Isotope effect of diffusion of 22Na+24Na+ in NaCl, KCl and KBr single crystals

I. Phys. Chem. Solids. 1974. Vol. 35. pp. 15-23.

Pergamon Press.

Printed in Great Britain

ISOTOPE EFFECT OF DIFFUSION 22Na‘-24Na’ IN NaCl, KC1 AND KBr SINGLE CRYSTALS

OF

F. NICOLAS, F. B~NIJ?RE and M. CHEMLA * Laboratoire d’Electrochimie, UniversitC Paris VI, 9, Quai St. Bernard, Paris S”. France (Received

24 November

1972; in revised form 1 June 1973)

Abstract-Accurate measurements of the diffusion coefficient and isotope effect for diffusion of Na’ have been carried out in the intrinsic range of NaCI, KCI and KBr single crystals. The technique which led to the more reliable results is described. This is based on the simultaneous use of the differences between the half-life times and the energy spectraof -y-radiationof the isotopes12Naand “Na after

diffusion into two adjacent samples.In NaCI, the isotope effect is found to vary very little with temperaturearound the meanvalue 0.75, while the isotope effect decreasesfrom 0.68at 772°Cdown to 044 at 574°Cin KCI, and from 064 at 693°Cdown to 0.52 at 601°Cin KBr. Comparisonof the vacancy pair contributions to the self-diffusion of Na’ and Cl-[11 in NaCl is attempted. Comparison of the isotope effect in the different matrixesshows an influence of the relative ion sizes in agreement with the theory of Le Claire[2]. 1.

INTRODUCTION

where M and D are the mass and diffusion coefficient of the two isotopes denoted by the subscripts 1 and 2 respectively. One can separate in the self-diffusion coefficient of the cation the contribution of the free vacancies, D,,, and that of the vacancy pairs, Dp+. So, the experimental diffusion coefficient is given by:

A number of works[3] have been devoted to the study of the transport properties in the alkali halide crystals, especially diffusion and ionic conductivity. The improved accuracy of the basic data allows to go further and more confidently in their analysis. Thus, while transport of the ions is mainly due to the free vacancies, it is known that vacancy pairs are also operative and it is the aim of recent works[l, 4-81 and of the present one to define the extent of their contribution. While contribution of the pairs to the anion diiTusion can be evaluated from diffusion coefficient measurements [ 1,4-61, that to the cation diffusion is not so well defined. Making use of theoretical calculations of Tharmalingam and Lidiard [9], BCnitre et al. [I] supposed that the two have comparable values. On the other hand, the experiments of electron&ration of Na’ into NaCl performed by Nelson and Friauf [7] indicate a vacancy pair contribution to the diffusion of Na’ which would be much larger than the vacancy pair contribution to the diffusion of Cl-. An a priori possible method to investigate this question is the study of the isotope effect of selfdiffusion of the cation. The isotope effect can be defined in terms of the mass change as:

D+=D,++Dp+

and the experimental isotope effect by: E=cufAK+(l-a)f’AK’ with: g=- D,+ D+

and f and f’ the correlation factors (10) for difTusion via free vacancies and vacancy pairs respectively, and AK and AK’ the corresponding sharing factors of the kinetic energy of the jumping ion with its neighbours. Thus, equation (3) shows that the isotope effect involves many parameters. We have measured the isotope effect of diffusion of the isotopes “Na and 24Na: (1) in NaCl, to attempt to evaluate the contribution of the vacancy pairs, and, (2) in KC1 and KBr, in order to investigate the influence on AK of the mass of the jumping ion relative to those of the neighbouring ions.

E=($-*)/(Jg-1) 15

16

F. NICOLAS. 2. EXPERIMENTAL

F.

B~NI~RE

METHODS

(a) General outlines The procedure earlier reported [ I] for the measurement of the diffusion coefficient of a single tracer concerning the diffusion conditions and sectioning has been used. A mixture of the two isotopes “Na and 24Nais deposited by sublimation at the interface of two Harshaw single crystals smoothed by accurate microtoming. The crystals are sectioned after diffusion in a given number of layers. The powder taken out by the blade is directly collected in the bottles of the counting device, filled with a solution to the same level; thus, all samples are counted according to the same geometrical conditions. Distribution of the specific activity of each isotope is given by: A,,, = Aa exp (- x’l4Dt)

(5)

where A,,, is the specific activity at the depth x from the interface, A0 the activity at x = 0, t the time of diffusion and D the diffusion coefficient. From the properties of radioactive decay of “Na and “Na we can calculate separately the activities “A and 24A of the two tracers. Then the relative change in the diffusion coefficients “D and 24D is readily obtained from the following equation derived from[5]: = Constant- [q)ln(“A).

(6)

(b) Counting Several methods were successively tried. They are described in a detailed paper[l l] and we shall focus on the one finally adopted. This is based on the counting of the y-rays emitted by ‘?Na and 24Na, each on one of its respective peaks with a twochannels well-type scintillation counter. The tracers are deposited such as the ratio of the counted activities 24Al”A be roughly equal to 2 at the beginning of the counting. The diffusion depth is divided into 20 layers, “A decreasing from 10” countslmin for the first slice down to 10’ for the last one. The first sections are diluted to give about lo5 counts/min. Two crystals are used in every experiment and the two times 20 samples obtained from sectioning of both crystals are settled simultaneously on the automatic counter and counted together with “Na and 14Nastandards: the first crystal is counted from the 1“ to the 201hslice and the second one inversely. Owing to the automatic sample changer and the

and

M.

CHEMLA

readout of the data on a teletypewriter the samples may be counted night and day. Each sample is counted for 5 times 4 min just after the experiment and for the same time 12 days after the experiment when the activity due to “Na is practically zero. A number of preliminary tests have been performed to check the reliability of the gamma counting. The ratio of the activities selected by the two channels has been determined as a function of time and as a function of the total amount of tracers of long half-life time in the range of practical interest for our experiments. This showed the crucial influence of temperature and the counting system has been settled in a special room whose temperature is maintained at 18 + 0.5”C. The same has been performed on mixtures of different amounts of ‘?Na and “Na (null effects). In all cases the accuracy is consistent with the Gaussian statistics. Let “A(O,O) and 2JA(0,0) be the specific activities of the first slice counted at the time chosen as the initial time. Activities in the slice of abscissa s, “A(s, t) and ‘“A(.~, t) are counted a time t after the initial time for the first crystal. These are corrected for the radioactive decay according to the equations: “A(x, 0) = ‘?A(x, t) exp (+“At) ‘“A(x, 0) = “A(x, t) exp (+‘JAt). The activities in slice .Y of the second crystal, “A’(.u, - t) and “A’(s, - t) are counted a time r before the initial time. Corrections for radioactive decay is then operated according to: “A’(x, 0) = “A’(x, - t) exp (-“At) ‘JA’(x, 0) = ‘JA’(x, -t) exp (-24hf). The decay constants “A and “A are known exactly for the pure isotopes and these are checked in our experiments owing to standards counted periodically in the chosen spectral ranges. Thus, the presence of any radioactive impurity would be detected. At last, a “‘Cs standard is also introduced on the automatic sample changer in order to check the influence of long time drifts. The isotope effect is deduced from the measurement of the variation of the ratio “A(x, O)/“A(x, 0) as a function of x, as also from the variation of 22A’(~,0)/‘4A’(x, 0). This gives two values for the isotope effect. If there were a time dependent error (due to, for instance, some drift acting differently on the spectral ranges used for discrimination of ‘2A

17

Isotope effect of diffusion of 21Na*-*4Na+

and ‘“A, or a radioactive impurity of short life) it would appear as a discrepancy between the isotope effects measured in the twin samples. (c) Separation

of *‘A and “A

To separate ‘?A and 24A the differences between the half-life times and between the energy spectra are both used. The half-life times of “Na and 24Na are respectively equal to 2.6 yr and I5 hr. Thus, a counting performed after 12 days solely gives the activity of “Na. Taking into account the slight variation of 22A due to its slow radioactive decay and to electronic drifts, if any, owing to the standards, it is therefore possible to separate the activity due to 24Na from the total activity measured just after sectioning. Second, the two tracers have two quite different spectra of y-ray emission (Fig. 1). *jNa has its main peak at 0.511 MeV while 24Na has a peak at 2.71 MeV which interferes very little with the end I I I I of the spectrum of “Na. The two channels of the 0.02 O-04 0.06 apparatus are centered upon these peaks as is x2, inn? shown on Fig. 1. Thus “A is also accurately deter- Fig. 2. Penetration plots (lnz2Avs x2) for the diffusion of mined since there is only a small contribution due Na’ in NaCI. Each division on the ordinate is I. The anneto ‘INa to be substracted on the 2.71 MeV peak. alingtimes are: 602.1”C: 138000set; 670°C:60500sec. Separation of “A and 14A, calculation of the half-life time, diffusion coefficient and isotope I effect are performed by the least squares method on a computer according to equations (5) and (6). The diffusion profiles plotted as Ln”A vs xz for diffusion of Na’ in NaCl are shown on Figs. 2 and 3. Some corresponding plots of Ln’*A/‘4A vs Ln”A

0 511 I

/ I I I I

I

I I I I I

I 37

I I 1 I I

271 'I : I I I !

1 E.

MN

I 0.2

I 0.4 2 X,

I 0.6

mm2 Fig. 1. y-ray emissionspectraof the isotopes22Na(in full line) and 24Na(in dotted line). The energy bandsdiscrimi- Fig. 3. Penetrationplots for the diffusion of Na’ in NaCl. nated are indicated, the two channelsbeing respectively Each division on the ordinate is 1. The annealing times centered around the 0.511MeV peak of 2zNa and the are: 731°C: 51900 set; 736.2”C: 45100 set; 743~9°C: 67700set; 751’C: 45900set; 755*7’C:54500sec. 2.71 MeV peak of “Na.

JPCS Vol. 35 No. I-B

18

F.

NICOLAS,

F. B~NI~RE

and

M.

CHEMLA

and the computed straight lines are shown on Fig. 4 for NaCl and on Fig. 5 for KCl:

638O R

(d) Accuracy The measurements of the self-diffusion coefficients of 22Nain NaCl agree within 1 per cent with those previously determined [ 11.The recent results of Rothman et al. [8] are also in excellent agreement with these values. However, the isotope effect is a small effect far more difficult to determine. While light errors on temperature, diffusion time and sectioning would not be very sensitive to the accuracy of the isotope effect, on the other hand, the slightest error in counting would have a crucial influence on the result. The conclusion of a detailed analysis of the experimental errors [ 1I] is that the main errors are those due to the statistical fluctuations. It has been verified[ 111by measuring 1000 times the same sample having an activity of 3 x 10’ counts/mm that the results given by the counter obey exactly the Gaussian statistics. Thus, according to the most elementary laws of statistics, any number of counts A is determined within the limits A +a with a confidence range of 85 per cent. The error bars represented on Figs. 4 and 5 are

Ln =A

Fig. 5. Plots of In “A/“A vs In *2A for the diffusion of Na’ in KCI. Each division on the ordinate is 0.01; each division on the abscissa is 1.

strictly equal to the statistical fluctuations, that is: S(,” 22A,24A)= f (22A-Ii?+ ?“A.I,?), Since A decreases roughly from 2~ 10’ (in the first slices after dilution) down to 2 x lo”, the errors become more important as the shift between the isotopes is increasing. One may notice that the computed straight lines cross about 85 per cent of the experimental bars, as it must be. Taking account of all the experimental points used in the determination of the isotope effect the standard deviation of the isotope effect is typically u=3 per cent. However, the reproducibility is sometimes a little less and the present isotope effects can be given with an accuracy of 6 per cent. (e) Comparison method [8, 141

Fig. 4. Plots of In 22A /*‘A vs In “A for the diffusion of **Na and *‘Na in NaCl. Each division on the ordinate is 0.02; each division on the abscissa is 2. Error bars are strictly equal to the statistical fluctuations i/G.

with

Peterson

and

Rothman’s

The improvement of the measurements of isotope effects for diffusion in solids is mostly due to the famous works of Peterson and Rothman. The main difference between their method and the present one lies in the counting process. In the P.R. method the total gamma activity above the lower level discriminator set below the 0.5 11 MeV peak of “Na is counted. On the other hand. we use a

Isotope effect of diffusion of 2*Nat-24Na*

two-channel analyser discriminating respectively the 0.5 11 MeV peak of “Na and the 2.7 I MeV peak of 24Na (Fig. 1). It is of interest to compare the accuracy of both methods. In P.R. method, for a given amount of tracers, a larger total activity is counted. This depresses the statistical errors but separation between “A and 24A lies then solely in the difference of the half-life times of radioactive decay. On the other hand, for the same amount of tracers, our method leads to higher statistical errors but, at the same time, greatly avoids the influence of radioactive impurity, as well as the interplay between the errors on the activities of “Na and 24Na. At last, in the case when a slow sliding of the gamma spectra would occur, the effect would be less in our method where the y-rays of each isotope are selected between two symmetrical discriminators, the part lost on one side being compensated on the other. 3. EXPERIMENTAL

RESULTS

Fig. 6. Diffusion coefficientsof Na’ in KC1 (curve B) and K’ in KC1 (curve A),

All measurements have been performed in the temperature intrinsic ranges of NaCI, KCI and KBr Harshaw pure single crystals. (a) Diffusion coeficients The diffusion coefficients of “Na+ in KC1 and KBr are reported in the Tables 2 and 3. The selfdiffusion coefficients of Na’ in NaCl are fitted by the previously [ l] determined equation D = 33 exp (- 1.975/k7’) cm’/sec. The results in KCl, plotted (Fig. 6) as log D against l/T together with the cation self-diffusion coefficients determined by Beniere et a/.[41 are fitted by the equation:

-7

-8 a -z -9

-10

D = 92.4 exp (-2+01/kT) cm’lsec. -111

The results in KBR, plotted (Fig. 7) together with the cation self-diffusion coefficients calculated from the data of Dawson and Barr [ 121and Chandra and Rolfe [ 133 are fitted by the equation: D = 4.23 exp (- 164/kT) cm’/sec.

1 I.0

I I.1 l/TX

I I.2

103

Fig. 7. Diffusion coefficients of Na* in KBr (curve B) and K+ in KBr (curve A).

change, i.e. E (equation (l)), together with the results obtained in RbCl by Peterson and (b) Isotope effects Rothman[ 141 in Table 4. Concerning self-diffusion in NaCl, the values obOur measurements of the isotope effect are reported under the form of the relative change in the tained in the temperature range 602-756°C show no diffusion coefficients, i.e. (“0-‘“D)/“D, in the Ta- noticeable influence of temperature within our bles 1,2 and 3 for NaCI, KC1 and KBr respectively. limits of error. The extreme values are equal to 0.68 These are also reported in terms of the mass and 0.79 and the mean value is equal to 0.75.

F. NICOLAS,

20

F. B~ZNI~RE

Table 1. Relative change in the diffusion coefficients of “Na+ and “Na* into NaCl pure single crystals

22D-24D 24 D

=D-“D

Temperature

='D This work

o-3 796.6 755.7 751 743,9 736.2 731 718.8 683.8 670 648.1 633.7 621.6 602.1 589.3

Ref. [8]

=D-=D

7 D Ref. [16]

0.0216

0.032 0.033 0.035 0.030 0.033

0.0232

0.0255 0.028 1 0.0325

0.0299 0.0269 0.0309

0.033

0.032

0.0277

Table 2. Diffusion coefficient and isotope effect for the diffusion of ‘2Na+ and “Na+ in KC1 pure single crystals

D x 10’” cm2/sec

0°C 771, 750 726, 709, 703, 681, 670 654, 636, 632

9 1 3 2 2 6 45

% 573, 7

192.7 122 72, 9 46, 7 38, 2 24, 2 16.9 11.4 7,41 6, 17 2, 53 2,00 1, 07,

'=D- 24D 7 0.030 0.028 0.024 0,022

0.018 0.019

Table 3. Diffusion coefficient and isotope effect for the diffusion of “Na* and “Na+ in KBr pure single crystals 8T

D x IO" cm’lsec

693, 5 658, 5 630, 4 625, 9 60099

123, 62, 31, 29, 14,

7 8 4 1 6

'*D/D D 0.029 0.027 0.023

and M. CHEMLA Table

4.

Measured

isotoue

effect

E=

(Ifv)/($&l)inNaCl:KCIandKBrand ,I , RbCI’(Ref. [14]) RbCl (Ref. [14])

eoc

NaCl

KC1

KBr

772 756 751 750 744 736 731 710 707 693 678 670 658 602 601 595 587 574

0.72 0.75 0.79 0.68 0.74

0, 68

-

-

0,;3

-

-

-

-

1 0.73 0.74 L -

-

-

0.54 -

0.48, -

-

0.41 044

0.64 0.60 0.52 -

0.36 0.38 -

-

= 0.29 -

good agreement with the present results. At the time these experiments were carried out, Rothman et al.[8] performed similar measurements. These authors obtained diffusion coefficients of Na’ in NaCl in excellent agreement with ours. However, concerning the isotope effects, while Rothman et al. report results in rather good agreement with the present ones at low temperature (E =0=695 at 621.6”C), a departure appears at high temperature (E = 0.522 at 751*2”C, close to our upper temperature considered). We carried out new experiments using new tracers but we still obtained the same results (i.e. E = 0.75) as those already published [ 111. On the other hand, the results relative to KC1 and KBr show a decrease of the isotope effect as temperature decreases. Such a variation had already been reported in the case of heterodiffusion of 12Na’, 24Na+in RbCl by Peterson and Rothman [ 141. This influence of temperature could be explained by the fact that in heterodiffusion more than one single jump frequency are involved in the diffusion process, making the correlation dependent.

factor temperature

4. DISCUSSION

The pioneer work (where the action of an electri- (a) Isotope effect of self-&fusion cal field was used in order to amplify the shift be-The theoretical expression of equation (3) for the tween both isotopes) was performed by Chemla [ 151 isotope effect involves five parameters, some of who reported a value close to 1. More recently, them being connected. Barr and Le Claire [ 161 reported a value of 0.73 in With the present value E = 0.75 20.05 of the

Isotope effect of diffusion of 21Nac-*4Na’ isotope effect we can combine other experimental data known in NaCl with a fairly high accuracy. These are: D, = D,., + D,,+, the diffusion coefficient of “Na’ in pure NaCl[8, 171; D- = D,.- + D,-. the diffusion coefficient of “Cll in pure NaC1[17]; D,-, the vacancy pair contribution to the anion diffusion (given by the diffusion coefficient of ‘LCll in NaCl highly doped by divalent cations)[l7]; D,,, the diffusion coefficient calculated from the ionic conductivity (T by application of the Nernst-Einstein relation:

/-

Furthermore, D,,- and D,,- are related (3) through the equation:

I I

D+ A=Dp-

f:v: f’v’

where f! and f’ are the correlation factors involved in the diffusion of the cation and anion, respectively, by the means of vacancy pairs and vi and vl the corresponding jump frequencies into the pair. Compaan and Haven [ 101 and Howard [ 181 have calculated f: (and hence fl) as a function of the ratio vi/v:. It results therefore a relation between the ratio D,JDpm and f!. This is represented on Fig. 8. Concerning v: and v!, one just has theoretical estimates performed by Tharmalingam and Lid&d [91. At last, the factor AK has been calculated for self-diffusion by free vacancies in NaCl by Brown et al. [19] who obtain AK = 0.998. However, a thorough analysis of the isotope effect still remains difficult because of uncertainties about some parameters involved in the theoretical relation: E=cufAK+(l-a)f’AK’. As a matter of fact, (Y,relative contribution of the vacancy pairs to the diffusion of the cation, has not yet been definitely determined, the value of AK from Brown et al. results from a simplified model, and there are no available data about AK’. Nevertheless, the consistency of two limiting values for (Ywith the isotope effect can be examined: (1) Making use of the calculation of v:/v’ from Tharmalingam and Lidiard [9] leads to the following estimate (1) D,,. = D,, and (Y= 0.94. But it is diffi-

21

I

06

03

0.2

01

0.06

0 03 0.02

0.01

I’

Fig. 8. Ratio of the diffusion coefficient of the cation by vacancy pairs (0~‘) to that of the anion (Op-) as a function of the correlation factor for the cation diffusion in a NaCl type ionic crystal by the means of vacancy pairs. (From the correlation factors calculated by Howard [ 181). cult to accept this result because it would entail, first, a value close to 1 for f, and second, an isotope effect E > 0.94. We conclude that the theoretical estimate of v:lvL is not consistent with the observed isotope effect and that it led to underestimate the extent of the contribution of the vacancy pairs to the cation diffusion [ 11. (2) On the other hand, by using the classical value f = 0.78 [lo] one obtains: D,,+ = D, + D,m - 0.780,. For the temperature range under consideration (602-756”(Z) this gives: 0*69<(r
22

F. NICOLAS,

F. B~NI~RE

and M. CHEMLA

We turn now towards the mea,surements in KC1 and KBr which give some useful information about the factor AK. (b) Isotope

effect for heterodifusion

Table 4 shows the isotope effect for diffusion of the same isotopes in NaCl, KCI, KBr and RbCl[ 141. This comparison clearly shows the decrease of the isotope effect as the radius of the host-cation increases. This effect could be explained by the influence of the relative ion sizes on AK. According to Mullen[20] this factor represents the part of the kinetic energy kept by the jumping ion, the other part being transmitted to the neighbouring ions which relax to allow the jump of the tracer. Le Claire [2] in a simple theory which well accounts for the isotope effect of self-diffusion in metals assumes that the kinetic energy of each ion is proportional to its displacement in the direction of the jump. This leads in metals to the relation: AK=(l+nd/r)-‘=

[ 1+;I(1 - A WI]-’

(7)

where n is the number of relaxed neighbours, d the distance of relaxation of the atoms around a vacancy in the direction of jump and 2r the jump length. The absolute value 11- AVfl represents approximately the relaxation volume relative to the molecular volume. This equation holds in metals because all atoms have same mass. On the other hand in ionic crystals cations are surrounded by anions of different mass. Moreover, the term (lA Vfl would there intend to represent the relaxation only of the anion neighbours surrounding a cation vacancy[21]. Though this term has been calculated theoretically by Faux and Lidiard [22] for each ion, the conductivity measurements under pressure of Beyeler and Lazarus[23] only give on overall volume change for the formation of a Schottky pair. Nevertheless, equation (7) can be useful for comparing the AK for diffusion of the same ion in different matrices. Let us consider the host cation of radius r. (in full line on Fig. 9) substituted by the impurity cation of smaller radius rz (in dotted line). It is to be expected that d’, the parameter of relaxation of the anions (in dotted line) neighbouring the vacancy and the impurity, will be higher than d, the relaxation parameter around the host cation. Then according to equation (7) one should have:

I Fig. 9. Schematic representation of the relaxation of the ions in an ionic crystal. The anions in full line neighbouring the vacancy and the host cation of radius rO are relaxed of the length d in the direction cation-vacancy. When the host cation is substituted by a cation of smaller radius r2 (in dotted line) the anions are relaxed of the length d’. Table 5. Values of the experimental isotope effect at 690°C in NaCI, KCI, KBr and RbCl compared with the ratio of ra, ionic radius of Na’, to r,, ionic radius of the host cation

b/r? (6;‘)

NaCl

KCI

KBr

RbCl

1

0, 71

0, 71

0, 63

0, 75

0, 63

0, 64

0, 37

tion radius (Na’, K’ and Rb’, respectively) to the diffusing cation radius (Na’), we do observe that:

Thus, it seems that the variation of the isotope effect for diffusion of the same ion in the different matrices can be qualitatively accounted for by the theory of Le Claire. As at the present time the numerical calculations of AK are just starting, this could give a way for calculating the factors AK and AK’ in self- and heterodfiusion by making use of the theory of Le Claire in the ionic crystals. 5. CONCLUSION

Considering the experimental results of the isotope effect reported in Table 5 for the same temperature together with the ratio of the host ca-

The isotope effect for self-diffusion in NaCl is found to be nearly independent on temperature and equal to 0.75 kO.05. Evaluation of the vacancy pair

Isotope effect of diffusion of Z2Naf-24Na+ contribution to the cation diffusion from the isotope effect seems to be difficult at the present time. Further calculations of the factors AK and AK’ and of the jump frequencies V: and v! would be necessary for a thorough analysis of the experimental isotope effect. Inversely, a direct experimental determination of the vacancy pair contribution combined with the diffusion, conductivity and isotope effect data could lead to the determination of the correlation factor f’ and of the sharing factors AK and AK’. Concerning heterodiffusion, the isotope effect depends on the relative ion sizes. For the alkali halides so far considered, i.e. NaCI, KCI, KBr and RbCI, it is maximum when the diffusing and host cations are identical and decreases when the host cation radius increases. This effect shows that the theory of Le Claire may be extended to the ionic crystals. Acknowledgements-We are very grateful to Dr. A. D. Le Claire and to Dr. S. J. Rothman for numerous correspondence and helpful discussions. REFERENCES F., Beniere M. and Chemla M., J. Phys. Chem. So/ids 31, 1205 (1970). 2. Le Claire A. D., Phil. Mag. 14, 1271 (1966). 3. See the reviews by Barr L. W. and Lid&d A. B., Physical Chemistry, an Advanced Treatise, Vol. 10, p. 151. Academic Press, New York (1970); Benibre 1. Btniere

23

F., Physics of Electrolytes, Vol. 1, p. 203. Academic Press, London (1972). 4. Ben&e M., Ben&e F. and Chernla M., J. Chim. Phys. 67, 1312 (1970). 5. Fuller R., Marctuardt C., Reilly M. and Wells J. C., Phys. Rev. 176,-1036 (1%8). 6. Barr L. W. and Dawson D. K.. Atom. Enerav -- Res. Estab. Rept. RG 234 (1%9). 7. Nelson V. C. and Friauf R. J., J. Phys. Chem. Solids 31, 825 (1970). 8. Rothman S. J., Peterson N. L., Laskar A. L. and Robinson L. C., J. Phys. Chem. So/ids 33,1061(1972). 9. Tharmalingam K. and Lid&d A. B., Phil. Mag. 6, 1157 (l%l). 10. Compaan K. and Haven Y., Trans. Faraday Sot. 52, 786 (1956). II. Nicolas F., Thesis, Paris (1971). 12. Dawson D. K. and Barr L. W., Phys. Rev. Lett. 19,844 (1%7). 13. Chandra S. and Rolfe J., Can. J. Phys. 49,2098 (1971). 14. Peterson N. L. and Rothman S. J., Phys. Rev. 177, 1329 (1%9). 15. Chemia M., Thesis, Paris (1954). 16. Barr L. W. and Le Claire A. D., Proc. Brit. Ceram. sot. 1, 109 (1964). 17. B&i&e F., Thesis, Orsay (1970). 18 Howard R. E., Phys. Rev. 144, 657 (1%6). 19: Brown R. C.. Worster J.. March N. H.. Perrin R. C. and Bullough R., Phil. iag. 23, 555 (l!k’l). 20. Mullen J. C., Phys. Rev. 121, 1649 (l%l). 21. Le Claire A. D., Private Communication. 22. Faux I. D. and Lidiard A. B., Z. Naturforsch. 2% 62 (1971). 23. Beyeler M. and Lazarus D., Z. Naturforsch. Ma, 291 (1971).