Lattice dynamics of mixed alkali halides. I. Negative-ion substitution

Phonon frequencies of one-mode behavior of mixed alkali-halide crystals with negative-ion substitution have been calculated as a function of wave-vector and molar-mixing ratio. The mixed-crystal model utilizes a pseudo-unit-cell as a basic repetitive unit incorporated into the breathing-shell model of lattice dynamics. The fundamental repetitive unit cell for the mixed crystal $A{B}_{1-x}{C}_x$ consists of $A$ ions at their normal lattice site and with nearest neighbors consisting of a fractional amount of $B$ and $C$ ions proportional to their mixing molar ratio. The model parameters are deduced from physical observables such as the elastic constants, long-wavelength transverse-optical mode frequency, and the high- and low-frequency dielectric constants of the parent compounds. Room-temperature calculations of one- and two-phonon densities of states and dispersion relations of $\mathrm{K}{\mathrm{Cl}}_{1-x}{\mathrm{Br}}_x$, $\mathrm{K}{\mathrm{Cl}}_{1-x}{\mathrm{I}}_x$, and $\mathrm{K}{\mathrm{Br}}_{1-x}{\mathrm{I}}_x$ are found to agree with the available Raman scattering, infrared reflection, and absorption data.