Lattice dynamics and thermal expansion of quartz

The mechanism of the thermal expansion and the \ensuremath{\alpha}-\ensuremath{\beta} phase transition of quartz are jointly studied within the framework of a lattice-dynamical treatment using the pair-wise potential by Tsuneyuki et al. [Phys. Rev. Lett. 61, 869 (1988)]. This shows that the essentially anomalous thermal expansion of quartz originates from the low-frequency phonon modes most of which have negative Gr\"ueneisen coefficients. The main factor driving the \ensuremath{\alpha}-phase structure variation at heating is the rotation of the ${\mathrm{SiO}}_{4}$ tetrahedra towards their \ensuremath{\beta}-phase positions. The volume variation follows this process thus keeping the static pressure small. The model reveals that at $T>430\mathrm{K}$ a number of the phonons have imaginary quasiharmonic frequencies being governed by a double-well potential. This result does not suggest any large-scale lattice instability, and just indicates that the relevant vibrations are essentially anharmonic and that the actual crystal structure is of a dynamically averaged character. The contribution of such modes to the free energy has been included by the extension of the quasiharmonic theory proposed by Boyer and Hardy [Phys. Rev. B 24, 2577 (1981)]. Then the accurate free-energy optimization with respect to all the structural parameters provides the \ensuremath{\alpha}-quartz structure at $T<{T}_{c}.$ We reveal that there is no free-energy minimum in the \ensuremath{\alpha} structure at $T>{T}_{c}\ensuremath{\approx}850\mathrm{K},$ but it exists in the \ensuremath{\beta} phase at $850\mathrm{K}