Multiple Scattering

In Rutherford scattering experiments, a metal foil of finite thickness is bombarded by alpha particles. At what thickness of foil do you have to worry about multiple scattering? Some simple calculations will be explored here to suggest that with the gold foil thicknesses around 1 micrometer in the historical experiments, there was very little multiple scattering. Using the concept of cross section, the fraction of the particles scattered can be taken to be the effective area of the scatterer compared to the total area.

Gold, for example, has a nuclear radius about 7 fermis and an atomic radius of about 0.13 nm. The geometrical cross section of the nucleus is about 154 fm² or 1.54 barns. The cross sectional area of the atom is then about 3.45 x 10⁸ times the area of the nucleus. You would then expect that if you fired a neutron into a gold atom, the probability of hitting the nucleus would be about one in 345 million!

One simple approach to estimating the probability for multiple scattering it to suggest that for a given scattered particle, the remaining targets would have to cover 1% of the area for you to have a 1% probability of a second scattering event. Using the area ratio, you would project 3.45 x 10⁶ layers of atoms to cover 1% of the area with nuclei, assuming a random distribution. At a nominal distance between layers equal to the atomic diameter, this projects to about 900 micrometers in thickness.

For alpha particle scattering, the cross section is much larger than the geometric cross section because of the Coulomb force. For 6 MeV alpha particles, the cross section calculation for scattering above 30° gives 15730 fm² = 157.3 barns. The ratio of atomic area to this area is about 3.38 x 10⁶. Using the same simple approach, it would take about 3.38 x 10⁴ atomic layers to scatter 1%, which projects to a thickness of about 9 micrometers.

This simple approach suggests that the 1 micrometer thickness of the gold foils used in Rutherford experiments was thin enough to minimize concern about multiple scattering.