Rutherford's nuclear model

The setup. Rutherford, Geiger and Marsden aimed a narrow beam of fast $\alpha$-particles (helium nuclei, charge $+2e$) from a radioactive source at a gold foil about $10^{-7}\ \text{m}$ thick — only a few hundred atoms deep. A circular screen coated with zinc sulfide, which flashes visibly wherever a charged particle strikes it, surrounded the foil so that particles scattered at any angle could be counted.

Three observations, three conclusions. About 99% of the $\alpha$-particles passed through with little or no deflection — so an atom must be mostly empty space, not a solid ball of matter. A small fraction deflected through noticeable angles — so somewhere inside the atom sits a concentration of positive charge strong enough to repel a positive $\alpha$-particle. And about 1 in 8000 bounced back at angles greater than $90°$, some almost retracing their path — a full reversal of a fast, heavy particle can only happen if it meets something both small and massive, since a light or spread-out target could never turn it around. Rutherford named this concentrated core the nucleus.

How small is small. The rarity of the sharp back-scattering event pins down the nucleus's size relative to the atom. If the nuclear radius is on the order of $10^{-15}\ \text{m}$ and the atomic radius on the order of $10^{-10}\ \text{m}$, the linear ratio is about $10^{-5}$, and since a large-angle deflection requires the incoming particle to pass close to the nucleus, the chance of it doing so scales with the nucleus's cross-sectional (roughly volume-like) share of the atom:

where: $r_{\text{nucleus}}$ and $r_{\text{atom}}$ are the nuclear and atomic radii, and $V_{\text{nucleus}}/V_{\text{atom}}$ is the fraction of the atom's volume occupied by the nucleus. A ratio this tiny is exactly the kind of number that makes a 1-in-8000 hit rate believable — most particles simply never come close enough to that vanishingly small target.

Every piece of Rutherford's model traces directly back to one of the three observations, and the logic connecting evidence to inference is worth holding onto for exam answers.

Step 1. Deflection through a large angle requires a strong repulsive force, and Coulomb repulsion between two positive charges only grows large when the particles come very close together. So a sharp deflection can only happen for $\alpha$-particles that pass near a concentrated pocket of positive charge — proving such a pocket exists.

Step 2. A full reversal (deflection past $90°$) is a much stronger demand than a mere deflection: it requires the target to be both positively charged and essentially immovable compared to the incoming $\alpha$-particle. A light or diffuse target would simply get pushed aside, not throw the $\alpha$-particle backward. This is why the observation also tells us the nucleus carries nearly all the atom's mass, not just its positive charge.

Step 3. Because only about 1 in 8000 particles met this fate, the region capable of producing such an encounter must be a very small target compared to the whole atom — consistent with the volume ratio of about $10^{-15}$ worked out above. Everything else — the 99% that sailed through undeflected — is simply particles that never passed near this tiny core, confirming that the rest of the atom's volume is essentially empty.

Step 4. Putting the three pieces together gives Rutherford's postulates: nearly all the mass and all the positive charge sit in a minute central nucleus; electrons revolve around it in orbits, held by electrostatic attraction, much as planets orbit the Sun under gravity; and the nucleus is so small relative to the atom that the atom is overwhelmingly empty space.

Question. In the gold-foil experiment only about 1 in 8000 $\alpha$-particles was deflected by more than $90°$. Taking the nuclear radius as $10^{-15}\ \text{m}$ and the atomic radius as $10^{-10}\ \text{m}$, show that this observation is numerically consistent with the relative sizes of the nucleus and the atom, and state what the same data reveal about where the atom's mass is concentrated.

Solution.

Step 1. Find the linear size ratio: $\dfrac{r_{\text{nucleus}}}{r_{\text{atom}}} = \dfrac{10^{-15}}{10^{-10}} = 10^{-5}$. The nucleus is about a hundred-thousandth the size of the atom along any one dimension.

Step 2. Convert to a volume (three-dimensional) ratio, since the chance of a head-on approach scales with the space the nucleus actually occupies: $\left(10^{-5}\right)^{3} = 10^{-15}$.

Step 3. Compare with the observed fraction. A volume ratio of $10^{-15}$ means an $\alpha$-particle has an extremely small chance of passing close enough to the nucleus to be strongly deflected — consistent with only 1 in 8000 (about $1.25\times10^{-4}$) showing any large-angle deflection at all; the true fraction that comes close enough to reverse direction is smaller still, matching the tiny volume ratio in order of magnitude.

Step 4. Because these rare, sharp reversals happen only for essentially head-on encounters with a stationary, heavy target, the same data show that nearly the entire mass of the atom — not just its positive charge — is concentrated in this same tiny nuclear volume. A nucleus of radius $\sim 10^{-15}\ \text{m}$ inside an atom of radius $\sim 10^{-10}\ \text{m}$ (volume ratio $\sim 10^{-15}$) is consistent with the observed 1-in-8000 large-angle scattering rate, and confirms that both the charge and the mass of the atom are concentrated in the nucleus.

Competitive stretch

The same reasoning explains why Rutherford's model, despite its success, could not survive scrutiny for long: classical electromagnetic theory says any accelerating (orbiting) charge must radiate energy continuously. An electron circling the nucleus is always accelerating, so it should lose energy, spiral inward, and collapse into the nucleus in a fraction of a second — yet atoms plainly do not collapse. This gap, along with the model's silence on why hydrogen emits light only at specific wavelengths, is what Bohr's model was built to close.

• Saying the gold-foil experiment "discovered the electron" — it discovered the nucleus; the electron was already known from Thomson's work.
• Believing most $\alpha$-particles were deflected — in fact about 99% passed straight through; only a small fraction deflected at all, and just 1 in 8000 by more than $90°$.
• Treating Rutherford's model as fully correct — its key limitation is that it cannot explain atomic stability or the hydrogen line spectrum, which is exactly why Bohr's model followed it.
• Mixing up nuclear size ($\sim 10^{-15}\ \text{m}$) with atomic size ($\sim 10^{-10}\ \text{m}$), or forgetting to cube the linear ratio when comparing volumes.

Ask the tutor: want to see exactly how the classical spiral-in time is estimated, or how Bohr's quantised orbits fix the stability problem — ask the AI tutor to walk through either one.