Anharmonic Contribution to Momentum-Flux Operator for a Lattice

From the general expression for the momentum-flux or pressure tensor, valid for all phases of matter, the momentum-flux operator is derived for a lattice through the cubic anharmonic contribution. The result is compared with that of classical elasticity. The comparison is exact, even including nondiagonal contributions, and provides an alternative derivation for the second- and third-order elastic constants in terms of lattice constants. Finally, in the phonon representation it is shown that the diagonal part of the pressure tensor reduces to the simple form ($\frac{1}{V}$) ${\Sigma }_{\mathrm{k}s}\hslash {\omega }_{\mathrm{k}s}(N_{\mathrm{k}s}+\frac{1}{2}){{\gamma }_{\mathrm{k}s}}^{\mathrm{ij}}$, where the ${{\gamma }_{\mathrm{k}s}}^{\mathrm{ij}}$ are defined as generalized Grüneisen parameters.