Perpendicular neutron recoil and phonon absorption ($\hslash \omega ={\hslash }^2{Q}^2/(2m_n)$)

Define transfer gained by the neutron as $\mathbf{Q}={\mathbf{k}}_f-{\mathbf{k}}_i$. If it is perpendicular to ${\mathbf{k}}_i$, expansion of the free-neutron kinetic energy gives $E_f-E_i={\hslash }^2{Q}^2/(2m_n)$. Absorbing one phonon requires this positive energy to equal $\hslash \omega$. A periodic phonon branch may therefore intersect a nonperiodic recoil parabola in an extended-zone representation. The sign of $\mathbf{Q}$ is opposite to the common sample-transfer convention and must be declared.