= Solution
A <phonon> is one quantum of a collective harmonic <normal mode> of a crystal. A mode labelled by <wavevector> and branch has energies $E_m=\hbar\omega(m+1/2)$, with bosonic occupation statistics. The <crystal momentum> $\hbar\mathbf k$ is defined modulo a <reciprocal lattice vector>; it is not generally the mechanical momentum of one atom. The <phonon dispersion relation> assigns its <frequency> and <polarization> to the <wavevector> and branch.
<Inelastic neutron scattering> can measure this relation in a single crystal. Determine the incident neutron <energy> and <wavevector>, scatter at a chosen angle, and analyze the outgoing <energy> by time of flight or a spectrometer analyzer. <Energy> and crystal-momentum conservation give
$$
E_i-E_f=\pm\hbar\omega_s(\mathbf k),\qquad
\mathbf k_i-\mathbf k_f=\mathbf G\pm\mathbf k.
$$
The signs distinguish <phonon> creation and annihilation with consistent momentum convention. Repeating measurements at different momentum transfers maps peaks in the dynamic scattering response into <phonon> branches. The measured intensity depends on occupation, scattering factors and the projection of the mode <polarization> along the momentum transfer; a weak or missing peak need not mean that the corresponding mode does not exist.
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