Crystallographic parameterisation of distortions in the SOD framework in the sodalite and helvine groups: An analysis in condensed normal modes of an aristotype phase

Abstract

AbstractCrystallographic distortions in the alternating aluminium and silicon tetrahedral framework of sodalite (Na8Al6Si6O24Cl2), and beryllium and silicon in helvine (Mn8Be6Si6O24S2), (framework designated SOD) are described in terms of a set of condensed normal mode amplitudes and phases derived from an ideal tetrahedron of a theoretical aristotype phase. For a sodalite-structured hettotype phase in space group $P{\overline 4} 3n$, these normal modes transform as the irreducible representations A1, E(α) and T1(z) of point-group ${\overline 4} 3m$, where to a good approximation A1 acts as a pure breathing mode, E(α) as a polyhedral distortive mode and T1(z) as a rigid unit rotation about the unique ${\overline 4}$ axis of the T-site under consideration. Parameterisation of the mode amplitudes in terms of low-order polynomials as a function of thermodynamic variable permits the crystal structure of sodalite-structured phases to be accurately interpolated at intermediate values of the thermodynamic variable. Published data for the high-temperature behaviour of sodalite have been re-analysed in terms of mode amplitudes which accurately reproduce the temperature dependence of the bond lengths, bond angles and the Al–O–Si inter-polyhedral angle. Full expressions for these derived structural parameters in terms of mode amplitudes and the lattice parameter are tabulated and agree with experimental results to within one estimated standard deviation of the experimental parameter. The potential for mode decomposition in lower symmetry SOD framework crystal structures is illustrated by deriving an aristotype structure for tugtupite (Na8Al2Be2Si8O24Cl2) at room temperature in space group $I{\overline 4}$.