e(°>) = -2--- — --- (3.36)
e0 £ (“ )
where £. is the macroscopic field, e is made up of a high frequency component, £«,, arising from electronic displacement and a static component, e0, which arises from nuclear and electronic displacements.
3.5.3 P olarisation M odels
The simplest method for including polarisation is that of the point polarisable ion (PPI) model introduced by Lyddane and Herzfield (1938) in an early lattice dynamic study. It assumes that each ion of polarisability a can develop a point dipole moment ja in the presence of an electric field E,
U = ccE (3.37).
The major inadequacy with this model is the neglect of the interdependence of ionic polarisabilities and short-range forces, which leads to an overestimation of polarisation energy. When polarisation occurs there is a distortion in the electronic charge clouds of ions. Since short-
range repulsion arises from charge cloud overlap the two must be related. In essence the short-range forces serve to dampen the polarisation effects.
A more reliable description of polarisation is given by the 'shell model' developed by Dick and Overhauser (1958), which allows short-range interactions between shells. In this simple mechanical description of polarisation each ion has a massless shell of charge y representing valence shell electrons, and a core of mass m representing the nucleus and core electrons, where m is the total ion mass. The two are coupled via a harmonic spring so that the dipole arises from core-shell displacements. Thus, by allowing shell displacements, the effects of polarisation and short-range forces are coupled together.
The polarisability for an ion is given by:
2
« = £ ■ (3-38)
where the spring constant K is determined from the core-shell interaction as a function of separation distance r,
4> (C) = Kr2 (3.39).
This model has proved most successful in modelling dynamical and defect properties of ionic halides and oxides (Catlow and Mackrodt, 1982), but it is not without its limitations. First, two more parameters y and K need to be derived and, as yet, this can only be done by fitting to known experimental data. Secondly, this fitting sometimes produces positive shell charges for cations. This paradox has partially been rationalised by Bilz et al. (1975) in terms of overlap of cation and anion nearest neighbours.
A further development has been the introduction of a 'breathing' shell model to overcome the inadequacies of the central force model, which fails
to reproduce Cauchy violations. Refinements by Schroder (1966) allow for the spherically symmetric distortion of shells and enable violations, C12 < C44, to be predicted for cubic binary oxides. Violations in the reverse sense, as in AgCI, are accounted for by allowing ellipsoidal deformations as introduced by Sangster (1974).
3.5.4 Rigid-ion Model
The rigid-ion model takes no explicit account of electronic polarisation and hence the high frequency dielectric constant, £««, is unity. One polarisation effect, i.e. dispersion can, however, be modelled, if van der Waals coefficients are used in the description of the potential. Nevertheless, rigid-ion potentials are very poor in giving a description of the dynamical properties of a lattice since these are influenced strongly by polarisation.
Sometimes, however, it is necessary to use these potentials as in the case of molecular dynamics. Here, inclusion of polarisability by a shell model would make an already cpu-demanding process more expensive by up to an order of magnitude. It has also been argued that, provided the static dielectric constant eq is correctly reproduced by adjusting the short-range
repulsive parameters (A and p), then the rigid-ion model is satisfactory (Catlow, 1977 and Catlow and Norgett, 1973). Indeed, a pilot MD study of C a F2 using shell model potentials gives similar results to those using optimised rigid-ion potentials (Dixon and Gillan, 1980b).
3.6 Derivation of Potentials
The methods used to calculate the potential parameters for the short- range and shell model interactions can be separated into two categories: firstly, empirical fitting, and, secondly, theoretical methods, i.e. electron gas and ab-initio Hartree-Fock calculations.
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3.6.1 Empirical Methods
A starting set of parameters for A, p, C, K and y are chosen. These are refined using a least squares fitting procedure until the best possible match is found between calculated and experimentally determined properties such as elastic and dielectric constants and lattice energy.
Although this method is often used there are two disadvantages: firstly, the reliance on the availability of experimental data and, secondly, the potentials derived are those for perfect lattice spacings. This is often not the case in defect studies, particularly for interstitial configurations, where inter- nuclear distances could be very much smaller than those in the perfect lattice. In such cases, reliability is then dependent on the accuracy of the analytical potential function over a wide range of separations.
3.6.2 Theoretical Methods
Other methods, such as electron gas and ab initio methods, have been developed to calculate interionic energies for a range of separations. Of the two, electron gas is the more approximate, whilst the ab initio is restricted by its use of computer time.
Electron gas methods calculate interactions between the charge densities of two ions at a series of separations. The total interaction energy is given by:
E = Eeiec + Eke + Eex + Ecorr + E^jsp
(3.40)
where Eejec is the electrostatic energy, Eke is the kinetic energy, Eex the exchange energy, Ecorr the short-range correlation energy in the region of electron density overlap and Edisp is the longer ranging dispersive energy. Edisp cannot be calculated by electron gas methods, but from perturbation theory (Salem,
1960).
The interaction energies are calculated based on methods of Wedepohl (1967) and Gordon and Kim (1972, 1974). The approach treats the electron densities as degenerate Fermi gases. The densities for isolated ions are calculated using closed shell Hartree-Fock wavefunctions and the total density was originally found by summing these energies. However, the work of Mackrodt and Stewart (1977, 1979) on ionic materials, has shown the importance of calculating wavefunctions in the Madelung field appropriate to each crystal i.e. ions are now placed in an effective external field.
The computer code Wedepohl (Harker, 1980) is used in the electron gas method and has subsequently been developed to fit to different potential functions (Harding and Harker, 1982).
The ab initio techniques (not employed in this thesis) use Hartree-
Fock molecular orbital methods on 'super molecules’ rather than isolated ions. Interatomic potentials are found from calculations performed as a function of internuclear separation. The method is described by Mackrodt et ai. (1980) and has been used most recently by Sim (1988) in studying potentials for a-quartz. Electron correlation is not included in the Hartree- Fock approximations, but could be calculated either by empirical fitting or by adapting the ab-initio calculations to use m ulti-configurational wavefunctions derived from configuration interaction, Cl, or multi- configurational self-consistent field, MCSCF, calculations (Szabo and Ostlund, 1984). The limitation of these calculations is the extreme demand on computer time. Should shell parameters also be required, these must be fitted empirically to static and dielectric constants. Therefore, a full parameterisation involves a combination of techniques.
Chapter 4