Statistical Mechanics of Defect-Containing Solids. II. Ionic Crystals

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Abstract
The cluster expansions for the partition function and defect distribution functions derived previously are studied in detail for the case of ionic crystals with the object of calculating activity coefficients, defect concentrations, and defect distribution functions at low defect concentrations. A diagram classification procedure analogous to that in the Mayer theory of ionic solutions is used to obtain nondivergent expansions for defect activity coefficients and distribution functions. The discreteness of the lattice requires some modification of the diagram summation techniques employed in the solutions theory. The theory of association of defects of the sort considered by Lidiard and Teltow is formulated more precisely in terms of the defect distribution functions. The formal multicomponent expressions are studied in more detail for the case of Schottky defects and impurity ions in a sodium chloride lattice. The results parallel those of the Mayer ionic-solution theory, the principal difference being that the Debye—Hückel potential of average force appearing in the final Mayer expressions is everywhere replaced by Ae2 exp(—κRξ)/RD, where A and ξ are structure and concentration dependent and go to unity in the continuum limit. As an example for the case of activity coefficients, calculations of the contributions from cycle diagrams and terms of next lowest order in concentration have been made for divalent impurity ions and cation vacancies in sodium chloride. The pair correlation function for oppositely charged defects and the degree of association have been calculated for the doped crystal. The theory reduces to that of Lidiard in the limit of zero concentration but differs at finite concentrations. However, calculation of the contribution of ``triangle diagrams'' to the activity coefficients indicates that below 500°C the expansions do not converge rapidly enough to be of value at concentrations of experimental interest because of the low dielectric constant. It was found that in the range of temperature and composition for which the theory converged, the parameters A and ξ differed little from unity.