# Rutherford cross section in CM frame

1. Jan 11, 2013

### jocke_x1

Hi

I need to plot the (differential) Rutherford cross section as a function of the scattering angle in the CM frame. the reaction is a typical projectile on stationary target. I know the projectile energy in the lab, θcm and the masses. I am confused over which energy to use though.

I know that

$\frac{dσ}{dΩ}=\left(\frac{zZe^{2}}{16\pi\epsilon_{0}E}\right)^{2}*csc(\frac{\theta_{cm}}{2})^{4}$

In the following m and M are the masses of projectile and target respectivly and Tp is the projectile kinetic energy in the lab frame.

Should I use the projectile energy converted to CM?
$E=\left(\frac{M}{m+M}\right)^{2}*T_{p}$

The kinetic energy of the CM frame?
$E=\left(\frac{m}{m+M}\right)*T_{p}$

The total kinetic energy in the CM frame?
$E=\frac{\sqrt{2MT_{p}+\left(m+M\right)^{2}}}{m+M}$

or something else?

I have looked in several textbooks and googled but have just become more confused since I have seen E defined as "total kinetic enrgy","projectile energy" or just "energy of the particles".

Any help to clear this up would be greatly appreciated.