Soft modes and elastic strain at the tetragonal-to-monoclinic phase transition in antifluorite and related structure types

Group-theoretical normal-mode analysis of the antifluorite $A_{2}B{X}_6$ structure in space group $\frac{P4\mathrm{}}{\mathrm{mnc}} \left(D_{4h}^6\right)$ identifies two effective modes with $E_g$ symmetry. Either mode may condense independently of the other at the phase transition to space group $\frac{P{2}_1}{n} \left(C_{2h}^5\right)$. One mode produces a displacement of the $A^{+}$ ions, the other a rotation of the $B{X_6}^{2-}$ octahedra. Lattice distortion thereby results from the coupling between order parameter ($Q$) and elastic strain ($e$). Linear and quadratic symmetry invariant coupling in terms of free energy results in $e_{13}\propto Q$ and, independently, $e_{22}-e_{11} \propto Q^2$. The linear coupling produces a discontinuous change in soft-mode frequency at $T_c$, whereas the change in specific heat is caused primarily by the quadratic coupling. Measurements on ${\mathrm{K}}_2$Te${\mathrm{Br}}_6$ reveal that one $E_g$ mode condenses at the second-order phase transition to $\frac{P{2}_1}{n}$ at 400 K and follows a Landau critical power law with $e_{13} \propto {(400\mathrm{}\mathrm{K}-T)}^{\frac{1}{2}}$, leaving $e_{22}-e_{11}=0\mathrm{}$ for $359<T<\mathrm{}400\mathrm{}$ K. The symmetry-conserving softening of the octahedral rotation $E_g$ mode takes place at 359 K, with $e_{22}-e_{11} \propto (359\mathrm{}\mathrm{K}-T)$. The incompatible symmetry of the $E_g$ and the $A_{1g}$ mode derived from the higher-temperature $\mathrm{Fm}3m \left(O_h^5\right)$ to $\frac{P4\mathrm{}}{\mathrm{mnc}}$ phase transition results in a sharp change in the tetragonal $c-a$ dimension at the tetragonal-to-monoclinic transition as further octahedral rotation becomes blocked. Evidence for the two $E_g$-mode displacement fields is found in the literature for all elpasolite $A_2{B}^{\prime }{B}^{\prime \prime }{X}_6$ and cryolite $A_{3}B{X}_6$ structures determined in space group $\frac{P{2}_1}{n}$. Formation of an additional phase by the antifluorites ${\mathrm{K}}_2$Se${\mathrm{Br}}_6$ and ${\mathrm{K}}_2$Sn${\mathrm{Cl}}_6$ in space group $\frac{C2}{c} \left(C_{2h}^6\right)$ on cooling from $\frac{P4\mathrm{}}{\mathrm{mnc}}$ to $\frac{P{2}_1}{n}$ results from two octahedral rotations due to the $E_g$-mode basis vectors having order parameters of equal magnitude. The orthorhombic pseudosymmetry in $\frac{C2}{c}$ is caused only by the $e_{12}$ strain component. The coupling found in ${\mathrm{K}}_2$Te${\mathrm{Br}}_6$ is shown to apply generally to related structure types.