Rutherford cross section in CM frame

[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, θ_cm 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 T_p 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.

 

[mark726]

Hi there,

Thank you for your question. The energy that you should use in the Rutherford cross section formula depends on the specific scenario that you are studying. Let's go through each of the options that you have listed and discuss when it would be appropriate to use them.

1. Projectile energy converted to CM: This option would be appropriate if you are interested in studying the scattering of the projectile particles in the CM frame. This would be useful if you want to analyze the scattering process from the perspective of the projectile particle.

2. Kinetic energy of the CM frame: This option would be appropriate if you want to study the scattering process from the perspective of the CM frame. This would be useful if you want to analyze the overall behavior of the scattering process without focusing on the individual particles involved.

3. Total kinetic energy in the CM frame: This option would be appropriate if you are interested in studying the overall energy of the system, including both the projectile and target particles. This would be useful if you want to understand how the total energy of the system affects the scattering behavior.

In general, the choice of energy to use in the Rutherford cross section formula will depend on your specific research question and the perspective from which you want to analyze the scattering process. I recommend discussing this with your advisor or colleagues to determine the most appropriate energy to use for your particular scenario.

I hope this helps clarify things for you. Good luck with your research!