Undergraduate · 20 min

Criticality: how a fission chain grows or dies

Track the neutron population generation by generation for three values of the multiplication factor k and discover what “critical” really means.

Goal

Define the neutron multiplication factor k, predict the long-term fate of a chain for any k, and link k to reactor control.

Apparatus and reagents

A fissile sample of $\ce{^{235}U}$ modelled as a neutron population multiplying by k each generation.

Procedure

Start with k = 0.80 and follow the curve: count the generations needed for the population to fall below one neutron.
Switch to k = 1.00: the population is flat. This is a controlled reactor — each fission triggers exactly one more.
Switch to k = 1.25 and read the markers at generations 5 and 10; extrapolate what happens by generation 45.
Estimate how long a real generation lasts (~10⁻⁸ s for prompt neutrons) and judge why a supercritical excursion is uncontrollable without engineered feedback.

What to observe

k < 1 gives geometric decay ( $N_n = N_0 k^n$ ): the chain self-extinguishes however large the sample.
k = 1 exactly holds the population constant; even k = 1.01 grows without bound — criticality is a razor’s edge.
At k = 1.25 the population multiplies by ~25000 in 45 generations — in real time that is microseconds.

Explanation

Fission of $\ce{^{235}U}$ by a neutron yields two fragments plus on average 2.4 neutrons. Whether a chain survives is set by k, the average number of neutrons from one fission that cause another: some escape the sample, some are captured in $^{238}\ce{U}$ or the moderator. Enriching the fuel, shaping the geometry (critical mass) and absorbing excess neutrons with control rods all act on k. In reactors, delayed neutrons (~0.7 % of the yield) stretch the effective generation time, which is what makes k ≈ 1 controllable at all.

Chemists behind it

Otto Hahn