Almost all of it is empty

Alpha particles fired at gold foil mostly sail through, and about one in eight thousand comes straight back. That single fact puts all of the positive charge in a volume a hundred billion times smaller than the atom — and hands the whole of chemistry to the electrons outside it.

1897 – 191315 min

By the end you should be able to

Explain why Thomson’s atom was a serious, quantitative model and what it accounted for
Say why a large deflection cannot be built out of many small ones
Estimate the size of a nucleus from the energy of the particle that bounces off it
State what the nuclear atom settles about chemistry, and what it leaves wide open

Start in 1904, with an atom that works. Thomson had found the electron in 1897 — a particle he could pull out of any gas, always the same, and almost two thousand times lighter than a hydrogen atom. That is the first evidence anyone ever had that an atom has parts. Atoms are electrically neutral. So if there are negative electrons inside them, there must be an equal positive charge somewhere too. And here is the difficulty: nobody had ever detected a positive particle inside an atom. Cathode rays gave you electrons readily. Nothing gave you their opposite number. So Thomson made the modest assumption, which is very often the right move. The positive charge is not a particle at all. It is a diffuse sphere, roughly the size of the whole atom, with the electrons sitting inside it. He called them corpuscles; the currants-in-a-pudding nickname came later, and slightly unfairly. It is easy to hear this as a placeholder. It was not. It was a specific, quantitative model, and Thomson worked out its consequences with some care.

What that atom bought you: Neutral matter. The charges balance by construction, which is what any model has to manage before anything else. Ions, with a mechanism. Remove an electron and what remains is a positive ion. Chemists had been writing Na⁺ and Cl⁻ for decades on purely electrochemical evidence, without any account of what the charge physically was. Now there was one. Why the electron is universal. Every element yields the same particle, so every element must contain them. That is a unification of the same kind as phlogiston’s, and this one was right. A first stab at periodicity. This is the part usually left out. Thomson calculated the stable configurations of electrons inside his positive sphere and found they arrange into concentric rings, with each ring holding only so many before a new one starts. He noted in 1904 that this might be the origin of the periodic table’s repeating pattern. It is a genuine prediction, from a genuine model, aimed at exactly the fact this act exists to explain. The reason the model could be killed is the reason it was worth having: it said something definite enough to be checked.

To test a model of the atom you need something you can throw at one, and it has to be a probe the electrons cannot deflect — otherwise you learn about the electrons instead. Radioactivity had just supplied exactly the right thing, for nothing. An alpha particle is a helium nucleus. Rutherford established that in 1908 by collecting alphas in an evacuated tube, letting the charge neutralise, and photographing the spectrum of the gas that resulted: helium, unmistakably. That work brought him the Nobel Prize — in Chemistry, which he found extremely funny. He remarked that he had studied many transformations with various half-lives, but the quickest he had ever met was his own instantaneous transformation from a physicist into a chemist. It is heavy. About 7,300 times the mass of an electron. An electron can no more deflect an alpha than a fly can deflect a bowling ball, which means the whole electron cloud is transparent to it and every deflection observed is a deflection by positive charge. It is charged, and fast. Two units of positive charge, arriving with several million electronvolts — around a million times the energy of a chemical bond. That is what lets it get close enough to a nucleus to be turned round. Nobody designed this. The right probe fell out of a radium salt two decades before anyone built a machine that could have made one.

The experiment

On a diffuse reflection of the α particles

Hans Geiger and Ernest Marsden, in Ernest Rutherford’s laboratory · 1909 · The University of Manchester

The question: Alpha particles passing through thin metal foil emerge with the beam slightly blurred. Is any fraction of them turned through a large angle — more than a right angle, back towards the source?
The apparatus: A radium source in a lead block, giving a collimated beam of alpha particles. A metal foil or plate. A screen of zinc sulphide, which flashes when an alpha strikes it, viewed through a microscope. A darkened room, and eyes given half an hour to adapt before counting begins. Marsden was an undergraduate; Rutherford suggested the search as something to keep him occupied.

Theory predicted

On Thomson’s atom, nothing at all. A positive charge smeared over the whole atom can deliver only a feeble sideways impulse — at these energies about 0.012° per atom — and the deflections from successive atoms point in random directions, so they accumulate as a random walk rather than a sum. Passing through 11,684 atoms of gold gives a spread of roughly 0.9°. Reaching ninety degrees that way is not improbable. It is arithmetically excluded.

They measured

Alpha particles came back. Marsden found them on the first afternoon of looking. Against a thick platinum reflector, roughly one incident particle in eight thousand was returned through more than a right angle — a small number, and an infinitely larger one than zero.

How sure could they be? The famous “one in eight thousand” deserves a footnote. It is a 1909 figure, measured against a platinum plate thick enough that a particle got many chances at a close approach, and thick enough for the count to be approaching saturation. It is not the single-scattering rate for one thin foil, which depends on thickness, on the metal, and on the alpha’s energy: for 3.0 µm of gold at 7.7 MeV the simulation computes about 1 in 8,230. The argument is completely untouched by the difference — the prediction it destroys is zero — but a number that has been repeated for a century is worth knowing the provenance of.

Why it mattered

The positive charge of an atom cannot be spread out. It has to be concentrated somewhere small enough and dense enough that a single encounter can turn a fast, heavy particle right round. Nothing else in the atom is capable of it.

You cannot build a right angle out of hundredths of a degree

This is the step that makes the result decisive rather than merely surprising, and it is worth doing slowly. Small deflections do not add up. They cancel. Each atom the alpha passes pushes it in some direction, and the directions are uncorrelated. A thousand pushes of one unit each do not give a thousand units; they give about thirty-two, the square root. That is a random walk, and it is why a beam through foil comes out slightly blurred instead of sprayed everywhere. Run the numbers for Thomson’s atom. Spread gold’s positive charge over the atom’s own radius and the largest possible kick from one atom is about 0.012°. A 3.0 µm foil is roughly 11,684 atoms deep. Square-root of that is about 108, so the beam should emerge spread by about 0.9° — which is exactly what was already known and not in dispute. And the chance of reaching ninety degrees that way is around 10⁻⁴¹⁷². Rutherford did this calculation himself and got about 10⁻³⁵⁰⁰ for his case. The exponent is not the point and neither man cared about its third digit. The point is that this is not a rare event. It is not an event. So there must be a single violent encounter. One atom, one approach, one deflection of ninety degrees or more. For that you need a field strong enough to reverse a particle carrying 7.7 MeV, and an inverse-square field only reaches that strength if you can get very close to the whole of the charge at once. Which means the charge must be very small.

The reason a spread-out charge is so feeble is worth seeing directly, because it is the same piece of electrostatics that makes gravity inside the Earth fall to zero at the centre. Inside a uniform ball of charge, only the charge beneath you pulls. Everything further out cancels. So as an alpha penetrates Thomson’s positive sphere, the force on it does not climb — it falls, reaching zero at the very centre. The hardest shove is available at the surface, and the surface is the edge of the atom, which is a long way from anything. Concentrate the same charge into a point and nothing cancels. The field goes as 1/r² all the way in, with no floor, and a particle aimed close enough meets an arbitrarily large force. There is no upper limit on the deflection at all: aim dead-on and it comes straight back. That is the whole argument, and the simulation below is that one sentence with a slider on it. The charge is the same in both cases. Only the volume it occupies changes.

A beam of alpha particles crossing gold foil, with the positive charge concentrated. Watch the detector arc: almost nothing reaches it, and then something does.

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Two thousand alphas a second, which is roughly the rate Geiger sat counting in the dark. The beam is drawn as a band because the overwhelming majority go straight through and are indistinguishable from one another; only particles deflected past ten degrees get a track of their own. About 1 in 8,230 is thrown back past ninety degrees, in red. The atom really is mostly nothing — to an alpha particle.

The same beam, the same foil, the same total charge — now spread over the whole atom, as Thomson had it. Then drag the slider back down and watch where the tracks reappear.

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Thomson’s atom, run identically. The largest deflection one atom can deliver is 0.012°, the detector arc stays empty, and it stays empty however long you leave it. This is the rare case where a theory does not predict a small number — it predicts none, and one observation is enough. Sliding the charge radius back down is the entire 1911 argument, in one control.

It was quite the most incredible event that has ever happened to me in my life. 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.
Ernest RutherfordFrom a lecture at Cambridge in 1936, printed in *The Background to Modern Science* (1938). It is a recollection given twenty-five years after the fact, not something he wrote at the time — the 1911 paper is entirely sober. The line is genuine and it is his; it is just not a contemporary record of the moment.

The experiment

The laws of deflexion of α particles through large angles

Hans Geiger and Ernest Marsden · 1913 · Manchester

The question: Rutherford’s 1911 formula makes four separate quantitative predictions. Do all four hold?
The apparatus: The same scintillation counting, done properly: a rotating microscope and screen so the angle could be varied continuously, foils of gold, silver, copper and aluminium, and mica absorbers to slow the alphas and vary their energy. Observers counted in shifts, in the dark, for over a year.

Theory predicted

The number scattered into a given direction should go as 1/sin⁴(θ/2); as the square of the nuclear charge of the foil; in proportion to the foil’s thickness; and inversely as the square of the alpha’s energy.

They measured

All four. The angular law was followed from 5° out to 150°, over which the counting rate changes by a factor of about 250,000, with the product of rate and sin⁴(θ/2) staying constant to within the counting statistics. Gold scattered roughly thirty times as strongly as aluminium, which is close to (79/13)². The thickness and energy dependences came out as predicted too.

Why it mattered

This is what promotes a striking anecdote into a measurement. One backscattered particle rules out Thomson; a law that holds over five and a half orders of magnitude in rate, across four metals and a range of energies, establishes the form of the field — an inverse-square field from a charge small enough to be treated as a point at these distances.

What the 1911 paper does not claim

The picture in most people’s heads — a tiny sun with electrons orbiting it — is not in Rutherford’s paper, and reading the paper is a useful corrective. He writes “central charge”, not “nucleus”. The word nucleus does not appear in the 1911 argument; it arrives a year or two afterwards. He says the sign cannot be determined. In as many words: the scattering law comes out identically for a positive or a negative central charge, so his experiment cannot distinguish them. He assumes positive for other reasons, and says so. There is nothing about orbits. No shells, no radii, no arrangement of the electrons at all. He needs them to be outside the central charge and that is the whole of his commitment. And the charge itself was only roughly known. Rutherford put it at about half the atomic weight in units of the electronic charge, which gives around 100 for gold. The true answer is 79, and pinning it exactly — nuclear charge equals atomic number, exactly, for every element — took Moseley’s X-ray work in 1913. A paper that establishes a scattering law and refuses to speculate beyond it is not a lesser achievement than the solar-system picture. It is why the picture survived being wrong in almost every detail.

What this hands to chemistry

An alpha of 7.7 MeV stops 29.5 femtometres from the centre of a gold nucleus, so the nucleus is smaller than that. A gold atom’s radius is 144,000 femtometres. That is a ratio of about 4,874 in radius, and about one part in 10¹¹ in volume. So, for chemistry: Everything an atom can touch is electrons. Size, shape, colour, hardness, every bond that has ever been formed. Two atoms approaching each other never come remotely near enough for their nuclei to matter directly; what meets is two clouds of charge. The nucleus contributes exactly two things. How many electrons the atom carries, and how hard it holds them. Both are the same number — its charge. Which is why isotopes are chemically the same. Change the mass and keep the charge, and you change essentially nothing a chemist can measure. That is also why the table had to be ordered by charge rather than by weight, and why the handful of pairs Mendeleev had to place out of mass order were never really anomalies. And it is why this act exists. If chemistry is the electrons, then the arrangement of the electrons is the whole subject. Rutherford’s experiment says nothing whatever about that arrangement.

Problem

How big can the nucleus be?

An alpha particle carrying 7.7 MeV of kinetic energy is fired straight at a gold nucleus. It slows, stops, and comes back. At the instant it stops, all of its kinetic energy has become electrostatic potential energy. How close does it get? Give the distance in femtometres. (These are the simulation’s default settings, so you can check yourself against it.)

Charge of an alpha particle: z = 2
Charge of a gold nucleus: Z = 79
Kinetic energy: 7.7 MeV
Coulomb constant: e²/4πε₀ = 1.440 MeV·fm

You might think

The gold foil experiment showed that atoms are mostly empty space.

Actually

It showed that the mass and the positive charge occupy almost none of the volume, which is not the same claim. The electrons are spread through the rest of it, and they are not a thin garnish: they are what stops your hand going through a table, they are the entire volume of every molecule, and they are the whole of chemistry. An alpha particle passes through them because it is about 7,300 times heavier than an electron and travelling at a twentieth of the speed of light — it goes through the electron cloud roughly as a bowling ball goes through a swarm of flies, barely noticing. The emptiness the experiment demonstrates is emptiness to a fast alpha particle. To another atom, an atom is not empty at all; it is completely full of the only thing that matters.

1897Thomson identifies the electron: the same particle from every gas, and far lighter than any atom.
1904Thomson’s model — electrons in a diffuse sphere of positive charge — with rings that close off, and a suggested origin for periodicity.
1906Rutherford notices that alpha beams are slightly blurred by passing through mica, and starts asking why.
1909Geiger and Marsden find alphas coming back from a metal reflector. About one in eight thousand, off thick platinum.
1911Rutherford derives the scattering law, and shows it needs a central charge occupying almost none of the atom.
1913Geiger and Marsden test the law over a factor of 250,000 in counting rate, and across four metals.
1913Moseley: nuclear charge equals atomic number, exactly. The table’s order is a nuclear fact.
1913Bohr puts the electrons in quantised orbits — and inherits an atom that classical physics says cannot exist.

Where this leaves us

Chemistry belongs to the electrons. The nucleus is a point of charge sitting at the centre, contributing a count and a grip and nothing else that a chemist will ever measure directly. Which makes the outstanding question sharp, and rather uncomfortable. The rows of the periodic table are two, eight, eight, eighteen. Something about the electrons must produce those numbers. And there is nothing whatsoever in the gold foil experiment to suggest that the electrons have any structure at all — Rutherford needs only that they are outside the nucleus. Bohr will propose orbits in 1913, and the proposal will be brilliant and unjustifiable in equal measure: it works, and the only defence for its central assumption is that it works. But there is a measurement that shows the structure without proposing anything, and it is almost embarrassingly direct. Take an atom. Pull off one electron and write down what it cost. Pull off the next and write that down too. Keep going until the atom is a bare nucleus. The numbers do not climb smoothly. They step — and they step in exactly the places the periodic table says they should.