Rutherford Scattering

In 1909 a few alpha particles fired at a thin gold foil bounced almost straight back, which Rutherford called as incredible as a shell rebounding off tissue paper. A diffuse cloud of charge could never do that; only a tiny, dense, charged nucleus could turn a fast alpha around. Each alpha follows a hyperbola in the nucleus's Coulomb field, and how sharply it bends is fixed by its impact parameter, the sideways miss-distance of its incoming line. Aim almost dead-on and the repulsion flings it back through a large angle; pass wide and it barely swerves, the deflection set by $\cot(\theta/2) = 2b/D$, where $D$ is the head-on distance of closest approach, larger for a heavier nucleus or a slower alpha. The top panel fires a beam at a spread of impact parameters and traces the orbits, the near-axis ones whipping around, the outer ones grazing past; the dashed circle is $D$, the closest a head-on alpha ever gets. The real triumph is statistical: counting how many scatter into each angle gives the differential cross section $d\sigma/d\Omega = (D/4)^2/\sin^4(\theta/2)$, the steep curve in the bottom panel, and its stubborn large-angle tail is the fingerprint of a point nucleus. Slide the energy and nuclear charge to squeeze or stretch the orbits, and pick an impact parameter to read its angle and closest approach.