Nuclear relaxation time in mixed sodium salts

Grunzweig, ]. Zamir, D. Zak, ]. 1963

Physica 29 50-62

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS *) by]. GRUNZWEIG, D. ZAMIR **) and

J.

ZAK

Department of Physics, Technion - Israel Institute of Technology, Haifa, Israel

Synopsis Reduction of up to 50% was observed in nuclear magnetic resonance measurements of relaxation time T 1 of 23Na in mixed, single phase sodium halide crystals. Van Kranendonk's theory of quadrupole nuclear relaxation mechanism, based on the point-charge model, has been slightly simplified at TlfJ > 1, and then applied to this problem. The calculations predict a much smaller reduction. Within the accuracy of the measurements, all relaxation transients show only a single relaxation time present which implies strong spin-spin coupling of all nuclei contributing to the signal, irrespective of the fact that some of its Zeeman levels have been shifted by static quadrupolar effects.

1. Introduction: Po und-) was the first to investigate, both theoretically and experimentally, the nuclear spin-lattice relaxation mechanism that arises due to coupling between the electric quadrupole moment of the nucleus and the timedependent gradient of the electric field, generated by the thermal vibrations of the charges outside the nucleus. In cases where the nucleus has an appreciable quadrupole moment, this mechanism often becomes the dominant mode of nuclear relaxation. This is the case, for example, in many of the alkali halides. Van Kr a.nen do n ks) developed a detailed model, applicable to ionic crystals, where the field is due to the neighbouring ions, and worked out the transition probabilities for the case of a NaCl-type lattice and a field due to point charges, placed at the lattice sites of the six nearest neighbours. The calculations have been extended since to the case of CaCl-3) and ZnS-type 4) lattices. The results, however, were rather disappointing, as in the majority of cases the calculated transition probabilities were much smaller than those experimentally measured; in some cases, e.g. that of 23Na in NaCl, by several orders of magnitude. *) Supported in part by a grant from the Executive Committee of the General Federation of Labour In Israel. **) Part of a thesis submitted by D. Zamir to the Department of Physics, Technion, in partial fuIfillement of the requirements for his degree of Doctor of Science, in June 1960. Present address: Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, N.Y., U.S.A.

-

50 -

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

51

Koch el a e vb) attempted to improve the model by taking into consideration the contribution of the optical branch of the vibration spectrum, disregarded byVan Kranendonk. Kochelaevassumedasimplespectrum and showed that, in many cases, the resulting effect is of the same magnitude as that due to the acoustical branch. Thus the reductions of the relaxation time T'; produced, as compared with Van Kranendonk's theory, were slight. More recently, Wikner, Blumberg and Ha h nv) have considered the effect of the dipole fields brought about by the polarization of the ions by the lattice vibrations in the optical branch. Their calculated results for the alkali halide crystals agree very well with the experimental data. However, there are certain doubts as to the validity of some of their assumptions. They assume that all the modes of the optical branch contribute. As they themselves point out, the main contribution should be due to modes of k ,....",0 and the short wave modes should not contribute appreciably, as then the dipole field decreases. Therefore, in the calculation of their contribution to the rate of the nuclear relaxations, one should take into consideration only a part of the total number of modes, perhaps 1~' It is very difficult to decide where exactly to place the cut-off, but it seems fairly certain that this model overestimates greatly the effect of ionic polarizability. Two slightly different attempts were that of Yoshida and Moriya 7). who assumed that the electric field is due to the distortion of the covalent bonds, and that of Kondo and Yarn ash it a") who considered the effect of the overlap of electron shells of neighbouring ions. However, their results in most cases disagreed appreciably with experiment. Therefore we felt that it was worthwhile to accumulate additional information about the relaxation mechanism of ionic crystals, in the hope that this might contribute to the final solution of the problem. We proposed to measure the rate of nuclear relaxations in slightly deformed crystals, in order to investigate the gradient of the electric field in the vicinity of the lattice sites. However, the sodium halides that were available did not lend themselves to measurements under externally applied stresses, as they deformed plastically under quite small pressures. Therefore we introduced strains into the crystal by means of alloying. We have used and developed further Van Kranendonk's point-charge model in order to discuss our experimental results, in spite of the criticism voiced above, as we think that this approach is instructive and basically correct in its description of the mechanism involved. We are fully aware of the crudeness and limitations of the model employed, but the more elaborate models at present hardly justify the additional work entailed. In the following we shall derive a simplified form of Van Kranendonk's theory, applicable at high temperatures (Tie = T* 1), and shall use it to discuss the results of our measurements of relaxation times on mixed sodium crystals.

>

52

J.

GRUNZWEIG, D. ZAMIR AND

J. ZAK

2. Theory. First we shall simplify the Van Kranendonk model by assuming that only the central nucleus vibrates, its six neighbours being at rest. Such an assumption is justified at high temperatures (T 0), when the neighbours vibrate in opposite phases, because then the relative displacement of two neighbours is of the same order of magnitude as the displacement of one nucleus. In Van Kranendonk's notation (in future we shall refer to his paper as K):

>

(1)

Here 11, is the relative displacement, 81, the displacement of the nucleus i and So the displacement of the central nucleus. By this assumption the transition probabilities at high temperatures in K's theory become very simple indeed: (2)

where a is the lattice constant, v the velocity of sound, d the mass density ofthe crystal. Qmu and Vii are defined according to Cohen and Reif 9) and Xi is the i-th component of the displacement So of the central nucleus. We calculated the transition probabilities at T* ~ 1 by using formula (2). They are larger than the WI and W2 obtained by Kranendonk (K equations 59, 60), as one would expect, due to the simplification of the model, but they differ from his by less than a factor of 1.5. From (2), by using the transformation properties of the VII' one obtains the angular dependence of W(m, m + f-t):

W(m, m

+ tt) =

_ 9 IQmill 2 NiI,Ntt" 82 VII' 82 VII" T*2}.:; DII/I!DIIII" _._- - 3 2 8 na 3v d i,i N;, 8x1,8x1 OX1,OX1

-

(3)

p'p."

here No = 1, N±l = N±2 = vi and D ll iI , are the elements of the fivedimensional irreducible representation of the rotation group. N ow in order to compute the experimentally observable relaxation times T 1, be it in the transient or saturation method, one has to make certain additional assumptions as to which spectral lines are being observed, the strength of the spin-spin relaxation rates, etc. In the case of a system with more than two energy levels, one obtains a single relaxation time T 1 only if one postulates a single spin temperature throughout the investigated levels; that is, spin-spin relaxation rates are much more rapid than the spinlattice relaxation. Gorter 10) and also Hebel and Slichter 11) 12) obtained in such a case that ~ (E1, - Ej)2 W(i, i) 1,1

(4)

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

53

where E t is the energy of the z-th level. In the case of interest, 23Na nucleus with spin 1= j, and under assumption of a purely quadrupolar mechanism of relaxation, (4) reduces to -

1

TI

= ~ (WI

+ W2),

where WI

W2

=

W(- j, - !)

=

W(-!, j)

=

= W (t,!) and W(-!, t)

a result also obtained by detailed consideration of population dynamics 6) 13) 14). In cases where the spin-spin relaxation rates are comparable with or smaller than the spin lattice ones, there is no unique relaxation time, Ts. We shall now assume that for a cubic crystal equation (4) is applicable. Substituting W's from eg. (3) into eq. (4) we find that T 1 is isotropic for nuclei in surroundings of cubic symrnetry t"). Mi eh er s) using Van Kranendonks' model and ego (4) proved this for the special case of NaCl- and 2nStype lattices. We shall now attempt to calcute the relaxation time Tl for 23Na in mixed crystals. We shall assume the point-charge model for a crystal distorted owing to substitution of certain ions of different diameter. The case of a single Br ion in the N aCl. lattice is represented schematically in fig. 1. CI

Fig. I. Deformation of the NaCl cell clue to replacement of one Cl," ion by a Br- ion.

The deformed crystal cell. is tetragonal. by assumption and we use the following notation: si-displacement of the Na ion s2-displacement of the Cl ions at a distance 2a from Br sa-displacement of the Cl ions at a distance -yl2a from Br N ow we develop the potential of the point charges (following K) and calculate the derivatives in (3) for the case of an assembly of randomly orientated tetragonally deformed crystal cells. We expect this model to be

J.

54

GRUNZWEIG, D. ZAMIR AND

J.

ZAK

applicable to the case of the polycrystalline alloyed specimens containing a small percentage of substituted atoms. The result of relaxation time Ti for such a tetragonally deformed crystal cell is obtained by averaging the relaxation time (4) over the orientations and expressing the gradients in terms of the deformations. 1 1 -T = --2{~[W(m,m+I)+ W(m,m-l)]+4~[W(m,m+2) + W(m,m-2)]} = 1

2~m

m

til.

n

-

+

9(eQ)2T*2 (21 3) 64na 3vsdz j2(21 _ 1)

(a8z4

4V)2

cs

a {I -

2I qJz

128

-

21 qJs

8732

2

+ 735 !PI}

(5)

where Bt

!Pi

=-, a

V) is the derivative .. in undeformed crystal.

a4 (-8z4

e

The last expression is the relaxation time of nuclei at sites adjoining those occupied by substituted ions. We can regard the sample as consisting of two types of nuclei - of N 1 nuclei in a cubic environment and of N 2 in a tetragonal one, and by assuming again a common spin tenperature, one can calculate the common measurable relaxation time (T l)m. We obtain 1

d

c

- = T-1 +Ti(Tl)m

(6)

where

T 1 is relaxation time of nuclei in a cubic environment Ti is relaxation time of nuclei in a tetragonal environment

The numerical terms in eq. (5) should not be taken too seriously. However, it is to be noted, and this seems difficult to reconcile with experiment, that Tl depends on the sign of !P2 and cps. In order to make T', independent of the sign of qJ2 and CPs, !PI has to be at least O. I and CP2 and CPs ten times smaller. In a similar manner, one can calculate the effect of distortion due to two or more SUbstituted neighbours that are present in crystals containing a large fraction of strange atoms. However, such a refinement of the model is not justified by the results obtained for crystals containing a small percentage of strange ions. The effect of dipole fields as considered by Wikner, Blumberg and Ha.hri") can be similarly taken into account. However, the relative change of the relaxation time due to deformation will be the same as in formula (5). There are two other factors that can influence the relaxation time - the

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

55

polarizibilities of the strange ions and the change of the Lorentz factor due to the change of the symmetry of the mixed crystal. The first of them does not explain the decrease of the relaxation time in mixed crystals, because if Cl is exchanged for Br the relaxation time decreases; but in the opposite case, when Br is exchanged by Cl, it will increase in disagreement with the experiment. The Lorentz factor is not known in mixed crystals and therefore we cannot consider the influence of its change. In view of this and our previous comment in the matter, we felt unable to compute this contribution in a satisfactory manner.

3. Experimental. The majority of specimens used were polycrystalline. They were prepared from chemicals of reagental purity by thorough mixing and melting of the mixtures. These were cooled rapidly to prevent segregation and then ground and compressed into cylinders of 1.2cm. diameter and about 2.0 em length. The cylinders were annealed for several hours at temperatures SO to 1aaoc below their melting points and then cooled quickly. Their composition, homogeneity and absence of lattice distortion were checked by means of X-ray diffraction. The nuclear magnetic resonance apparatus consisted of a standard PoundKnight marginal oscillator. The relaxation times were measured by "cooling after saturation". A careful check was made each time to detect whether the RF intensity was sufficiently low so as not to affect the measured relaxation times T 1 . Due to the sluggishness of the apparatus and the unfavorable signal-tonoise ratio, we preferred to work at liquid air and sclid-COj, temperatures. 4. Results. The measurements of T'; of z3Na at liquid air temperature for the mixed crystal system NaCl/NaBr and NaBr/NaI are shown in fig. 2 and fig. 3 respectively. The replacement of a part of the halogen ions by halogens of different types reduces T 1 of 23Na appreciably. This is to be expected as the distortion in the vicinity of the foreign ions alters the electric field the 23N a nuclei see. As all the above mentioned mixed crystals contain bromine and the effect may be due specifically to bromine, we also carried out measurements on T 1 of z3Na in bromine-free alloy systems (fig. 4), as well as its dependence on the magnetic field H o fig. 5. In the NaCI/KCl mixed crystals, the signal strength decreases sharply with the increase of KCl content, and we were able to measure only on crystals containing comparatively little KCl, but even then we obtained a definite reduction of T 1 . Similar results were obtained in the case of the NaCl/NaCN system, but here the effect is no doubt partially due to the nonspherical shape of the CN ion, which probably cannot rotate freely at such low temperatures. The curves of 7\ in the NaCl/NaBr and in.the NaI/NaBr system are similar.

56

J.

GRUNZWEIG, D. ZAMIR AND

J.

ZAK

T, 200

se s

100

I 50

o Na.C~

CONCENTRATION

or

Ncl.Br

Fig. 2. Relaxation time Tl of 23Na in NaCljNaBr mixed crystals. Circles denote measurements on single crystals. To

150

.eo

100

o

10

20

30

40

Nal

ATOMIC

CONCENTRATION

or

NdBl'"

Fig. 3. Relaxation time T 1 of 23Na in NaBrjNaI mixed crystals.

57

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

200

se,

\

\ \

\50

r-:::t-

T= 770K

\NOC1+ NoCN

100

\ \t-.

50

o NQC~

10

15

2Q

ATOM'C PERCENT Of KCl (OR NOCN)

Fig. 4. Relaxation time Tl of 23Na in NaCljNaCN and NaCljKCl mixed crystals. 200

I---:.:..--H a

. u

150

o '" ~

~100

50

1000

3000 MAGNETIC f1oLO IOER5TEO)

--

4000.

5000

Fig. 5. Ts: of 23Na as a function of the magnetic field in NaBrjNaCl mixed crystals: a) 0.95 NaCl + 0.05 NaBr b) Pure NaBr c) 0.20 NaCl + 0.80 NaBr.

58

]. GRUNZWEIG, D. ZAMIR AND

J.

ZAK

They show a sharp decrease, then a long plateau and eventually a slow increase towards the value of T 1 for the pure NaBr. The power of substituted halogens to bring about a reduction of T 1 is not proportional to the differences between the original and the substituted ion radii. We also measured T 1 as function of crystal orientation in single crystals, in pure NaCl, 0.97 NaCl 0,03 NaBr and 0.92 NaCl 0.08 NaBr, In no case did we detect any dependence of T 1 on the crystal orientation. As mentioned before, the signal recovery curves approximated closely to a straight line when plotted against time on semi-log paper. T 1 was also determined in a few mixed crystals at about 2000 K and at room temperature. The results show that T 1 varies as the reciprocal of the sq uare of temperature, as is to be expected in a Raman effect at temperatures about and above the Debye temperature. Literature search shows that there are not many single phase alloys of the type Na X/Na Y, favorable from the signal intensity point of view. Alloys of type Na X/M X are not so convenient to work with, but probably approximate more closely the theoretical model used, as the strange ion does not adjoin directly the Na ion.

+

+

S. Discussion; Experiment shows that alloying of the order of 20% reduces T1 of 23Na nearly by a half. Further change of composition does not affect T 1 appreciably. A similar initial rapid decrease and then a plateau was reported in the plot of signal intensity of 23N a as function of compositio1116) 17) • • It is a very convincing assumption that substitution of strange ions will give rise to appreciable local lattice distortions and hence changes in ,the crystalline electric field. According to the model used in section 2 I the quadrupole relaxation mechanism is a function of several parameters, and in order to be able to discuss the results, we make some additional gross simplifications. We neglect the changes in all parameters arising on alloying, except for displacements of the point charges. We shall now try to justify this. The average lattice constant a is a linear function of composition, The changes in a alone should produce a monotonic change in T 1 and this can be taken into consideration by calculating (T l)corr = T l' a- 13

The change in elastic properties and in the Debye temperature (both closely connected) are more difficult to estimate. At present there seems to be only one exhaustive study of the effect of composition on the elastic properties of alloys of non-metals, the work of Sundra Rao on mixed potassium-chromium alums-e), where the moduli change linearly with composition. It is not certain whether this assumption is correct in our

59

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

case, but if we employ it in the absence of better information 19), this assumption can be incorporated in the same way as that of the average lattice parameter. This procedure would not appreciably affect the discussion of the results. We assume further that in spite of the quadrupole (static) shift of energy levels due to strains, all 23Na nuclei which contribute to the observed signal, even if only by one of their three transitions, remain in thermal equilibrium at all of its levels. This is due to the fact that the level shifts are gradual, as we proceed towards the center of distortion, and can be bridged by fluctuations in the local magnetic dipole field (fig. 6). All our measurements show, within the admittedly appreciable experimental error, the presence m

-- ---

-t W,

W.

W,

_ - - r - - - - ~c._ - - -

----vb-

W

Vb

-+--'---y- - - -

-

----

,

w,

W.

-

_ _+'

OISTANC, FROM STRAIIG, ATOM

Fig. 6) Schematic shift of energy levels of 23Na due to introduction of a strange ion into the lattice. Wl and W2 correspond to transitions P 1 and P2 respectively; fie. VII> Ve to interactions arising from spin-spin coupling.

of one single relaxation time. In the case of 1= i this will happen when all the spins are at one common spin temperature, either due to spin coupling or if WI f'>j W 2. Back extrapolation of the signal intensity to time t = 0 (which corresponds to the removal of the saturating RF. field), gave zero signal. This means that there was no fast decay component which the slowresponding apparatus might have missed. There is no obvious reason why both transition probabilities WI and W 2 should be equal, except by coincidence. Hence we assume this as an indication, until we obtain evidence to the contrary, that the vast majority of 23Na spins remain coupled together. However, it should be pointed out that so far the only reported independent measurement of WI and W 2 in nuclear magnetic resonance is that

60

J.

GRUNZWEIG, D. ZAMIR AND

J.

ZAK

on 23Na in NaN0 3 by Goldburg 20) who found, for a certain crystal orientation, that W 2jJiV 1 = 0.90 ± 0.05. We now attempt to estimate the effect of alloying on the spin-lattice relaxation time, by means of the model developed in section 2 together with all our simplifying assumptions. From equations (5) and (6) we find that the displacements of nuclei tpl, ep2, epa in the vicinity of the substituted ion that are required to explain the observed reduction of 1\ by half are so large as unlikely to be the correct explanation. Neglecting for the moment tp2 and tpa, we find that our model requires, for a 50% reduction of T 1, a displacement of the nearest neighbour tpl of about 0.3. From the theoretical calculations published 21) 22) 23) we can estimate with a fair degree of confidence the displacements in the lattice to be expected. For example, for the NaCljKCl alloy Hard y 23) obtained, for the displacement of the nearest neighbour (here Cl) epl = 0.05, and epa = 0.015 for the displacement of the next-nearest neighbour (i.e. Na). It appears that displaced-paint-charges model is inadequate. What other model could account for these results? It has been suggested *) that the effect might be due to cross relaxation between the Na spin system and, say, that of 79Br and/or 81Br isotopes which, in turn, have a very short T 1. Such a mechanism could be very efficient 24). As the gyromagnetic ratios of 81Br are close to that of Na, and as furthermore the levels are shifted in alloying, this question must be seriously investigated. However, the effect is present in alloys not containing bromine, or any nuclei, whose gyromagnetic ratio is close to that of 23Na, e.g. in NaCljKCl and NaCljNaCN. Further, in the case of a cross relaxation mechanism, T 1 depends on the difference in frequency of the two interacting transitions and hence on the static magnetic field H 0 24). This is so in the case of mixed crystals containing bromine, but with increasing He, T 1 saturates and for H o of the order of 5000 gauss, where all our measurements were made, T 1 is independent of u; (fig.S). It appears that the two systems are decoupled in strong fields, and cross relaxations are not effective any more. Furthermore, as mentioned above, in a high magnetic field the cross relaxation mechanism should be enhanced by slight shifting of the Na or Br Zeeman levels. Now, an isolated Br> ion in a mainly NaCl (or NaI) matrix is placed at positions of nearly perfect cubic symmetry; hence its static quadrupole level shift should be zero and thus should not contribute as strongly to the cross relaxation mechanism as a Br- ion does in a primarily NaBr crystal, containing a small Cl(or 1-) admixture. In the latter case any Br- ion adjoining a guest ion has its levels shifted and hence would be better coupled to the 23N a spin system. However, the experiment shows that a small admixture of NaBr to NaCl *) Private communication from Professor Schumacher, whom we wish to thank here.

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS

61

or NaI reduces T 1 very strongly, while the addition of NaCI or Nal to NaBr is not as effective. A further piece of evidence is provided by the measurements of T 1 as function of the heat treatment of pure crystals. Quenching gives rise to internal strains. A decrease of T 1 was observed on quenching, though it is smaller than that due to alloying. TABLE I The effect of heat treatment on relaxation time Crystal NaBr NaBr Nal Nal

I

Treatment Annealed Quenched Annealed Quenched

I

Tl of 23Na

90 80 65 48

± ± ± ±

10sec 10 esc 10sec 10 esc

In the light of the above evidence, it is clear that the reduction of T 1 is due to changes in the electric field gradient induced by lattice strains. However, our initial assumptions of a point-charge model and a constant, isotropic antishielding factor y is too great a simplification, and y might change appreciably, as the electron clouds are distorted due to lattice strains. This conclusion is confirmed by other experiments as well. Bloembergen and Tay lcr s") determined Vvx and V XY as function of strain in NaCI, by saturation measurements in the presence of ultrasonic vibrations, and concluded that for 23Na, V xx measured agreed closely with that calculated on the assumption of the point-charge model and the "static" value of y(y,....., 5 for 23Na). However, changes in V XY were about three times too large and of the wrong sign, as compared with the simple model. More recently, Le m a.n o vw) determined the changes of the electric field gradient by studying the line shifts in elastic compression of single crystals of NaC!. It is surprising that he succeeded, as NaC! deforms plastically at very low stresses, while Lemanov claims to have been able to reach pressures of 60 kg cm- 2 in elastic compression. His results and those of Bloem bergen and Taylor are compared with the point-charge model in table 2, where 8n and 844 are the components of the 4th order elastoelectric tensor defined in ref. 25. The disagreements in sign should be noted. In passing it might be pointed out that in the study of nuclear relaxations due to the quadrupole Raman mechanism, as discussed in detail in the previous section, one can employ a 6th order elasto-electric tensor, similar to the 4th ordertensorintroduced by Sh ulman, Wyl uda and An der scn s") and by Bloembergen and Tay lo r s") in the study of line broadening. It appears that more elaborate models are necessary, which would take into account the electronic structure of the ions in a crystal lattice and the changes caused by deformations and vibrations. The electric field gradients so calculated should be able to account better for the observed relaxation times, both of pure and alloyed crystals, than the crude, semi- phenomenological model used by us.

62

NUCLEAR RELAXATION TIME IN MIXED SODIUM SALTS TABLE II The Elas to-electric tensor s for 28Na in NaCl

I

Point charge model

I

Ultrasonic experiment

25;

I

Static compression

36)

in stat coul- em-t

Su

+ 1.17

544

s .4/511

X 1015

-0.5

-

±2 X 101. +0.55 X 101&

+0.75

-0.25

1.3

I

X

1015

y(25Na) = 4.5

However, the attempts so far 7) 8) using tractable wave functions have been disappointing. More defined and elaborate calculations are necessary. As pointed out by B'lo em b er g e n sv) the ionic (point charge) model is satisfactory in problems involving cohesive energy, but internal electric field gradients of interest here depend mainly on valence orbitals, which do not contribute strongly to the binding energy of the crystal. The measurement of relaxation time as a function of alloying is a much more stringent test of electron distribution than the measurements of line shifts as it involves higher derivatives. Received 17-4-62 REFERENCES 1) Pound, R. V., Phys. Rev. 7ft (1950) 685. J., Physiea20 (L954) 781. 3) Das, T. P., Roy, D. K. and Ghosh, Roy S. K., Phl's. Rev. 104 (1956) 1568, ibid 113 (1959 1696. 4) Mieher, R. L., Phys, Rev. Letters 4 (1960) 57. 5) Ko ohe l a e v, B. N., Zh. eksper, tear. Ftz. 37 (1959) 242. 6) Wf c k n e r, E. G., Blumberg, W. E. and Hahn, E. L., Phys, Rev. 1UI (1960) 631. 7) Yoshida, K. and Morlya, T., J. Phys. Soc. Japan 11 (1956) 33. 8) Ko n d c, J. and Yamashita, ].,]. Phys, Chern. Solids 10 (1959) 245. 9) Cohen, L. and Reif, F., Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, New York (1958), Vol. S. 10) Gorter, C. J., Paramagnetic Relaxations (Elsevier Publishing Co. New York, 1947). 11) Hebel, L. C. and Sllehter, C. P., Phys. Rev. 113 (1959) 1504. 12) Grunzweig,]. and Genossar, J. - to be published. 13) Andrew, E. R., Swanson, K. M. and Williams, B. Ro, Proc, Phys. Soc. 77 (1961) 36. 14) Zamir, D., Thesis, Israel Institute of Technology, Haifa (1960). 15) Zak, ]., Bull. res. COUlIC, Israel 10 (196L) 28. (6) .I3loembergen, N., and Rowland, T. J., Acta Met. 1 (1953) 73l. 17) Kawamura, H., Otsuka, E. and Lsh i w a t ar i, K. J., J. phys. Soc. Japan 11 (1956) 1064. (8) Sundra, Rao, R. V. G., Current Sci. (India) 16 (1947) 91. 19) Zeuer, C., Acta Cryst. 2 (1949) 163. 20) Go ld b u r g, W. r., Bull. Am. phys. Soc." (1959) 251. 21) Matt, N. F. and Li ttleton, M. J., Trans. Faraday Soc. 34 (1938) 485. 22) Bassani, F. and Thomson, Ro, Phys, Rev. 102 (1956) 1264. 23) Hardy, J. R. J., J. Phys. Chern. Solids 15 (1960) 39. 24) SChumacher, R., Phys. Rev. 112 (1958) 837. 25) Bloembergen, N. and Taylor, E. F., Phys. Rev. H:l (1959) 431. 26) Le m a n o v, V. V. Zb. eksper. teor. Fiz, ~O (1961) 775. 27) Shulman, R. G., Wy l ud a, B.]. and An d e rs o n, P. W., Phys. Rev. 107 (1957) 953. 2) Van Kr a n e n d on k,