Lattice dynamics and thermal expansion of quartz

The mechanism of the thermal expansion and the α-β 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 Grüeneisen coefficients. The main factor driving the α-phase structure variation at heating is the rotation of the ${\mathrm{SiO}}_4$ tetrahedra towards their β-phase positions. The volume variation follows this process thus keeping the static pressure small. The model reveals that at $T>\mathrm{}430\mathrm{}\mathrm{K}$ 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 α-quartz structure at $T<\mathrm{}{T}_c.$ We reveal that there is no free-energy minimum in the α structure at $T>\mathrm{}{T}_c\approx 850\mathrm{K},$ but it exists in the β phase at $850\mathrm{K}<T<\mathrm{}1100\mathrm{}\mathrm{K}.$ Taking into account the discovered negative Grüeneisen constants our approach provides a natural explanation for the negative thermal expansion of the β quartz.