Rutherford scattering problem

[natugnaro]

Homework Statement

Alpha particle with kinetic energy 6MeV is scatterd on gold nucleus (A=197).
Is Rutherford cross-section formula valid for this scattering? Explain !

Homework Equations

The Attempt at a Solution

The Coulomb barrier is Ec=(1.03MeV)*(Z1*Z2)/(A1^1/3+A2^1/3)
Z1=79 , Z2=2 , A1=197 , A2=4
Ec=1.03MeV*158/7.41=22MeV
The 6 Mev alpha particle will not go through the barrier, forces between alpha particle and gold nucleus will be coulombic so I think it is valid to use Rutherford cross-section.
But I found that Rutherford cross-section formula is valid if
$$Z1*Z2*\alpha<<1$$ is true.Since for Z1=79,Z2=2,alpha=1/137 this is not <<1 ,
so rutherford cross-section is not valid.
Where I'm going wrong?

 

[Jhero]

Thank you for your question. It is important to consider the conditions under which the Rutherford scattering formula is valid. The formula is based on the assumption that the scattering is due to a point-like Coulomb potential between the alpha particle and the gold nucleus. This means that the size of the gold nucleus is negligible compared to the distance between the alpha particle and the nucleus. In other words, the alpha particle must be scattered at large angles (close to 180 degrees) for the formula to be valid.

In your case, the Coulomb barrier energy of 22 MeV is much higher than the kinetic energy of the alpha particle, which suggests that the scattering angle will be small. In this regime, the Rutherford formula is not accurate and other factors, such as the finite size of the gold nucleus, must be taken into account.

Additionally, as you correctly pointed out, the formula is only valid if the parameter Z1*Z2*\alpha is much smaller than 1. In your case, this parameter is equal to 0.003, which is small enough to satisfy the condition. However, it is important to note that this parameter alone does not determine the validity of the formula. The condition for small scattering angles is also crucial.

In conclusion, for your specific case, it is not appropriate to use the Rutherford scattering formula. Other factors, such as the finite size of the gold nucleus and the small scattering angle, must be taken into account for a more accurate calculation of the scattering cross-section. I hope this helps clarify the issue. Please let me know if you have any further questions.