Dynamical model for the polar-incommensurate transition in BaMnF4

Using a Landau-Khalatnikov approach we calculate the temperature dependence of the dielectric constant and of the coupled normal modes in incommensurate phase transitions such as that displayed by BaMn${\mathrm{F}}_4$. It is assumed that the phase mode is relaxational in the incommensurate phase and satisfies a Debye relaxation equation. Two results of interest are generated: first, the temperature dependence of the inverse relaxation time is given by ${\tau }^{-1\mathrm{}}\mathrm{}\left(T\right)=\left[\frac{(T_{\mathrm{I}}-T)}{T_{\mathrm{I}}}\right]{\tau }_0^{-1\mathrm{}}+{\tau }_1^{-1\mathrm{}}$, in agreement with experimental results for both BaMn${\mathrm{F}}_4$ and ${\mathrm{Ba}}_2$Na${\mathrm{Nb}}_5$ ${\mathrm{O}}_{15}$; and second, a new relationship between the dynamical parameters ${\tau }_0$ and ${\tau }_1$ and the temperature width of stability for the incommensurate phase, $T_{\mathrm{I}}-T_{\mathrm{II}}$ is derived: $\frac{(T_{\mathrm{I}}-T_{\mathrm{II}})}{T_{\mathrm{I}}}=C\frac{{\tau }_0}{{\tau }_1}$, where $C$ is a numerical constant of order 1.0 to 6.0 in the anisotropic case. Using experimental values of $\frac{1}{2}\pi {\tau }_0=(1.7\mathrm{}\pm 0.7)\times {10}^{11}$ Hz and $\frac{1}{2}\pi {\tau }_1=(6.7\mathrm{}\pm 0.4)\times {10}^8$ Hz for BaMn${\mathrm{F}}_4$, we calculate $T_{\mathrm{I}}-T_{\mathrm{II}}=6\mathrm{}\pm 4$ K, in good agreement with the recent measurement of 8.2 K by Scott, Habbal, and Hidaka; similar estimates for ${\mathrm{Ba}}_2$Na${\mathrm{Nb}}_5$ ${\mathrm{O}}_{15}$ predict a large incommensurate temperature region, which is also in good accord with experiment.