Elasticity of carbon allotropes. II. Modified anharmonic Keating model for hexagonal diamond

Abstract

The parameters of the modified Keating model deduced for cubic diamond are used to predict the elastic constants and zone-center optic mode frequencies of hexagonal diamond. A geometrical difference between the two structures at the third-neighbor level requires an adjustment to the ${\ensuremath{\beta}}^{*}$ parameter that results in excellent agreement between the model and the three observed Raman frequencies. The lower symmetry and extra degree of freedom associated with the hexagonal unit cell permit exploration of three distinct r\'egimes: quasi-cD (same bond lengths, atomic volume and $c/a$ ratio), equal bondlengths with increased $c/a$ ratio, and unequal bondlengths with unchanged $c/a$ ratio. Numerically there is little difference between the predictions in each case. The quasi-cD case implies first an isotropic compressibility consistent with the observed constancy of the $c/a$ ratio over a wide range of pressure and second no lifting of the triply degenerate Raman frequency which is consistent with experiment but slightly at variance with a first-principles calculation. All three cases yield a bulk modulus slightly smaller than the fitted value for the cubic allotrope, and similar values for the pressure derivatives of the elastic constants and of the optic mode frequencies.