Structural phase transformation inK2SeO4

Successive phase transformations in ${\mathrm{K}}_2$Se${\mathrm{O}}_4$ at $T_i=130\mathrm{}$ K and $T_c=93\mathrm{}$ K were studied by the neutron-scattering technique. The superlattice reflections in the intermediate phase were found to be incommensurate with the lattice periodicity. The wave vector characterizing the reflections is ${\overrightarrow{\mathrm{q}}}_{\delta }=\frac{(1-\delta ){\overrightarrow{\mathrm{a}}}^{*}}{3}$ with $\delta =0.07\mathrm{}$ at 122.5 K. The deviation δ decreases with decreasing temperature with an apparently discontinuous jump to zero at $T_c$. Below this temperature, the crystal remains commensurate and is known to be ferroelectric. The incommensurate-commensurate transition and the simultaneous occurrence of the commensurate phase and the spontaneous polarization are discussed using a Landau-type expansion of the free energy in which a term proportional to $Q^3\left({\overrightarrow{\mathrm{q}}}_{\delta }\right)P_z\left({\overrightarrow{\mathrm{q}}}_{3\delta }\right)$ plays an essential role in driving the incommensurate-commensurate phase transformation and in inducing the spontaneous polarization. Here, $Q\left({\overrightarrow{\mathrm{q}}}_{\delta }\right)$ is the amplitude of the primary atomic displacements with wave vector ${\overrightarrow{\mathrm{q}}}_{\delta }$ and $P_z\left({\overrightarrow{\mathrm{q}}}_{3\delta }\right)$ is the polarization wave with wave vector ${\overrightarrow{\mathrm{q}}}_{3\delta }=3\mathrm{}\delta \left(\frac{{\overrightarrow{\mathrm{a}}}^{*}}{3}\right)$ and becomes the macroscopic polarization below $T_c$. Above $T_i$, a ${\Sigma }_2$ optic-phonon branch along ($\xi ,0,0\mathrm{}$) shows a striking softening and ${\omega }_j\left(\overrightarrow{\mathrm{q}}\right)$ for $\overrightarrow{\mathrm{q}}\sim (\frac{1}{3},0,0)$ tends to zero at $T_i$. The softening results from a temperature-dependent decrease of the interlayer forces with ranges $\frac{a}{2}$ and $a$ ($a$ is one unit-cell length along the $a$ axis) in the presence of strong and persisting forces with a range $\frac{3a}{2}$. The intensities of the soft phonon were measured about different reciprocal-lattice points and were used to determine the nature of the soft-phonon mode and suggest a coupled translation of potassium ions with rotational motion of Se${\mathrm{O}}_4$ groups to be the origin of the lattice instability.