# $\alpha$-particles scattering at Al foil

I am currently reading myself into the topic of scattering experiments and (differential cross sections) and stumbled across the following problem which I really dont know how to solve (this is $$\textbf{not}$$ any kind of homework assignement, I just want to solve this problem :D).

An $$\alpha$$-particle with energy $$E_\alpha=7\,$$MeV scatters at an Al-foil. Now I have to show that the integrated cross section $$\sigma_{int}(\theta,180^\circ)=\int_0^{2\pi}\int_\theta^{\pi}\frac{d\sigma}{d\Omega}\sin\theta'd\theta'd\phi$$ for a particle thats scattered in the range $$[\theta,180^\circ]$$ is given by

$$\sigma_{int}=\pi\left(\frac{Z_{Al}Z_\alpha\hbar c}{2E_\alpha}\right)^2 \cot^2\left(\frac{\theta}{2}\right).$$

To start things of I dont even know how to calculate the differential cross section in this case? Is this Rutherford-scattering?

Any help or advice is very much appreciated!

• Yes this is Rutherford Scattering. You can first try and find a relation between the impact factor and the scattering angle. Oct 26 '19 at 9:16
• To be clear, any non-relativistic Coulomb-driven scattering is Rutherford. It goes even further, since Rutherford derived the scattering cross section for a $1/r^{2}$ potential, so one can apply it, with suitable prefactors, to gravitational interactions as well. Oct 26 '19 at 15:53

## 1 Answer

I will not provide a complete derivation but I'll give you the key steps involved. As I said in my comment you should first try to find the relation between impact factor and scattering angle. That will be $$b \propto \cot(\theta/2)$$ To obtain the result you'll have to apply conservation of angular momentum and Newton's second law.

Next you can find the differential cross section: $$\frac{d\sigma}{d\Omega} = \frac{b}{\sin(\theta)}{\frac{db}{d\theta}}$$

Hope this helps.