What did the gold foil experiment show?

Almost all alpha particles went through the foil nearly undeflected, so the atom is mostly empty space, yet about one in 8000 was deflected by more than 90°, some bouncing straight back. In Thomson’s model the positive charge is spread through the whole atom and its field is far too weak for such deflections. Rutherford concluded that the positive charge and almost all the mass sit in a tiny nucleus.

How are impact parameter and scattering angle related?

Under Coulomb repulsion the alpha follows a hyperbola, and the impact parameter b (the perpendicular distance of the incoming line from the nucleus) satisfies b = (D/2)cot(θ/2), where D = kzZe²/E is the head-on closest approach. Smaller b gives larger deflection. For gold at 5 MeV D ≈ 45 fm, so a 90° deflection needs b ≈ 23 fm, while the atom is about 10⁵ fm across; hitting such a small target is very unlikely.

Why does the Rutherford formula go as 1/sin⁴(θ/2)?

Particles scattered near θ come from a ring of radius b and width db with area 2πb·db. Differentiating b = (D/2)cot(θ/2) and dividing by the solid angle 2π sinθ dθ gives dσ/dΩ = (D/4)²/sin⁴(θ/2). The rate falls by about five orders of magnitude from 15° to 150°, and it scales with foil thickness, with Z² and as 1/E², all confirmed by experiment.

How does scattering give the size of the nucleus?

The larger the angle, the closer the alpha gets: r_min = (D/2)(1 + 1/sin(θ/2)), equal to D head-on. If large-angle data still follow Rutherford’s formula, the alpha felt only the Coulomb force at r_min, so the nucleus is smaller than that. Gold only gives an upper bound of about 30–45 fm; with light nuclei such as aluminium and faster alphas the large-angle data deviate, showing the nuclear force, and the radius comes out at a few fm.

Why is the Thomson model so far off?

Thomson’s atom spreads its positive charge through a sphere of about 0.1 nm, so one atom deflects an alpha by at most about 0.01°. Crossing a foil means passing about a thousand atoms, and the random kicks add up to an rms angle under about 1°. With a Gaussian distribution, deflections of more than a few degrees are negligible and beyond 90° the probability is below 10⁻³⁰⁰⁰. Switch the model on the counting tab to see the two curves.

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🎯Rutherford Scattering Simulator - alpha trajectories, gold foil, 1/sin⁴(θ/2)

Repeat the 1909–1913 gold foil experiment of Geiger and Marsden: 5–8 MeV alpha particles hit gold, silver, copper or aluminium foil and follow exact Coulomb hyperbolas. On the trajectory tab you drag the impact parameter and read the scattering angle and closest approach, or fire a random beam and watch nearly everything pass straight through while a rare few bounce back. On the counting tab you swing a detector arm, count with Poisson statistics and plot the rate on a log scale against Rutherford’s 1/sin⁴(θ/2) and the Thomson model. The third tab turns closest approach into an upper limit on nuclear size and shows, qualitatively, the anomalous scattering from aluminium at high energy.

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How to use Rutherford Scattering Simulator

1Choose a foil and alpha energy, then drag the impact parameter b on the Trajectories tab to see θ and r_min
2Switch to the random beam, press Fire and watch the fraction deflected past 10° or bounced back; compare with Thomson’s model
3On the Counting tab drag the detector arm and count, or measure a whole set of angles from 15° to 150°
4Check the log-scale plot and the (N/t)·sin⁴(θ/2) column against Rutherford’s formula
5On the Closest approach tab vary energy and foil to see the size limit and aluminium’s deviation

FAQ

What did the gold foil experiment show?: Almost all alpha particles went through the foil nearly undeflected, so the atom is mostly empty space, yet about one in 8000 was deflected by more than 90°, some bouncing straight back. In Thomson’s model the positive charge is spread through the whole atom and its field is far too weak for such deflections. Rutherford concluded that the positive charge and almost all the mass sit in a tiny nucleus.
How are impact parameter and scattering angle related?: Under Coulomb repulsion the alpha follows a hyperbola, and the impact parameter b (the perpendicular distance of the incoming line from the nucleus) satisfies b = (D/2)cot(θ/2), where D = kzZe²/E is the head-on closest approach. Smaller b gives larger deflection. For gold at 5 MeV D ≈ 45 fm, so a 90° deflection needs b ≈ 23 fm, while the atom is about 10⁵ fm across; hitting such a small target is very unlikely.
Why does the Rutherford formula go as 1/sin⁴(θ/2)?: Particles scattered near θ come from a ring of radius b and width db with area 2πb·db. Differentiating b = (D/2)cot(θ/2) and dividing by the solid angle 2π sinθ dθ gives dσ/dΩ = (D/4)²/sin⁴(θ/2). The rate falls by about five orders of magnitude from 15° to 150°, and it scales with foil thickness, with Z² and as 1/E², all confirmed by experiment.
How does scattering give the size of the nucleus?: The larger the angle, the closer the alpha gets: r_min = (D/2)(1 + 1/sin(θ/2)), equal to D head-on. If large-angle data still follow Rutherford’s formula, the alpha felt only the Coulomb force at r_min, so the nucleus is smaller than that. Gold only gives an upper bound of about 30–45 fm; with light nuclei such as aluminium and faster alphas the large-angle data deviate, showing the nuclear force, and the radius comes out at a few fm.
Why is the Thomson model so far off?: Thomson’s atom spreads its positive charge through a sphere of about 0.1 nm, so one atom deflects an alpha by at most about 0.01°. Crossing a foil means passing about a thousand atoms, and the random kicks add up to an rms angle under about 1°. With a Gaussian distribution, deflections of more than a few degrees are negligible and beyond 90° the probability is below 10⁻³⁰⁰⁰. Switch the model on the counting tab to see the two curves.

Features

Analytic hyperbolic orbits with b = (D/2)cot(θ/2), animated by angular momentum, with asymptotes and closest approach
Random beam statistics for large-angle scattering, with the Thomson uniform-sphere model for comparison
Draggable detector arm, scintillation flash animation and Poisson counting
Log-scale count rate vs angle: Rutherford 1/sin⁴(θ/2) against Thomson’s multiple small-angle scattering
Adjustable source rate, foil thickness, detector area and distance, with solid angle and expected rate
Closest approach r_min(θ), an upper bound on nuclear radius and the nuclear-force anomaly (qualitative)

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