How Does a Nuclear Reactor Work?
Split one uranium nucleus and you release fifty million times more energy than burning one atom of coal. The hard part was never the energy. It was holding the chain reaction at exactly one.
A very complicated kettle
Junior level — plain language, no maths
Here is the part nobody tells you first: a nuclear power station makes electricity by boiling water. Steam spins a turbine, the turbine spins a generator, and the generator makes the electricity - exactly like a coal plant. The only unusual thing is the kettle.
Inside that kettle, uranium atoms are being split in half. Fire a neutron - a tiny neutral particle - at a uranium-235 nucleus and it wobbles, stretches and tears apart. The two pieces fly away from each other at enormous speed, and when they crash to a halt in the surrounding metal, all that speed becomes heat. One split releases about fifty million times more energy than burning a single atom of coal. A fuel pellet the size of your fingertip holds as much energy as a tonne of coal.
But the real trick is what else comes out. Each split also throws out two or three fresh neutrons, and each of those can split another nucleus. One becomes three, three become nine, nine become twenty-seven. That is a chain reaction, and left alone it doubles again and again in a fraction of a second. What separates a power station from a disaster is that the reactor is built to sit at exactly one: every split causes precisely one more split, for ever, steadily.
Keeping it there takes two things. Control rods made of boron slide in between the fuel and swallow spare neutrons - push them in and the reaction dies down, pull them out and it picks up. And the water is not just there to boil. Neutrons come out of a fission far too fast to split anything else, and bouncing off water molecules slows them down to the speed where they work. Which means the water is part of the reaction: lose the water and, in a Western reactor, the chain reaction stops on its own. That is not a safety system. It is physics doing the job for you.
Things worth knowing
- One gram of uranium-235 releases about as much energy as three tonnes of coal. A single fuel assembly the size of a person powers roughly a thousand homes for a year.
- About 0.65% of the neutrons come out seconds late rather than instantly. Without that tiny delay the reactor's power would double in milliseconds and no control system on Earth could keep up. Reactors are controllable because of a rounding error in the physics.
- You can stop the chain reaction in a second, but not the heat. Immediately after shutdown a reactor still makes about 7% of its full power from radioactive decay - that leftover heat is what melted the cores at Fukushima, days after the fission had stopped.
k, reactivity and the delay that makes control possible
Student level — the core equations
Everything reduces to one number. The multiplication factor \(k\) is how many fissions each fission causes in the next generation. Below one and the reaction dies away - the reactor is subcritical. Above one it grows exponentially - supercritical. Exactly one, and the power holds steady: critical, which despite the word is the normal, boring, everyday operating state of every reactor on the planet.
Reaching \(k = 1\) is a design problem before it is a control problem. A neutron born in fission carries about 2 MeV and is far too fast to split another U-235 nucleus efficiently: the fission cross-section at that energy is tiny. Slow it to thermal energies, around 0.025 eV, and the cross-section rises by a factor of several hundred. That is what the water is for. It takes roughly twenty collisions with hydrogen nuclei to thermalise a neutron, and only then is it useful. Remove the water and you do not just lose cooling, you lose the moderator - and \(k\) falls below one by itself. This negative void coefficient is the single most important safety property of a light-water reactor, and its absence in the RBMK design is the reason Chernobyl was possible.
Now the delay. Most fission neutrons appear within \(10^{-14}\) s, and with a prompt neutron lifetime \(\Lambda \approx 2\times10^{-5}\) s, a reactor at \(k = 1.001\) driven by prompt neutrons alone would double its power every 14 milliseconds. Nothing mechanical could follow that. But about \(\beta = 0.65\%\) of neutrons are emitted seconds later by the decay of fission fragments, and those delayed neutrons dominate the timing whenever the reactivity stays below \(\beta\). The reactor period becomes tens of seconds instead of milliseconds, and a human with a lever can run it.
Which sets the one line you must never cross. Reactivity \(\rho = (k-1)/k\) below \(\beta\) means the delayed neutrons are still in charge, and control is comfortable. Reach \(\rho = \beta\) and the reactor is prompt critical: it no longer needs the delayed neutrons at all, the period collapses to milliseconds, and the power excursion is over before any rod can move. That is the boundary, and every reactor control system exists to keep a wide margin from it.
Key Formulas
| Multiplication factor | \(k = \dfrac{\text{fissions in generation } n+1}{\text{fissions in generation } n}\) | |
|---|---|---|
| Reactivity | \(\rho = \dfrac{k-1}{k}\) | often quoted in pcm = 10⁻⁵ |
| Power growth | \(P(t) = P_0 e^{t/T}\) | T = reactor period |
| Delayed fraction | \(\beta \approx 0.0065\) | 0.65% of neutrons, seconds late |
| Stable period | \(T \approx \dfrac{\beta-\rho}{\lambda\rho}\) | λ ≈ 0.077 s⁻¹ |
| Prompt critical | \(\rho \geq \beta\) | period collapses to Λ/(ρ−β) |
| Energy per fission | \(\approx 200\ \text{MeV} = 3.2\times10^{-11}\ \text{J}\) | |
| Decay heat | \(\dfrac{P}{P_0} \approx 0.066\,t^{-0.2}\) | t in seconds after shutdown |
Things worth knowing
- A thermal neutron has a fission cross-section on U-235 of about 585 barns; a 2 MeV fast neutron, about 1.2 barns. The moderator is not an accessory - without it, natural or low-enriched uranium simply cannot sustain a chain reaction.
- One watt of thermal power is about 3.1 × 10¹⁰ fissions per second. A 3 GW core is running roughly 10²⁰ fissions every second, and holding the count to within a fraction of a per cent.
- As fuel heats up, Doppler broadening of the U-238 resonances absorbs more neutrons, so reactivity falls. This feedback is prompt, unavoidable and always negative - it acts within milliseconds, long before any control system notices.
Six factors, point kinetics, and the feedbacks that decide whether a design forgives you
Scholar level — full mathematical depth
01Where the 200 MeV comes from
The binding energy per nucleon peaks near iron at about 8.8 MeV and falls to roughly 7.6 MeV at uranium. Splitting a heavy nucleus therefore moves its nucleons to a more tightly bound configuration, and the difference - about 0.9 MeV per nucleon across 236 nucleons - appears as kinetic energy. Roughly 168 MeV of the 200 goes into the two fission fragments, which stop within micrometres of where they were born, which is exactly why a fuel pellet heats itself rather than radiating its energy away. Another 5 MeV arrives as prompt neutrons, 7 MeV as prompt gammas, and about 20 MeV as delayed beta and gamma emission from fission products - the last of these being the decay heat that a shutdown reactor cannot switch off.
02The six-factor formula
For a finite reactor, \(k_{\text{eff}} = \eta f p \varepsilon P_{\text{FNL}} P_{\text{TNL}}\): the reproduction factor \(\eta\) (neutrons produced per thermal neutron absorbed in fuel), the thermal utilisation \(f\) (fraction of thermal absorptions occurring in fuel rather than moderator, cladding or control poison), the resonance escape probability \(p\), the fast fission factor \(\varepsilon\), and two non-leakage probabilities. Control rods act on \(f\), moderator density acts on \(p\) and on leakage, fuel temperature acts on \(p\) through Doppler. This decomposition is what lets a designer see which physical knob moves which term, and it is why the same reactivity change can be achieved by very different physical means with very different time constants.
03Cross sections and why moderation is not optional
The U-235 fission cross-section follows an approximate \(1/v\) law at low energy, reaching about 585 barns at 0.025 eV, against roughly 1.2 barns at 2 MeV. Between these lies the resonance region, where U-238 has enormous, narrow capture resonances; a neutron slowing through them can be captured and lost, which is what \(p\) measures. Spatial self-shielding in the fuel lumps and the geometry of the lattice are therefore not details but the design itself: heterogeneous lattices exist precisely so that neutrons do their slowing down in the moderator, away from the U-238 resonances, and return to the fuel already thermal.
04Point kinetics, and the meaning of the prompt-critical line
The point kinetics equations, \(\frac{dP}{dt} = \frac{\rho-\beta}{\Lambda}P + \sum_i \lambda_i C_i\) with \(\frac{dC_i}{dt} = \frac{\beta_i}{\Lambda}P - \lambda_i C_i\), contain the whole story. For \(\rho \ll \beta\), the prompt term is negative and the precursors act as a flywheel: the asymptotic period is set by \(\lambda\), tens of seconds, and there is even a fast prompt jump \(P \to P_0\beta/(\beta-\rho)\) before it. At \(\rho = \beta\) the prompt term vanishes, and beyond it the reactor is critical on prompt neutrons alone with period \(\Lambda/(\rho-\beta)\) - microseconds to milliseconds. This is why reactivity is quoted in dollars, one dollar being \(\beta\): the unit exists to make the boundary impossible to overlook.
05Feedbacks, and Chernobyl as an equation
Doppler broadening of the U-238 resonances is prompt and always negative: hotter fuel means broader resonances, more capture, less reactivity, and it acts in milliseconds. The moderator coefficient is more interesting. In a light-water reactor, boiling reduces moderation, so voids reduce \(k\) - a negative void coefficient, self-limiting. The RBMK used graphite as moderator and water only as coolant, and since water is a neutron absorber there, boiling increased reactivity. On 26 April 1986 a xenon-poisoned core, an operating point far below the permitted power, most control rods withdrawn, and graphite-tipped rods that briefly added reactivity on insertion combined into a positive feedback loop that went prompt critical in about four seconds. Every element of that was a property of the design, not of the operators alone.
06What comes after shutdown
Insert the rods and fission stops within a second, but the fission products keep decaying: \(P/P_0 \approx 0.066[t^{-0.2}-(t+t_0)^{-0.2}]\), giving about 7% of full power at one second, 1.3% after an hour and 0.4% after a day. For a 3 GW core that is 200 MW immediately after shutdown - the thermal output of a small power station, and it must be removed no matter what else has failed. Fukushima Daiichi lost its ability to remove that heat, not its ability to stop the reaction. Everything that has happened in reactor design since - passive circulation, gravity-fed cooling, cores that can dry out without melting - is an answer to that one term in the equations.
Key Formulas
| Six-factor formula | \(k_{\text{eff}} = \eta f p \varepsilon P_{\text{FNL}} P_{\text{TNL}}\) | |
|---|---|---|
| Point kinetics | \(\dfrac{dP}{dt} = \dfrac{\rho-\beta}{\Lambda}P + \sum_i\lambda_i C_i\) | |
| Precursors | \(\dfrac{dC_i}{dt} = \dfrac{\beta_i}{\Lambda}P - \lambda_i C_i\) | |
| Prompt jump | \(P \to P_0\,\dfrac{\beta}{\beta-\rho}\) | for a step of ρ < β |
| Prompt period | \(T = \dfrac{\Lambda}{\rho-\beta}\) | Λ ≈ 2 × 10⁻⁵ s |
| Reactivity in dollars | \(\$ = \dfrac{\rho}{\beta}\) | |
| Thermal cross-section | \(\sigma_f \approx 585\ \text{b at }0.025\ \text{eV}\) | 1.2 b at 2 MeV |
| Decay heat | \(\dfrac{P}{P_0} \approx 0.066\left[t^{-0.2}-(t+t_0)^{-0.2}\right]\) | |
Things worth knowing
- A natural fission reactor ran at Oklo in Gabon about 1.7 billion years ago, when natural uranium was still 3% U-235. It cycled on and off for hundreds of thousands of years, moderated by groundwater that boiled away and returned.
- Reactivity is measured in dollars, where one dollar equals β. It is possibly the only unit in physics invented specifically so that a number greater than one is unmistakably catastrophic.
- Xenon-135 has a capture cross-section of 2.6 million barns - the largest known. It builds up after a power reduction and can leave a reactor unable to restart for a day, a state operators call a xenon pit.