I am trying to understand the dependence of a differential cross-section on $$\sigma(\theta) = \left(\frac{s}{ \sin \theta}\right)\left|\frac{ds}{d\theta}\right|,$$ where $s$ is the impact parameter and $\theta$ is an angle that is between the scattered and incident directions.
Any explanation as to what is going on would be a big help.
I can't give a complete answer but I'm sure someone will. For a low energy projectile, lower than the first excited state of the target, where the projectile is neutrons, this is just the elastic or inelastic scattering cross-section. The neutron scatters like a billiard ball off the stationary nucleus. Applying conservation of momentum and conservation of energy you can calculate the cross-section. As the energy increases more channels open. You can excite the nucleus into a higher energy level which will decrease the energy of the outgoing neutron. Other scattering mechanisms become possible with increase in energy.
