Physical momentum versus crystal momentum of acoustic phonons in a crystal lattice

From the statistical mechanics together with the kinetic theory of the pressure of phonons in a crystal we have established the general relation of ${\mathit{p}}_{\mathit{q}}=n\gamma \hslash \mathit{q}$ for the physical momentum (or anharmonic momentjm) of an acoustic phonon of wave vector $\mathit{q}$ in $n$ dimensions, $\gamma$ being the Grüneisen constant. This is only valid under the condition of broken translational symmetry. For then the anharmonic interaction would enable a $\mathit{q}\ne 0$ phonon to couple with and hence acquire a mate of acoustic phonon of $\mathit{q}=0$, resulting in a composite of the same $\mathit{q}$. This composite can then carry a physical momentum relative to the wall confining the crystalline medium via its momentum-carrying, $\mathit{q}=0$ mate. On the other hand, if translational invariance is maintained, a $\mathit{q}\ne 0$ phonon is incapable of any physical momentum relative to the inertial wall frame, with or without anharmonic coupling; it can only carry the more familiar “crystal momentum” which is the kind that is transported like sound intensity within and hence relative to the equilibrium crystalline medium. An elementary, semiclassical derivation of the “crystal momentum” is presented and discussed versus the “anharmonic momentum.” Some applications of the latter are presented.