= 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 $E_n=\hbar^2k_n^2/(2m_n)$, so the measured transfers are
$$
\boxed{\mathbf Q=\mathbf k_i-\mathbf k_f,\qquad
\hbar\omega=\frac{\hbar^2}{2m_n}(k_i^2-k_f^2).}
$$
For <phonon> creation, momentum conservation reads $\mathbf Q=\mathbf G+\mathbf q$ and the neutron loses energy $\hbar\omega_{\nu}(\mathbf q)$. 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.
Back to article page
