Entropy of a point defect in an ionic crystal

The problem of calculating the entropy of formation of a point defect in an ionic crystal is reexamined in some detail. We use two methods of calculation, which we term the embedded-crystallite and the Green-function methods. In both methods, the crystal is divided into an inner region, which contains the defect and a set of its neighbors, and an outer region. In the embedded-crystallite method, the entropy is calculated directly from the determinants of the force-constant matrices for perfect and defective crystals, restricted to the defect region. The Green-function method, which is expected to be more accurate, exploits a reformulation of the entropy expression in terms of the Green functions of the perfect crystal and the change of the force constants in the defect region. The principal feature of our calculations is the examination, for the first time in ionic crystals, of the convergence of the calculated entropy as the size of the defect region is increased. Associated problems, such as the correct inclusion of the long-range Coulomb contributions and the accuracy of the Green functions, have also been addressed. The model numerical calculations which we present are for the case of Frenkel (vacancy and interstitial) defects in Ca${\mathrm{F}}_2$. The majority of the calculations have been performed with the use of a rigid-ion potential, which allows a simpler discussion of the technical problems, but we have also investigated the effect on the results of using a realistic shell model. We show that the convergence of the results is strongly affected by previously unrecognized fluctuation effects coming from the boundary of the defect region; these effects are specific to ionic crystals. It is demonstrated that the boundary effects can be eliminated with the use of a simple substraction technique, and that the resulting entropy values can be extrapolated to infinite region size with an uncertainty of no more than a few tenths of Boltzmann's constant. It is pointed out that in a finite crystal there is a contribution to the entropy of charged defects from the structure of the physical surface; this contribution cancels out, however, for neutral sets of defects. We discuss previous work on this problem in the light of our findings.