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 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
See in general the monograph of K. R. Atkins, Liquid Helium (Cambridge University Press, Cambridge, 1959). H. S. Sommers, Phys. Rev. 88, 113 (1952). B. N. Esel'son and N. G. Beremiak, Sov. Phys. JETP 3,568 (1956). T. R. Roberts and S. G. Sydoriak, Phys. Rev. 118,901 (1960). V. P. Peshkov and V. N. Kachinskii, Sov. Phys. JETP 4, 607 (1957). A. K.Sreedhar and J. G. Daunt, Phys. Rev. 117,891 (1960). D. H. N. Wansink, K. W. Taconis and F. A. Staas, Physica 22,449 (1956). J. G. Daunt, R. E. Probst, H. L. Johnston, L. T. Aldrich and A. 0. Nier, Phys. Rev. 72, 502 (1947); J. Chem. Phys. 15, 759 (1947). J. J. M. Beenakker, K. W. Taconis and Z. Dokoupil, Phys. Rev. 78, 171 (1950). D. H. N. Wansink and K. W. Taconis, Physica 23, 125 (1957). H. London, G. R. Clarke and E. Mendoza, Proc. sec. Symp. liquid and solid We, (Ohio State University press, Columbus, 1960) p. 148; Nature 185, 349 (1960); International Inst. of Refrigeration, Commission 1, London Sept. 1961, p. 337; Phys. Rev. 128,1992 (1962).
18
Z. Dokoupil, G. Van Soest, D. H. N. Wansink and D. G. Kapadnis, Physica 20,1181 (1954); Z. Dokoupil, D. G. Kapadnis, K. Sreeramamurthyand K. W. Taconis,Physica 25, 1369 (1959).
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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
44
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K. W.TACONIS A N D R. DE BRUYN OUBOTER
[CH.U
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).
76 76
L.D. Landau and E. M.Lifshitz, Statistical Physics (Pergamon Press, London, 1958). R. L. Garwin and H. A. Reich, Physica 24, S 133 (1958).
F. A. Staas, K. W. Taconis and W. M. Van Alphen, Physica 27, 893 (1961). I. Pomeranchuk, Zhur. Eksp. Teor. Fiz. SSSR 20, 1919 (1950). 79 J. L. Baum, D. F. Brewer, J. G. Daunt, D. 0. Edwards and A. S. McWilliams, Phys. Rev. Letters 3, 127 (1959); Proc. VII Int. C o d . Low Temp. Phys. (Toronto, 1960) p. 610. 80 S. G. Sydoriak, R.L. Mills, E. R. Grilly, Phys. Rev. Letters 4,495 (1960); Proc. Sec. Symp. liquid and solid aHe (Ohio State University Press, Columbus, 1960); R.L. Mills, E. R.Grilly and S. G. Sydoriak, Ann. Phys. 12, 41 (1961). 81 G. K. Walters and W. M. Fairbank, Bull. Am. Phys. SOC.2, 183 (1957); Proc. first Symp. on liquid and solid SHe (Ohio State University Press, Columbus, 1957) p. 220. 82 L. Goldstein, Phys. Rev. Letters 5, 104 (1960); Phys. Rev. 122, 726 (1961); 128, 1520 77
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
88 89
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
98
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 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
=
+
(<St>>
- TS,
(3.40)
112
[m.m,5 3
D. H.DOUGLASS JR.A N D L. M. FALICOV
where 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
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.
(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 ~ 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 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 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
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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 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
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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
F. London, Proc. Roy. SOC.(London) A152,24 (1935). E. Hirschlaff, Proc. Cambridge Phil. SOC.33, 140 (1937). J. Daunt, S. Keeley and K. Mendelssohn, Phil. Mag. 23, 264 (1937). 4 H. London, Proc. Roy. SOC.(London) A176,522 (1940). 5 H. Welker, Z. Physik 114, 525 (1939). 6 W. Meissner and R. Ochsenfeld, Naturwiss. 21, 787 (1933). 7 V. L. Ginzburg, Zhur. Eksp. Teor. Fiz. SSSR 14, 134 (1946). V. L. Ginzburg, Usp. Fiz. Nauk 48, 25 (1952). V. L. Ginzburg, Fortsch. Physik 1, 25 (1952). lo L. D. Landau, J. Phys. (USSR) 11,91 (1947). l1 J. Daunt and K. Mendelssohn, Proc. Roy. SOC.(London) A 185, 225 (1946). l 2 B. B. Goodman, Proc. Phys. SOC.(London) A66,217 (1953). l3 A. Brown, M. a man s k y and H. A. Boorse, Phys. Rev. 92, 52 (1953). l 4 H. Koppe, Ann. Physik 1,405 (1947). R. Glover and M. Tinkham, Phys. Rev. 108, 243 (1957). l 6 P. L.Richards and M. Tinkham, Phys. Rev. Letters 1, 318 (1958). l7 M. A. Biondi, A. T. Forrester and M. P. Garfunkel, Phys. Rev. 108, 497 (1957). J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106,162 and 108,1175 (1957). lG H.Frohlich, Phys. Rev. 79, 845 (1950). 2o J. Bardeen, Phys. Rev. 80, 567 (1950). a1 J. Bardeen, Phys. Rev. 97, 1724 (1955). 22 M. J. Buckingham, Phys. Rev. 101, 1431 (1956). 23 L. N. Cooper, Phys. Rev. 104, 1189 (1956). 1
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D.H.DOUGLASS JR. AND L. M. FALICOV
[CH.III
N. N. Bogoliubov, D. N. Zubarev and Yu. A. Tserkovnik, Dokl. Mad. Nauk SSSR 117,788 (1957). 25 L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 34, 735 (1958) (transl. Soviet Phys. JETP 7, 505 (1958)). 28 I. Giaever, Phys. Rev. Letters 5, 147 (1960). 27 I. Giaever, Phys. Rev. Letters 5, 464 (1960). 28 I. Giaever, Proc. 7th Int. Cod. on Low Temp. Phys., p. 327 (Univ. of Toronto Press, 1961). 28 J. Nicol, S. Shapiro and P. H. Smith, Phys. Rev. Letters 5,461 (1960). 30 L. Van Hove, Physica Suppl. 26,200 (1960). 31 N. N. Bogoliubov, V. V. Tolmachev and D. V. Shirkov, New method in the theory of superconductivity (Moscow, 1958) (Consultants Bureau, New York). 32 N. R. Werthamer, H. Suhl and T. Soda, Proc. 8th Int. Cod. on Low Temp. Phys. (1962)to be published. 33 P. W. Anderson, Phys. Rev. 110,827and 112,1900(1958). 34 L. N. Cooper, SuperconductivityConf., Cambridge (1959). 35 B. Muhlschlegel, Z. Phys. 155, 313 (1959). 38 E. Maxwell, Phys. Rev. 78,477 (1950). 37 Reynolds, Serin, Wright and Nesbitt, Phys. Rev. 78, 487 (1950). 38 J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955). 39 D. J. Thouless, Phys. Rev. 117,1256 (1960). 40 Geballe, Matthias, Hull and Corenzwit, Phys. Rev. Letters 6,275 (1961). 41 J. M. Rowell, A. G. Chynoweth and J. C. Phillips, Phys. Rev. Letters 9, 59 (1962). 48 J. M.Rowell, P. W. Anderson and D. E. Thomas, Phys. Rev. Letters 10, 334 (1963). 43 D. N.Zubarev, Usp Fiz. Nauk 71,71(1960)(transl.SovietPhys. Uspekhi 3,320(1960)). 44 J. W. Garland, Phys. Rev. Letters 11, 111 (1963);11, 114 (1963)and to be published. 45 J. C. Swihart, Phys. Rev. 116,45 (1959). 48 J. C. Swihart, IBM J. Research Develop. 6, 14 (1962). 47 G. M. Eliashberg, Zhur. Eksp. Teor. Fiz. SSSR 38, 966 (1960) (transl. Soviet Phys. JETP 11, 696 (1960)). 48 S. H.Liu, Phys. Rev. 125, 1244 (1962). 49 P. Morel and P. W. Anderson, Phys. Rev. 125,1263 (1962). 50 G. J. Culler, B. D. Fried, R. W. Huff and J. R. Schrieffer, Phys. Rev. Letters 8,399 (1962). 5 1 J. W.Garland, Proc. 8th Int. Conf. on Low Temp. Phys. (1962)to be published. 62 V. L. Pokrovskii, Zhur. Eksp. Teor. Fiz. SSSR 40, 641 (1961) (transl. Soviet Phys. JETP 13,447 (1961)). 53 V. L. Pokrovskii and M. S. Ryvkin, Zhur. Eksp. Teor. Fiz. SSSR 43, 92 (1962) (transl. Soviet Phys. JETP 16, 67 (1962)). 54 B. Rosenblum, G. Fischer and M. Cardona, Bull. Am. Phys. SOC.I1 8, 192 (1963). 55 V. L. Ginzburg and L. D. Landau, Zhur. Eksp. Teor. Fiz. SSSR 20, 1064 (1950). 56 V. L. Ginzburg, Dokl. Akad. Nauk SSSR 83, 385 (1952). 67 V. L.Ginzburg, Nuovo Cim. 2,1234 (1955). 58 V. L. Ginzburg, Zhur. Eksp. Teor:Fiz. SSSR 29, 748 (1955)(transl. Soviet Phys. JETP 2, 589 (1955)). 58 V. L. Ginzburg, Dokl. Akad. Nauk SSSR 110,358 (1956)(transl. Soviet Phys. Dokl. 3, 102 (1956)). 60 V. L. Ginzburg, Zhur. Eksp. Teor. Fiz. SSSR 34, 113 (1958) (transl. Soviet Phys. JETP 7, 78 (1958)) and Dokl. Akad. Nauk SSSR 118, 464 (1958) (transl. Soviet Phys. Dokl. 3,102(1958)). 61 L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 36, 1918 (1959) (transl. Soviet Phys. JETP 9, 1364 (1959)). 24
1
cn. 1111
THE SUPERCONDUCTING ENERGY GAP
191
L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 37, 1407 (1959) (transl. Soviet Phys. JETP 10,998 (1959)). 83 D. H. Douglass Jr, Phys. Rev. Letters 6, 346 (1961). 84 D. H. Douglass Jr, IBM J. Research Develop. 6, 44 (1962). 65 D. H. Douglass Jr, Phys. Rev. 132,513 (1963). 66 L. P. Rapoport and A. G. Krylovetskii, Dokl. Akad. Nauk SSSR 145, 771 (1962) (transl. Soviet Phys. Dokl. 7, 703 (1962)). 87 N. R. Werthamer, Phys. Rev. 132, 663 (1963). 88 L. Tewordt, Phys. Rev. 132,595 (1963). 8Q K. K. Gupta and V. S. Mathur, Phys. Rev. 121,107 (1961). 70 Y. Nambu and S. F. Tuan, Phys. Rev. 128,2622 (1962). 7 1 J. Bardeen, Rev. Mod. Phys. 34,667 (1962). 72 P. W. Anderson, J. Chem. Phys. Solids 11, 26 (1959). 73 A. A. Abrikosov and L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 35, 1558 (1958) (transl. Soviet Phys. JETP 8, 1090 (1958)). 74 A. A. Abrikosov and L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 36, 319 (1959) (transl. Soviet Phys. JETP 9, 220 (1959)). 75 A. A. Abrikosov and L. P. Gor’kov, Zhur. Eksp. Teor. Fiz. SSSR 39, 1781 (1960) (transl. Soviet Phys. JETP 12, 1243 (1960)). 76 R. H. Parmenter, Phys. Rev. 118,1173 (1960). 77 L. N. Cooper, Phys. Rev. Letters 6, 689 (1963). 78 D. H. Douglass Jr, Phys. Rev. Letters 9, 155 (1962). 79 P. G. de Gennes and E. Guyon, Physics Letters 3, 168 (1963). 80 N. R. Werthamer, Phys. Rev. 132,2440 (1963). 81 P. G. de Gennes (1963) to be published. 82 J. C. Fisher and I. Giaever, J. Appl. Phys. 32,172 (1961). 83 I. Giaever and K. Megerle, Phys. Rev. 122, 1101 (1961). 8 4 1. Giaever, H. Hart and K. Megerle, Phys. Rev. 126,941 (1962). 85 S. Shapiro, P. H. Smith, J. Nicol, J. Miles and P. F. Strong, IBM J. Research Develop. 6, 34 (1962). 86 B. Taylor, E. Burstein and D. Langenberg, Bull. Am. Phys. SOC.I1 7, 190 (1962). 87 J. Bardeen, Phys. Rev. Letters 6, 57 (1961). 88 M. H. Cohen, L. M. Falicov and J. C. Phillips, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 8Q R. E. Prange, Phys. Rev. 131, 1083 (1963). 90 W. A. Harrison, Phys. Rev. 123,85 (1961). Q1 M. H. Cohen, L. M. Falicov and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962). 92 J. Bardeen, Phys. Rev. Letters 9, 147 (1962). 83 B. D. Josephson, Physics Letters I, 251 (1962). 94 J. R. Schrieffer and J. W. Wilkins, Phys. Rev. Letters 10, 17 (1963). 95 B. N. Taylor and E. Burstein, Phys. Rev. Letters 10, 14 (1963). 98 J. R. Schrieffer, D. J. Scalapino and J. W.Wilkins, Phys. Rev. Letters 10,336 (1963). 97 Corak, Goodman, Satterthwaite and Wexler, Phys. Rev. 96, 1442 (1954); 102, 656 (1956). 9 8 Corak and Satterthwaite, Phys. Rev. 102,662 (1956). 99 M. A. Biondi, A. T. Forrester, M. P. Garfunkel and C. Satterthwaite, Rev. Mod. Phys. 30, I109 (1958). loo H. A. Boorse, Phys. Rev. Letters 2, 391 (1958). 101 L. N. Cooper, Phys. Rev. Letters 3, 17 (1959). 102 N. V. Zavaritskii, Zhur. Eksp. Teor. Fiz. SSSR 39, 1193 (1960) (transl. Soviet Phys. JETP 12,831 (1960)). lo3 D. E. Morris and M. Tinkham, Phys. Rev. Letters 6, 600 (1961). 62
192
D. H. DOUGLASS JR. AND L. M. FALICOV
[CH.111
D. E. Morris, Ph. D. Thesis University of California (1962) unpublished. J. Bardeen, G. Rickayzen and L. Tewordt, Phys. Rev. 113, 982 (1959). lo6P. L. Richards and M. Tinkham, Phys. Rev. 119, 575 (1960). Io7 M. A. Biondi and M. P. Garfunkel, Phys. Rev. Letters 2, 143 (1959). loSD. M. Ginsberg and M. Tinkham, Phys. Rev. 118, 990 (1960). logM.A. Biondi, M. P. Garfunkel and A. 0. McCoubrey, Phys. Rev. 108, 495 (1951). P. L. Richards (1963) private communication. ll1 P. L. Richards, Phys. Rev. Letters 7, 412 (1961). 112 C. J. Adkins, Proc. Roy. SOC.(London) A 268, 276 (1962). 113 D. M. Ginsberg and J. D. Leslie, IBM J. Research Develop. 6, 55 (1962). 114 J. D. Leslie and D. M. Ginsberg Phys. Rev. 133A, 362 (1964). 115 D. K.Finnemore and D. E. Mapother (1963) to be published. 116 T. Tsuneto, Phys. Rev. 118, 1029 (1960). lL7J. R. Schrieffer (1963) private communication. 118 H. E. BGmmel, Phys. Rev. 96, 220 (1954). 119 L. MacKinnon, Phys. Rev. 98, 1181 (1955). lZoJ. Bardeen and J. R. Schrieffer, Progress in Low Temperature Physics, ed. C. J. Gorter 3, p. 170 (North-Holland Publishing Co., Amsterdam, 1961). lZ1T. Tsuneto, Phys. Rev. 121, 402 (1961). lZ2R. W. Morse, T. Olsen and J. D. Gavenda, Phys. Rev. Letters3,15, erratum 193 (1959). lZ3P. A. Bezuglyi, A. A. Calkin and A. P. Korolyuk, Zhur. Eksp. Teor. Fiz. SSSR 39, 7 (1960) (transl. Soviet Phys. JETP 12, 4 (1960)). 124 R. W.Morse and H. V. Bohm, Phys. Rev. 108, 1094 (1957). 125 R. David and N. J. Poulis, Proc. 8th lnt. Conf. on Low Temp. Phys. (1962) to be published. lZ6H. R. Hart and B. W. Roberts, Bull. Am. Phys. SOC. I1 7, 175 (1962). 1 2 7 M. Levy, R. Kagaiwada and I. Rudnick, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 1 2 8 M. Levy and I. Rudnick, Bull. Am. Phys. SOC.I1 6, 501 (1961). lZ9 H. V. Bohm and N. H. Horowitz, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. E. R. Dobbs and J. M. Perz, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 131 N.H. Horowitz and H. V. Bohm, Phys. Rev. Letters 9, 313 (1962). 132 N. V. Zavaritskii, Zhur. Eksp. Teor. Fiz. SSSR 43, 1123 (1962) (transl. Soviet Phys. JETP 16, 793 (1962)). 133 D. H. Douglass Jr. and R. Meservey, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 134 P. Townsend and J. Sutton, Phys. Rev. 128, 591 (1962). 135 P. Townsend and J. Sutton, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. N. V. Zavaritskii, Zhur. Eksp. Teor. Fiz. SSSR 41, 657 (1961) (transl. Soviet Phys. JETP 14, 470 (1961)). 137 M. D. Sherrill and H. H. Edwards, Phys. Rev. Letters 6,460 (1961). 138 I. Giaever, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 139 1. Dietrich, Proc. 8th Int. Conf. on Low Temp. Phys. (1962) to be published. 140 S. Bermon and D. M. Ginsberg, Bull. Am. Phys. SOC.I1 8, 232 (1963). 1 4 1 D. H. Douglass, Phys. Rev. Letters 7, 14 (1961). 142 R. Meservey and D. H. Douglass Jr. (1963) to be published. 143 L. C. Hebel and C. P. Slichter, Phys. Rev. 113, 1504 (1959). 144 B. N. Brockhouse, T. Arase, G. Caglish, K. R. Roa and A. D. B. Woods, Phys. Rev. 128, 1099 (1962).
lo4 lo5
CH. 1111 l45 146
l47
I49 l50
151 152 153 154
THE SUPERCONDUCTING ENERGY GAP
D. J. Scalapino and P. W. Anderson, Phys. Rev. 133A, 921 (1964). C. B. Walker, Phys. Rev. 103, 547 (1956). J. C. Phillips, Phys. Rev. 113, 147 (1959). F. Reif and M. A. Woolf, Phys. Rev. Letters 9, 315 (1962). J. C. Phillips, Phys. Rev. Letters 10, 96 (1963). H. Suhl and D. R. Fredkin, Phys. Rev. Letters 10, 131 (1963). P. Smith, S. Shapiro, J. Miles and J. Nicol, Phys. Rev. Letters 6,686 (1961). A. H. Dayem and R. J. Martin, Phys. Rev. Letters 8,246 (1962). P. K.Tien and J. Gordon, Phys. Rev. 129, 647 (1963). P. W. Anderson and J. Rowell, Phys. Rev. Letters 10,230 (1963).
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
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10
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-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
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2
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3405
0-1
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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
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€37
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0 2
Fig. 9.
5
(0
15
20
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0 2
E39
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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
222
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
232
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
KEI YOSIDA
[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 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 , 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
274
KEI YOSIDA
[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)
[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. > 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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14 l5
342
C.WMBANDA.R.MIEDFM4
[CH. VI
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CH. VII
343
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117,1222 (1960).
'
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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 ecca + 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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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
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 ) = 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 * )= 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 %. 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 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
