Question 3.8: In neutron scattering experiments, neutrons can be thought o......
In neutron scattering experiments, neutrons can be thought of as hard objects consisting of three quarks that are roughly spherical in shape. Suppose that a high-speed neutron of radius r scatters off of an oxygen nucleus of radius R and that the differential scattering cross section for this reaction is dσ/dΩ = (r + R)^2/4. What is the total scattering cross section σ_s for the neutron?
The total scattering cross section σ is simply the differential scattering cross section dσ/dΩ integrated over all solid angles between 0 and 2π, or σ_s = \int_{0}^{2\pi}{d\sigma/d\Omega d\Omega } . Performing the required integration yields σ_s = π(r + R)^2. So, in this case, the total cross section is just the area of the circle where the neutron and the oxygen nucleus overlap, and the radius of this circle is R′ = r + R.