Rutherford scattering: how the gold foil experiment found the nucleus

Guide · Updated 2026-11-10

Why alpha particles bouncing off gold foil proved atoms have a tiny nucleus, with the formulas for closest approach and scattering angle and a worked example on silver foil.

Open the Rutherford scattering Free, in your browser, nothing uploaded.

Around 1910, a simple experiment changed our picture of the atom. Fast alpha particles were fired at a very thin sheet of gold, and while almost all of them went straight through, a tiny number came flying back. The rutherford scattering tool lets you repeat that experiment on screen, with a beam of particles or a single aimed shot, and compare the result with the older “plum pudding” model. It works at two levels: Class 9 for the ideas and Class 12 for the formulas. To see why matter is made of particles at all, start with states of matter. After the nucleus, the hydrogen spectrum shows how electrons sit in energy levels, and hydrogen orbital viewer shows the modern picture of where they are.

What it is

An alpha particle is a helium nucleus: two protons and two neutrons, with a charge of +2e (e is the charge of one proton). In J. J. Thomson’s plum pudding model, an atom was a ball of thinly spread positive charge with electrons scattered through it. Spread-out charge pushes weakly, so an alpha particle should barely be deflected. Ernest Rutherford’s team, Hans Geiger and Ernest Marsden, found otherwise. The only way to send a fast particle straight back is to pack all the positive charge, and nearly all the mass, into a tiny nucleus.

The push between the alpha particle and the nucleus is the electric (Coulomb) force, which weakens as 1/r². Three results follow from it.

Head-on closest approach. Fire the particle straight at the nucleus and it slows until all its kinetic energy K has turned into electric energy:

d = k × (2e) × (Ze) / K

Here Z is the number of protons in the nucleus and k e² = 1.44 MeV fm. One femtometre (fm) is 10⁻¹⁵ m, and an MeV is a million electronvolts, a unit of energy.

Scattering angle. If the particle is aimed off-centre by a distance b (the impact parameter), it is turned through an angle θ given by

tan(θ/2) = d / (2b)

At b = 0 it comes straight back (180°).

Closest approach for any aim. The particle never gets nearer than

r_min = (d/2) × (1 + √(1 + (2b/d)²))

which is d itself when b = 0, and grows as the aim moves further out.

How to use it

1. Choose Nucleus or Plum pudding at the top to pick the model of the atom.
2. Leave A beam selected and watch particles fly in at random heights. Three counters keep score: almost straight (under 10°), turned 10° to 90°, and bounced back (over 90°), each with a percentage. Pause stops the motion and Reset counts starts the tally again.
3. Pick a Foil (gold, silver, copper or aluminium) and move the Alpha particle energy slider, which runs from 1 to 10 MeV.
4. Switch the level to Class 12 to see the head-on distance worked out, along with a graph of angle against aim.
5. Choose One particle and slide the aim from 0 to 5 times d. The tool shows the path, the angle θ and the closest approach, with a dashed circle marking how close it got.
6. Work through the five Check yourself questions.

A worked example

Take silver foil (Z = 47) and alpha particles of 7.5 MeV.

Head-on distance: d = 1.44 MeV fm × 2 × 47 / 7.5 MeV = 18.0 fm (18.05 fm with the full constant).

Aim b = 2 d = 36.1 fm. Then tan(θ/2) = 1 / (2 × 2) = 0.25, so θ/2 = 14.04° and θ = 28.1°. The closest approach is (d/2)(1 + √17) = 2.56 d = 46.2 fm.

Aim b = 0.25 d = 4.5 fm. Then tan(θ/2) = 1 / (2 × 0.25) = 2, so θ/2 = 63.43° and θ = 126.9°: the particle comes back towards where it started. Its closest approach is (d/2)(1 + √1.25) = 19.1 fm, only a little more than the head-on distance.

Notice how sharply the angle depends on the aim. Moving in from 2 d to 0.25 d, an eighth of the distance, takes the deflection from under 30° to well over 90°. That is why the beam counters show so few bounce-backs.

Where it is used

• Measuring nuclear size. When fast particles get close enough to touch the nucleus, the results stop following the formula. The energy where this starts gives the size of the nucleus.
• Rutherford backscattering spectrometry. Labs fire helium ions at a material and measure the energy of the ones that bounce back. Heavier atoms return more energy, so the method shows which elements are in a thin film and how deep they lie.
• Particle physics. The same “fire things and count where they go” approach, with electrons at much higher energy, later showed that protons have smaller parts inside them.

Common mistakes

1. Mixing up b and the closest approach. b is how far the particle would miss if there were no push. The closest approach is always larger, because the push bends the path away.
2. Thinking a higher energy means more bounce-backs. Faster particles get closer, but they also bend less. In the counters, fewer particles come back at higher energy.
3. Forgetting the 2 for the alpha charge. The alpha particle carries 2e, so the formula has 2 × Z, not Z.
4. Reading the picture as true scale. The view is zoomed in hugely on one nucleus.
5. Using the formula when the particle touches the nucleus. For light nuclei and high energies, the strong nuclear force joins in and the simple result no longer holds.

Questions people ask

Why gold? Gold can be beaten into foil less than a micrometre thick, and its large charge (Z = 79) gives strong, easy-to-see deflections.

Which aim gives a 90° turn? Half the head-on distance: b = 0.5 d gives tan(θ/2) = 1, so θ = 90°.