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 A₂BX₆ structure in space group P4/mnc (D₄ₕ⁶) identifies two effective modes with Eg symmetry. Either mode may condense independently of the other at the phase transition to space group P2₁n (C₂ₕ⁵). One mode produces a displacement of the A⁺ ions, the other a rotation of the BX₆^2- 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 e₁₃∝Q and, independently, e₂₂-e₁₁ ∝Q². The linear coupling produces a discontinuous change in soft-mode frequency at Tc, whereas the change in specific heat is caused primarily by the quadratic coupling. Measurements on K₂TeBr₆ reveal that one Eg mode condenses at the second-order phase transition to P2₁n at 400 K and follows a Landau critical power law with e₁₃ ∝(400K-T)1/2, leaving e₂₂-e₁₁=0 for $359<T<400$ K. The symmetry-conserving softening of the octahedral rotation Eg mode takes place at 359 K, with e₂₂-e₁₁ ∝(359K-T). The incompatible symmetry of the Eg and the A1g mode derived from the higher-temperature Fm3m (Oₕ⁵) to P4/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 Eg-mode displacement fields is found in the literature for all elpasolite A₂B^'B^''X₆ and cryolite A₃BX₆ structures determined in space group P2₁n. Formation of an additional phase by the antifluorites K₂SeBr₆ and K₂SnCl₆ in space group C2/c (C₂ₕ⁶) on cooling from P4/mnc to P2₁n results from two octahedral rotations due to the Eg-mode basis vectors having order parameters of equal magnitude. The orthorhombic pseudosymmetry in C2/c is caused only by the e₁₂ strain component. The coupling found in K₂TeBr₆ is shown to apply generally to related structure types.