Induced Fission

SPEC 3.8.1.7, 3.8.1.8 - NUCLEAR INSTABILITY

Nuclear reactors, chain reactions, and safety systems

What is Induced Fission?

Definition

Induced fission occurs when a slow-moving (thermal) neutron is fired at a heavy nucleus (e.g. uranium-235), causing it to become extremely unstable and split into two smaller daughter nuclei.

Unlike spontaneous fission, induced fission requires an external trigger - a neutron with the right energy to be absorbed by the nucleus.

Key Concept

Thermal neutrons have low energy meaning they can induce fission. Neutrons with higher energy rebound away from the uranium-235 after collision and do not cause fission.

Fission Equation

_{92}^{235} U +_{0}^{1} n \to_{92}^{236} U^{*} \to_{56}^{144} Ba +_{36}^{89} Kr + 3_{0}^{1} n + energy

Products are two daughter nuclei and at least one neutron

Thermal vs Fast Neutrons

Thermal Neutrons

Energy: ~0.025 eV

Speed: ~10⁴ m s⁻¹ (order of magnitude)

In thermal equilibrium with surroundings

LOW energy = CAN induce fission

Slow speed allows nucleus to 'capture' neutron

Fast Neutrons

Energy: ~2 MeV

Speed: ~10⁷ m s⁻¹

Released directly from fission

HIGH energy = REBOUND away

Too fast to be captured by nucleus

Thermal Energy

E_{k} = \frac{3}{2} k T

where k = 1.38 × 10⁻²³ J K⁻¹

Why Thermal?

Relatively low velocities allow time for the nucleus to 'capture' the incoming neutron, forming the unstable U-236 nucleus.

Chain Reactions

Once one nucleus undergoes fission, the neutrons released can split further nuclei. If this fission can be sustained, a chain reaction is produced. The time for an emitted neutron to cause another fission is about 0.01 μs.

Multiplication Factor

k = \frac{neutrons in generation ( n + 1 )}{neutrons in generation n}

Conditionk ValueResult

Sub-criticalk < 1Reaction dies out

Criticalk = 1Steady state (reactor)

Super-criticalk > 1Runaway reaction

Critical Mass

Definition

The critical mass is the minimum mass of fuel required to maintain a steady chain reaction. Using exactly the critical mass means a single fission reaction follows the last.

A nuclear reactor needs a minimum amount of fuel to sustain a chain reaction. Anything less and the loss of neutrons from the surface will be too great and the chain reaction will stop.

Factors Affecting Critical Mass

Shape: Sphere minimises surface area (least neutron escape)

Purity: Higher U-235 enrichment → lower critical mass

Density: Compressed material → lower critical mass

Reflectors: Surrounding material reflects neutrons back

MaterialCritical Mass

Pure U-235 (with reflector)~15 kg

Pu-239~5-10 kg

Reactor grade U (3-5% enriched)Requires moderator for criticality

Uranium Enrichment

Natural Uranium

>99.2% U-238 (non-fissile)

≤0.72% U-235 (fissile)

Cannot sustain chain reaction without heavy water moderator

Enriched Uranium (LEU)

Typically 3-5% U-235

Used in most commercial reactors

Called "reactor grade"

Requires enrichment process to increase U-235 percentage

Role of U-238

The U-238 in fuel absorbs fission neutrons and so helps to control the rate of fission reactions. This is why reactor-grade fuel cannot produce a nuclear explosion - the U-238 prevents runaway reactions.

Fuel Rod Handling

Fuel rods are inserted into the reactor remotely to limit workers' exposure to radiation.

Thermal Nuclear Reactor

ComponentFunctionMaterials

Fuel RodsContain fissile materialEnriched uranium (U-235/238 mixture)

ModeratorSlow neutrons to thermal speedsGraphite or heavy water

Control RodsAbsorb neutrons to control rateBoron steel or cadmium

CoolantRemove heat from coreCO₂, water, helium, molten salt

Steel VesselContain high-pressure coolantThick steel

Concrete ShieldBlock radiation (biological shield)Concrete (2-3 m thick)

The Moderator

Function

The moderator slows down the neutrons released in fission reactions to thermal speeds through elastic collisions between the nuclei of moderator atoms and fission neutrons.

Why Light Elements?

The closer the moderator atoms are in size to a neutron, the larger the proportion of momentum transferred. Therefore the lower the number of collisions required to get neutrons to thermal speeds.

Water as Moderator

Water is often used as it contains hydrogen. It is also inexpensive and not very reactive. Graphite is also sometimes used.

ModeratorMass Number (A)Collisions to Thermalise

Water (H₂O)1 (H)~18

Heavy Water (D₂O)2 (D)~25

Graphite (C)12~115

Key Point

Lighter moderator nuclei (closer to neutron mass) transfer more energy per collision, requiring fewer collisions to slow neutrons to thermal speeds.

Worked Example01

Explain why graphite requires many more collisions than water to thermalise fast neutrons.

1

In an elastic collision, maximum energy transfer occurs when the two colliding objects have equal masses.

2

A neutron has mass number A = 1. Hydrogen also has A = 1 (equal masses), so a head-on collision transfers all the neutron's kinetic energy.

3

Carbon (graphite) has A = 12, which is much larger than the neutron's mass.

4

When a light object collides with a heavier object, only a small fraction of kinetic energy is transferred per collision.

5

Therefore, many more collisions with carbon nuclei are needed to reduce the neutron's energy to thermal levels compared to hydrogen.

Key formula:

\frac{E _{a f t er}}{E _{b e f ore}} = (\frac{A - 1}{A + 1})^{2}

— for carbon this gives 0.716 (28% loss), for hydrogen this gives 0 (100% loss possible).

Worked Example02

A neutron with kinetic energy 2.0 MeV collides head-on with a stationary carbon-12 nucleus in a graphite moderator. Calculate the kinetic energy of the neutron after the collision.

1

Identify masses: neutron m = 1, carbon M = 12

2

Apply energy ratio:

\frac{E _{a f t er}}{E _{b e f ore}} = (\frac{12 - 1}{12 + 1})^{2} = (\frac{11}{13})^{2}

3

Calculate:

(\frac{11}{13})^{2} = \frac{121}{169} = 0.716

4

Final energy:

E_{a f t er} = 0.716 \times 2.0 = 1.43

MeV

Answer1.43 MeV (28.4% energy loss per collision)

Control Rods

Function

Control rods absorb neutrons in the reactor in order to control chain reactions. The height of control rods can be adjusted to control the amount of energy produced.

They are made of materials which absorb neutrons without undergoing fission, such as boron and cadmium.

ActionEffectResult

Lower rods into coreMore neutrons absorbedk < 1, power decreases

Raise rods outFewer absorbedk > 1, power increases

Emergency shutdownAll rods fully insertedReaction stops

Fail-Safe: Electromagnets

The control rods are held by electromagnets. If the temperature in the core rises to an unsafe level, the current in these magnets is switched off and the control rods fall into the core by gravity, absorbing most of the neutrons and shutting off the reaction.

The Coolant

Function

The coolant absorbs the heat released during fission reactions in the core of the reactor. This heat is then used to make steam which powers electricity-generating turbines.

Required Properties

High specific heat capacity - can transfer large amounts of thermal energy

High thermal conductivity - efficient heat transfer

Low neutron absorption - doesn't affect chain reaction

Non-corrosive - protects pipework

Water as Coolant and Moderator

Sometimes water is both the coolant and moderator as it has a high specific heat capacity.

CoolantReactor TypeNotes

Water (H₂O)PWR, BWRAlso acts as moderator

Carbon DioxideAGR, MagnoxGas coolant, high temp

Helium gasHTGRInert, high temp operation

Molten saltMSRVery high heat capacity

Worked Example03

A nuclear reactor operates at 2400 MW thermal power. Each fission releases 200 MeV. Calculate the number of fission events per second.

1

Convert energy: E = 200 MeV × 1.60 × 10⁻¹³ J/MeV = 3.20 × 10⁻¹¹ J

2

Convert power: P = 2400 MW = 2.4 × 10⁹ W

3

Fission rate = P / E = (2.4 × 10⁹) / (3.20 × 10⁻¹¹)

4

Calculate: = 7.5 × 10¹⁹ s⁻¹

Answer7.5 × 10¹⁹ fissions per second

Worked Example04

If the mass defect in the fission of one nucleus of uranium-235 is 0.1276 u, calculate the energy in joules available by the fission of 1 kg of uranium-235. (Avogadro's number = 6.02 × 10²³ mol⁻¹, 1 u = 1.66 × 10⁻²⁷ kg)

1

Energy per fission: E = Δmc² = 0.1276 × 1.66 × 10⁻²⁷ × (3.0 × 10⁸)²

2

E = 0.1276 × 1.66 × 10⁻²⁷ × 9.0 × 10¹⁶ = 1.91 × 10⁻¹¹ J per fission

3

Number of atoms in 1 kg: N = (1000 / 235) × 6.02 × 10²³ = 2.56 × 10²⁴ atoms

4

Total energy: E = 2.56 × 10²⁴ × 1.91 × 10⁻¹¹ = 4.9 × 10¹³ J

Answer4.9 × 10¹³ J (equivalent to ~25 tonnes of coal)

Safety: Shielding

Around the nuclear reactor is a very thick concrete shielding (2-3 m), which blocks radiation from escaping and affecting workers in the power station.

Shielding Becomes Radioactive

The shielding may become radioactive after long term use because neutrons which escape the reactor may enter the shielding nuclei, causing them to become unstable and start experiencing beta-minus decay.

MaterialRadiation StoppedUse

Concrete (2-3 m)Neutrons, gammaBiological shield

SteelGammaPressure vessel, containment

LeadGammaLocal shielding

WaterNeutrons, gammaFuel storage pools

Defence in Depth

Multiple barriers: fuel pellet → cladding → pressure vessel → concrete containment

Emergency Shutdown

Emergency Shut-Down

In an emergency, the control rods are dropped into the reactor core entirely in order to stop fission reactions occurring as soon as possible by absorbing all the free neutrons in the core.

Automatic Safety System

For safety, any overheating in the reactor leads to automatic shut-down. The control rods are held by electromagnets. If the temperature rises to an unsafe level, the current in these magnets is switched off and the control rods fall into the core, absorbing most of the neutrons.

Decay Heat

Even after shutdown, radioactive decay continues producing heat (~7% of full power initially). Continued cooling is essential:

Immediate: ~7% of full power

After 1 hour: ~1.5%

After 1 day: ~0.5%

Radioactive Waste

During fission, two daughter nuclei are produced. These nuclei are usually extremely unstable and have very high activity, so need to be disposed of responsibly.

CategoryExamplesCharacteristics

High-Level (HLW)Spent fuel rodsCan stay radioactive for thousands of years. ~1000× more radioactive than ILW

Intermediate (ILW)Fuel containersLong half-lives, stored in 500L steel drums (~50,000 m³ so far)

Low-Level (LLW)Gloves, clothing, toolsShort-lived radioactivity, disposed of near surface after ~15 years

HLW Disposal Process

Waste removed and handled remotely to limit exposure

Placed in cooling ponds for up to a year (still producing heat)

Uranium and plutonium removed for recycling

Vitrified (encased in glass), placed in thick steel casks, stored in deep caverns in geologically stable locations

Worked Example05

Calculate the speed of a thermal neutron at temperature T = 300 K. (Neutron mass = 1.67 × 10⁻²⁷ kg, k = 1.38 × 10⁻²³ J K⁻¹)

1

Thermal kinetic energy:

E_{k} = \frac{3}{2} k T = \frac{3}{2} \times 1.38 \times 1 0^{- 23} \times 300

2

E_{k} = 6.21 \times 1 0^{- 21}

J

3

Using

E_{k} = \frac{1}{2} m v^{2}

, rearrange:

v = \frac{2 E _{k}}{m}

4

v = \frac{2 \times 6.21 \times 1 0 ^{- 21}}{1.67 \times 1 0 ^{- 27}} = 7.44 \times 1 0^{6}

5

v = 2.73 \times 1 0^{3}

m s⁻¹ ≈ 2700 m s⁻¹

Answer~2700 m s⁻¹ (order of magnitude 10³ to 10⁴ m s⁻¹)

Risks vs Benefits

Benefits

No polluting gases - no CO₂ during operation

Reliable for production of power

Far less fuel needed - only 1 kg of uranium gives as much power as 25 tonnes of coal

Small land footprint

Energy security - fuel stockpiled

Risks

Radioactive waste produced

Nuclear meltdown could have catastrophic consequences

Decommissioning problems at end of life

Proliferation - link to weapons

High costs - construction

Balancing Risks and Benefits

The risks and benefits of nuclear power stations must be balanced. Risks are minimised through safety aspects to make sure the benefits of nuclear power outweigh the risks. Expanding nuclear power could reduce CO₂ emissions by up to 30%.

Site Selection & Decommissioning

Considerations When Choosing Site

Transport facilities

A hard-rock site (geological stability)

Environmental aspects

Local centres of population

Local industry and commerce

Availability of cooling water

Decommissioning

Although nuclear power is a very good way of producing energy, there are problems when the reactor reaches the end of its useful life:

'Used' fuel elements stored under water in 'ponds' while they cool

Fuel is reprocessed to remove uranium and plutonium

Remaining toxic waste encased in glass blocks

Material stored underground for 50+ years before disposal

Practice

Question 1: A reactor operates at 3000 MW thermal power. If each fission releases 200 MeV, calculate the mass of U-235 consumed per day. (1 u = 1.66 × 10⁻²⁷ kg)

1

Energy per fission: 200 × 1.60 × 10⁻¹³ = 3.20 × 10⁻¹¹ J

2

Fissions per second: (3000 × 10⁶) ÷ (3.20 × 10⁻¹¹) = 9.375 × 10¹⁹ s⁻¹

3

Fissions per day: 9.375 × 10¹⁹ × 86400 = 8.1 × 10²⁴

4

Mass per fission: 235 × 1.66 × 10⁻²⁷ = 3.90 × 10⁻²⁵ kg

5

Total mass: 8.1 × 10²⁴ × 3.90 × 10⁻²⁵ = 3.2 kg

AnswerAbout 3.2 kg of U-235 per day

Practice

Question 2: Explain why reactor-grade uranium (3-5% U-235) cannot produce a nuclear explosion, even if control systems fail completely.

1

Low enrichment: Only 3-5% U-235 (rest is U-238 which absorbs neutrons)

2

U-238 absorbs fission neutrons, helping control the reaction rate

3

Requires moderator: Chain reaction only sustains with moderation

4

Self-limiting: If moderator damaged/expelled, chain reaction cannot continue

AnswerReactor fuel requires a moderator for criticality and contains U-238 which absorbs neutrons. Without a functioning moderator and with neutron-absorbing U-238 present, a nuclear explosion is impossible.

Practice

Question 3: Explain why concrete shielding around a nuclear reactor may become radioactive over time.

1

Neutrons escape from the reactor core

2

These neutrons may enter the nuclei of atoms in the concrete shielding

3

Neutron absorption causes these nuclei to become unstable (neutron-rich)

4

The unstable nuclei then undergo beta-minus decay to become stable

AnswerNeutrons escaping from the reactor are absorbed by nuclei in the shielding, making them neutron-rich and unstable. These nuclei then undergo beta-minus decay, making the shielding radioactive.

Summary

Key Concepts

ConceptDescription

Induced FissionThermal neutron + U-235 → U-236* → daughter nuclei + neutrons + ~200 MeV

Thermal NeutronsLow energy (~0.025 eV), speed ~10⁴ m s⁻¹, can be captured by nucleus

Chain Reactionk = 1 critical; requires critical mass (~15 kg pure U-235)

ModeratorSlows neutrons via elastic collisions (water, graphite, heavy water)

Control RodsAbsorb neutrons (B, Cd), held by electromagnets for fail-safe

CoolantRemoves heat (water, CO₂, helium, molten salt)

EnrichmentNatural U: 0.72% U-235; Reactor grade: 3-5% U-235

Key Exam Points

Use thermal energy formula:

E_{k} = \frac{3}{2} k T

U-238 absorbs neutrons, helps control reaction

Control rods fall by gravity if electromagnets fail (fail-safe)

Shielding becomes radioactive (neutron activation → β⁻ decay)

1 kg uranium = 25 tonnes coal equivalent