Lattice-dynamical model for the elastic constants and Raman frequencies in (${\mathrm{V}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Cr}}_{\mathrm{x}}$${)}_2$${\mathrm{O}}_3$

We report the first lattice-dynamical study of ${\mathrm{V}}_2$ ${\mathrm{O}}_3$ and (${\mathrm{V}}_{0.985}$ ${\mathrm{Cr}}_{0.015}$ ${)}_2$O ${}_3{}^{}{}_{}{}^{}$ in the long-wavelength limit using a rigid-ion model with an effective ionic charge and various short-range interactions. The parameters of the model were determined from nonlinear least-squares fits to the experimental elastic constants and the frequencies of Raman-active modes at room temperature. The validity of the model was established by verifying that the calculated elastic constants and Raman frequencies were in good agreement with experimental data for ${\mathrm{V}}_2$ ${\mathrm{O}}_3$ and (${\mathrm{V}}_{0.985}$ ${\mathrm{Cr}}_{0.015}$ ${)}_2$O ${}_3{}^{}{}_{}{}^{}$ as well as for ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ to which the same model was also applied. The effective ionic charge of each (${\mathrm{V}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Cr}}_{\mathrm{x}}$ ${)}_2$ ${\mathrm{O}}_3$ crystal was found to be considerably larger than that of ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ in accordance with the ionicity of ${\mathrm{V}}_2$ ${\mathrm{O}}_3$, estimated from the dielectric theory of Levine, being larger than that of ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$. The short-range force constants of (${\mathrm{V}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Cr}}_{\mathrm{x}}$ ${)}_2$ ${\mathrm{O}}_3$ were found to be quite different from those of ${\mathrm{Al}}_2$