Modelling Rutherford Scattering - in the school lab

[Grooze]

Hi,

New to the forum here! I'm teaching age 16-18 physics in the UK. I was wondering if anyone in here might have experience using equipment that looks like this:

https://www.eiscolabs.com/products/alpha-scattering

It's a physical analogue of Rutherford's famous experiment: roll ball bearings (representing alpha particles) down the slope, watch them get deflected by that cymbal-ish thing. Qualitatively, it's a lovely demonstration of the basic idea behind the result.

However, I was wondering if it might be possible to do a more mathematical treatment - has anyone ever tried to calculate the size of the 'nucleus' by counting backscattered particles? Intuitively I feel like we could get an upper limit on the size by just counting the fraction of particles that are backscattered over 90 degrees, but I haven't done the derivation yet, and I thought I might ask others first in case it's been done. I've looked at the original equation but it seems to have coulomb repulsion embedded in it, plus the fact that they had more than one nucleus - surely the maths will be simpler when it's just one object!

Thanks

 

[Charles Link]

See https://www.physicsforums.com/threa...a-rutherfords-experiment.965947/#post-6131309
The answer is the total cross section $\sigma_{total} \approx P_{scattered}/E=N \sigma_o$, where $N$ is the number of atoms, and $\sigma_o$ is the cross section of a single scatterer.
It needs to be determined what angle should be used to separate $P_{scattered}$ from $P_{unscattered}$, but I think that can normally done without too much ambiguity.
Note that for $P_{scattered}$, this is the number of scattered particles/second, and $E$ is the number of incident particles/m^2 per second. You can also do it without the time in the formulas and count the total particles.

 

[Grooze]

Thanks for this, really detailed. I suppose it makes more sense with ball bearings to just count totals and ignore time, and I suppose I can translate all the solid angles into 2d angles with this being a flat analogy.

 

[Charles Link]

In just 2-D, you would be measuring the width of the scatterer. Ideally you send in a stream of incident particles where the density is uniform and encompasses the target, e.g. $n=200$ particles/meter. If you observe scattered particles e.g.$N_{scattered}=20$, that means the width $w$ of your target is $w=N/n=.10$ meters.
It is a very similar calculation in 3-D, where the total area of the targets $\sigma_{total}=N_{scattered}/n_{incident \, per \, area}$.
Finally, the area of a single atom (scattering cross section) $\sigma_{atom}=\sigma_{total}/N_{total \, atoms}$.
It is somewhat arbitrary what angle $\theta$ you use for a cut-off to distinguish between scattered and unscattered particles, but it should be rather clear from looking at the data points what a practical value would be, in order to count as $N_{scattered}$.
See also https://en.wikipedia.org/wiki/Ruthe... is the elastic,and eventually the Bohr model.
For an introduction, the students really don't need the finer details, but the article provides some of those.