Scattering from nuclear dynamics

The calculations the lead to the description of the inelastic nuclear scattering are of quantum mechanical nature and are specified in Scattering theory for nuclear dynamics. These calculations lead directly to the observable scattering cross section, which covers both elastic and inelastic nuclear scattering: \begin{align} \label{eq:inel_final_nuc} %\lefteqn{\frac{d^2\sigma}{d\Omega dE_{\rm f}}} &= %ToDo: Make this boxed %\boxed{ \frac{d^2\sigma}{d\Omega dE_{\rm f}} &= \frac{k_{\rm f}}{k_{\rm i}} \sum_{j,j'} \frac{b_j b_{j'}}{2 \pi \hbar} \\ &\quad \times \int_{-\infty}^{\infty} \big\langle \exp(-i {\bf q} \cdot {\bf R}_j(0)) \exp(i {\bf q} \cdot {\bf R}_{j'}(t)) \big\rangle \exp(-i\omega t) \, dt \nonumber . %} \, . \end{align} It should be noted that this is a rather general result, valid for any kind of nuclear motion. The result can (and will often be) interpreted semiclassically, in the sense that \({\bf R}_j(t)\) is viewed as the classical position of the atoms or molecules under study.

In Scattering from lattice vibrations, we will describe the scattering resulting from vibrations of nuclei around their equilibrium positions in a lattice. We can, however, already see now that since the nuclear position enter (\ref{eq:inel_final_nuc}), we will need to make a series expansion of the complex exponentials in order to describe the effects of the small movements.