Critical mass

Why does a lump of uranium do nothing at all until it reaches a certain size, then everything at once?

Two half-lumps of uranium sit on a bench, inert. Push them together and something enormous happens — though nothing about a uranium nucleus changed when the halves touched, and each atom has the neighbours it always had at the same spacing. The phrase "critical mass" makes it sound as though a quantity of material has a magic threshold built in. It does not. So what actually flips?

Start with the single event. A slow neutron strikes a uranium-235 nucleus, which splits and releases, on average, about 2.4 neutrons. Notice the shape of that fact: each event produces more of the thing that causes events. That is the ingredient for a chain, but not sufficient, because a produced neutron does not necessarily go on to cause another fission.

So ask the accounting question. Of the fissions in one generation, how many fissions do they cause in the next? Call that number k. If k is below one, each generation is smaller than the last and any burst of neutrons dies away geometrically — there is a reaction, it simply extinguishes itself. If k exceeds one the population climbs geometrically. Between them, k exactly one: self-sustaining at unchanging intensity.

What can a neutron do besides cause fission? It can be absorbed without splitting anything — by an impurity, by uranium-238, by a control rod. Or it can simply leave, escaping through the outside surface before it meets a nucleus at all.

That last one is the whole answer to the question. Neutron production happens throughout the material, so it scales with volume. Escape happens across the boundary, so it scales with surface area. Volume grows as the cube of the radius and surface as the square, so the ratio of escape to production falls as the lump gets bigger. Below a certain size, so many neutrons walk out of the edges that k stays under one no matter how pure the material. Above it, enough stay inside. Critical mass is not a property of uranium — it is the size at which a sphere's surface stops leaking faster than its interior produces.

That reframing pays off immediately, because it predicts things the "magic quantity" story cannot. Squeeze the same mass into a smaller volume and the surface shrinks while the atoms crowd closer, so a compressed subcritical mass can become critical without adding an atom. Wrap it in a material that scatters escaping neutrons back inside and the figure drops sharply — for uranium-235 a bare sphere runs to roughly 52 kilograms, a well-reflected one to a fraction of that. Shape matters too: a sphere has the least surface per volume, so anything flatter needs more.

Now the second half of the question — why everything at once, with no gentle middle. Because k does not set a rate; it sets the base of an exponential. The population after n generations goes as k to the n, and neutron generations in fast material are measured in tens of nanoseconds. A k of 1.01 and a k of 0.99 differ by two per cent, but one decays to nothing and the other multiplies by e roughly every hundred generations — microseconds. The threshold looks infinitely sharp not because anything discontinuous happens at k = 1, but because we are watching the sign of an exponent, and a sign has no gradual version.

Which raises the obvious worry: if the margin is that tight, how is a reactor steerable at all? The answer is a genuinely lucky detail of nuclear physics. Not all fission neutrons appear instantly — around 0.65 per cent of them in uranium-235 come out of fission fragments that decay seconds later. A reactor is run so that the prompt neutrons alone give k just under one, and the chain is sustained only when the delayed fraction is included. The effective generation time is then set by those stragglers, seconds rather than nanoseconds. So the line that matters in operation is not criticality but prompt criticality, the point at which the delayed neutrons stop being the throttle.

The stubborn misconception is that criticality is a property of a substance — so many kilograms and it goes off. It is a property of a configuration: mass, density, geometry, purity, surroundings, and whether a moderator is present to slow neutrons into the range where uranium-235 fissions most readily. Quoted critical masses are shorthand for one specified arrangement, usually a bare sphere.

It is also worth separating two things "chain reaction" merges. k slightly above one means a reactor running up in power over seconds; growth on a microsecond scale requires the prompt neutrons alone to sustain the chain, a much higher bar. Most of what makes a reactor safe lives in the gap between those two conditions.

A boundary worth marking: elsewhere in this collection, nuclear binding energy explains where a fission's energy comes from, and herd immunity treats the same threshold algebra — a reproduction number crossing one — in an epidemic. This piece asks the geometric question neither of those does: why the threshold falls at a particular size, and why the transition looks instantaneous.

• Why a moderator makes a reactor critical with far less enriched fuel, and what slowing neutrons buys.
• How reactivity feedback — fuel heating and reducing k on its own — supplies a margin no operator has to.

Every term the collection defines is gathered in the glossary.