= Solution
<Inelastic neutron scattering> measures both <momentum> and <energy> exchanged with the <crystal>. Comparing incident and outgoing neutron <Wavevector>[wavevectors] gives $\mathbf Q=\mathbf k_i-\mathbf k_f$, while their <energy> difference gives a created or annihilated <Phonon>[phonon energy] $\hbar\omega$. <Crystal momentum> conservation identifies the reduced <Wavevector>[phonon wavevector] modulo a <Reciprocal lattice vector>[reciprocal-lattice vector]. Varying scattering geometry maps <Dispersion relation>[dispersion curves] and <Phonon polarization>[polarization-dependent] <Intensity>[intensities] across a substantial part of the <Brillouin zone>.
<Inelastic X-ray scattering> provides a complementary energy- and momentum-resolved probe, particularly when small samples or high-pressure environments favor photon beams. <Raman spectroscopy>[Raman] and <infrared spectroscopy> measure optical-frequency resonances subject to different symmetry selection rules. Photon <momentum> at these frequencies is usually small on the Brillouin-zone scale, so these methods mostly sample near the <Brillouin zone>[zone center] rather than an arbitrary <Wavevector>[phonon wavevector]. <Brillouin scattering> and <Ultrasound>[ultrasonic] measurements probe low-frequency acoustic motion and infer <Speed of sound>[sound velocities] or <Elastic modulus>[elastic constants].
<Heat capacity>[Heat-capacity] measurements constrain the thermally weighted <Phonon density of states>[vibrational density of states] and its low-frequency behavior, but do not by themselves reconstruct a unique wavevector-resolved dispersion. Elastic <X-ray diffraction>[diffraction] and temperature-dependent <displacement> factors also constrain mean-square motion, while energy-resolved inelastic measurements distinguish individual excitations directly. A monatomic scalar chain has only the <Acoustic phonon branch>[acoustic branch] derived below; it should not be assigned an <Optical phonon branch>[optical branch] merely because <Raman spectroscopy>[Raman] spectroscopy exists for more complicated <Crystal>[crystals].
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