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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.

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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.

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    $\begingroup$ Thank you for your answer. I think i should try numerically with your equation. $\endgroup$ – Exciting_Squid Oct 20 '18 at 15:59

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