Nuclear Fission, Chain Reactions, and Critical Mass

Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei (fission fragments) with the release of neutrons and about 200 MeV of energy per nucleus; when the neutrons induce further fissions, a chain reaction occurs, which is self-sustaining above a critical mass. In brief: the fission of 1.00 g of uranium-235 releases about 8.2 × 10¹⁰ J, about 2.8 million times as much energy as burning 1 g of coal.

Fission is the source of the energy in nuclear power plants and of the destructive power of the first atomic bombs. Its discovery in 1938 showed that the binding energy of the nucleus could be released on a large scale. This article explains how fission works, where the energy comes from and how it is distributed, what makes a chain reaction possible, how reactors are controlled and what critical mass means, with a table and examples.

What happens in fission?

When a uranium-235 nucleus absorbs a slow (thermal) neutron, it forms an excited uranium-236 nucleus, which becomes deformed, like a drop of liquid, and splits into two fragments of unequal mass, usually with the release of two or three neutrons. One of the many possible modes is:

23592U + 10n → 23692U* → 14156Ba + 9236Kr + 3 10n + energy

More than 400 different fission products are known. The fragments are radioactive, since they have a high neutron-to-proton ratio, and decay by a series of beta emissions. The average number of neutrons released in the fission of ²³⁵U is about 2.4. Nuclides that can sustain a chain reaction with slow neutrons are called fissile: ²³⁵U, ²³³U and ²³⁹Pu. Uranium-238 is fissionable by fast neutrons only, and is not fissile; it absorbs neutrons and is converted to plutonium-239 through two beta decays.

Where does the energy come from?

The binding energy per nucleon is about 7.6 MeV for uranium and about 8.5 MeV for the fragments with mass numbers around 90 and 140. The increase of about 0.9 MeV per nucleon, over 236 nucleons, gives about 200 MeV per fission. As a check using masses, the total mass of the products is about 0.215 u smaller than that of the reactants, and 0.215 × 931.5 MeV/u = 200 MeV. In terms of the mole: 200 MeV × 9.649 × 10⁷ kJ/mol per MeV = 1.9 × 10¹⁰ kJ/mol of nuclei, or 8.2 × 10⁷ kJ/g of uranium-235.

How is the energy distributed?

Most of the energy appears as the kinetic energy of the fragments, which is transformed into heat when they are slowed down in the fuel. About 10 MeV is carried off by neutrinos and is lost, so about 200 MeV can be used as heat.

What is a chain reaction?

Each fission releases neutrons that can cause further fissions. The multiplication factor k is the average number of neutrons from one fission that cause another fission:

• k < 1 (subcritical): the reaction dies out.
• k = 1 (critical): the reaction is steady, as in a nuclear reactor at constant power.
• k > 1 (supercritical): the number of fissions increases exponentially, as in a nuclear weapon.

For a time of about 10⁻⁸ s between generations of prompt neutrons, even a value of k only a little more than 1 would raise the power at an uncontrollable rate. Luckily, about 0.65 % of the neutrons in the fission of ²³⁵U are delayed neutrons, emitted seconds after the fission by some of the fission products, which slow down the response of a reactor to a small change and make control possible. Natural uranium contains only 0.72 % of ²³⁵U, and most neutrons are captured by ²³⁸U or escape, so a chain reaction is not possible without a moderator or enrichment.

What is critical mass?

Neutrons can escape from the surface of the fuel without causing fission. In a small piece of fuel, the ratio of surface to volume is large, and too many neutrons are lost. The critical mass is the smallest mass of fissile material for which a chain reaction is self-sustaining. It depends on the nuclide, the shape (a sphere is best), the density, the purity and the surroundings: a reflector of neutron-scattering material, such as beryllium or natural uranium, reduces the critical mass. For a bare sphere, the critical mass of ²³⁵U (highly enriched) is about 52 kg, of ²³⁹Pu about 10 kg, with a radius of about 9 cm and 5 cm. A weapon contains a supercritical mass assembled very fast, for example by implosion. A reactor can never explode like a bomb, because the fuel is too dilute (3–5 % ²³⁵U), although it can melt if cooling is lost.

How is a chain reaction controlled?

• Moderator: fast neutrons from fission (about 2 MeV) are slowed to thermal energies (0.025 eV), when the probability that they induce fission of ²³⁵U is about a thousand times larger. Light water, heavy water and graphite are used.
• Control rods: rods of boron, cadmium or hafnium absorb neutrons. Inserted deeper, they reduce k; withdrawn, they increase it.
• Enrichment: raising the proportion of ²³⁵U from 0.72 % to 3–5 % for reactors, to 20 % or more for research reactors and above 90 % for weapons.
• Coolant: removes the heat and keeps the fuel from melting.

Worked examples

Example 1: the energy from 1 g of uranium-235

1. n = 1.00 g / 235.04 g/mol = 4.25 × 10⁻³ mol, or 2.56 × 10²¹ atoms.
2. Energy per fission: 200 MeV × 1.602 × 10⁻¹³ J/MeV = 3.2 × 10⁻¹¹ J.
3. Total: 2.56 × 10²¹ × 3.2 × 10⁻¹¹ = 8.2 × 10¹⁰ J, which is 22.8 MWh.

1 g of coal gives about 29 kJ when burned, so the ratio is 8.2 × 10¹⁰ / 2.9 × 10⁴ = 2.8 × 10⁶. 1 kg of uranium-235 would release 8.2 × 10¹³ J, equivalent to about 20,000 tonnes of TNT (1 tonne of TNT = 4.184 × 10⁹ J).

Example 2: balancing a fission equation

²³⁵U + n → ¹⁴⁴Ba + ⁸⁹Kr + x n. The sum of the mass numbers on the left is 236, and 144 + 89 = 233, so x = 3 neutrons. The atomic numbers: 92 = 56 + 36 = 92 ✓.

Example 3: neutrons in a chain reaction

If k = 1.05 and 1000 neutrons start a generation, the number in the n-th generation is 1000 × 1.05ⁿ. After 100 generations (about 1 µs for a fast system) this would be 1000 × 131 = 1.3 × 10⁵, and after 500 generations 1000 × 4.0 × 10¹⁰. For k = 0.95 the number would fall to 1000 × 0.0059 = 6 after 100 generations.

Example 4: uranium enrichment

To make 1.00 kg of uranium enriched to 4.0 % ²³⁵U from natural uranium (0.72 %), with the tails at 0.25 %, the mass of natural uranium needed is found from the mass balance of ²³⁵U: F = P (x_p − x_t) / (x_f − x_t) = 1.00 × (0.040 − 0.0025) / (0.0072 − 0.0025) = 1.00 × 0.0375 / 0.0047 = 8.0 kg. About 8 kg of natural uranium is needed per kg of 4 % product.

How was fission discovered?

Enrico Fermi bombarded uranium with neutrons in 1934 and thought that he had made transuranium elements. In December 1938 Otto Hahn and Fritz Strassmann found barium among the products, a much lighter element, and could not explain it. Lise Meitner, who had been forced to leave Germany, and her nephew Otto Frisch explained it as the splitting of the nucleus in the liquid-drop model, and Frisch named it fission, in 1939. Hahn received the Nobel Prize in Chemistry in 1944 for the discovery, and Meitner’s contribution is now widely recognized (the element meitnerium is named after her). The first controlled chain reaction was achieved by a team led by Fermi in the Chicago Pile-1 reactor on 2 December 1942.

Which mistakes are common?

• Thinking that a reactor can explode like a bomb: the enrichment is far too low.
• Mixing up fissile and fissionable: only some nuclides sustain a chain with slow neutrons.
• Ignoring the moderator: slow neutrons are needed for ²³⁵U.
• Assuming that the critical mass is a fixed number: it depends on the shape and surroundings.
• Forgetting the radioactive fission products: they are the main waste problem.