Virtual laboratory/Physics/University-level physics/The chain reaction and critical mass

The chain reaction and critical mass

Each fission releases neutrons that trigger further ones. The multiplication factor k decides it: k<1 dies out (subcritical), k=1 sustains the reaction (reactor), k>1 produces an avalanche (supercritical). Set k and the number of generations, and you will see how a small excess multiplication grows exponentially.

The chain reaction and critical mass
Physics · Nuclear physics · university level
Multiplication factor kk1.00
Number of generations nn30
Each fission triggers on average 1.02 more (k=1.02k=1.02). After 100 generations the neutron count:
factor k
1.00
N/N₀ after 30 gen.
1.00
state
critical
🎯 Challenge: identify the regime
streak: 0

I will give you a multiplication factor k — guess the regime (no slider). The critical boundary lies at k = 1.

What just happened?

Each fission releases neutrons that can trigger further ones. The multiplication factor k tells how many offspring one fission has on average. The neutron count in a generation grows (or decays) like a power — which is why small changes in k have huge effects.

Geometric growth:

Nn=N0 k nN_n = N_0\, k^{\,n}

k<1k<1 — subcritical (dies out); k=1k=1 — critical (reactor, steady level); k>1k>1 — supercritical (avalanche). The reactor holds k=1k=1 with control rods that absorb neutrons.

Invariant — a sharp boundary:

At k<1k<1 the reaction dies out always, even if you start from a huge number of neutrons — it is only a matter of how many generations. k=1k=1 is the boundary: critical mass is the amount and geometry of material at which kk just reaches 1.

Why this matters for your exam:

This is an academic extension of the topic of nuclear reactions and fission. It links nuclear physics with the mathematics of geometric sequences — the same formula knk^n describes compound interest and population growth.

Remember: Nn=N0knN_n = N_0 k^n. k<1 dies out, k=1 reactor, k>1 avalanche. Critical mass = the threshold at which k=1.

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Theory & formulas

Multiplication factor

kk is the average number of fissions triggered by one preceding fission:

  • k<1k<1 — the reaction dies out (subcritical),
  • k=1k=1 — stable (reactor),
  • k>1k>1 — avalanche (supercritical).

Exponential growth

N=N0 knN = N_0\,k^n The number of fissions grows with generation nn. Even k=1.02k=1.02 after 100 generations gives ~7×. Critical mass is the amount and geometry of fuel at which k=1k=1 (fewer neutrons escape). The reactor holds k≈1k\approx1 with rods that absorb neutrons.

Symbols: kk — neutron multiplication factor, N0N_0 — initial number, NN — number after nn generations, nn — generation index.

Frequently asked questions

›

What is critical mass?

It is the amount and geometry of fissile material at which the multiplication factor k reaches 1 — that is, each fission triggers on average exactly one more. Below the critical mass too many neutrons escape outward and the reaction dies out.

›

How does a reactor hold k = 1?

With control rods made of a neutron-absorbing material (e.g. boron, cadmium). Inserting them lowers k (more neutrons absorbed); withdrawing them raises it. Feedback keeps k very close to 1, i.e. a steady power level.

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Why is a small excess in k so dangerous?

Because the neutron count grows like kⁿ — exponentially with the number of generations. Even k=1.02 after 100 generations gives ~7× growth, and a neutron generation lasts fractions of a second. That is why reactor control requires continuous, precise regulation.

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