Application of a Modified Random-Element-Isodisplacement Model to Long-Wavelength Optic Phonons of Mixed Crystals

The random-element-isodisplacement (REI) model has been modified to include the polarization field, thereby enabling it to predict the long-wavelength longitudinal optic frequencies of a mixed crystal. A criterion has been obtained to determine whether a given mixed crystal will exhibit "one-mode" or "two-mode" behavior. Under certain conditions the modified REI (MREI) model may also be made to work for mixed crystals with one-mode behavior. The so-called "virtual-crystal model" applicable to this type of crystals becomes a special case of the MREI model. Given the properties of the "pure" end-members $\mathrm{AB}$ and $\mathrm{AC}$, the MREI model then can determine whether $A{B}_{1-x}{C}_x$ is going to display one-mode or two-mode behavior, and can predict the concentration dependence of the TO and LO modes. A large number of examples of both types of mixed crystals is given.