Alpha-particle Scattering and Rutherford's Nuclear Model of Atom

The figure shows the apparatus that Geiger and Marsden used under Rutherford’s guidance — the experiment that revealed the atomic nucleus. At the centre of the drawing is a thin gold foil, shown as a small tilted square. Around it, drawn in perspective, is a shallow cylindrical drum. The inner curved wall of this drum is coated with zinc sulphide — that is the screen. A lead-shielded box containing a radioactive source of $\alpha$-particles sits below and to the left, outside the drum. A narrow beam of $\alpha$-particles emerges from the box, travels in a straight line, and strikes the gold foil. On hitting the foil, the $\alpha$-particles scatter in all directions. Those that hit the zinc-sulphide screen produce tiny flashes of light — scintillations — which can be viewed and counted through a microscope. The entire setup is placed inside a vacuum chamber (not shown in the figure) so that the $\alpha$-particles do not lose energy by colliding with air molecules.

The physical idea the figure teaches is simple but revolutionary: most $\alpha$-particles pass straight through the foil with little or no deflection, a few are deflected through small angles, and an extremely small number — about 1 in 8000 — bounce back almost the way they came. This could not happen if the positive charge and mass of the atom were spread out uniformly (as in Thomson’s plum-pudding model). The only explanation is that the atom has a tiny, dense, positively charged nucleus at its centre, and the rest is mostly empty space.

The Geiger-Marsden experiment, often called the gold-foil experiment, is the single most important experimental setup in the Atoms chapter because it forced a complete rewrite of atomic theory. Fig. 12.2 is a schematic of that setup, drawn as a 2D top-down view.

At the far left sits a radioactive source that emits $\alpha$-particles (helium nuclei, $^4_2\text{He}^{2+}$). These particles are not naturally collimated — they fly in all directions. To get a narrow beam, the source is surrounded by lead bricks with a single narrow slit cut through them. Only $\alpha$-particles that happen to travel straight through the slit emerge as a well-defined beam. This beam travels from left to right and strikes a thin gold foil at the centre of the diagram. The foil is only a few thousand atoms thick — thin enough that most $\alpha$-particles pass through it without hitting a gold nucleus head-on.

What happens after the foil is the entire point of the figure. The scattered $\alpha$-particles are detected by a ZnS screen mounted on a rotatable arm. When an $\alpha$-particle hits the zinc sulphide, it produces a tiny flash of light — a scintillation — which is observed through a microscope. The detector can be moved along a circular arc centred on the foil, so the experimenter measures the number of scintillations per minute at each scattering angle $\theta$.

The figure shows three distinct outcomes, drawn as trajectories leaving the foil:

• Small-angle scattering: Most $\alpha$-particles continue almost straight ahead, deflected by only a few degrees. These are the particles that passed through the empty space between atoms, experiencing only weak Coulomb repulsion from the distant positive charge of the gold nuclei.
• Large-angle scattering: A smaller number are deflected through angles like $30^\circ$, $60^\circ$, or $90^\circ$. These particles came close enough to a gold nucleus to feel a strong electrostatic repulsion.
• Backward scattering: A very tiny fraction — about 1 in 8000 — are scattered backward, toward the lower-left of the diagram, through angles greater than $90^\circ$ and even close to $180^\circ$. This was the shocking result. If the positive charge in an atom were spread out like a pudding (Thomson's model), an $\alpha$-particle could never be turned around; the maximum deflection would be tiny. Only a concentrated, massive positive nucleus could produce such a violent reversal.

The textbook uses this figure to derive the key formula for the distance of closest approach, $r_0$. At the moment an $\alpha$-particle is turned back (scattered through $180^\circ$), its entire initial kinetic energy $K$ has been converted into electrostatic potential energy at the point of closest approach to the nucleus:

Here:

• $K$ is the initial kinetic energy of the $\alpha$-particle (typically a few MeV).
• $Z$ is the atomic number of the target nucleus (for gold, $Z = 79$).
• $e$ is the elementary charge ($1.6 \times 10^{-19}\ \text{C}$).
• $2e$ is the charge of the $\alpha$-particle. …

The graph in Fig. 12.3 is a log-linear plot. The vertical axis, labelled "Number of scattered particles detected", is logarithmic — it runs from $10$ at the bottom to $10^7$ at the top. The horizontal axis is linear, showing the scattering angle $\theta$ in degrees from $0^\circ$ to $180^\circ$.

The experimental data points, shown as dots, trace a curve that starts extremely high near $10^6$–$10^7$ at the smallest angles (close to $0^\circ$). As $\theta$ increases, the number of detected particles drops steeply. By the time $\theta$ reaches about $90^\circ$, the count has fallen to roughly $10^3$–$10^4$. At the largest angles, near $180^\circ$, the curve levels off at around $10^2$ — a factor of ten thousand lower than at small angles. The solid curve drawn through these points is Rutherford's theoretical prediction, and it matches the data remarkably well across the entire range.

What this figure teaches. Before this experiment, the prevailing model was Thomson's "plum pudding" atom — a diffuse positive sphere with electrons embedded like raisins. If that model were correct, alpha particles would barely be deflected; the chance of a large-angle scatter would be negligible. Geiger and Marsden found the opposite: a tiny but real fraction of alpha particles bounced back at angles greater than $90^\circ$. Rutherford later said it was "almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

The graph makes this point visually. The steep drop with angle tells you that most alpha particles pass through with little deflection (small $\theta$, huge count), but a very small number suffer a near-head-on collision (large $\theta$, tiny count). That pattern is exactly what you expect if the positive charge is concentrated in a tiny, massive nucleus — not spread out.

The key formula. The theoretical curve in the figure comes from Rutherford's scattering formula. For a thin foil of thickness $t$, with $N$ atoms per unit volume, each containing a nucleus of atomic number $Z$, the number $N(\theta)$ of alpha particles scattered into a detector at angle $\theta$ (subtending a small solid angle $\Delta\Omega$) is:

where:

• $N_i$ = number of incident alpha particles
• $n$ = number density of target atoms (atoms per unit volume)
• $t$ = foil thickness
• $Z$ = atomic number of the target nucleus
• $e$ = elementary charge
• $\varepsilon_0$ = permittivity of free space
• $K_\alpha$ = kinetic energy of the incident alpha particle
• $\theta$ = scattering angle
• $\Delta\Omega$ = solid angle subtended by the detector

The crucial dependence is $N(\theta) \propto 1/\sin^4(\theta/2)$. At small $\theta$, $\sin(\theta/2) \approx \theta/2$, so $N(\theta) \propto 1/\theta^4$ — a very steep drop as $\theta$ increases. At $\theta = 180^\circ$, $\sin(90^\circ) = 1$, giving the minimum count. This inverse-fourth-power dependence is what produces the dramatic fall seen in the graph.