12.1: Cross sections and geometry

Most experiments in physics consist of sending one particle to collide with another, and looking at what comes out.

The quantity we can usually measure is the scattering cross section: by analogy with classical scattering of hard spheres, we assuming that scattering occurs if the particles ‘hit’ each other. The cross section is the apparent ‘target area’. The total scattering cross section can be determined by the reduction in intensity of a beam of particles passing through a region on ‘targets’, while the differential scattering cross section requires detecting the scattered particles at different angles.

We will use spherical polar coordinates, with the scattering potential located at the origin and the plane wave incident flux parallel to the $z$ direction. In this coordinate system, scattering processes are symmetric about $\phi$, so $\frac{d\sigma}{d\Omega}$ will be independent of $\phi$.

We will also use a purely classical concept, the impact parameter $b$ which is defined as the distance of the incident particle from the z-axis prior to scattering.

Figure $\PageIndex{1}$: Standard spherical coordinate geometry for scattering