Thermal (slow) neutrons are more easily absorbed than fast neutrons β their de Broglie wavelength better matches nuclear dimensions
Fission produces 2β3 prompt neutrons per event (average ~2.5 for U-235)
Products are neutron-rich and therefore radioactive (undergo Ξ²β» decay)
About 200 MeV of energy is released per fission, mostly as kinetic energy of fragments
Fissile materials: U-235 and Pu-239 are fissile (can be split by thermal neutrons). U-238 is fertile β it can absorb a neutron to become Pu-239 but is not itself fissile with thermal neutrons.
Chain Reactions
The 2β3 neutrons released in each fission can go on to trigger further fissions β creating a chain reaction.
The multiplication factor k (or k-effective) is the average number of neutrons from one fission that cause the next fission:
k value
Condition
Outcome
k < 1
Sub-critical
Reaction dies out
k = 1
Critical
Sustained, steady reaction
k > 1
Super-critical
Exponentially growing reaction
In a nuclear weapon, k >> 1 is achieved instantaneously. In a reactor, k is maintained at exactly 1 by control systems. The difference is in the speed and degree of supercriticality.
Critical Mass
The critical mass is the minimum mass of fissile material needed for a self-sustaining chain reaction. Factors affecting critical mass:
Enrichment: higher U-235 concentration β lower critical mass
Shape: a sphere minimises surface-to-volume ratio β fewer neutrons escape β lower critical mass
Reflectors: surrounding the core with a neutron reflector (e.g. beryllium) bounces neutrons back β reduces critical mass
Density: compressing the material increases density β more fissile nuclei per volume β reduces critical mass
Critical mass for bare U-235 sphere β 52 kg; with a reflector this drops to ~15 kg. For Pu-239, critical mass β 10 kg (bare sphere).
Nuclear Reactor Design
A thermal nuclear reactor has these key components:
Component
Function
Example material
Fuel
Fissile material undergoing fission
Enriched UOβ pellets (~3% U-235)
Moderator
Slows fast neutrons to thermal energies
Graphite, light water, heavy water
Control rods
Absorb neutrons to control k
Boron, cadmium, hafnium
Coolant
Removes heat from the core
Water, COβ, liquid sodium
Shielding
Absorbs radiation, protects operators
Concrete, steel, water
Control rods can be inserted further into the core to increase neutron absorption (reduce k) or withdrawn to increase the reaction rate. This maintains k = 1 for steady power output.
Nuclear Fusion Conditions & Confinement
For fusion to occur, nuclei must overcome the Coulomb barrier β this requires:
Temperature: ~10βΈ K (100 million kelvin) β so the plasma nuclei have sufficient kinetic energy to tunnel through the Coulomb barrier
Density: sufficient number of nuclei per unit volume for adequate collision rate
Confinement time: plasma must be held together long enough for enough reactions to occur
Lawson criterion: n Γ Ο > threshold value (where n = density, Ο = confinement time)
Magnetic confinement (tokamak): Plasma is held in a torus (doughnut shape) by powerful magnetic fields. The JET (UK) and ITER (France) tokamaks use this approach.
Inertial confinement: Lasers simultaneously compress and heat a small pellet of fusion fuel, creating conditions for ignition for a brief instant.
A plasma is the fourth state of matter β electrons are stripped from nuclei, producing a hot gas of charged particles. At fusion temperatures, hydrogen exists entirely as plasma.
E β 8.2 Γ 10ΒΉΒ³ J β 82 TJ per kg β roughly equivalent to 20,000 tonnes of TNT
Example 3: Control rod insertion
A reactor is producing too much power (k = 1.02). Explain the sequence of events when control rods are inserted.
1 Control rods are made of neutron-absorbing material (boron/cadmium). Inserting them removes more neutrons from the chain reaction.
2 Fewer neutrons are available to cause fission. The multiplication factor k decreases towards 1.0.
3 At k = 1.0, the reaction is critical: the number of fissions per second is constant, and power output stabilises.
Inserting control rods increases neutron absorption, reducing k from 1.02 to 1.00 and stabilising the power output at the desired level.
Example 4: Why thermal neutrons are needed for fission
Explain why fast neutrons from fission are not effective at inducing further fission in U-235, and how a moderator solves this problem.
1 Fast neutrons (E ~MeV) have very short de Broglie wavelengths. The nuclear cross-section (probability of absorption) for U-235 is small for fast neutrons.
2 Thermal neutrons (~0.025 eV) have much longer wavelengths comparable to nuclear dimensions. U-235's fission cross-section is ~1000Γ larger for thermal neutrons.
3 A moderator (e.g. graphite) slows down fast neutrons through elastic collisions without absorbing them significantly. Nuclei close in mass to the neutron (e.g. carbon) are most effective at energy transfer.
The moderator thermalises fast neutrons, dramatically increasing the fission cross-section of U-235 and making a self-sustaining chain reaction possible.
Q1. What is the role of the moderator in a thermal nuclear reactor?
Q2. If the multiplication factor k = 1, what is happening in the reactor?
Q3. Why does nuclear fusion require temperatures of about 10βΈ K?
Q4. What is the function of control rods in a nuclear reactor?
Q5. A tokamak uses magnetic fields to confine plasma. Why can plasma not simply be contained in a solid vessel?
Challenge 1. A 1000 MW nuclear power station uses U-235 fuel. If each fission releases 200 MeV and the overall efficiency (nuclear β electrical) is 35%, estimate (a) the fission rate per second, and (b) the mass of U-235 consumed per day.
(b) Fissions per day: 8.93 Γ 10ΒΉβΉ Γ 86400 = 7.71 Γ 10Β²β΄
Mass = (7.71 Γ 10Β²β΄ / 6.02 Γ 10Β²Β³) Γ 235 g = 12.81 mol Γ 235 g/mol = 3010 g β 3.0 kg/day
(For comparison, a coal power station of same output burns ~7000 tonnes of coal per day!)
Challenge 2. Compare the relative advantages and disadvantages of fission and fusion as energy sources under these headings: fuel availability, energy density, waste products, technical readiness, and safety.
Fuel availability: Fusion wins β deuterium is extracted from seawater (virtually limitless). U-235 is scarce (~0.7% of natural uranium).
Energy density: Fusion slightly better (~4Γ per kg of fuel); both are orders of magnitude above fossil fuels.
Waste: Fusion produces He-4 (inert, non-radioactive) and neutron-activated structural materials (shorter-lived than fission waste). Fission produces long-lived radioactive waste (tΒ½ up to thousands of years), requiring secure storage for centuries.
Technical readiness: Fission β proven, commercially operating worldwide since 1950s. Fusion β still experimental; no net-energy-producing reactor yet (as of 2026).
Safety: Fusion plasma is self-limiting β loss of confinement stops reaction immediately. Fission reactors require active cooling; loss-of-coolant accidents (e.g. Fukushima) can cause meltdown. No fission "explosion" is possible in a reactor due to low enrichment, but partial meltdown remains a risk.
Challenge 3. A neutron is produced with kinetic energy 2.0 MeV in a fission event. It must be thermalised (to ~0.025 eV) by a graphite moderator. Calculate the ratio of initial to final kinetic energies, and explain in terms of de Broglie wavelength why thermalised neutrons are more effective at inducing fission.
Ratio: E_initial/E_final = 2.0 Γ 10βΆ / 0.025 = 8 Γ 10β·
The neutron must lose a factor of 80 million in kinetic energy!
De Broglie wavelength: Ξ» = h/p = h/β(2mE)
Since Ξ» β 1/βE: fast neutron Ξ» is β(8 Γ 10β·) β 9000Γ shorter than thermal neutron Ξ».
Thermal neutrons (Ξ» β 1β2 Γ β 10β»ΒΉβ° m) are still much larger than nuclear dimensions (fm), but the cross-section for neutron-nucleus interaction scales with the effective target area.
More fundamentally: the U-235 nucleus has resonance capture energies, and the broad cross-section peak at thermal energies (~550 barns for U-235 at 0.025 eV vs ~1 barn at MeV) makes thermal neutrons ~550Γ more likely to be captured and induce fission.