experiment 040 · atomic physics · scattering

Rutherford Scattering

In 1911, Ernest Rutherford published his model of the atom based on the gold foil experiment. By firing alpha particles at a thin gold foil, he discovered that atoms contain a dense, positively charged nucleus. Here, you can visualize the hyperbolic trajectories of alpha particles repelled by a central Coulomb potential.

The Mathematics

The force on an alpha particle of charge $q = 2e$ by a nucleus of charge $Q = Ze$ is given by Coulomb's law:

$$ \mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q Q}{r^2} \mathbf{\hat{r}} $$

For a repulsive inverse-square force, classical mechanics predicts hyperbolic trajectories. The scattering angle $\theta$ depends on the initial kinetic energy $E$ and the impact parameter $b$ (the perpendicular distance from the initial velocity vector to the nucleus):

$$ b = \frac{z Z e^2}{4\pi\epsilon_0 m v^2} \cot\left(\frac{\theta}{2}\right) $$

Particles approaching with a small impact parameter $b$ experience a strong repulsive force and undergo large-angle scattering (including backscattering). This led Rutherford to conclude that the positive charge and most of the mass of an atom are concentrated in a tiny nucleus.

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