# Coherent inelastic single-phonon scattering from a powder sample -- Aluminum

The scattering intensity is given by \begin{aligned} {\left(\frac{d^2\sigma}{d\Omega dE_f}\right)}_{inc\pm 1} = & \frac{\sigma_{coh}}{4\pi} \frac{k_f}{k_i} \frac{(2\pi)^3}{v_0} \exp(-2W) \\ & \times \sum_s \sum_{\mathbf{\tau}} \frac{ \hbar^2 ( \mathbf{Q} \cdot \mathbf{e}_s )^2 }{2M\;E_s} \frac{1}{2} \left\{ \coth\left(\frac{\hbar \omega}{2k_B T}\right) \pm 1 \right\} \delta(E- E_s) \; \delta(\mathbf{Q} - \mathbf{q} - \mathbf{\tau}) \end{aligned} where $\mathbf{e}_s$ is the polarization of the phonon mode, $E_s$ is the energy of the phonon mode, $\mathbf{\tau}$ is a reciprocal lattice vector. The total cross section for a phonon mode of energy $E$ at $\mathbf{Q}$ can be deduced as $$\sigma_{\mathbf{Q}} = \frac{\sigma_{coh}}{4\pi} \frac{k_f}{k_i} \frac{\left( 2\pi \right)^3}{v_0} \exp(-2W) \frac{\hbar^2 ({\mathbf{Q}}\cdot {\mathbf{e}})^2}{2M E} \frac{1}{2} \left\{ coth\left(\frac{\hbar \omega}{2k_B T}\right) \pm 1 \right \} \frac{1}{2 k_i k_f Q}$$ The main input for this kernel is the energies and polarization vectors of phonon modes in a brillouin zone.

## MCViNE simulation results

Coherent scattering give the scattering spectra more features.

The experimental result is given in panel (a) and panels (b)-(e) show simulated data. In (b) only the incoherent elastic and incoherent single phonon scattering are included. This is the only plot with different, much lower maximum intensity -- it is scaled by the ratio of incoherent/coherent cross sections of aluminum, otherwise the intensities in this plot are barely visible. In (c) only the coherent elastic (powder diffraction) and the coherent single-phonon inelastic scattering are included. In (d), all of the kernels in (b) and (c) with the addition of a multi-phonon kernel using the incoherent approximation. In (e), all of the kernels in (d) are used with multiple scattering turned on. Comparison of (b) and (c) shows that coherent scattering gives rise to more features such as diffraction peaks and phonon dispersions. It is evident from comparing (c) and (d) that multiphonon scattering increases in intensity at higher Q. The most obvious difference in (d) and (e) is in the elastic line which shows that multiple scattering seems to contribute similarly to incoherent elastic scattering. The elastic line in (a) and (e) seems to show that the sample used in the experiment may contain traces of an additional phase, most likely from a surface layer of Al$_2$O$_3$.