A systematic derivation of the energy-flux operator for a three-dimensional lattice is given. The treatment is based on the general expressions for the energy flux which are valid for all phases of matter; a short derivation of these expressions, making no restrictions to two-body forces, is presented. The average energy flux is transformed to the phonon representation, and it is shown that the diagonal contribution from the harmonic forces has the familiar form $\Sigma {\mathrm{k}s}^{}{N}_{\mathrm{k}s}\hslash {\omega }_{\mathrm{k}s}{\mathrm{v}}_{\mathrm{k}s}$. There are, in addition, nondiagonal contributions to the energy flux, even in the harmonic approximation. The significance of these corrections is discussed. The contributions to the average flux from the anharmonic forces and from lattice imperfections are also treated. Finally, the problem of forming wave packets of the plane-wave normal modes to obtain an expression for the local energy flux is considered.

• Received 31 May 1963

DOI:https://doi.org/10.1103/PhysRev.132.168