Finding the number of particles scattered by a certain angle

Question

I'm trying to do the problem below, but it seems like there is incomplete information.

PROBLEM STATEMENT: In a scattering experiment, $10^6$ $\alpha$ particles are scattered at an angle of $4^{\circ}$. Find the number of $\alpha$ particles scattered at an angle of 6$^{\circ}$.

The reason it seems incomplete is because the number of particles scattered at an angle of $6^{\circ}$ would be dependent on the "intensity profile" of the incident beam, right?

By intensity profile, I mean the following (see figure for reference): Imagine that there are no particles other than those that are coming in at distance $s$. In such a case, $\it all$ particles would be deflected by a certain amount, say $\Theta$, and no particles would be deflected by any other amount. Therefore, it seems like the number of particles deflected by any given angle is a direct function of the profile of incident beam, which is not given in the problem statement. To put things another way, the problem statement gives me the information that $10^6$ particles were located at distance $s$, assuming that $\Theta = 4^{\circ}$. It does not tell me how many particles were located at any distance other than $s$. I am assuming a classical picture, which is therefore completely deterministic.

Answer 1

The problem doesn't depend on the spatial distribution of the beam as long as the target is larger than the beam spot and uniform in areal density.

Why?

In the real world the target is a thin sheet of something¹ and you only care (to a first approximation) about the one nucleus the alpha comes closest to.

The nuclei of the target can be assumed to be uniformly distributed and beam variation happen on a much (much!) larger spacial scale² so the impact parameters ($s$) are effectively uniformly random as well.

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¹ Gold leaf in Rutherford's experiment.

² At JLAB we could measure beam intensity variations at the $10^{-5}$ meter level, but that is still huge compare to inter-atomic spacing. In Rutherford's experiment it would have depended on the size of the collimator.