= Solution
For <photon absorption> or emission, <electron> <energy> changes by $\pm\hbar\omega$ and crystal <wavevector> by $\pm\boldsymbol q$ modulo a reciprocal lattice vector. Optical <photon> momentum is normally tiny on the <Brillouin zone> scale, so interband optical transitions are approximately vertical on an energy-wavevector diagram. A direct-gap <semiconductor> can therefore absorb a <photon> between valence and conduction states near the same <wavevector>; an indirect transition usually needs an additional <phonon> or another source of momentum. Occupation and optical matrix elements impose further restrictions.
Within one ideal parabolic band of a clean metal, such a small <photon> momentum cannot normally supply the required finite <energy> difference. Intraband optical absorption then relies on momentum relaxation by impurities, phonons or boundaries, as in the finite-scattering-time Drude response. Interband transitions can still occur. The first panel shows the near-vertical direct transition.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2008-electron-scattering-kinematics.png]
{title=Energy and momentum transfer in a direct optical transition, acoustic-phonon emission and optical-phonon emission}
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