Elasticity of carbon allotropes. I. Optimization, and subsequent modification, of an anharmonic Keating model for cubic diamond

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The inner elastic constants of single-crystal diamond at the second and third order characterize the response of the crystal to the internal displacement of its component sublattices, of which there are two, either alone or coupled to external strain. These constants feature in the decomposition of the macroscopic elastic constants at second and third order, and give rise to one linear and three quadratic independent internal strain parameters. All these constants have been obtained via the implementation of a Keating potential notionally restricted to two-body and three-body interactions between nearest neighbors. Four harmonic parameters have been optimized to reproduce the second-order elastic constants, the frequency of the (triply degenerate) optical mode at the zone center and the internal strain parameter. The resulting fit is excellent and also accounts very well for the frequencies of the TO modes at the X and L critical points. Excessively large frequencies predicted for the TA modes at these points are shown to be due to a particular four-body interaction that cannot be separated elastically from the three-body bond-bending interaction. The assignment of a fifth parameter allows all the critical point frequencies to be well fit, the largest discrepancies being 3% and 6% for the TA modes. One of six anharmonic parameters is shown to be statistically insignificant. The remaining five are fit to the pressure derivatives of the second-order elastic constants and to the various stress derivatives of the frequency of the zone-center modes. These parameters are used to predict the values of all the third-order elastic and inner elastic constants, and of the quadratic internal strain parameters. Finally the Keating strain is redefined so that the parameters of the model no longer depend on the dimensions of the unit cell chosen to describe the structure. New expressions are obtained for all elastic constants and the optimized parameters are appropriately modified.