The magnetic susceptibilities of NiCl2 · 6H2O

The magnetic susceptibilities of NiCl2 · 6H2O

Physica 69 (1973) 67-75 t~) North-Holland Publishing Co. THE MAGNETIC SUSCEPTIBILITIES OF NiCI 2 • 6H20 * A. I. HAMBURGER** and S. A. F R I E D B ...

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Physica 69 (1973) 67-75 t~) North-Holland Publishing Co.

THE MAGNETIC

SUSCEPTIBILITIES

OF NiCI 2 • 6H20 *

A. I. HAMBURGER** and S. A. F R I E D B E R G

Carnegie-Mellon University, Pittsburgh, Pa. 15213, USA

Received 17 April 1973

Synopsis The magnetic susceptibilities of monoclinic single crystals of the antiferromagnet NiCI2 • 6H20 (TN = 5.34 K) have been measured between 1.2 and 4.2 K and from 14 to 20 K. The preferred spin axis below TN was found to lie in the ac plane at an angle of 10.54-1.5 ° from the a direction. At 1.24 K, Z[I ~ Z*II = 0.0012 cgs/mole and extrapolates nearly to zero as T---~ 0 K. Z~ and Zc.,. are nearly equal and only slightly temperature dependent below 4.2 K. They extrapolate to a value of ~0.069 cgs/mole. Z, ii, Zb, Zc~ obey Curie-Weiss laws in the H2 range which can be fitted with a spin hamiltonian for S = 1 having an isotropie splitting factor g = 2.224-0.01, D = -- 1.54-0.5 k, E = 0.264-0.40 k and a molecular field interaction zd = 6.0 k. Assuming the anisotropy below Tr~ to be predominantly uniaxial and of the single-ion type, these results yield an anisotropy field Ha, = 10 kOe at T = 0 K. Z.L(0) then gives a nearest-neighbor interaction zd = 6.3 k or an exchange field, HE = 85 kOe. The energy gap in the spin-wave spectrum at k = 0 implied by Ha. and HE corresponds to TA~ ~ 6 K.

I. I n t r o d u c t i o n . It has b e e n e s t a b l i s h e d t h r o u g h m e a s u r e m e n t s o f the specific heat1), single-crystal m a g n e t i c susceptibility 2' 3), p r o t o n r e s o n a n c e 4' 5), a n d anti-

ferromagnetic

r e s o n a n c e 6) t h a t

NiClz'6H20

becomes

antiferromagnetically

o r d e r e d b e l o w a N 6 e l t e m p e r a t u r e T N = 5.34 K . X - r a y studies 7) o f this m o n o c l i n i c c r y s t a l at r o o m t e m p e r a t u r e s h o w t h e s p a c e g r o u p t o b e C 2 / m . T h e u n i t cell c o n tains t w o N i ++ i o n s a n d has t h e d i m e n s i o n s a = 10.23 A, b = 7.05 A , c = 6.57 A w i t h fl = 122010 '. E a c h N i ÷+ i o n is c o o r d i n a t e d w i t h t w o C1- i o n s a n d f o u r H z O m o l e c u l e s , the C1- i o n s o c c u p y i n g o p p o s i t e vertices o f a d i s t o r t e d o c t a h e d r o n . N e u t r o n d i f f r a c t i o n s ) s h o w s t h a t t h e s t r u c t u r e is essentially u n c h a n g e d d o w n , at least, to 4.2 K a n d , thus, t h r o u g h T N. T h e o r i g i n a l susceptibility d a t a 2, 3) w e r e o b t a i n e d a l o n g the b a n d c axes as well as a l o n g a d i r e c t i o n in t h e a c p l a n e o r t h o g o n a l to t h e c axis, t h e so c a l l e d a ' d i r e c t i o n . B e l o w TN, Xb a n d Xc a r e n e a r l y e q u a l a n d essentially t e m p e r a t u r e i n d e p e n d e n t w h i l e X,, falls w i t h t e m p e r a t u r e s u g g e s t i n g t h a t t h e p r e f e r r e d d i r e c t i o n * Work supported by the National Science Foundation and the Office of Naval Research. ** Permanent address: University of S~o Paulo, S~.o Paulo, Brazil. 67

68

A. 1. H A M B U R G E R

A N D S. A. F R I E D B E R G

of antiferromagnetic spin alignment is close to if not coincident with a'. It was noted3), however, that Xa, does not extrapolate to zero as T --+ 0 K. When neutron diffraction studies of the ordered spin structure were performedg), the preferred direction was found to lie in the ac plane but 22.5 ° away from the a' direction at (a - 10°). Antiferromagnetic resonance observations 6) gave similar results, locating the preferred direction at (a - 7°). They also indicated that the dominant source of anisotropy in NiC12 • 6 H 2 0 was of the single-ion type and provided estimates of the spin-hamiltonian parameters D and E which describe the combined effect of the crystalline field and spin-orbit interaction producing such anisotropy. In this paper we wish to report new measurements of the single-crystal susceptibilities of NiC12 • 6 H 2 0 between 1.2 and 4.2 K and from 14 to 20 K. The objects of this work are 1) to locate independently the preferred direction of spin alignment below TN; 2) to determine XH and Z± and their temperature dependence in the ordered phase; 3) to examine the anisotropy of Z above TN; and 4) to compare the single-ion anisotropy and exchange constants needed to fit the data above and below T N. 2. Experimental details. Magnetic susceptibilities were measured by an ac mutual inductance method at 275 Hz. The technique and most of the characteristics of the apparatus used in this laboratory for such measurements, have been described elsewhere1°). The single-crystal specimens of NiC12" 6 H 2 0 weighed between 500 and 1000 mg and were grown from aqueous solutions of the reagent grade salt. They were oriented using the data of Groth 11). Particular care was taken to insure accurate alignment of the crystal axes with the direction of the oscillatory measuring field. Specimens were attached with G E 7031 lacquer to lucite wedges cut to appropriate angles and their orientation checked with a goniometer. They were then mounted inside cylindrical containers made o f C8 resin. Two such cylinders were attached to opposite ends of a rod of C8 resin to form a rigid assembly 122 m m in length and 17 mm in maximum diameter. This holder was connected to a rod which was manipulated from the top of the cryostat so that the specimens could be moved vertically through the mutual inductance coils. A standard sample, suspended beneath the holder, could also be placed within the coils for calibration purposes. Clearances were such as to permit the holder axis and that of a crystal specimen to deviate from the vertical direction by no more than 0.5 ° . Allowing for errors in the mounting of a crystal, we estimate that orientation of our specimen is accurate to within ___1°. Measuring coils and specimens were immersed directly in baths of liquid 4He, between 1.2 and 4.2 K, and H2, between 13.8 and 20.2 K. Temperatures were determined from the measured vapor pressure in terms of the 1958 4I-/e scale ~z) and the 1951 NBS scale 13) for equilibrium liquid H2.

THE M A G N E T I C SUSCEPTIBILITIES OF NiCI2 • 6H20

69

3. R e s u l t s a n d d i s c u s s i o n . B e l o w TN, t h e s u s c e p t i b i l i t y w a s m e a s u r e d a l o n g d i r e c t i o n s in t h e a c p l a n e w h i c h m a k e t h e f o l l o w i n g a n g l e s w i t h t h e a a x i s ; 8 °, 9 °, 10 °, 11 °, 12 ° a n d 32 ° ( a ' ) . T h e r e s u l t s a r e s h o w n i n fig. 1. T h e s u s c e p t i b i l i t i e s a t ( a - 3 2 °)

i.e. a', a g r e e w e l l w i t h p r e v i o u s l y r e p o r t e d v a l u e s 2' 3) a n d m a y b e c o m p a r e d w i t h t h e m i n fig. 2. T h e s u s c e p t i b i l i t i e s f o r all o t h e r d i r e c t i o n s b e t w e e n a ' a n d t h e a axis {

0.03

l

from

o

o

32"(o')

o:. 12" I1= x

I

~

Deviation

i

an

,~-

,

~ C

+~o

IO°+_l *



o

+

E

l

0 axis:

8 ° + I°

t~x

0.02

o +

~

o o

o



~Q

• x

x

0.01

0

A

....

I....

• .....

L .....

J......

z.o

,.o

i......

L-

1

s.o

4.0

T(K) Fig. 1. Magnetic susceptibilities ofNiCl2 • 6H20 below 4.2 K measured in the ac plane at various angles from the a axis. The preferred direction of spin alignment is found to be a--10.5 ~ 1.5 ° all. The direction a--32 ° -~ a" was originally thought to be preferred. The temperature-independent susceptibility is indicated by the dashed line.

O. lO

I

I

]

l

I

]

0.08 C L Direction

x

"5 E

0.06



b

0

Q'

GII --

Fiippen

z~ Hoseda

0.04 -

e{oL

et ol.

/ o

0.02

o.1z~°

--

° - ~ z x ~oI" ° /

+

+

.v¢

0

I +~" "'+i

0

1.0

~L

3.0

[

1

I

5.0

T(K) Fig. 2. Magnetic susceptibilities of NiCI2 • 6H20 below 4.2 K measured along the principal axes o~l, b, and c± as well as a'. Typical values from refs. 2 and 3 are shown for comparison. Z=lt is an average over measurements nominally at 10 °, 11 ° and 12 ° from the a axis in the ac plane.

70

A. !. H A M B U R G E R

A N D S. A. F R I E D B E R G

are seen in fig. 1 to be much smaller than X,, at low temperatures. In the (a - 11 °) direction, a value of 0,0012 cgs/mole is reached at 1.24 K. Inspection o f the data of fig. 1 suggests that the extremal susceptibility occurs between (a - 10°) and (a - 11°). Noting that the precision of specimen orientation is about ___1° and that other uncertainties equivalent to an additional I ° occur in data for (a - 8 °) and ( a - 10°), we take as the direction of preferred spin alignment [ a - ( 1 0 . 5 ° _+ 1.5°)]. This direction, which we designate as all, agrees within the combined experimental uncertainties with the neutron-diffraction result9), [a - (9 ° + 1°)], and that from antiferromagnetic resonance6), [a - (7 ° +__ 3°)]. When a small net paramagnetic temperature-independent contribution is subtracted from Za,~, the remainder extrapolates to zero at 0 K as should the parallel susceptibility of a simple antiferromagnet. We estimate the temperature-independent diamagnetic susceptibility to be - 1.42 x 10-4 cgs/mole. Using the measured departure of the g-factor (see below) from the spin-only value, the Van Vleck temperature-independent paramagnetic susceptibility is calculated to be +2.16 x 10 -4 cgs/mole. The net result is +0.74 x 10 -~ cgs/mole which is significant at the lower temperatures (see fig. 1). In fig. 2 we also show the results of measurements along the b axis as well as along another direction in the ac plane, c . , which is orthogonal to all and to b. Xc~. and Zb coincide within experimental error, and are nearly temperature independent at a value of 0.069 cgs/mole. This agrees with the values of Zb and gc reported by Haseda et aL2), some of which are shown for comparison in fig. 2. The values of Zb and Xc found by Flippen and Friedberg 3) are slightly larger and rise at the lowest temperatures suggesting that their specimens may have contained some paramagnetic impurity. In the liquid H2 region, 13.8 to 20.4 K, we find that ZaH, Zc~_and Zb are well described by the Curie-Weiss law. Least-squares fits of the data give: 1.232 ± 0.012

cgs

Za,,

T + (7.50 ± 0.12) mole

Zc~ =

1.240 + 0.017 cgs -T + (8.11 ± 0.20) m o l e '

Zb =

1.227 ± 0.017

cgs

(1)

(2)

(3)

T + (8.37 ± 0.35) mole" Fig. 3 shows the data for this region plotted as ( z T ) -~ versus T -x. The fitted Curie-Weiss law, in the form (zT) -~ = C-1(1 + O/T), is shown as a straight line in each case. This plot is convenient a 4) because it yields directly the reciprocal of the Curie constant C = Ng21t~S(S + l)/3k when we let T -1 ~ 0 and thus emphasizes the coincidence of C for the three crystal directions. The splitting factors obtained from C, taking the spin of the Ni ÷ ÷ ion to be S = 1 are gall = 2.219

T H E M A G N E T I C S U S C E P T I B I L I T I E S O F NiCI2 ' 6 H 2 0

71

_ 0.008; gcx = 2.227 +__0.009; and gb = 2.215 -t- 0.012. Within the experimental uncertainty, the splitting factor is thus isotropic and equal to g = 2.22 __ 0.01. In order to analyze further the data taken above TN we snan assume that each of the equivalent Ni + + ions in the crystal can be described by a spin hamiltonian of the form: o~° = D S 2 + E(S~ - S 2) + gli n H " S + 2 z S S " ,

(4)

where S = 1, g = 2.22, and the z axis lies along the all direction. D and E measure, respectively, the contributions of the axial and rhombic parts of the crystal field to the zero-field splitting o f the Ni + + ion ground state. The term 2 z J S . represents the molecular-field approximation to the sum of the exchange interactions, each of strength J, of the given ion with its z equivalent nearest neighbors. The principal susceptibilities of a system of N such interacting ions are given by Watanabe 15). For k T >> IDI, [El, and zIJI they approximate the Curie-Weiss law with effective Weiss constants: Oa,~ = (D + 4zd)/3k, Ob = - ( O - 3E - 8zJ)/6k, Oc± = --(D + 3E -- 8zJ)/6k.

(5) (6) (7)

A value for z J is easily calculated using the relation OAv = (0,,, + Ob + 0~_~)/3 = 4zJ]3k.

(8)

Referring to eqs. (1), (2) and (3) we find that the average of the measured O's is 8.0 K which agrees reasonably well with a reported powder value 16) of about 7 K and another value from crystal measurements 2) of about 10 K. From our result and eq. (8), we find z J / k = 6.0 +_ 0.4 K, where the sign is antiferromagnetic. From the additional relations O/k = --3(0AV

-

Oa,l)

(9)

and E / k = Ob - 0 ~ . ,

(10)

we find D = - 1 . 5 + 0.5 k or - 1 . 0 2 + 0.33 cm -1 and E = 0.26 + 0.40 k or 0.18 + 0.27 cm -1. These values of D and E agree well with those deduced from antiferromagnetic resonance data 6) taken below TN, namely, D = --1.2 + 0.2 era- 1 and E = 0.15 + 0.2 c m - 1. Let us consider now some properties of the antiferromagnetic phase. The neutron-diffraction work of Kleinberg 9) shows that the ordered spin structure below T~ belongs to the magnetic space group I~2]e. Within an (001) plane, each spin is antiferromagnetically aligned with respect to four nearest neighbors. Neighboring spins on adjacent (001) planes are also antiferromagnetically aligned.

72

A . I . H A M B U R G E R A N D S. A. F R I E D B E R G

Date and Motokawa 6) have estimated the dipolar contribution to the anisotropy field for this structure and found it to be nearly uniaxial and equal to 0.84 kOe. This is smaller by an order of magnitude than the single-ion effect, as we shall see. F r o m the analysis of their antiferromagnetic resonance results, Date and Motokawa conclude that the magnetocrystalline anisotropy in ordered NiC12 • 6 H 2 0 is orthorhombic. The most obvious consequence of this departure from uniaxial symmetry for the low-field susceptibility with which we are concerned might be a difference between those values measured in the two directions perpendicular to the preferred axis. However, above TN the fitted value of E is smaller than the experimental error and such a difference has not been clearly established. Below TN we see in fig. 3 that, although there may be some tendency for Zc± to be slightly below Xb at the lowest temperatures, the difference is small and near the limit set by experimental error. To a good approximation, Zc± and Zb behave like the perpendicular susceptibilities of a uniaxial antiferromagnet and may be analyzed on that basis. r

i

F

[

I

I

I

b o.o_~//

1.2

0

F:

0.8

C/,

Data in H a range with

T

f i t t e d Curie-Weiss Law

' : ' (~

I.-X

X--T" 0.4

o

~

~

o.o2

~

.~.--~

o.o4

I+

2)

~

0.06

±T ( K -J )

Fig. 3. Magnetic susceptibilities o f NiC12 • 6I-{20 in the liquid H2 region (T > TN = 5.34 K) measured along the all, b, and c.L directions and plotted as ( z T ) - 1 versus T-~.

Assuming the anisotropy energy to vary with the angle q~ between the sublattice magnetization and the preferred axis as K ( T ) sin 2 q~, it may be shown 17) that at T = 0 K the single-ion mechanism gives K(O) = - D N S ( S

- ½),

(11)

where N i s the number o f N i + + ions per unit volume and S = 1. Letting D = - 1.5 k, the value deduced from our results above TN, we find K(0) = 6.3 x 107 erg/ mole. Defining an anisotropy field as Ha, = K / M o , where Mo = 1/2Ng,u~S = 6200 erg/Oe mole, and substituting this value of K(0), we obtain Ha, = 10 kOe. This will outweigh the estimated dipolar contribution, 0.84 kOe, mentioned earlier. For a uniaxial two-sublattice antiferromagnet, in either the molecular field approximation or the lowest-order spin-wave approximation~8), one finds

THE MAGNETIC SUSCEPTIBILITIES OF NiCI2 ' 6H20

73

1

xl(0) - A + H R/2Mo

(12)

where d = 4zJ]Ng2p 2. For antiferromagnets with high N6el points, usually A ~ H~,,/2M o and the latter term may be ignored. In the present instance, however, the anisotropy term is not completely negligible and we estimate it using the above value of Ha.. The dipolar contribution may be roughly compensated by the effects of orthorhombicity and, therefore, has not been included. Extrapolating %c± to 0 K, we take X.(0) = 0.069 cgs/mole and find A = 1/X_L(0) -- H~J2Mo = 14.49 -- 0.83 = 13.66 Oe2mole/erg. Thus, from the definition of A, we conclude that z J = 6.3 k which agrees very closely with the value obtained for T > T N. It should be noted that eq. (12) contains only the nearest-neighbor interaction in spite of the fact that intrasublattice or next-nearest neighbor coupling is explicitly included in the model for which it is derived. On the other hand, the latter interactions are ignored in the derivation of eqs. (5), (6), and (7). The agreement of the values of z J obtained in these two ways thus suggests that nearest-neighbor exchange is the dominant interaction in NiCI 2 • 6H20. An effective exchange field is defined in terms of the quantity A by the relation H E = A M o . Using the value of A obtained above, one finds H E = 85 kOe, as compared with H E = 86 _+ 1 kOe from antiferromagnetic resonance6). The energy gap in the spin-wave spectrum of a two-sublattice antiferromagnet at zero wave vector in the absence of an applied field is expressed in terms of a " g a p temperature", TAr- It arises from the presence of anisotropy is' 19,2o) and depends on H E and A~, as follows: kTAE = gI.tB[Ha.(2HE +

nan)3 42-.

(13)

Substituting values for g, H~n, and HE given above, we find TAr = 6.4 K. This gap is measured in antiferromagnetic resonance experiments as a critical field at which the antiferromagnetic phase becomes unstable with respect to the spin-flopped phase. At 0 K, we obtain Ho = [H,,(2HE + Ha,)] ~.

(14)

Using our values of HE and Ha,, one finds H e = 43 kOe which compares well with the experimental result H c = 40 ___ 1 kOe corresponding to TAE = 5.8 K. Recently Becerra and Paduan u*) have determined the spin-flop boundary for NiCI2 • 6HuO by a direct high-field susceptibility method. At 1.1 K they find a critical field of 39.6 kOe. Anderson and Callen 2a) have pointed out that, because of the first-order character of this phase transition, the critical field measured by many techniques may actually not be that of eq. (14) but rather U ° = [nan(En~ - n~n)Y.

(15)

In systems with small exchange fields the difference between He and H ° can be

74

A.I. HAMBURGER AND S. A. FRIEDBERG

significant. With the previously used constants, eq. (15) gives H ° = 40 kOe at 0 K which is quite close to the measured value. It was noted by Robinson and Friedberg ~) that the heat capacity of NiC12 • 6 H 2 0 could be approximated rather well from their lowest temperature, 1.4 K, to about 4.3 K at 0.8 TN by spin-wave theory in the weak anisotropy limit. Donaldson and Edmonds z3) extended the measurements down to 0.7 K, by which point the presence of an energy gap in the spin-wave spectrum had become clearly evident. They fitted the data with a spin-wave model 19) of a uniaxial two-sublattice antiferromagnet, determining structure-dependent constants in the theory by normalization to measured values at temperatures where the heat capacity is proportional approximately to Z 3. The best fit was obtained for TA~ = 4.0 + 0.5 K. The temperature dependence of the parallel susceptibility should reveal the presence of an energy gap at a higher temperature 19, so) than the heat capacity. We have, therefore, fitted our data for Za,, with the same simplified spin-wave model. While determination of constants by normalization is not practical in this case, the constants are the same as those appearing in the expression for the heat capacity and are thus available. In fig. 4 we show the Z,,, results together with theoretical curves for three different values of TAE. An excellent fit of the data up to about 4 K or 0.75 Tr~ is obtained with TAE = 4 K, the same value needed to fit the heat capacity over a comparable range. - - ~ 0.03

--7

I - - - - T - - I ~

Parallel

'

--

I

Susceptibility

T/

TAE = 5,"3K

•,



Expertmen! (Xou)

-0-

--

Spin-Wove Theory

/

,

/~r TAE = 5.8K./

m / "-'//e"

::

TAE:4 K

to

X

0.01

j / . e -

0

I

2

TN = 5.34 K

5

4

T (K)

Fig. 4. Parallel susceptibility of NiClz • 6H20 below TN together with theoretical curves calculated for a simple spin-wave model with several values of the gap temperature, TA~. The fact that the spin-wave model gives a consistent description of the heat capacity and parallel susceptibility to such a high temperature is quite striking. It is probably associated with the relatively large energy gap which suppresses the excitation o f spin waves and, hence, their interaction until T is close to TN. The discrepancy between the value TAE = 4 K obtained by fitting these temperaturedependent quantities and TAE = 5.8-6.4 K found essentially at 0 K by measurements of antiferromagnetic resonance, spin-flopping and Z_L appears to be signifi-

T H E MAGNETIC SUSCEPTIBILITIES OF NiC12 • 6H20

75

cant. However, it is not nearly so large as that originally suggested by Donaldson and Edmonds whose preliminary antiferromagnetic resonance experiment had indicated that TAE = 7.7 K, for reasons now understood6). As a possible source of lhe discrepancy, they considered, qualitatively, the modification of the excitation spectrum produced by spin wave-phonon coupling. We prefer, at this point, however, not to speculate until a more realistic spin-wave model of NiC12 • 6H20 has been developed. Such a model would have to include BriUouin-zone effects, the erthorhombic anisotropy, and the possible two-dimensional character of the spin system resulting from weak exchange interaction between (001) layers of Ni ++ ions. A c k n o w l e d g e m e n t s . One of the authors (S. A. F.) wishes to express his deep appreciation to Professor C. J. Gorter for introducing him to the study of lowtemperature magnetism. We join our colleagues in honoring Professor Gorter as scientist, teacher, and friend. Travel expenses for A. I. H. were provided by the Conselho Nacional de Pesquisas of Brazil and are gratefully acknowledged. REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

Robinson, W. K. and Friedberg, S. A., Phys. Rev. 117 0960) 402. Haseda, T., Kobayashi, H. and Date, M., J. Phys. Soc. Japan 14 (1959) 1724. Flippen, R. B. and Friedberg, S. A., J. appl. Phys. Suppl. 31 (1960) 338S. Sugawara, T., J. Phys. Soc. Japan 14 (1959) 1248. Spence, R. D., Middents, P., E1 Saffar, Z. and Kleinberg, R., J. appl. Phys. 35 0964) Date, M. and Motokawa, M., J. Phys. Soc. Japan 22 (1967) 165. Mizuno, J., J. Phys. Soc. Japan 16 (1961) 1574. Kleinberg, R., J. chem. Phys. 50 (1969) 5690. Kleinberg, R., J. appl. Phys. 38 (1967) 1453. Schriempf, J. T. and Friedberg, S. A., J. chem. Phys. 40 (1964) 296. Groth, P., Chem. Kryst. II (1908) 247. Brickwedde, F. G., Van Dijk, H., Durieux, M., Clement, J. R. and Logan, J. K., J. Nat. Bur. Stand. 64A (1960) 1. Woolley, H. W., Scott, R. B. and Brickwedde, F. G., J. Res. Nat. Bur. Stand. 41 (1948) Danielian, G., Proc. Phys. Soe. 80 (1962) 981. Watanabe, T., J. Phys. Soc. Japan 17 (1962) 1856. Haseda, T. and Date, M., J. Phys. Soc. Japan 13 (1958) 175. Kanamori, J. and Minatono, H., J. Phys. Soc. Japan 17 (1962) 1759. Kubo, R., Phys. Rev. 87 (1952) 569. Eisele, J. A. and Keffer, F., Phys. Rev. 96 (1954) 929. Keffer, F., Handbuch der Physik, S. Fli~gge, ed., Springer (Berlin, 1966), Vol. 18 Pt. 2, Castilla Becerra, C. and Paduan, A., Phys. Letters (to be published) (1973). Anderson, F. B. and Callen, H. B., Phys. Rev. 136 (1964) A1068. Donaldson, R. H. and Edmonds, D. T., Proc. Phys. Soc. 85 (1965) 1209.

854.

Res. 379.

p. 1.