Python Exercise

Possible Python Scattering Exercise

This exercise is designed to introduce students some computer skills that they probably haven't seen much before: reading and parsing of large data files, and binning of data. It is also designed to give students a stronger sense of how the theoretically derived diffential cross-section is connected in a straightorward way to data from scattering experiments. It also is meant to reinforce basic notions of solid angle.

Consider three runs with different energies in which alpha particles are incident on gold foil. The data from each run consists of 500,000 data points, giving the individual scattering angles, (\(\phi\), \(\theta\)).

Run #1 (E = 7.75 MeV)
Run #2 (E = 28.182 MeV
Run# 3 (E = 46.5 MeV)

The students are given the luminosity of the beam of alpha particles, the density and thickness of the gold foil, and the area of the incident beam. I'm envisioning two basic questions/problems:

Show that Run #1 is consistent with results predicted by the Rutherford differential cross-section.
Estimate the size of the nucleus.

In analyzing the data, students can make the following assumptions:

Classical scattering
Gold nuclei are heavy enough to be consdered fixed in position
Electrons are light enough that they play no role in scattering

These assumptions just say that the the results should be governed by the usual Rutherford scattering differential cross-section if the alpha particles don't penetrate the nucleus. The "data set" has been fabricated with a realisitic nuclear radius, and an extremely simple model of a uniformly charged nucleus.

Here are some results for the three runs in which the observed number of counts in bins of width \(\pi/22\) are plotted vs. \(\theta\), along with the results predicted by the Rutherford differential cross-section. The two plots for each "run" differ only in the vertical scales. The "B" plots allow closer inspection of the lower number of counts at large scattering angles.

Run #1 Plot A, Plot B
\(E = 7.75\, \mbox{MeV}\) gets "head-on" alpha particles to about \(4\times R_\text{nucleus}\); Rutherford scattering model works well.
Run #2 Plot A, Plot B
\(E = 28.182\, \mbox{MeV}\) gets "head-on" alpha particles to about \(1.2\times R_\text{nucleus}\), so Rutherford scattering model still works well. At this higher energy, there are fewer large angle scattering events.
Run #3 Plot A, Plot B
\(E = 46.5\, \mbox{MeV}\) gets "head-on" alpha particles well into nucleus Looks like Rutherford scattering, until you look closely at larger angles. There is a complete absence of scattering events above about 89^o, and a slight excess below this value. From this, you can deduce the nuclear radius.

Simple Python script to bin data: binData.py