Modelling the structural variation of quartz and germanium dioxide with temperature by means of transformed crystallographic data

The pseudocubic (PC) parameterization of O(4) tetrahedra [Reifenberg & Thomas (2018 ▸). Acta Cryst. B74, 165–181] is applied to quartz (SiO(2)) and its structural analogue germanium dioxide (GeO(2)). In α-quartz and GeO(2), the pseudocubes are defined by three length parameters, a (PC), b (PC) and c (PC), together with an angle parameter α(PC). In β-quartz, α(PC) has a fixed value of 90°. For quartz, the temperature evolution of parameters for the pseudocubes and the silicon ion network is established by reference to the structural refinements of Antao [Acta Cryst. (2016), B72, 249–262]. In α-quartz, the curve-fitting employed to express the non-linear temperature dependence of pseudocubic length and Si parameters exploits the model of a first-order Landau phase transition utilized by Grimm & Dorner [J. Phys. Chem. Solids (1975) ▸, 36, 407–413]. Since values of tetrahedral tilt angles about 〈100〉 axes also result from the pseudocubic transformation, a curve for the observed non-monotonic variation of α(PC) with temperature can also be fitted. Reverse transformation of curve-derived values of [Si+PC] parameters to crystallographic parameters a, c, x (Si), x (O), y (O) and z (O) at interpolated or extrapolated temperatures is demonstrated for α-quartz. A reverse transformation to crystallographic parameters a, c, x (O) is likewise carried out for β-quartz. This capability corresponds to a method of structure prediction. Support for the applicability of the approach to GeO(2) is provided by analysing the structural refinements of Haines et al. [J. Solid State Chem. (2002) ▸, 166, 434–441]. An analysis of trends in tetrahedral distortion and tilt angle in α-quartz and GeO(2) supports the view that GeO(2) is a good model for quartz at high pressure.