Effects of low-lying phonon modes on the anomalous temperature dependence of the elastic constants of rutile and quartz

Abstract

It is shown that the inclusion of the anharmonicity of low-lying dispersive phonon modes accounts for the unexplained anomalies of the temperature dependence of the second-order elastic constants C 11, C 13, C 44, C 66 and C 110L of rutile in a broad temperature range below 600 K, and of the constants C 11 and C′ 66 of quartz below 75 K. The relative variations of C 33 and C 44 of rutile were measured in the range 4–300 K where earlier data were incomplete. It is found that the quasi-harmonic formulation extended to include low-lying phonon modes reproduces the constant temperature coefficient β of the fractional variation of the elastic constants of quartz and of rutile at cryogenic temperatures. It also accounts for the change in β to a less negative value which occurs in rutile at temperatures above about 200 K. Best fits to the experimental data yields estimates for the C ij effective Gr üneisen parameters of both materials.