Problem: The cross section

Imagine a beam of neutrons arriving randomly over a surface of area \(A\) perpendicular to the beam, with an arrival rate of \(N\) neutrons per second. In a semi-classical approximation, you can consider each neutron to be point shaped. Now, on the surface we place one nucleus with an effective radius of \(2b\). Assume that each neutron hitting the nucleus is scattered and all other neutrons are left unscattered.

Question 1

Calculate the neutron flux.

Solution

The flux is \( \Psi = N / A \).

Question 2

Calculate the probability for one neutron to hit the nucleus.

Hint

Consider the area of the nucleus (\(a\)) versus the area of the beam (\(A\)).

Solution

The area of the nucleus perpendicular to the beam is \( a = 4 \pi b^2 \). The probability for hitting is thus \(p = a/A = 4\pi b^2 / A\).

Question 3

Show that the scattering cross section of the nucleus is \(\sigma = 4 \pi b^2\).

Solution

The number of neutrons scattered per second must (in average) be \(n_{\rm scatt} = N p\). Using the definition for cross section we reach \(\sigma = n_{\rm scatt} / \Psi = 4 \pi b^2 \).