Librational displacements of silicate tetrahedra in response to temperature and pressure

Recently it has been concluded that the SiO₄ silicate tetrahedra in crystals behave as rigid bodies. This conclusion is based on analyses of the atomic displacement factors of Si and O atoms obtained from single crystal diffraction experiments wherein the amplitudes of atomic vibrations are ascribed to translational, librational and screw-correlated modes of motion for the entire SiO₄ group. If the displacement ellipsoids are considered to represent time averaged quadratic surfaces of equal configurational potential energy about the mean position of an atom, then an analysis of the these displacements should provide detailed information about the SiO₄ group and the crystal.

The apparent SiO bond lengths recorded for silicates over a range of temperatures are typically either invariant or exhibit a contraction with increasing temperature. A rigid-body thermal analysis was completed for the tetrahedra in nine silicates whose structures have been determined over a range of temperatures from 15 K to 1250 K and whose tetrahedra seem to behave as rigid units. The coordinates provided by the analysis yield bond lengths and polyhedral volumes corrected for the librational motion of each silicate tetrahedron. The bond lengths and volumes estimated for tetrahedra with four bridging oxygens seem to increase with temperature at a faster rate than those with four nonbridging oxygen atoms. Those for tetrahedra with two or three nonbridging oxygen atoms tend to increase at an intermediate rate. An analysis of the rigid-body motion of coordinated polyhedra yields a simple but accurate expression for correcting bond lengths for thermal vibrations.

Observed anisotropic displacement parameters for Si and O atoms indicate that the SiO₄ tetrahedra in quartz behave as rigid bodies. A configurational potential energy curve, constructed from the librational components of the rigid body motion of the tetrahedra, shows a double well for α quartz and a single well for β quartz when plotted as a function of the displacement of the O atom with temperature. The configurational energetics of α and β quartz are examined with a theoretical potential energy function based on parameters obtained from molecular orbital calculations. The calculations indicate that the temperature behavior of a quartz is governed by the energetics of the SiOSi angle, in contrast to β quartz which is governed by the energetics of the SiO bond. The mechanism of the α ⇌ β transition is examined in terms of the experimental and modeled configurational potential energy curves. Evidence for the proposal that π bonding is the driving mechanism for the transition is lacking.

Structural and volume compressibility data for α-cristobalite were determined by single crystal X-ray diffraction methods for pressures up to ~1.6 GPa, where cristobalite undergoes a reversible phase transition. The bulk modulus was determined to be 11.5(7) GPa with a pressure derivative of 9(2). The SiOSi angle shows a greater decrease than observed for quartz and coesite while the SiO bond lengths and the OSiO angles remain essentially unchanged. The responses of V/V₀ and SiOSi angle to pressure for the silica polymorphs are compared and it is found that the percentage decrease in the volume is linearly correlated with the percentage decrease in the SiOSi angle, regardless of the framework structure type. A mathematical modeling of the energies of the structural changes that are induced by pressure suggests that the contribution to the total energy ascribed to Si0Si angle bending terms is the same in quartz and cristobalite.