Crystal Stability ofAX2Compounds in Terms of Three-Ion Interactions

By taking into account three-ion exchange interactions in solids of $A{X}_2$ compounds, it is shown how the observed stability relations can be explained on a quantitative basis. The analysis is an extension of those given earlier for stability of rare-gas crystals, of solids of the alkali halides, and of II-VI and III-V compounds whose ions are isoelectronic with rare-gas atoms. Of the compounds $A{X}_2$, $A$ denotes an element of columns II, IV, or VI of the periodic table, $X$ a corresponding element of the columns I, II, VI, or VII. All ions considered are isoelectronic with rare-gas atoms. As before, the stability analysis is based on a first-and second-order perturbation calculation, starting from complete ionicity in zeroth order of approximation, with Gaussian-type effective electron wave functions for the ions. The structures considered are (the ideal lattices of) fluorite, two types of rutile, anatase, cadmium chloride, cadmium iodide, cuprite, quartz, and cristobalite; a comparison is made between the static lattice energies of these structures. The effect of polarization energy on crystal stability is considered in detail in the framework of the Born-Mayer model; Madelung constants for all structures were determined on the basis of the Bertaut method. The theory accounts for all observed stability relations; except that for the compound Ti${\mathrm{O}}_2$ the fluorite structure is found to be more stable than a rutile lattice on the basis of closed-shell electron configurations of the Ti ion. In particular, a quantitative explanation is given for the difference in lattice energy between the cadmium iodide and cadmium chloride structures (6,3 coordination) and between the $\beta$-quartz and $\beta$-cristobalite lattices (4,2 coordination).