Correction factor for my formula obtained by Rutherford scattering results

Question

I did an experiment on Rutherford scattering , finding the number of counts at an angle $\theta$. The problem is that I had a single sensor i.e. I measured the counts at a single position , and the scattering occurs over a cone. I was thinking of integrating my obtained function, $f(\theta)$ over the area of a hoop and so obtaining :

$N(\theta) = \int_{0}^{2\pi}f(\theta)d\phi$

Where $d\phi$ is the angle subtended by part of the hoop. Therefore I get just

$N(\theta) = 2\pi f(\theta)$

Because surely

$f(\theta) = \frac{N(\theta)}{d\phi}$

Answer 1

The angle $d\phi$ that your detector subtends (out of a possible $2\pi$) changes with $r$; if the circumference of the "hoop" is $2\pi r \sin\theta$, and the dimension of the detector perpendicular to the $\theta$ direction is $d$, then

$$d\phi = \frac{d}{2\pi R \sin\theta}$$

where $R$ is the distance to the detector.

After that, your argument works - so the fraction $F$ of counts you observe is

$$F = \frac{d}{2\pi R\sin\theta}$$

and the conversion from $f(\theta)$ to $N(\theta)$ is

$$N(\theta) = \frac{f(\theta)}{F}=\frac{2\pi R\sin\theta}{d}f(\theta)$$

The term $\frac{2\pi R}{d}$ is something that will be constant for the apparatus; you are left with a $\sin\theta$ scaling term. The approximation will obviously break down when the detector dimension gets large compared to the size of the cone... i.e. for very small values of $\theta$ you have to be careful.