Physics · Module 24
Ten thousand times smaller than the atom it sits in, and holding a million times the energy of any chemical bond. This module is about how we found it, what holds it together, why some of them fall apart on a schedule nothing can alter, and why the Sun has been burning for four and a half billion years without running low.
In 1909 the accepted picture of the atom was J. J. Thomson's: an even, spread-out ball of positive charge, about 10−10 m across, with electrons embedded in it like raisins in a cake — which is why it is still called the plum-pudding model. It was a perfectly reasonable model. Nobody had any evidence against it.
Ernest Rutherford set two students, Geiger and Marsden, to fire alpha particles — helium nuclei, fast and heavy — at a very thin gold foil, and count where they went. On the pudding model the answer was easy to predict: the positive charge is smeared out thinly over the whole atom, so the electric field anywhere inside it is weak, and a fast alpha ploughing through will be nudged by a fraction of a degree at most. Thousands of tiny nudges, all in random directions, add up to a small blur. Every alpha should go essentially straight on.
Almost all of them did. But Rutherford had asked Marsden to check, as a matter of thoroughness, whether any came backwards.
About one alpha particle in every eight thousand bounced back off the foil — deflected through more than 90°, some of them straight back the way they had come.
What does that force you to conclude about the atom?
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 Rutherford, recalling 1909
To turn a fast alpha around you need an enormous force, and to get an enormous force out of Coulomb's law (Module 13) you need to get very close to a large charge. Here is why that forces the charge to be tiny. If the positive charge is spread through the whole atom, then as soon as you are inside the ball, most of it is behind you and pulls the other way; only the part still ahead of you counts, and that part shrinks as you go in. You can never get close to all of it at once, because you are always in the middle of it. The only way to have the whole charge on one side of you and very near is for the whole charge to occupy almost no room at all. Rutherford worked backwards from the fraction that bounced and arrived at a size: the positive charge of a gold atom is packed into a region under 10−14 m across, ten thousand times smaller than the atom.
Try to hold that ratio in your head, because it is the fact this whole module rests on. If a nucleus were a marble on the centre spot of a cricket ground, the atom would reach the boundary. Everything in between is empty in the strong sense: not "filled with thin gas", but nothing at all except the standing waves of Module 22. Every solid object you have ever touched is, by volume, about 99.9999999999 per cent nothing.
Now notice where the picture misleads, because this matters. The boundary is not a wall and there is nothing solid at it — the atom simply fades out where the electron's standing wave dies away. And if you conclude from all that space that things should fall through each other, you have the wrong stopper: your hand is not held up by nuclei meeting nuclei, it is held up by the electron waves of one surface refusing to overlap those of another (Module 22). Alphas can cross the empty space; your finger cannot.
Switch to Thomson's model and fire a few thousand. Every single one goes essentially straight through, because with the charge spread out there is nowhere to get close to a lot of it. Switch back to Rutherford's and the rare violent bounces appear. Geiger and Marsden's result was not a small anomaly to be explained away; it was flatly impossible on the old picture.
The last ledger box is doing something worth noticing. It reports the closest an alpha got to the centre before being turned around. That distance is an upper limit on the nuclear size — if the nucleus were bigger than that, the alpha would have hit it and the scattering would not match Coulomb's law. The scattering does match, so the nucleus must be smaller. Rutherford got the nuclear size out of nothing but energy conservation and the inverse-square law.
Now a problem that should be obvious and is usually skated over.
A uranium nucleus contains 92 protons, all positive, all packed into a space about 15 femtometres across — a femtometre (fm) being 10−15 m, a millionth of a billionth of a metre. Put two protons 2 fm apart and Coulomb's law gives a repulsion of about 58 newtons. Fifty-eight newtons is the weight of a six-kilogram bag of rice — applied to an object of mass 1.7 × 10−27 kg. That is a colossal force at that scale. Multiply it up over 92 mutually repelling protons and the thing should fly apart instantly, violently, and permanently.
It does not. So there is another force, and it must be:
This is the strong nuclear force — the third force this course has met, counting properly: gravity, electromagnetism (Module 20 showed you electricity and magnetism are one), and now this. A fourth exists, and this module will introduce it without naming it until now: the weak force, which turns a neutron into a proton in beta decay. Four forces; that is the full list. The strong force is not an inverse-square law; it switches on hard at about 1 fm and is essentially gone by 3 fm, and at very short range it turns repulsive, which is why nuclei do not collapse to a point.
Its short range explains the shape of the chart of the nuclei — the plot of every known nucleus by its proton count and neutron count, which you are about to meet.
Here is the measurement that turns all of this into energy.
Take a helium-4 nucleus: two protons and two neutrons. Weigh a proton, weigh a neutron, add up two of each. Now weigh the helium nucleus itself.
It is lighter. Measurably, reliably lighter — by about 0.7%. Nothing leaked out: the four particles really did weigh that much, and the object they became really does weigh less. The mass left as energy the instant they snapped together — which is not a bookkeeping trick but the same fact read twice, from either side of E = mc².
Module 20 tells you what to do with a mass discrepancy. E = mc², where E is the energy the missing mass would release. The missing mass is the binding energy. Read that in two directions. Going one way: it is the energy you would have to supply to tear the nucleus apart into free protons and neutrons. Going the other: it is the energy that came out when they first snapped together — like the height you must climb a hill equals the height you fall coming down. Assembling a helium nucleus from its parts releases 28.3 MeV — about 7 MeV for each of its four nucleons.
Compare that with chemistry. Burning carbon releases about 4 eV per atom. Nuclear processes release millions of electron-volts per nucleus. That factor of a million is the whole difference between a coal fire and a reactor, and it comes from two things multiplied together. The strong force is about a hundred times stronger than the electric force between the same two particles — and it acts across a femtometre rather than the tenth of a nanometre a chemical bond spans, while electrical energy goes as 1/r. A hundred thousand times closer in, times a hundred times stronger: that is the factor of a million.
Now do this for every nucleus and plot the binding energy per nucleon — how tightly each individual particle is held. That single curve explains both nuclear power and the Sun.
The curve rises steeply from hydrogen to iron, peaks at about A = 56, and then falls slowly all the way to uranium. Height on the curve means tightly bound — and a tightly bound nucleon has already given up its energy. A nucleon that moves higher gives up the difference.
Which processes release energy?
Both, and for one reason: energy comes out whenever nucleons end up more tightly bound than they started, and the peak at iron is the most tightly bound place there is. Light nuclei get there by fusing; heavy nuclei get there by splitting. Iron-56 sits at the top of that curve — the top of the graph and the bottom of the energy well at the same time, because high on the curve means the nucleus has already given up as much energy as it ever will. (Nickel-62 is marginally more tightly bound, but for our purposes the peak is iron.) Both can do neither. They are the ash of the universe.
Which is why a massive star, fusing its way up the curve — hydrogen to helium to carbon to oxygen to silicon — comes to a dead stop at iron. Fusing iron would consume energy rather than release it, so the furnace that has been holding the star up against its own gravity switches off in seconds. The core collapses, and the star explodes. The iron in your blood was made in the last day of a dying star. And everything heavier than iron — silver, gold, uranium — was made in the explosion itself or in the collision of neutron stars, because those elements cost energy to build and only a catastrophe can pay.
A nucleus that is off the valley of stability will change until it gets there. There are three classic ways — in the table, Z is the proton count (the atomic number) and A is the total number of nucleons.
| Decay | What is emitted | What changes | What stops it |
|---|---|---|---|
| Alpha | A helium-4 nucleus: two protons and two neutrons, bound very tightly, which is why this particular clump comes out | The proton number Z (which decides the element) drops by 2, and the total nucleon count A drops by 4. Heavy nuclei shedding bulk. | A sheet of paper, or the dead outer layer of your own skin. That makes an alpha source almost harmless from outside — and the most damaging of the three once it is inside you, because being stopped so easily means it dumps all its energy into a few cells instead of passing through. This is why radon, an alpha-emitting gas that seeps out of some rocks and collects in unventilated rooms, is a genuine health hazard, and why "stopped by paper" must never be read as "harmless". |
| Beta | An electron (together with an antineutrino — a nearly massless, chargeless particle that carries off some of the energy and is almost impossible to detect), created at the instant of decay — it was not sitting inside waiting | A neutron turns into a proton — the weak force does this conversion: Z rises by 1, A unchanged. A nucleus with too many neutrons correcting its ratio. | A few millimetres of aluminium. |
| Gamma | A very high-energy photon — Module 18's electromagnetic wave, at the far end of Module 18's spectrum | Nothing. The nucleus was left in an excited state by a previous decay and drops to its ground state, exactly as an atom does in Module 22, but with a million times the energy gap. | Centimetres of lead, and even then only partly. |
Alpha decay is worth a moment, because Module 23 already explained it. An alpha particle inside a heavy nucleus is trapped behind a barrier of electrical repulsion far higher than its own energy. Classically it can never get out, and uranium would last for ever. It escapes by tunnelling. Tunnelling gives each nucleus a fixed, small probability of escaping in any given second, and the half-life is simply how long until that probability adds up to a half: wait one half-life and there is a 50-50 chance any particular nucleus has gone. Because tunnelling probability depends exponentially on the barrier, tiny differences in nuclear size and alpha energy produce half-lives ranging from microseconds to billions of years. Polonium-214 has a half-life of 164 microseconds; uranium-238 has one of 4.5 billion years. The difference in alpha energy between them is less than a factor of two. That is what an exponential does.
A particular nucleus has a half-life of 5 000 years. This one has already been sitting in a rock for 20 000 years without decaying.
How likely is it to decay in the next 5 000 years?
Exactly 50%. This is the strangest and most important thing about radioactivity, and it is easy to miss because "half-life" sounds like an ageing process.
Nuclei do not age. There is no internal clock, no wearing out, no accumulated strain. Each nucleus, at every instant, has exactly the same probability of decaying in the next second as it had on the day it formed — and as every other identical nucleus has. A uranium atom that has sat in a rock since the Earth condensed is not a tired old uranium atom; it is indistinguishable in every measurable way from one made this morning.
So why does a lump of it decay on such a reliable schedule? Because of numbers. Any real sample contains something like 1020 nuclei. Each one is a private, unpredictable event, and yet the total is as regular as anything in physics — for the same reason that an insurance company cannot tell you when you will crash your car but can predict the national claims figure to within a percent: large numbers of independent events smooth out. Notice where the comparison stops working, though: a driver does age, and insurers charge a seventeen-year-old differently from a forty-year-old. Nuclei have neither history nor age. Take from the analogy only the part about large numbers, not the part about risk changing over time.
Run it once with 20 nuclei and once with 4000, and watch the difference. With twenty, the measured half-life bounces around wildly and the "curve" is a lurching staircase. With four thousand it is a smooth exponential and the measured half-life lands within a few percent of the true one. The law of radioactive decay is a law of large numbers, and the underlying events are irreducibly random — this is Module 21's probability, showing up in a rock.
Because the schedule cannot be altered — not by heat, pressure, chemistry, or anything else anyone has ever tried — a radioactive nucleus is a clock that started when the sample formed and cannot be reset.
Carbon-14, half-life 5 730 years, is made continuously in the upper atmosphere by cosmic rays and taken up by every living thing. When something dies it stops taking in fresh carbon and the C-14 in it starts running down. Measure how much is left and you have the date. The method runs out after about nine half-lives, roughly 50 000 years, because beyond that too little of the original carbon-14 survives to measure — which still covers most of human prehistory.
Uranium-235 and 238, half-lives 700 million and 4.5 billion years, decay through long chains ending in different isotopes of lead. Measure the ratio of uranium to lead in a zircon crystal and you get the age of that crystal. The oldest ones on Earth come out at about 4.4 billion years, which sets a floor under the planet's age but not a ceiling — the surface gets remelted and recycled. For the Earth's actual birthday, you date something that has not been recycled: meteorites from the same original cloud. Those give 4.54 billion years, to about one per cent.
Potassium-argon dating of volcanic ash layers is what dates hominid fossils, because the ash above and below a fossil bed can be dated even when the fossil cannot.
Uranium-235 is on the right-hand slope of the binding-energy curve. Split it roughly in half and the two fragments sit much closer to the iron peak, so energy comes out: about 200 MeV per nucleus, which is around fifty million times what you get from burning one carbon atom.
What makes it a technology rather than a curiosity is that fission is triggered by a neutron — and produces more of them. A U-235 nucleus that absorbs a slow neutron becomes briefly U-236, wobbles, and splits, throwing out two fragments and, on average, about 2.4 fresh neutrons. Each of those can go on to split another nucleus.
Everything then depends on a single number, which physicists call k: how many of the neutrons from one fission go on to cause another fission, on average.
Three things fix k: the concentration of U-235, whether the neutrons are being slowed down, and the size and shape of the lump. Hold the first two constant and the third is pure geometry: neutrons are lost by escaping through the surface, and they cause fissions in the volume. Volume grows as the cube of the size and surface only as the square, so a bigger lump loses proportionally fewer. Below a certain size — the critical mass — too many escape and k stays under 1 no matter what.
The bench counts what actually happens: how many fissions occur in each generation, and what the ratio between consecutive generations turns out to be. Nothing tells it what k should be — k is what comes out.
The two devices are not the same machine at different settings. Reactor fuel is uranium enriched to 3–5% U-235 — meaning 3–5% of its atoms are U-235 rather than the far more common U-238. A weapon needs above 90%, and the enormous industrial effort of getting from one to the other is the main reason reactor fuel is not a route to a weapon. A neutron straight out of a fission is far too fast to be captured efficiently by U-235. A reactor therefore surrounds its fuel with a moderator — water, or graphite — whose light nuclei slow the neutrons down over many collisions until they are slow enough to be absorbed. That slowing takes time, and it is why a reactor's chain reaction runs on a timescale of seconds rather than nanoseconds. Take the moderator away and the surviving fast neutrons cannot keep a chain going at all in 3–5% fuel: it is the low concentration of U-235, not the shape of the machine, that closes off the fast chain. A reactor therefore cannot produce a nuclear explosion, however badly it fails — though, as below, it can still destroy itself.
What makes control possible at all is a subtlety worth knowing. About 0.65% of the neutrons from fission are not released immediately but seconds later, from the decay of certain fission fragments. A reactor is run so that it is subcritical on the prompt neutrons alone and reaches k = 1 only with the delayed ones included. That turns the response time of the whole machine from nanoseconds to seconds, which is what makes it controllable by mechanical rods moved by motors. Without that 0.65%, nuclear power would be impossible to regulate.
There are two dangers, and the second is the one most people have never heard of. The chain reaction itself can still run away: if k rises above 1 on the prompt neutrons alone, the power can multiply faster than any rod can move, and the reactor wrecks itself. That is what happened at Chernobyl in 1986 — not a nuclear explosion, but a power surge that destroyed the core in seconds. The second danger outlives the first. Even after the chain is stopped dead, the accumulated fission fragments go on decaying and go on producing heat — several per cent of full power at first, falling slowly over days. If cooling fails, that heat alone is enough to melt the core. Fukushima's reactors shut down correctly within seconds of the earthquake; what destroyed them was the loss of cooling for the decay heat afterwards.
The other way to the iron peak is upward, from hydrogen. Four hydrogen nuclei combine, through a chain of steps, into one helium-4, releasing 26.7 MeV. The Sun does this about 1038 times a second, converting roughly four million tonnes of its mass into energy every second — and has done for four and a half billion years.
But there is a serious problem with that sentence, and it took until 1928 to solve.
The resolution has a lovely structure to it, and the next bench draws it. Two effects fight each other. The number of protons with a given energy falls off steeply as the energy rises (Module 4, sharpened). The probability of tunnelling at a given energy rises steeply as the energy rises (Module 23). Multiply the two and you get a sharp peak at an energy far above the average but far below the barrier — the Gamow peak. Almost all of the Sun's fusion happens in that narrow window, and it involves a vanishing fraction of the protons.
The bench shows that the fusion rate depends extraordinarily steeply on temperature — raising the core temperature by 10% raises the rate by about half as much again (the rate goes roughly as T4 near solar conditions).
Now suppose the Sun's core were briefly to get a little hotter than usual. What happens?
The Sun is a thermostat. If the core heats up, the gas pressure rises, the core expands against gravity, and expansion cools it — Module 4's gas laws, doing exactly what they did in a bicycle pump. The fusion rate drops back. If the core cools, it contracts, contraction heats it, and the rate climbs. The core settles at whatever temperature makes the pressure of its gas exactly balance the weight of the star pressing down — and the fusion rate at that temperature is what keeps the gas that hot.
That feedback loop has held the Sun's output steady to within a fraction of a per cent for billions of years, which is the reason there has been time for anything to evolve on this planet.
Now switch the bench to the fuel budget and do the arithmetic that follows. The Sun has about 2 × 1030 kg of mass, of which roughly a tenth is hot and dense enough to take part: 2 × 1029 kg of usable fuel. Fusing hydrogen to helium converts about 0.7% of that into energy (Module 20, cashed in at last), so 1.4 × 1027 kg turns into energy. Now use E = mc²: multiply by 9 × 1016 and you get about 1.3 × 1044 joules. Divide that by the Sun's measured output of 3.8 × 1026 watts and you get 3 × 1017 seconds — around ten billion years, of which four and a half have gone.
Do the same sum with any chemical fuel and you get a few tens of thousands of years. In the nineteenth century, before anyone knew about nuclei, this was a genuine crisis: Lord Kelvin calculated the Sun's maximum possible age from gravitational contraction and got about 20 million years, and then told the geologists and Darwin that they were wrong, because their evidence needed hundreds of millions. He was not being unreasonable; he was missing E = mc², which would not exist for another forty years. The rocks were right.
The Sun's power output is 3.8 × 1026 watts, which is 3.8 × 1026 joules every second. Use E = mc² backwards: m = E/c². With c² = 9 × 1016, that is about 4.2 × 109 kg per second — four million tonnes. Every second. Since you started reading this module, the Sun has lost more mass than the total weight of every human being alive.
Then work out how much of the Sun that is. Its total mass is 2 × 1030 kg; over its whole 4.5-billion-year life so far it has lost roughly 6 × 1026 kg, which is about 0.03% — a fraction so small that the Earth's orbit has barely noticed. The Sun is enormously wasteful and enormously rich.
| Idea | What it says |
|---|---|
| The nucleus | Found by Rutherford's back-scattered alphas. Ten thousand times smaller than the atom, and holding essentially all its mass. Matter is overwhelmingly empty. |
| The strong force | Attractive, charge-blind, and dead beyond a few femtometres. Its short range against electricity's long range is why heavy nuclei need surplus neutrons and why there is a heaviest element. |
| Binding energy | A nucleus weighs less than its parts. The missing mass is mc² of binding energy — millions of eV per nucleus against a few eV for chemistry. |
| The curve | Binding energy per nucleon peaks at iron-56. Fuse below it, split above it, and either way energy comes out. Iron is the ash. |
| Decay | Alpha (tunnelling out), beta (a neutron becoming a proton), gamma (a photon from an excited nucleus). Alpha decay's enormous range of half-lives is an exponential at work. |
| Half-life | Nuclei do not age. Each has the same decay probability every second, for ever. The exponential curve of a sample is a law of large numbers, and it is what dates rocks and bones. |
| Fission | 200 MeV per nucleus, and 2.4 fresh neutrons. Everything depends on k, and k is mostly geometry: surface loses, volume gains. |
| Fusion | The Sun's protons have a five-hundredth of the energy they need and get through by tunnelling. The Gamow peak sits between a falling Boltzmann tail and a rising tunnelling probability. Gravity and pressure make the whole thing a thermostat, good for ten billion years. |