Problem: Simulation of incoherent scattering

We will now excercise the rule of weight transformations, as shown in the weight master equation, \(f_{\rm MC} w_j = P\), from the Monte Carlo simulation of neutron instrumentation page. First, consider a thin sample of an incoherent scatterer, area \(A\), thickness \(t\) with \(d\Sigma / d\Omega = \rho \sigma_\text{inc} / (4 \pi)\), where \(\rho\) is the number density per unit volume and \(d \Sigma / d \Omega\) is the differential scattering cross section per unit volume.

Question 1

Show from the equation for the differential scattering cross section on the Basics of neutron scattering page, that the scattering probability for a given neutron ray is \(P=\sigma_\text{inc} \rho t\). Calculate the value for V, which has \(\rho^{-1} = 13.77\) Å\(^{-3}\).

Question 2

In a simulation, we choose to focus the neutron rays into an area of \(\Delta \Omega\) in the following way: (a) pick a random direction inside \(\Delta \Omega\), (b) scatter all incident neutrons. Argue that the weight factor adjustment should be \(w = \rho \sigma_\text{inc} t \Delta \Omega / (4 \pi)\).

Question 3

For a general sample, the cross section per unit volume reads \((d \Sigma / d \Omega) ({\bf q})\). Argue that the weight factor adjustment will be \(w = (d\Sigma / d\Omega) ({\bf q}) t \Delta \Omega / (4 \pi)\), also if the cross section varies with \({\bf q}\) across \(\Delta \Omega\).