Conceptual meaning of differential cross section with respect to theta

Question

By doing a seminar paper about Compton scattering I came across two different differential cross-sections: $\frac{d\sigma}{d\Omega}$ and $\frac{d\sigma}{d\theta}$ where $\theta$ is the scattering angle. Comparison of the two is shown in the graph below (graph is made by myself so the labels on the y-axis might not be accurate). What I am looking for is a conceptual meaning of $\frac{d\sigma}{d\theta}$ (and the difference between the two). I would also like to know if there is a specific name/term used for $\frac{d\sigma}{d\theta}$ or is it also called differential cross section? $\frac{d\sigma}{d\theta}$ has two peaks, while $\frac{d\sigma}{d\Omega}$ has a minimum. What is the interpretation of this graph?

Answer 1

I'll start with a quick reminder of the relevant variables. In spherical coordinates, you have $d\Omega = \sin\theta d\theta d\phi$. For the case of Compton scattering, you have cylindrical symmetry, so you can integrate out the $\phi$ dependence which is constant. This means that the two quantities that you are plotted are related by: $$ \frac{d\sigma}{d\Omega} = \frac{1}{2\pi\sin\theta}\frac{d\sigma}{d\theta} $$ This is consistent with the graph as the $\frac{d\sigma}{d\theta}$ curve goes to $0$ at $\theta=0,\pi$ due to the extra $\sin\theta$ factor, and gains a rough factor of $6$ at $\theta=\pi/2$.

In general, the term differential cross section could refer to any differential with respect to different quantities (energy, momentum etc.) so it still applies here.

For the actual interpretation of the graph, the $\frac{d\sigma}{d\Omega}$ is more transparent physically since an isotropic scattering would correspond to a flat curve. The two peaks corresponds to the cases where the particle is either unimpeded $\theta=0$, either is backscattered $\theta=\pi$, which tend to dominate.

Hope this helps, and tell me is you find any mistakes.