X-ray/neutron scattering from crystals and liquids is well-described by the scattering theory to give the (dynamic) structure factor which is a function of momentum and energy:
$$S(\mathbf{k},\omega)= \int d\omega \, e^{i \omega t}\langle \hat{\rho}^{\dagger}(\mathbf{k}, t) \hat{\rho}(\mathbf{k}, 0)\rangle$$
Often, the energy-dependence is ignored (because you can't measure it) and one says that they measure the (static) structure factor
$$S(\mathbf{k}) = \int d\omega \,S(\mathbf{k},\omega)$$
From here, people model the temperature dependence of the structure factor by (quoting wikipedia)
$$S(\mathbf{k}, T\neq0) = S(\mathbf{k}) \, \langle\textrm{exp}(i\mathbf{k}\cdot \mathbf{u}) \rangle^2 \sim S(\mathbf{k}) \textrm{exp}(-k^2 \langle u^2 \rangle)$$
Where the term $\langle\textrm{exp}(i\mathbf{q}\cdot \mathbf{u}) \rangle^2 $ is the Debye-Waller factor that is temperature-dependent and comes from atoms moving randomly at finite temperature ($u$ is the average displacement which depend on temperature).
But, because the number of atoms hasn't changed, the total amount of scattering is conserved, and we need the following relation
$$\int d\mathbf{k}\, S(\mathbf{k}) \textrm{exp}(-k^2 \langle u^2 \rangle)= \int d\mathbf{k}\, S(\mathbf{k}) = \int d\mathbf{k} \,d\omega\, S(\mathbf{k},\omega)$$
The introduction of the Debye-Waller factor however, causes the integral to decrease, and it is commonly said that the rest of the integral comes from inelastic scattering that has a $1- \textrm{exp}(-k^2 \langle u^2 \rangle)$ dependence to exactly balance things.
My question is, where is the energy going? Specifically, where exactly in $(\mathbf{k},\omega)$-space is it located?
For example, is it going to increased Compton scattering off of the atoms, into more phonons, or both? Is it going to high or low $k$,$\omega$? Is there anisotropy? I'm having trouble locating a general answer to this, but I am sure it should have been answered decades ago. Ideally I would like to see how these processes give back the missing $1- \textrm{exp}(-k^2 \langle u^2 \rangle)$ term