 Solution

A neutron can exchange both energy and crystal momentum with a lattice vibration. Determine its incoming and outgoing wave vectors ${\mathbf{k}}_i,{\mathbf{k}}_f$ from neutron wavelengths and directions. Its nonrelativistic kinetic energy is $E_n={\hslash }^2{k}_n^2/(2m_n)$, so the measured transfers areFor phonon creation, momentum conservation reads $\mathbf{Q}=\mathbf{G}+\mathbf{q}$ and the neutron loses energy $\hslash {\omega }_{\nu }(\mathbf{q})$. For phonon annihilation the corresponding phonon energy is gained by the neutron. Reduce $\mathbf{Q}$ by a reciprocal lattice vector $\mathbf{G}$ to obtain the phonon wave vector, and trace a branch by repeating this measurement at several reduced wave vectors.

A triple-axis spectrometer provides a suitable arrangement: a monochromator selects the incident wavelength, the oriented sample defines the momentum transfer through the scattering angles, and an analyzer crystal selects the final wavelength before detection. A time-of-flight spectrometer instead uses pulse timing and flight paths to obtain the energy change. In either case instrumental energy and angular resolution must be separated from intrinsic line widths. Nuclear scattering from atomic displacements couples neutrons to phonons; the one-phonon intensity contains a polarization factor involving $\mathbf{Q}\cdot {\mathbf{e}}_{\nu }$, atomic scattering lengths, and thermal occupation factors. Intensities can therefore differ even at equivalent phonon frequencies.