Structural phase transformation inK2SeO4
Successive phase transformations in ${\mathrm{K}}_{2}$Se${\mathrm{O}}_{4}$ at ${T}_{i}=130$ K and ${T}_{c}=93$ 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 ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{\ensuremath{\delta}}=\frac{(1\ensuremath{-}\ensuremath{\delta}){\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}}^{*}}{3}$ with $\ensuremath{\delta}=0.07$ at 122.5 K. The deviation \ensuremath{\delta} 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 ${Q}^{3}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{\ensuremath{\delta}}){P}_{z}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{3\ensuremath{\delta}})$ plays an essential role in driving the incommensurate-commensurate phase transformation and in inducing the spontaneous polarization. Here, $Q({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{\ensuremath{\delta}})$ is the amplitude of the primary atomic displacements with wave vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{\ensuremath{\delta}}$ and ${P}_{z}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{3\ensuremath{\delta}})$ is the polarization wave with wave vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}_{3\ensuremath{\delta}}=3\ensuremath{\delta}(\frac{{\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}}^{*}}{3})$ and becomes the macroscopic polarization below ${T}_{c}$. Above ${T}_{i}$, a ${\ensuremath{\Sigma}}_{2}$ optic-phonon branch along ($\ensuremath{\xi},0,0$) shows a striking softening and ${\ensuremath{\omega}}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\sim}(\frac{1}{3},0,0)$ 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.