A THEORETICAL STUDY OF INTRINSIC AND EXTRINSIC DEFECT PROPERTIES OF ALKALI HALIDES

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A THEORETICAL STUDY OF INTRINSIC AND EXTRINSIC DEFECT PROPERTIES OF ALKALI

HALIDES

C. Catlow, J. Corish, K. Diller, P. Jacobs, M. Norgett

To cite this version:

C. Catlow, J. Corish, K. Diller, P. Jacobs, M. Norgett. A THEORETICAL STUDY OF INTRINSIC AND EXTRINSIC DEFECT PROPERTIES OF ALKALI HALIDES. Journal de Physique Colloques, 1976, 37 (C7), pp.C7-253-C7-259. �10.1051/jphyscol:1976761�. �jpa-00216921�

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JOURNAL DE PHYSIQUE Colloque C7, supplkment au no 12, Tome 37, Dkcembre 1976, page C7-253

A THEORETICAL STUDY OF INTRINSIC

AND EXTRINSIC DEFECT PROPERTIES OF ALKALI HALIDES

C. R. A. CATLOW ( I ) , J. CORISH ( 2 ) , K. M. DILLER, P. W. M. JACOBS(3) and M. J. NORGETT Theoretical Physics Division, AERE Harwell, Oxfordshire, U. K.

R6sum6. - Une etude theorique detaillee des energies de formation, ainsi que des migrations d e defauts simples et complexes dans seize halogenures alcalins de structure NaCl est effectuk. Deux types de potentiels interatomiques differents basks sur le modele de couche ont kt6 employes dans ce travail. I1 y a un certain nombre de resultats qui sont importants pour la comprehension des phe- nom6nes de transport dans ces substances :

a) Les energies d'activation pour les lacunes cationique et anionique sont tres proches, et il en rksulte que leur contribution relative a la conductivite est independante de la temperature ;

b) Les complexes & trois lacunes ne contribuent pas h la conductivitk, par contre les paires de lacunes contribuent

a

la diffusion de f a ~ o n significative ;

c) L'energie d'Arrhenius du transport par mecanisme interstitiel colineaire n'est pas beaucoup plus grande que celle des lacunes, et il est possible que la courbure (de log (oT) vs. T-1) observk

a

hautes temperatures soit due a la contribution des interstitiels.

Comme test de fiabilitt des potentiels employes, nous avons Bgalement 6tudie un certain nombre de systemes avec des ions mono-valents occupant un site non centre. L'accord de ce travail avec I'experience est encourageant, sans 6tre completement satisfaisant.

Abstract. - We have made a detailed theoretical study of the energies of formation and migration of simple and complex defects in the sixteen alkali halides with the rock-salt structure. This investi- gation uses two sets of interatomic potentials based on a simple shell model. There are a number of important consequences for understanding transport phenomena in these substances :

a) The activation energies for cation and anion vacancies are very similar so their relative contribution to conductivity is temperature independent ;

b) trivacancies will not contribute to the conductivity but a significant contribution to diffusion from vacancy pairs is confirmed ;

c) the Arrhenius energy for transport by the interstitially collinear mechanism is not much greater than that for vacancies and this contribution may explain the high temperature curvature in plots of log (oT) vs. T-1.

As a particularly demanding test of our new potentials, we have also studied a range of systems where a univalent substitutional ion may occupy an off-centre site. There is encouraging, but not wholly satisfactory agreement with experiments.

1. Introduction. - The energies of formation, migration and association of points defects in ionic crystals have been investigated experimentally by a variety of techniques for about 45 years. Nonetheless, there remain a number of outstanding problems even for the simplest alkali halides ; there are still relatively few defect energies which may be said to be known with high accuracy. (For recent reviews of the subject see Fuller [I] and Corish and Jacobs [2].) There have in addition been various attempts t o calculate values for defect energies, although earlier calculations [3, 41 used models which are in varying degrees unsatisfactory. There have been recent advances in the applica-

(1) Department of Theoretical Chemistry, Oxford, U. K.

( 2 ) Department of Chemistry, University College, Dublin,

Ireland.

(3) Department of Chemistry, University of Western Ontario, London, Ontario, Canada.

tion of improved lattice models and in numerical techniques. Thus we report here an extended series of calculations of the defect energies which govern diffusion and ionic conductivity in those alkali halides with the rock-salt structure. This should make a consi- derable contribution to the interpretation of experimental data on ionic transport in these materials. And because our calculation can provide detailed informa- tion o n the positions of ions surrounding a defect, we also present results for a series of univalently substituted systems that may show interesting off- centre effects.

2. Models and method of calculation. - The minimum requirement for a successful calculation of defect energies is an adequate potential model, compatible with the physical properties of the lattice. The shell model [5] used in this work has already been applied successfully in similar calculations o n various mate-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1976761

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C7-254 C. R. A. CATLOW, J. CORISH, K. M. DILI ,ER, P. W. M. JACOBS AND M. J. NORGETT

rials 16, 71. It is superior to the previously used polari- sable-point-ion model as it can simulate correctly not only equilibrium properties (correct lattice constant and cohesive energy) but both the elastic and dielectric properties of an ionic material. The latter is particularly important as the effective charge of the defect provides the major perturbation and the conse- quent relaxation is due mainly to the dielectric response of the crystal.

We have recently derived two improved sets of pair- potentials for the alkali halides [8] based on the shell model. Although we have neglected the small many- body effects, which we know are unimportant in defect calculations [7], our potentials still reproduce a much wider range of crystal data than earlier potentials.

In particular, a more realistic second-neighbour interaction combines electron-gas [9] estimates of the repulsive potential with increased van der Waals forces deduced from crystal data. The first set of potentials have second-neighbour interactions specific to each material. Subsequently, we found a more flexible form of interaction would allow a coherent description of the anion-anion and cation-cation interactions consistent with data on the complete halide series.

The two sets of potentials have substantial differences but give very similar defect energies. We used the earlier and simpler set in most calculations reported here. Crystal data give details of the potentials only in a particular (equilibrium) configuration. Calcula- tions of energy changes as ions relax about a defect constitute a more stringent test of any proposed lattice model.

The calculation of the relaxation about the defect and the defect energy has been fully described elsewhere [3,7,10]. We have used the standard HADES package generally applicable to defects in ionic crystals. Following methods first suggested by Mott and Littleton [I 11, the lattice is divided into two regions. In region I, which immediately surrounds the defect, ions interacting by a pair-potential are relaxed to zero-force using an adaptation of the Newton- Raphson procedure. The inner region contains typi- cally 100 ions ; it is surrounded by an outer region I1 which is treated as a dielectric continuum. Defect energies are essentially unchanged when the inner region is further expanded if the dielectric properties of the lattice model match the continuum properties.

Results are given here only for the three most thorou- ghly investigated alkali halides : NaCl, KC1 and KBr.

Data for the complete set of alkali halides studied will be published elsewhere.

3. Calculations for intrinsic defects. - The calculated energies of formation of Schottky defects are given in table I ; they are in good agreement with the experimental values. Tables I1 and 111 contain activation energies for both cation and anion vacancy motion with the appropriate experimental results. We have used a symmetric saddle point after confirming that

Schottky defect formation energies (eV)

Calculated value

Substance Potential 1 Potential 2 Experimental values

- - - -

NaCl 2.32 2.54 2.30 [12],2.4-2.5 [13],

2.20 [14], 2.75 [15], 2.44 1161

KC1 2.50 2.56 2.49 1171, 2.44 [IS],

2.59 [19], 2.30 [20], 2.42 1151, 2.54 [I61 KBr 2.28 2.45 2.39 [18], 2.37 [21], 2.53 [22], 2.33 [23]

Cation vacancy activation energies (eV)

Calculated value

Substance Potential 1 Potential 2 Experimental values

- - -

NaCl 0.68 0.66 0.75 [12], 0.65 [13], 0.80 [14], 0.71 [15], 0.69 [16]

KC1 0.69 0.71 0.76 [17], 0.74 [IS],

0.73 [19], 0.67 [20], 0.75 1151, 0.73 [16]

KBr 0.62 0.67 0.73 [18], 0.67 [21],

0.65 [22], 0.66 [23],

Anion vacancy activation energies (eV)

Calculated value

Substance Potential 1 Potential 2 Experimental values

- - - -

NaCl 0.72 0.71 1.12 [14], 1.59-1.62 [15], 0.77 [16],

KC1 0.69 0.69 0.89 [17], 1.11 [IS],

0.99 [19], 1.30 [20], 0.85 [16], 0.52 [24], 0.6 [25]

KBr 0.63 0.66 1.20 [IS], 0.92 [21],

1.22 [22], 1.08 [23]

the most direct path for the ion jumping into a vacancy has the lowest activation energy. The calculated and experimental results for the cation vacancy are clearly in good agreement. The experimental anion vacancy results show a wide variation which we consider in our discussion. The calculated anion activation energies are very close to the cation results.

Interstitial migration, by a variety of possible mechanisms, may also contribute to ion transport in alkali halides. In a direct interstitial jump, the migrating ion passes through the face-centred position directly to a neighbouring cell. In the alternative interstitialcy mechanism, the migrating ion displaces a lattice ion of the same type. The displaced ion is removed to an interstitial position in one of the adjacent corner- sharing cells. This replacement may take place in a

<

11 1

>

collinear direction or in a non-collinear

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DEFECT PROPERTIES OF ALKALI HALIDES C7-255

Frenkel energies and activation energies for the collinear interstitialcy mechanism (eV)

Frenkel energies Activation energies

Substance Cation Anion (potential 1)

Potential 1 Potential 2 Potential 1 Potential 2 Cation Anion

- - - - - - -

NaCl 3.21 3.50 3.85 4.33 0.29 0.16

KC1 3.24 3.61 3.41 3.71 0.38 0.28

KBr 2.75 3.40 3.11 3.58 0.37 0.24

direction. Our calculations show that in every case the interstitialcy migration has a lower activation energy than the direct process and, of the possible interstitialcy mechanisms, the collinear displacement is preferred. The activation energies for this process are in table IV with the appropriate Frenkel defect formation energies.

However, the contribution of a particular mechanism to ion transport depends on the appropriate defect concentration as well as the activation barrier. The basic equilibria governing the concentrations xv+, xv- of vacancies and x,+, xl- of interstitials are

where Gs is the Schottky defect and GF+ the appropriate Frenkel defect formation free energy. In an intrinsic material, with dominant Schottky disorder, then

so that the appropriate Arrhenius energies are

for the vacancies and EF+ - $Es

+

AE,+ for the interstitials. Es and EF+ are the calculated Schottky and Frenkel energies and AEv* and AEI& the appropriate activation energies.

These Arrhenius energies for the movement of anion and cation vacancies and of anion and cation interstitials by the interstitialcy collinear mechanism are shown in table V. The Arrhenius energies for the

Arrhenius energies for the migration of intrinsic defects (eV)

Arrhenius energy (potential 1) Vacancy Migration Interstitial Migration Substance Cation Anion Cation Anion

- - - -

cation and anion vacancy movement are almost equal while the energies for interstitialcy collinear migration are larger by 0.2 to 0.5 eV. The consequences of these results in the interpretation of ionic conductivity and diffusion data will be discussed later.

Since the presence of vacancy pairs and trivacancies has been invoked to explain the transport properties of alkali halides, we have made the necessary calculations of their formation and activation energy to determine the probable role of these aggregates.

If G, and GT are the Gibbs free energies of formation of the vacancy pair and trivacancy from isolated vacancies, then the concentrations x, of pairs and xT of triplets, relative to the intrinsic concentration xv of vacancies, are

xP/xV = ZP exp(- { G,

+

$ Gs } / k ~ ) (3.5) and

xT/xv =

zT

exp(- { GT

+

^{Gs }/kT)} ^(3.6)

where Z,, Z, are the symmetry factors for pairs and triplets respectively.

Neglecting the small vibrational and configurational entropies, and equating enthalpies and energies at low pressure, then the approximate conditions for the existence of appreciable concentrations of pairs and triplets are

The binding energies E: and E: are conventionally positive for stable defect aggregates and have the opposite sign to the energies of formation of the aggregates from isolated vacancies.

Our calculations for trivacancies confirm that the expected linear configuration has the lower energy.

Vacancy pair formation and migration energies (eV) Activation

energies Formation Binding Anion Cation Substance energy energy Eschottky jump jump

NaCl 1.84 1.88 2.34 2.85 ^- ^- ^- ^- ^- ^-

KC1 1.94 1.94 2.37 2.44 ^NaClKC1 ^1.421.53 ^0.900.97 ^1.161.25 ^0.890.80 ^0.900.79

KBr 1.76 1.77 1.98 2.21 KBr 1.42 0.86 1.14 0.71 0.71

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C7-256 C. R. A. CATLOW, J. CORISH, K. M. DILLER, P. W. M. JACOBS AND M. J. NORGETT

However, the calculated binding energies are always alternative potential interpolated between the Lif -Li+

much less than Es ; there will thus be an insufficient and MC-M+ Buckingham potentials concentration of trivacancies to contribute to trans-

port processes. Our results for the vacancy-pair (Vii(r) = Aii exp(- r/pii) - cii/r6) ; formation and activation energies are in table V1 ; we have used the well-known geometric-mean rule the Arrhenius energies are compared with the contri- for both repulsive and attractive parts of this inter- butions estimated from anion and cation diffusion action.

studies in table VII.

TABLE VII

5. Discussion. - 5.1 INTRINSIC ^DEFECTS. ^-The satisfactory agreement of the calculated Schottky energies and the most recent experimental estimates Calculated and experimental vacancy pair

able

I) is a good indication o h h e reliability of the Arrhenius energies (eV) potentials. Trial calculations with alternative potentials have emphasized the sensitivity of Schottky

Substance Calculated Experiment defect energies to the form of crystal potential.

- - -

The cation vacancy activation energies are also in

NaCl 2.32 anion diffusion 2.37 [26], 2.54 [16]

cation 2,35 [271, [161 good accord with measured values. However, the KCI 2.33 anion diffusion 2.62 [171,2.39 [16] experimental estimates of anion vacancy activation

cation diffusion 2.65 [16]. energies depend sensitively on the manner of their

KBr 2.13 anion diffusion 2.60 [21] derivation.

4. Calculation for extrinsic defects. - We have investigated the off-centre behaviour of Lif substituted in the twelve salts with other cations but present here in table VIII only the results for KC1 and KBr host crystals. We have also studied the F- substitutional impurity in a number of alkali halides and report results for F- in NaBr and KI in table IX.

Calculations for a foreign-ion, say Li+ in MX, required only the single additional interaction Li -M that is not found in the potentials that describe the set of sixteen halides. In this case, we have used potentials calculated using the electron-gas method [9].

We have for comparison made calculations with an

Thus early experimental data on the conductivity of alkali halide crystals were analysed by assuming that the mobile species were cation vacancies, present as extrinsic defects at low temperatures because of divalent cation doping and generated as the intrinsic Schottky disorder of the crystal at high temperatures.

Problems with this simple model arise because of curvature of the plots of log (aT) against T-' (where o is the specific conductance) at both low and high temperatures. The association of M2+ ions with cation vacancies [28] explains the curvature in the extrinsic region but the intrinsic region has not so far been satisfactorily analysed. Allnatt and Jacobs [29]

suggested that the curvature was due to the onset of conduction by anion vacancies. Although initial

TABLE VIII

Defect energy (eV) and displacements (units of anion cation separation) for Lif/KC1 and Li+/KBr systems Energy for

symmetric

<

^{1 11}

>

Relaxation

<

¹⁰⁰

>

Relaxation System Potential configuration Energy Displacement Energy Displacement

- - - - - - -

LiC/KCI El gas - 0.98 - 0.95 0.12 - 0.99 0.10

G. M. - 0.98 - 1.05 0.13

Li+/KBr El gas - 0.90 - a.91 0.08 - 0.90 0.07

G. M. - 0.89 - 0.86 0.0 1

Defect energy (eV) and displacements (units of anion cation separation) for the NaBr/F- and KI/F- systems Energy for

symmetric

<

¹¹¹

>

Displacement

<

¹¹⁰

>

Displacement

System configuration Energy Displacement Energy Displacement

- - - - - -

NaBr/F-

-

1.10 - 1.05 0.01 - 1.10 0.09

KI/F- - 1.39 - 1.21 0.03 - 1.28 0.02

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DEFECT PROPERTIES OF ALKALI HALIDES C7-257 computer analyses based on this premise gave anion

vacancy activation energies in KC1 close to those deduced from diffusion studies, subsequent more detailed investigations of conductivity consistently yielded unrealistically large values for the anion vacancy mobility [20, 231. Various proposals to account for the shape of the conductivity curve include : trivacancies 1301 which our calculations show are unimportant ; Frenkel defects on the cation or both sublattices 120, 311, or ionic transport along dislocations [23]. These analyses have generally yielded activation energies for anion vacancy migration in NaCl, KC1 and KBr that are higher than the cation value by at least 0.3 eV ; indeed the diffe- rence is much larger if the model is limited to vacancy conduction alone.

Our calculations do not support such a conclusion ; we find that the activation energies for anion and cation vacancy migration are very similar, as of course are the Arrhenius energies (Table V).

Because of the near equality in Arrhenius energies (AE = 0.04-0.05 eV for NaCl and only 0.01-0.02 eV for KC1 and ~ b r ) the ratio of cation and anion con- tributions to conductivity will be temperature independent. These two-mechanisms alone would give a nearly linear conductivity curve which could scarcely be resolved in the most thorough non-linear least squares analysis. This result is supported by results of measurements where anion motion has been studied by tracer diffusion [16, 17, 271. Moreover, conductivity measurements in which the anion contribution is enhanced by doping with divalent ions suggests more comparable activation energies for anion and cation vacancies [19]. Finally, studies of the formation of colour centre aggregates by anion vacancy migration [24, 251 all indicate lower values of the anion vacancy activation energy in reasonable accord with our calculated results,

As an alternative explanation of the curvature of the Arrhenius plot, we may expect a contribution to conductivity from the migration of cation or anion interstitials or from both of these defects. The concentration of Frenkel defects is small but they have a very high mobility by the collinear interstitialcy mechanism. The Arrhenius energies for conduction by this step are compared with those for a vacancy conduction in table V and the differences would appear sufficient to account for the observed curvature of the conductivity plot.

Equation (3.7) is the condition for the existence of appreciable numbers of vacancy pairs. The binding energies in table VI are only 0.2-0.3 eV less than half the Schottky energy ; vacancy pairs will thus contribute significantly to diffusion. The activation energies for the anion and cation jumps of the pair are very similar, as they are for the simple vacancies. In general, the vacancy pair contribution in both anion and cation diffusion will depend on the activation energy for the slower jump. This calculated Arrhenius energy

is compared in table VII with activation energies deduced from cation 127, 161 and anion diffusion [26, 16, 17, 211. The experimental results are in good general agreement with our calculations.

In summary, the results of our calculations of intrinsic energies confirm that vacancy migration is the predominant transport mechanism in alkali halides. However, the relative contribution from anion and cation vacancies in the intrinsic region is temperature independent. The high mobility of interstitials allows transport by these defects, with an Arrhe- nius energy generally only a few tenths of an electron volt greater than for the vacancy mechanism. This is clearly a likely explanation for the high-temperature curvature of the Arrhenius plots. It does not preclude additional complication from a possible temperature dependence in the defect energies. We have also confirmed that vacancy pairs make an important contribution to diffusion but trivacancies will play no part in conduction.

5.2 UNIVALENT SUBSTITUTIONAL IMPURITIES. - A small monovalent ion substituted for a larger ion may adopt an off-centre configuration and give rise to paraelectric phenomena (see the review by Bridges [32]).

The prediction of such behaviour is a severe test of a lattice model. Earlier studies [33, 341 have been somewhat unsatisfactory in that the models have been specifically adjusted to avoid problems with the calculation, particularly those associated with polarisation catastrophies. Our use of the shell model avoids this difficulty and our potentials were derived directly from crystal data and applied without arbitrary modification. We have studied the relaxation of the lattice about the substitutional ion when the lattice is constrained to a symmetric configuration and when the foreign ion is displaced along

<

11 1

>, <

110

>

or

<

100

>

directions.

The systems, Li/KCl and Li/KBr, have energy minima for both

<

11 1

>

and

<

110

>

off-centre displacements of the foreign ion but these minima have almost the same energy as the symmetric on- centre configurations. Both systems were studied with two potentials differing only slightly in the details of the Li+-Kf interaction. These two calculations predict different absolute minima. These potentials are not yet refined to the point where accurate predic- tions of true absolute minima may be made. However, we can identify those systems with off-centre minima that may show paraelectric effects. Li/KCl certainly has an off centre

<

11 1

>

configuration although the experimental results for Li/KBr are somewhat contradictory [32].

For substitutional F - (Table IX) the displacements are small except for NaBrlF- which shows a more substantial displacement with a

<

¹¹⁰

>

off-centre configuration. The energy of this state is equal to that of the symmetric defect configuration but the displacement explains the observed properties of this

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C7-258 C. R. A. CATLOW, J. CORISH, K. M. DILLER, P. W. M. JACOBS AND M. J. NORGETT system [35]. Gongora and Luty [36] and Wahl and

Luty [37] have recently found a

<

¹¹⁰

>

off-centre displacement in KI/F and RbI/F. Our results for KI/F identify an energy above that for the symmetric configuration. Thus, as with the Li* substitutional defects, it seems that the models are sufficiently accurate to locate off-centre minima but they are not able to find an absolute minimum when various configurations have very similar energies.

Acknowledgment. - J. Corish thanks the Chemical Physics Centre of the University of Western Ontario for the award of a Visiting Fellowship and P. W. M. Ja- cobs thanks the North Atlantic Treaty Organization for a Grant, both of which facilitated the progress of this research. The authors are grateful to their respective computing centres for their co-operation.

We thank Brenda Parker for help with some of the calculations.

References

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H. and Slifkin, L. M. (Plenum Press, New York) 1972, chap. 2.

CORISH, J. and JACOBS, P. W. M., <(Surface and Defect Properties of Solids ^D,Vol. 2, Chap. 7, The Chemical Society, London, 1973.

BOSWARVA, I. M. and LIDIARD, A. B., Phil. Mag. 16 (1967) 805.

BOSWARVA, I. M. and SIMPSON, J. H., Can. J. Phys. 51 (1973) 1923.

DICK, B. G. and OVERHAUSER, A. W., Phys. Rev. 112 (1958) 90.

FAUX, I. D. and LIDIARD, A. B., Z. Naturf. 26A (1971) 62.

CATLOW, C. R. A., FAUX, I. D. and NORGETT, M. J., J. Phys. C 9 (1976) 419.

CATLOW, C. R. A., DILLER, K. M. and NORGETT, M. J., AERE Harwell Report TP 672.

WEDEPOHL, P. T., Proc. Phys. Soc. 92 (1967) 79.

LIDIARD, A. B. and NORGETT, M. J., cc Computational Solid State Physics ⁾⁾ed. Herman, F., Dalton, N. W.

and Koehler, T. R. (Plenum Press, New York) 1972, p. 385.

[ I l l Mom, N. F. and LITTLETON, M. J., Trans. Farad. Soc.

34 (1938) 485.

[12] LAREDO, E. and DARTYGE, E., J. Chem. Phys. 53 (1970) 2214.

[I31 ALLNATT, A. R., PANTELIS, P. and SIME, S. J., J. Phys. C 4 (1971) 1778.

[14] PERSHITS Ya. N. and VEISMAN, V. L., Fiz. Tverd. Tela. 12 (1970) 317.

1151 NADLER, C. and ROSSEL, J., Phys. Status Solidi (a) 18 (1973) 71 1.

[16] BENI~RE, M., CHEMLA, M. and BBNI~RE, F., J. Phys. &

Chem. Solids 37 (1976) 525.

[17] FULLER, R. G., MARQUARDT, C. L., REILLY, M. H. and WELLS, J. C., Phys. Rev. 176 (1968) 1036.

[18] PERSHITS, Ya. N. and PAVLOV, E. V., Fiz. Tverd. Tela. 10 (1968) 1418.

[19] CHANDRA, S. and ROLFE, J., Can. J. Phys. 48 (1970) 412.

[20] JACOBS, P. W. M. and PANTELIS, P., Phys. Rev. B 4 (1971) 3757.

[21] BARR, L. W. and DAWSON, D. K., Report R-6234, AERE Harwell, Oxon, UK (1969).

[22] CHANDRA, S. and ROLFE, J., Can. J. Phys. 49 (1971) 2098.

[23] BROWN, N. and JACOBS, P. W. M., J. Physique Colloq. 34 (1973) C9-437.

1241 MATSUYAMA, T. and HIRAI, M., J. Phys. Soc. Japan 27 (1 969) 1526.

[25] DAWSON, R. K. and POOLEY, D., Solid State Commun. 7 (1969) 1001.

[26] BARR, L. W., MORRISON, J. A. and SCHROEDER, P. A., J . Appl. Phys. 36 (1965) 624.

[27] NELSON, V. C. and FRIAUF, R. J., J. Phys. & Chem. Solids 31 (1970) 825.

[28] TELTOW, J., Ann. Phys. (Leipzig) 5 (1949) 71.

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DISCUSSION L. SLIFKIN. - 1) IS it not possible that double

jumps of the vacancy can contribute to the curvature of the Arrhenius plot at high temperatures. Such effects can occur in metals ; perhaps they also are found in alkali halides.

2) It is surprising that the migration energies of vacancies vary so little among the various fcc alkali halides.

CATLOW/JACOBS, et a[. - 1) This seems unlikely in the alkali halides although we have not done any calculations to test it.

2) We don't think this too surprising because of the major contribution to the activation energy from the coulomb term which is roughly constant. The experimental data seem to support i.e. his feature of our results.

R. J. F R I A ~ . - It is possible that the lattice expansion at high temperature loosens the lattice and there- fore lowers the formation, and perhaps also the migration energies. This might contribute to curvature of the Arrhenius plot at high temperatures. Have you considered this effect ?

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DEFECT PROPERTIES OF ALKALI HALIDES C7-259 JACOBS et al. - Yes, we have begun calculations that the linear path was the one that required less on the effect of lattice expansion on the defect energies, energy. No, the calculation is a quasi-harmonic one using the quasi-harmonic approximation. Some pre- without vibrational effects.

l~minary results are given in the table below.

A. LAFORGUE. - Can you clarify the following Effect of Lattice Strain on Schottky Energy points about the calculations : is it proposed that the

jumping ion follows a straight line ; are the vibra- Lattice at Lattice subject to

tional energies taken into account ? Substance OK 1 %expansion

- - -.

P. W. M. JACOBS. - Yes, we used the symmetrical NaCl 2.54 eV 2.47 eV saddle point, subsidiary calculations having shown KC1 2.56 eV 2.52 eV