Incommensurate instability and lattice dynamics of potassium selenate within a semiempirical rigid-ion model

The lattice dynamics of potassium selenate is analyzed using a rigid-ion model with the selenate groups reduced to rigid bodies. The interatomic forces have been adjusted only using static structural data. The number of adjustable parameters varies from two to five. Such a simple model is already sufficient to reproduce semiquantitatively the phonon dynamics of the real system. In particular, the model exhibits the lattice instability leading to the existence of an incommensurate phase. The characteristics of the resulting soft mode agree with those observed experimentally. The calculated eigenvector, in excellent agreement with the experimental one, is rather insensitive to the details of the interactions. This explains the strong similarities of the incommensurate modulations in most ${\mathit{A}}_2$ ${\mathit{BX}}_4$ compounds. On the other hand, the form of the soft-phonon branch strongly depends on the force model. It is sufficient to fit the model to the static structure observed at 145 K instead of the one at room temperature, to provoke a conspicuous softening of the branch. The branch minimum is specially sensitive to some potassium-oxygen interactions. The relative size of the cations plays an essential role in the origin of the incommensurate instability. For comparison the results of a similar analysis for ${\mathrm{Cs}}_2$ ${\mathrm{SeO}}_4$ are presented. In this case, the unstable or ‘‘soft’’ character of the lowest ${\mathrm{\Sigma }}_2$ branch disappears.