= Solution
In <inelastic scattering>, the probe transfers <energy> as well as <momentum> to the specimen. A neutron, X-ray or <electron> can create or annihilate a <phonon>, while electronic or magnetic excitations provide other channels. Measuring the incident and outgoing energies resolves the transfer $\hbar\omega$; their wavevectors determine <momentum> transfer $\hbar\mathbf Q$. In a crystal, phonon <momentum> is conserved modulo a <reciprocal lattice> vector. Resolved spectra therefore probe excitation dispersions and lifetimes, rather than only average atomic positions.
<Thermal diffuse scattering> arises from thermally populated lattice displacements and their correlations. The equilibrium lattice produces sharp Bragg peaks, but displacements redistribute part of the scattering into diffuse intensity. To one-phonon order a mode contribution contains a displacement projection and thermal occupation factors of the form
$$
I_\nu(\mathbf Q)\propto\frac{|\mathbf Q\cdot\mathbf e_{\mathbf q\nu}|^2}{\omega_{\mathbf q\nu}}
\left[n_B(\omega_{\mathbf q\nu})+\frac12\right],
$$
with atomic scattering weights, phases and geometrical factors understood. Stokes and anti-Stokes energy-resolved components carry $n_B+1$ and $n_B$ respectively; the displayed combination describes their summed scale. At high <temperature>, $n_B\simeq k_BT/(\hbar\omega)$, so soft modes and their <polarization> strongly influence the pattern. Zero-point motion also contributes at low <temperature>. An energy-integrating <diffraction> image can record this inelastic intensity as a diffuse background without resolving its small <energy> transfers. Static disorder gives additional diffuse scattering, so diffuse intensity alone does not establish a thermal mechanism.
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