Lattice dynamics ofα-quartz

The phonon dispersion curves, the temperature dependence of the Debye temperature, and the static elastic, dielectric, and piezoelectric constants of $\alpha$-Si${\mathrm{O}}_2$ have been calculated from a modified rigid-ion model with an effective charge, with short-range central forces for the two Si-O interactions, for the four nearest-neighbor O-O interactions, for the four second-neighbor O-O interactions, and with three-body type angle-bending forces between nearest-neighbor O ions sharing a common Si-ion, and between Si-ions of adjacent tetrahedra and the linking O ion. The model involves 23 parameters which are reduced to 8 independent parameters by the six equilibrium conditions and by 15 constraints on the first and second derivatives of the interaction potential corresponding to the inner and outer O-O interactions. Using explicit expression for the elastic constants and for the optical modes at zero wave vector, the eight free parameters were determined from a least-squares fit of the six elastic constants and of the 16 Raman and transverse ir frequencies. The effective charge of the Si ion is found to be 0.96 electronic charges, indicating a strong degree of covalency and/or screening. The model gives good over-all agreement for the elastic and optical data, rough agreement for the dielectric constants, and the correct sign and order of magnitude of the peizoelectric constants. The dispersion curves along [001] and the temperature dependence of the Debye temperature agree well with available experimental data, with moderate discrepancies occurring for the two acoustic branches and at low temperatures, respectively.