Fission, the chain reaction and the reactor

Splitting a nucleus

In 1938 Otto Hahn and Fritz Strassmann shot neutrons at uranium and found barium in what was left. This was a surprise, since barium has only about half the mass of uranium. Lise Meitner and Otto Frisch explained it: the uranium nucleus had split in two. They called it nuclear fission, a word from biology for the division of a cell.

A slow neutron strikes a uranium-235 nucleus, which becomes uranium-236 in an excited state. It wobbles like a drop of liquid and divides. One of many ways it can split is

The masses are $235.043930\ \text{u}$ and $1.008665\ \text{u}$ before. After, they are $140.914411\ \text{u}$, $91.926156\ \text{u}$ and $3 \times 1.008665\ \text{u}$. The loss is $0.186033$ u, so $Q = 173.3$ MeV for this way. There are many other ways, and the fragments are never of equal mass. Counting the energy of the later decays of the fragments too, one fission of uranium-235 gives about $200$ MeV.

Why about $200$ MeV? Use the curve of the first course. Uranium has $B/A$ near $7.6$ MeV, and the fragments, with mass numbers near $95$ and $140$, have about $8.5$ MeV. The gain is $0.9$ MeV for each of $236$ nucleons, which is about $212$ MeV. Almost all of it appears as the kinetic energy of the two fragments. They stop in the fuel in a very short distance, and the energy becomes heat.

How much energy is that?

One kilogram of uranium-235 has $\dfrac{1000\ \text{g}}{235\ \text{g/mol}} \times 6.022 \times 10^{23} = 2.56 \times 10^{24}$ nuclei. At $200$ MeV each, the energy is

Coal gives about $3 \times 10^{7}$ J/kg when it burns. So $1$ kg of uranium-235 gives as much energy as about $2.7 \times 10^{6}$ kg of coal, or $2700$ tonnes. That is a million times more, as the energy of the nucleus is a million times that of a chemical bond.

Which neutrons split uranium?

Natural uranium is $99.3\ \%$ uranium-238 and only $0.72\ \%$ uranium-235. The two behave differently.

• Uranium-235 splits easily when it takes in a slow neutron, one with the energy of the thermal motion of the atoms, about $0.025$ eV. A slow neutron stays near the nucleus longer, so it is more likely to be captured.
• Uranium-238 splits only when hit by fast neutrons of more than about $1$ MeV, and slow neutrons it captures without splitting. It then becomes plutonium-239 after two beta decays. Plutonium-239 also splits with slow neutrons, which is why it is used as a fuel.

A fission frees two or three neutrons, with an average of about $2.4$ for uranium-235. They are fast, so they must be slowed to split more uranium-235. For a reactor that uses ordinary water, the fuel is enriched, so that $3\ \%$ to $5\ \%$ is uranium-235.

The chain reaction

Each fission gives two or three new neutrons, which can cause more fissions. The multiplication factor $k$ is the number of neutrons in one generation, divided by the number in the one before.

• $k < 1$, subcritical. The reaction dies out.
• $k = 1$, critical. The reaction goes on steadily. This is the working state of a reactor, with the power staying the same.
• $k > 1$, supercritical. The number of neutrons, and the power, grow as $k^g$ after $g$ generations. If it is not held, as in a bomb, a huge amount of energy is released in a very short time.

Neutrons are lost by leaking out of the surface and by being absorbed without fission. A bigger piece of fuel has a smaller surface for its volume, so it loses fewer. Below a critical mass, too many neutrons leak out and $k$ is less than $1$.

Set $k = 1.0$. How many neutrons are there after $10$ generations? Now try $k = 1.2$ and $k = 0.8$.

The nuclear reactor

A reactor holds a chain reaction at $k = 1$ and takes out the heat. These are its main parts.

• Fuel: rods of uranium oxide, enriched in uranium-235, in thin metal tubes.
• Moderator: a material that slows the fast neutrons to thermal energies. Water, heavy water and graphite are used. The neutrons lose energy in collisions with light nuclei. A neutron can give all its energy to a hydrogen nucleus in one head-on collision, because the masses are equal, but it loses very little to a heavy nucleus.
• Control rods: rods of boron or cadmium, which absorb neutrons. Pushing them in lowers $k$. Pulling them out raises it. They are also used to stop the reactor.
• Coolant: water, heavy water, carbon dioxide or liquid sodium. It takes the heat from the core to a heat exchanger.
• Steam generator, turbine and generator: the heat makes steam, which turns a turbine joined to an electric generator.
• Shield and containment: thick concrete and steel keep the radiation and the radioactive material inside.

Why a reactor can be controlled. About $0.65\ \%$ of the neutrons from uranium-235 fission come out a few seconds late, from the decay of some of the fragments. These delayed neutrons make the average time between generations long enough, about a tenth of a second, so that the control rods can follow any change. Without them, a tiny rise of $k$ above $1$ would grow in a tiny fraction of a second.

Other fuels. A breeder reactor uses the plutonium-239 that is made from uranium-238. India has also planned to use thorium-232, which is common in the country. It takes a neutron and becomes uranium-233 after two beta decays, and uranium-233 is a fuel too.

Worked example: a power plant

A reactor makes $1000$ MW of heat. The plant is $33\ \%$ efficient. Find the electric power, the number of fissions in each second and the mass of uranium-235 that is used up in a day.

1. The electric power. $0.33 \times 1000\ \text{MW} = 330\ \text{MW}$.
2. The fissions each second. One fission releases $200 \times 1.602 \times 10^{-13}\ \text{J} = 3.20 \times 10^{-11}$ J. So the rate is $\dfrac{1.0 \times 10^{9}\ \text{W}}{3.20 \times 10^{-11}\ \text{J}} = 3.12 \times 10^{19}\ \text{s}^{-1}$.
3. In a day. $3.12 \times 10^{19} \times 86\,400 = 2.70 \times 10^{24}$ fissions. The mass is $\dfrac{2.70 \times 10^{24}}{6.022 \times 10^{23}} \times 235\ \text{g} = 1052\ \text{g}$, about $1.05$ kg.

A coal plant of the same heat would burn $\dfrac{1.0 \times 10^{9}\ \text{W} \times 86\,400\ \text{s}}{3 \times 10^{7}\ \text{J/kg}} = 2.9 \times 10^{6}$ kg, which is $2900$ tonnes, each day. Nuclear fuel is far less to carry. But the used fuel is highly radioactive, because the fragments are, and it must be stored with great care.

Keep the efficiency the same and double the heat. What happens to the fuel? Now lower the energy of one fission from $200$ MeV to $180$ MeV. Does the fuel go up or down?

Practise with new numbers

Try it in Python

The program finds what a plant of $1000$ MW of heat burns in a day, and compares it with coal.

Common mistakes

• Saying that slow neutrons are the same as the thermal speed of the fuel. They are slow compared with the fission neutrons, at about $0.025$ eV.
• Using heavy materials as the moderator. A neutron loses little energy in a collision with a heavy nucleus. Light nuclei do the job.
• Mixing up control rods and the moderator. The rods absorb neutrons. The moderator slows them.
• Saying that $k = 1$ means no fission. It means the same number of fissions in each generation.
• Taking the electric power as the heat of the core. The heat is the electric power divided by the efficiency.
• Using MeV as joule. Multiply by $1.602 \times 10^{-13}$ to get joule.

Remember these

• One fission of uranium-235 gives about $200$ MeV and $2$ or $3$ neutrons. $1$ kg gives $8.2 \times 10^{13}$ J.
• Uranium-235 splits with slow neutrons. Uranium-238 needs fast neutrons, and with slow ones it breeds plutonium-239.
• $k < 1$ is subcritical, $k = 1$ is critical and $k > 1$ is supercritical.
• Reactor parts: fuel, moderator (slows neutrons), control rods (absorb neutrons), coolant, shield.
• Fissions each second: $\dfrac{P}{E}$. Fuel in a day: that number times $86\,400$, over $N_A$, times $235$ g.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why is a moderator made of light nuclei, such as hydrogen or carbon, and not of lead?

2. In a reactor the multiplication factor is $1.00$. The control rods are pushed in a little. What happens to $k$ and to the power?

3. About how much energy does the fission of one nucleus of uranium-235 release, and where does most of it go?

4. A neutron source feeds a block of fuel with $k = 0.9$. Starting with $1000$ neutrons in one generation, how many are there in the next one?

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on nuclear energy, fission and the reactor.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on fission.
• AME2020 atomic mass evaluation (Wang and others, Chinese Physics C, 2021), for the atomic masses.