# Coulomb Potential Scattering

### From FSUPhysicsWiki

We now consider the scattering of an electron from the Coulomb potential. This problem is important because it is relevant to the famous scattering experiment by Rutherford that showed that the atomic nucleus only makes up a very small fraction of the total size of an atom. We will first use the Born approximation to find the cross section for a Yukawa potential,

which reduces to the Coulomb potential when We will then find the exact cross section for the Coulomb potential, starting with the Schrödinger equation.

## Scattering From a Yukawa Potential in the Born Approximation

As alluded to earlier, let us first consider the Yukawa potential,

Within the Born approximation, the scattering amplitude is

For the elastic scattering . Therefore,

and thus

The differential cross section is then

## Exact Coulomb Scattering Cross Section

When we are considering scattering due to the Coulomb potential, we can not neglect the effect of this potential at large distances because it is only a potential.

We will work in parabolic coordinates, which are related to Cartesian coordinates by

The Schrödinger equation in parabolic coordinates is

Recall that, for a spherically symmetric potential, the scattering amplitude is a function of only; therefore, we will seek solutions that are independent of

We will look for solution of the form,

The Schrödinger equation then becomes

We now assume a series solution,

The recursion relation for the coefficients is then

where

Recall that the confluent hypergeometric function _{1}*F*_{1} is given by

The recursion formula for its coefficients is

Comparing this to what we obtained earlier, we find that Φ(ξ) is

where is a c-number.

The full wave function is thus

In the limit, the confluent hypergeometric function is approximately

We may use this to rewrite in the limit of large ξ as

In the same limit, the full wave function is

where is

The differential cross section is thus

Note that this is the same result that we would obtain in the limit of the Yukawa potential using the Born approximation, as well as from a classical calculation.

## The Gamow Factor

We will now determine the relative probability of finding a particle at the origin to that of finding a particle in the incident beam, simply given by

Since the probability density at the origin is just

At large distances, on the other hand, the probability density is

Using the fact that

this becomes

We therefore obtain

If we now define the velocity,

then this becomes

The quantity that appears in the exponential is known as the Gamow factor. The Gamow factor (or Gamow-Sommerfeld factor), named after its discoverer George Gamow, is measure of the probability that two nuclear particles will overcome the Coulomb barrier in order to undergo nuclear reactions, such as nuclear fusion. Classically, there is almost no possibility for protons to fuse by overcoming the Coulomb barrier, but, when George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.