Lattice dynamics and thermal expansion of quartz

The mechanism of the thermal expansion and the alpha-beta phase transition of quartz are jointly studied within the framework of a lattice-dynamical t reatment using the pair-wise potential by Tsuneyuki ct al. [Phys. Rev. Lett . 61, 869 (1988)]. This shows that the essentially anomalous thermal expans ion of quartz originates from the low-frequency phonon modes most of which have negative Grueneisen coefficients. The main factor driving the alpha-ph ase structure variation at heating is the rotation of the SiO4 tetrahedra t owards their beta-phase positions. The volume variation follows this proces s thus keeping the static pressure small. The model reveals that at T>430 K a number of the phonons have imaginary quasiharmonic frequencies being gov erned by a double-well potential. This result does not suggest any large-sc ale lattice instability, and just indicates that the relevant vibrations ar e essentially anharmonic and that the actual crystal structure is of a dyna mically averaged character. The contribution of such modes to the free ener gy 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-e nergy optimization with respect to all the structural parameters provides t he alpha-quartz structure at T< T-c. We reveal that there is no free-energy minimum in the alpha structure at T>T-c approximate to 850 K, but it exist s in the beta phase at 850 K<T<1100 K. Takings into account the discovered negative Grueneisen constants our approach provides a natural explanation f or the negative thermal expansion of the beta quartz.