Estimating the size of an atomic nucleus
This is a slightly simplified approach
If Rutherford’s scattering rule works then we know that the only force acting on the alpha is the electrostatic repulsion of the nucleus.
The electrostatic force depends on distance. The distance is measured from the centre of the nucleus.
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We can work out the closest an alpha particle gets to the centre of the nucleus before coming back. We know that the nucleus can’t be bigger than this distance (assuming we can ignore the size of the alpha).
The energy of the alpha particle depends on the isotope. We can choose the energy of the alphas we use but only from the isotopes that actually exist. We want to use the highest energy alpha available where Rutherford’s scattering rule still holds. This alpha gets as close as you can get to the edge of the nucleus without hitting it. The ‘actual’ size will always be smaller than this.
Animations explaining how you generally get better estimates for the size of the nucleus, the higher the energy of the alpha.The calculation uses the conservation of energy
When the alpha particle is a long way from the nucleus we can think of it having a certain kinetic energy. When the alpha particle is as close to the nucleus as it's going to get then we assume that it's stopped so all its kinetic energy has been stored in the electric field between the alpha and the nucleus.
Animations explaining how to calculate an upper limit for the size of a nucleus using the closest approach of an alpha particle.If we solve this equation we get an answer of around 10-14 metres for the radius of a typical nucleus. A better figure is around 10-15 metres (which you can get from scattering electrons and looking at the pattern).