Method of Calculation

6

.

Interionic Potentials

As in the static simulations, the pair potentials used are of the form:

V jj(r) = ZjZje2/r + Ayexpf-r/py) - Cy,!* (6.1)

where i, j denote ionic species, e is the electronic charge and Zj, z} are the charges on the species in units of |e|.

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Ions were again assigned their full formal charges of +3 and -1 for the long-range Coulombic interaction. Short-range interactions were developed using a rigid-ion model, as explicit inclusion of the effects of electronic polarisability would greatly increase the computer time requirements of the MDS. The short-range parameters were derived by the empirical fitting method. The initial parameters for the F '-F - potential were those used in the static calculations and, for the La3+-F _ interaction, were those derived from an electron gas calculation. The A and p parameters of the La3+-F ' interaction and the C parameter of the F - - F' interaction were subsequently fitted to bulk crystal properties, namely the static dielectric constant (the high frequency constant is necessarily 1), elastic constants, basis strains and the lattice energy. The La3+-La3+ interaction was not fitted and was the same as that used in the static calculations. The final potential parameters are given in Table 6.1 and throughout the MDS study the cut-off for the direct lattice sum was taken at 1.2 lattice units.

Comparisons between experimental bulk properties and those calculated using this rigid-ion model are listed in T a b le 6.2. The components of the static dielectric constant are reproduced to ~3%. In defect modelling studies, it is important to obtain an adequate description of the dielectric response of the lattice to charged defects and such excellent agreement gives confidence in the ability of the potential to represent a real system. Thus, despite using rigid-ion potentials, this feature may allow a partial representation of the effects of electronic polarisation - a point argued successfully in the MDS work on fluorites by Dixon and Gillan (1980b) and Gillan and Dixon (1980).

A further encouraging point comes from the comparison of the elastic constant data. At the time these potentials were derived, the only available experimental data were the values of Laiho et al. (1983) from a study at 300K. The errors in the calculated constants were 30% (C-n), 27% (C 12), 31% (C i3), 41% (C33), 35% (C44), 33% (C66). However, for the later data of

1 9 0

Ngoepe et al. (1986) and Ngoepe (1987), (examples at 1 2 0 0K are given in Table 6.2), the fits were closer with elastic constants being reproduced to 13% (C n ), - (C12), 27% (C13), 27% (C3 3), 18% (C44), 13% (C66). The fit continues to improve when comparisons are drawn with other data from the Ngoepe study at higher temperatures.

6.2.2 Box Size and Geometry

The basic simulation 'box', to which periodic boundary conditions are applied, contains 648 ions and is built from 27 unit cells in a 3a x 3b x 3c arrangement. This is the largest box that can currently be used by the MD program whilst still retaining the hexagonal symmetry of the individual unit cell. Each unit cell contains 6 La3+ and 18F- ions and so, of the 648 ions in the box, 486 are F* ions distributed over the inequivalent F" sub-lattices. The simulation is at constant volume, with lattice constants a = 7.185'A and a = 7.351 A. The energy of the system is also conserved and, as a result of the periodic boundary conditions, so is the particle density.

6.2.3 Details of Trajectory Calculations

Ion trajectories were calculated using the computer code 'FUNGUS' (Walker, 1982). The main details of the simulations are given below.

♦ Separate simulations were performed at 1 2 0 0K, 1500K and 1700K - temperatures which should all be within the superionic region and still be below the melting temperature at 1766K. A preliminary simulation with a much smaller 'box' of only 96 ions showed no evidence of superionic behaviour at 1000K. A repeat simulation for the larger box was not, therefore, considered to be worthwhile.

♦ A time-step of 5 x 10' 15 s was used for all simulations. Each simulation began with an equilibration period. The system was considered to have reached equilibrium when,

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a (T )~ % of target temperature, 7 n

where o is the estimated standard deviation and N is the number of particles in the simulation box. Following

equilibration, trajectory data were calculated and stored for analysis.

The first run at 1200K was initiated using the perfect crystal coordinates as the 'start-up' configuration. This was the preferred method of assignment because the static calculations had indicated the formation of both Frenkel pairs and bound Schottky quartets. Since experiment is still open on this, it was considered best to allow the system to evolve its own defect structure during the equilibration process. It should be noted, however, that MD with periodic boundary conditions cannot generate Schottky disorder. For the trajectory calculations at the two higher temperatures the start-up data were each taken to be the equilibration configurations from the previous temperature.

The details of each simulation are given in Table 6.3. Each time- step required 0.59 seconds of cpu time on the CRAY-1 S with ~8 hours of Cray time being used in total on the simulations, excluding the time used for the data analysis.