In a diffraction experiment, most of the scattering is elastic scattering where the wavelength of the incident wave is the same as the wavelength of the scattered wave, $|\vec{k}|=|\vec{k}'|$. In inelastic phonon scattering, some of the energy and momentum of the incoming wave is transferred to or from a phonon. The conservation of momentum in this case is,

$$\hbar\vec{k} = \hbar\vec{k}' +\hbar\vec{G} \pm\hbar\vec{K}_{ph}.$$

Here $\hbar\vec{k}$ is the momentum of the incoming wave, $\hbar\vec{k}'$ is the momentum of the scattered wave, $\hbar\vec{G}$ is the momentum that has been transferred to the lattice, and $\hbar\vec{K}_{ph}$ is the momentum of a phonon. The phonon can either be emitted $+\hbar\vec{K}_{ph}$ or absorbed $-\hbar\vec{K}_{ph}$. Inelastic scattering causes weak diffraction peaks near the main diffraction peaks, called satellite peaks. Assuming that $\vec{G}$ has already been determined by analyzing the diffraction peaks that occur due to elastic scattering, the phonon wave vector is,

$$\hbar\vec{K}_{ph}= \hbar\vec{k}' - \hbar\vec{k} - \hbar\vec{G}.$$

The frequency of the phonon can be determined by the conservation of energy. Inelastic phonon scattering can be performed with x-rays or with neutrons. If neutrons are used, the energy of the incident neutrons is $\frac{\hbar^2k^2}{2m_n}$ and the energy of the scattered neutrons is $\frac{\hbar^2(k')^2}{2m_n}$, where $m_n$ is the mass of a neutron. The energy of the phonon is,

$$\hbar\omega_{ph} = \frac{\hbar^2(k')^2}{2m_n} - \frac{\hbar^2k^2}{2m_n}.$$

If the inelastic phonon scattering is performed with x-rays, the energy of the phonon is,

$$\hbar\omega_{ph} = \hbar ck' - \hbar ck, $$

where $c$ is the speed of light.

Each satellite peak that can be detected provides one experimental point on the $\omega \text{ vs. }k$ phonon dispersion relation.

These are expensive experiments because a neutron source with sufficient flux for this experiment requires a nuclear reactor, and an x-ray source with sufficient energy resolution to perform this experiment requires a synchrotron.