Rutherford scattering
Alpha particles onto a thin foil, counted at an angle: the 1/sin⁴(θ/2) law is what a point nucleus gives, and a head-on alpha says how small it is.
Nuclear & quantum — Radioactive decay, the absorption of radiation, photons, and energy levels in the atom.
Experiments already set up on it
The 1/sin⁴ law
The headline: counts against 1/sin⁴(θ/2) is straight — a point charge.
- A point charge gives dσ/dΩ ∝ 1/sin⁴(θ/2)
- So counts against 1/sin⁴(θ/2) is a straight line through the origin
- Which is the evidence that the positive charge is concentrated in something tiny
Does the metal matter?
The cross-section goes as Z², so gold scatters far more than aluminium.
- dσ/dΩ ∝ (Z₁Z₂)², so the nuclear charge counts twice over
- A heavy nucleus repels an alpha much harder, and much further out
How big is a nucleus?
A head-on alpha stops where all its energy has become potential energy.
- An alpha turned right back got closest of all
- All its kinetic energy became electrical potential energy there
- Which puts an upper bound on the size of the nucleus — tens of femtometres
What you can measure
Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.
- Timet (s)
- Counts so farN
- Count rateR (1/s)
- Rate the formula predictsR_e (1/s)
- Detector angleθ (°)
- 1/sin⁴(θ/2)isin4
- Cross-section per steradiandsdo (m^2)
- Solid angle of the detectorOmega
- Closest approach, head-ond (fm)
What you can change
Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.
- Energy of the alphasE (MeV)
- Alphas hitting the foil, per secondR₀ (1/s)
- Thicknessτ (um)
- Detector angleθ (°)
- Half-angle of the detectorα (°)
- Count fort_g (s)
The equations behind it
What rutherford scattering is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.
The nucleus
- Nuclide notation — A is the nucleon number, Z the proton number.
- Mass–energy equivalence
- Energy from a mass defect
- Nuclear radius — R₀ ≈ 1.2 fm.
- Binding energy per nucleon — The curve peaks at iron-56, which is why both fusion and fission release energy.
Also in Nuclear & quantum
Radioactive decay
A source counted with a Geiger tube, decay by decay, with the √N scatter of real counting statistics and a background that does not decay.
Absorption of radiation
Alpha, beta and gamma through absorbers of your choosing: only one of the three is absorbed exponentially, and the counter still does not subtract the background.
Photoelectric effect
Light on a metal plate and a retarding voltage turned up until the current stops: the stopping voltage is found rather than calculated, and brightness changes nothing about it.
Muon decay & time dilation
Cosmic-ray muons counted at any altitude you like: far more arrive than a muon’s lifetime allows, and only a slow-running clock accounts for it.
Electron diffraction
Electrons accelerated through a voltage and diffracted by graphite: the rings move as 1/√V, which is de Broglie’s λ = h/p made visible.
Atomic spectra
Hydrogen through a grating spectrometer: each line is an electron falling between two levels, and a graph of them gives the Rydberg constant.