# Conceptual meaning of differential cross section with respect to theta

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?

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 $$\theta=\pi/2$$.
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.