Formal elasticity of four carbon allotropes: I. The inner elastic constants, internal strain tensors, and zone-centre optic mode frequencies and their pressure dependences

The symmetry structure of the six families of inner elastic constants (derivatives of the free. energy with respect to one, two, or three components of the relative sublattice displacement) that enter the elasticity, through the third order, of the cubic and hexagonal diamond allotropes and of the hexagonal and rhombohedral graphite allotropes is analysed in detail. This is followed by derivations (i) of the forms of the linear and quadratic internal strain tensors, (ii) of the expressions for the zone-centre optic mode frequencies and eigenvectors, and (iii) of the effective inner elastic constants that determine these frequencies in arbitrarily strained crystals together with (iv) the derived pressure dependences of those frequencies.