Atomic Models

Thomson's plum-pudding model. Once the electron was identified as a light, negatively charged particle inside the atom, J. J. Thomson proposed the simplest picture consistent with a neutral atom: a sphere of positive charge, roughly the size of the whole atom, with electrons studded through it like seeds in a watermelon. Both the positive charge and essentially all the mass were assumed to be spread evenly across the atom's volume, with no special structure at the centre.

The alpha-scattering experiment. Rutherford, working with Hans Geiger and Ernest Marsden, aimed a narrow beam of fast, positively charged alpha particles at a gold foil only a few hundred atoms thick, with a rotatable fluorescent screen around it to record where the particles ended up. Three results came out of the run: almost every alpha particle went straight through as if the foil were not there; a small fraction bent through modest angles; and a tiny fraction — about 1 in 20,000 — came back at angles greater than $90°$, some almost reversing direction entirely. Thomson's spread-out-charge model had no way to produce that last group at all: a diffuse positive sphere simply cannot deliver the kind of sharp, close-range shove needed to turn a fast, heavy alpha particle backward.

The nuclear model. To account for all three results at once, Rutherford proposed that an atom is mostly empty space, with virtually all of its positive charge and mass packed into an extremely small central region — the nucleus — while much lighter electrons occupy the much larger volume around it. A slow head-on approach to a nucleus, or a close pass by one, is rare, which is exactly why large deflections were rare in the data; but on the few occasions it happened, the concentrated charge and mass could deliver a strong enough electrostatic shove to send the alpha particle backward.

How small is the nucleus? Rutherford's data let him estimate the nucleus at about $r_{\text{nucleus}} \approx 10^{-15}\ \text{m}$, compared with an atomic radius of about $r_{\text{atom}} \approx 10^{-10}\ \text{m}$ — the nucleus is roughly a hundred-thousandth the size of the atom it sits inside.

where: $r_{\text{nucleus}}$ is the radius of the nucleus, $r_{\text{atom}}$ is the radius of the whole atom, and the ratio $10^{-5}$ means the nucleus's linear size is one hundred-thousandth of the atom's — which is why the overwhelming majority of alpha particles pass through a foil of atoms without meeting any obstacle at all.

The three scattering results only make sense together if the positive charge and the mass of the atom are concentrated in something very small and very dense. Undeflected passage tells us most of an atom's volume is empty — an alpha particle travelling through it meets essentially nothing. Small-angle deflection tells us there is some charge concentration nearby to nudge a passing particle a little. Large-angle and back-scattering — the rare, decisive result — can only be produced by a close encounter with a region carrying both a large positive charge, to electrostatically repel the positive alpha particle strongly enough, and a large mass, so that the much lighter alpha particle rebounds instead of simply pushing the region aside. No version of Thomson's diffuse-sphere model can generate that combination anywhere in the foil, which is why the large-angle events alone were enough to overturn it. Rutherford's nuclear model explained all three observations, but it created two new problems the moment it was taken seriously. First, it pictured electrons revolving around the nucleus somewhat like planets around the sun, held in orbit by electrostatic attraction. Classical electromagnetic theory says that any charged particle undergoing accelerated motion must continuously radiate energy — and a particle moving on a circular path is always accelerating, since its direction of motion keeps changing even if its speed does not. An orbiting electron losing energy this way should spiral inward and crash into the nucleus in a tiny fraction of a second, meaning the model predicts every atom should be unstable. Second, the model gives no reason why atoms emit light only at specific, sharply defined wavelengths — the line spectrum of hydrogen — rather than a continuous spread of colours as a classical orbiting charge would produce. Both gaps are genuine limitations of the nuclear model as Rutherford left it, and both became the starting problems Bohr's model was built to solve.

Question. In the alpha-scattering experiment, the nucleus is estimated at $r_{\text{nucleus}} \approx 10^{-15}\ \text{m}$ and the atom at $r_{\text{atom}} \approx 10^{-10}\ \text{m}$. (a) Find the ratio of nuclear to atomic radius. (b) Use it to explain, in terms of geometry, why only about 1 in 20,000 alpha particles are deflected through large angles.

Step 1. Divide the two radii: $\dfrac{r_{\text{nucleus}}}{r_{\text{atom}}} = \dfrac{10^{-15}}{10^{-10}} = 10^{-5}$. The nucleus is about a hundred-thousandth the linear size of the atom.

Step 2. Convert the linear ratio to a volume comparison, since it is volume (space occupied) that determines how "crowded" the atom is: $\left(\dfrac{r_{\text{nucleus}}}{r_{\text{atom}}}\right)^3 = (10^{-5})^3 = 10^{-15}$. The nucleus occupies only about one part in a thousand trillion of the atom's volume.

Step 3. Interpret this alongside the experimental count. A ratio of $10^{-15}$ for volume means an alpha particle's path through the foil almost never comes near enough to a nucleus to feel a strong repulsion — consistent with the observation that large deflections are rare (about 1 in 20,000), even though the two numbers are independent measurements rather than one being calculated from the other. The nucleus is roughly $10^{-5}$ times the atom's radius, so the atom is overwhelmingly empty space, which is exactly why undeflected passage is the overwhelming outcome and large-angle scattering is rare.

Competitive stretch

A related exam favourite: if the nucleus were scaled up to the size of a marble (about $1\ \text{cm}$ radius), the ratio $10^{-5}$ says the atom's radius would scale up to about $1\ \text{cm}/10^{-5} = 1000\ \text{m}$ — the atom would be as wide as a small town, with the marble-sized nucleus sitting at its centre and nothing but empty space (occupied only by the tiny orbiting electrons) filling the rest. This scale comparison is a common way examiners test whether students actually grasp how empty an atom is, beyond just quoting the ratio.

• Calling Thomson's model the "nuclear model" — the nuclear model is Rutherford's; Thomson's model has no nucleus at all, only uniformly spread positive charge.
• Thinking large-angle deflections in Rutherford's experiment were common — they were about 1 in 20,000, and it is precisely their rarity, alongside their sharpness, that proves the nucleus occupies a vanishingly small volume.
• Believing Rutherford's model explains atomic stability or line spectra — it explains neither; both are explicit, named limitations that motivated Bohr's model.
• Mixing up which particle is fired and which is struck — positively charged alpha particles are fired at the gold foil; it is not electrons being scattered.

Ask the tutor: want to work through why classical electromagnetism forces an orbiting electron to spiral inward, or see how the marble-and-town size comparison is built step by step?