Lattice Dynamics and Elasticity of Stressed Crystals

A conservative system composed of a collection of interacting ions plus externally applied forces is considered. The mechanical problem of motion is defined for such a system, and the equilibrium and the translational and rotational invariance conditions discussed. It is shown that the second equilibrium condition of Born and Huang, namely that the stresses must vanish in the equilibrium configuration for the infinite lattice model, is not a requirement of the theory. The dynamical matrices, whose eigenvalues are simply related to the phonon frequencies, are shown to be of the same form for a homogeneously strained crystal as for an ideal unstrained crystal. The elastic constants of a homogeneously strained crystal are calculated by the method of homogeneous deformation, and also by the method of long waves. The quantities which are observed in the measurement of sound velocities in strained crystals are the effective elastic constants, which differ from the elastic constants by terms involving the stress components. These effective elastic constants are also calculated in the lattice theory, and their properties are discussed.