Rutherford's Gold Foil Experiment: How the Nucleus Was Found

The gold foil experiment is one of the most retold stories in physics, and one of the most compressed. It was not a single afternoon, it did not use only gold, and the 1911 paper that explained it did not describe electrons orbiting a nucleus. What it did do was show that almost all of an atom's mass and positive charge sit in a region tens of thousands of times smaller than the atom. That idea, summed up as “some came back”, is the theme of our Ernest Rutherford sticker.

What did people think atoms looked like before 1911?

J. J. Thomson, who discovered the electron in 1897, proposed in 1904 a model in which electrons sit inside a sphere of spread-out positive charge Plum pudding model, Wikipedia^[1]. It became known as the plum pudding model, although Thomson and his colleagues never used that analogy Plum pudding model, Wikipedia^[1]. In such an atom no region holds enough concentrated charge to turn a fast alpha particle sharply. Any deflection should be the sum of many tiny nudges, adding up to at most a few degrees Rutherford scattering experiments, Wikipedia^[2].

What did Geiger and Marsden actually observe?

The experiments ran from 1906 to 1913 at the University of Manchester, carried out by Hans Geiger and the undergraduate Ernest Marsden under Ernest Rutherford's direction Rutherford scattering experiments, Wikipedia^[2]. Alpha particles were detected one at a time: each one hitting a zinc sulfide screen made a faint flash of light, which the experimenter counted through a microscope in a darkened room Rutherford scattering experiments, Wikipedia^[2].

In their 1909 paper, Geiger and Marsden reported a “diffuse reflection” of alpha particles Geiger & Marsden 1909^[3]. Using radium C as the source, they found that roughly 1 in 8,000 of the alpha particles striking a platinum reflector came back onto the screen on the same side Rutherford scattering experiments, Wikipedia^[2] Geiger & Marsden 1909^[3]. They tested several metals, and gold became the favourite because it can be beaten into extremely thin foil Rutherford scattering experiments, Wikipedia^[2]. One in 8,000 sounds small, but for a Thomson atom it should have been effectively zero.

How did Rutherford explain the result?

Rutherford's 1911 paper in the Philosophical Magazine took the opposite assumption to Thomson's: the atom's charge is concentrated in a tiny central region, surrounded by a compensating charge spread through the rest of the atom Rutherford 1911^[4] Rutherford model, Wikipedia^[5]. A single close encounter with that central charge can then turn an alpha particle through any angle up to 180°. He treated the alpha particle as a point charge moving under Coulomb repulsion, which gives a hyperbolic path, and worked out how many particles should arrive at each angle Rutherford scattering, Wikipedia^[6].

Some details are often misremembered. Rutherford took the central charge to be positive, but a negative charge would have fitted his scattering model just as well; the sign was settled in 1913 Rutherford scattering experiments, Wikipedia^[2]. He estimated the central charge of gold at about 100 units, having actually suggested 98 as half the atomic weight of 196 Rutherford model, Wikipedia^[5]. And he did not propose a model for the electrons; that came from Niels Bohr, who worked in Rutherford's laboratory Rutherford model, Wikipedia^[5]. In 1913 Antonius van den Broek suggested that the nuclear charge equals the atom's place in the periodic table, an idea Rutherford's group quickly took up Rutherford model, Wikipedia^[5].

The Rutherford scattering formula

In modern notation, the number of particles scattered into a small solid angle around angle θ is set by the differential cross-section Rutherford scattering, Wikipedia^[6]:

dσ/dΩ = (zZe² / (16πε₀E_k))² · 1/sin⁴(θ/2)

Here z = 2 is the alpha particle's charge number, Z is the nucleus's (79 for gold), e is the elementary charge (1.602 × 10⁻¹⁹ C), ε₀ is the vacuum permittivity (8.854 × 10⁻¹² F/m), E_k is the alpha particle's kinetic energy in joules, and dσ/dΩ is in m² per steradian. The formula assumes non-relativistic motion and a nucleus heavy enough that its recoil can be ignored. The striking feature is the 1/sin⁴(θ/2) factor: the rate drops steeply with angle but never reaches zero, which is why some particles come back.

In 1913 Geiger and Marsden tested this prediction in detail Geiger & Marsden 1913^[7]. They confirmed that the count rate at angle Φ was proportional to cosec⁴(Φ/2), to the foil thickness, to the square of the central charge and to 1/(mv²)², the inverse square of the alpha particles' kinetic energy Rutherford scattering experiments, Wikipedia^[2].

How small is the nucleus? A worked estimate

In a head-on collision the alpha particle stops when all its kinetic energy has turned into electrostatic potential energy. The stopping distance is

r_min = zZe² / (4πε₀E_k)

and for a glancing path the impact parameter b, the sideways offset of the incoming line from the nucleus, fixes the angle through

b = (r_min/2)·cot(θ/2)

Take a speed of 2 × 10⁷ m/s and the alpha mass 6.644 × 10⁻²⁷ kg, as in a standard textbook example Rutherford scattering experiments, Wikipedia^[2]. Our own arithmetic gives E_k ≈ 1.33 × 10⁻¹² J (about 8.3 MeV) and r_min ≈ 2.7 × 10⁻¹⁴ m, or 27 fm. The alphas never touch the nucleus, so this is only an upper limit; the true radius of a gold nucleus is about 7.3 fm Rutherford scattering experiments, Wikipedia^[2]. Rutherford's own 1911 bound was below 3.4 × 10⁻¹⁴ m Rutherford model, Wikipedia^[5], against an atom about 10⁻¹⁰ m across Rutherford scattering experiments, Wikipedia^[2]. Particles scattered by 90° pass at b = r_min/2, about 14 fm, which is why such large deflections are so rare.

What holds tens of protons together at that scale is the strong force, covered in what holds the nucleus together. The alpha particles themselves came from radioactive decay, the subject of our post on Marie Curie's discoveries, and the link between alpha energy and half-life is the Geiger–Nuttall law on our Geiger–Nuttall poster.

Quick FAQ

Was the foil really gold?

Often, but not only. Several metals were tested, including platinum in the 1909 reflection measurement Rutherford scattering experiments, Wikipedia^[2].

Did Rutherford discover the planetary atom?

No. His 1911 model concentrated the charge and mass at the centre and did not describe electron orbits Rutherford model, Wikipedia^[5]. A related Saturn-like model had been proposed by Hantaro Nagaoka in 1904 Rutherford model, Wikipedia^[5].

Does the formula still work?

For a point-like Coulomb target, yes. It breaks down for light nuclei and for energies high enough that the projectile reaches the nucleus, where the strong force takes over Rutherford scattering experiments, Wikipedia^[2]. More about nuclei is in the Particle Physics collection.

References

1. Wikipedia, “Plum pudding model”. https://en.wikipedia.org/wiki/Plum_pudding_model
2. Wikipedia, “Rutherford scattering experiments”. https://en.wikipedia.org/wiki/Rutherford_scattering_experiments
3. H. Geiger and Ernest Marsden (1909), “On a diffuse reflection of the α-particles,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 82, 495–500 (DOI 10.1098/rspa.1909.0054; metadata verified via Crossref). https://doi.org/10.1098/rspa.1909.0054
4. E. Rutherford (1911), “The scattering of α and β particles by matter and the structure of the atom,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 21, 669–688 (DOI 10.1080/14786440508637080; metadata verified via Crossref). https://doi.org/10.1080/14786440508637080
5. Wikipedia, “Rutherford model”. https://en.wikipedia.org/wiki/Rutherford_model
6. Wikipedia, “Rutherford scattering”. https://en.wikipedia.org/wiki/Rutherford_scattering_(physics)
7. H. Geiger and E. Marsden (1913), “The laws of deflexion of α particles through large angles,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 25, 604–623 (DOI 10.1080/14786440408634197; metadata verified via Crossref). https://doi.org/10.1080/14786440408634197

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.