SL & HL Nuclear and Quantum Physics BETA E.4

Fission

This topic covers the mechanics of nuclear fission, focusing on how splitting heavy nuclei releases immense binding energy, how self-sustaining chain reactions are controlled in reactors, and the management of high-level nuclear waste.

Part of IB Physics (2025-2030 syllabus) — Standard and Higher Level.

Key points

Nuclear fission occurs when a heavy, unstable nucleus (such as $^{235}_{92}\text{U}$ ) captures a neutron and splits into two lighter, more stable daughter nuclei, releasing several fast neutrons and a large amount of energy.
The energy released in a fission reaction arises from the mass defect: the total rest mass of the products is less than the total rest mass of the reactants, converted to energy via $E = \Delta m c^2$ .
A self-sustaining chain reaction requires that the neutrons produced in one fission event go on to trigger at least one subsequent fission event, which is dictated by having a mass of fissile material equal to or greater than the critical mass.
Moderators (such as water or graphite) slow down the fast neutrons produced in fission to thermal energies, as slower neutrons have a significantly higher probability of being captured by $^{235}_{92}\text{U}$ nuclei.
Control rods made of neutron-absorbing materials (such as boron or cadmium) are inserted into or withdrawn from the reactor core to adjust the rate of fission by regulating the neutron flux.
The daughter nuclei resulting from fission are typically highly unstable, neutron-rich isotopes that present a significant hazard because they undergo beta-minus ( $\beta^-$ ) decay and remain radioactive for long periods, requiring long-term high-level waste storage.
Thick concrete and steel shielding surrounds the reactor core to absorb escaping neutrons and gamma radiation, protecting workers and equipment from ionizing radiation.

Subtopic by subtopic

Nuclear fission and energy release

Induced nuclear fission begins when a heavy nucleus such as $^{235}_{92}\text{U}$ captures a slow neutron, forming an unstable compound nucleus that deforms and splits into two lighter daughter nuclei together with two or three free neutrons. One possible reaction is

$^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3\,^{1}_{0}\text{n}$

Energy is released because the products sit higher on the binding-energy-per-nucleon curve than the parent: their nucleons are more tightly bound, so the total rest mass of the products is less than that of the reactants. The mass defect $\Delta m$ becomes energy through:

$E = \Delta m c^2$

This is typically about $200\ \text{MeV}$ per fission, millions of times the energy per atom released by any chemical reaction. Most of this energy appears as kinetic energy of the recoiling fragments, which is transferred to the surroundings as thermal energy.

Balance nuclear equations by conserving nucleon number $A$ and proton number $Z$ on both sides.
Calculate the energy released either from rest masses using $E = \Delta m c^2$ or from the difference in total binding energy between products and reactants.

Chain reactions

Each fission of $^{235}_{92}\text{U}$ releases two or three fast neutrons, and these can strike other fissile nuclei. If, on average, at least one neutron from each fission induces a further fission, the process is self-sustaining: a chain reaction.

The average number of neutrons from one fission that go on to cause another fission decides the behaviour:

fewer than one — subcritical: the reaction dies away;
exactly one — critical: a steady fission rate, the operating condition of a power reactor;
more than one — supercritical: the rate grows exponentially, as in a weapon or briefly during reactor start-up.

Neutrons are lost by escaping through the surface of the fuel or by absorption in non-fissile material such as $^{238}_{92}\text{U}$ . Because the surface-to-volume ratio falls as a sample gets larger, a lump of fissile material below the critical mass leaks too great a fraction of its neutrons to sustain the chain, while one at or above the critical mass does not.

You should be able to describe these three criticality conditions, link critical mass to neutron escape, and explain why natural uranium (which is mostly $^{238}_{92}\text{U}$ ) must usually be enriched in $^{235}_{92}\text{U}$ before it can sustain a chain reaction.

Nuclear reactors (control rods, moderators, shielding)

A thermal fission reactor brings together several components, each with a distinct job:

fuel rods of enriched uranium;
a moderator;
control rods;
a coolant;
shielding.

To induce fission in $^{235}_{92}\text{U}$ , a neutron must be captured, but the neutrons ejected during fission are fast neutrons with kinetic energies around $2\ \text{MeV}$ , and at these energies the probability (cross-section) of capture is low. To raise this probability they must be slowed to thermal energies (about $0.025\ \text{eV}$ ) through elastic collisions with light nuclei in the moderator.

Momentum conservation shows that collisions with nuclei of similar mass to the neutron, such as the protons in water or deuterium in heavy water, transfer kinetic energy most efficiently, slowing the neutrons over a short distance; graphite is also widely used.

Control rods of boron or cadmium absorb neutrons: operators withdraw or insert them to hold the chain reaction exactly critical, or drive them fully in to shut the reactor down.

The coolant (often the same water that acts as moderator) carries thermal energy from the core to a heat exchanger, where steam is raised to drive turbines and generators. Thick concrete and steel shielding surrounds the core, absorbing escaping neutrons and gamma radiation to protect workers.

Be ready to state the purpose of every component, and in particular to distinguish the moderator (slows neutrons so fission becomes more likely) from the control rods (absorb neutrons to reduce the fission rate).

Fission products and nuclear waste

The daughter nuclei produced in fission are almost always neutron-rich: heavy nuclei need a higher neutron-to-proton ratio for stability than medium-mass nuclei do, and the fragments inherit the parent's ratio.

Lying above the band of stability, fragments such as strontium-90 and caesium-137 stabilise through successive $\beta^-$ decays, emitting ionizing radiation, and with half-lives of around $30$ years they keep spent fuel dangerously radioactive long after it leaves the reactor.

These decays release thermal energy even after the main chain reaction has been halted by the control rods — the decay-heat problem. Spent fuel rods must therefore be cooled continuously in storage pools for years; only then can the high-level waste be encased in vitrified (glass) blocks and transferred to deep geological repositories designed to isolate it for thousands of years.

You should be able to:

explain why fission products are unstable and decay by $\beta^-$ emission;
explain why cooling must continue after a reactor shuts down;
outline how high-level waste is treated and stored.

Formulae

E = \Delta m c^2

To calculate the energy released during a fission reaction when the change in rest mass (mass defect $\Delta m$ ) between the reactants and products is known.

E_{\text{released}} = \text{Total } BE_{\text{products}} - \text{Total } BE_{\text{reactants}}

To determine the energy released in a fission event using the binding energies ( $BE$ ) of the initial heavy parent nucleus and the final product nuclei.

Definitions

Nuclear Fission: The process in which a heavy nucleus splits into two smaller, lighter nuclei of approximately equal mass, accompanied by the release of neutrons and energy.
Critical Mass: The minimum mass of a fissile isotope required to sustain a self-supporting nuclear chain reaction.
Moderator: A material in a nuclear reactor used to slow down fast-moving neutrons to thermal speeds without absorbing them, increasing the probability of further fission.
Control Rod: A movable rod composed of neutron-absorbing material (such as cadmium or boron) used to regulate the rate of the chain reaction in a nuclear reactor.
Chain Reaction: A self-sustaining sequence of fission events in which neutrons released by one fission go on to induce fission in other fissile nuclei.
Shielding: Dense material, typically thick concrete and steel, surrounding a reactor core to absorb neutrons and gamma radiation and protect people and equipment.

Worked examples

An induced fission reaction is represented by the equation: $^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3^{1}_{0}\text{n}$ The rest masses of the particles involved are: $m(^{235}_{92}\text{U}) = 235.0439\ \text{u}$ $m(^{141}_{56}\text{Ba}) = 140.9144\ \text{u}$ $m(^{92}_{36}\text{Kr}) = 91.9262\ \text{u}$ $m(^{1}_{0}\text{n}) = 1.0087\ \text{u}$ Calculate the energy released in this single fission event in millions of electronvolts ( $\text{MeV}$ ). Take $1\ \text{u} = 931.5\ \text{MeV c}^{-2}$ .

1
Step 1: Calculate the total mass of the reactants before the fission event. $m_{\text{initial}} = m(^{235}_{92}\text{U}) + m(^{1}_{0}\text{n}) = 235.0439\ \text{u} + 1.0087\ \text{u} = 236.0526\ \text{u}$
2
Step 2: Calculate the total mass of the products after the fission event. $m_{\text{final}} = m(^{141}_{56}\text{Ba}) + m(^{92}_{36}\text{Kr}) + 3 \times m(^{1}_{0}\text{n})$ $m_{\text{final}} = 140.9144\ \text{u} + 91.9262\ \text{u} + 3(1.0087\ \text{u}) = 232.8406\ \text{u} + 3.0261\ \text{u} = 235.8667\ \text{u}$
3
Step 3: Calculate the mass defect ( $\Delta m$ ). $\Delta m = m_{\text{initial}} - m_{\text{final}} = 236.0526\ \text{u} - 235.8667\ \text{u} = 0.1859\ \text{u}$
4
Step 4: Convert the mass defect into energy in units of $\text{MeV}$ . $E = 0.1859\ \text{u} \times 931.5\ \text{MeV u}^{-1} \approx 173.166\ \text{MeV}$

Answer: $1.73 \times 10^{2}\ \text{MeV}$

The core of a nuclear power station produces a steady thermal power of $1.2\ \text{GW}$ . Each fission of a $^{235}_{92}\text{U}$ nucleus releases, on average, $200\ \text{MeV}$ of energy. Calculate the number of fission events occurring each second, and hence estimate the mass of $^{235}_{92}\text{U}$ that undergoes fission in one day. Take the mass of one $^{235}_{92}\text{U}$ atom to be $235\ \text{u}$ and $1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$ .

1
Step 1: Convert the energy released per fission into joules: $E = 200 \times 10^{6} \times 1.60 \times 10^{-19} = 3.20 \times 10^{-11}\ \text{J}$ .
2
Step 2: Divide the thermal power by the energy per fission to find the fission rate: $R = \dfrac{1.2 \times 10^{9}}{3.20 \times 10^{-11}} = 3.75 \times 10^{19}\ \text{s}^{-1}$ .
3
Step 3: Multiply by the number of seconds in a day to find the total number of fissions: $N = 3.75 \times 10^{19} \times 86\,400 = 3.24 \times 10^{24}$ .
4
Step 4: Find the mass of a single uranium-235 atom: $m = 235 \times 1.66 \times 10^{-27} = 3.90 \times 10^{-25}\ \text{kg}$ .
5
Step 5: Multiply the number of fissions per day by the mass of one atom: $M = 3.24 \times 10^{24} \times 3.90 \times 10^{-25} = 1.26\ \text{kg}$ .

Answer: $3.75 \times 10^{19}$ fissions per second; about $1.26\ \text{kg}$ of $^{235}_{92}\text{U}$ per day

In one fission reaction, a $^{235}_{92}\text{U}$ nucleus absorbs a slow neutron and splits into $^{140}_{54}\text{Xe}$ and $^{94}_{38}\text{Sr}$ , together with two free neutrons. The average binding energies per nucleon are $7.59\ \text{MeV}$ for $^{235}_{92}\text{U}$ , $8.29\ \text{MeV}$ for $^{140}_{54}\text{Xe}$ and $8.61\ \text{MeV}$ for $^{94}_{38}\text{Sr}$ . Estimate the energy released in this fission event.

1
Step 1: The total binding energy of the reactants comes from the uranium nucleus alone, since a free neutron has zero binding energy: $BE_{\text{reactants}} = 235 \times 7.59 = 1783.65\ \text{MeV}$ .
2
Step 2: Calculate the binding energy of the xenon fragment: $BE_{\text{Xe}} = 140 \times 8.29 = 1160.6\ \text{MeV}$ .
3
Step 3: Calculate the binding energy of the strontium fragment: $BE_{\text{Sr}} = 94 \times 8.61 = 809.34\ \text{MeV}$ .
4
Step 4: Add the fragment binding energies, noting that the two emitted neutrons contribute nothing: $BE_{\text{products}} = 1160.6 + 809.34 = 1969.94\ \text{MeV}$ .
5
Step 5: The energy released is the gain in total binding energy: $E = 1969.94 - 1783.65 = 186.29\ \text{MeV} \approx 186\ \text{MeV}$ .

Answer: $\approx 186\ \text{MeV}$

Common mistakes

×Confusing the roles of the moderator and the control rods. Remember: moderators slow down neutrons to make fission more likely, whereas control rods absorb neutrons to stop them from causing further fission.
×Forgetting to account for the incoming neutron on the reactant side, or the multiple emitted neutrons on the product side, when calculating the mass defect $\Delta m$ .
×Stating that the binding energy per nucleon decreases. Fission products actually have a higher binding energy per nucleon than the parent nucleus, which is why they are more stable and why energy is released.

Exam tips

✓When asked to **distinguish** between control rods and moderators, structure your answer with separate, clear sentences for each component, using technical terms like 'neutron absorption' for control rods and 'elastic collisions to reduce kinetic energy' for moderators.
✓If a question requires you to **explain** why fission products present a disposal hazard, explicitly state that they are highly radioactive, neutron-rich isotopes that emit ionizing radiation (specifically $\beta^-$ particles) and remain hazardous for very long times, so they need secure long-term storage.
✓Do not round intermediate values in mass-defect calculations. Keep all decimal places provided in the data booklet/prompt until the final conversion step to avoid severe rounding errors.

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