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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!

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    $\begingroup$ Yes this is Rutherford Scattering. You can first try and find a relation between the impact factor and the scattering angle. $\endgroup$
    – Manvendra Somvanshi
    Oct 26, 2019 at 9:16
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    $\begingroup$ 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. $\endgroup$
    – Jon Custer
    Oct 26, 2019 at 15:53

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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.

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