Elastic response for coupled external and internal strains

Alternative methods of deriving the elastic response are presented for crystals where the external lattice strains are coupled to internal atomic or molecular displacements. A lattice dynamical method using a long-wave expansion shows how the effective second-order elastic constants are related to the bare elastic constants via the optical or rotational susceptibility and the coupling between strains. Different results suited to different limiting types of internal motion are presented. A previous generalised thermodynamic method giving similar results is extended to yield expressions for the effective third- and fourth-order elastic constants as well. The methods provide direct and efficient ways of analysing lattice dynamical and thermodynamic models involving internal strain, as demonstrated for previous models of KCN and sym-triazine.