Fission & Fusion · 26.1
Chain Reactions & Criticality
A slow neutron drifts into a uranium-235 nucleus, which swells, wobbles, and tears itself in two — flinging out 200 MeV and, crucially, two or three fresh neutrons. If each generation of neutrons breeds at least one successor, the reaction feeds itself. That single bookkeeping number decides between a power station and a bomb.
Build the model
Connect the measurement to the mechanism.
Induced fission starts when ²³⁵U captures a neutron, forming excited ²³⁶U* that splits into two mid-mass fragments plus, on average, 2.5 fresh neutrons and about 200 MeV. The capture works best for slow (thermal) neutrons: the fission cross-section of ²³⁵U is roughly a thousand times larger at 0.025 eV than at the 2 MeV the neutrons are born with, so fast neutrons must be slowed by a moderator before they are useful. The fate of the chain is summarised by the neutron multiplication factor k — the number of neutrons in one generation per neutron in the previous one.
With k < 1 the population dies away (subcritical), with k = 1 it holds steady (critical — exactly what a reactor wants), and with k > 1 it grows exponentially (supercritical). Whether k reaches 1 depends on geometry: neutron production scales with volume but leakage scales with surface area, so too small a lump loses too many neutrons — the minimum self-sustaining amount is the critical mass. How fast anything happens is set by the mean generation time τ, the average delay between one fission and the next it causes.
- Simple definition
- A chain reaction is fission sustained by its own neutrons; the multiplication factor k (neutrons per neutron of the previous generation) decides everything — k < 1 dies away, k = 1 runs steady, k > 1 grows exponentially, generation after generation.
- Example
- A reactor at steady power sits at k = 1.000: of the ~2.5 neutrons each fission releases, about 1.5 are absorbed or leak away, and exactly one, on average, causes the next fission.
The captured neutron excites the nucleus past its breaking point. The fragments land near the top of the binding-energy curve — that climb is the 200 MeV — and the spare neutrons are the seed of the next generation.
One typical channel of many — A and Z balance
Repeated multiplication is all a chain reaction is. Ten generations at k = 2 turn one neutron into a thousand; at k = 0.9 the same ten generations lose two-thirds of the population.
k = neutrons this generation ÷ neutrons last generation
Fissions happen throughout the volume, but neutrons only escape through the surface. Since volume grows faster than surface, there is a size below which leakage always wins and k cannot reach 1 — the critical size.
Bigger lump → smaller fraction of neutrons escapes
Why slow neutrons win
A thermal neutron lingers near the nucleus far longer than a 2 MeV one, so it is far more likely to be captured — the ²³⁵U fission cross-section is about a thousand times larger. A moderator (water, graphite) slows fission neutrons by elastic collisions; light nuclei work best because a neutron hands over the most energy to a partner of similar mass, the way one billiard ball can stop dead on another.
The three regimes of k
Subcritical (k < 1): each generation is smaller; a burst of neutrons fizzles out. Critical (k = 1): every generation replaces itself; power is constant. Supercritical (k > 1): exponential growth — at k = 1.01 the population doubles every 70 generations, at k = 2 every generation. A reactor spends its life trimmed to k = 1; a weapon slams pieces together to reach k ≈ 2 with fast neutrons before the assembly blows itself apart.
Generation time sets the clock
Exponential growth is measured in generations, but wall-clock speed is k and τ together. Prompt neutrons in a thermal reactor take ~0.1 ms from birth to captured; in a bare fast assembly τ is ~10 ns. The same k = 1.01 that is leisurely in one system is catastrophic in the other — which is why lesson 26.2 will make so much of the rare neutrons that arrive late.
Change one variable at a time
Make the relationship visible.
Each dot is one neutron generation: N(g) = kᵍ from a single starting neutron. k decides the shape; τ decides the wall-clock speed. Try k = 1.05 at τ = 0.1 s (a controlled reactor) versus the same k at 0.01 µs (a fast assembly) — same curve, timescales apart by ten million.
Regimesupercritical — exponential growth
After 12 generations1.8 × N₀ · in 1.2 ms
Doubling time14.2 generations = 1.42 ms
One neutron → a million283 generations = 28.32 ms
Catch the common trap
Explain before calculating.
A reactor is held at exactly k = 1.00. Generation after generation, its neutron population…
Choose an answer to test the model.
Worked examples
State the rule, substitute, then check units.
EasyIn a small supercritical assembly, k = 2 and the mean generation time is τ = 10 ms. Starting from a single neutron, how many neutrons exist after 10 generations, and how long did that take?
- N = N₀kᵍ = 1 × 2¹⁰ = 1024 neutrons.
- Each generation takes τ = 10 ms, so t = 10 × 10 ms = 0.10 s.
- Every neutron came from a fission worth ~200 MeV — the energy release is doubling right along with the population.
Answer1024 neutrons after 0.10 s
MediumA reactor drifts to k = 1.005 with an effective generation time of 0.1 s. How many generations, and how many seconds, until the power doubles?
- Doubling needs kᵍ = 2, so g = ln 2 / ln k = 0.693 ÷ ln(1.005) = 0.693 ÷ 0.004988 ≈ 139 generations.
- t = gτ = 139 × 0.1 ≈ 14 s.
- Fourteen seconds is ample time for control rods to respond — a 0.5% excess in k is a gentle drift, not a runaway, thanks to the long effective τ.
Answer≈ 139 generations ≈ 14 s to double
HardIf every nucleus in 1.0 kg of pure ²³⁵U fissions, releasing 200 MeV each, how much energy is that? Compare with coal at 32.5 MJ per kg.
- Nuclei: N = (1000 ÷ 235) × 6.02 × 10²³ = 2.56 × 10²⁴.
- E = 2.56 × 10²⁴ × 200 × 10⁶ × 1.60 × 10⁻¹⁹ J ≈ 8.2 × 10¹³ J.
- Coal equivalent: 8.2 × 10¹³ ÷ 3.25 × 10⁷ ≈ 2.5 × 10⁶ kg — one kilogram of fissioned uranium matches about 2500 tonnes of coal.
Answer≈ 8 × 10¹³ J — roughly 2500 tonnes of coal from one kilogram
ChallengingExplain (a) why a lump of fissile material below a certain size can never sustain a chain, and (b) roughly how many collisions with hydrogen a 2 MeV fission neutron needs to reach thermal energy (0.025 eV).
- (a) Fission neutrons are produced throughout the volume (∝ r³) but escape through the surface (∝ r²). Shrink the lump and surface loses grow relative to production, so the escape fraction rises until k < 1 no matter what. The break-even size is the critical size; below it a chain always fizzles.
- (b) Colliding with a proton (equal mass), a neutron loses on average a factor e of its energy per collision: n ≈ ln(E₀/E) = ln(2 × 10⁶ ÷ 0.025) = ln(8 × 10⁷) ≈ 18.
- About 18–20 collisions suffice in water — but hundreds would be needed in carbon and thousands in lead, which is why moderators are made of the lightest practical nuclei.
AnswerLeakage ∝ surface beats production ∝ volume below the critical size; ~20 collisions with hydrogen thermalise a fission neutron