PLENARY SESSION.Computer Modelling of Ionic Crystals

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PLENARY SESSION.Computer Modelling of Ionic Crystals

C. Catlow

To cite this version:

C. Catlow. PLENARY SESSION.Computer Modelling of Ionic Crystals. Journal de Physique Collo- ques, 1980, 41 (C6), pp.C6-53-C6-60. �10.1051/jphyscol:1980615�. �jpa-00220053�

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PLENARY SESSION.

Computer Modelling of Ionic Crystals

C R. A. Catlow

Department of Chemistry, University College London, 20 Gordon Street, London WClH OAJ, England

RBsumC. - Nous resumons l'ttat actuel des calculs atomiques des dkfauts dans les cristaux ioniques. La methodo- logie de base des calculs est dkcrite. Une attention particulikre est donnee aux facteurs qui determinent la fiabilitt:

des calculs ; dans ce contexte, nous presentons une breve revue des potentiels interatomiques employts B l'heure actuelle dans les calculs atomistiques. Les difftrents types de simulation et ^lesprogrammes actuellement disponibles son[ dCcrits. Nous illustrons l'application des mCthodes par des exemples empruntCs aux domaines de la non- stcechiomktrie, dommage d'irradiation et conduction superionique.

Abstract. - In this paper we summarize the present status of atomistic simulation of defects in ionic crystals. The basic methodology of the calculations is described. Particular attention is given to the factors which determine the reliability of the calculations ; in this context, we present a short review of the interatomic potentials presently employed in atomistic calculations. The different types of simulation, and the available computer codes are briefly reviewed. We then illustrate the application of the methods, with examples taken from the fields of non-stoichiometry, radiation damage and superionic conduction.

1. Introduction. - In the last ten years, the use of atomistic computer simulations has become a major feature of research into the properties of ionic materials. The methods have been applied to a diverse range of problems - a point illustrated by a number of papers presented in this conference. Two areas of application have, however, dominated : first, computer modelling of transport properties - d 8 u - sion, conductivity (including superionic phenomena) and gas migration in ionic crystals; secondly, the study of defect aggregation in grossly defective phases (e.g. U 0 2 + , and Fe, -,O), or solid solutions e.g.

YF,/CaF2. The previous ionic crystals conference reported a number of such studies. Since then, however, the methods have been extended to several other fields including radiation damage phenomena [I, 21, surface studies [3] and the modelling of dislocation core structures [4].

This article aims to give an account of the scope and limitations of this expanding field. We will describe the basic methodology of atomistic simulations, paying particular attention to the factors neces- sary for a reliable calculation. The different simulation codes will be reviewed; and we will emphasize the diversity of the applications with examples reported recently in the literature.

2. Lattice simulation techniques. - All the computational methods discussed in this review have in common the use of the Born model to simulate the ionic solid. In addition the calculations all refer to static lattices, i.e. they include no representation of vibrational properties. In section 3, however, we will

illustrate the relationship of the results obtained from static simulations to those produced by dynamic simulation codes.

Static simulations are, in general, concerned with predicting structures and calculating the energetics of perfect and defective lattices. The prediction of structures is achieved by adjusting the configurations of all the ions concerned until they are at equilibrium -a procedure demanding high efficiency in the nume- rical minimization methods. The calculations of energetics use classical summation techniques. Effi- ciency in the computational procedures is again essential. A special feature of ionic crystal calculations is the treatment of the Coulomb summation using the Ewald method [5] ; these replace the slowly converging real-space summations of the long-range interactions by rapidly convergent sums in reciprocal space.

An explicit atomistic simulation of an entire crystal is clearly impossible. Thus a general feature of all atomistic calculations (including those outside the range of our present study, e.g. simulations of metals and molecular crystals) is the use of a procedure which we will describe as the two-region strategy. In these calculations, explicit simulation of the lattice is confined to a region immediately surrounding the defect where the forces exerted by the defect are strong ; the structure and energy of this region are determined exactly using specified pair potentials. In contrast, the response of the remainder of the crystal is treated by approximate methods which use information on the macroscopic response functions, elastic or dielectric, of the crystal. The latter methods are acceptable

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980615

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C6-54 C. R . A. CATLOW

for the outer regions remote from the defect, where the defect forces are weak.

The two-region method plays a central role in all defect calculations. We therefore devote the next section to a more detailed discussion of its methodology.

2.1 DEFECT SIMULATIONS USING << TWO-REGION )>

METHODS. -Given the division of the defective crystal described above, we may formally write down the following expression for the total energy, E, of the system :

E = E , ( x )

+

E 2 ( x , a )

+

E3(a) !I>

where x are the coordinates of all ions within the inner region I, surrounding the defect, and a are the coor- dinates of the ions in region I1 which extends to infinity. El is therefore the energy arising purely from interactions in region I ; E,, entirely from inter- actions within region 11, and E, is the energy of inter- action between the two regions. Defect calculations thus involve the determination of the equilibrium values of x and a of the terms E l , E, and E3.

The treatment of the inner region presents, a t least in principle, no difficulties. The term El is evaluat- ed exactly using the specified iriteratomic potentials ; while determination of the equilibrium value of x simply requires adjusting the coordinate of all ions until the forces acting on them are zero (I). However, the calculation of El and x, although conceptualy simple, may involve major computational effort.

In particular, the determination of the coordinates, x,

. needs sophisticated, efficient minimization proce- dures [8], which have played a vital role in the develop- ment of the simulation methods.

In contrast to the exact treatment of region I, the determination of a, E, and E3 uses approximate methods. First, as discussed above, the values of the coordinates, a, are generally determined using simple theories employing macroscopic response functions.

The most important example is the Mott-Littleton [9]

approximation, which calculates the displacement of these coordinates in region I1 by assuming that they are purely a dielectric response to the effective charge of the defect and that the response can be kalculated using the macroscopic dielectric constant. The resulting expression for the displacements is deduced from the expression for the polarization P, at a distance r from the defect :

where q is the defect charge, and E is the static dielectric constant of the crystal.

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of the inner region is not exactly equivalent, when the contribution of E, is included, to minimization of the total energy with respect t o the coordinate x - a point discussed by Norgett 16, 71. The errors are, however, minimal, and force balance procedures are adopted in most computer codes, owing to their greater simplicity.

Analogous theories have been developed for the displacements in the outer region assuming that these are due to the elastic forces exerted by the defect;

in this case the response is written in terms of the elastic constants of the solid [lo].

Having determined, a, further simplifications are then made in the evaluation of E2 and E,. First, the harmonic approximation is assumed for the outer region - an approximation consistent with Mott- Littleton and analogous methods for calculating the displacements in this region. It can be shown that, given that the outer region is in equilibrium, we can write

This procedure simplifies the expression for the total energy by removing any explicit dependence on E,.

It now remains to evaluate E2 and (dE,/da) for given values of x and for values of a calculated by e.g. the Mott-Littleton method. The general procedure is to take an inner portion of region I1 (region 11'), for which the terms are evaluated by explicit surnma- tion of the interactions with all ions within the inner region. For the remainder of region 11, E, and its derivative can be evaluated assuming that the perturbation of the outer by the inner region can be represented by a single point, e.g. the total effective charge of the defect, acting at the centre of region I. Analy- tical expressions can then be used for the two terms.

In summary, the steps involved in defect calculations are first the determination of the equilibrium configuration of the inner region generally by a force-balance minimization procedure ; second the evaluation of the displacements in the outer region by a Mott-Littleton or analogous approximation.

E, is calculated explicitly ; the harmonic approxirna- tion is used to remove explicit dependence of the total energy on E,; E, and (dE,/da) are then cal- culated using the approximations outlined above.

The uncertainties introduced into the calculated defect energies by these methods will be considered in the next section where we give a general account of the factors influencing the accuracy of defect simulations.

2.2 THE RELIABILITY OF SIMULATION TECHNIQUES. -

2.2.1 Errors arising Jiom the two-region method. -

The obvious question here concerns the sensitivity of the calculated energies to the size of region I. This problem has been extensively investigated for the case of point defects in cubic crystals [ l l , 12, 131.

The results suggest that with a Mott-Littleton treatment of region 11, we may achieve convergence (i.e;

insensitivity of the calculated defect energies to further expansion of region I) with three or four shells of ions surrounding a point defect - that is a region containing typically 50-60 ions. And we may be confident of the adequacy of calculations of point

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defects and small clusters with inner regions containing 100 ions. We should note, however, that once the calculated defect energy has converged in the sense defined above, small irregular oscillations, typically of 0.01 -0.05 eV, occur as the region is further expanded.

This may lead to difficulties if we wish to investigate problems where changes in energy of

-

0.01 eV are important. A good example is provided by the study of off-centre displacements of substitutional cation impurities in alkali halide crystals - a long standing field of study in defect physics. In a detailed theoretical survey of this problem [14] we found that the energy change accompanying e.g. a ( 11 1 ) displacement of a Li+ ion from the regular lattice site to its equilibrium position is

-

0.05 eV. If significant results are to be obtained in calculations on such problems, identical inner regions must be used in all the calculations involved. This will ensure that any errors arising from the two-region approximation will be self-cancelling. The same procedure may be used in all cases where relative energies of different defects or defect configurations are required.

Where absolute energies are needed. the uncertaintv u

of

-

0.05 eV must remain from the two-region approximation, for any calculations on a manageable size of region I.

The limited number of studies that have been performed on non-cubic materials suggest that convergence may give rise to greater difficulties in these systems. Detailed studies of the defect energies as a function of the size of region I for defects in A1203 [15, 161 found that the calculations did not converge adequately until -- 150-200 ions were present in the inner region. The difficulty is almost certainly due to the use of spherical inner regions which are appropriate for the isotropic cubic systems but not for non- cubic materials. The problem will require further attention if calculations on non-cubic systems are to achieve the same accuracy as those for cubic crystals.

The above discussion of the convergence properties of defect calculations also demonstrates that the Mott-Littleton approximation, which was used in calculating the displacements in region 11, is in general adequate. Significant errors in the approximation would lead to a continued systematic variation in total calculated defect energies with expansion of region I. The omission of the elastic forces exerted by the defect on region I1 appears therefore to be unimportant for defects in ionic crystals. This point was confirmed by a study of Catlow, Faux and Nor- gett [17] in which the elastic term was incorporated following the procedure discussed by Hardy and Lidiard [lo] and Faux and Lidiard [18]. The study showed that the inclusion of this term had a negli- gible effect on the calculated energies.

The two region method appears therefore to pro- vide a completely adequate method of simulating defects provided care is taken in specifying the inner regions. The reliability of the calculations then beco-

mes entirely dependent on the lattice potentials. These will be discussed in the next section.

2.3 LATTICE AND DEFECT POTENTIALS. - The potential models used in the defect simulations reported to date, have the following features in common.

(i) The crystal has been described by the fully ionic model. Such models are clearly acceptable for halides and oxides of the alkali and alkaline earth metals. They are, however, more questionable when applied to transition metal or actinide compounds.

Successful simulation studies have, however, been reported for e.g. FeO [19,20], ZnO [21] and UO, [12].

(ii) The short range interactions between ions are represented by two-body, central force models, for which we use simple analytical functions (commonly of the Born-Mayer and Buckingham form). The errors caused by the approximati&s inherent in these potentials are again of minor importance in strongly ionic crystals, e.g. the alkali halides ; they will become more significant when the mzthods are applied to

systems with a greater tlegree of covalence.

(iii) Ionic polarization must be included in a reliable simulation. Without a satisfactory representation of this effect it is not possible to describe correctly the response of the crystal to the electrostatic perturbation provided by a charged defect. Polarization can be simulated simply and effectively by the shell model [22], originally developed for treatment of the lattice dynamical properties of ionic crystals. This model describes the ionic dipole in terms of the displacement of a mass-less electron shell from a core, the two being connected by an harmonic spring constant. Despite the crudeness of this simple mechanical model, shell-model potentials allow us to simulate accurately the dielectric properties of ionic crystals.

Indeed the replacement of earlier point polarizable ion (PPI) treatments by shell model potentials (see e.g. Faux and Lidiard [18]) was one of the major advances in the development of reliable simulation techniques ; the incorrect description of the dielectric response of the crystal given by the earlier PPI methods led to erroneous defect energies as well as to instabi- lities in several calculations.

All potentials of the type described above contain variable parameters, both in the analytical representation of the short range potentials and in the shell model treatment of ionic polarization. Careful attention must be given to the methods used in deriving these parameters, as the adequacy of the parameters will determine the reliability of the simulations using the potentials. Two types of procedure have been adopted to date. The first, purely empirical parameterization~ adjust the variable parameters in order that the model reproduces as closely as possible the observed macroscopic crystal data - that is the crystal structure, the cohesive energy and the elastic and dielectric constants.

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C6-56 C. R. A. CATLOW

This empirical procedure undoubtedly gives a good representation of the potential at interionic separations close to those which occur in the perfect crystal. It is, however, more questionable if the potential is to be used for defect configurations (e.g. interstitial~) in which certain interionic spacings are very different from the perfect lattice values. For this reason Mackrodt and Stewart [23] have devoted considerable attention to the development of non- empirical procedures for obtaining short-range potential parameters. Their approach follows the electron gas methods described by Wedepohl[24] and Gordon and Kim [25]. They use simple procedures closely related to earlier Thomas-Fermi approaches, to calculate the interaction between charge densities representing the interacting ions, the densities being obtained from Hartree-Fock calculations on the isolated ions. Mackrodt and Stewart have shown that it is important to use ionic charge densities obtained from calculations in which the ion is present in a Madelung potential appropriate to its crystal envi- ronment.

Both types of parameterization have been extensively used. In many cases [13, 231 the parameters for the two types of potential are very similar. Further- more, there is, in general, an encouraging measure of agreement between defect energies calculated with empirical and non-empirical models, although definite differences do occur in certain cases. At present it seems to be most satisfactory to obtain both types of potential - a procedure which we employed in our recent detailed study of defect energies in A1203 [15, 161.

This completes our account of the basic features of the calculations, and of the factors limiting their reliability. We continue with a review of presently available atomistic simulation codes, before discussing applications of the techniques.

2.4 SIMULATION CODES. - In this, as in other areas of computational physics, there has been an increasing tendency to develop general computer programs capable of performing a whole class of simulations, often with little or no programming required of the user. In this section we will describe the types of atomistic simulations for which such codes have been written. Where a general program is available, we give its name.

A) Perfect lattice simulations (PLUTO) [26]. -

This code calculates unit cell structures and lattice energies of ionic crystals. A trial structure is proposed ; then using the specified interatomic potentials the coordinates of the ions and the unit cell dimensions may be adjusted so as to find the configuration of minimum energy. The lattice energy of this structure is then calculated; in addition, calculations may be performed of the elastic and dielectric constants of the crystal. This class of simulation has two important applications in the field of defect physics. First, the

I

efficiency of presently available codes permits calculations on very large unit cells. This enables us to investigate (( defect super-cells )>. i.e. large super- lattices containing arrays of point or extended defects.

An important application of this type has been reported for the compounds Ti,O, and Ti,Og [15, 271 which contain regularly spaced arrays of planar defects known as shear-planes; reference will be made to these calculations in the following section.

A second important application of the program follows from its ability to calculate structural, elastic and dynamical properties of a crystal. This facility may be exploited in a reverse sense, i.e. the program may be used to determine interatomic potentials using a least squares fitting procedure, from measured structural, elastic and dielectric data. Indeed, the use of the perfect lattice simulation code in the reverse sense has played a major role in the development of empirically parameterized interionic potentials for the more complex low symmetry ionic crystals, e.g.

Al2O3, Ti02.

B) Point defect simulations (HADES) [26]. - TO date this is by far the most widely applied type of ionic crystal simulation. The energy to introduce point defect or defect aggregates into a perfect crystal is calculated. The response of the lattice to the defect is obtained using the two-region strategy with a Mott-Littleton treatment of the outer region. The feasibility of rapid large scale calculations here is due in a large measure to the development of efficient procedures for handling the summation and minimization problems. Here we should emphasize the contributions of Norgett [6, 71 who wrote the initial general HADES package. This program, originally limited to cubic crystals, has been generalized in the latest HADES I11 package [16], to handle crystals of any symmetry. These modifications necessitated the development of methods for treating an anisotro- pic region I1 ; the original Mott-Littleton formalism was confined to crystals with dielectric isotropy.

Calculations using the HADES program have enjoyed considerable success. The methods have proved of great value in obtaining reliable defect formation, interaction and migration energies as will be seen from the paper of Jacobs [28].

Jacobs concentrates on applications to simple materials. However, complex problems and structures can now be' handled ^-a point which is also apparent from the contributions to this conference [29, 301 and from the applications discussed later in this paper.

C) Surface simulations (MIDAS) [31]. ^-TWO types of calculation are involved here. First, simulation of the structure and energies of perfect surfaces.

Summation techniques related to those used in the HADES and PLUTO programs have been developed by Stewart and Mackrodt [23] and by Tasker [31], [3] in order to handle this problem. Stewart and

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Mackrodt [23] have considered the additional problem of surface defects. Their work and that of Tasker also illustrates the applications of the class of simulation techniques.

D) Dislocation simulations. - These calculations calculate the structure and energies of dislocation cores. Interactions between dislocations and point defects may also be calculated. The two region strategy is used, although problems have been encounter- ed in developing fully satisfactory treatments of the outer displacements and of the region Ilregion I1 boundary. For further details and for applications of the methods we refer to Puls et al. [4] and Granzer et al. [32].

A diverse range of computer codes are therefore available in the field of atomistic simulations. The diversity of scientific problems that can be investigated using the techniques will be apparent from the following discussion of applications which concen- trate on recent studies using HADES and PLUTO.

3. Applications. ^-Three successful studies are discussed in this section. The first concerns the defect structure of TiO, - a long-standing problem in the field of non-stoichiometry ; in the second we consider a detailed mechanistic problem arising in the study of the radiolysis of SrF,. Finally we illustrate the application of the methods to a problem in the growing field of superionic conduction.

3.1 SHEAR PLANE STABILITIES IN TiO,. ^-It has been known for several years that the non-stoichiometric phase Ti0,-,, accommodates deviations from the stoichiometric composition, not, as occurs in the majority of non-stoichiometric compounds, by forming point defects or point defect aggregates, but by creating extended planar defects known as shear planes [33]. In these planar faults, sharing between oxygen octahedra is replaced by face-sharing ^- a structural modification which lowers the oxygen to metal ratio. Alternatively, shear-plane formation may be seen as mode of eliminating point defects.

This latter concept is illustrated diagrammatically in figure 1 - a diagram which we should emphasize does not refer to the real shear plane structure in Ti0,-,. We imagine that the vacancies formed on reduction are aligned as illustrated in figure la.

Shearing of the lower half of the crystal as shown in the diagram can superimpose oxygen atoms on the vacancies. The latter defects are therefore eliminated, but at the cost of creating a fault characterized by a change in packing on the cation sub-lattice. This fault, illustrated in figure lh, is a schematic representation of a shear plane.

The observation of shear planes poses two funda- mental problems which were until recently unanswer- ed. The first concerns the factors which determine whether shear-plane or point-defect structures dominate in a non-stoichiometric compound. The observa-

c a t i o n

0 oxygen v a c a n c y

(oxygen i o n s a t c o r n e r s of m e s h )

s h e a r p l a n e ( d i r e c t i o n o c a t i o n r e l a x a t i o n s shown bv a r r o w s )

Fig. 1. - Hypothetical structure containing aligned vacancies shown in figure la. Direction of shear marked. Shear results in superposition of starred oxygen atoms on vacancies giving shear plane shown in figure lb.

tion of shear-planes is in fact restricted to a small group of non-stoichiometric compounds, i.e. Ti0,-,, YOz-, and WO,-,. The majority of non-stoichiometric compounds incorporate compositional variation by forming point-defects or point-defect aggregates. Is it therefore possible to identify some special factor responsible for stabilizing shear planes in the small group of oxides in which they form ? The second problem concerns the relationship between point and extended defect structures. Can point defects co-exist with shear-planes ? And, if so, what are their natures (vacancy or interstitial) and their concentrations ?

Application of the HADES, PLUTO and MIDAS programs has largely solved these problems. A series of calculations [15, 271 of shear plane formation ener- gies have shown that cation relaxation in the vicinity of the shear plane is an essential factor in stabilizing these defects. The results of the computer simulation studies indeed suggest that shear planes will only form in those oxides which permit such extensive relaxations, without which the repulsion between the cations in the plane destabilizes the extended defect.

Strong evidence ;favouring our proposal is provided first by the observation of Tilley [34] that those compounds in which shear planes form have high values of the static dielectric constant, t o ; materials with a high value of E, would be expected to permit large cation relaxations. More direct evidence is obtained from diffraction studies on compounds such as Ti,O, [35] and Ti,O, [36] which contain ordered arrays of shear planes. Examination of the structures of these compound reveals large relaxations (- 0.3-0.4

A)

of the cations neighbouring the planes.

To investigate the question of the relationship between extended and point defect structures we have compared our estimates of shear plane energies with those of point defects. Calculations using HADES I11 find anion vacancies to be the more stable point defects in reduced rutile (although we should note

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C6-51 C. R. A. CATLOW

that certain experimental studies have suggested the alternative cation interstitial model). A comparison of the calculated vacancy energy with that of the shear planes shows that the latter defects were stable by

-

^{2 eV}^(,). Energies of this magnitude, although suggesting that shear planes will dominate the defect structures of the reduced phase, do not preclude significant point defect concentrations. Indeed simple thermodynamic considerations suggest that point defects will exist in equilibrium with shear planes, and that at sufficiently low deviations from stoichiometry and high temperatures the planar defects will dissociate into vacancies. Thus, if the above value of 2 eV for the energy of the shear plane relative to the point defects is used in a thermodynamic analysis of this equilibrium, we predict that at e.g. 1 000 K point defects will dominate in TiO,

-,

at compositions nearer stoichiometry than x = ; for higher values of x, shear planes will be the major defect species. The prediction is in line with recent electrical conductivity studies of Baumard et al. [37] which showed behaviour characteristic of point defects in the near-stoichiometric region, with clear evidence for defect aggregation, presumably into extended defects, for higher values of x at temperatures greater than 1 000 K.

Further details of the work summarized above are available elsewhere [15,27]. The point to be emphasiz- ed here is the role played by the calculations in iden- tifying cation relaxations as the crucial factor in stabilizing shear planes and in elucidating the nature of extended-point defect equilibria. In the next section we will see how the theoretical techniques have proved to be of equal value in investigating a detailed mechanistic problem arising in the study of irradiated alkaline earth fluorides.

3 . 2 PERTURBED F-CENTRES IN IRRADIATED ALKA- LINE EARTH FLUORIDES. - X- and electron irradiation of the alkaline earth fluorides (of which SrF, has been most intensively studied) leads to formation of F and V, centres whose properties have been studied principally by Hayes and coworkers 138, 39, 401 using optical and spin-resonance spectroscopy. An intriguing feature of this work is the observation of perturbed F centres apparently of trigonal symmetry. The model most compatible with the ENDOR data attributes the perturbation to an anion interstitial in the next-nearest neighbour (n.n.n .) interstitial sites with respect to the F centre as illustrated in figure 2.

Why are F centre interstitial complexes with this structure formed ? F centre formation involves displacement of fluorine atoms from their lattice sites.

We have shown [l] that these atoms will capture electrons to become F - interstitial ions. The question remains why the displaced F- ion is detected only

(') The energies are given per eliminated oxygen Ion.

@

Fig. 2. - Tetragonal F centre-interstitial complex.

in the trigonal site with respect to the F centre. We offer a simple explanation : the F centre formation mechanism (whose nature we have discussed elsewhere [l 11) results in F centre-interstitial pairs with a variety of separations. We suggest that those interstitial~ more remote than the site illustrated in figure 2 cannot be detected by the spin-resonance techniques while those complexes in which the interstitial is nearer to the F centre than in the trigonal structure in figure 2 are destroyed by recombination. The instability to recombination is not, we propose, a property of the F centre interstitial complexes themselves; these species will be stable owing to the high energy of displacing an electron from the anion vacancy - a process which must accompany recombination of these pairs. But the ionized complexes, in which the electron has been removed from the vacancy will clearly be thermodynamically unstable to recombination, as the complex is now simply a vacancy-interstitial pair. Ionization of the F centres will, however, take place constantly during irradiation. We propose therefore that all complexes in which the interstitial is nearer to the F-centre than in the trigonal structure are destroyed by recombination of the ionized species; for these complexes, we suggest that there is no kinetic barrier to their recombination even at very low temperatures of the radiation damage experiments. To explain the stability of the trigonal complex we postulate that there exists an appreciable kinetic barrier to recombination of the ionized complex.

The argument successfully explains the observation of the trigonally perturbed F centres. Additional support is, however, needed for an essential compo- nent of the reasoning, namely the assumption of kinetic barriers to the recombination of certain of the vacancy-interstitial pairs. Simulations using the HADES program have proved to be of great value here. A detailed examination was made of the mecha-

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nisms of recombination of a variety of vacancy- interstitial pairs in the three alkaline earth fluorides.

For complexes closer than the trigonal pairs, either no barrier to recombination was found, or the barrier was of the order of the zero point motion energy of the F - ion in these compounds; in this case it will not prevent recombination. For the trigonal pair, barriers of

-

0.4 were calculated ^-sufficient to prevent recombination at low temperatures, and resulting therefore in stability of the ionized traps.

With the aid of further calculations we have been able to develop detailed models of radiation damage processes in the alkaline earth fluorides [I]. We should note that the calculations have also been of considerable value in discussing radiation damage in the alkali halides [2]. The calculations referred to in this section have, provided perhaps the best illustra- tion of this use of the techniques in investigating a detailed mechanistic problem.

3 . 3 SUPERIONIC HIGH TEMPERATURE FLUORITES. ^- Fluorite structured compounds, e.g. BaF,, PbF,, SrCl,, all show a diffuse phase transition manifested by a ktype specific heat anomaly at

-

^200-300^OC

below the melting point. Above this transition temperature the materials are superionic conductors due to exceptionally high anion mobility. A detailed discussion of the properties of superionic fluorites is presented in this conference by Hayes. Here, we wish to stress the role which calculations have played in advancing our understanding of two important features of these compounds : the first concerns the extent of disorder in the superionic phase, and the second the mechanisms of ion migration.

Earlier discussions of the superionic fluorites assumed high levels of disorder in the high temperature structures. The term sub-lattice melting has frequently been used to describe the structural transformation accompanying the phase transition to the superionic state. Since the intrinsic disorder of all fluorite materials is of the anion Frenkel type, such models would imply high concentrations of anion interstitial defects. An immediated objection to this hypothesis follows, however, from the fact that anion Frenkel formation energies in fluorites are high ( - 2-3 eV) when compared with enthalpy change accompanying the diffuse transition (generally

-

0.1 eV) [41]. The answer advanced to this objection is that the interstitial formation energy is reduced by defect interactions ^-an argument which is indeed the basis of all phenomenological theories of the phase transition [42, 431. Using the PLUTO program we may test this hypothesis by calculating the energetics of a structure in which vacancy-interstitial interactions are maximized. Such calculations have shown that defect interactions do indeed reduce the interstitial formation energy; but that the latter still remains at

-

^{1 eV}^-that is an order of magnitude larger than the enthalpies of the diffuse transition.

We conclude that models based on very high levels of interstitial disorder are not acceptable for these materials. The conclusion is supported by additional calculations using the PLUTO code which investigated the variation of elastic constants with interstitial concentration 1441. The results were compared with experimental values for the change of the elastic constant through the diffuse transition, deduced from the Brillouin scattering data of Catlow et al. 1441. The comparison suggests relatively low interstitial concentrations. And the review of Hayes [45] will indeed show that models based on limited disorder for the high temperature phase are becoming increasingly favoured by the experimental data.

Calculations using the HADES code allow us to investigate migration mechanisms - a point which will be clear from the review of Jacobs [28]. For the fluorites, calculations have shown that anion vacancies are highly mobile with activation energies of

-

0.3 eV [l 11. This result clearly explains the superionic behaviour of the high temperature materials, as the diffuse phase transitibn evidently involves gene- ration of appreciable, if not massive, levels of anion disorder. Dynamical properties of superionics can, however, be investigated in more detail using a further simulation technique, namely molecular dynamics -

a method developed for the simulation of liquid systems. Molecular dynamics simulations follow the time evolution of a system of particles by solving the classical equations of motion for the particles using specified pair potentials. The work of Dixon, Richard- son and Gillan [46] on the high temperature fluorites has proved complementary to the static calculations discussed above. Analyses of the results of the simulations supports the models of limited interstitial disorder and suggests that vacancy migration as discussed above plays a major role in effecting ion transport at high temperatures. It appears that interstitial migration could also make an appreciable contribution.

Thus the use of both static and dynamic simulation methods is making a large contribution to our understanding of the complex problems posed by the superionic fluorites. And we-believe that the concerted use of several simulation techniques may play an increasingly important role in solid state theory in future years.

4. Conclusions. - The success enjoyed to date by the simulation techniques should make their use a general and routine procedure in the analysis of experimental studies of the defect properties of ionic crystals.

In view of this success we will conclude by specu- lating whether the methods can be extended to other areas of solid state research, e.g. minerals and semi- conductors. The essential factor in determining the feasibility of such extensions is our ability to develop adequate interatomic potentials for the materials

(9)

C6-60 C. R. A. CATLOW

simulated. Covalent crystals will unquestionably Acknowledgments. ^-I would like to acknowledge require models of greater sophistication than the two the benefit I have received from discussions with body centrai-force potentials discussed in this article. Drs. M. J. Norgett, A. B. Lidiard, P. W. Tasker and The challenge is to develop models which both ade- A. M. Stoneham (AERE, Hamell), J. R. Walker quately simulate the solid and can feasibly be incor- and R. James (UCL), and W. C. Mackrodt and porated into efficient computer codes. The success R. Stewart (ICI Corporate Laboratory).

enjoyed by the simulation of ionic crystals suggests that such studies would be worth while.

DISCUSSION

Question. ^-H. J. STOCKMANN. Reply. - C. R. A. CATLOW.

I would like to point out the fact that exactly the Your observation supports the type of model I defect configuration you showed on your slide on proposed. It would be interesting if you could find SrF, (i.e. F--interstitial

+

vacancy or F-center at a direct evidence on the structures of vacancy-interstitial certain distance) has been observed experimentally pairs.

in the compounds CaF,, SrF,, BaF, (see contribution page C6-381 to this conference). The annealing enthalpy is about 0.2 eV in all three compounds.

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