Lattice dynamical calculations of anisotropic temperature factors of atoms in quartz,and the structure of β-quartz

Temperature factors for oxygen and silicon atoms in β-quartz were calculated on a Born-von Karman lattice dynamical model of an ordered structure. The calculated thermal ellipsoids were in excellent agreements with those of the previous structure refinements of the order model, for both magnitudes and orientations of the principal axes. The temperature factors are contributed mainly by the soft optic modes in Γ-M and the lowest-lying acoustic modes along Γ-A, which are also strongly temperature-dependent. The cusp-shaped temperature dependence of mean square displacements, 〈u²〉, of oxygen atom, observed previously around the α-β transition, are resulted from the softening of these modes. The temperature-dependent modes in Γ-A were also found to cause diffuse scattering extending along ±c^* of the fourth hexagon of the hk0 reciprocal lattice plane. The negative expansion known in β-quartz were interpreted in terms of asymmetrical forces exerting on oxygen atoms in Si-O-Si bending modes. In β-quartz, librational motions of oxygen atoms around Si-Si lines with large amplitudes, whose center is just on the β-position of high symmetry, must be possible under the condition that bending tetrahedral O-Si-O angles is energetically more favourable than compressing or stretching Si-O bonds.