falcoz, R. Maillard, T. Mikumo, A. Papineau, L. Schecter and J. Thirion, J. Phys. to be published. 100 R. Beurtey, Thesis Paris (1963). 101 L. D. Schearer, F. D. Colegrove and G. K. Walters, Phys. Rev. Letters 10, 108 (1963). 102 A. Kastler, J. Phys. Radium 11, 255 (1950). 103 G. C. Phillips, R. R. Perry, P. M. Windham, G. K. Walters, L. D. Schearer and F. D. Colegrove, Phys. Rev. Letters 9, 905 (1962). 104 C. Ryter, Phys. Rev. Letters 5, 10 (1960); Physics Letters 4, 69 (1963). 105 R. T. Schumacher and J. L. Hall, Phys. Rev. 125,428 (1962). 108 C. Robert, private communication. 107 C. D. Jeffries, Cryogenics 3,41 (1963). lo8 A. Abragam, Cryogenics 3, 42 (1963). lo9 R. Orbach, Phys. Rev. 126, 1349 (1962). 110 R. J. Elliott and K. W. H. Stevens, Proc. Roy. SOC.A219, 387 (1953). 111 F. N. H. Robinson, Physics Letters 4, 180 (1963). 1~ J. Combrisson, J. Jkatty and A. Abragam, Compt. Rend. 257,3860 (1963). e9
CH. VIE] 118 114
115 118 117
DYNAMIC POLARIZATION OF NUCLEAR TARGETS
W. G.Clark, G. Feher and M. Weger, Bull. Am. Phys. SOC.8,468 (1963). G.Clark and G. Feher, Bull. Am. Phys. SOC.7, 613 (1962). M.Goldman and A. Landesman, Phys. Rev. 132,610(1963); M. Goldman, private communication. F. E.Pretzel, D. T. Vier and E. G. Szklasz, Phys. Chem. Solids 19, 139 (1961). J. A. Cowen and D. E. Kaplan, Phys. Rev. 124, 1098 (1961).
449
CHAPTER IX
THERMAL EXPANSION OF SOLIDS BY
J. G. COLLINS
AND
G. K. WHITE
C.S.I.R.O. DIVISION OF PHYSICS, SYDNEY,AUSTRALJA
CONTENTS: 1. Introduction, 450 - 2. Theory, 451. - 3. Experimental methods, 457. - 4.
Dielectricsolids,460. - 5. Metals, 464. - 6. Glasses and diamond-structure solids, 471. 7. Superconductors, 473. - 8. Summary, 476.
1. Introduction Thermal expansion is the dimensional change which occurs with change in temperature. It is clearly related to dilatation which may result from altering other parameters such as pressure, magnetic field, etc., and it reveals something about the dependence on volume of the energies of various interaction processes in solids. Many years ago Griineisenl showed that the coefficient of thermal expansion for a Debye solid should be approximately proportional to the specific heat. He and his collaborators proved experimentally that, for a large number of solids, this was true, at least down to temperatures as low as about 30,f3 being the Debye characteristic temperature. However, it was not possible to detect very small length changes with dilatometers in use at the time or by X-ray methods, so that expansion coefficients were not determined down to temperatures in the region of particular interest below The era since World War I1 has seen greatly increased cryogenic facilities and interest in materials. Measurements on copper by Bijl and Pullan 2, and by Rubin et al.3, each suggested that Griineisen’s rule was not well obeyed at temperatures at and below &O, although the sensitivity of the measurements still left details rather uncertain and there were obvious discrepancies between the two sets of observations. Theoreticians such as Barron4. and Blackmans-8 investigated models of the lattice and predicted departures from Griineisen’s rule (Born9 and Dayall0 had previously predicted this
he.
References p. 477
450
m, 8 21
THERMAL EWANSION OF SOLIDS
451
for the alkali halides). Physicists have also become interested in determining volume changes in the superconducting -+ normal transition in some metals. The general result has been the devising of much more sensitive methods of measurement: optical leverll. 12, 3-terminal capacitance1Sy differential transformerl4, as well as some improvements in the older interferometric method. By detecting length changes of the order of 10-8 cm, expansion coefficients have now been determined, with varying degrees of certainty, down to temperatures near &8. Among other things, this has allowed the direct determination of the electronic contribution to the thermal expansion in a number of metals16. Already considerable data have been obtained below the temperatures of &8 with these techniques; this brief review is an attempt to correlate and discuss such data. 2. Theory
2.1.
GRUNEISEN y
Thermal expansion is due to anharmonicity in the potential energy of a crystal. A perfectly harmonic crystal has potential energy proportional to the second power of the ionic displacements only, and would show no thermal expansion. Such a system is mechanically unstable5, and no such crystals occur. Little is known about the specificvalues of anharmonic terms in a particular lattice, and thermal expansion is usually studied within a quasi-harmonic approximation, which neglects anharmonic terms but instead allows the interaction constants of the harmonic theory to be volume dependent (e.g.ls*17). The coefficient of volume expansion of a solid is
p='(">v aT p , and the isothermal compressibility
xT=-L(!!)f v ap
T
It follows from thermodynamics that fi/xT
= (aS/aV), = - a2F/aVaT,
(1)
which relates thermal dilatation to changes in the entropy S or the Helmholtz free energy F of the system. Since F and S are additive functions of state, this Rcfcences p . 477
452
[CH.M,5 2
J. 0. COLLINS AND 0. K.WHITE
formulation embraces contributions to the thermal expansion not only from the lattice but also from conductionelectrons,and from magnetic interactions. For the moment we are concerned only with the lattice; other contributions to the thermal expansion are discussed in 0 2.4. In the quasi-harmonic approximation the lattice of N ions is represented by a system of 3N loosely-coupled harmonic oscillators. The free energy of the system is 3N
3N
F ( K T ) = U ( V ) +C$Fio,(V) I=1
configurational
+ k T z l n ( 1 - e-fim*’kT), i=1
zero point
(2)
thermal
where o1is the frequency of the ithlattice mode. The Griineisen relation
follows from equations (1) and (2) if 3N
3N
Here
ahai
yi
=
-( dlnv),’
and C,is the contribution to the specific heat C, from the ithmode exp (ti o,/kT) C,= k(hm,) k T [exp (hwi/kT)- 112 ’ At high temperatures (T>> e), all modes contribute equally to the specific heat, and y = y m is the simple average
For many solids y is approximately constant at high temperatures with a numerical value of about two. At very low temperatures ( T << a), where the specific heat obeys a Debye Ta-law, y approaches a limiting value 3N
3N
where vi is the velocity of the ithmode. In this limit we can treat the lattice as an elastic continuum. If we assign to each mode a characteristic temperature di = (h/k)qmoi,where qm is the maximum allowed wave-vector for the References p . 477
CH.
=,I 21
THERMAL EXPANSION OP SOLIDS
453
continuum, we can rewrite eq. (7a) as
Here 3N
e i 3= ( 3 ~ ) - l C e ; ~ ; i=l
Bo is the familiar Debye temperature calculated from the velocity of elastic waves at T = 0. The limiting values yo and ym are two of a set of general averages of the yt defined by y(s) = Cyiv;/Cv: = - (1/s)(a1n7/aln v), i
i
where ?is the sth moment of the frequency distribution, and is well known in the theory of specific heats. In this notation y ( - 3) = yo, and y(0) = y m (see Barron et ~ 1 . for 1 ~ details). Although y is the ratio between an averaged anharmonic and an averaged harmonic property of a solid, it is not a unique index of crystal anharmonicity. A corresponding parameter occurs, for example, in the Mie-Griineisen equation of state*. Slaterlg also calculated an index ys from the variation of isothermal compressibility with volume
while yet another y has been introduced into the theory of lattice conduction as a measure of the strength of phonon-phonon couplingls. Although each of these parameters is proportional to the “strength” of anharmonic forces in a crystal, they are not in general the same quantity20.
-
2.2 THEORETICAL MODELS
The calculation of y from lattice dynamical models is fraught with difficulties and uncertainties due to lack of detailed knowledge of the inter-particle forces and of their volume dependence in a solid. Born9 and Dayall0 first showed that y should be temperature dependent in calculations for ionic models with the rocksalt structure. Born’s potential consisted of Coulomb plus repulsive terms between all ions, and, although he made some drastic approximations5, his result that ym 21 2.2, yo N 1.5 is typical of more recent calculations for similar models. Barron * has considered a face-centred-cubic solid with a central-force potential between nearest neighbours only. Without specifying the potential further (thus precluding a calculation of the absolute References p . 477
454
I. 0. COLLINS AND 0. K. WHITE
m.m,9 2
magnitude of y) he showed that for this model ym - yo H 0.3. At the same time his high and low temperature expansions for y indicated that any variation of y with temperature would occur around T/BM 0.2. Barron4 also used a Mie-Lennard-Jones “6-12” potential in the central-force model to calculate y for inert-gas solids. He found that ym = 3, y m - yo ‘v 0.15 when interactions between all neighbours were included. This is rather less than the difference predicted by the nearest-neighbour calculation. Both Barrons and Blackmanet have studied the rocksalt structure using a potential V(r) = Ze2/r + A/#‘.
*
The Coulomb interactions were taken between all ions, and the repulsive terms were taken variously between nearest-neighbours or between all neighbours. Although the calculations differed in detail, for values of n between about 8 and 10, they gave yoo between 1.5 and 2.0 with a drop of about 25 % to yo. The models were almost isotropic in their elastic behaviour, but individual values of yi varied widely (- 0.5 < yi < 3.5). The negative values for yt came from shear modes associated with the modulus 1244, indicating that increasing pressure decreased the shear stiffness in certain directions in the crystal. This led Blackman? to consider the possibility of negative dilatation arising from purely vibrational motion. From a study of ionic lattices with the zinc blende and the rocksalt structures he suggested that a negative dilatation was theoretically possible, and would be favoured by a loose-packed lattice and by a very low shear modulus 1244. A more phenomenological approach was used by Horton21 to study the thermal expansion of copper. He introduced four independent force constants to describe central-force interactions between nearest and next nearest neighbours, and expressed y in terms of these constants and their volume derivatives. The parameters were determined from measured values of the elastic moduli and their temperature variation, which was assumed to come solely from the volume change of the lattice. Horton then evaluated y and found reasonable agreement at all temperatures with values obtained from recent expansion measurements3122. Ganesan and Srinivasan set up models using non-central forces for the ionicsolids MgO23 and CaF,24. The force constants were fixed by experimental values for the elastic moduli, and for Raman and infrared frequencies; the volume derivatives of the force constants were estimated indirectly. In both cases they predicted a rise in y with falling temperature. SheardzS,,and later CoUins26, treated a solid as an anisotropic, elastic References p . 477
THERMAL EXPANSION OF SOLIDS
455
continuum, and calculated y from measured values of the pressure derivatives of the elastic constants. These are related to the volume dependence of the lattice modes by Yi=-4+
(a
ci/aP)T
. Y
2XT
the yi were averaged over the lattice spectrum of the anisotropic continuum using eqs. (4) and (5). The model, which is based on the method of long waves, ignores both dispersion and the discrete structure of the lattice, and cannot represent adequately the behaviour of high frequency modes. It has, however, produced values of y ( T ) which are in reasonable agreement with experiment for a number of metals and ionic solids. Daniels27 has pointed out that yo can be calculated directly from eq. (7b) by differentiatingthe expression for Bo given by de Launay28 and introducing experimentalvalues for ctj and acij/aP.This method is a variant of Sheard’s, and it yields the same values for yo.
2.3. ANISOTROPIC MATERIALS The thermal dilatation of anisotropic materials is specified by the expansion tensor ail. This is defined as the temperature derivative of the elastic strain tensor qij, measured at constant stress nil, i.e. ail = (8 ?,a /
T),*
This is related to derivatives of the free energy (2) by
vaij= - a 2 F / a O @ -
c c sij&,y&’cr, 3N
=
3
r = l kl=l
where
Y&) =(- alnwr/a~~& expresses the dependence of lattice frequency on strain, Cr is the Einstein function (5) for the rthmode, and s is the fourth-order elastic compliance tensor for the lattice. By analogy with the isotropic case (4) an averaged strain derivative can be defined as
so that 3 kl=l
References p . 477
456
J. G. COLLINS AND G. K. WHITE
=,Q 2
[m.
Eq. (8) gives a linear relation between components of the expansion tensor and the specific heat for any class of crystal. Since a is a symmetric tensor it may be referred to principal axes; in the Voigt notation, the principal coefficients are given by 6
VaJC, = C s U y j , j= 1
i =1,2,3.
(9)
In general the at are related to the variation of lattice frequencies under both tensile and shear strains. But for cubic, hexagonal, orthorhombic and some tetragonal crystals the compliances sii( j = 4, 5, 6) in (9) are zero, and the principal coefficients are unaffected by changes in the frequency spectrum due to shearing. The volume coefficient of expansion p is the trace of the expansion tensor, so that
When sij ( j= 4,5,6) = 0, a single Gruneisen parameter can be defined as the average value of yi weighted according to the linear compressibility along the i& axis,
Griineisen and Goensz8 first studied the thermal expansion of anisotropic materials, and they developed a special case of eq. (9) for the two principal coefficients of expansion of the hexagonal metals Zn and Cd. There do not appear to have been any attempts to calculate y theoretically for non-cubic materials, probably because of a lack of stimulus from experiment. Sheard’s method cannot yet be applied to anisotropic materials since no data are available for the variation of elastic moduli with uni-axialstress. 2.4. ELECTRONIC AND MAGNETIC CONTRIBUTIONSTO THE THERMAL EXPANSION
The conduction electrons in a metal give rise to a measurable expansion at low temperatures30-32. This can be interpreted on a naive kinetic picture as a change in volume of the lattice required to maintain a constant electron “pressure” when the temperature changes. More precisely, the entropy of the conduction electrons S, = $ n 2 k 2 T V N ( E F ) , References p . 477
CH. M,8 31
THERMAL EXPANSION OF SOLIDS
457
where N(EF) is the total density of states at the Fermi energy per unit volume This can be exof crystal, gives rise to a volume expansion coefficient /Ie. pressed in terms of the electronicspecific heat Ceand an electronic Gruneisen parameter ye as P e = Ye C e XTIV, with ye = 1 (d In N (EF)/a In V)T .
+
-
The expansion term Pe should be comparable in magnitude with the lattice since , at these temperatures C, C,. term below &I For a Fermi surface S(EF) of arbitrary shape, the density of states is
A change in the volume of the crystal has two effects on N (EF): it changes the position of the Fermi level EF due to the variation in dimensions of the unit cell, and it changes the shape of the energy surfaces if the coupling between the conduction electrons and the lattice varies. Only the geometrical effect operates for free electrons and ye = 3, but real metals show a wide variation from this figure (Table 2). Apart from the formal analysis of (a In N / a h V )by Varley32, no attempts have been made to evaluate ye for real metals. If entropy Smis contributed by magnetic interaction, it will give rise to an additional term Pm in the expansion coefficient (1). Provided that the magnetic entropy can be expressed in the form F ( E m / T ) ,where Em is an interaction energy, p,,, can be related to the magnetic contribution to the specific heat Cmby a “magnetic Gruneisen parameter” ym33, where
3. Experimental Methods 3.1.
EXPERIMENTAL TECHNIQUES
Techniques which have been applied to determining expansion coefficients at low temperatures include the following (roughly in order of increasing sensitivity): a) X-ray diffractio113~-3~: Changes in lattice parameter a have been which have measured with standard deviations in Aa/a of about 4 x allowed the determination of expansion coefficients of Cu, Al and KCI References p. 477
458
J. G. COLLINS AND G. K. WHITE
[CH.IX,8
3
down to temperatures T 1:h e ; here probable errors reach a few per cent so that this is the useful lower limit of X-ray methods. b) 2-terminal electrical capacitance2137: A length change causes an alteration in a 2-tenninal capacitance. This is part of an oscillator circuit whose resonant frequency therefore alters. Changes in resonant frequency were measured with sufficient accuracy to give a detection limit of less than cm. Using specimens 5 cm long, sensitivities of in AZ/Z were achieved 37. c) Optical interferometer39381 39: Detectionlimitshave been about lO-'cm or a few thousandths of a fringe order. The Toronto group38J9, using specimens 5 cm long, have determined changes in length with an uncertainty AZ/l, as small as 2 x lo-'. d) Optical lever119 40s 41 and optical grid12: These take various forms and, as may be judged from the pioneer work of R.V. Jones42, are capable of very great sensitivity.Those developed in Zurich for determiningisothermal length changes in the superconducting-normal transitionlf 41 and latterly for determining thermal expansionl2, have detection limits of about cm and use specimens nearly 10 cm long. e) Differential or variable transformer149 43: At Iowa State University a sensitivity of about 10-8 cm has been achieved so that with specimens 10 cm long the limit of AE/l is also about f ) 3-terminal capacitance13: This is based on advanced techniques for compariqg small 3-terminal capacitors in ratio-transformer bridges, developed largely by Thompson and co-workers44.It does not share the obvious disadvantage of the 2-terminal arrangement which includes unstable lead capacitances in the measuring circuit. In its present form capacitances of the order of 10 pF are compared with a detection limit of pF, which corresponds to a limit as small as 10- cm. Electrical sensitivities of less than lo-' pF can be achieved but whether a corresponding improvement in detection limit to 10- cm is possible remains to be seen. None of the three last named methods can normally be used at full sensitivity at room temperature, so that they have little advantage in terms of sensitivity over interferometric techniques except in the low temperature region, where the ultimate in sensitivity is required 3.2. ANALYSIS At temperatures above &8 it is usual to calculate the average linear expansion coefficient corresponding to a discrete temperature change, AT, and associated length change, AE, as a = Al/lAT. References p . 477
=.rx,B 31
l'HERM4L EXPANSION OF SOLIDS
,
459
Near the lower limits of observation, expansion coefficients are usually too small and vary too rapidly to use this analysis. Therefore it is more usual to plot a graph of 1 or 1 - lo as a function of temperature and differentiate the smooth curve graphically (or algebraically) to obtain cr(T). For normal metals at the lowest temperatures (T<&8) we expect (cf. specific heat) 45 a N aT bT3 (+ CT5 ...) , whence A l / I = A T 2 + B T 4 ( + C T 6 +...).
+
+
Graphical differentiation may be used to give u, and an a/T versus T 2 plot used to obtain coefficients a and b (see Fig. 1,upper curve). Alternatively the
Fig. 1. Analysis of typical expansion data for palladium16: Upper curve shows a/T versus Ta, lower curve shows Al/lT2 versus Ta.Analysis indicates that Y (3.7 i 0.2) X T + (4.2 f 0.3) X 10-l1T8 ( T < 8" K). (Y
l ( T )curvemay be extrapolatedto zero and values of Al/TZ= (Z-lo)/T2 then obtained and plotted against T 2 (Fig. 1 lower curve), from which coefficients A, Bare obtained; the latter is probably the more reliable method. Similarly, for dielectric solids we plot AZJT4 versus T2, from which coefficients b and c may be obtained. When sufficiently reliable data, free from any systematic errors, are available, relations of the above type are best fitted by machine computation with a least-squares method. References p . 477
460
J. G.COLLINS AND G.K. WHITE
[CH.IX,8 4
4. Dielectric Solids
4.1. IONICSOLIDS Since 1960 the alkali halides of the rocksalt structure have received particular attention in the low temperature range. This recent work (including that of Srinivasan and Salimaki at rather higher temperatures) is as follows: (i) LiF - Yates and Panter46 from 270 to near 40"K, 50(30-10"K). (ii) NaCl- Rubin et ~ ~ 1(290-20" . ~ 7 K), Meincke and Graham48(2857"K), White49150 (283,90-60,304" K), Srinivasan51(6OO-10OoK),46 (270-30" K). (3) NaI - 48 (285-7" K), 493 50 (283,90-60,304"K). (iv) KBr - 46 (270-20" K), 48 (290-7" K), 49, 50 (283, 90-60, 3 0 4 " K), Srini~asan5~ (600-100" K), Salimaki53 (670-100" K). (v) KC1 - 46 (270-30" K), 48 (290-8" K), 499 50 (283, 90-60, 30-4" K), 52 (6W100" K), 53 (670-100" K), Rubin et ~1.54(290-20" K) and Schuele36 (104, 87, 40, 30" K). (vi) KI - 46 (270-20" K), 50 (283, 90-60, 3 0 4 " K), Srinivasan55t 56 (430-140" K). (vii) RbI - 36 (175-25" K). 499
In most instances the lower limit was set by the sensitivity of the measurement, with the result that accuracy suffers near this extremity. Bearing this in mind, agreement in the case of KC1 and NaCl amongst most investigators is quite good, as evidenced by Fig. 2. In this figure, for the sake of clarity, all values of y for NaCl and KC1 have not been included; in particular, some of the most scattered points, taken near the respective authors' lower limits of observation, have been omitted. Data from47-50~54 differ generally by less than 2 % above 25" K. For LiF and RbI, the data from46 and36 respectively do not extend to the region where we expect y to have its limiting value yo. Data for KBr and KI are not included in Fig. 2 as their y-values exhibit a very similar behaviour to KC1; likewiseNa1behaves like NaCl and is omitted for the sake of clarity. In Table 1 are listed some meaa values for the linear coefficients of alkali halides; the limits are our assessment of possible inaccuracies, judged from the spread of values from various references or from stated errors. Values of a and y at 15 and 10" K are taken from Meincke and Graham48 and White49150. Limiting values of yo (see Table 1 and hatched areas in Fig. 2) are taken49 from length changes observed below 8" K using the analysis outlined above (5 3.2), i.e.plotting All T4 versus T2,obtaining coefficientB and References p . 477
CH.IX,8 41
461
THERMAL EXPANSION OF SOLIDS
comparing this with coefficient of T3 in the expression for the heat capacity. An unusual expansion anomaly has been reported for KI: Srinivasan55, using an interferometric technique, and later Viswamitra and Ramaseshan 56 using X-rays, found that values of the linear expansion coefficient did not increase monotonically with temperature between 100 and 300" K but passed
00001
002
01 2
0.2
10
2 01
rho Fig. 2. Variation of Griineisen y for representative alkali halides. Expansion data are from 048, x 47.54, A 48, 0 46, 061. v VsS.
through marked maxima and minima. Yates and Panter46found no evidence of this in their sample. Two other ionic crystals, MgO and CaF,, have been measured, largely as a result of prognostications23124 that yo might be appreciably greater than the y observed at normal temperatures (see Q 2.2). Experiments49 indicate that y decreases slightly on cooling for CaF,, the percentage decrease (see Table 1) being rather similar to that for the sodium halides; for MgO there t. appears to be little change in y between 300" K and about 20" K (w
AS)
t Note adaedinproofi CsBr (bee) also shows little change in y which equals ca. 2.1 at room temperature and at liquid helium temperatures6". References p . 477
TABLE 1 Values of linear expansion coefficient (per deg K), Griineisen parameter and elastic constants (dynes/cm8 at room temperature) for dielectric sotids
P Q w
v
eo 10" a 10s a 106 a 106 a 106a 10s a
Y Y Y Y YO YO Yo0
10-11 c44 10-11c11 10-11 cia
(" K)
LiF
NaCl
NaI
KBr
KCl
KI
RbI
CaFa
MgO
740
321
164
174
235
132
115
507
946
(285°K) 32.6 f0.4 (150°K) 19.0 k0.3 ( 80°K) 6.1 kO.1 ( 25°K) O.12afO.01 ( 15°K) 0.027&0.002 (10°K) (285°K) 1.60 ( 80°K) 1.60 ( 25'K) 1.65 &O.l ( 10°K) (Oh) 1.6s kO.1 1.89 (calc) (calc) 1.79 6.37 11.37 4.76 0.56 1.40 0.52
+
9
39.5 f0.2 45.3 kO.1 38.1 ~ t 0 . 3 36.7 f 0.2 40.5 k0.5 18.8 10.4 39.5 kO.1 33.1 f0.2 30.8 f 0.1 35.5 k0.5 32.3 f O . l 36 20.5 f O . l 31.2 &0.1 25.9 &0.2 21.6 f 0.1 29.5 k0.3 29 5.3 1.32-fO.05 7.6 k0.2 4.3 f0.1 1.7 f 0.1 7.4 f0.2 7 0.14 0.22 iO.01 2.4 10.1 0.83 iO.03 0.24 5 0.01 2.2 i O . 1 0.026 0.060 f0.002 0.67 4~0.020.16 f0.01 0.053 k 0.02 0.50 k0.03 1.90 1.52 1.57 1.72 1.47 1.45 1.49 1.43 1.80 1.43 1.67 1.45 1.34 1.40 1.oo 1.2s 0.90 1.4-1.5 0.70 0.93 0.7 1.3 1.4e 1.2 0.95 1.oa 0.36 0.34 0.41 0.93 f 0.03 0.95 fO.10 0.30f0.04 0.32 & 0.03 0.28 & 0.03 ? 1.2 f 0.2 1.09-1.23 0.314.53 0.14 1.51-1.61 1.061.37 1.25 1.28 0.73 0.505 0.63 0.276 3.37 15.5 0.364 4.83 3.01 3.46 4.03 28.9 2.71 2.54 16.4 1.27 0.91 0.58 0.66 8.57 0.45 0.407 5.3 0.26 0.24 0.145 0.15s 0.11 0.21 0.13s 0.54 t 0.34 - 0.42 -0.51 1.39 1.44 2.85 2.68 3.10 3.87 1.64 0.65
%;:A
-
P
8
f
8
P
i
-
'ii
$4 OD1
P
CH. M,$41
TWERMAL EXPANSION OF SOLIDS
463
Specific heat data used in compiling the y-values are from Morrison et Na' and K ' halides and MgO, Clusius et al. 80 for LiF and RbI, Huffman and Norwood61for CaFz.Elastic constants are from the following: LiF 62, NaCl63, NaI 64, KBr 66, KC1, KI 66, RbI 67, MgO 68, CaF,al.
~1.67-69 for
4.2. DISCUSSION OF IONIC SOLIDS
The values of y (5") for the alkali halides with rocksalt structure show two systematic trends, firstly a similarity of behaviour among various halides of each metal, and secondly, evidence of a steady increase in ym- yo going from Li' to Na' ... to Rb' salts. This apparent predominance of the alkali ion in determining inter-ionic forces has been commented on before in relation to specific heats69 and elastic constants67; for example, the ratio of elastic stiffness, C44/c11, and the elastic anisotropy, (cll - C ~ ~ ) / ; ? C ~ ~ , listed in Table 1 illustrate the dominant role of the alkali ion. The elastic behaviour can be understood qualitatively on the Born model, in which repulsive forces between ions largely determine the elastic stiffness (e.g.70971). In lithium halides the small size of the Li' ion relative to any anion means that there are comparable contributions to the compressional stiffness cll, and to both shear stiffnesses cq4 and +(cl1-cl2), from overlap between nearest neighbours, and from overlap between secondneighbour halide ions. In Na', K' and Rb' halides, however, the cation radius becomes progressively larger, and second-neighbour halide ions become more separated, resulting in both c44 and +(cl1-clZ) becoming progressively smaller relative to cll. At the same time the decrease in second-neighbour interaction leads to c44 decreasing faster than +(ell -c12), and the anisotropy increases. The thermal expansion can also be understood qualitatively on this picture. The increase in cation size - or in anisotropy - results in yi varying more widely for different directions and polarizations in the crystal, with the lowest values of y, always being associated with c4,-shear rnodesS*7 ~ 2 5 .As c44/c1 becomes relatively smaller with increase in cation size, these low values of yr are more heavily weighted at low temperatures, and yo (eq. (7a)) decreases. On the other hand, y, is a simple arithmetic average of the yr, and remains fairly close to 1.5 for all the halides. For purposes of comparison, Table 1 includes estimates of yo and y m calculated from the pressure derivatives of the elastic constants2~~26~ 36 (see 6 2.2). The crystal structure and elastic properties of MgO suggest that its expansion behaviour should closely resemble that of the Li' halides, a References p . 477
464
J. G.COLLINS AND G.K.WHITE
[a. M,4 5
conclusion also drawn by Barron et ~ 1 . ~from 2 heat capacity data. The observed expansion (Table 1) confirms this view. There is no evidence for the suggestion that yo > y m in Mg049nor in [email protected] fact, in the latter case, yo < yoc, to much the same extent as in Na' halides, as we might expect from similarity of elastic anisotropy ratios. 4.3. INERT-GAS SOLIDS
Little experimental evidence is yet available to show how y varies with temperature for the solidified inert gases, at least for temperatures TKO. Data have been obtained on argon73 and down to 20" K by X-ray methods, indicating that y has maximum values of about 2.8 and 2.5 (near 60" K) for A and Kr respectivelyt ;in each case at 20' K, which is only about $0, y has fallen to nearly 2.0. Barron's calculations suggest that y should not change by more than 0.3, and that the major variation should occur near i d . It will be of interest to see whether data below 9 0 will confirm the trend of these results. 5. Metals
5.1. ISOTROPIC ELEMENTS Table 2 lists the metallic elements for which expansion data have been obtained at temperatures below h e . References to sources of this data are given in the column listing a,. Data at 290 and 80" K are taken generally from the extensive compilation of Corruccini and Gniewekso. Copper has received more experimental attention than any other element, and as a result the behaviour of the expansion coefficient is fairly well established. As Cu is monovalent, with a small electronic heat capacity, viz. C, 21 0.69 x 10-3 TJ/g,at, we expect the electronic expansion coefficient to be small and therefore difficult to determine accurately. Two investigators43~49have established that a contains terms in T and T3 at the lowest temperatures, and that the linear term leads to a value of ye not significantly different from the free-electron value of (Table 2). In Fig. 3 are shown experimental values of y, the lattice component yl where it differs appreciably from observed y, and likely limiting values of yI (= yo) and ye. For most other elements examined, a clear-cut analysis into T- and T 3terms has been possible. Exceptions are silver and tungsten, for which a, seems to be small and experimental data are not adequate, and chromium t Note a&ed in proof: data for xenon have been reported by Eatwell and Smith (Phil. Mag. 6, 461 (1961)) and Sears and Klug (J. Chem. Phys. 37, 3002 (1962)).
References p . 477
P
TABLE2 Values of the linear expansion coefficient and Griineisen parameter for cubic metals and some polycrystalline samples of h.c.p. metals. Room temperature values, and electronic and lattice contributions at low temperatures are listed.Values at 290" and 80°K are taken generally from Corruccini and Gniewekso
b
$
5
'tl Q
Y
290"K 106 a 19.2 23.0
cu
16.7
Fe Mo
Nb
11.6 5.1 7.0
Ni Pb
12.9 29.0
Pd
11.7 8.9 6.6
Pt Ta V
W cf. Cr polycryst. Cd polycryst. co polycryst. Mg single crystal Mg polycryst. Re
7.75 4.6 31.1 12.6 25.3 25.6 6.6
T<&e lolo ae/T
9.1 11 8.1 1.5 1.7 29.5 4.3 22 25 38 20 10 37 22 11
290°K 101' q/TS
-
49
f 0.349 f 1 f 0.543 f 0.249850 f 0.5*3 f 1.5l5y49 f 0.876 f 2 77 f 10 l2 f 2 49 f 6 78 f 2 0 12 f 2 15.49 & 2 75 f 1 77 10.5 f 2 75 39 f 2 77 0.3 f 0.378 - 35 f 5 1 5 1.5 & 3 75 25 & 1 49 12 f 2 76 12 f 2 79 9 f 1 7 5
2.6 2.2 3.0 2.7 2.9 1.0 0.38 2.2 15 0.96 165 140 4.3
f 0.2 f 0.2
f 0.1 f 0.1 f 0.2 0.1 f 0.1 f 0.2 f 3 & 0.07 f 12 f20 f 0.5 5.9 f 0.5 3.5 f 0.3 3.2 f 0.5 0.7 & 0.1 0.45 & 0.1
*
-
35 f 5 0.9 f 0.1 4.2 =t 0.4 2.9 f 0.3 0.7 f 0.1
80°K
Y
Y
T<&0 YO =
Yl
Ye
-
2.4 2.3s
2.3 2.5
2.2 4=0.1 2.65 f 0.15
2.00
1.94
1.69 4= 0.05
0.7
f 0.2
1.8 f 0.1
1.7 1.7 1.6
1.4-1.6 1.6 1.5-1.6
1.5 =t0.4 1.4 & 0.3 1.3 f 0.2
2.1 1.5 1.5
=t0.2 f 0.3 k 0.2
1.86 2.65
1.7-1.8 2.68
1.65 & 0.10 2.7 f 0.2
2.0 1.7
f 0.1 f 0.5
2.3 2.5 1.7
2.3 2.5 1.5-1.7
2.3 3.0
f 0.2 f 0.3
1.5
4= 0.2
2.1 f 0.1 2.4 f 0.2 1.3 & 0.1
1.2 1.6
1.2 1.5
1.0 f 0.2 1.5 f 0.3
-
2.2 1.9 1.53 1.55 2.4
-
2.1 1.8
1.48 1.51
-
-
2.0 1.5 2.0 1.4 2.4
f 0.5 f 0.2 f 0.2 f 0.1 f 0.4
1.65 0.2 - 8.5 0.5 1.9 1.4 1.4 3.5
f 0.1 f 0.2 f 1.0 1.0 & 0.1 f 0.2 f 0.2 & 0.4
466
[a. o(., B 5
J. 0. COLJJNS A N D 0. K. WHITE
for which the lattice term ol, is dwarfed by a term in T,either electronic or magnetic in origin. Generally y or yl decreases on cooling; the amount of the decrease, y290-~0, ranges up to 0.3,but in most instances uncertainties of 0.2 still exist in the yo values listed. Only in aluminum and platinum is there any evidence of yo being larger than yzgo. For comparison with values listed in Table 2, estimates of yo and ym based on pressure derivatives of elastic
468
[CH.M,5 5
J. 0. COLLINS A N D 0. K.WHITE
which has a negative value, -7.75 x 10-13 cmZ/dyn for zinc. On warming the rapid increase in all ceases at the comparatively low temperature of 100" K because all the appropriate modes are excited; above 100" K all actually decreases as the vibrations in the direction normal to c themselves produce a cross-contraction, Magnesium is relatively isotropic, having c/a = 1.623; as expected, values of sll are only slightly greater than sg3, and recent data79 show that below 30" K [xII is larger than cyI, whereas at normal temperatures it is slightly smaller. Below 10" K, these linear coefficients for Mg can be represented by all = (1.0 & 0.2) x L X ~= (1.3
& 0.2) x
whence B = +(a11
lo-'
T t- (2.1 & 0.2) x lo-'' T 3
T + (3.3 f 0.3) x lo-" T'
+2q)
= (1.2 & 0.2) x lo-' T
+ (2.9 f 0.3) x lo-''
T'
.
The average electronic term agrees with that found for polycrystalline magnesium by A n d r e ~ 7(see ~ Table 2), but the lattice terms do not. Beryllium has c/a = 1.568, so that anisotropy is more marked, as shown in Fig. 5 , with the "softer" direction now being perpendicular to the hexad
CH.M,5 51
THWMAL EXPANSION OF SOLIDS
461
5.2. ANISOTROPIC METALS
Little startling new information has come forward concerning the anisotropic metals since the review by Childssl in 1953. New data for the expansion coefficients all (parallel to the hexad or c-axis) and aI (normal to the c-axis) of the hexagonal elements Be, Yt, Zn, TI have been obtained down to temperatures in the vicinity of & 8,but the Iimited sensitivity (All1NN 10-6) makes the data below 0 rather dubious. In Fig. 4 are data for zinc which extend
Fig. 4. Linear coefficient of thermal expansion of zincs2 (00 = 322" K) parallel (cq) and normal (aL) to the hexad axis; aL for cadmium is also shown49 (00 21 208" K).
the earlier data of Griineisen and Goens29 and agree quite well with theirs at temperatures above 50" K. Zinc and cadmium are metals whose axial ratios, c/a, are far from the ideal ($)* = 1.633 expected for hexagonal-close-packed spheres. Both crystals are extended in the c-direction (c/a = 1.86 for Zn, 1.89 for Cd) so the elastic compliance is much greater in this direction, e.g. for zinc the compliance s33 (11 to c ) is 28.4 x 10-13 cm2/dyn and sll(1 to c) is only 8.4 x 10-13. Thus atomic vibrations in this direction generally are of lower frequency and are excited first as the solid is warmed29. This causes a rapid expansion along the c-axis, accompanied by a marked contraction normal to the c-axis?, this contraction taking place via the cross-compliance sI3,
t
cf. the two-dimensional model used by Barron5 to illustrate negative expansion.
References p . 477
468
55
J. G. COLLINS AND G. K. WHITE
[CH. XX,
which has a negative value, -7.75 x cm2/dyn for zinc. On warming the rapid increase in all ceases at the comparatively low temperature of 100" K because all the appropriate modes are excited; above 100" K all actually decreases as the vibrations in the direction normal to c themselves produce a cross-contraction. Magnesium is relatively isotropic, having c/a = 1.623; as expected, values of sll are only slightly greater than s33,and recent data79 show that below 30" K all is larger than a*, whereas at normal temperatures it is slightly smaller. Below 10" K, these linear coefficients for Mg can be represented by
+ (2.1 f 0.2) x T + (3.3 f 0.3) x T
all = (1.0 f 0.2) x
aI = (1.3 whence
0.2) x
lo-" T 3 lo-" T 3
+
iii = $(aIr 2a,) = (1.2
0.2) x lo-' T
+ (2.9
0.3) x lo-" T 3 .
The average electronic term agrees with that found for polycrystalline magnesium by Andres76 (see Table 2), but the lattice terms do not. Beryllium has c/a = 1S68, so that anisotropy is more marked, as shown in Fig, 5 , with the "softer" direction now being perpendicular to the hexad
to
Fig. 5. a,
Linear expansion coefficients, q and for beryllium*2 (#o = 1160" K).
axis. As expected, a* >aII ;data are insufficient to indicate whether all might become negative below 80" K. An interesting feature in these materials is the behaviour of y. For the relatively isotropic Mg, y computed from /?= (all 2a,) is sensibly constant (Table 2 and Fig. 3). Data for Zn50882, lead to a rise in y below 35"K(&O), so that y -2.8 for T 5 20"K; for beryllium82,the contrary occurs
+
References p . 477
a. m,8 51
469
THERMAL EXPANSION OF SOLIDS
and y shows a marked decrease below & 8. These trends do seem to lie outside the range of experimental error. On the other hand, polycrystalline cadmium (Table 2) shows no such obvious trend, neither does Co (c/a= 1.621.63), nor Re (c/a= 1.615). However, internal constraints or preferred orientation might prevent the polycrystalline samples from displayingaverage characteristics of single crystals. There is an obvious need for closer study of such hexagonal-close-packed single crystals as Co, Ti, Zr, Be, as well as representatives of other anisotropic structures, e.g. Bi, Sb (rhombohedral), Sn, In (tetragonal), Se, Te (hexagonal) for which no low-temperature values have yet been reported. a-Uranium (orthorhombic) is a highly anisotropic element for which measurements83 indicate negative expansion coefficients below 40” K in the “a” and “6” directions, but positive coefficients in “cYydirection. Earlier dataB4onpolycrystalline specimens also suggested that the volume coefficient might be negative below 40”K.
5.3. UNUSUAL METALS Certain metals exhibit behaviour at low temperatures which does not fit into
-2.0
I
Fig. 6. Linear expansion coefficient for chrorniuml6and two Fe + 36 %Ni alloys : 0 Nil0 36, as received,Wiggin and Co. (UK)49, x Invar, annealed, Carpenter Steel Co. (USA) aB. References p . 477
470
[a. =,B 5
I.0. COLLINS AND G. K.WHITB
+
the patterns discussed. Two of these are chromium and Fe 36 % Ni alloy (invar or nil0 36), for which large negative coefficients, proportional to temperature, are observed (Fig. 6). Over the range of observations shown in Fig. 6 these may be represented by 01=
-
a=-
3.8 x lO-'T
(Cr 3)15,
9.1 x lo-* T + 5 x lo-'' T 3
(nil036)~'
and a=-
10.6 x 10-BT
+ 5 x 10-11T3
(in~ar)~~,
and may be compared with iron or nickel in Table 2, e.g. 01
= 0.3 x lo-* T
+ 1 x 10-l' T 3
(Fe) *
It has been suggested15.85 that this behaviour in chromium is due to a large electroniccontributionarising fromits band-structure:experimental evidence on specific heats, susceptibility, etc., points to the Fermi edge in chromium lying near the bottom of a steeply rising density-of-statescurve86, a condition which could lead to ye being negative on Varley's analysis. However, it has been pointed out87 that magnetic interactions might be more important in Cr and also in invar: chromium is antiferromagnetic with a Nee1 point at 310" K and invar is ferromagnetic ( T , N 480" K). In each case the magnetic ordering temperature is very sensitive to pressure, dTc/dp being about -7 deg/1000 atm for chromium88 and -3.6 deg/1000 atm for i n ~ a r ~ ~ . Such a strong sensitivity of the exchange interaction to pressure implies that ym = - ah EJaln V is large and that therefore the magnetic contribution,,3,/ to the expansion coefficient might be large; a negative sign for j?, would be expected from the sign of aT,/ap. Gadolinium is another ferromagnetic material for which dTc/dp is relatively large and negative89, - 1.2 deg/1000 atm, and preliminary observations by Andres76 indicate a negative expansion coefficient in this w e also; however, gadolinium is anisotropic, being hcp with c/a = 1.590, so data for polycrystalline samples must be treated with reserve. Other theoretical arguments which may be pertinent to these negative coefficients have been advanced, one involving a magneto-strictive mechanismQOand the other (applied to FeNi, FePt, FePd alloys) involving relative occupancy of two different spin states91. The behaviour observed in dilute CuMn alloys seems a clear case of magnetic contribution to expansion. An alloy containing about 0.5 % Mn showed an expansion anomaly33 at liquid helium temperatures similar in form to that observed in the specific heatD2. The proportionality between the References p . 477
CH. D(,8 61
471
THERMAL EXPANSION OF SOLIDS
anomalous expansion relative to pure copper, Aa, and the anomalous heat capacity, ACpled to a value of ym = 3 -Aa-V / A C - x= 3.2
from which it was concluded that the exchange interaction between the Mn spins varies inversely as the third power of the volume ($2.4).
6. Glasses and Diamond-Structure Solids Thus far we have dealt with solids whose expansion coefficients depart in varying degree from the predictions of Griineisen's rule, and we have only encountered negative volume coefficients in isolated metals where magnetic interactions are probably responsible. However, there are a large number of isotropic materials which have been shown to exhibit negative dilatation; these are glasses and diamond-structure elements. Whether the occurrence has a common origin, e.g. in the tetrahedral bonding, is still open to conjecture. Negative expansion coefficients have also been observed in both crystallographic directions for single crystals of ice and heavy ice. 6.1. GLASSES
It has long been known that vitreous silica has a negative coefficient of expansion below about 200" K, and recently Gibbons93 has shown that this remains so down to 5" K. White49950 has observed the expansion of both a borosilicate ("Pyrex" type) glass and pure vitreous silica ("Spectrosil") from 30" K down to 2" K, and finds that below 10" K their coefficients are not very Merent in magnitude, being
- 2.0 x 1 0 - ' O ~ ~ a = - 3.0 x lo-'' T 3 a=
and
("Pyrex") ("Spectrosil")
.
At higher temperatures the presence of the Na20, A1203, B,O,, alter the expansion characteristics enormously, but they appear to have a much smaller effect in the low temperature region. It would be interesting to know whether the differencethat does appear at low temperaturesis primarily due to replacement of Si atoms by B or Al atoms in the linked chain-like structure, or to the effect of Na ions filling the structure. This might be solved by obtaining data on other glasses such as GeOz or SiOz +B203. The general negative expansion characteristic is presumably associated with the same transverse vibrational modes which increase the heat capacity above that expected hferences p . 477
472
J. 0. COLLINS AND G . K.WHITE
[a. M,8 6
from the elastic constants94*96.Down to 3" K it appears that transverse optical modes have not been frozen out, and it has been suggested that these may arise from transverse vibrations of the 0-atoms in the linked Si-0-Si structure. It is to be hoped that data will be forthcoming shortly on the expansion of hexagonal silica (quartz) at low temperatures, as well as on the expansion of other glasses. 6.2. DIAMONLGTRUCTURE SOLIDS
Solids for which negative expansion coefficients have been reported include silicon ((120" K)96397, germanium (<48" K)98, InSb ((55" K)96998, a-Sn (<45" K) 98, CdTe (<72' K) 98, GaAs (<55" K) 98, ZnSe (<64" K) g8, diamond ((90" K)98, measurements generally extending down to 20 or 30" K. Daniels27 has pointed out that the measured pressure derivativesof the elastic constants for Ge and Si lead to limiting values of yo equal to +0.49 and $0.25 respectively. The most recent data on germanium99 show that the expansion coefficient is indeed positive below 15" K, and that y appears likely to have a limiting value of 0.5 f O . l (see Fig, 7). Gibbons' data on silicon also indicate a rising y below 65" K, where y has its minimum value
Fig.7. Variation of Grtineisen parameter with reduced temperature for germanium (00 = 380" K), silicon (00 = 650" K) and vitreous silica = 495" K). Continuous curves in main figure are from Gibbons 9 6 ; points are from 90. Inset shows recent extended data for vitreous silica49. References p . 477
CH.IX,8 71
THERMAL EXPANSION OF SOLIDS
473
of about -0.4. We wonder whether positive values of y (and of volume coefficient) will occur for all diamond-structure solids in the continuum limit (see also Daniels loo),and whether the same might be true for the glasses below 2" K, say, when optical modes may be expected to be frozen out and the specific heat should decrease rapidly to the value predicted by elastic constants (€Jelastic 21 495, cJ: Bo -N 395" K94). 6.3. ICE Crystalline ice101 can exist in either a hexagonal, tridymite-type structure, with c/a 2: 1.632, or in a cubic, cristobalite-type structure. Each 0-atom is tetrahedrally surrounded by four others, with one H-atom lying along each 0-0 bond, nearer to one 0-atom than to the other but in a random sequence throughout the crystal. Dantl's102 recent data for the expansion coefficients of single crystals of hexagonal ice and heavy ice from room temperature down to 20" K indicate that they are isotropic, and have negative volume coefficients of expansion below 63" K. There is no marked difference between H,O and D,O, which suggests that vibrations of the tetrahedrally linked 0-atoms play a larger role in the negative expansion than do transverse vibrations of the H- and D-atoms. There is a structural similarity between ice and diamond-structure solids, and we presume that the tetrahedral bonding is again responsible for the negative dilatation. 7. Superconductors
Since the electron gas makes an appreciable contribution at low temperatures to the free energy of a metal, we expect differences in behaviour in the superconducting (s) and normal (n) states. Thermodynamically the volume difference between states may be related to the pressure dependence of the critical field H,, as may also the difference in expansion coefficients or difference in bulk modulus K (e.g., Shoenberglos, p. 75):
and at T = T,, for example,
References p . 477
474
3 . 0 . COLLINS AND 0. K. WHITE
[CH.M,8 7
Olsen and his collaborators have determined volume or length changes in a number of superconductors41~ 1 0 4 4 0 7 when a magnetic field isothermally destroys the superconductivity. Much of their work has been concerned primarily with understanding superconductivity rather than expansion. However, it has led to estimates of the electronic parameter ye as well as parameters of direct concern in the mechanism of superconductivity, e.g. dHc/dp,d In Tc/dIn V,d In N(0)A/d In V ,etc. Likewise direct measurementsof dH,/dp have also led to estimates of yelo*. The considerable uncertainty in many such estimates of ye104~105~108 may arise from experimental difEculties (particularly for “hard” superconductors), or from sensitivity of the analysis to the assumed form of the critical field-temperature relationship. An alternative method of finding ls-ln is to determine the expansion curves, i.e. I(s) in zero field (T < To),Z(n) in zero field ( T > T,), as shown for lanthanum76 in Fig. 8. Below the transition temperature Z(n) may be found by extrapolating the normal curve (dashed curve in Fig. 8) or by determining it in a fixed field which exceeds the critical field at all temperatures; then Z, - I, is merely the difference between the two curves. This has been used in the case of not-very-pure samples of Ta, V, Nb77 where the method involving a changing field was unsatisfactory. For soft superconductors there is agreement between different methods. In lead, for example, both Olsen and Rohrer413104 and White78 observed (I, - &,)/Is 2: (16 i1) x 10-8 for T < 2” K. Limiting values of dH,/dp deduced from these data using eq. (10) are as follows: (dHc/dp)T-o = - 7 x (dH,/dp),,,
lo-’
Oe/dyn]cm2 ,
= - 11 x 10-90e/dyn/cm2,
and the latter leads by eq. (1 1) to
AK/K = 5 x For comparison direct measurements of the effect of pressure on the critical fieldlog and changes in elastic constants110 have given (dH,/dp),,,,
= - (7.9 It 0.2) x lo-’ Oejdyn/cm2
and AKIK = 4.0 x It has generally been observed that 1, - 1, is positive; in many transitionelement superconductors for which the electronic component of expansion is appreciable near the temperature T,, Z,(T) decreases slightly in warming from near 0”K up to T, (e.g., La in Fig. S), whereas I,, (T) of course References p. 477
CH. Ix,
Q 71
THBRMAL EXPANSION OF SOLIDS
475
monotonically increases with T. Vanadium 77 is a puzzling exception as 1, ( T ) increases more rapidly than 1, (T) so that 1, - I,, < 0 and (dH,/dp) is positive. Since recent data111 on the specific heat of superconductors indicate that the lattice contributionmay be less in the s- than in the n-state, it is interesting to see whether this is also the case for the thermal expansion. Lead is the only instance so far where a reasonable estimate has been made78, and q(s) (T 4" K) does seem to be about 20 % smaller than aI(n). Finally thermal expansion and related thermodynamic data have given
-=
Fig. 8.
Linear thermal expansion of lanthanum rod (Andres'").
values for the volume dependence of the interaction in the Bardeen-CooperSchrieffer theory107. This theory states
k ~ ,= 1.14hoexp[- 1/N(O)A], whence
K/O = 0.85 exp [- l/N(O)A] .
Rohrer's data on "soft" superconductors have led him to conclude that106 dluN(O)A/dIn I/ = 2 to 3. However, for transition-element superconductors such as Ta, Nb, V, Ru, Re106*107it is apparent that this volume dependence is much smaller, so that the interaction mechanism responsible for superconductivityappears to be rather different. Referencesp . 477
476
I. G . COLLINS A N D G.K.WHITE
[CH.M,8 8
8. Summary Thermal expansion measurements on cubic solids below 8 have generally confirmed the picture presented by the lattice models of Barron, Blackman and others. The Griineisen parameter y of isotropic metals decreases by not more than 20 % on cooling, whereas in ionic solids this decrease may be much greater; ya, - yo is greatest for those ionic solids in which the shear stiffness is least relative to the compressionalstiffness. Further measurements of high accuracy are needed in the region below & 8 to determine more clearly the relation between yo, pressure derivatives of elastic constants and the frequency spectrum and to establish y(T) sufficiently accurately that parameters y(-2), y(O), etc., may be obtained. In anisotropic materials incomplete data are yet available regarding y and its relation to the elastic moduli, although the qualitative behaviour of the expansion coefficients in hexagonal-close-packed metals seems clear. Reliable values for the electronic contribution to expansion have been obtained for many transition elements but are still lacking for W, Ti, Zr, Os, Ru, Mn, as for most monovalent metals; for these latter ye should approach the free electron value of # most closely. For Al and many transition elements ye exceeds this value by a factor of two to three. No theoretical attempts have yet been made to assess the quantitative relation between ye and the density of states in specific metals. Glasses and diamond-structure solids present an interesting problem as they display negative expansion coefficients at moderately low temperatures. At least one of the diamond-structure solids shows a return to positive behaviour in the low temperature limit and others seem likely to do the same thing, but whether glasses (and ice) will also is unknown. Equally uncertain are which vibrational modes in glass are responsible for the negative behaviour. Finally magnetic materials and superconductorspresent a host of problems but existing data show clearly that accurate measurements, with and without magnetic fields, made on samples of suitable purity can give useful information concerning the volume dependence of the interaction energies.
Acknowledgements We are grateful to many people who have communicatedtheir data and ideas to US before publication: Drs. K. Andres, T. H. K. Barron, R. H. Carr, P. G. Klemens, P. P. M. Meincke, D. Schuele, C. S. Smith and R. Srinivasan. References p , 477
CH.m]
THERMAL EXPANSION OF SOLIDS
477
We are also indebted to the United States Air Force Office of Scientific Research for supporting research on expansion in this laboratory. REFERENCES
E. Griineisen, Handb. d. Phys. 10, l(1926). D. BY1 and H. Pullan, Physica 21,285 (1955). T. Rubin, H.W. Altman and H. L. Johnston, J. h e r . Chem. SOC.76, 5289 (1954). T. H.K.Barron, Phil. Mag. 46,720 (1955). T. H. K.Barron, Annals of Phys. 1, 77 (1957). M.Blackman, Proc. Phys. SOC.B 70,827 (1957). M.Blackman, Phil. Mag. 3,831 (1958). M. Blackman, Proc. Phys. SOC.74,17 (1959). M.Born, Atomtheorie des festen Zustandes (Teubner, Leipzig, 1923). lo B. Dayal, Proc. Indian Acad. Sci. A 20, 138 (1944). l1 P. Grassmann and J. L. Olsen, Helv. Phys. Acta 28,24 (1955). K. Andres, Cryogenics 2,93 (1961). G.K.White, Cryogenics 1, 151 (1961). l4 R. H. Carr and C. A. Swenson, Cryogenics 4 (1964). Is G.K.White, Nature 187,927(1960); Phil. Mag. 6,815 (1961). l6 G.Leibfried and W. Ludwig, Solid State Physics 12,276 (1961). l7 T. H. K. Barron and M. L. Klein, Phys. Rev. 127,1997 (1962). la T. H. K. Barron, M. L. Klein, A. J. Leadbetter, J. A. Morrison and L. S. Salter, Proc. VIII Int. Conf. Low Temp. Phys. (Butterworth, London, 1963) p. 415. J. C. Slater, Introduction to Chemical Physics (McGraw-Hill, 1939). ao T. H. K. Barron, Nature 178,871 (1956). 21 G.K.Horton, Can. J. Phys. 39,263 (1961). 22 G. K. White, Proc. VII Int. Conf. Low Temp. Phys. (University of Toronto Press, 1961)p. 685. as S. Ganesan, Phil. Mag. 7, 197 (1962). 24 S. Ganesan and R. Srinivasan, Can. J. Phys. 40,91 (1962). F. W. Sheard, Phil. Mag. 3, 1381 (1958). J. G. Collins,Phil. Mag. 8,323 (1963). 27 W.B. Daniels, Phys. Rev. Lett. 8, 3 (1962). 3. de Launay, Solid State Physics 2, 220 (1956). 2e E. Gruneisen and E. Goens, Zeits. fur Phys. 29,141 (1924). 2. Mikura, Proc. Phys. Math. SOC.Japan 23, 309 (1941). 31 S. Visvanathan, Phys. Rev. 81,626 (1951). 38 J. H.0.Varley, Proc. Roy. SOC. A 237,413 (1956). G. K.White, J. Phys. Chem. Solids 23, 169 (1962). 34 R. 0. Simmons and R. W. Ballufii, Phys. Rev. 108,278(1957). 3g B. F.Figgins, G. 0. Jones and D. P. Riley, Phil. Mag. 1, 747 (1956). 36 D.E.Schuele, Case Institute of Technology, Ph. D. Thesis (1962)to be published. s7 P. N. Dheer and S. L. Surange, Phil. Mag. 3, 665 (1958). 38 D. B. Fraser and A. C. Hollis Hallett, Proc. VII Int. Conf. Low Temp. Phys. (University of Toronto Press, 1961) p. 689. s8 P. P. M. Meincke and G. M. Graham, Roc. VIII Int. Conf. Low Temp. Phys. (Butterworths, London, 1963)p. 401. 40 E. Huzan, C. P. Abbiss and G. 0. Jones, Phil. Mag. 6,277 (1961). 41 3. L. Olsen and H. Rohrer, Helv. Phys. Acta 30,49 (1957). a
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R. V. Jones, J. Sci. Instr. 38,37 (1961); R. V. Jones and I. C. S. Richards, J. Sci. b t r . 36,m (1 959). 4s R. H. Cam, Iowa State Univ., Ph.D. Thesis (1962). 44 A. M. Thompson, I.R.E. Trans. on Instrumentation 1-7,245 (1958). 46 T. H. K. Barron, Proc. VII Int. Cod. Low Temp. Phys. (University of Toronto Press, 1961) p, 655. 46 B. Yates and C. H. Panter, Proc. Phys. SOC.80,373 (1962). 47 T. Rubin, H. L. Johnston and H. W. Altman, J. Phys. Chem. 65,65 (1961). 48 P. P. M. Meincke and G. M. Graham, Proc. VIII Int. Conf. Low Temp. Phys. (Butterworths,London, 1963) p. 399; also P. P. M. Meincke, private communication (1963). 49 G. K. White, VIII Int. Conf. Low Temp. Phys. (Butterworths, London, 1963) p. 394. G. K. White, unpublished; also Phys. Letters 8, 294 (1964); Cryogenics 4,2 (1964). R.Srinivasan, J. Indian Inst. Sci. 37, 232 (1955). 62 R. Srinivasan, J. Indian Inst. Sci. 38,20 (1956). 63 K. E. Salimitki, Ann. Acad. Sci. Fennicae A 6, No. 56 (1960). 64 T. Rubin, H. L. Johnston and H.W. Altman, J. Phys. Chem. 66,948 (1962). 65 R. Srinivasan, Proc. Ind. Acad. Sci. A 42, 255 (1955). 66 M. A. Viswamitra and S. Ramaseshan, Acta Crystall. 15,513 (1962). 57 J. A. Morrison, D. Patterson and J. S.Dugdale, Can. J. Chem. 33, 375 (1955). 68 W. T. Berg and J. A. Morrison, Proc. Roy. Soc. A 242,467 (1957). T.H.K.Barron, W. T. Berg and J. A. Morrison, Proc. Roy. Soc.A 250, 70 (1959). 6o K. Clusius, J. Goldrnann and A. Perlick, Zeits. fiir Naturf. 4a, 424 (1949). a D. R. HulTman and M. H. Norwood, Phys. Rev. 117,709 (1960). s2 C. V. Briscoe and C. F. Squire, Phys. Rev. 106, 1175 (1957). W.C. Overton and R.T. Swim,Phys. Rev. 84, 758 (1951). O4 R. N. Claytor and B. J. Marshall, Phys. Rev. 120, 332 (1960). O5 J. K. Galt, Phys. Rev. 73, 1460 (1948). 66 M. H. Norwood and C. V. Briscoe, Phys. Rev. 112,45 (1958). 67 K. Reinitz, Phys. Rev. 123, 1615 (1961). 88 M. A. Durand, Phys. Rev. 50,449 (1936). T. H. K. Barron, W.T. Berg and J. A. Morrison, Proc. Roy. Soc.A 242,478 (1957). K. S. Krishnan and S. K. Roy, Proc. Roy. SOC.210,481 (1952). H. B. Huntington, Solid State Physics 7, 213 (1958). T. H. K. Barron, W.T. Berg and J. A. Morrison, Proc. Roy. SOC.A 250, 70 (1959). 78 E. R. Dobbs, B. F. Figgins, G. 0.Jones, D. C. Piwcey and D. P. Riley, Nature 178, 483 (1956). 74 B. F. Figgins and B. L. Smith,Phil. Mag. 5,186 (1960). 75 K. Andres, Proc. VIII. Int. Conf. Low Temp. Phys. (Butterworths, London, 1963) p. 397. 78 K. Andres, Phys. Letters 7,315 (1963); also Phys. Rev. Lett. 10,223 (1963). 77 G. K. White, Cryogenia 2,292 (1962). 76 G. K. White, Phil. Mag. 7, 271 (1962). 79 R. D. McCammon and G. K.White, unpublished. 80 R. J. Corruccini and J. J. Gniewek, N.B.S. Monograph 29 (US. Government Printing office, Washington). 81 B. G. Childs, Rev. Mod. Phys. 25, 665 (1953). 88 R. W. Meyerhoff and J. F. Smith, J. App. Phys. 33,219 (1962). 88 C. S. Barrett, M. H. Mueller and R. L. Hitterman, Phys. Rev. 129,625 (1963). 84 H. L. Laquer, USAEC Report 3706, Los Alamos (1952); Phys. Rev. 86,803 (1952). G. K. White, Aust. J. Phys. 14, 359 (1961). 88 M.Shimizu, T.Takahashi and A. Katsuki, J. Phys. Soc. Japan 17,1740 (1962). 42
CH.Ixl
THBRMAL EXPANSION OF SOLIDS
479
D. Gugan, private communication(1962). P. W. Bridgman, Proc. h e r . Acad. Arts. Sci. 68, 33 (1933). 88 L. Patrick, Phys. Rev. 93, 384 (1954); S. H. Liu, Phys. Rev. 127, 1889 (1962). 90 D. S. Rodbell, Phys. Rev. Lett. 7, 1 (1961). 81 R. J. Weiss, Roc. Phys. SOC.82, 281 (1963). 92 J. E. Zimmerman and F. E. Hoare, J. Phys. Chem. Solids 17,52 (1960); J. de Nobel and F. J. du Chatenier, Physica 25, 969 (1959). 98 D. F. Gibbons, J. Phys. Chem.Solids 11,246 (1959). 04 P. Flubacher, A. J. Leadbetter, J. A, Morrison and B. P. Stoicheff, J. Phys. Chem. Solids 12, 53 (1959). 95 0. L. Anderson and G. J. Dienes, Conference on Non-Crystalline Solids (Wiley, New York, 1960) p. 449. 96 D. F. Gibbons, Phys. Rev. 112, 136 (1958). B7 S. I. Novikova and P. G. Strelkov, Fiz. Tverd. Tela 1, 1841 (1959). 08 S. I. Novikova, Fiz. Tverd. Tela 2,43 (1960); ibid2, 1617 (1960); ibid 2,2341 (1960); ibid 3, 178 (1961); also /bid 5, 2138 (1963). 99 R. D. McCammon and G. K. White, Phys. Rev. Lett, 10,234 (1963). 100 W.B. Daniels, Proc. Int. Conf. Semiconductor Physics, Exeter (1962) p. 482. 101 K. Lonsdale, Proc. Roy. Soc.A 247,424 (1958). 102 G. Dantl, Zeits. fur Phys. 166, 115 (1962). 103 D. Shoenberg, Superconductivity (Cambridge University Press, 1952). 104 J. L.Olsen and H. Rohrer, Helv. Phys. Acta 33, 872 (1960). 105 H.Rohrer, Helv. Phys. Acta 33, 675 (1960). 108 K.Andres, J. L. Olsen and H.Rohrer, LBM Journal 6,84 (1962). 107 E. Bucher and J. L. Olsen, Proc. WI Int. Cod. Low Temp. Phys. Qutterworths, London, 1963) p. 139. 106 C.A.Swenson, IBM Journal 6,82 (1962); see also Phys. Rev. 123,1106,1115 (1961). 109 M.Garfinkel and D. E. Mapother, Phys. Rev. 122,459 (1961). 110 0. A. Alers and D. L. Waldorf, IBM Journal 6,89 (1962). 111 C. A. Bryant and P. H. Keesom, Phys. Rev. Lett. 4,460 (1960). 67
CHAPTER X
THE 1962 3He SCALE OF TEMPERATURES BY
T. R. ROBERTS, R. H. SHERMAN
AND
S. G. SYDORIAK*
UNIVERSITY OF CALIFORNIA, L o S &AMOS SCIENTIFIC LABORATORY, Los h m o s , NEWMEXICO AND
F. G. BRICKWEDDE PENNSYLVANIA STATEUNIVERSITY, UNIVERSITY PARK,PENNSYLVANIA CONTENTS: 1. Introduction, 480. - 2. Brief historical review of earlier determinations of the 3He vapor pressuretemperature relation, 482. - 3. Plan for development of the 1962 3He scale, 488. - 4. The 1961 L.A.S.L. intercomparisons of the 3He and 4He vapor pressures, 489. - 5. The determination of the critical pressure and temperature of 3He, 491. - 6. Calculation of a vapor-pressure equation below 2°K using a theoretical equation from statistical mechanics, 492. - 7. Extension of the vaporpressure equation to the critical point, 493. - 8. The 1962 3He vapor-pressure scale, 494. - 9. An evaluation of the 1962 3He scale, 495. - 10. Thermodynamic properties of 3He consistent with the 1962 3He scale, 505. - 11. Corrections to the measured pressure of a 3He vapor-pressure thermometer, 509. - 12. Conclusion and considerations for the future, 511.
1. Introduction
The 1962 3He Scale of Temperatures is a practical, or secondary, scale on which temperatures from 0.25" K to the critical temperature of 3He (3.324' K) are defined by the vapor pressure of pure liquid 3He in equilibrium with pure 3He vapor at the temperature in question. The International Committee on Weights and Measures, at its October, 1962, meeting in Sbvres, France, recommended the Scale for the calculation of temperatures, with the designation T62, from measurements of the vapor pressure of 3He. The
*
Work performed under the auspices of the U.S.Atomic Energy Commission.
References p . 512
480
CH. X, 8 11
THE 1962 3 H e SCALE OF TEMPERATURES
481
relation between temperature and vapor pressure, defining the Scale, is an analytic equation, eq. (18) in this chapter. This equation is, we believe, the best available representation in analytical form over the range 0.25" to 3.324' K of the vapor pressure of 3He on the thermodynamic, Kelvin, scale. A table of 3He vapor pressure, Ps,at millidegree intervals and an inverse table of T as a function of P3 are presently available as a Los Alamos Scientific Laboratory report These tables and three companion papers describing the new scale measurements2, derivation3 and evaluation4 are being submitted for publication. The 1962 3He Scale was made by plan to agree closely with the 1958 4He Scales. The two Scales are supposedly in like agreement with the thermodynamic Kelvin Scale which Scale it is intended both helium scales should reproduce. The 4He Scale is usable at higher temperatures (5.2 O K ) than the 3He Scale because of the higher critical temperature of 4He, but the 3He Scale is usable at lower temperatures. For vapor pressures below one mm Hg the corresponding temperature on the 3He Scale is about one-half that on the 4He Scale (see Table 3). Below the 4He I-point (2.172' K), use of a closed, or sealed, 4He vaporpressure bulb as a thermometer is complicated by two effects of the creeping Rollin film that are difficult, practically, to evaluate accurately. The evaporating film generates a mass flow of vapor that condenses in the liquid holder at the bottom causing: (1) a pressure gradient in the vapor tube and (2) a Kapitza temperature discontinuity at the solid wall-liquid helium interfaces across which the heat of condensation flows to the refrigerant. There is a great variability in the rate of film flow over a given surface because the rate is strongly influenced by minute amounts of impurity (e.g. solid air) on the surface. The heat leak due to the refluxing film also prevents reaching temperatures much below 1' K in an apparatus having a 4He vapor-pressure thermometer. With liquid 3He there is no Rollin film. This gives the !jHe vapor-pressure thermometer an important advantage for the measurement of temperatures below the &point. However, the isotopic purity of the helium filling the 3He vapor-pressure thermometer has a practical importance that does not usually exist for the 4He vapor-pressure thermometer, a disadvantage of the 3He thermometer. In some cases the lower thermal diffusivity of liquid 3He below 2" K may be a disadvantage when determining the temperature of an object from a helium bath or bulb pressure. These and other problems, attendant upon the use of He vapor-pressure thermometers, are discussed in Sections 4 and 11 and in ref.2. References p . 512
T.R. ROBERTS, et 01.
482
[a. x, B 2
2. Brief Historical Review of Earlier Determinations of the 3He Vapor Resure Temperature Relation
-
sHe was first liquefied and its vapor pressure measured in the Los Alamos Scientific Laboratory, by Sydoriak, Grilly, and Hammela in 1949. At the time, the liquefaction of 3He had special theoretical significance because doubt existed before this that SHe was condensable as a liquid. It had been suggested that the "zero-point" vibrational energy of 3He in the liquid state would exceed the van der Waal's cohesive energy' and prevent condensation as aliquid. It was suggested also that the Fee-Dirac statistics, which govern SHe, might enhance the kinetic energy of 3He atoms in a liquid state sufficiently to prevent condensations. However, measurements at Yale9 on dilute solutions of SHe in 4He were analyzed to indicate a possible boiling point of 2.9" K for pure 3He. The various theories of liquid 3He and the temperature dependence of the vapor pressure and liquid entropy have been discussed in earlier volumes of this series by Hamme110 and by Grilly and Hammel11. In retrospect, it can be seen that the zero-point vibrational energy of 3He is not as large as had been anticipated. Not only was the magnitude of the zero-point vibrational energy of liquid 4He from which the zero-point energy of liquid 8He was estimated uncertain, but the molecular volume of liquid 3He was afterwards found to exceed that of liquid 4He by about seventy per cent 129 1s. This large molecular volume reduces the zero-point vibrational energy of liquid 3He. The properties of a Fermi-liquid with intermolecular forces have since been found to be quantitatively quite different from those of a dense ideal gas upon which the earlier thinking was based. This may be seen by comparing the calculated Fermi degeneracy temperature of 5" K for an ideal gas having the density of liquid SHe with the experimentaldegeneracy temperature of 0.45' K. It is interesting and significant, also, that before it had been demonstrated that SHe could be liquefied, de Boer and Lunbeckl4 applied de Boer's "quantum-theoryyylaw of corresponding states15to the determination of the vapor pressure of liquid 3Heand obtained values in fair agreement with later observed data. Thus, de Boer and Lunbeck predicted the critical temperature of SHe would be between 3.2" and 3.5" K, and the critical pressure between 0.9 and 1.2 atmospheres. The experimental data are 3.324" K and 1.15 atm. The de Boer and Lunbeck value for Lo, the heat of vaporization of liquid 3He at 0' K,. was 21.4 joules/mole, whereas the most recent value derived from the experimentalvapor-pressuredata is 20.6 joules/mole (see Section 10). References p . 512
QI.x,021
THB 1962 *HeSCALE OF TEMPERATURB~
483
The de Boer quantum-theory law of corresponding states makes use of a quantum parameter A* =h/cr’dme’ in addition to the “reduced” variables for temperature, pressure, and volume. Here m is the mass of a molecule and cr’ and e’ are defined by the molecular-interactionpotential
q(r) = 4.5’[(a‘/r)12 - ( ~ ’ / r ) ~ ] . A* for 3He is 3.05,whereas for substances of large molecular weight, for which quantum effects are small, A* is << 1. From a curve of Lo values, derived from experimental vapor-pressure data for a number of low temperature condensable gases, plotted against their characteristic A* values, de Boer and Lunbeck obtained, by extrapolation, for liquid 3He: L0=21.4 joules/mole. Substitution of this value of Lo in the theoretical ideal vaporpressure equation,
LO RT
InP, = - -+ 3 h T + i , ,
in which i, is the “chemical constant” neglecting nuclear spin and m 4.63 i f P is in mm Hg,gives the de Boer-Lunbeck vapor-pressure equation
2.57
InP3(mmHg) = - -+ 31n T
T
+ 4.63.
(3)
In 1950,Abraham, Osborne and Weinstock (A.O.W.)IS, at the Argonne National Laboratory, measured the vapor pressure of liquid 3He by comparison with 4He, more accurately than it had been measured before (at L.A.S.L.), covering the range 1.025’ to the critical temperature (3.324’ K). Temperatures were obtained from the vapor pressure of 4He and were on a scale advanced by A.O.W., which was based on the 1948 4He Scale17 with Kistemakerls “corrections” as applied by A.O.W. In 1957, Sydoriak and Roberts19, at the Los Alamos Scientific Laboratory measured the vapor pressure of 3He from 0.45”to 1” K,using ferric ammonium and chromic methylamine alum magnetic thermometers to measure the temperature. The calibration constants of the magnetic thermometers were determined by Sydoriak and Roberts by calibration against the vapor pressure of 3He at several temperatures between 1.1’ and 2.5” K, using the A.O.W. (1950) intercomparisonsof 3He and 4He vapor pressures and the L55 and 55E 4He Temperature Scales2% 21 for getting temperatures from the vapor pressures of 4He. In 1961, Sydoriak and Shermana, at the Los Alamos Scientific Laboratory, compared the vapor pressures of 3He and 4He, from 0.9”to the critical temperature, in an apparatus designed to minimize the errors arising R&ernrces p . 512
484
T.R. ROBERTS,
[CH.X , 5 2
et al.
from the Rollin film in the *He bulb and pressure sensing tube. These measurements, undertaken for the determination of the 1962 3He Scale, are discussed in Section 4. Temperatures were expressed on the 1958 4He Scale&. From these experimental vapor-pressuredata, 3He vapor pressure-temperature equations were deduced. The objectives were: (1) the calculation of temperatures from 3He vapor-pressure measurements and (2) the determination of the entropy of liquid 3He and its dependence on temperature. Interest in the entropy arose from an interest in the removal at low temperatures of the degeneracy of nuclear spin orientation in liquid 3He. Later, in 1954, the removal of the spin degeneracy was determined by magnetic susceptibility measurements2a and since then the interest in the entropy of the liquid has shifted to the investigation23of the relation between the spin degeneracy and the entropy of the liquid. The 3He vapor-pressure data were fitted to the theoretical equation, from statistical thermodynamics, for a monatomic gas
LO
lnP3 = -- +31nT + l n [ o ( 2 n n ~ ) ~ k ~ h+- ~ ] RT T
T
-L RT I S L d T + &jVLr$) 0
sat
dT
+
(4)
E,
0
where one formulation of the gas imperfection term, E , is
and where P3 is the vapor pressure of liquid SHe, Lo the molar heat of vaporization at 0" K, S, and V, the molar entropy and volume of the saturated liquid, rn and Q the mass and nuclear spin degeneracy of a 3He atom, and B is the second virial coefficient in the equation of state for the saturated vapor, eq. (13). The third term on the right side of eq. (4) is io,the chemical constant for 3Heincluding nuclear spin. For 3He, Q = 2 and io = i,,, t In 2 since eq. (4) assumes complete spin degeneracy in the vapor with a vapor spin-entropy of R In 0. Eq. (4) results from equating the statistical-mechanics expression for the Gibbs free energy of the saturated vapor with the Gibbs free energy of the saturated vapor. In the development of the 1962 3He Scale the following constants were used: gas constant R =8.31470 joules/mole deg, atomic weight of 3He =3.0162 g/mole, and io = 5.31733 for pressures in mm Hg at 0" C (density 13.5951 g/cm3) at standard gravity (980.665 cmlsecz). References p . 512
CH. x, 0 21
THE
485
1962 sHe SCALE OFTJWPBRATLJ~
In 1950, A.O.W. fitted their 3He vapor-pressure data to the equation A
In P3 = - - + $In T + C +f(T) T for which it was assumedf(0" K) = 0. Using the method of least squares, the constantsA and C were evaluated andf( T )was found adequately represented by a single term, DT3. The A.O.W. equation on their "corrected" 1948 4He Scale is 2.2518
In P3 (mm Hg,0°C)= - + $In T + 4.4116 + 0.000 695T3. (7) T For Lo = A * R, A.O.W. obtained 18.7 joules mole-1, and, from their vapor-pressure equation, 3.195" K for the normal boiling point of 3He, and 3.35 f 0.02" K for the critical temperature, corresponding to a critical pressure of 890 f 20 mm Hg. Assuming complete spin degeneracy in the liquid A.O.W. obtained: SL(loK) = 8.76 joules mole-1 deg-1 (a corrected value; see ref.24) and S, (0" K) = R In 2 1.76 f0.46 =7.52 f0.46 joules mole-1 deg-1. In 1953, Weinstock, Abraham and Osborne24 recalculated the entropy of the liquid, S, from their 1950 vapor-pressure data 16 assuming S, extrapolated to R In 2=5.76 joules mole-' deg-1 at 0" K. This was before measurements of the magnetic susceptibility, x, of liquid 3He by
+
o ' 0
0.2
0.4
0.6
0.0
1.0
1.2
1.4
1.6
1.0
2.0
T I'Kf
Fig. 1. Showing the dependence of the magnetic susceptibility x of liquid 3He on T. For the validity of Curie's Law: xT/C = 1. For an ideal, degenerate Fermi-Dirac "gas": xT/C is proportional to T (i.e. x = const) at very low temperatures. The full-line curve is for an ideal, Fermi-Dirac gas having a degeneracy temperature of 0.45"K.In the region of validity of Curie's Law, the nuclear spin entropy of 3He is R l n 2 . In the region of spin degeneracy, near 0' K, the spin entropy, S,, is proportional to T; So = (XT/C)RIn 2. References p . 512
fa.x, 0 2
T.R. ROBERTS, et PI.
486
Fairbank, Ard and Walters22 in 1954 showed that the nuclear spin degeneracy is removed gradually as the temperature is decreased below 1" K in liquid BHe and that this approximates what would be expected for the susceptibility of an ideal Fermi-Dirac gas having a degeneracy temperature of 0.45"K (see Fig. 1). The nuclear spin component of the entropy of liquid 3He in Fig. 1 is (xT/C)R In 2. Before the magnetic susceptibility measurements of Fairbank, Ard and Walters, T. C. Chen and F. London25 fitted the A.O.W. 3He vapor pressuretemperature data between 1" and 2.5" K to the theoretical vapor-pressure equation, eq. (4), incorporating the assumption that the entropy of liquid 3He approaches zero at 0" K like SL=(constant) x T. They wrote for the entropy of the liquid SJR = aT + bT2
+ cT3 + dT4 + sat
The constants of this equation were evaluated using the A.O.W. 1950 data. Chen and London obtained the equation 2.6620 hP3(mmHg,00C) = - - 2.5111T T
+
+ 5.3250 - 0.581 49T -+
- 0.01536T2 + 0.121 25T3 - 0.02786T4
(9) which fits the A.O.W. data between 1" and 2.5"K as well as eq. (7). The Chen and London value for Lo is 22.1 joules mole-1. Fig. 2 illustrates the different assumptions that were made about the 0 K, before the manner of the extrapolation of the entropy of liquid 3He to ' measurements22 of the magnetic susceptibility of liquid SHe,made in 1954, indicated how the nuclear spin entropy of liquid 3He might approach zero. In 1957, Sydoriak and Roberts published'@the following vapor-pressure equations for 3He, valid for the range 0.45"to the critical temperature, on the 55E and L55 4He Scales21.20 of temperature which were current at that time lnP3 = - 2.538 53/TE+ 2.3214ln TB+ 4.8153 - 0.20644TE+
+ 0.0864OTi - 0.009 19T: and
(10)
+
+
In P3 = - 2.52608/TL+ 2.3214h TL 4.8153 - 0.20046TL
+ 0.081 83Tf - 0.008 50T:. References p . 512
(11)
CH.x, 0 21
TEIE 1962 *He SCALE OF TEMPERATURES
487
These equations found general use for the calculation of temperatures from SHe vapor-pressure measurements before the adoption of the 19623He Scale. After the adoption of the 1958 4He Scale, approximate T5*3He temperatures were derived in some papers from 3Hevapor-pressure data by adding to the TE or TLcalculatedfromeq.(10)or(11),theAT's,(T58T 5 , J or (T58-Z'L55). This approximation does not give good agreement with the 1962 3He Scale (see Fig. 12 and ref.S). Eqs. (10) and (11) are not of the proper form for extrapolation to 0" K because the entropy of liquid SHe calculated from them does not approach zero like S, = (constant) x T.This is a consequence of the coefficient of In T
Fig. 2. Illustrations of the different extrapolations to 0"K of the entropy (SLIR) of liquid SHe assumed by Abraham, Osborne and Weinstock16 (1950), Weinstock, Abraham and Osborne24 (1953), and Chen and London25 (1953). The curve for the nuclear spin component (&/It) of the liquid entropy is a graph of ( x T / C )In 2, see Fig. 1.
in eqs. (10) and (11) not being 3 and the constant term differing from io. Below 0.45", the calculated vapor pressures would be expected to increase in percentage error. However, for temperatures above 0.45" ,where eqs. (10) and (11) were expected to fit the available vapor-pressure data, this failure of S, to approach 0" properly is not important. In Volume I of Progress in Low Temperature Physics, Hamme110 has discussed the calculation of the entropy of liquid SHe from vapor-pressure and thermodynamic data. The experimental specific heat data on liquid 3He were used for the calculation of SL(T)-SL(0.5"), and the 1957 L.A.S.L.le vapor-pressure data from 0.45" to lo, and the A.O.W. data from 1" to 2" were used to calculate SL(O.So). The normal boiling point of *He calculated from eq. (10) is 3.189" and References p. 512
488
T.R. ROBERTS, et
[CH.X, 5 3
al.
from eq. (ll), 3.190'. The critical temperatures for a critical pressure of 875 mm Hg are 3.327 and 3.329', respectively. Lo calculated from the coefficient of 1/Tin eq. (10) is 21.10 joules/mole. 3. Plan for Development of the 1962 'He Scale
Following the adoption by the International Committee on Weights and Measures of the 1958 4He Scale&,Sydoriak, Roberts and Sherman made a proposal26 to the VII-th International Conference of Low Temperature Physics held at the University of Toronto in 1960 for a new 3He vaporpressure scale to be based on the 1958 4He Scale and on various thermodynamic properties of 3He. The proposed procedure was similar to that described for the TEand TL3He scalesl@,eqs. (10) and (1 l), except that newly available specific heat data27-29 could be included, making the use of a calculated "spin entropy" term 23 unnecessary. In addition, a different conversion of magnetic to thermodynamic temperatures was being studied for the paramagnetic salts used in the vapor-pressure measurements19 intended for extending the scale below 1" K. The proposal was favorably received by members of the Conference, with some reservations as to the feasibility of including the vapor-pressure data obtained with an iron-alum thermometer. Subsequently, incorporation of any data obtained with a paramagnetic salt thermometer into the derivation of the Scale was abandoned except for measurements of the specific heat of liquid 3He using a cerium magnesium nitrate thermometer27. A different procedure for establishment of the low temperature end of the new 3He Scale was described and discussed30 in a report to the Fourth Symposium on Temperature, Its Measurement and Control in Science and Industry, Columbus, Ohio, 1961. By this procedure, the thermodynamic consistency of the helium vapor-pressure data with the experimental data for other thermodynamicproperties of 3Hecan be examined.The method showed that the A.O.W. vapor-pressure data could not be combined with the 1958 4He Vapor-Pressure Scale to yield a 3He vapor-pressure equation consistent with all the 3He data in the range from 1"to 2' K. A detailed discussion of the inconsistency, which is equivalent to several millidegrees, is given in refs.80 andal. Because of this inconsistency, new intercomparisons of the 3He and 4He vapor pressures were made2931 in an improved apparatus designed to minimize errors due to the reflux of the 4He filmin the pressure transmitting tube. Only these new vapor-pressure data, which are the subject of Section 4, were used in deriving the 1962 3He Scale. The A.O.W. data above the I-point a
References p . 512
CH. x, 5 41
THE
1962 $He SCALE OFTEMPERATURES
489
were used to check the final scale. The 1961 3He-*He vapor-pressure data, also, were found31 not to be consistent thermodynamically with data on other properties of 3He, although the inconsistency and scatter were less than that found using the A.O.W. P3 data (see Sections 9.3 and 9.4). Therefore, a method of development of the 1962 3He Scale was adopted that resulted in a 3He Scale in close agreement with the 1958 4He Scale of Temperatures. This method is summarized in Sections 6 and 7.
4. The 1961 L.A.S.L. Intercomparisons of the 3He and 4He Vapor Pressures Fig. 3 is a schematic diagram of the part of the 1961, L.A.S.L. apparatus used by Sydoriak and Sherman2for the intercomparison of the 3He and 4He vapor pressures, in which the helium samples to be intercompared were
--T
Fig. 3. A schematic diagram of the apparatus used for comparison of the vapor pressures of 3He and 4He. 3He was condensed in enclosures A, B and C, and 4He in D. Enclosure C is at same temperature as D, but the Kapitza thermal resistance and flow of heat of condensation of refluxing 4He out from D into A make the temperature of C and D higher than A and B. The heavy lines are copper and thin lines are tempered inconel, a poor thermal conductor.
condensed, and a uniform temperature was established. This part of the apparatus was thermally isolated from its surroundings and equalization of the temperature was established by conduction through copper walls, one mm thick. The design eliminates that difference in temperature between the liquid 3He and 4He samples being intercompared that could arise from the Kapitza thermal resistance if there were a flow of heat between the 3He and 4He samples. Such a flow of heat can arise with different designs of apparatus by reason of the refluxing of 4He vapor in the pressure transmitting tube. 3He was condensed in the enclosures A, B and C, and 4He in D. The intercomparison of vapor pressures was made using enclosures C and I>, and as enclosure C is surrounded by D and the heat of condensation of refluxing 4He in D is carried away through the outside walls of D, the 3He and 4He References p . 512
490
T.R. ROBERTS, et al.
[ax, .6 4
samples were at the same temperature. The vapor pressures of liquid 3He in enclosures B and C were in good agreement above the 4He A-point where there is no iilm flow (see Fig. 4). Below the I-point the temperatures of the enclosures B and C differed as Fig. 4 shows; the difference in temperatures was measured by the difference in vapor pressures. At temperatures below the tpoint a pressure gradient in the 4He pressure transmitting tube (2.7 mm bore) was maintained by the flow of the refluxing vapor over the length of the creeping film. The source of heat evaporating the film was a liquid 4He bath with which the 4He pressure-transmitting tube was in thermal contact at a temperature about 0.1 deg above the temperature in D. The rate of reflux of 4He in space D was measured by the rate of
Fig. 4. Graph to show the magnitude and the temperature dependence of the A T that arises from the Kapitza thermal resistance and the flow of the heat of condensation of re5uxing 4He in enclosure D (in Fig. 3) out to the liquid BHe in enclosure A. TB and TCare the temperatures in enclosuresB and C, derived from vapor-pressuremeasurements.
evaporation of 3He from the space A, and the pressure gradient in the 4He pressure-transmitting tube was calculated. Fig. 5 shows that the ATcorrection for the pressure gradient in the *He tube was small above lo, but below 1" increased rapidly when the temperature was further reduced. The rate of film creep is very sensitive to the cleanliness of the surface over which the creep occurs. Condensation of foreign gases as solids greatly increases the creep-rate; in one case, a 9-fold increase in the creep-rate was observed to take place over a 7-day period. Where necessary, correctionswere made to the measured 4He pressures for the pressure gradient in the pressure-transmitting tube, the rate of reflux being measured in each case by the rate of evaporation from A. The intercomparisons of the vapor pressures of 3He and *He in 1961 exReferencesp . S12
CH. X,
9 51
THE
1962 aHe s c m OFTEMPERAIURES
491
tended from 0.9" to the critical temperature of 3He. The scatter of the data is indicated by the scatter of points in Fig. 6. When the estimated probable errors in the intercomparisons are expressed as temperature intervals, these amount to about f 0.6 millidegree at temperatures below 1.3".
' h
-4/
i
0-UNCORRECTED 0 - CORRECTED FOR FILM REFLUX
Fig. 5. Graph to show the effect on a temperature measurement of the pressure gradient in the 4He pressure transmitting tube in Fig. 3, due to refluxing of the 4He carried up the tube by the creeping film.Tc and TDare the temperatures calculated from the vapor pressures of *He and 4He in the enclosures C and D in Fig. 3.
For comparative purposes vapor pressure values on TS2and T5*for liquid 3He and 4He, respectively, will be found in Table 3. 5. The Determination of the Critical Pressure and Temperature of 3He
A determination of the criticalpressure of 3He was made2 at L.A.S.L. in 1961 using the same apparatus (see Fig. 3) that was used for the intercomparisons of the vapor pressures of 3He and 4He (see Section 4). The increase of pressure of 3He in enclosure C was measured as increasing amounts of 3He were transferred to C from a storage container, the temperature of C being held constant. The temperature of C was determined from the vapor pressure of liquid 4He in D. When C was filled with a single phase, additions of 3He resulted in an increase of pressure in C, but when two phases (liquid and vapor) were present small additions resulted in no change of pressure. This permitted the determination of the 2-phase range in C as a function of T (the vapor pressure of 4He) and the 2-phase equilibrium pressure. After a correction for a 0.031 mole per cent 4He impurity in the 3He the following values were obtained for the critical pressure and temperature of 3He: 873.0 f 1.5 mm Hg (at 0" C) Pressure : References p. 512
[CH.x, 9 6
T.R. ROBERTS, et al.
492
Temperature: 3.3240f0.0018" K on T,,Scale from 4He vapor pressure in enclosure D Temperature: 3.3246 f0.0017" K on TS2Scale, calculated from eq. (18) and P, = 873.0 mm Hg. The uncertainty of f 1.5 mm in the critical pressure is determined by the uncertainty in locating the limiting isotherm for the 2-phase region.
6. Calculation of a Vapor-Pressure Equation below 2" K Using a Theoretical Equation from Statistical Mechanics
At temperatures lower than 0.9" K, the lowest temperature of the useful Pa vs P4 intercomparison measurements, the 1962 3He Scale is based on an equation relating the vapor pressure, temperature, and other thermodynamic quantities. This equation used was obtained from eq. (4). For E , the form used by van Dijk and Durieux20 was employed: E
=In(PVG/RT)
- 2B/VG - (3C/2V:),
(12)
where V, is the molar volume of the saturated vapor and B and C are the second and third virial coefficients in the equation of state PV,/RT = 1 + B/VG + C p g
.
(13)
Dominant terms in eq. (4) below 1" K are Lo and (in contrast to 4He) the entropy integral. Several years ago, before specific heat measurements were extended to very low temperatures, it was suggested10 that for any arbitrary temperature, Tm,the entropy integral term in eq. (4) could be expressed as an equation with two unknown coefficients as T
T
T'
where C,,,is the specific heat of the saturated liquid. It is convenient to take 1" for T,. Substitution of eqs. (14) and (12) in eq. (4), and multiplication by T yields an equation whose terms can be regrouped to give an equation of the form
j.
RF(P,,T) = (L o - C,,,dT)
+ SL(l")-T
(15)
0
in which all terms in F(P3,T), and hence F(P3,T) also, can be evaluated numerically between 0.9" and 2" using empirical data on the properties of 3He, including the 3He vapor-pressure data, P3( T ) . The right side of eq. (15) References p . 512
CH. X,
0 71
THE 1962 *He SCALE OFTEMPERATURES
493
is linear in T and contains two constants: 1
Lo - JC,,,dT
and &(lo).
0
These constants were evaluated by the method of least squares using the empirical data, from 0.9" to 2", to numerically evaluate F(P3,T) for each 3He-4He vapor-pressure intercomparison between 0.9" and 2". The expanded form of F(P3, T ) is given in Section 9.2 and the details of procedure and calculation are given in refs.lOj30 and3. Using values of 1 (LO
- JCsatdT) 0
and &(lo), eq. (15) can be extended to temperatures lower than 0.9" for the calculation of 3He vapor pressures, P3(T), since all terms in eq. (15), except the term In P3, can be expressed as functions of T with numerical coefficients, evaluated from data on the properties of 3He, including C,,, for the liquid. The temperature 0.2" was arbitrarily selected as the lower limit for the extension of F(P3,T).At this temperature, 0.2", the vapor pressure of 3He is only 1.2 x 10-5 mm Hg. Eq. (15) for the vapor pressure of 3He in its expanded form (see Section 9.2) is a theoretical and exact equation, as is eq. (4). To distinguish this theoretically exact equation from the equation derived from it by substituting in F(P3,T) empirical data and empirical equations in T for properties and terms in F ( A , T ) , eq. (15) with these empirical substitutions has been differently written as RF,(P,, T ) = a bT. (16)
+
Fx(P3,T) in its expanded form is given in eq. (19) of Section 9.2. Eq. (16) has been named the "Empirical Thermodynamic Equation" (ETE) for the vaporpressure of 3He, and the 3He vapor-pressure scale of temperatures it represents, the ETE Scale. Since C,,, data were not available for liquid 3He above 2", the upper limit of the ETE Scale is 2". Its lower limit is 0.2". The effect of random and possible systematic errors in the ETE Scale, and a comparison of this ETE Scale with the 1962 3He Scale are discussed in Section 9.2 and refs.4 and 32.
7. Extension of the Vapor-Pressure Equation to the Critical Point The 3He vapor-pressure equation and scale of temperatures was extended References p . 512
T.R. ROBERTS, et al.
494
[m.x, B 8
from 2" to the critical temperature 3.3240°, making use of the ETE equation, eq. (16), which is valid for the range 0.2" to 2", in the following manner: (a) An equation was written for In PSas a function of T having terms in T-1, In T, TO, T, T Z T3 , and T4 with constant (numerical) coefficients
1nP3 = C-,T-'
+ C,,lnT + C, + C, T + C2Tz+ C, T 3+ C 4 T 4 . (17)
The numerical coefficients C-1,C1, and C, in this equation were taken without change from the terms in T-1, In T and TO in the ETE equation, eq. (16). These are the dominant terms in eq. (17) at temperatures below 1". (b) Keeping the coefficients C-1,Cln, COfixed as determined above, the coefficients CI,CZ,C3 and C4 were evaluated by a method of least squares using all the 1961 L.A.S.L. 3He-4He vapor pressure intercomparisons from 0.9"to the critical point. Temperatures on the 1958 4He Scale were calculated from the 4He vapor pressures, A method of multiple variable least squares by W. E.Demings4,was used in which consideration a n a l y ~ i33, s ~developed ~~ was given to the occurrence of error in the measurement of P3 and in the "measurementyyof T on the 1958 4He Scale but not to possible errors in the 1958 Scale itself. The vapor-pressure equation for 3Hethat results from this 2-step procedure is valid from 0.2" to the critical temperature 3.3240'. This equation is
InP, = - 2.49174/T + 2.24846111 T + 4.80386 - 0.28600lT+ 0.198608T2 - 0.050223 7 T 3 + 0.00505486 T4.
+
(18)
This equation defines the 1962 3He Vapor-Pressure Scale of Temperatures. The fit of eq. (18) to the A.O.W. and the 1961 L.A.S.L.3He-4He vaporpressure intercomparisons is discussed in Section 9. Also considered in Section 9 is the thermodynamic consistency of eq. (18) for the vapor pressure of 3He with data for other thermodynamic properties of 3He. 8. The 1962 3He Vapor-Pressure Scale The 1962 3He Scale of Temperatures is defined by the 3He vapor-pressure equation, eq. (18), and extends from 0.25" to the critical temperature, 3.3240" K. The 1962 3He Scale was constructed on the basis of the 1958 4He Scale through reference to measurements of the vapor pressure of 4He. There have been no measurements of the vapor pressure of 3He on the absolute thermodynamic scale except for a measurement at a single temperature by W. E. References p . 512
Keller 86. Measurements of absolute thermodynamic temperatures have been made for some 4He vapor pressures. These can be referred to 3He vapor pressures through the use of the data on the intercomparison of the vapor pressures of 3He and 4He. Their agreement with the temperatures calculated from eq. (18), which is satisfactory, is discussed in Section 9.5. The close agreement of the 1962 SHe Scale with the 1958 4He Scale can be seen in Fig. 6 in which temperatures T62calculated from eq. (18) are compared to T5,,’sdeduced from isothermally compared 4Hevapor pressures and the 4He vapor-pressure tables defining the 1958 4He Scale5. Thispractical equivalence of the 1962 3He and 1958 4He Scales was recognized by the Advisory Committee on Thermometry of the International Committee on Weights and Measures. Quoting from the minutes36 of the meeting of the Advisory Committee in Skvres, France, September 26 and 27, 1962: “I1 a estimd que I’Echelle 3He 1962 doit dgalement Ctre recommandde pour l’usage gdndral, avec la ddsignation T6*. “Les deux dchelles T,, (I’Echelle 4He 1958) et Tsfpeuvent Ctre utilisdes concurrement dans le domaine oa elles sont valables. Cependant, quand il s’agit de l’adoption de cette nouvelle tchelle SHec o m e partie de l’E.I.P.T., on doit prendre soin d’kviter toute ambiguitd dans le domaine de recouvrement avec l’dchelle 4He.” The recommendation by the Advisory Committee of the 1962 3He Scale was approved in October, 1962, by the International Committee on Weights and Measures meeting in Skvres. The International Committee requested the United States and Russian Governments to take steps to make high purity 3He available internationallyfor vapor-pressure thermometry, and to prepare and distribute known mixtures of 3He and 4He for the calibration of or the checking of the calibration of apparatus for measuring the isotopic purity of 8He. The 1962 8He Scale was presented37 at the VIIIth International Conference on Low Temperature Physics in London, September 17-21, 1962. 9.
An Evdwtion of the 1962 3He Scale
9.1 FITOF THe INPUT 1961 L.A.S.L. VAPOR-PRESSURE DATA
One factor in evaluating the 1962 3He Scale is the fit of the input vaporpressure data to the Scale. Figure 6 shows the deviations of the observed data from the final scale as T6,(Pa)- Tsg(P4).The standard deviation of the data from the scale is 0.25 millidegree and the maximum deviation over the full range is 0.6 mdeg. The data may not scatter completely randomly. For References p . 512
496
T.R. ROBERTS,
[CH.X, 8 9
et al.
example, the data points just below 2" are all below the T5*scale and the points just above 2" are ail above the T,, scale. Hence, if one wished to obtain the vapor pressure of 3He which is most probably isothermal with a given 4He vapor pressure, a slightly better value (in terms of consistency 0.a
h
0.4 W
a
0 W
n
i
d
c
t
G
-Os4 0
0
1
-0.t
Fig. 6. Circles are the deviations in tenths of a millidegree of the 1962 SHe Scale from the input (P3,P4) data of Sydoriak and Sherman2: A T = T62 (P3) - T58 (P4). The solid line is the deviation ofthe T62 3He Scale, eq. (18), from the Experimental Thermodynamic Equation (ETE) Scale, eq. (16), in the range of validity of the ETE Scale.
with the observed data) may be obtained by drawing a smooth curve through these data points or by using direct interpolation equations as discussed in ref.2. However the overall fit of the Scale to the input data is very satisfactory in comparison with the errors of measurement of the two vapor pressures. 9.2 FITOF THE EXPERIMENTAL THERMODYNAMIC EQUATION (ETE) SCALE Below 0.9" K the T62scale was evaluated by examining its fit to the ETE scale described in Section 6 and ref.3. As shown by the full line curve in Fig. 6, temperatures calculated from eqs. (18) and (16) are in good agreement; nowhere below 2" do the scales differ by more than 0.4 millidegree. Equation (18) is therefore in effect an ETE scale from 0.2" to 2" and an empirical scale above 2". The effect of possible errors in the various terms of the equation for the ETE scale, over the temperature range from 0.2 to 1.0", should also be considered in evaluating the TS2scale. The ETE scale was obtained from a least squares fit of the 3He-4He vapor-pressure intercomparisons between 0.9 and 2.0" K to eq. (16). The References p . 512
CH. x,
491
THE 1962 aHeSCALE OFTEMPERATURES
8 91
+oat0
-
+0.005
91 \
I
m
c”
-& -If: -
Q
o
Y
I
L~-OD05
Q
L?
-0.010
0
1.5
1.0
0.5
2.0
7’tnK)
Fig. 7. Deviations from the ETE Scale, eq. (16). The circles are deviations from the calculated from eq. (19) for each input (Ps, T.58) fitted equation of the function Fz (Pa,TSS) data point. The central cross-hatched area is the 95 % prediction interval38for a prediction of F(P3, T ) from eq. (16) as calculated from the k values for a and b given in eqs. (20) and (21). The outermost solid curves represent a change in Fz(P3, T )corresponding to a one mjUidegree change in temperature and are calculated as 0.001 T(d In P3/dT.ht.
function F’(Ps,T) is calculated for each experimental Ps, T,,-ObSerVatiOn; deviations of this “observed” value of F,(Ps,T) from the fitted value, [ ( u f b T,,)/R], from eq. (16) are shown as circles in Fig. 7. The function F,(Ps,T) is derived in refs.a and 30 as F,(P,,T)=?’[+hT--nP,
+ i o +f,(~L)-fx(Csat)+&],
(19)
where io is the chemical constant including the nuclear spin degeneracy;
f,( VL) is an empirical power series representing the theoretical and exact integral term,
T n
f( vL) = (
J
1 / ~ v, ( d ~ d ~ d~) , ~ 0
involving the molar liquid volume, VL;f,(Csat) is an equation for the theoretical and exact double integral term of eq. (14),
j.1
f(CS8J = ( l / R T ) dT’ (C,,,/T”)dT” 1
1
based on an empirical power series fitted between 0.2 and 2.0” to data for References p . 512
[a. x, 9 9
T.R. ROBERTS, et al.
498
Csat,the specific heat of the saturated liquid; and
E
is the gas imperfection
term, eq. (12). The least squares values for the coefficients a and b of eq. (16), and their equivalent expressions from eq. (15) are a = Lo-
1
C,,,d T = 17.4459 f 0.0058joule mole-'
(20)
0
and
b = SL(l.Oa)= 9.0098 f 0.0038jo~lemole-~ deg-l
.
(21)
The plus-or-minus values for a and b were found from the fit of the data points to the straight line, a+bT. Mood38 has given an equation for the prediction interval for a single prediction of Fx(P3,T ) from eq. (16) for any value of T.The ETE Scale is just the set of Ps values obtained by solving eq. (19) for In Pa at any given temperature using the predicted value of F,(Ps,T). Hence a prediction interval for F,(Ps,T) can be expressed as an equivalent prediction interval in Ps by neglecting the contribution of possible TABLE1 Effect of possible systematic or random errors on the ETE Scale, eq. (16). Systematic or random changes in the constants a and 6 of eqs. (20) and (21) and of temperatures on the ETE scale below 1" K are listed for the following arbitrary, but plausible, cases: Case 1) the fit of the input (P3,T58) data expressed as the 95 per cent confidence limita8 for the prediction of a value of In P at any single temperature (see Fig. 7); Case 2) random errors of i 3 per cent in fz (Gat); Case 3) random errors of f 3 per cent in e, eq. (12); Case 4) a systematic error of - 3 per cent in all G a t values, and hence in fi(Gat); 3 per cent in all values of E ; Case 5) a systematic error of Case 6) the difference between the empirical function, fs ( VL), and the numerically inte-
+
T
grated values of the thermodynamic term, f( VL) =
VL (dP/dVsat dT; and 0
Case 7) the increase of all values of input temperatures between 0.9' and 2.0" by adding 0.002"to each TSS. case I)
2) 3) 41 5)
joules _ moledeg
i0.006
& 0.004
0.069 0.073
- 0.060 - 0.076
0.094
- 0.026
6)
7)
References p . 512
AT (mdeg) at
46
Aa joules mole
_
1.0"K & 0.4 0.0 f 0.2 0.2 0.2 - 0.1
~
0.8"K
0.6"K
0.4"K
0.2"K
& 0.3 f 0.1 f 0.1 0.5 0.4 - 0.1 1.7
i0.3 i 0.2
i 0.2 f 0.4 0.0
i0.1
0.0 0.8 0.5 - 0.1 1.5
1 .o 0.6 0.0 1.2
f 0.4 0.0 1 .o 0.5 0.0 0.7
CH.x, I91
THE
1962 3He SCALE OFTEMPERATURES
499
errors in the other terms on the right hand side of eq. (19). The 95 per cent prediction interval for a single prediction of Fx(P3,T) [with a 95 per cent probability of containing the statistically "true" value of F'(P3, T)] is shown as the central cross-hatched area in Fig. 7. The outermost curves in this figure show the change in Fx(P3,T) corresponding to a one millidegree change in temperature. The same prediction intervals have been converted to equivalent temperature scale errors as listed in Table 1. The effects of random errors of rt 3 per cent inf,(C,,,) and E on individual temperatures below 1" are also shown in Table 1. Systematicerrors infx(Cs,,), E and infx( VL)would be compensated between 0.9" and 2.0" by the least squares process of fitting the (Pa,T5J data to eq. (16). The scale below 1" would be skewed by such errors as is shown in Table 1 for assumed 3 per cent systematic errors in all values of C,,, or E. Table 1 also shows the systematic errors below 1" resulting from use of the approximate empirical function,f,( VL),instead of the graphically integrated values of the true thermodynamic function, f(V,) [see Table 4 of ref.31. Another possible source of systematicerror is a smooth deviation of the 1958 4He Scale from the thermodynamic Kelvin Scale. The effect on the ETE scale of adding two millidegrees to each scale input temperature is listed in Table 1. The overall effect on the 3He scale of random errors in the specific heat of liquid SHe, the virial coefficient equations, and the 1958 4He Scale might amount to as much as three millidegrees. D ~ r i e u has x ~ ~analyzed the 1962 3HeScale for the effects of possible errors and has reached similar conclusions. 9.3 FITOF THE ARGONNE LABORATORY VAPOR-PRESSURE DATA The 3He-4He vapor-pressure intercomparisons of Abraham, Osborne and Weinstock16 have been used in deriving all previous 3He temperature scales, but these were rejected for the TSzScale because of apparent thermodynamic inconsistency (see refs. 30 and 3l) with recent precise measurements of the latent heat of vaporization of 3He by the same workers28*29.The fit of the A.O.W. vapor-pressure data to the T62Scale is shown in Fig. 8. Above 1.9" the agreement is within 0.5 mdeg. Below 1.5" the A.O.W. data deviate systematically from the 1962 Scale indicating either that their 3He vapor pressures were lower or that their 4He pressures were higher than the data of Sydoriak and Sherman2. For encircled points in Fig. 8, the 4He vapor pressure was measured in a bulb fastened to the 3He bulb. The 4He pressure may have been high because of insufficient correction for effects due to the refluxing superfluid film.For the open crosses and dots, the 4Hepressure was measured in the pumping line in a warm part of the apparatus. The corrected References p . 512
so0
T.R. ROBERTS, et -1
,L
I
al. 1
I
I
+
P, FROM I BULB SERIES
e
+
-71 1.0
I
Q
II U
8
m
BATH BULB BATH BULB
I
I
I
I
1.5
21)
2.5
3.0
r(w
I
Fig. 8. The deviation in millidegrees of the 1950 Abraham, Osborne and Weinstocklo (Ps,P4) data from the 1962 sHe Scale: Tea (Ps) - Tse (P4). The uncircled crosses and small dots are data points below the 2-point for which the 4He cryostat bath pressure was published as P4.
4He pressure may have been high because the effluent gas was significantly colder than the tip of the pressure sensing tube, thus increasing the thermomolecular pressure difference (see Section 1I .2). 9.4.
FITOF HEAT-OF-VAPORIZATION DATA
Latent heat of vaporization, L, data may be used to test the thermodynamic consistency of the temperature scale. The equations used may be either L = T(S0 - SL)
(22)
with the vapor entropy, SG,calculated from the Sackur-Tetrode equation, or
L = T(vG - vL)(dP/dT)snt (23) from the Clausius-Clapeyron equation. The use of these equations for testing 4He Scales of temperature has been discussed extensively; see van Dijk and Durieux20, Durieux39, Berman and Mate4O, and Keller 41. The liquid entropy at 1.0" K (or at any other temperature) may be calculated from eq. (22) and data for C,, &(l.Oo)
= SG(T)
- ( L / T )-
i
(C"JT')dT'.
(24)
1.0
The Argonne (W.A.O.) latent heat measurements2*were undertaken in order to determine the entropy of liquid 3He in this way. At one stage of the derivation of the 1962 SHe Scale, it was proposeda0to use all the W.A.O. latent-heat data and eq. (24) to get an average value for Referencesp . 512
CH. X,
8 91
THE
1962 3He SCALE OF TEMPERATURES
501
SL(l.Oo),which is just the coefficient, byin eqs. (16) and (21). Then, values of (u/R)=F'(P3, T)-(bT/R) from eq. (16) were computed. for every vaporpressure datum point between 0.9" and 2.0" K. This two-step method failed to converge on a stable, consistent pair of constants a and b, and led to the decision to undertake the new L.A.S.L. intercomparisons of the vapor pressures of 3He and 4He. Analysis of the new intercomparisons31 by this same two-step method also failed to yield a consistent set of a and b values, although these data showed much less scatter, as shown in Fig. 9. The latent-
s t A
2.13
I-
n
p
2 . 1 2 ~
i
SS DATA SL(l.Ool = 1.0720 R
1 1
I
1.0
I
I
I
d
l
1.5
1
1
I
I
2.0
i I
T (OK)
Fig. 9. Demonstration of the remaining thermodynamic inconsistency between eqs. (1 6) and (24) involving the 3He latent-heat data, (L,P3), of Weinstock, Abraham and Osborne28329; the (Pa,P4) intercomparisons of Sydoriak and Sherman2; and the 1958 4He Scales, T58 (P4).The indicated value of SL(1 .O") is the average of nine values calculated from eq. (24) for the W.A.O. (L,P3) data and is about 1 per cent lower than the T6Z scale value for that entropy, eq. (21). The solid circles are the individual values of the theoretical constant, u/R, in eq. (16) calculated for each (Pa, T58) data point as F, (Ps, T58)- [SL(l.Oo)T58]/R using the indicated average value of SL(1.0").
heat data were not used to determine the 1962 3He Scale; instead a and b values were obtained by a least squares fit of the 3He vapor-pressure data to eq. (16) using for T the T,, values that corresponded to the 4He vapor pressures observed concurrently with the 3He vapor pressures. The 1958 W.A.O. latent-heat data may be used to test the 1962 3He Scale in the manner explained by Durieux39. In terms of an apparent heat of vaporization, La (defined as the heat necessary to evolve a mole of gas outside of a calorimeter), eq. (23) becomes References p . 512
T.R. ROBERTS, et al.
502
[CH.X, $ 9
La = T ~G(dP/dT),at =
- R ( l + B/VG + C / V z )[dlnP/d(l/T)],,,
= R T2Z(d In PIdT),,,
(25)
+
+
where the compressibility coefficient, Z = ( P VG/RT)=(I B/VG C/V&) from the virial form, eq. (13), of the equation of state, As shown in Fig. 10, the observed values of La average about 0.5 % higher than the values calculated from the 1962 3He Scale using eq. (25), a deviation
4
I
0
c
4 1
dl
,
,
,
0-0.5 I .o
1.2
1.4
I.6
T
[
1.8
, 2.0
[I 22
(OK)
Fig. 10. Deviations of values of La, the apparent heat of vaporization (see Section 9.4), observed by Weinstock, Abraham and Osborne2*.29 from values of La calculated for the 1962 3He Scale from eq. (25) using eqs. (18), (13), (27) and (28).
considerably in excess of the estimated 0.1 % accuracy29 of the La measurements. D ~ r i e u xcalculates ~~ that the La values of Berman and Mate40 for 4He average 0.76 % higher than the values calculated from the 1958 4He Scale between 2.2" and 3.0"K. These differences may be due in part to a deviation of the TS2and T,, scales from the true thermodynamic temperature scale. An estimate of such a difference may be obtained by neglecting the variation of (d In PldT),,, and Z in eq. (25). In this case 6T/T=.3(dLa)/La. From the dashed curve drawn through the 3He data of Fig. 10 this estimate of 6T varies from 3 mdeg at 1.2" K to 1 mdeg at 2.0" K, a not unreasonable range of deviations. For the 4He data, this approximation yields values from 8 mdeg at 2.2"K up to 11 mdeg at 3.0" K which are much larger than appear reasonable for the departure of the 1958 4He Scale from the thermodynamic scale. An error in the temperature scale also would cause both (d In P/d T),,, and Z to change. References p . 512
X,
9 91
THE
503
1962 3He SCALE OFTEMPERATURES
9.5. FITOF GAS THERMOMETER, ISOTHERM AND ACOUSTIC INTERFEROMETER MEASUREMENTS
In principle, gas thermometer, isotherm and acoustic interferometer measurements of the absolute temperature associated with a 4He vapor pressure may be used to check the 1962 3He Scale by directly interpolating the 4He vapor pressure to an isothermal 3He vapor pressure. The direct interpolation equations and tables described in ref.2 are usable for this interpolation of 4He vapor pressures. Since the 1962 3He Scale has been made to agree so closely with the 1958 4He Scale, at least in terms of the (Ps, P4) data of ref.2 (see Fig. 6), little really significant new information would be expected from this kind of comparison. Van Dijk42 has critically reanalyzed all published gas thermometer and isotherm measurements and compared them with the 1958 4He Scale. The majority of these measurements between 1.5 and 3.3" seem to indicate that the thermodynamic temperature may be several millidegrees higher than TS8, although the data scatter over a range of about f 10 millidegrees from TS8. Two preliminary acoustic interferometer results of Cataland and Plumb43 indicate thermodynamic temperatures that are 3 f 2 mdeg higher than T,, at 2.0 and 2.2". The isotherm measurements of Keller35144 are generally conceded to be the most accurate in the liquid helium temperature range, and are the only data for gaseous 3He. During the derivation of the 1962 3He Scale, the isotherm data of Keller were reanalyzed30133by Deming's meth0d3~of least squares adjustment with errors in more than one measured variable. For the 3He isotherm measurements the quantity minimized by the least squares W,( Po-P,>2 Wn(N0-N,)2] where the independently adjustment was observed variables are Po, the 3He gas pressure, and No, the molar density. The fitted function was
I[
F(P,,N,) = P,
+
- N,RT(l
+ BN, + CN:)
=0
(26)
where B and C are the second and third virial coefficients, T is the isotherm temperature for a normabed set of data, and P, and N, are the calculated or adjusted values 'of P and N. The individual weights, W, and W,,, were calculated from Keller's assignment of errors in the various observed quantities. The results of the reanalysis of Keller's gaseous 3He data35 were fitted to empirical equations for the 3He second and third virial coefficients. These equations are References p . 512
and
I=. x, 5 9
T.R. ROBERTS, et al.
504
B = (4.942 - 270.986/T)cm3/mole
(27)
C = (2866/,/T)cm6/mole2.
(28)
The deviations of both sHe and 4Heisotherm temperatures from T62 and T5, are shown in Fig. 11. Keller used a 4He vapor-pressure thermometer for all but one of his isotherms, but his observed P4's have been interpolated to Ps's by use of the direct interpolation equations of ref.2. The weighted average ofthe isotherm temperaturesis 1.50 f 1.O mdeg above the corresponding T,,
f I
2
3 TPK)
1
4
4
6
Fig. 11. Deviations of temperature scales from Keller'ss6~44isotherm temperatures, T ~ Mas,reanalyzed in refs.4 aO and 3a. (Tes - Ti=)for 4Heisotherms; w (Tea - TI=) for 8He isotherms; 0 , IJ (TSE - TI,) for 4 H e and SHe isotherms, respectively. To get Tes,Kellers's 4He vapor-pressure thermometer readings were converted to isothermal P8's by direct P4-to-P~interpolation equations (see ref.a). Lengths of vertical bars for the Tm scale deviations are equal to the standard deviation for Tiso as calculated in the analysis of the isotherm data. The Tea scale deviations include the standard deviation of the conversion from P4 to equivalent Pa. The T6S and TSLI bars terminate in solid and open triangles, respectively.
values and 1.52 f 1.2 mdeg above the T62 values. For one isotherm 3He containing about 0.25 % 4He was condensed in the vapor-pressure bulb. The corrected average pressure for pure 3He was 197.62 f 0.02 mm Hg (T62=2.1550" K) at an isotherm temperature of 2.1537 f 0.0010" K. It was concluded that nothing would be gained by trying to base a *He scale more directly on these isotherm temperatures since the T5,scale is based on these data. Moreover, the T62 scale, as it has been set up to agree with the T,, scale, adequately expresses the experimental 8He-4He vapor-pressure relation for most practical purposes. References p . 512
a.x, I 101
THE 1962 BHe SCALE OF TEMPERATURES
505
10. Thermodynamic Properties of 3He Consistent with the 1962 3He Scale Table 2 gives values of Ps,the vapor pressure of saturated liquid SHe, at ten millidegree intervals on the 1962 3He Scale of Temperatures. These vapor pressures were calculated from the Scale defining equation, eq. (18). Table 3 gives a number of other quantities which are consistent with the 1962 3He Scale, including, for comparison, values of 4He vapor pressures from the 1958 4He Scale. The concentration derivativeof In P,(d In P/dX)T,x= ,where Xis the mole fraction of sHe, is useful for correcting for the 4Heimpurityin a SHe vapor-pressure thermometer.This correction is discussed in Section 11.1. The boiling point of SHe on the 1962 3He Scale, at a pressure of 760 mm Hg at 0" C and standard gravity, is 3.1905'. The table values for Csat,the specific heat of the saturated liquid, were calculated from the empirical equationS fitted to specific-heat data for the derivation of the T,, scale. The calculated values agree with the experimental data to f 1 %; the deviations are given in Table 2 of ref.3. Values for S,, the entropy of the saturated liquid, were calculated from the relation
SL(T)
SL(1.0')
+
1
(C,,JT')dT'
(29)
1.0
using the above-mentioned C,,, equation and the 1962 SHe Scale value of SL(l.Oo), given in eq. (21). Values for the second and third virial coefficient were calculated from eqs. (27) and (28) while V,, the molar volume of the saturated vapor, was calculated from eq. (13). These calculated values of V, agree to within 1 % with the smoothed fit to the experimental data of Ken45 up to 2.8". The values of V,, the molar volume of the saturated liquid, are taken from the table of Kerr and TaylorlS. The values of L,the heat of vaporization, are calculated from eq. (23). The specific heat of liquid SHe under a pressure of a few cm Hg has been measured46 down to a temperature of 0.015' K. The empirical equation for C,, the specific heat at a constant pressure of 0.12 atm, is 7.09Ts for O
+
1 4 C,,,dT
0
R+
C,dT = 0.331 f 0.013joules/mole.
0
This value is in excellent agreement with the value 0.333 f0.010joules/mole References p . 512
TABLE 2
h
h s 'P
3He vapor pressures on the 19623HeScale, at 0" C and standard gravity, 980.665 cm/sec2 The units of pressure are microns (10-3 mm) of mercury below 1" K and millimeters of mercury at higher temperatures 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0,012 1.877 28.115 159.224 544.490 1381.771 2892.496 5304.397
0.024 2.636 34.546 183.339 604.337 1498.789 3089.381 5603.862
0.046 3.633 42.086 210.139 668.902 1622.766 3295.508 5914.815
0.084 4.921 50.864 239.81 1 738.402 1753.928 3511.105 6237.478
0.144 6.561 61.017 272.546 813.059 1892.506 3736.398 6572.071
0.239 8.619 72.686 308.540 893.094 2038.728 3971.613 6918.813
0.382 11.173 86.022 347.992 978.729 2192.821 4216.976 7277.923
1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90
8.842 13.725 20,163 28.360 38.516 50.822 65.467 82,638 102.516 125.282
9.267 14.295 20.900 29.285 39.646 52,178 67.068 84.501 104.660 127.724
9.704 14.881 21.655 30.229 40.799 53.558 68.694 86.391 106.833 130.197
10.156 15.484 22.428 31.193 41.973 54.961 70.345 88.309 109.035 132.701
10.622 16.102 23.220 32.177 43.169 56.389 72.022 90.254 111.266 135.236
11.102 16.737 24.029 33.181 44.388 57.840 73.726 92.228 113.527 137.803
2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90
151.112 180.184 212,673 248.757 288.613 332.425 380.383 432.686 489.549 551.203
153.870 183.276 216.117 252.570 292.813 337.031 385.414 438.164 495.495 557.642
156.661 186.403 219.597 256.420 297.053 341.679 390.489 443.687 501.488 564.131
159.485 189.564 223.113 260.309 301.333 346.368 395.608 449.256 507.531 570.672
162.342 192.760 226.665 264.236 305.653 351.100 400.771 454.871 513.622 577.264
1M
617.907
624.866
631379
638.945
646.066
T62
2 tv
0.07
0.08
0.09
0.592 14.304 101.179 391.106 1070.189 2355.017 4472.71 1 ' 7649.620
0.891 18.105 118.319 438.087 1167.698 2525.542 4739.044 8034.120
1.308 22.673 137.610 489.145 1271.483 2704.626 5016.198 8431.641
11.597 17.388 24.857 34.206 45.629 59.316 75.455 94.229 115.818 140.401
12.106 18.056 25.704 35.252 46.893 60.817 77.21 1 96.258 118.138 143.031
12.631 18.741 26.571 36.319 48.179 62.342 78.993 98.315 120.489 145.692
13.170 19.443 27.456 37.407 49.489 63.892 80.802 100.402 122.870 148.386
165.232 195.990 230.255 268.202 310.013 355.874 405.978 460.534 519.762 583.907
168.155 199.256 233.881 272.206 314.414 360.690 41 1.230 466.242 525.951 590.602
171.112 202.557 237.544 276.249 318.855 365.549 416.526 471.998 532.189 597.349
174.102 205.894 241.244 280.331 323.337 370.450 421.868 477.801 538.477 604.149
177.126 209.266 244.982 284.452 327.861 375.395 427.254 483.651 544.815 611.002
653.241
660.472
667.757
675.098
682.496
9
P
a "X
M 1
0
TABLE 3 P
Thermodynamic properties of 3He and vapor pressures of 4He consistent with the 1962 3He Scale
I r
(" K)
(microns Hg at 0" C and std. g)
(deg-1)
0.0121 28.115 0.2812 544.49 2 892.5 11.445 8 842.4 120.00 20 163 625.02 38 516 2 155.4 65 467 5 689.9 102 516 12 466 151 112 23 767 212 673 40 466 288 613 63 304 380 383 93 733 489 549 132 952 617 907 182 073
73.32 21.04 10.57 6.650 4.721 3.613 2.908 2.425 2.077 1.815 1.611 1.448 1.317 1.210 1.122
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
2.8 3.O
(at X = 1)
joules
0
0.23 0.47 0.70 0.88 0.92 0.92 0.91 0.89 0.88 0.86 0.81 0.78 0.75 0.73 0.71
joules 0
2.728 3.146 3.477 3.812 4.222 4.753 5.414 6.201 7.079 7.983
3.703 5.731 7.071 8.117 9.010 9.824 10.605 11.378 12.159 12.951
-1 350 - 672.5 - 446.7 - 333.8 - 266.0 - 220.9 - 188.6 - 164.4 - 145.6 - 130.6 - 118.2 - 108.0 - 99.3 91.8 85.4
6409 4 532 3 700 3 204 2 866 2 616 2 422 2 266 2 136 2 027 1932 1850 1 777 1713 1655
103 x 107 887 x 103 68 274 16 909 6 776.6 3 476.7 2 060.7 1 338.9 925.5 667.8 496.6 376.7 288.8 221.o 163.1
* Here X is the 3He mole fraction. The 4He mole fraction, (1 - X ) , is used to make the impurity correction.
36.8346 36.7809 36.7203 36.7197 36.7849 36.9073 37.0897 37.3517 37.7151 38.201 38.854 39.680 40.734 42.124 44.049 46.818
20.56 24.39 27.97 31.42 34.61 37.51 40.08 42.29 44.07 45.34 45.99 45.91 44.94 42.84 39.11 32.25
T.R. ROBERTS, et a1.
508
[a€. x, 5 10
obtained by graphical integration of the smooth curve through recent specific heat data given by Strongin, Zimmerman and Fairbank47. From the empirical equation for Csatused for the T62scale,
s
1.0
Csat dT = 2.784 4 0.056joules/mole.
0.2
Thus from the fitted constant, a, in eq. (20) the value of Lo, the heat of vaporization at absolute zero, is 20.56 0.07 joules/mole. The entropy of the saturated liquid can be assumed to be approximately equal to
1
(C,/T')dT'
0
from the 0.12 atm C, equation of Anderson et aZ.4egiven above. From this is 4.05 f 0.17 joules/mole deg. Strongin et al.47also approximation SL(0.23") have computed the entropy from their extrapolation of specific heat data to 0" K. Their value for the entropy at 0.23" is 3.97 f 0.17 joules/mole deg. These values are in good agreement with the value 4.02 f 0.25 joules/mole deg obtained by Weinstock, Abraham and OsborneZ8,who used eq. (22) to calculate S, (1.5") from their three separate latent heat measurements at that temperature. The equivalent of eq. (29) was used by W.A.O. to compute other entropies in the range of the then available specific heat data. The value of
Fig. 12. Graphs of differencesbetween old and new *He and 4He temperature scales.
The SHe curve, TL(Ps)- Tea (Pa),is a graph of the difference between eqs. (11) and scaleS0and the Ts8 scale6 &en in (18). The 4He curve is the difference between the TLSS Table 6 of ref.6. It was intended in their derivations that the TLscale would reproduce the TL55 scale, and that the Tea scale would reproduce the TSSscale. The difference P4) intercomparison data between the two curves is explained principally by the new (Ps, of Sydoriak and Sherman2 used in deriving the TSS3He Scale.
References p . 512
a. x,g 111
1962 BHeSCALE OFTEMPERATURES
509
S, (0.23') consistent with the 1962 3He Scale, calculated from eq. (29), is SL (0.23") =4.09 f0.10 joules/mole deg, allowing a f 2 % Limit of error
for the specifk heat integral term. The good agreement among these values is gratifying but it may be fortuitous. For example, the 3He temperature scale used by Weinstock et ~1.28to assign temperatures to observed 3He vapor pressure was the TL scale, eq. (11). The deviations of the TL scale from the T62 scale are shown in Fig. 12 and range from +6 to -3 mdeg. At 1.5", however, the two scales are in good agreement as to the temperature. Thus the value of T used in computing both S, and LIT in eq. (24) would give essentially the same value for liquid entropy at T m 1.5" on the TL and TSz scales. If, however, a value of L calculated from eq. (23) is used in eq. (24) instead of a W.A.O. experimentalvalue of L, significantly different values are obtained for SL depending upon which of the two scales, TL or T62, is used for the calculation of (dPa/dT),,, since there is a difference of about 0.4 per cent in these derivatives on the two scales as inferred from the slope of the AT curve in Fig. 12 (see, for example, ref.48). 11. Corrections to the Measured Pressure of a 3He Vapor-Pressure Thermometer
Sydoriak and Sherman2 have discussed corrections to be applied to an observed pressure as measured in the laboratory in order to obtain the vapor pressure at the surface of liquid 3He. Three of these corrections are briefly discussed here. 11.1. CORRECTION FOR THE 4He IMPURITY IN 3He
The presence of 4He in liquid BHe thermometers lowers the vapor pressure below that of pure 3He.Most of the3He availablefor purchase up to the present has contained significant quantities of4He; even the 3He used for the T62scale input data measurements2 contained about 0.04 % 4He. Speciallypurified 3He containing less than 0.01 % 4He is being made available for purchase for research requirements such as vapor pressure thermometrythrough the Division of Research of the United States Atomic Energy C o m m i s s i ~ n ~ ~ . The correction for small amounts of 4He may be calculated from the approximate relation: P3- Px w (1 - X)Px(d In Px/dX)T,x=i
(30)
where Px is the observed vapor pressure for mole fraction, X, of 3He, and Pa is the vapor pressure of pure SHe. The derivative (dlnPx/dX),,x=l may be interpolated from a curve drawn through the values of this deRr;ference#p . 512
T.R. ROBERTS, et
510
[CH.x,8 11
al.
rivative listed in Table 3. Although very little data has been published for dilute solutions of 4He in 3He, the liquid phase diagrams from the data of Sydoriak and Roberts60 and Esel'son and Berezniak61 indicate that (dPx/dX), is probably a constant for X20.9. Between 0.6" and 2.0" the Table 3 entries were calculated from the smooth table of ref.50 assuming that (dP,/dX),,,=, =(Ps-P,.,)/O.l. Below 0.6" the table entries have been assumed to go linearly to 0 at 0". At these temperatures liquid mixtures undergo phase separation, and, when this happens, the vapor pressure is independent of changes in X. Above 2.0", the values were calculated from Raoult's law: (dlnPx/dX)T,x=l= l--(P4/P3), since that law is in good agreement with all existing vapor pressure data for X20.89 and T 2 1.7". The correction to a temperature measurement using a 3He sample containing 0.1 per cent 4He ranges from 0.02 mdeg at 0.4" to 0.71 mdeg at 3.2". 11-2. CORRECTION FOR THE THERMOMOLECULAR PRESSURE RATIO The phenomenon of thermal transpiration maintains a pressure difference between the warm and cold ends of a vapor pressure sensing tube. The ratio of the pressure, P,,at the cold end of a tube to the pressure, P,, at the warm end was expressed for 4He by Weber and Schmidt62 as a function of the T(P,,) CK) 0.3
100
d.E
-
u)
w
W
10-
0.5
04
0.6
I,
019
I!O
/.I
h
r(P,)- T I P , ) -
0.7 0.8 0.9 1.0
l!3 I!4 l!S
AT
a
(I
W
9
-1 %
I-
-
-
I I00
PWARM (MICRONS HO)
Fig. 13. Magnitude of the thermal transpiration correction in the apparatus of Sydoriak and Sherman$. The observed pressure, Pw,at the warm end of the vapor-pressure sensing tube, with a vapor-pressure scale of temperatures yields T(P,) as a first approximation to the true liquid temperature, T(Pc).The figure shows the correction, AT, to be subtracted from T(Pw)for the thermal transpiration in the pressure-sensing tube of ref.2 as calculated from eq. (31). Although the ratio of the cold pressure, Pc, to Pw is the same for 8Heand 4He,d T/dP is not the same for the two isotopes and the function AT(Pw) is therefore not the same.
References p . 512
CH. x, Q 121
THE
1962 *He SCALE OF TEMPERATURES
511
warm and cold temperatures, T, and T,, and a compositevariable, rP,, where r is the radius of the pressure sensing tube,
Y,
1 T , +0.181311n In-Po =-hipw 2 Tw Y,
- 0.15823
+ 0.1878 + 0.412 84111 Y, + 1.8311 + + 0.1878 Yw + 1.8311
Y , + 4.9930 Y, + 4.9930
where Y,= (273.15/T)'"47 rPJ13.42, where i = c or w. Experimental measurements 53 have shown that the ratio PJP, for 3He between liquid helium temperature and room temperature does not differ within experimental error from P,/P, calculated from eq. (31) for rPwvalues down to 0.005 cm-mm Hg. The correction amounts to 1 per cent in pressure at rP, =0.2 cm-mm Hg. Detailed tables of the correction are given in ref.58 and a simple graphical representation of the correction for vapor-pressure thermometry expressed as a temperature correction in d d e g r e e s is shown in Fig. 13.
PRESSURE CORRECTIONS 11.3. HYDROSTATIC Where they are significant, temperature and vapor-pressure differences due to differences in hydrostatic pressure in the liquid may be minimized by design. For precision measurements a small correction may be necessary for the hydrostatic head of the vapor column. The magnitude of this correction is discussed in ref.2. For the portion of the vapor in the liquid helium temperature range, virial coefficients given by eqs. (27) and (28) may be used to calculate the vapor density. For higher temperatures, the second virial coefficients tabulated by Kilpatrick, Keller and Hamme154 are available, although densities calculated from the ideal gas law will probably be satisfactory for most purposes.
12. Conclusion and Considerations for the Future This summary of the development of the 1962 3He Vapor-Pressure Scale has emphasized marked differences between liquid 3He and 4He. One is in the low temperature variation of the liquid entropy. The quantum mechanical exchange forces in liquid 3He lead to appreciable correlation of the nuclear spins in the liquid at temperatures as high as 1" K. The difficulty in fitting the entropy curve with a simple analytical expression has led to difficulty in attaching theoretical significance to certain terms in empirical vapor-pressure equations at temperatures below the temperature of fitted data. References p . 512
512
T.R. ROBERTS, @td.
[CEX
The superfluid transition of liquid 4He makes accurate isothermal intercomparisons of 3He and 4He vapor pressures difficult below the 4He I-point. The 1961 intercomparison data of Sydoriak and Shermana, however, are considered to be valid to a few tenths of a millidepee. The 1962 3He Scale, which is based on these intercomparisons,is believed to be in agreement with the 1958 sHe Scale to within two or three tenths of a millidepee over the full range of the intercomparison data, from 0.9' to the critical point, 3.324" K. The 1962 3He Scale has been recommended by the International Committee on Weights and Measures for the calculation of temperatures from measurements of the vapor pressure of 8He. A lower limit for practical vapor-pressure thermometry is 0.25" K corresponding to a 3He vapor pressure of 0.24 microns of Hg. A number of independent measurements, including measurements of heats of vaporization of SHe and 4He, precise isotherm measurements, and preliminary acoustical interferometer measurements, indicate that both the 1962 SHeScale and the 1958 4He Scale may be two or three miUidegrees lower than true thermodynamic temperatures between I" and 3.3" K. When more accurate data on the thermodynamicproperties of SHe and 4He are available together with better determinationsof absolute thermodynamic temperatures, the precise intercomparison data will allow both the 8He and the 4He scales to be adjusted consistently and simultaneously. 3He and *He scales, thermodynamically consistent between 1' and 3", may be extended by precision paramagnetic salt thermometry above 3" using 4He and below 1" using 8He. Also, the scales may be extended downward by thermodynamic calculations, using coefficients evaluated between 1" and 3" as in the development of the 4He Scale. Further measurements on W e that 1962 3He Scale and the TLss would be helpful in improving the Scale are measurements of the vapor pressure and thermomolecular pressure ratio down to temperatures of 0.25", more accurate specific heat measurements on the liquid below 2",and specific heat, latent heat, and high precision vapordensity measurements above 2" K.
REFERENCES 1
a 3 4
R. H. Sherman, S. G. Sydoriak and T. R. Roberts, Los Alamos Scientific Laboratory Report LAMS-2701,July, 1962; to be published t. S. G . Sydoriak and R. H. Sheman. To be publishedtvtt. S. G . Sydoriak, T, R. Roberts and R. H. Sherman. To be published?$f t . T. R. Roberts, R. H. Sherman and S. G. Sydoriak. To be published f.
t, t t
See footnotes to references on p. 514.
CH. XI
THE 1962 8He SCALE OF TEMPERATURES
513
F.G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement and J. K. Logan, J. Research Nat. Bur. Standards 64A, 1 (1960). 6 S. G. Sydoriak, E. R. Grilly and E. F. Hammel, Phys. Rev. 75,30(1949). 7 F. London and 0. K. Rice, Phys. Rev. 73, 1188 (1948). 8 See also, L. Tisza, Physics Today 1, 26 (1948). 9 H.A. Fairbank, C. A. Reynolds, C. T. Lane, B. B. McInteer, L. T. Aldrich and A. 0. Nier, Phys. Rev. 74,345 (1948). 10 E. F. Hammel, Progress in Low Temperature Physics, Vol. I, edited by C. J. Gorter (North-Holland Publishing Co., Amsterdam, 1955), p. 78. 11 E. R. Grilly and E. E Hammel, Progress in Low Temperature Physics, Vol. 111, edited by C. J. Gorter (North-Holland Publishing Co., Amsterdam, 1961), p. 113. 12 E. R. Grilly, E. F. Hammel and S. G. Sydoriak, Phys. Rev. 75, 1103 (1949). 18 E. C. Kerr and R. D. Taylor, Annals of Phys. 20,450 (1962);see also R. H. Sherman and F. J. Edeskuty, Annals of Phys. 9, 522 (1960). 14 J. de Boer and R. J. Lunbeck, Physica 14,510 (1948). 16 3. de Boer, Physica 14, 139 (1948). 16 B. M.Abraham, D. W. Osborne and B. Weinstock, Phys. Rev. 80,366 (1950). 17 H. van DGk and D. Shoenberg, Nature 164, 151 (1949). 18 J. Kistemaker, Physica 12,272 and 281 (1946). 19 S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). 20 H. van Dijk and M. Durieux, Physica 24, 1 (1958). 21 J. R. Clement, Low Temperature Physics and Chemistry, edited by J. R. Dillinger (University of Wisconsin Press, Madison, Wisconsin, 1958), p. 187 (Proceedings of the Fifth International Conference on Low Temperature Physics and Chemistry, Madison, 1957). 8s W.M.Fairbank, W. B. Ard and G. K. Walters, Phys. Rev. 95,566 (1954); W.M.Fairbank and G. K. Walters, Suppl. Nuovo Cimento 9,297(1958). 88 L. Goldstein, Phys. Rev. 96,1455 (1954); 102,1205 (1956). 24 B. Weinstock, B. M. Abraham and D. W. Osborne, Phys. Rev. 89,787 (1953). 25 T. C. Chen and F. London, Phys. Rev. 89, 1038 (1953). 26 S. G. Sydoriak, T. R. Roberts and R. H. Sherman, Proceedings of the VIIth International Conference on Low Temperature Physics (University of Toronto Press, Toronto, 196l), p. 717. 27 D. F.Brewer, A. K. Sreedhar, H. C. Kramers and J. G. Daunt, Phys. Rev. 125, 836 (1959). 28 B. Weinstock, B. M. Abraham and D. W. Osborne, Suppl. Nuovo Cimento 9, 310 (1958). 89 D.W.Osborne, private communication. 80 T. R. Roberts, S. G. Sydoriak and R. H. Sherman, Temperature, Its Measurement and Control in Science and Industry, Vol. 3, Part 1, edited by F. G. Brickwedde (Reinhold Pub. Corp., New York, 1962), p. 75. 31 R.H. Sherman, T. R. Roberts and S. G. Sydoriak, Suppfkment au Bulletin de Vlnstitut International du Froid, Annex 1961-5,p. 125 (Proc. of Meeting of Commission 1 of the International Institute of Refrigeration, London, 1961). 32 M.Durieux. To be published tt. 58 T. R. Roberts, S. G. Sydoriak and R. H. Sherman, see ref.g1, p. 115. 84 W. E. Deming, Statistical Adjustment of Data (John Wiley and Sons, New York, 1943). 35 W. E.Keller, Phys. Rev. 98,1571 (1955). 36 J. A. Hall, Sixikme Rapport du Cornit6 Consultatif de Thermom6trie au Cornit6 International des Poids et Mesures. To be published tt. 5
t, t t
See footnote to references on p. 514.
514 97
38 89 40
41
42 43 44
45 46
47 48 49
50
51 62
53 54
t
T.R. ROBERTS, et a[.
[CH.X
S. G. Sydoriak, T. R. Roberts and R. H. Sherman, Proceedings of the VIIIth International Conference on Low Temperature Physics (Butterworth and Co., London, 1963), p. 437. A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1950), p. 299. M. Durieux, Thesis, Leiden (1960). R. Berman and C. F. Mate, Phil. Mag. 3,461 (1958). W. E. Keller, Nature 178, 883 (1956). H. van Dijk, Progress in Cryogenics, Vol. 2, edited by K. Mendelssohn (Academic Press, Inc., New York, 1962), p. 125. G. Cataland and H. H. Plumb, see ref.37, p. 439. To be published ++. W. E. Keller, Phys. Rev. 97, 1 (1955); “Errata”, Phys. Rev. 100, 1790 (1955); Temperature, Its Measurement and Control in Science and Industry, Vol. 11, edited by Hugh C. Wolfe (Reinhold Pub. Corp., New York, 1955), Chap. 6, p. 99. E. C. Kerr, Phys. Rev. 96, 551 (1954). A. C. Anderson, W. Reese and J. C. Wheatley, Phys. Rev. 130,495 (1963). M. Strongin, G. 0. Zimmerman and H. A. Fairbank, Phys. Rev. 128, 1983 (1962). B. M. Abraham, D. W. Osborne and B. Weinstock, Phys. Rev. 98,551 (1955). G. R. Grove and W. J. Haubach, Jr., see ref.87, p. 441 ; for further information, write Gaseous Isotope Sales, Monsanto Research Corporation, Mound Laboratory, Miamisburg, Ohio,U S A . S. G. Sydoriak and T. R.Roberts, Phys. Rev. 118, 901 (1960). B. N. Esel’son and N. G. Berezniak, B u r . Eksp. Teor. Fiz. SSSR 30, 628 (1956) [transl. Soviet Phys. JETP 3,568 (1956)J. S. Weber and G. Schmidt, Leiden Comm. 246c; Rapp. et Commun. 7e Congr. int. du Froid, la Haye-Amsterdam, (1936). T. R. Roberts and S. G.Sydoriak, Phys. Rev. 102, 304 (1956). J. E. Kilpatrick, W. E. Keller and E. F. Hammel, Phys. Rev. 97, 9 (1955).
Refs.’, as3 and4 have been submitted for publication in the Journal of Research of the U.S.National Bureau of Standards - A. Physics and Chemistry. ttRefs.%3,82,36 and 43 have been submitted for publication in Prods-Verbaux des Sknces, 6e Session (1962) du Comitd Consultatif de Thermomdtrie aupres du Comite International des Poids et Mesures (Gauthier-Wars, Paris, France).
AUTHOR INDEX Abbis, C. P., 477 Abelks, F., 244,264 Abragam, A., 320, 342, 392,446, 447, 448 Abraham, B. M., 59,94,96,447,483,485, 487,499,500,501,502,508,509,513,514 Abrikosov, A. A., 136, 137,185,191 Adenstedt, H., 261 Adkins, C. J., 162,172,192 Adler, J. G., 189 Aharoni,A., 383 Aldrich, L. T., 72,93,513 Alekseevskii, N. E., 199,200,237,260,263 Aleonard, R., 345,362,382 Alers, G. A., 479 Alers, P. B., 261 Allen, J. F., 1,36 Altman, H. W., 450,460,477,478 Ambegaokar, V., 189 Ambler, E., 6,36,446 Andelin, J. P., 370,382 Anderson, A. C., 96,514 Anderson, 0. L., 479 Anderson,P. W., 110, 120, 135, 162, 181, 182, 183, 184, 186, 190, 191, 193, 254, 256,264,309,341,382 Andres, K., 468,470,477,478,479 Andrew, E. R., 448 Andronikashvili, E. L., 11,22,32,36,37 Arase, T.,182,183, I92 Ard, W. B., 486,513 Arens, J., 446 Arnold, G. P., 295 Arp, V., 223,224,227,231,247,262 Arrott, A., 227,247,251,262,263,292,295 Artman, J. O., 447 Atkins, K. R., 3, 4,24, 32,36,37,93 Ayant, Y., 370,382,383 Azbel’, M. Ya., 447 Bailey, C. A., 447 Bailyn, M., 222,250,262 Baird, D. C.,259 Baker, G. A., 298,340,341
Baker, J. M., 448 B a l l d , R. W., 477 Baratoff, A., 189 Barbier, J. C., 345,382 Bardeen, J., 98, 99, 102, 114, 133, 134, 147, 149,180,189,190,191,192 Barieau, R. E., 260 Barret, C. S., 478 Barron, T. H. K., 450, 453, 454, 463, 464, 467,477,478 Bar’yakhtar, V. G., 383 Baurn, J. L., 90,96 Bean, C. P., 243,263 Becker, G., 262 Becker, J. J., 224,243,263 Beenakker, J. J. M., 51, 54, 55, 58, 60,61, 65, 66, 68, 69, 70, 72, 77,93,94,95,96 Behrendt, D. R., 270,294 Behringen, R. E., 229,262 Beljers, H. G., 446 Bellemans, A., 71,94 Belov, K. P., 268,294 Bendt, P. J., 23,37 Benoit, H., 447 Benson, C. B., 21,37 Benson, F. A., 448 Berezniak, N. G., 50,54,55,78,93,94,95, 96,510,514 Berg, G. J. van den, 195,202,218,233,259, 260,261,262,263 Berg, W. T., 463.464,478 Berlincourt, T. G., 210,262 Berrnan, R., 500,502,514 Bermon, S., 177,189,192 Bertaut, F., 345,346,348,351,382 Bethe, H., 261,297,306,331,340,341 Betz, F., 446 Beurtey, R., 448 Bezuglyi, P. A., 170,192 Bingen, R., 71,94 Biondi, M. A., 160,161,189,191,192 Bitter, F., 263 Bizette, H., 342
515
516
AUTHOR INDEX
Blablidze, R. A., 95 Blackman, M., 450,454,477 Blandin, A., 253,256,264 Bleaney, B., 446 Blewitt, T. H.,208,221, 261 Bloch, F., 259,298,305,340,448 Bloembergen, N., 401,446,447 Bloom, M., 447 Blue, M. D., 233,263 Blum, P.,345,382 Blumberg, W. E., 446 Boakes, D., 382 Bogoliubov, N. N., 99, 102, 108, 110, 148, I90 Bogoyavlenskii, I. V., 96 Bohm, H. V., 171,192 Bommel, H. E., 167,192 Bonner, J. C., 341 Boorse, H. A., 98,189,191,205,261 Borelius, G., 216,217,261 Borghini, M., 446,447,448 Born, M., 450,453,477 Borovik-Romanov, A. S., 338,343 Bowen, K. D., 446 Bowers, R., 6,36,222,262 Boyle, W. S., 259 Bozorth, R. M., 268,274,293,294,295,341 Bragg, W. L., 297,340 Brailsford, A. D., 253,264 Brewer, D. F., 90,94,96,513 Bnckwedde, F. G., 513 Bridgman, P. W., 479 Briscoe, C. V.,478 Brockhouse, B. N., 182,183,192,383 Brockman Jr., K. W.. 448 Broek, J. van den, 341,342 Brogden, T. W. P., 417,448 Brout, R., 247,251,264 Brown, A., 98,189 Brown, J., 447 Brown, J. B., 5,6,36 Brown, M. R., 324,342 Browne, M. E., 196,223,224,227,228,231, 247,260,262 Bryant, C. A., 479 Bucher, E., 479 Buckingham, M. J., 95,99,189 Burge, E.J., 6,36 Burger, J. P.,264 Burget, J., 447 Bums. G., 277,294 Burstein,E., 146,150, 151,177, 187,191
Busey, R. H., 327,342 Bijl, D., 450,477 Cable, J. W.,265, 270, 272, 273, 274, 285, 291,293,294,295,332,333,343 Cagliih, G., 182,183,192 Cape, J. A., 210,261 Capel, W.H., 261 Cardona, M., 190 Careri, G., 77,96 Caroli, B., 244,259,264 Carr, R. H., 477,478 Carson, J. W.,383 Carter, W.S.,326,342 Carver, T. R., 447 Casimir, H. B. G., 259 Cataland, G., 514 Catalano, E., 308,338,341,343 Catillon, P., 446 Chamberlain, O., 446,448 Chaminade, R., 448 Chandrasekhar, B. S., 4,36 Chapellier, M., 448 Chapman, A. C., 230,263 Chari, M. S.R., 197,213,214,260,261 Chase, C. E., 13,18,36 Chen, T.C., 486,487,513 Chester, G. V., 95 Child, H. R., 291,294,295 Chitds, B. G.. 467,478 Chisholm, R. C., 327,332,342,343 Chopra, K.L., 5,6,36 Christenson, E. L., 205,208,261 Chynoweth, A. G., 181,182,190 Clark, C. W.,260 Clark, W. G., 449 Clarke, G. R., 72,81,93 Claytor, R. N., 478 Clement, J. R., 240,263,513 Clogston, A. M., 211, 261, 264, 339, 343, 383 Clougherty, E. V., 479 Clusius, K., 463,478 Cohen, E. G. D., 71,94 Cohen, M.H., 147,148,191 Colegrove, F. D., 448 Coles, B. A., 324,342 C o b , B. R., 227,262 Colliings, E. W., 223,227,228,262 Collins, J. G., 454,477 Coltman, R. R., 208,221,261 Combrisson, J., 447,448
AUTHOR INDEX
Cooke, A. H., 316, 328,339,341,342,343, 446,447 Cooper,L.N., 98, 99, 102, 113, 139, 189, 190,191 Coopersmith, M., 247,251,264 C o d , W. S., 154,191 Coremwit, E., 190,211,261,264 Comer, W. D., 268,294 Corruccini, R. J., 244,464,465,478 Coustham, J., 446 Cowen, J. A., 449 Crane, L. T.,243,263 Croft, A. J., 199,259 Cuelenaere,A. J., 262 Culler, G. J., 120,121, 181,190 Culvahouse, J. W., 447 Daane, A. H., 265, 268, 293, 294, 318, 341 Dabbs, J. W. T., 446 Danielian, A,, 303,341,342 Daniels, J. M., 446,447 Daniels, W. B., 455, 472, 473, 477, 479 Dantl, G., 473,479 Das, P., 64,85,86,91,94,96 Dash, J. G., 27,37, 59, 67, 68, 78,94 Date, M., 342 Daunt, J. G., 2, 6, 11,36, 50, 51,59, 71, 72, 85, 87, 88, 89, 90,93, 94,95,96, 98,189, 513 David, R., 95, 171,192 Davis, D. D., 274,294 Dayal, B., 450,453,477 Dayem, A. H., 185,193 Dearborn, E. F., 346,382 De Boer, J., 61,64,94, 195,259,482,513 DeBruyn Ouboter, R., 51, 54, 55, 58, 60, 61,64,65,66,68,69,70,85,86,91,94,96 De Faget de Casteljau, P., 264 De Gemes, P. G., 139, 191, 288, 295, 446 De Haas, W. J., 194,195,213,259,261,343 Dekker, A. J., 251,264 De Launay, J., 455,477 Delsing, A. M. J., 9,36 Deming, W. E., 494,503,513 DeNobel, J., 197, 198, 213, 214, 224, 241, 260, 261,263,479 De Vroomen, A. R., 218,249,250,262,264 Dheer, P.N., 477 Dickson, C. C., 6,36 Dieke, G. H., 383,447 Dimes, G. J., 479 Dietcrle, B., 446
517
Dietrich, I., 177, 186,192 Dillinger, J. R., 513 Dillon Jr., J. F., 368,370,379,381,383 Dmitrenko, I. M., 263 Dobbs, E. R., 171,192,478 Dokoupil, Z., 59,60,72,77,93,95,340,343 Domb, C., 334,340,341,343 Domenicali, C. A., 205,208,246,261,264 Dost, H., 446 Douglass Jr., D. H., 139,176,177,178,179, 180,189,191,192 Douglas, R. L., 371,383 Drewes, G. W. J., 328,342 Dreyfus, B., 383 Du Chatenier, F. J., 198,224,241,260,263, 479 Duffus, H. J., 446,447 Duffy, W. T.,329,331,342 Dugdale, J. S., 199, 200, 202, 260, 463, 478 Dunnington, F. G., 259 Durand, M. A., 478 Durieux, M., 448, 492, 499, 500, 501, 502, 513,514 Duyckaerts, G., 330,343 Dyson, F. J., 298,305,316,341 Edeskuty, F. J., 513 Edmonds, D. T., 316,339,341,343,383 Edwards, D. O., 85,87,88,89,90,95,96 Edwards, H. H., 177,192 Ehrenfest, P., 61,94 Eisenstein, J. C., 330,343 Eliashberg, G. M., 120,190 Elliot, J. F., 294 Elliot, R. J., 276,283,292,294,295,448 Elliot, S. D., 59,94 Em, U., 279,285,294 Erb, E., 447 Esel'son, B. N., 6,36,50, 54, 55,61,64,78, 93,94,95,96,510,514 Essam, J. W., 341 Estle, T. L., 447 Ewald, H., 447 Euatty, J., 448 Fairbank,H. A., 59, 79, 94, 95, 508, 513, 514 Fairbank, W. M., 17,36,64,91,94,95,96, 486,513 Falcoz, A., 448 Falge, R. L., 239,261 Falicov, L. M., 147, 148,191
518
AUTHOR INDEX
Fallot, M., 341,382 Faughnan, B.W., 446 Faulkner, E. A., 199,259 Feher, G., 447,449 Ferrell, R. A., 189 Feynman, R.P.,31,32,37,79,95 Figgins, B. F., 477,478 Filby, J. D., 263 Finn,C. P.B.,316,341,447 Finnemore, D. K., 164,192 Fischer, G., 190 Fisher, J. C., 140,191,246,263 Fisher, M. E., 307,341 Fiske, M.D., 200,260 Flournoy, J. M., 448 Flubacher, P.,479 Fokkens, K.,78,79,95 Foner, S.,341 Ford, N., 448 Forestier, H., 344,382 Forrat, F., 345,346,348,351,382 Forrester, A. T., 189,191 Forstat, H., 327,342 Franck, J. P.,240,262 Franken, B., 233,260 Fraser, D. B., 477 Fredkm, D. R.,184,193 Freeman, A. J., 277,294 Fried,B. D., 120,121,181,190 Friedberg, S.A., 243,263,327,342 Friedel, J., 223,225,229,239,248,253,255, 262,264 Fritz, J. J., 330,342 Frohlich, H., 189 Fukuroi, T., 198,205,233,260,261 Gaidukov, Yu. P., 199, 200, 233, 237, 260, 263 Galkin, A. A., 170,192 Galleron, G., 447 Galt, J. K.,383,478 Gammel, J., 341 Gamzemlidze, G. A. J., 22,28,37 Ganesan, S., 454,477 Garfinkel, M.,479 Garfunkel, M. P.,160, 161, 189, 191, 192, 259 Garland, C. W., 244,263 Garland,J. W.,117,121,122,166,190 Garrett, C.G. B., 322,342 Garton, G., 382 Galwin, R. L.,77,%
Gaudet, G., 261 Gaunt, P.,243,263 Gavenda, J. D., 169,170,192 Geballe, T.H., 190,330,343 Geller, S., 345, 346, 348, 349,382,383 Gerasimenko, V.I., 447 Gemtsen, A. N., 196,213,231,236,245, 260,261,263 Gersteiu, B. C., 341 Gerstenberg, D., 225,226,261 Giaever, I., 99,140,144,146,173,177,178, 181,190,191,192 Giansoldati, A., 224,262 Giauque, W. F., 260,327,330,342,343 Gibbons, D. F., 471,472,479 Gille, M. A., 383 Gilleo, M. A., 348,349,382 Ginsberg,D. M., 161,163, 164,177,189, 192 Ginzburg, V.L.,27,33,37,97,124,125, 189,190 Glover, R.,98,157,164,189 Gnjewek, J. J., 244,464,465,478 Goens, E., 456,467,477 Gold, A. V.,217,218,262 Goldemberg, J., 446 Goldman, J. E., 227,243,262,263 Goldman, M., 449 Goldmann, J., 463,478 Goldring, G., 383 Goldstein, L.,91,%, 513 Goodings, D. A., 335,343 Goodman, B. B., 98,154,156,l89,191 Gordon, J. P.,193, 339,343 Gor’kov, L. P., 99, 125, 127, 131, 136, 137, ia5,190,191 Gorter, C. J., 34,37,61,64,94,95,218,262, 293,340,341,342,343,446,448,513 Garter, E. W., 382 Gossard, A. C., 314,342 Graham Jr., C. D., 268,294 Graham, G. M., 460,477,478 Grassman, P.,477 Green, R. W., 273,294 Griffel, M., 341,342 Griffiths, R.B., 330,337,343 Grilly, E. R.,90,96,448,482,513 Grove, G. R.,514 Griineisen, E., 194,213,259,262,450,456, 467,477 GuenSult, A. M., 250,264 Gugan, D., 200,260,479
AUTHOR INDEX
Guggenheh, E. A., 95 GUiOt-GUillain, G., 344,382 Guptn, K. K., 132,191 Guptill, E. W., 95 Gustafsson, G., 224,262 Guthrie, G. L., 243,263 Guyon, E., 139,191 G$man, H. M.,342 Hake, R. R., 210,261 Hall, H. E., 31,37 Hall, J. A., 513 Hall, J. L., 448 Hall, L. A., 383 Hammel, E. F., 448,482,481,511,513,514 Hanak, J. J., 294 Handel, J. van den, 342 Hanking, B.M.,383 Hansen,W.W.,448 -ding, G. O., 95 Harris, A. B., 370,378,383 Harrison, W. A., 147,191 Hart, E.W., 229,248,262 Hart, H.R., 146,171,181,191,192,447 Hartmans, R.,342 Harvey, A. F., 448 Haseda, T., 329,330, 331,342,343 Has, W.P.A,, 229,230,262 Hatton, J., 199,259,447 Haubach Jr., W.J., 514 Hayes, W.,448 Hebel, L. C., 181, I92 Hcdgmck, F. T., 205,206,221,223,227, 228,230,261,262 Heer, C.V.,59,71,94 Hegland, D. E., 293 Hein, R. A., 239,261 Heisenberg, W.,297,304,340 Henderson, J. W.,381,382 Herbert, G. R., 5,6,36 Herington, E. F. G., 95 Herlin, M. A., 198,259 Hermann, K. W.,294 Heroux. L., 447 Herpin, A., 285,294,346,382 Hill, E. D., 341,342 HirscW, E., 189 Hitterman, R. L., 478 Hoare, F. E., 241,250,263,479 Hofman, J. A., 308,312,341 Holland-Nell, U.,210,261 Hollis-Hallett, A. C., 20,21,37,477
519
Horowitz, N. H., 171,192 Horton, G. K., 454,477 Huang, K., 71,94 Huff, R. W., 120,121,181,190 Huffman, D. R., 463,478 Huiskamp, W.J., 341,342,343,446 Hull Jr., G. W.,190 Hulthen, L., 306,337,338,341 Huntington, H. B., 478 Huzan, E., 477 Hwang, C.,447 Ikeda, T., 198,233,260 Ising, E., 297,340 Ivantsov, V. G., 94 Jaccarino, V., 314,339,341,343,383 Jackson, L. C.,6,36 Jacobs, J. S., 215,231,236, 247,260,263 Jeffers, H. R., 240,263 Jeffries, C. D., 392,441,446,447,448 Jennings, L. D., 265,268,293,318,341,342 Jetter, L.K., 294 Johansson, C. H., 216,261 Johnston, H. L.,72,93, 450,460,477, 478 Jones, D. A., 448 Jones, G. O., 477,478 Jones, H., 217,262 Jones, R. V., 458,478 Josephson, B. D., 149,186,191 Kachinskii, V. N..42,93 Kagaiwada, R., 171,192 Kaganov, M . I., 61,64,94 Kalinkina, I . N., 343 K W , A., 224,240,262 Kan, L. S., 198,259 Kanda,E.,331,343 Kanda, S., 329,342 Kapadnis, D. G., 59,60,93,342 Kapitza, P. L., 1,36 Kaplan, D. E., 383,449 Kaplan, J., 383 Kaplan, T. A., 276,288,294,295 Kastler, A., 448 Kasuya, T., 294 Kabuki, A., 478 Kaufman, L., 479 Kaufmann, A. R., 263 Kedzie, R. W., 447 Keeley, S., 189 Keesom, A. P., 95
520
AUTHOR INDEX
Keesom, P. H., 244,263,479 Keesom, W. H., 61, 94, 95, 194, 216, 259, 261 Keffer, F., 262,278,294 Keller, W. E., 67,68,94,495,500,503,504, 511,513,514 Kellers, C. F., 86,95 Kemp, W. R. G., 197,211,260 Kerr, E. C., 58,59,94,505,514 Keyston, J. R. G., 94 Khalatnikov, I. M., 77,95 Khan, G. A,, 327,342 Kidder, J. N., 17,36 Kido, K., 330,343 Kikuchi, R., 247,251,263 Kilpatrick, J. E., 511,514 King, J. C., 59,79,95 Kint, L. van der, 446 Kip, A. F., 223,224,227,231,247,262 Kistemaker, J., 483,513 Kister, A. T., 55,95 Kitano, Y.,285,294 Kittel, C., 196,223,224,227,228,229, 247, 250, 258, 260, 262, 286, 287, 294, 311, 341,383 Kjekshus, A., 221,260 Klauder, J. R., 263 KIein, M. L., 453,477 Klemens, P. G., 95 Knight, W.D., 196,223,224,227,228,247, 260 Knook, B., 202,205,218,260,261 Kobayasbi, H., 331,342,343 Kobayashi, K., 243,263 Koehler, W. C., 265,270,272,273,274,285, 291,293,294,295 Koerts, W., 328,342 Kohler, M., 218, 261,262 Kohn, W.,289,295 Koppe, H., 98, I89 Korolyuk, A. P., 170,192 Korringa, J., 231, 236,245,263 Kouvel, J. S., 233, 250,259,263,264 Kramers, H. A., 382 Kramers, H. C.,79,91,95,96,513 Krishnan, K. S., 478 Kronquist, E., 224,262 Krylovetskii, A. G., 132, 180, I91 Kubo, R., 227,262,341 Kunzler, J. E., 263,383 Kuper, C. G., 28,37 Kurti, N., 6,36
Kutsischvili, G. R., 446 Lamarehe, G., 261 Landau, L. D., 27, 37, 74, 94, 96, 98, 124, 125,189,190,297,340 Landesman, A., 446,449 Lane, C.T.,198,260,513 Langenberg, D., 146,191 Laquer, H. L., 478 Lazenby, R., 328,342 Lasheen, M. A., 342 Lazarev, B. G., 6,12,36,61,94,95,1 8,25 Leadbetter, A. J., 453,477,479 Leask, M. J. M., 314,334,341 Le Craw, R. C., 383 Lee, D. M., 95 Lee, T. D., 71,86,94 Leech, J. W., 342 Leeden, P. van der, 198,200,260 Lefever, R. A., 383 Le Guillerm, J., 243,263 Legvold, S., 265,268,270,271,273, 274, 293,294,318,341 Leibfried, G., 477 Leifson, O., 446,447 Le Pair, C., 58,60,64,65,66,68,69,80,81, 85,86,91,94,95,96 Leslie, D. H., 210,261 Leslie, J. D,., 161,163, 164,192 Levitin, R. Z., 268,294 Levy, M., 171,192 Lifshitz, E. M., 27,37, 61, 64,94,96,447 Linde, J. O., 195, 196, 205, 213, 216, 259, 260,261 Lingelbach, R., 261 Linhart, P. B., 95 Lips, E., 343 Lipshultz, F. P., 86,95 Liu, S. H., 120,190,479 Livingstone, J. D ,243,263 Livingstone, R., 447 Logan, J. K., 240,263,513 London, F., 59, 72, 95, 97, 189, 486, 487, 513
London, H., 81,93,97,189 Londsdale, K., 479 Lorien, J., 383 Los, G. J., 260 Loutchikov, V. I., 447 Love, W. F.,233,263 Lubbers, J., 329,331,342 Ludwig, W.,477
,
AUTHOR INDEX
Lugt, W. van der, 229,230,262 Lunbeck. R. J., 482,513 Lutes, 0. S.,231,232,263 Liithi, B., 383 Lynton, E. A., 77,95 Lyons, D. H., 288,295 Mac Causland, M. A. H., 447,448 MacDonald, D. K. C., 197, 198, 199, 202, 210, 217, 218, 221, 222, 250, 259, 260, 262, 264 MacKinnon, L., 167,192 MacKintosh, A. R., 292,295 Maillard, R., 448 Maki, K., 189 Makinson, R. E. B., 214,261 Manchester, F. D., 240,262 Mangum, B., 314,341 Manuel, A. J., 342 Mapother, D. E., 164,192,479 Maroni, P., 382 Marshall, B. J., 478 Marshall, W., 241,250,263, 326,341,342, 343 Martin, D. L., 239,240,262,263,264 Martin, R. J., 185,193 Marudin, A. A., 95 Mataresse, L. M., 342 Mate, C. F., 500,502,514 Mathur, V. S.,132,191 Matthias, B. T., 190,211.261,264,341,383 Matthys, C. J., 190, 216, 261 Maxwell, E., 113,190 Maxzi, F., 342 McCammon, R. D., 478,479 McCoubrey, A. O., 161,192 McHargue, C. J., 294 McInteer, B. B., 513 McKim, F. R., 328,342 McNeeley, D. R., 327,342 McWilliams, A. S.,85,87, 88,89,90,95,96 Megerle,K., 140, 144, 146, 173, 177, 178, 181, I91 Meincke, P. P. M., 460,477,478 Meissner, W.,97,189, 194, 259,260 Mellink, J. H., 34,37 Mendelssohn, K., 2,4,6,11,18,36,98,189, 198,259,446 Mendoza, E.. 72,81,93,198,205,210,259 Mkriel, P., 285,294, 346,382 Meservey,R., 37, 176, 177, 179, 180, 189, 192
521
Meijdenberg, C. J. N. van den, 80,81,95 Meyer, A. J. P., 239,263 Meyer, H., 370,378,383 Meyerhoff, R. W., 478 Miedema, A. R., 329, 330, 331, 341, 342, 343,447 Mikumo, T., 448 Mikura, Z., 477 Miles, J., 146, 177, 185,191,193 Miljutin, G., 327,342 Miller, P. D., 448 Miller, R. E., 341 Mills, R. L., 90, 96 Misener, A. D., 1,36 Mitui, T., 243,263 Miwa, H., 276,292,294,295 Montgomery, H., 339,343 Montroll, E. W., 95 Mood, A. M., 498,514 Morel, P., 120, 181,190 Morgan, L., 341 Morris, D. E., 156, 179, 180,191,192 Morris, D. P., 224,262 Morrison, J. A., 453,463,464,477,478,479 Morse, P. M., 264 Morse,R. W., 169,170,171,192 Motchane, J. L., 447 Mott, N. F., 217,223,262 Mueller, M. H., 478 Muhlschlegel, B., 113,190 Muir, W. B., 205,206,221,230,261,262 Murray, R. B., 327,342 Murty, C. R. K., 326,342 MutB, Y.,205,260,261 Myers, H. P., 224,262 Nagamiya, T., 227,262,276,285,294, 341 Nagata, K., 285,294 Nakhimovich, N. M., 260 Nambu, Y.,132,191 NCel, L., 224, 262, 297, 312, 340, 341, 345, 356,358,381,382 Neganov, B. C.,447 Nereson, N. G., 295 Nesbitt, L. B., 113,190 Nettel, S.J., 288, 294 Nicol, J., 99, 146, 177, 185, 190,191,193 Niels-Hakkenberg, C. G., 79,95 Nielsen, J. W., 346,348,382,383 Nier, A. O., 72,93,513 Niewodniczanski,H., 205,261 Nikitin, S. A., 268,294
522
AUTHOR INDEX
Norbury, A. L.,195,260 Nordheim, L.,218,262 Norwood, M.H., 463,478 No&, H., 341 Novikova, S. I., 479 O’Brien, M. C. M., 448 Ochsenfeld, R.,189 Odenhral, M., 447 Ohtsuka, T.J., 331,343 Olsen, C. E., 295 Olsen, J. L., 474,477,479 Ohen, T.,169,170,192 Onsager, L., 29,3 1,37,297,340 Orbach, R.,389,446,447,448 Orton, J. W., 446 Osborne, D. W.,59,94,96, 483,485, 487, 499,500,501,502,508,509,513,514 Overhauser, A. W.,241,253,263,264,392, 395,446 Overton, W. C., 478 Owen, J., 196, 223, 224, 227, 228,231, 247, 260,262,324,342,446 Packard, M. E., 448 Palma, M. U., 328,342 Palma-Vittorelli, M.B., 328,342 Pan,S. T.,263 Panter, C. H., 460,461,478 Papineau, A., 448 Pappztlardo, R., 382,383 Parfenov, L. B., 447 Parmenter, R. H., 138,191 Paskin, A., 308,312,341 Patrick,L.,479 Patterson, D., 463,478 Pauthenet, R., 345,382,383 Pearson, R. F., 382 Pearson, W.B., 197,205,210,217,218,221, 222,260,261,262 Pedko, A. V.,268,294 Peekers, W., 223,262 Pekisen, R. G.,383 Pellam, J. R., 78,95 Perlick, A., 463,478 Perry, R. R., 448 Perahan, P. s., 447 Pen, J. M.,171,192 W o v , V. P., 14, 18,36,37,60,65,93,94 Peter, M., 211,261,264,339,343,383 Petricek, V., 447 Phillips, G. C.,448
Phillips, J. C., 147, 148, 181, 184, 190,191. 193 Phillips, N. E., 338,343 Piercey, D. C.,478 Piette, L. H., 448 Pinch, H. L.,330,342 Pines, D., 114,190 Plumb, H. H., 514 Pokrovskii, V. L., 122,169,190 Poll, J. D., 9,36 Polluntier, R., 223,262 Pomeranchuk, L. J., 57,70,74,90,94,96 Portis, A. M., 401,447 Postma, H.,341 Potters, M . L.,249,250,264 Potts, R. B., 95 Poulis, N. J., 171, 192, 229, 230, 262, 340, 342 Pound, R.V.,401,430,447,448 Range, R. E., 147,189,191 Pretzel, F.E., 449 Price, P. J., 95 Prigogine, I., 71,W Prince, E., 382 Probst, R. E., 72,93 Proctor, W. G., 392,446 Provotorov, B. N., 401,405,447 Pryce, M.H.L., 320,342 Ptucha, T.P., 59,76,77,94,95 Pugh, E. W., 227,262 Pullan, G. T.,261 Pullan,H., 450,477 Purcell,E.M., 401,447
Ramaseshan, S., 461,478 Rapoport, L.P., 132,180,191 Raynal, J., 448 Read,S., 448 Rebka Jr., G. A., 447,448 Reddeman, H.,261 Redfield, A. a.,401,447 Redlich, O., 55,95 Reese, W.,514 Reich, H. A., 77,96 Reif, F., 184,193 Reinitz, K., 478 Remeika, J. P., 314,341 Rempel, R. C., 448 Renss, J., 96 Reynolds, C. A., 113,190,513 Rhoda, B. L., 274,294,341 Rice, 0.K., 513
AUTAOR INDEX
Richards, J. C. S.,478 Richards, P.L., 157,159,161,162,189, I92 Rickaymn, G., 192 Riley, D. P.,477,478 Rimek, D. M., 205,261 Roa, K.R.,182,183,192 Robert, C., 359,382,448 Roberts, B. W., 171,192 Roberts, L. D., 446 Roberts, L. M., 342 Roberts, T.R., 50,51, 52, 54,55,56,57,58, 59, 60, 61, 64,65, 93, 94, 96, 483, 486, 488,510,512,513,514 Robinson, F. N. H., 430,447,448 Robinson,W. K., 327,342 Rocher, Y. A., 293,295 Rodbell, D. S., 479 Rodrigue, G. P., 382 Roe,W. C.,268,294 Rogers,J. S.,189 Rohrer, H., 474,475,477,479 Rohschach, H. E., 198,259 Rollin, B.V., 447 Rosenberg, H. M., 197,212,213,260 Rosenblum, B., 190 Rosset, J., 383 Roubeau, P.,446,448 Rowell, J. M., 181, 182, 183, 184, 186, 188, 190, I93 Rowlinson, J. S.,95 Roy, S. K.,478 Rubin, T., 450,460,477,478 Ruby, R.H., 447 Ruderman, M.A., 229,250,258,262,286, 287,294 Rudnick, I., 171,192 Rushbrooke, G. S.,341 Ryan, D., 382 Ryan,F. M., 227,262 Ryter, C.,447,448 Ryvkin, M. S., 122,190
Sacha, J., 447 Sakudo, T.,228,262 Salimiiki, K.E., 460,478 Salinger, G. L., 96 Salter, L. S.,453,477 Sanders Jr., T. M., 447 Sanderson, J. T.,448 Sandiford, D. J., 95 Sanikidze, D. G., 61,94 Sapp, R. C., 447
.
523
Sato, H., 247,251,263,383 Satterthwaite, C., 154,191 Sauerwald, F., 210,261 Scalapino, D. J., 151, 152, 182,183,191, 193 Schearer, L. D., 448 Schecter, L., 448 Scheffers, H., 260 Scheil, E., 224,240,262 Schieber, M.,383 Schmidt, G., 510,514 Schmit, J. L., 231,232,263 Schmitt, R.W., 200,215,231,236,246,247, 260,263,264 Schmugge, J. T.,441,447 Schrieffer, J. R.,98, 99, 102, 120, 121, 150, 151, 152, 166, 181, 182, 183, 188, 189, 190,191,192,193 Schubnikow, L., 327,342 Schuele, D. E., 460,477 Schultz, C., 446,448 Schumacher, R. T.,448 Scott, P.L., 446 Seidel, G., 244,263 Seki, H., 6,36 Sekula, S. T., 208,221,223,261,262 Serin,B., 113,190,259 Serres, A., 382 Seymour,E. F. W., 199,230,259,263 Shapiro, G., 446.448 Shapiro, S., 99,146,177,185,189,190,191, 193,447 Sharnoff, M.,448 Sheard, F. W., 95,454,477 Sherman, R. H., 483,488,496,499,501, 508,509,510,512, 513,514 Sherrill, M. D., 177,192 Sherwood, R. C., 211,261,264,383 Shibuya, Y.,205,261 Shimizu, M., 478 Shinozaki, S. S.,383 Shirkov, D. V., 102,190 Shoenberg, D., 473,479,513 Shvets, A. D., 94,95 Severs, A. J., 372, 376,383 Silcox, J., 243,263 silin,v. P., 94 Silverman, J., 244,263 Simmons,R. O., 477 Simon,M., 71,94 Skochdopole, R. E., 341,342 Slater, J. C.,453,477
524
AUTHOR INDEX
Slichter, C. P., 181,192,447 Smit, J., 294 Smith, B. L., 478 Smith, J . F., 478 Smith, P. H., 99, 146, 177, 185, 190, 191, 193 Smith, P. L., 261 Smoluchowski, R., 263 Soda, T., 103, I90 Solomon, I., 359,382,401,447, 448 Sommerfeld, A., 217,218,261 Sommers, H . S., 50, 51, 55, 67, 68,93,94 Sonder, E., 223,262 Specht, H., 262 Spedding, F. H., 265,268,270,271,273, 274,293,294,318,341,342 Spence, R. D., 326,327,342 Spohr, D. A., 212,261 Squire, C. F., 478 Sreedhar, A. K., 50,51,93,96,197,211, 260,513 Sreeramamurthy, K., 59,60,93 Srinivasan, R., 454,460,461,477,478 Staas, F. A., 72,73, 78,79, 82,93,95,96 Stanton, R. M., 341 Stapleton, H . J., 447 Star, C., 263 Star, W. M., 261 Steele, W . A., 18,36 Steenland, M . J., 446 Steffens, F., 223,262 Steven, D. H., 448 Stevens, K. W . H., 276,277,294, 326, 328, 342,446,448 Stevenson, R. W . H., 324,342 Steyert, W . A., 96 Stoicheff, B. P., 479 Stoner, E. C., 255,262,264 Stout, J . W., 61,94,260,308,327,332,341, 342,343 Strandberg, M. W. P., 446 Strandburg, D. L., 271,294 Strelkov, P. G., 479 Strong, P. F., 146,177, I91 Strongin, M., 508,514 Struckov, V. B., 14,36 Sugawara, T., 229,230,262,447 Suhl, H., 103,184,190,193,211,261,383 Surange, S. L., 477 Sutton, J., 176, 177, 186,192 Swartz, B. K., 51,56, 57,61, 64,94 Swenson, C. A., 477,479
Swihart, J. C., 118, I90 Swim,R. T., 478 Sydoriak, S. G., 50, 51, 52, 54, 55, 58, 59, 60,64, 65,90, 93,96,482,483, 486,488, 496,499,501,508,509,510,512,513,514 Sykes, M. F., 312, 334,340,341,343 Szklasz, E. G., 449 Taconis, K. W., 51, 54, 55, 58, 59, 60,64, 65, 66, 68, 69, 72, 73, 77, 78, 79, 80, 81, 82,85,86,91,93,94.95,96,448
Taglang, P. J., 239,263 Takahashi, T., 478 Tanuma, S., 221,262 Taran, Y.V., 447 Tauer, K. J., 308, 312,341 Tawara, Y., 205,261 Taylor, B. N., 146, 150, 151, 177, 187, 191 Taylor, G., 327,342 Taylor, K. N . R., 268,294 Taylor, M. A., 264 Taylor, R. D., 59, 78,94, 505,513 Teale, R. W., 383 Teaney, D. T., 339,343 Tedrow, P. M., 86,95 Templeton, D. H., 447 Templeton, I . M., 205, 210, 217, 218, 221, 222,260,261,262 Terrier, C., 342 Teutsch, W.B., 233,263 Tewordt, L., 132,191, I92 Thirion, J., 446,448 Thoburn, W . C., 294 Thomas,D. E., 182,183,184, I90 Thomas, J., 370,382,383 Thomas, J. G., 198,205,210,259 Thompson, A. M., 458,478 Thompson, E. D., 341 Thorsen, A. C., 210,261 Thouless, D. J., 116,190 Tien, P. K., 193 Tinkham,M., 98, 156, 157, 159, 161, 164, 179,189,191,192,372,376,383 Tisza, L., 513 Tkachenko, V. K., 18,37 Tolhoek, H . A., 446 Tolmachev, V . V., 102,116,190 Tournier, R., 243,263 Townsend, P., 176,177,186,192 Trapeznikov, O., 327,342 Troka, W., 446 Tsai, B., 342
AUTHOR INDEX
Tserkovnik, Yu.A., 190 Tsubokawara, I., 314,341 Tsuneto, T., 166,167, I92 Tuan, S. F., 132,191 Turner, E. H., 383 Ubersfeld, J., 447 Ukei, K., 329,342 Unruh, W., 447 Urushadze, G. I,, 383 Uruyu, N., 342 Vajer, Z., 383 Valentiner, S., 224,262 Van Alphen, W. M., 96, Van Baarle, C., 262 Van Dijk, H., 448,492,500,503,513,514 Van Houten, S., 343 Van Hove, L., 102,190 Van Iersel, A. M. R., 95 Van Itterbeek, A., 223,262 Van Kempen, H., 329,331,341,342,343 Van Leeuwen, J. M. J., 71,94 Van Rongen, H. J. M., 261 Van Rossum, L., 446 Van Soest, G., 59, 60, 77,93,95 VanVleck, J. H., 278, 293, 294, 295, 326, 341, 361,362,364,366,382,383,446 Van Wieringen, J. S., 446 Varley, J. H. O., 457,477 Vassel, C. R., 264 Verdone-Thuilier, J., 383 Verkin, B. J., 263 Veyssit, J. J., 243, 263 Veyssid-Counillon, M., 383 Vier, D. T., 449 Vignos, J., 95 Villain, J., 276,294 Villers, G., 383 Vinen, W. F., 11, 12, 14,18,36,37 Visvanathan, S., 477,478 Viswamitra, M. A., 461,478 Vogt, E., 225,226,261 Voigt, G., 194,259 Voogd, J., 194,259 Voorhoeve, W. H. M., 340,343 Wachtel, E., 224,240,262 Wakiyama, T., 268,294 Waldorf, D. L., 239,263,479 Walker, C. B., 184,193 Walker, E. J., 95
525
Walker, L. R., 339,343,370,379,381,383 Wallingford, E., 261 Walsh, W. M., 339,343 Walters, G. K., 64,91,96, 448,486,513 Wansink, D. H. N., 50, 51, 59, 60, 72, 73, 74,93,94 Waring, R. K., 6,36 Wasscher, J. D., 95,342 Watabe, A., 288,295 Watanabe, H., 383 Watanabe, T., 330,331,342,343 Watson, R. E., 271,294 Wayne, M., 447,448 Weaver, H. E., 448 Webb, R. H., 446 Webber, R. T., 212,261 Weber, S.,510,514 Wedgwood, F. A., 292,295 Weger, M., 449 Weil, L., 243, 263 Weinstock, B., 59,94,96,483,485,487,499, 500,501,502,508,513,514 Weinstock, H., 86,95 Weiss, P., 297, 308, 312,340 Weiss, R. J., 479 Weiss, T. J., 341 Welker, H., 97,189 Wernick, J. H., 383 Werthamer, N. R., 103, 132, 139, 190, 191 Wexler, A., 154,191 Wheatley, J. C., 96,447,514 White, G. K., 197, 202, 211, 260, 460, 471, 474,477,478,479 White, R. L., 374,381,382,383 Wickersheim, K. A., 374,381,383 Wiebes, J., 91,95,96 Wilkins, J. W., 150, 151, 152, 182, 183, 188, 191 Wilkinson, H., 312,341 Wilkinson, M. K., 265, 270, 272, 273, 274, 285,293,294,332,333,343 Wilks, J., 95 Williams, E. J., 297,340 Williams, H. J., 211, 261, 264,383 Williams, J., 224,262 Wilson, A. H., 217,262 Windham, P. M., 448 Winkel, P., 9,36 Winter, J. M., 446 Wolf, W. P., 314, 316, 328, 334, 341, 342, 361,362,366,382,383,446,447 Wolfe, H. C., 514
526
AUTHOR
Wolfenskin, L., 448 WoB, P. A., 254,264 Woll Jr., E. J., 288,294,295 W o w E. O., 265,270,272,273,274,285, 291,293,294,295,332,333,343 Wood, D. L., 382 Wood, D. W., 334,343 Wood, P. J., 341 Woods, A. D. B., 182,183,192 Woolf, M. A., 184,193 Wright, W. H., 113,190 Wucher, J., 264 Wyatt, A. F. G., 314,334,341
Yakel Jr., H. L., 294 Yang, C. N.,71,94 Yates, B., 460,461,478
mmx Yntema, G.B., 198,260 Yoshimori, A., 276,294 Yosida,K., 227, 229, 241, 248, 249, 257, 258,262,264,276,288,294,295,341
Zakadze, D. S., 32,37 Zavaritskii, N. V., 156, 172, 177, 186, 191, 192
Zemsky, M.,98,189
Zener,C., 278,294 Zenove’va, K. N., 60,65,94 Zharkov, V. N., 77,78,94,95 Ziman, J. M., 95,196,246,260 Zinunerman, J. E.,241,243,250, 263,479, 508,514
Zubarev, D. N., 117,190 Zucker, I. J., 341
SUBJECT INDEX Anisotropic material, thermal expansion 45% anisotropicmetals, thermal expansion 467 anisotropy energy 276,284 anisotropy hamiltonian 277,278 antiferromagneticchain,susceptibility 331 antiferromagnetism in special lattices 326 antiferromagnets 322ff, 336 apparent heat of vaporization 501 atomic clusters 27
De Haas-VanAlphen effect 230 density of state function 108, 181 deuterium, solid 423 deuterons, dynamic polarization 423ff diamond structure solids, thermal expansion 472 dimensionality of a lattice model 300 dipolar interaction 392 dipole-dipole hamiltonian 392 direct process 388 dirty superconductors 123, 135ff domains 311 domainsize 232 Dyson’s formula 306 dysprosium, anisotropy energy 278 -, magnetic structure 269, 270
Bardeen-Cooper-Schrieffermodel 11Iff Bardeen-Pines interaction 119 Bogoliubov canonical transformation 108,149 boiling point of 8He 51ff Bow-Einstein excitations 100 Calciumfluoride, dynamic polarization 425 cerium, electronic relaxation time 419 -,ion 387 Clausius-Clapeyron equations 62,92,500 cobalt-tutton salts, magnetic properties 335s collective excitations 166 combined ferro- and antiferromagnetism 31W comparison of the vspour pressures of aHe and4He 489 cooperpain 102ff copper, thermal expansion 464 critical point of mixing aHe-4He 88 Critical pressure of 8He 491 Critical temperature of sHe 491 crystalline fields 369 CurieWeiss constant 309 De Boer-Lunbeck vapour pressure equation 483 De Boer’s quantum theory law of corresponding states 482 Debije temperature 453
Electrical magnetoresistance 215 electrical resistivity of diluted alloys of transition metals 200 electron phonon interaction 117 electronic contribution to the thermal expansion 456 electronic paramagnetism 385ff electronic specilk heat 307 electronic thermal resistivity 211, 213 electron spin resonance 227 electron tunnelling, microscopic theory 146 -, in Superconductors 14off elementary domains, observation of 368 elementary excitation in a superconductor 98 Eliashberg interaction 121, 136 empirid thermodynamic equation 493 energy gap 286 energy gap equation 114 energy gap in superconductors, acoustic attenuation 167 -, anisotropy of 161, 169, 186 -, at zero temperature 157 -,magnetic field dependence 123, 159, 173
527
528
SUBJECT INDEX
-, normalized 129 -, parameter 105, 137
-, photon excitations across 157 -,reduced 129 -, temperature dependence 159, 173 entropy of conduction electrons 456 entropy of liquid He 91, 92, 505 -, of solid He 91, 92 equilibrium of a two phase system 61 erbium, magnetic structure 272, 273 europium ferrites 365, 366 excess Gibbs function 39, 57 exchange hamiltonian 280 exchange integral 304, 309ff exchange interaction 252, 275ff, 285, 320ff, 326 -, anisotropy of 320 exchange resonance 377 expansion tensor 456 Fermi-Dirac excitations 100, 101 Fermi function 111 Fermi sphere 291 -, surface 289,457 ferrimagnetic model 354 ferromagnetic metals 334 ferromagnetic resonance 377 ferromagnets 31W, 333ff -, influence of a magnetic field 316, 317 -, magnetic susceptibility 302, 311ff, 333 -, specific heat 312ff fountain pressure 75 free energy of a spin system 280,281 freezing point of SHe4He mixtures 84ff Friedel’s condition 256 Gadolinium, magnetic structure 268 garnet-type ferrites, magnetic interactions 349, 353ff, 360 -, magnetic moments 355 -, preparation 346 giant thermoelectric power 221, 250 Gibbs-Duhem equation 51 Ginzburg-Landau theory 125 glasses, thermal expansion 471 gold wires, electrical resistivity 199 Gor’kov’s theory 131ff Gorter’s number 34 Griineisen constant, magnetic 457 Griineisen relation 452 Hall coefficient 233,234
Hall effect, in noble metal based alloys 233ff -, in transition metal based alloys 237ff Hartree approximation 280 heat conductivity in 3He 75 3He, entropy of liquid 487 4He impurity in 8He 50Y 3He, magnetic susceptibility of liquid 485, 486 8He vapour pressure equation 484 sHe-4He vapour pressures, intercomparison 493,496 Heisenberg model 298, 304ff -, thermodynamic properties 304 helium, boiling point 49, 55 -,dewpoint 49 -,heat conductivity 76, 77 -, melting pressure 90 -, mixing heat 70 -, solid state 41, 84ff Helmholtz free energy 133 holmium, anisotropy energy 279 -, magnetic structure 270, 271 hydrogen nuclei, dynamic polarization 421,422 -, relaxation time 421 hydrogen, solid 423 hydrostatic pressure corrections 511 hyperfine interaction 391ff, 397 -, in metals 395 Ice, thermal expansion 473 ideal vapour pressure equation 483ff independent spin packets 415 inert-gas solids, thermal expansion 464 inverse spin temperature 403 ionic solids, elastic behaviour 463 -, thermal expansion 460 king model 284,298ff -, critical properties 300 -, thermodynamic properties 299ff isothermal 3He vapour pressure 503 isotropic elements, thermal expansion 464 Jeffries’ method 416 Kapitza thermal resistance 489 Keesom-Ehrenfest relations 61 Knight shift 228 Kohn anomaly 289 Kramers doublets 387
SUBrnlNDEx
Lambda temperature 69 lambda transition 40,59 Landau criterion 26 Landau relation 29 latent heat of vaporization 500 lattice ground state energy 306 lattice specilk heat 307ff layer type antiferromagnets 325ff leakage 412 Lennard-Jones parameters 39 linear chains 306, 326, 329 linear chain antiferromagnets 330ff, 337 linear expansion coefficient 462 lithiumfluoride,dynamicpolarization 424 London limit 125,132 London’s thermomechanical relation 75 Lorem parameter 213 Magnetic contribution to the thermal expansion 456 magnetic Gibbs free energy 125, 134 magnetic Helmholtz free energy 125, 133 magnetic remanence 231 magnetic specific heat 238ff, 308,309,371 magnetic susceptibility,of antiferrornagnets 303, 310, 322, 328 -, of a pure metal 222 -, of the Heisenberg model 305 magnetoresistance 206 melting point of 3Ht+He mixtures 84ff metals, electrical resistivity 198 -, thermal expansion 464ff molecular field coefficient 357 molecular field theory 246ff,280,283,328, 357, 367 molecular interaction equation 483 Neodymium, ion 419
-, relaxation time 419 Nernst’s heat theorem 69 Norbury’s rule 195 normal modes of excitation 100 nuclear magnetic resonance 228 nuclear paramagnetism 390 nuclear polarization 396, 398 -, measurements 435ff nuclear resonance, experimental technique 430,431 nuclear spin diffusion 411 nuclear spin lattice relaxation 391 nuclear spin ordering of *He 90 Orbach process 389
529
osmotic pressure in He 72ff, 80 Overhauser effect 395ff overshoot method 9 Pair creation operator 104 pair destruction operator 104 paramagnetic susceptibility 357 persistent currents 23 phase separation diagram 64 phase separation of He mixtures 87, 89 phonon bottle-neck 389 phonons 120, 386 Pippard limit 132 polarization 385 -, enhancement 409 proton polarization, in LMN 417 -, in plastics 416 protons, polarized targets 432ff, 440 Provotorov’s theory 405 pseudo potential method 71 pseudo quadrupole interaction 283 Quadrupole-quadrupole interaction 283 quasi harmonic approximation 451 quasi particle 105, 107, 386 -, free energy 11Iff Raoult’s law 510 rare earth ferrites 361 rare earth gallates 362 -, crystalline fields 363 rare earth garnets 344 -, crystal structure 348 -, magnetization at absolute zero 354 -, magnetostatic properties 349 -, spectroscopic investigations 373ff rare earth ions, magnetic interaction 361 -, nuclear specific heat 371 rare earth metals, conduction band 267 -, crystal structure 266ff -, magnetic moments 265 -, magnetic properties 318 Rayleigh disk 23 Redfield’s theory 401ff refrigeration cycle with He mixtures 81 relaxation hamiltonian 397 ripplon excitation 28 rocksalt structure, potential 454 Rollin film 481 rotational cooling 442 Rudermann-Kittel interaction 229, 250, 286ff
530
SUBJBCTINDEX
Samarium femtes 365,366
saturated resonance 390 saturation parameter 406 scattering of conduction electrons 252 screw structure 269,275ff second sound llff, 14,79 short range ordering in He I 67 single particle excitations 166 solid effect theory 399fF, Mff, 411 spec& heat, of a binary compound 308 -, of liquid He 505 -, of liquid He mixtures 67 specific heat, singularity 301 -, of solid He mixtures 87, 88 -, of some pure metals 244 specific resistance of some copper alloys 202 specific resistivity 248, 249 spin density matrix 402 spin diffusion 393ff -, equation 394, 411 spinel-type femtes 353 spin-flip scattering 167, 215 spin lattice relaxation 387ff, 403 spin polarkation 257 spin-spin correlation coefficient 434 spin systems, homogeneous 411 -, inhomogeneous 414 spin temperature 402 spin waves 305,371, 372 -, antiferrornagnets 338 -, thermal conduction 373 spontaneous magnetization 302, 303, 349,357 -, temperature variation 350 statistical mechanics of *He 57 srratificationof liquid He mixtures 64, 67 superconductingdensity of states 109fT superconductingground state 102 superconductors, specific heat 154 -, thermal conductivity 156 -, thermal expansion 473 superexchange interactions 354 supeduid, equation of motion 74 superfluid flow 74
superfluid He,dry friction 28 -,filmtransfer 2ff -, flow through capillaries 6,25 -, flow through wide channels 1lff -, oscillating disks 20, 31 -, oscillating spheres 20 -, rotation of 23, 31 superimposed films of metals 138ff superleak 75 Terbium, anisotropy energy 278
-, magnetic structure 268 thermal expansion, tensor 456
-, volume coefficient 451
thermal magnetoresistance 215 thermal resistance 211ff thermal transpiration 510 thermoelectric power 219 thermomechanicalel€& 1, 17 thennoosmotic relation in He mixtures 75 thulium, magnetic structure 274 transition metal - transition metal alloys 253ff tunnelling current 149 tunnelling hamiltonian 147ff turbulence centers in liquid He 19 two fluid model 40,75,97 Unsaturated illm flow 6 Vinen’s equation 33 virial coefficients 503 virtual bound state 253ff viscosity of He mixtures 78 vortex wave resonance 31 vortices 29fF W e b theory 297,300 Yosida interaction 248ff Yttrium ferrites, hyperline structure 359 -, magnetic interactions 356 -, spontaneous magnetization 359 -, susceptibility 358
Zero point energy of He 39,482
