Rutherford scattering formulalaw

Ernest Rutherford, skeptical of the complex structure of the atom, tasked Hans Geiger and Ernest Marsden with bombarding gold foil with alpha particles. Everyone expected the particles to pass through, only slightly scattering. But Marsden noticed: about one in 8000 particles bounced back! Rutherford was amazed and later compared it to a 15-inch shell ricocheting off tissue paper. He realized that almost all the mass of the atom is concentrated in a tiny nucleus. In 1911, he derived a formula relating the number of scattered particles to angle, energy, and nuclear charge. This was a purely classical result, perfectly explaining the experiment; only later were quantum corrections needed for light nuclei.

Imagine you are throwing tennis balls into a fog where a bowling ball is hidden. Almost all balls will fly past, only slightly changing trajectory. But those that fly straight at the ball will bounce back. That is exactly how Geiger and Marsden saw the rare alpha particles reflected by the gold foil. Rutherford explained this with a formula: the probability of strong deflection increases with the nuclear charge and decreases with the particle energy. The nucleus is a super-dense 'pea' inside an almost empty atom.

Rutherford's formula is the classical differential cross-section for elastic Coulomb scattering of a non-relativistic point particle on a stationary point nucleus. In the center-of-mass system: dσ/dΩ = (Z₁Z₂e²/(16π ε₀ E))² · 1/sin⁴(θ/2). It is applicable when the energy is high enough to neglect electron screening, but low for relativistic and nuclear effects. At high energies or for light nuclei, quantum mechanical corrections (Mott formula) and relativistic generalizations are required.