The total cross section for a system of particles

We have seen above that the total cross section can be written as a sum of the coherent and incoherent cross sections. In general, each of these cross sections can have an elastic and an inelastic part, giving rise to four terms:

\begin{equation}\label{dummy260518738} \frac{d^2\sigma}{d\Omega dE_{\rm f}} = \displaystyle\sum_j \frac{d\sigma_j}{d\Omega}\biggr|_{\rm inc} \delta(\hbar\omega) + \frac{d\sigma}{d\Omega}\biggr|_{\rm coh} \delta(\hbar\omega) + \displaystyle\sum_j \frac{d^2\sigma_j}{d\Omega dE_{\rm f}}\biggr|_{\rm inc} + \frac{d^2\sigma}{d\Omega dE_{\rm f}}\biggr|_{\rm coh} . \end{equation}

where each of the four terms will contain both a nuclear and a magnetic contribution. We will in the remainder of these notes concentrate upon the two coherent scattering processes, unless explicitly noticed.

Experimental considerations

The distinction between coherent and incoherent scattering is very important. In most types of experiment you will aim to minimize the incoherent cross section, which creates a uniform background, and maximize the coherent cross section, which generates the features you intend to study. A typical strong source of incoherent scattering is hydrogen, \(^1\)H, where the incoherence is due to a strong spin dependence of the interaction between the neutron and the proton.

Inelastic incoherent scattering can, however, be used to study dynamic processes; mostly the motion of hydrogen. This type of scattering is not discussed further in this version of the notes.