= Solution
For annihilation of a <phonon> $(\mathbf q,\omega)$,
$$
\boxed{E_f=E_i+\hbar\omega,\quad\mathbf k_f=\mathbf k_i+\mathbf q+\mathbf G.}
$$
For <phonon> creation,
$$
\boxed{E_f=E_i-\hbar\omega,\quad\mathbf k_f=\mathbf k_i-\mathbf q+\mathbf G.}
$$
The signs follow whether the <neutron> gains or loses the <phonon>'s <energy> and <crystal momentum>. These are one-phonon coherent <inelastic neutron scattering> conditions, with $\mathbf G$ accounting for lattice periodicity.
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