Fission and fusion

Induced fission of uranium-235

A uranium-235 nucleus can absorb a slow neutron and form an unstable uranium-236 nucleus, which may split into two smaller nuclei and additional neutrons. The neutron has to be a thermal neutron, one moving at everyday molecular speeds, since uranium-235 captures those far more readily than fast ones. Only nuclei on the heavy side of iron sit low enough on the binding-energy curve for their fragments to end up better bound, so only they release energy by splitting; splitting a light nucleus would require an input of energy.

A typical split turns uranium-235 plus a neutron into barium-141 and krypton-92 plus three neutrons. Check that both nucleon number and proton number are conserved: proton number gives 92 = 56 + 36, and nucleon number gives 236 = 141 + 92 + 3. The products have a greater binding energy per nucleon, so their total mass is lower, and the mass difference is released mainly as kinetic energy. A typical fission releases about 200 MeV. Those fragments carry too many neutrons for their new size and sit above the stable band, so they decay by β⁻ in chains, a fact whose consequences fill the next lesson.

Energy from the masses

The energy calculation follows one method every time. Total nuclear mass before, minus total after, times 931.5. The products of any release reaction weigh less than the ingredients, and the difference leaves as kinetic energy of the products. For fission the sum uses the fuel nucleus, the incoming neutron, both fragments and the freed neutrons. The question supplies the masses.

The chain and the critical mass

Each fission's spare neutrons can induce further fissions, which is the door to a chain reaction. Left alone in a large enough mass of fuel, one fission becomes three, becomes nine, growing by powers. Held so that exactly one neutron per fission goes on to cause another, the chain ticks over at a steady rate, which is what a reactor is built to do.

Describe the drawing

Left, uncontrolled: each fission frees three neutrons and each finds a nucleus, so one becomes three becomes nine becomes twenty-seven, the frame filling faster each generation. Right, controlled: absorber rods eat two of every three, and the chain ticks over at exactly one, steady power instead of a bomb. The difference between a reactor and a weapon is that one number.

Whether a chain can sustain itself at all is a question of size. In a small lump, too many neutrons reach the surface and escape before meeting a nucleus; the chain fizzles. The critical mass is the minimum amount of fissile material in which, on average, one neutron per fission is retained to fission again, and it is not a constant of the material alone: shape, density, enrichment and any surrounding reflector all move it. Below it a chain is impossible; at and above it, a chain can be sustained.

Fusion of light nuclei

At the curve's other end, light nuclei release energy by fusion, and the steep left slope makes each fused nucleon worth more than a fissioned one, so kilogram for kilogram fusion yields several times more. The standard reaction fuses deuterium and tritium into helium-4 plus a neutron, releasing 17.6 MeV.

The condition is the approach distance. Two nuclei have to come within a few femtometres for the strong force to act, and both of them are positive, so this is the closest-approach problem from earlier in the unit, now working against us. Temperatures of order 10⁷ K raise the collision energy and so cut the distance of closest approach, and in a star the crushing pressure keeps the density, and with it the collision rate, enormous. That much is the answer an exam question is after.

The arithmetic does not close there, though, and it is worth seeing why. At 1.5 × 10⁷ K the average nucleus carries only about 2 keV, while the electrical hill between two protons runs to several hundred keV. Practically nothing in a star has the energy to get over it, so a purely classical treatment predicts a fusion rate hopelessly below what stars actually deliver. What rescues it is quantum tunnelling, the same barrier-penetration effect that lets alpha particles leave a nucleus.

So the full picture combines the two. The fastest nuclei in the tail of the speed distribution get close, and tunnelling carries them the rest of the way through a barrier they were never going to climb. Stars manage it, but not easily: the Sun's core releases only a few hundred watts per cubic metre, and it is sheer size that makes the total colossal. Building a reactor that sustains it remains one of engineering's great open problems.