Ionic conductivity of silver chloride single crystals

Ionic conductivity of silver chloride single crystals

J. Phys. Chem. Solids, 1972,Vol.33, pp. 1"/99-1818. PergamonPrcss. PrlntedinGreatBritain IONIC C O N D U C T I V I T Y OF SILVER C H L O R I D E S I ...

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J. Phys. Chem. Solids, 1972,Vol.33, pp. 1"/99-1818. PergamonPrcss. PrlntedinGreatBritain

IONIC C O N D U C T I V I T Y OF SILVER C H L O R I D E S I N G L E CRYSTALS J. CORISH* and P. W. M. JACOBS

Department of Chemistry, University of Western Ontario, London, Ontario, Canada (Received 27 D e c e m b e r 1971)

Abstract-The ionic conductivity of five single crystals of silver chloride has been measured from room temperature up to 25°C below the melting temperature. The experimental data for two of these crystals, one from the Harshaw Co. with total divalent impurity cation concentration of 4.8 x 10-6 and the other doped with Cd 2+ at a concentration of 31-2 × 10-6, have been subjected to a non-linear least squares computer analysis to evaluate the thermodynamic parameters governing the formation, migration, and interaction of the point defects. The model used in this analysis includes migration of interstitial cations by both collinear and non-collinear interstitialcy mechanisms and includes longrange Coulombic interactions in the Debye-Hiickel approximation. The parameters obtained have been tested by using them to predict (i) the conductivity of the remaining three crystals; (ii) the diffusion coefficient of silver in silver chloride; and (iii) conductivity isotherms. The quality of the non-linear least squares fits and the generally satisfactory nature of the above predictions lead to the conclusion that confidence may be placed in the numerical values of the parameters with the exception of the enthalpy and entropy of migration by the interstitialcy non-collinear mechanism. Possible reasons for the difficulty in resolving accurately this contribution to the total conductivity are discussed. 1. INTRODUCTION

MEASUREMENTS of the ionic conductivity of simple ionic crystalline solids provide information concerning the thermodynamic parameters which govern the formation, migration, and interaction of the point defects present in these materials. Although silver chloride has been the subject of several such investigations there has never been a thorough study and analysis of the temperature-dependence of its ionic conductivity. Early measurements of conductivity isotherms made by Ebert and Teltow[1] between 200°C and the melting point covered most of the temperature range in which useful data can be obtained by this method. However, the impurity content of their purest samples has since been estimated [2] to have been in the range 20-30 ppm. MiJller[2] measured the temperature-dependence of the conductivity over an extended

temperature range but with relatively few points by modern standards. Furthermore his analysis, which was based on a simple model which neglected the association of divalent cation impurities and cation vacancies, was confined to the temperature range 50-250°C and the parameters obtained have been shown [3] to yield a poor extrapolation of the conductivity to higher temperatures. The most recent detailed investigation has been carded out by Abbink and Martin [4, 5] who again measured conductivity isotherms. Their analysis included impurity-vacancy complex formation and long range Coulomb interactions which were treated by the Debye-HiJckel theory as introduced by Teltow[6] and discussed in detail by Lidiard [7]. At temperatures above 298°C their results are subject to rather large uncertainties which reach 5-10 per cent at 372°C [5]. These uncertainties arise principally from a timedependence of their conductivity measurements at constant temperature, which they

*Present Address: Department of Chemistry, University College, Belfield, Stillorgan Road, Dublin 4, Ireland. 1799

1800

J. CORISH and P. W. M. JACOBS

conclude is due to the diffusion of impurities from the electrodes into their samples. The data used in their analysis extends from 150 to 370°C only. The purpose of the work described in this paper was to make accurate conductivity measurements on both pure and doped silver chloride single crystals from room temperature to within 25°C of the melting point. The curves obtained have a very high density of points and the defect parameters have been obtained as best-fit values using a non-linear least squares computer program. This is the technique introduced by Beaumont and Jacobs[8] for KC1 and since applied and improved in other investigations [9, 10]. A further reason for undertaking this type of analysis on silver chloride, in which Frenkel defects on the silver sub-lattice are now known to predominate[l 1], is the fact that there is some doubt as to the complete adequacy of a model involving only Schottky defects in accounting for the intrinsic conductivity of some alkali halides[10, 12-14]. It was thus of interest to see if the equivalent model would be adequate for a Frenkel conductor. For silver chloride there is also the question of the mechanism of the movement of the interstitial silver ions. The direct interstitial jump has been excluded [6, 15] on the basis of the large amount of energy required to overcome the ion displacements involved and the process is expected to proceed instead by the interstitialcy mechanism[16]. Measurements of the diffusion coefficient of silver in silver chloride by Compton and Maurer[17, 18] and by Reade and Martin[19], when compared with the ionic conductivity, provide evidence for the interstitialcy mechanism. Friauf[3] has carried out a careful analysis of his own and other available data for both the diffusion of silver in silver chloride and the ionic conductivity. This shows that the correlation factor at high temperatures is consistent with the simultaneous operation of the interstitial collinear and interstitial non-collinear mechanisms for

the movement of the silver ions. Abbink and Martin[5] were unable to distinguish these two types of interstitiaicy jumps from their conductivity measurements but the inclusion of both collinear and non-collinear jumps is shown to be a necessary feature in the analysis of the present data. 2. EXPERIMENTAL AND RESULTS

Complete conductivity data were obtained for five single crystals. These crystals were stored in darkness and also maintained in the dark while measurements of their conductances were being made. One crystal was supplied by the Harshaw Chemical Company and the other four; one pure, two doped with cadmium, and one manganesedoped sample, were grown from the melt at the University of North Carolina*. After growth these latter crystals were annealed in sealed ampoules containing an atmosphere of 80 torr triple-distilled chlorine and 680 torr high purity nitrogen which before mixing with the chlorine, had been passed through a trzp cooled in a mixture of dry ice and acetone. The anneal procedure involved increasing the temperature to 368°C at a rate of not more than 17°C/hr and the crystals were then cooled to room temperature at not more than 13°C/hr. All the crystals had dimensions of approximately 1.0 × 1.0 × 0.3 cm and were prepared for use by microtoming and the application of pure graphite powder electrodes as has been described previously[10]. These operations as well as the mounting of the crystals in the conductance cell were carried out under red light. The conductance cell was essentially the same as that used by Allnatt and Jacobs [20] except that pure-graphite discs were placed between the crystal and the platinum electrodes. These discs prevented any indentation of the rather soft silver chloride by the *We are indebted to Prof. L. Slifkin and C. B. Childs of the University of North Carolina who kindly gave us these crystals.

SILVER CHLORIDE SINGLE CRYSTALS

P t + 13% Rh thermocouple wires brazed to the backs of the platinum electrodes. Also it was found by experience that the use of these graphite discs in conjunction with rubbed graphite on the crystals provided an excellent and reproducible contact which necessitated hardly any pressure being applied to the sample. The cell constants were determined by micrometer measurements made on the crystals both before and after each run and these showed that there was practically no distortion of the crystals during the conductivity measurements. Lead corrections were applied and these were determined as a function of temperature for each cell before the conductivity runs. The technique developed for obtaining the conductivity data involved firstly heating the crystal slowly from room temperature to 380°C while making accurate measurements of the conductivity and temperature at 25 ° intervals. There then followed a period of anneal, usually for one or two days, until reproducible results were obtained on thermal cycling between 380 and 420°C. For the doped crystals the rough conductivity curve from the anneal temperature down to room temperature was also rechecked to ensure that complete dissolution of the impurities had occurred. Following this preparative period, which usually took about one week to complete satisfactorily, the temperature was increased to the highest value at which conductivity measurements were to be made. Readings of temperature and conductivity were then taken as the temperature was lowered by small intervals (4 to 2 °) which were chosen to provide a fairly equal spacing on a 1/T scale. Each measurement, which was repeated to check for any thermal drift, was made only after adequate time (approximately 1 hr) had elapsed from the time at which the temperature controller had been reset. The full conductivity curves established in this way for each crystal were found to be in excellent agreement with their respective rough outlines as given by the measurements

1801

made during heat-up but only the data points taken in the final downward run were used in the analyses. The conductance measurements were made using a W a y n e - K e r r transformer ratio arm bridge (Model B221 or B642). The frequency-dependence of the conductance was checked during each run and the lowest frequency in the range 104/27r Hz to 20 kHz which would ensure that polarization resistance was negligible was used. Measurements at other than the standard operating bridge frequency were made using a W a y n e - K e r r A F Signal Generator S 121 and Waveform Analyser A321. The full conductivity curves for the five crystals are shown in Fig. 1. The curves are numbered in order of increasing impurity content in the crystals with curve 1 therefore showing the conductivity of the purest sample which contained a concentration of divalent impurities equal to about 0-3 x 10-6. Curve 2 shows the results for the Harshaw crystal

b

0

~T

5

I

1.6

I

2.0

I

I

2.4

io3/7-,

2.8

I

3.2

K -~

Fig. 1. Complete conductivitycurves for the five silver chloride crystals.The concentrationsof divalentimpurity ions in these crystals are: crystal 1,0-3 × 10-s; crystal 2, 4.8 x 10-e;crystal3, 6.9 x 10-e MnZ+;crystal4, 31.2 × 10-e Cd2+; crystal 5, 1200x 10-6 Cd2÷. The nature of the impurities in crystals 1 and 2 is unknown.

1802

J. C O R I S H

and P. W. M. J A C O B S

which had a total divalent impurity content of approximately 5 × 10-6. Curve 3 is from a crystal doped with manganese at a concentration of about 7 × 10-6. The crystals represented by curves 4 and 5 were both cadmiumdoped at concentrations of 31 and 1200× 10-6 respectively. It is evident that the curves show a good deal of character; even when drawn on a large scale they exhibit no straight portions and are well suited to a non-linear least-squares analysis. Each curve is composed of about 140 data points and extends from room temperature to about 430°C i.e. to 25 ° below the melting point of the silver chloride. There were two reasons why the

measurements were not continued to higher temperatures. Test-runs showed that the crystals became very plastic when their temperature approached nearer to the melting point and thus distorted easily. Also the resistance of the crystals became so small at these high temperatures that the lead corrections, which were typically about 2.5 l~ and were known to +1 per cent, became a significant part of the total resistance thus increasing the uncertainty in the measurements. Figure 2 shows the individual data points in the temperature region above 315°C for the three purest crystals and also for the most heavily doped sample. Data points for crystal

z~

I-5

o

%t~ % o% %

o 0

0

% %a O0 D ~ & o

~

0

T

o

1.0

[] 0

T

0 o

E t-

D

°o

O IE

% t2~ O

Ib

O

~A

% O

O

%

O

o % O

D A

O

05--

O

£3 •

o

o0% o ~, Oo D

I-4

I

I

1.5

1.6 103/T,

K -I

Fig. 2. H i g h - t e m p e r a t u r e region o f t h e c o n d u c t i v i t y o f plots for four o f the silver chloride crystals: A = crystal l, • = crystal 2, [] = crystal 3, a n d O = crystal 5.

SILVER CHLORIDE SINGLE CRYSTALS

4 are not included since even at these high temperatures their departure from intrinsic behaviour, as established by crystals 1, 2 and 3, is evident. The agreement between the measurements on these three crystals is excellent and the conductivity line, as has been reported previously[2,5] is seen to exhibit positive curvature. When plotted on an even larger scale slight systematic differences between the data obtained from each of the three crystals at the highest temperatures measured are discernable and this will be commented upon later. Crystal 5 is seen never to become truly intrinsic in the temperature range shown. The dip below the intrinsic line is characteristic of a cationic Frenkel conductor in which the interstitials are much more mobile than the cation vacancies. In the temperature range 210-290°C our data for crystal 1 is in excellent agreement with the conductivity of pure silver chloride as reported by Abbink and Martin[5]. Below this temperature region any differences are clearly attributable to differences in the impurity contents of the crystals. At temperatures above 290°C disagreements in the data are also evident and at 352°C the value of trT given by Abbink and Martin is some 6 per cent lower than that of our purest crystal. This observation is completely consistent with these authors' own conclusion, to which we have already referred, that impurities diffused from the electrodes into their crystals at high temperatures. Further evidence to support their explanation is the fact that at 372°C, the highest temperature at which they made measurements, their data once again agrees to within 2.5 per cent with that for our crystal 1. This behaviour would be expected in view of the decreasing importance of the effect of the impurity content on t h e conductivity as the temperature is increased in the intrinsic region.

1803

silver sublattice at concentrations given by the Frenkel defect equilibrium expression. In addition divalent cation impurity ions are assumed to occupy substitutional lattice sites and to form associated pairs (called complexes) with cation vacancies on nearestneighbour cation sites. Long range Coulomb interactions between the defects are allowed for using the Debye-HiJckel approximation [7]. If all concentrations are expressed as site fractions then the equations resulting from this model are

flclfici = K~ = 2 exp (--gF[kT)

(1)

cJflClfm(C--Ck) = K~ = 12 exp (--ga/kT). (2) The Frenkel defect equilibrium is given by equation (1): ca and ci are the concentrations of cation vacancies and silver interstitials respectively and gF = he--TsF is the change in the Gibbs free energy accompanying the formation of the Frenkel defect pair, apart from the configurational entropy contribution, hF and SF being the corresponding enthalpy and entropy. If c is the total concentration of divalent cation impurity a fraction of which is associated to give complexes of concentration Ck then equation (2) expresses the equilibrium for this complex formation ga = ha--Tsa is the Gibbs free energy of association, apart from the configurational entropy contribution, ha and sa being the corresponding enthalpy and entropy. In the standard article on this subject[7] ga, ha and s8 are replaced by 4, -- X and - ~ respectively. The activity coefficientsfx,fi a n d f u which correspond to cation vacancies, interstitial silver ions and substitutional divalent impurity cations respectively are given by

-

f l = f i = f u = exp {--e2K/2ekT(1 + KR)} (3)

3. ANALYSIS OF DATA

The basis of the model used in the analysis is the existence of Frenkel defects in the

where e is the charge on the proton, K-1 is the D e b y e length and R is the distance of closest

1804

J. C O R 1 S H and P. W. M. J A C O B S

approach as defined by Lidiard [7]. The permittivity e was evaluated by non-linear leastsquares fitting of the data of Smith [2 I] for the dielectric constant of silver chloride to the equation E/e0 = 12.46810+214.4514 exp (--2.280431 × 10a/T)

(4)

where e0 is the permittivity of free space. Values of the Debye length were calculated from the formula

cy of the species ur, has in all cases been set equal to the frequency of the longitudinal optical modes in silver chloride at the centre of the Briilouin zone; this frequency is given by Vijayaraghavan et al.[22] as 5.80x 1012 sec -1. Any differences will be incorporated in the migration entropy A s r which is related to Agr by Ag,. = A h ~ - - T A s ~ , Ah~ being the migration enthalpy, go is the Onsager-Pitts mobility correction [2 3 ] go = 1--[e~K/3ekT(l + ~ ) × ( X/2+ KR) ]

(5)

K'-' = 47re 2 ~, c f l 2 a 3 ~ k T

(1 +KR) (9)

J

cj denotes a sum over the concentrations of J the interacting species: in this model E c~= 2

in which the values of R used were R = 0 . 5 ~ / l l a when c~ > c and R = 2 a when c~ < C.

in which u~ is the mobility of species r. The mobility is given by

Three mobile species are included in the model. They are: cation vacancies for which s~ = 4, interstitial silver ions moving by the interstitialcy collinear mechanism for which s~ = 2 and finally interstitial silver ions moving by the interstitialcy non-collinear mechanism for which s,.= 6. The mobility parameters corresponding to each of these types of motion carry the subscripts 1, ic and inc respectively. The direct interstitial jump is omitted for energetic reasons [15] and the partially backwards non-collinear interstitialcy jump is omitted for the reasons advanced by Weber and Friauf[3]. A number of calculations were also carried out in which only a single interstitialcy mechanism was allowed for in addition to the movement of the cation vacancies. The method of calculation used is essentially that devised by Beaumont and Jacobs [8]. The electroneutrality condition for the crystal

u , . = g o ( s r e a 2 u r / k T ) exp ( - - A g ~ ] k T )

c l - - c i = c--ct¢

cl + c~+ c-- ck = 2c~. a is the nearest-neighbour anion-cation distance and is calculated from least-squares fitting of the expansion data of Fouchaux and Simmons[11]. The equation used, in which the temperatures refer to a scale based on a standard value of a at 20°C, is a = 2.7748 × 10-8{1.0+0.31143 × 10-4(t-20) -0-3784 × 10-8(t--20) 2 + 0.76713 × 10-I°(t--20)3} cm.

(6)

The specific conductance, tr, is given by o - = ~, c,.eu~/2a 3

(7)

r

(8)

sr is the symmetry number for species r and Agr is the Gibbs free energy change accompanying the motion of the species to the col of the energy barrier in the direction of motion. In the absence of more detailed information the effective vibrational frequen-

(10)

is combined with equations (1) and (2). ci and ck are eliminated and the following cubic equation in c~ results. cl a + c12[K2 -- c~ ( KF + c / K 2 ) -- K F [ K2 = O.

(11)

SILVER CHLORIDE SINGLE CRYSTALS

The omission of primes on the symbols for the equilibrium constants denotes their corresponding concentration products. In calculating K~ from K~, R = 0 . 5 V ~ a in equation (3) but in calculating K2 from K~, R = 2a. Equation (11) is solved by Newton's method for assumed values of the parameters and log o-T is calculated. Refinement of the parameters is then continued until !

~o = ~ [ (log o - T ) e a l c . - (log o'T)expt.] 2

(12)

is a minimum, the sum being taken over all the data points. In this investigation a thorough fitting of the data from crystals 2 and 4 has been carried out since these data were considered most suitable for analysis. The defect parameters obtained were mutually consistent and they were then used to predict the conductivity of crystals 1, 3 and 5. The parameters were further tested by calculating (i) the diffusion coefficient of silver in silver chloride as a function of temperature and (ii) a number of conductivity isotherms, o-/o-0 as a function of c, and comparing these results with the experimental data available from the literature. 4. RESULTS OF THE ANALYSIS

(a) Analysis of the conductivity data The conductivity curves shown in Fig. 1 for the five crystals examined indicate that data from crystals 1 and 5 are least suitable for computer analysis. The impurity content of crystal 1 is so small that the conductivity curve has only a very short extrinsic region and thus the parameters which depend on this region may be expected to be poorly defined. By a similar argument data from crystal 5, which has virtually no intrinsic region,, could not be expected to produce accurate values for the intrinsic parameters. Since the data for crystals 2 and 3 are quite similar crystal 2 was chosen with crystal 4 for detailed analysis. The parameters calculated for these two crystals using the basic model are shown

1805

in Table 1. It should be emphasized that these two sets of parameters are not averages but actual best fits that have resulted from an exceedingly thorough analysis which involved many trials with different sets of starting parameters. The criteria used to establish the final minima were; lowest value of ~pand maximum number of sign changes in (log o-Texpt.--log o-T~a~c.), as well as the ability of the parameters to make the various predictions which were mentioned briefly in the previous section. The values in Table 1 have been rounded off but the full eight digits provided by the computer program were used for all the calculations to be described. The parameters which refer to the impurity-vacancy association, ha and sa, may be expected to differ since they depend on the nature of the impurity present. The other parameters are in good agreement with the exception of those relating to the non-collinear interstitiaicy motion, Ah~,c and As~,,c; these differences will be discussed later. A plot of the difference between experimental and calculated values of log o-T for crystal 2 is shown in Fig. 3. The calculated values for log o-T are from the parameters I of Table 1. With the exception of a small number of points the fit is within ± 0.5 per cent in trT over the entire temperature range which includes 120 data points. The differences between the calculated and experimental values appear to be quite random and contain fifty-four well distributed sign-changes. (b) Calculation of further conductivity curves The conductivity curves for the remaining three crystals were now predicted using each of the sets of parameters given in Table 1. To do this use was made of the facility in the minimization routine which allowed the leastsquares fitting to be effected with any chosen number of the parameters fixed. The results of the calculations of the conductivity curve for the most heavily doped crystal 5 are shown in the form of difference plots in Fig. 4. The broken line results from using the parameters

1"468

1"475

[

hr (eV)

I1

Parameters

4-8

10"30 31"2

9"95

0'287

0"288

sFlk IO"cAh, (eV) --0"501

--0'498

Asdk 0-056

0'048

A/hc(eV) --2-82

--2"65

0"308

0"688

As~c/k Ah~,c(eV) --0"147

6-40

--0"292

--0"331

Asi,clk h,,(eV)

--1"06

-- 1"94

salk

2"8

1'3

4

2

120 146

Number of lO:~o's.a. Crystal points

©

.~

.~

e'~

,,.t

Table 1. Thermodynamic parameters for the formation, migration and interaction of defects in silver chloride as o 70 obtained fi'om conductance measurements. The symbols have been defined in the text. Crystal 2 is a Harshaw crystal and ~ crystal 4 is one doped with Cd "+ ions. o's.a, is the standard deviation ~:

¢.,.

oo

SILVER CHLORIDE SINGLE CRYSTALS I03/T, 1.6

2.0

K-i

2.4

T

1807

2.8

T

r

3.2

r-----

' - - ' ~ - -

2

~"

0

g'

-2

%

i

-4

300

400

200

I00

50

I, °C

Fig. 3. Plot of 103[log (drT)exot. - l O g (o'T)ea,e.] vs. I031T for crystal 2. The calculated values of log o-T are from the best least squares fit to the experimental data for crystal 2 and correspond to the set of parameters designated I in Table 1.

103/T, 1.6

2.0

L 12 - -

K -I

2.4

I

,

2.8

I

'

I

5.2

'

I

'

,, • -- -n

ILl-

!iVi'

bv

b W

bv

b

0

LL'

-4

0

0

-8 -12

:1 4OO

J

I 500

z

I

,

200 t,

I

i

I00

50

°C

Fig. 4. Plots of 10a[ log (o'T)e.,ot. -- log (o'T)c,t,..] vs. I03/T for crystal 5. The continuous line represents the difference between the experimental values of log o'T and those predicted from parameter set 11 (from crystal 4) with only c variable. The broken line represents the difference between the experimental values of log o-T and those predicted from parameter set I (from crystal 2) except that Ah, and As, were changed to values appropriate to Cd z+ and then only c was allowed to vary.

1808

J. CORISH and P. W. M. JACOBS

from crystal 2 and the full line is from those the same reason the effect of the association for crystal 4. Since both crystals 4 and 5 parameters was expected to be minor and were cadmium doped the minimization em- consequently they were left fixed at the values ploying parameter set II involved one free given by the Harshaw crystal 2. The poor fit parameter only; i.e. c, the concentration of around the knee may be ascribed to the impurity. An approximate starting value for combined effects of keeping the formation c was found by comparing the conductivity parameters hF and se fixed together with the of crystal 5 in the extrinsic vacancy-controlled program's inability to define c accurately region with that of crystal 4, from which the when it has only a short extrinsic region to parameters originated, taking impurity-vacan- work with. cy association into account. Before using parameter-set / the values of h,, and s, were (c) Prediction of diffusion data changed to those appropriate to divalent One of the advantages in obtaining a cadmium, taking these two values from set II. complete set of parameters such as those in The minimization was then again carried out Table ! is that they can be used to predict with only c variable. As is evident from Fig. 4 the diffusion coefficient of silver as a function both sets of parameters were successful in the of temperature. This was done using the extremely severe test of predicting the con- modified Nernst-Einstein relation ductivity of the heavily doped crystal 5 with all parameters fixed but c. The calculated and D,. = 2aaf r(kT/e ") o',. (i 3) experimental points agree to within _ 3 per cent over the entire temperature range in where Dr is the diffusion coefficient of species which the conductivity was measured, The r and fr the corresponding correlation factor. computed values for the impurity content of Since the mobility parameters for each type of crystal 5 were 1201 and 1154 x 10-6 respec- mobile species have been distinguished, it tively from the two minimizations carried out was possible to calculate the individual using parameters I and II. contributions from each of these species, In calculating the conductivity curve for tr,., to the specific conductance. These values crystal 3, which was manganese doped, the of or,. were then substituted in equation (13), values for h,, and s, were also allowed to vary along with the correlation factors for each as well as c since differences in curvature in species, to give their individual contributions the association regions of the conductivity to the diffusion coefficient. The values used curves for crystals 2 and 3 were apparent for the correlation factors were obtained (Fig. 1). The remaining eight parameters were from Compaan and Haven[24, 25] and were held fixed. The predicted conductivity curves 0-78146, 0.3333 and 0-7273 for the vacancy, were in the same good agreement in the high collinear interstitialcy and non-collinear interand low temperature regions as was found for stitialcy mechanisms respectively. crystal 5 with a slightly worse deviation in a The values of the silver diffusion coefficient short region about the knee. The converged calculated as a function of temperature by the values for h,, s J k and c were --0.266eV, method described are shown as a solid curve - 0 . 2 3 0 and 6.9x 10-6 respectively. For crys- in Fig. 5. The most recently available experital 1 the agreement between the predicted and mental measurements of DAg by Weber and experimental curves was also good at high Friauf[3] and Reade and Martin[19] are also temperatures but was poor around the knee. shown for comparison. The agreement beSince there is hardly any extrinsic region it tween these workers in the temperature range was not surprising that difficulty was experi- in which their data overlap is good and these enced in finding the correct value for c. For data are also in good agreement with our

SILVER CHLORIDE SINGLE CRYSTALS

1809

the computed line and the experimental data is seen to extend almost to 148°C, the lowest temperature at which the experimental measurements were made. The difficulties in obtaining accurate diffusion data increase as the temperature is lowered and the authors estimate the relative uncertainty in their values of D at 148°C as 20 per cent. A second diffusion curve similar to the one shown in Fig. 5 was calculated from parameters II with the concentration of impurity again chosen at 5 x 10-6. It too was found to be in good agreement with the experimental data. ._= § O

i-O

i71

o~

-I 1.4

I

~

I

I

1.6

1.8

2.0

2.2

103/T.

°1 2.4

K-'

Fig. 5. Logarithm of the diffusion coefficient of silver in AgCI as a function of reciprocal temperature. The continuous line was calculated as described in the text from parameter set I of Table 1 for a crystal with an impurity content of 5 × 10-e. • experimental points from Weber and Friauf[3]; O experimental points from Reade and Martin [ 19].

calculated line. In the lower temperature region the diffusion rate is expected to depend on the impurity content of the crystal. The data reported by Reade and Martin [19] were obtained from a single crystal supplied by the Harshaw Chemical Company. In an attempt to match their experimental conditions as closely as possible our line was calculated using parameters I with an impurity content of 5 x 10-e which we estimate to be the approximate impurity content of their crystal. The computed concentration of divalent impurity in our own Harshaw crystal was 4.8 × 10-6 (Table 1). The agreement between

(d) Prediction of conductivity isotherms The analysis of conductivity isotherms has been the most common method used to evaluate the formation and migration parameters for defects in substances such as silver chloride which exhibit Frenkel disorder. A comprehensive review of the theory and earlier experiments has been given by Lidiard [7]. More recently Abbink and Martin[4, 5] have made careful measurements of both the impurity concentrations and relative conductivities of a number of cadmium-doped silver chloride single crystals at a series of temperatures. Since these comprehensive data of high quality were available for comparison purposes, we considered it worthwhile to calculate conductivity isotherms for silver chloride from our parameters. Certain of the anticipated general features of the isotherms for silver chloride may be deduced by a careful examination of Fig. 1. If we consider a temperature of 181.4°C, which corresponds to 103/T = 2-2K -1, then it is evident that small additions of divalent impurities, as in crystals 2 and 3, have caused the ionic conductivity to decrease relative to the pure crystal 1. This observation is explained by taking into account the Frenkel equilibrium expression given by equation (1). The divalent impurities added increase the vacancy concentration and necessitate a corresponding decrease in the concentration of the interstitial silver ions. Since these

1810

J. C O R I S H and P. W. M. J A C O B S

interstitial ions are the more mobile of the Frenkel defect species the conductance of the crystal is diminished. However, should the concentration of the divalent impurities be further increased to the levels in crystals 4 and 5 then the conductivity is seen to increase again and for the most heavily doped crystal to be several times larger than that of the purest crystal. These increases in o- occur because a silver chloride crystal containing this level of impurity has already, at the temperature being considered, passed from the intrinsic region where defect concentrations are governed by the Frenkel equilibrium to the extrinsic or impurity controlled region. Here its conductivity depends principally on the vacancy concentration and thus increases with the impurity content of the crystal. From these simple considerations we may conclude that the conductivity isotherms for silver chloride, at least within a certain temperature range, will contain a well-defined minimum. The impurity concentration at which this

minimum occurs is temperature-dependent (Figs. 6 and 7). These show, at 192-0 and 298.3°C respectively, the conductivity isotherm points calculated from parameter sets I and II and also the experimental data of Abbink and Martin [5]. At 192.0 the value of tr0 used was that given by Abbink and Martin [5] since their purest sample contained only 0.I × 10-6 of divalent impurities. Our own purest crystal, with a concentration of 0.3 x 10-6 was already some 4 per cent lower in o-T. For the data of Fig. 7 however, the value of o-0 was taken as our own experimental value for crystal 1 since 298.3°C is in the region where the Abbink and Martin values are less reliable and crystal 1 is intrinsic. The agreement between the calculated and experimental data in Figs. 6 and 7 is good with most of the calculated points well within the scatter evident in the experimental data. The uncertainty in c is estimated by the authors to be ___5 per cent. The successful prediction of these isotherms proved to be very sensitive O n o o o

B,x 0 B

El &

b~ -g

O~

O

Z Bat:) O

"5

~o

,P

0

I

I

I

I

2

4

6

8

concentration

c

x

I0 4

Fig. 6. Comparison of calculated and experimental conductivity isotherms for AgCl at 192.0°C. O Experimental data of Abbink and Martin[5]; A Calculated from parameter set I; [] calculated from parameter set II.

SILVER CHLORIDE SINGLE CRYSTALS

1811

1.2 i.l b~ 0 o

b

l.O

0

o

> o

0.9 --0

.~ 0.8

o~

'b _%

OZ~ 0 [] A

o

2 0.7

--

0

~, 0

o

0

0.6

0

I

I

2

I

4

concentration

6

I

8

c x 10 4

Fig. 7. Comparison of calculated and experimental conductivity isotherms for AgCI at 298'3°C. © Experimental data of Abbink and Martin[5l; A calculated from parameter set I; [] calculated from parameter set II.

to certain of the parameters and this test was probably the most stringent of those to which the parameters have been subjected because of the wide concentration range ( 0 - 8 0 0 x 10-6 ) covered. 5. DISCUSSION

(a) The data In a detailed analysis of the kind carded out here it is essential that data of the highest possible quality be used. We believe that this criterion has been satisfied although such a pronouncement is of necessity rather subjective. A model which includes both collinear and non-collinear interstitialcy jumps requires eleven parameters, two more than the Schottky defect model used for alkali halides [9, 10]. Several of the parameters are highly correlated so that the non-linear least squares problem is a formidable one. The average number of data points per run was 140 and each run took about a month to complete. It is impossible to display the data on a large

scale over any extended range but in Fig. 2 the scatter of individual points about a hypothetical best line for that crystal is typically 0.2 per cent in trT. Individual runs on different crystals show comparable reproducibility within the short intrinsic region which occurs at around IOa/T = 1"6K-1. At higher temperatures there is a gradual systematic decrease in o'T for crystals 2 and 3, when compared with that for the purest crystal 1, which amounts to a maximum value of 2 per cent at the highest temperature shown. For crystal 4, for which c = 31-2 x 10-8 and which never becomes quite intrinsic, this deviation in trT increases to --11 per cent at the highest temperature. It is difficult to make a quantitiative assessment of the probable errors. Errors in these measurements are of two kinds, those affecting every point on a conductivity curve and those affecting individual points. The first kind, which include errors made in measuring the crystal with a micrometer, irregularities

1812

J. CORISH and P. W. M. JACOBS

in the shape of the crystal, and imperfect contact with the electrodes, will result only in a translation of the conductivity curve parallel to itself. They will affect the entropies but not the enthalpies. That such errors can together be as large as 0-5 per cent in trT was shown by one experiment in which the crystal was removed from the conductance cell, remicrotomed to different dimensions, new graphite electrodes applied, and the crystal then replaced in the cell and its conductivity remeasured over a 30° temperature range. Thus in assessing run-to-run reproducibility a parallel translation of conductivity curves of this order of magnitude should be allowed for. The run-to-run reproducibility observed in the intrinsic region is in fact better than this figure but both above and below the short intrinsic range the curves diverge because of concentration effects. Superimposed on such constant errors are those associated with each individual point. The temperature was measured to 0.01°C at each electrode and these were in thermal contact with two opposite faces of the crystal through a thin graphite layer. Usually these two readings agreed to better than 0-1°C but differences up to 0.2-0-3°C (due to thermal gradients in the furnace) were tolerated as this small temperature gradient does not affect cr which corresponds to the mean temperature. Small constant thermocouple errors would translate the whole curve and affect the entropies. Fluctuating errors are again seen to be small because of the excellent reproducibility on thermal cycling during the same run and from run to run. The largest probable source of error lies in the measurement of or. The accuracy of the bridg e is stated by the makers to be better than ___0-25 per cent up to 10 kHz, and better than ±1.0 per cent up to 20 kHz. Measurements were made at the lowest frequency compatible with the avoidance of polarization, most of the readings being taken at frequencies below 10 kHz. However, the results depicted in Fig. 2 were all taken at 2 0 k H z because polarization

effects are largest at the highest temperatures and the smallness of the random scatter shows that the conductance measurement is the largest source of error and that the bridge specifications are probably somewhat conservative. One final estimate of the random errors is available. The mean deviation [Oexot.-- o-c~lc.1/ o-c~lc, for a typical best fit, actually that shown in Fig. 3, is 0-25 per cent. If the model is perfectly correct and the parameters are correctly assigned then this is the mean of the random experimental errors associated with each point, assuming that all the error is in o- and none in T. If the model is imperfect, or the parameters are imperfectly assigned, then the mean deviation is an upper limit to the mean of the random experimental errors. (b) The analysis The reliability of the computing method for analysis of conductivity data has recently been subjected to a thorough test[10]. It was shown that a set of perfect-fit parameters can be located by the program to a good degree of accuracy with the exception of c and As1, which are strongly correlated. This is especially true when the degree of association p is small as it was in the test case. A separation of these parameters may, however, be achieved by comparing the analyses of the data from a number of crystals[10]. In the case of silver chloride we feel that the problem has been largely solved by the application of the following two tests. Firstly the values of o- at a constant temperature in the extrinsic region were plotted against c ( l - - p ) . As a straight line is expected both o- and c ( 1 - p ) can be multiplied by the same power of 10. This feature of the plot (see Fig. 8) has the great merit of spreading the numbers out over a range of 0-10 instead of over more than three decades. At 103/T = 2.7K -1 crystal 1 is not extrinsic and so only four points appear on line a. The good straight lines obtained show that the computed concentrations are consistent although they may all still be wrong by

SILVER CHLORIDE SINGLE CRYSTALS

6

--

1813

S 2

4 b 4

2

3

_

b

0 0

I

I

I

I

I

2

5

4

5

6

c (l-p) Fig. 8. Specific conductance o- at constant temperature as a function of the concentration of free vacancies c( 1 - - p ) . Line a, 97.2°C; line b, 60"2°C. o- is expressed in ohm -1 cm -1 and since a linear dependence on c(1 - p ) is expected, both o- and c(l - p ) may be multiplied by the same power of I0 so that all the points lie in the same decade. The numbers next to each point designate the crystal to which the numerical values refer.

a constant factor which would be absorbed in the other parameters, principally Asl. The second test involved the use of isotherms. One of the great advantages of using the parameters to predict isotherms is that this calculation decouples c from As1. At high impurity concentrations the isotherms may be expected to depend principally on the values of Ahl, As1, Ah, and As,. Figure 9 shows plots of o-/o-0 vs. c at 267-3°C calculated from four sets of parameters obtained during the fitting of the temperature-dependence of the crystal 4 data. The relevant parameters are listed in Table 2. These sets were selected because the fits were all closely similar but displayed a reasonable range in As1. Comparison of the calculated isotherms with the experimental data of Abbink and Martin[5] shows very clearly that subset B is to be preferred. Subset B is from parameter set 1I of Table 1. Particularly in view of the excellent agreement between the values of As~ obtained independently from crystals 2 and 4 using this method we

conclude that the computed concentrations for all the crystals are not only internally consistent but also substantially correct. (c) Models Our discussion so far has dealt exclusively with the interpretation of the conductivity data in terms of a simple model. In this model contributions to the conductivity from the movement of cation vacancies and from interstitial silver ions, which move by both collinear and non-collinear mechanisms, are considered. Although the correlation factors obtained by comparison of conductivity and diffusion data are consistent with this interpretation[3] a separation of the two types of interstitialcy motion has not been made previously on the basis of conductivity data alone. Previous investigators have regarded the interstitials as moving by some kind of 'average' interstitialcy mechanism which had no real physical meaning. In such a treatment the plot of uT vs. l/T, where u is the mobility

1814

J. CORISH and P. W. M. JACOBS

_ _

A

Z~

b > .t.O

Z~ 0

2m

• D

tO ¢.3

0

._>

0



c~ o ~7

o v

0

~7

o v



D Ix

t--

F"

.0"

0

I

I

2

4

concentration

C

I

I

G

8

x I0 4

Fig. 9. Conductivity isotherms at 239"7°C calculated from four sets of parameters obtained during the analysis of data for crystal 4, compared with the experimental data of Abbink and Martin [5]. The four sets had closely similar values for all the parameters except As~ and these values, and those of other relevant parameters are designated A, B, C, D in Table 2. • Experimental data[5]; [] = B; A = A , O = C and V = D when these differ significantly from B. f o r this a v e r a g e i n t e r s t i t i a l c y m o t i o n , w o u l d b e e x p e c t e d t o e x h i b i t a k n e e if t h e t w o k i n d s o f i n t e r s t i t i a l c y m o t i o n w e r e to b e d i s t i n guished. A b b i n k and Martin[5], who un-

successfully sought such evidence, concluded that their data was of insufficient accuracy a n d c o v e r e d t o o s m a l l a t e m p e r a t u r e r a n g e to provide an answer.

Table 2. Values o f the vacancy and association parameters used in the calculation o f the isotherms shown in Fig. 9. Each set shown is a sub-set o f parameters that resulted from an attempted least squares fit to the data for crystal 4 Sub-set

ha, eV

sa/k

Ahl, eV

As~/k

A B C D

--0.2921 --0.2916 --0.2905 --0.2914

--1.0209 --I .0547 -1.2163 --1.1775

0.2875 0.2871 0.2907 0.2876

--0-3891 --0.5014 --0.5487 --0-6630

SILVER C H L O R I D E S I N G L E CRYSTALS

Although the parameters in Table 1 appear to describe adequately the creation and movement of defects in silver chloride it was felt to be necessary to check that the inclusion of the two types of interstitialcy mechanism was an essential feature. To do this we carried out a number of analyses on the data from crystals 2 and 4 using a program based on a simpler model which included only one type of interstitialcy motion. Since our calculation demands that the nature of the mechanism be specified we chose to use the more probable collinearjump. All attempts to derive mutually consistent sets of parameters from the data for these two crystals using this simple model were unsuccessful. The more heavily doped crystal 4 gave a good fit, comparable in quality to that obtained using the original model, but with several differences in the parameter values. But values of ~, which resulted from analyses of the crystal 2 data for the simple model were between one and two orders of magnitude worse than had been obtained previously, while some of the parameters consistently converged to physically absurd values. Furthermore, the predictions of the temperature-dependence of the diffusion coefficient of silver and of conductivity isotherms, based on the best-fit parameters for crystal 4 (which gave a reasonable ~p) were poor compared to those calculated from the complete parameters sets I and II of Table 1. We conclude that the inclusion of non-collinear interstitial jumps is essential if conductivity data at reasonably high temperatures for reasonably pure crystals are to be interpreted satisfactorily. In a number of investigations a change of slope has been observed in plots of log D vs. l / T , where D is the diffusion coefficient of an impurity cation in silver chloride [ 19, 26-30]. Various explanations of this effect have been offered including (i) diffusion of complexes at low temperatures and of single Cd 2+ ions by an unidentified mechanism at high temperatures [19]; (ii) diffusion by Cd '-'+ complexes at all temperatures with the break at 320°C

1815

caused by a change in the association process from an impurity controlled one to one involving intrinsic vacancies [31] and (iii) for both Cd 2+ and Mn 2+ diffusion by interstitial M 2+ at low temperatures and by M 2+ complexes at high temperatures[29, 32]. Mobile complexes being neutral will not contribute to the conductivity but interstitial divalent cations will change the conductivity of a silver chloride crystal doped with that cation in two ways. Firstly, the extra cation vacancy produced with each M ~-÷ interstitial will affect the conductivity, directly in the extrinsic region, and indirectly in the intrinsic region by reducing the number of interstitial silver ions in the Frenkel defect equilibrium expressed by equation (1). The degree of association of the substitutional cations and cation vacancies will also be affected. Secondly, if the impurity interstitials are mobile then their motion will contribute directly to the conductivity. Our data was therefore analyzed on two further models: both models allowed for a Frenkel defect equilibrium involving impurity cations but in the first model, in order to keep the number of parameters at 13 rather than 15, the interstitial impurity cations were assumed to be immobile, whereas in the second version they were allowed to migrate (by the direct interstitial mechanism) and thus contribute to the conductivity. The results of these calculations are summarized below. The calculation proved to be somewhat unstable and although several converged sets of parameters were obtained these invariably included some absurd values. Furthermore the fits were very insensitive to the values of the enthalpy and entropy of formation of the impurity Frenkel defect pair and also depended on the initial values chosen for these parameters. This behaviour indicates the presence of, at most, a small number of impurity interstitials, a conclusion in agreement with that of Laskar and Slifkin from their diffusion data[29]. We conclude that, although it might be possible to detect the presence of impurity interstitials from conductivity

1816

J. C O R I S H

and

measurements, any effects due to this cause are too small to affect our present measurements to any significant extent. (d) The parameters In Table 3, all the available values for the thermodynamic parameters for defects in silver chloride are compared. The early data of Ebert and Teltow [ 1] has not been included since their analysis was carried out without the inclusion of long-range Coulombic interactions between the defects. Miiller[2], whose values are shown in column I, also neglected these interactions but analyzed data from a temperature range in which they are less important. Abey and Tomizuka[33] have studied the pressure-dependence of the conductivity of AgCI but as they derive only two of the parameters being compared in Table 3, these results have been omitted. All the mobility entropies in the Table are referred to the standard vibrational frequency used in this paper, namely 5.8× 10I~ sec -1. The values in columns I and II have been

P. W . M . J A C O B S

calculated from the original references. The only previous values, other than the theoretical ones of column VII, for the interstitiaicy collinear and non-collinear parameters are those in columns III and IV, which were obtained by Weber and Friauf[3] from the analysis of their own diffusion data coupled with the conductivity data from references [2] and [4]. Weber and Friauf[3] quote their results as the frequencies ~T which correspond to the microscopic diffusion coefficient dr for species r, 1

dr = -~ pr)kr

2

(14)

where hr is the actual jump distance. The frequencies quoted by these authors have been used in conjunction with our equation (8). taking due account of appropriate symmetry numbers, to obtain the entropy values given here. The two sets of parameters obtained by our analysis of the data for crystals 2 and 4, using the original model, are shown in

Table 3 *. Summal3' of values for defect parameters in AgCl

Parameter

he, e V sr/k Ahl, eV

Asi/k Ahlc, e V

Astelk Ah,,c, eV

Asi,,c/k A h , eV~:~

As/k~;qt

!

I1

I11

IV

V

VI

Vll

Miiller R e f . [2]

Abbink and Martin't R e f s . [4, 5]

Weber and Friauf-~ R e f . [3]

Weber and Friauf§ R e f . [3]

Present work** Crystal 4

Present work Crystal 2

Present work Crystal 4

Theoretical valuesit Ref. [15, 34]

1-25 6-0 0.35 0"25 ----0.16 -- 0.45

1"55 12.24 0.27 - 1'19 ----0.055 -- 2.54

----0.114 - - 1 "64 0.260 -- 0-952 ---

----0.008 -- 3"76 0.132 -- 3-57 ---

1-520 11.66 0.302 --0"191 ----0-082 - - 2-43

1.475 10-30 0.287 --0"501 0.056 - - 2"82 0.308 -- 0-147 ---

! .83 -----0-1 to0-4 -< 0.8 ----

1.468 9.95 0-288 --0"498 0.048 - - 2"65 0.688 6-40 ---

VIII

* A l l m o b i l i t y e n t r o p i e s q u o t e d a r e r e f e r r e d t o t h e s t a n d a r d v i b r a t i o n a l f r e q u e n c y u s e d in t h i s p a p e r , i.e. 5 ' 8 × 1012

sec -~. tCalculated by the adthers ~:Calculated by Weber and §Calculated by Weber and ** C a l c u l a t e d u s i n g a m o d e l ttFormation enthalpy from :~:~'Average' i n t e r s t i t i a l .

f r o m t h e d a t a o f r e f e r e n c e s [4] a n d [5]. F r i a u f [ 3 ] f r o m t h e d a t a o f M i i l l e r [2] u s i n g a v a i l a b l e d i f f u s i o n d a t a . F r i a u f [ 3 ] f r o m t h e d a t a o f A b b i n k [4] u s i n g a v a i l a b l e d i f f u s i o n d a t a . which did not include non-collinear interstitialcy jumps. Y a m a s h i t a a n d K u r o s a w a [ 3 4 ] ; m o b i l i t y e n t h a l p i e s f r o m H o v e [ 15].

SILVER C H L O R I D E S I N G L E CRYSTALS

columns VI and VII for convenient reference. The parameters derived from the model involving a single (collinear) interstitialcy mechanism shown in column V are seen to be inreasonable agreement with those obtained by Abbink and Martin, column II, with the exception of As~. Neither of these As1 values is correct because of the neglect of the non-collinear interstitialcy mechanism. Both the enthaipy and the entropy for the 'average' interstitialcy motion are closer to the corresponding values derived for collinear jumps in the complete model (columns VI and VII). This may be expected as the collinear mechanism dominates in the lower part of the intrinsic region. There is good agreement between all the parameters obtained by the independent fitting of the data from crystals 2 and 4, with the exception of those relating to the non-collinear interstitialcy motion. These parameters (columns VI and VII) are also in reasonable agreement with the rather limited theoretical results (column VIII). The differences between the two sets of inc parameters show up clearly in the predictions of the conductivity of crystal 5 shown in Fig. 5. While both sets predict the conductivity equally well the deviations have opposite signs in the hightemperature region. The value of Ah~n~ from crystal 4 is in moderate agreement with the values derived by Weber and Friauf[3], columns III and IV, but the corresponding entropies agree poorly. The ratio of interstitialcy non-collinear to collinear jump frequencies is given by

w~.clw~c= exp {(Asi.e-As~c)/k}

(15)

The values of this ratio for crystals 2 and 4 come out to be 8500 and 14.5, respectively. The first figure seems much too large and so we incline towards favouring the values from column V I I as more likely although it is apparent that the inc parameters are known with far less precison than the others. The principal reason for this is that the mechanism

1817

makes a significant contribution to the conductivity only at high temperatures and the corresponding parameters are thus much harder to' define by least squares fitting than those which influence the conductivity over a wider temperature range. There are three further factors, in addition to the onset of the inc mechanism, that influence the detailed shape of the intrinsic part of the conductivity curve. Firstly there is the possibility of a small contribution to tr from Schottky defects [11]. Secondly, there is the increase in association which occurs both with increasing temperature, at constant impurity content, and with increasing c at constant T. (The former may sound surprising at first but it occurs because the increase with temperature of the cation vacancy concentration via the Frenkel equilibrium, outweighs the decrease in/(2 that occurs because Aga is a negative quantity.) Thirdly there is the inadequacy of the Debye-Hiickel model for treating the long-range Coulombic interactions. Allnatt and Loftus[35] have shown recently that evaluation of the cycle diagrams [36], that in solution give the Debye-Hiickel approximation, leads to values of the activity coefficient f for an ionic crystal which are smaller than those predicted by the D e b y e Hfickel formula (3). The magnitude of this discrepancy is approximately 8 per cent in f a t the highest temperature employed. The system responds to a decrease i n f b y an increase in c and thus long-range interactions always lead to an increasingly positive curvature (i.e. concave upwards) in the conductivity curve. The net result of the other two effects (association, Schottky defects) is hard to predict since it involves the interplay of three defect equilibria. Qualitatively one can see that the concentration dependence of tr at high temperatures could be connected with both these factors. All these three effects on o--underestimating the Coulombic interactions between the defects, association, and the possible occurrence of a significant number of Schottky

1818

J. C O R I S H and P. W. M. J A C O B S

d e f e c t s - a r e but minor perturbations and should have little effect on the parameter values that have resulted from our analysis, with the exception of the inc parameters. Association has been included in our model and so the poor agreement between the inc parameters for crystals 2 and 4 leads us to suppose that some additional factor is influencing to some extent the high-temperature conductivity of silver chloride. Acknowledgements--We are indebted to the National Research Council of Canada for their support of this work. We should also like to thank Prof. Ailnatt and Dr. Loftus for allowing us to make use of their results prior to publication.

REFERENCES 1. E B E R T 1. and T E L T O W J., Ann. Phys. 15, 268 (1955). 2. M 0 L L E R P., Phys. Status Solidi 12, 775 (1965). 3. W E B E R M. D. and F R I A U F R. J., J. Phys. Chem. Solids 30, 407 (1969). 4. A B B I N K H. C., Thesis, Iowa State University (1964). 5. A B B I N K H. C. and M A R T I N D. S Jr., J. Phys. Chem. Solids 27, 205 (1966). 6. T E L T O W J.,Ann. Phys. 5, 63, 71 (1949). 7. L I D I A R D A. B., In Handbuch der Physik, (Edited by S. Flugge), Vol. XX, Part 2, pp. 246-349, Springer -Verlag, Berlin (t 957). 8. B E A U M O N T J. H. and J A C O B S P. W. M., J. chem. Phys. 45, 1496 (1966). 9. A L L N A T I " A. R., P A N T E L I S P. and S I M E S. J., J. Phys. 4, 1778 (1971). 10. J A C O B S P. W. M. and P A N T E L I S P., Phys. Rev. B 4,3757 (1971). 11. F O U C H A U X R. D. and S I M M O N S R. O., Phys. Reo. 136, A1664 (1964). 12. A L L N A T T A. R. and P A N T E L I S P., Solid State Commun. 6, 309 (1968). 13. BROWN N. and J A C O B S P. W. M., to be published.

14. F U L L E R R. G. and R E I L L Y M. H.,J. Phys. Chem. Solids 30, 457 (1969). 15. H O V E J . E.,Phys. Rev. 102, 915 (1956). 16. S E I T Z F.,Acta crystallogr. 3, 355 (1950). 17. C O M P T O N W. D., Thesis, University of Illinois (1955). 18. C O M P T O N W. D. and M A U R E R R. J., J. Phys. Chem. Solids 1, 191 (1956). 19. R E A D E R. F. and M A R T I N D. S. Jr.,J. appl. Phys. 31, 1965 (1960). 20. A L L N A T T A. R. and J A C O B S P. W. M., Trans. FaradaySoc. 58, 116 (1962). 2 I. S M I T H G. C., Materials Science Center Report No. 51, Cornell University (1962), unpublished. 22. V 1 J A Y A R A G H A V A N P. R., N I C K L O W R. M., S M I T H H. G. and W I L K I N S O N M. K., Phys. Rev. B 1, 4819 (1970). 23. PITTS E., Proc R. Soc. Lond. A217, 48 (1953). 24. C O M P A A N K. and H A V E N Y., Trans. Faraday Soc. 52, 786 (1956). 25. C O M P A A N K. and H A V E N Y., Trans. Faraday Soc. 54, 1498 (1958). 26. H A N L O N J. E.,J. chem. Phys. 32, 1492 (1960). 27. B A T R A A. P., L A S K A R A. L., B R E B E C G. and S L I F K I N L. M. In Diffusion Processes, Proc. Thomas Graham Sym., Univ. of Strathclyde (1969) (Edited by J. N. Sherwood, A. V. Chadwick, W. M. Muir and F. L. Swinton), Gordon and Breach, New York (1970). 28. B A T R A A. P., L A S K A R A. L. and S L I F K I N L. M.,J. Phys. Chem. Solids 30, 2053, 2061 (1969). 29. L A S K A R A. L. and S L I F K I N L. M.,J. Nonmetals, in press. 30. S U P T I T Z P. and W E 1 D M A N N , Phys. Status Solidi 27, 631 (1968). 31. F R I A U F R. J.,J. Phys. Chem. Solids 30, 429 (1969). 32. S A W Y E R E. W. and L A S K A R A. L., Bull. Am. Phys. Soc. 16, 363 (1971). 33. A B E Y A. E. and T O M I Z U K A C. T.,J. Phys. Chem. Solids 27, 1149 (1966). 34. Y A M A S H I T A J. and K U R O S A W A T., In Photographic Sensitivity Vol. I, Proc. Symp. Hakone, Japan (1953) (Edited by S. Fujisawa), Maruzen, Tokyo (1956). 35. A L L N A T T A. R. and L O F T U S E. J., Crystal Lattice Defects, in press. 36. A L L N A T T A. R. and C O H E N M. H., J. chem. Phys. 40, 1871 (1964).