# Cumulative distribution function for Compton scattering

Just for reference, this is the cross section for Compton scattering.

$$\frac{d\sigma}{d\Omega} \propto\left(\frac{E'}{E}\right)^2\left(\frac{E'}{E}+\frac{E}{E'}-\sin^2\theta\right)$$

Currently I am integrating $$d\sigma$$ numerically to obtain scattering probabilities per angular bin. However, numerical integration would not be necessary if I had a cumulative distribution function.

Is there a closed form cumulative distribution function for Compton scattering?

Noting that $$d\Omega=\sin\theta\,d\theta\,d\phi$$, we have $$d\sigma\propto \left(\frac{E'}{E}\right)^2 \left(\frac{E'}{E}+\frac{E}{E'}-\sin^2\theta\right)\sin\theta \,d\theta\,d\phi$$
Let $$g(\theta)$$ be the following antiderivative. $$g'(\theta)= \left(\frac{E'}{E}\right)^2 \left(\frac{E'}{E}+\frac{E}{E'}-\sin^2\theta\right)\sin\theta$$
Then $$\begin{multline*} g(\theta)=-\frac{\cos\theta}{R^2} +\log\bigl(1+R(1-\cos\theta)\bigr)\left(\frac{1}{R}-\frac{2}{R^2}-\frac{2}{R^3}\right) \\ {}-\frac{1}{2R\bigl(1+R(1-\cos\theta)\bigr)^2} +\frac{1}{1+R(1-\cos\theta)}\left(-\frac{2}{R^2}-\frac{1}{R^3}\right) \end{multline*}$$
where $$\begin{equation*} R=\frac{E}{mc^2} \end{equation*}$$
Then the cumulative distribution function $$F(\theta)$$ is $$F(\theta)=\frac{g(\theta)-g(0)}{g(\pi)-g(0)},\quad0\le\theta\le\pi$$