# How can I calculate scattering cross section of Yukawa potential classically?

https://en.wikipedia.org/wiki/Yukawa_interaction#Classical_potential

Here is classical form of Yukawa potential.

I want to calculate classical scattering cross section of this potential analytically(theoretically).

Is there any analytic solution of this?

Thank you.

I suspect that an analytical solution doesn't exist. The usual technique for computing classical scattering cross-section (at a fixed energy $$E$$) involves first calculating the scattering angle $$\Theta$$ as a function of the impact parameter $$b$$. This is done by calculating the integral $$\Theta = \pi - 2 \int_{\rho}^\infty \frac{(b/r^2)\, dr}{\sqrt{1 - (b^2/r^2) - U(r)/E}},$$ where $$U(r)$$ is the potential and $$\rho$$ is defined as the distance of closest approach: $$\frac{U(\rho)}{E} + \frac{b^2}{\rho^2} = 1.$$ The integral above is ugly enough for a power-law potential like $$U(r) \propto r^n$$; to the best of my knowledge it can only be evaluated for certain values of $$n$$. The prospect of dropping $$U(r) = \alpha e^{-\mu r}/r$$ into that integral and trying to evaluate it does not fill me with joy; and I'm pretty sure that the equation defining $$\rho$$ is a transcendental equation (i.e., no closed-form expression) in this case.