A continuum model for acoustic and optical phonons in heterostructures

A continuum description of dispersive acoustic and optical modes in a cubic, elastically isotropic and inhomogeneous binary crystal is given, based on a simple model for the Lagrangian density. Putting the total Hamiltonian in the form of a sum over classical oscillators which can be quantized in the usual way, gives rise to continuity conditions at an interface which are the standard ones for acoustic modes but which highlights the effect of the discontinuity of mass across the boundary for optical modes. Unless this discontinuity is very small the general boundary condition for optical modes is u=0.