Case Study 1: The Rutherford Scattering Experiment — How We Learned the Atom Has a Nucleus

The Context: Manchester, 1909

In the first decade of the twentieth century, the internal structure of the atom was one of the great unsolved problems in physics. J.J. Thomson's discovery of the electron in 1897 had established that atoms contained negatively charged components, and the neutrality of atoms demanded a compensating positive charge. But how was the positive charge distributed?

Thomson's own model — the "plum pudding" — envisioned a uniform sphere of positive charge roughly the size of the atom ($\sim 10^{-10}\,\text{m}$) with electrons embedded throughout, like raisins in a pudding. This was not idle speculation; it was the best theory consistent with existing evidence. The model predicted that a fast, heavy charged particle passing through matter would experience many small deflections from the diffuse charge distribution, producing a Gaussian angular distribution with a characteristic width of only a few degrees.

Ernest Rutherford, a New Zealander working at the University of Manchester, had been studying radioactivity since the 1890s. By 1908, he had identified alpha particles as helium nuclei (charge $+2e$, mass $\approx 4\,\text{u}$) and realized they could serve as probes of atomic structure — tiny, fast, positively charged bullets that would interact electromagnetically with the atom's charge distribution.

The Experiment

Rutherford tasked two members of his group — Hans Geiger (a German physicist who would later invent the Geiger counter) and Ernest Marsden (an undergraduate student, only 20 years old) — with a systematic study of alpha particle scattering from thin metal foils.

The Apparatus

The experimental setup was elegant in its simplicity:

1. Source: A small quantity of radon gas (${}^{222}\text{Rn}$, an alpha emitter) sealed in a glass tube, producing a collimated beam of alpha particles with energies around 5.5 MeV.

2. Target: A thin gold foil, chosen because gold is highly malleable (producing very thin, uniform foils) and has a high atomic number ($Z = 79$), maximizing the scattering signal. The foil thickness was approximately $0.4\,\mu\text{m}$, or about 1300 atomic layers.

3. Detector: A zinc sulfide (ZnS) screen viewed through a low-power microscope. Each alpha particle striking the screen produced a tiny flash of light (a scintillation) visible to the dark-adapted eye. The detector could be rotated around the foil to measure the scattering at different angles.

The entire apparatus was enclosed in an evacuated chamber to prevent the alpha particles from being stopped by air.

The Measurement

Geiger and Marsden spent months in a darkened room, hunched over the microscope, counting individual scintillations. This was painstaking work — each measurement at a given angle required counting hundreds to thousands of flashes to accumulate adequate statistics. At small angles, the count rate was high and data collection was relatively fast. At large angles, the rate dropped precipitously, requiring long counting sessions.

They measured the number of scintillations as a function of angle from about $5°$ to $150°$. The results, published in 1909 and more comprehensively in 1913, were startling.

The Data

The key observations were:

• Most alpha particles passed through the foil with little or no deflection, consistent with the Thomson model.
• The angular distribution fell off very steeply with increasing angle — but not as a Gaussian. Instead, it followed a $\sin^{-4}(\theta/2)$ law over several orders of magnitude.
• About 1 in 8000 alpha particles scattered at angles greater than $90°$, with a few bouncing almost straight back ($\theta \approx 180°$).

The large-angle scattering was the bombshell. In the Thomson model, the probability of a single scatter exceeding $90°$ was absurdly small — roughly $10^{-3500}$, according to Rutherford's own calculation. Even compounding many small deflections through 1300 atomic layers could not produce the observed rate.

Rutherford's Analysis

Rutherford published his analysis in 1911, in a paper titled "The Scattering of $\alpha$ and $\beta$ Particles by Matter and the Structure of the Atom" in the Philosophical Magazine. His reasoning is a model of physical insight.

The Key Insight

Rutherford realized that only a concentrated charge could produce large-angle scattering in a single encounter. If the positive charge of the atom were concentrated in a region much smaller than the atom itself — a "nucleus" — then an alpha particle passing close to this nucleus would experience an intense Coulomb field capable of deflecting it through large angles.

The Calculation

Treating the scattering as a classical two-body Coulomb problem (justified because the de Broglie wavelength of a 5.5 MeV alpha particle is $\lambda \approx 6\,\text{fm}$, much smaller than the distance of closest approach $d_0 \approx 41\,\text{fm}$ for gold), Rutherford derived the scattering cross section:

$$\frac{d\sigma}{d\Omega} = \left(\frac{a}{2}\right)^2 \frac{1}{\sin^4(\theta/2)}$$

where $a = kz_1z_2e^2/(2T)$.

Comparison to Data

The formula predicted:

1. A $\sin^{-4}(\theta/2)$ angular dependence — confirmed over four orders of magnitude.
2. A $Z^2$ dependence on the target atomic number — confirmed by comparing gold, silver, copper, and aluminum targets.
3. A $T^{-2}$ dependence on the alpha energy — confirmed by varying the source-to-foil distance (which changes the effective energy due to air absorption) and by using different radioactive sources.
4. A linear dependence on foil thickness (for thin foils where multiple scattering is negligible) — confirmed.

The agreement was spectacular. Rutherford's formula described the data from $5°$ to $150°$ without a single adjustable parameter (the only inputs being the known charges, energy, and target thickness).

The Size of the Nucleus

From the absence of deviations at the largest angles, Rutherford could set an upper limit on the nuclear size. For the alpha not to "touch" the gold nucleus at $\theta = 180°$, the distance of closest approach $d_0 = kz_1z_2e^2/T = 41\,\text{fm}$ must exceed the sum of the nuclear radii. Since the data obeyed the point-charge formula even at the largest angles, the gold nucleus must be smaller than about $41\,\text{fm}$.

Today we know the gold nuclear radius is about $R \approx 1.21 \times 197^{1/3} \approx 7.0\,\text{fm}$, comfortably smaller than $d_0$. The atom ($\sim 10^{-10}\,\text{m} = 10^5\,\text{fm}$) is thus about 14,000 times larger than its nucleus — the atom is overwhelmingly empty space.

The Legacy

Immediate Impact

Rutherford's nuclear model immediately raised a new problem: if electrons orbit a tiny positive nucleus, classical electrodynamics predicts they should radiate energy continuously and spiral into the nucleus in about $10^{-11}$ seconds. The atom should be unstable. This paradox was resolved by Niels Bohr in 1913 with his quantum model of the hydrogen atom, which postulated stable orbits with quantized angular momentum — the first step toward quantum mechanics.

Modern Descendants

The Rutherford scattering experiment launched several enduring legacies:

Scattering as a method. The idea of probing structure by shooting particles at a target and measuring the angular distribution became the central paradigm of subatomic physics. Hofstadter's electron scattering (1950s) mapped nuclear charge distributions. Deep inelastic scattering at SLAC (1960s) revealed quarks inside protons. The LHC at CERN continues this tradition today, probing structure at the smallest accessible scales.

Rutherford Back-Scattering (RBS). The original experiment lives on as a standard analytical technique in materials science. MeV ion beams are used to determine the elemental composition and depth profile of thin films and surface layers, exploiting the same $Z^2$ dependence that Rutherford derived in 1911.

The cross section concept. The differential cross section $d\sigma/d\Omega$ — the effective area presented by the target per unit solid angle of the detector — became the universal language for describing all scattering processes, from nuclear reactions to particle physics.

Lessons for the Modern Student

1. Simple experiments can reveal profound truths. The Geiger-Marsden apparatus was technologically primitive — a radioactive source, a thin foil, and a zinc sulfide screen. Yet it overturned the prevailing model of the atom.

2. Anomalies matter. The large-angle scattering was only 1 in 8000 — a tiny signal. Most physicists would have dismissed it as a background effect. Rutherford recognized its significance because he understood the theoretical expectation quantitatively.

3. Quantitative prediction distinguishes models. Both the Thomson and Rutherford models predicted "some scattering." Only when the predictions were made quantitative — the angular distribution, the energy dependence, the $Z$ dependence — could the data distinguish between them.

4. Classical physics can describe quantum systems when conditions are right. The Rutherford formula is a classical calculation, yet it describes quantum scattering from a Coulomb potential exactly (the quantum-mechanical Coulomb cross section, derived by Mott, reduces to the Rutherford formula in the appropriate limit). This is a special property of the $1/r$ potential and is not true in general.

Discussion Questions

1. Why did Rutherford use gold foil rather than a lighter element? What would change if aluminum were used?

2. Geiger and Marsden counted individual scintillations by eye. What modern detector technology would you use to repeat the experiment today? What advantages would it offer?

3. At what alpha particle energy would you expect the Rutherford formula to break down for gold? What new physics would this reveal?

4. The Rutherford formula has no quantum mechanics in it — it is purely classical. Yet it gives the correct quantum-mechanical result for Coulomb scattering. Explain why this is not a contradiction. (Hint: Consider the relationship between the de Broglie wavelength and the distance of closest approach.)