Nuclear Chain Reaction

Equivalent forms

criticality

k < 1 \text{ subcritical}, \quad k = 1 \text{ critical}, \quad k > 1 \text{ supercritical}

fission example

n + {}^{235}U \to {}^{141}Ba + {}^{92}Kr + 3n + 200\,\text{MeV}

growth rate

N(t) = N_0\, e^{(k-1)t/\tau}

The entire difference between a power plant and a weapon is whether a single dimensionless number sits at 1.000 or 2.

Symbol	Name	SI unit	Dimension	Range
$N_n$	neutrons in generation noutput	—	1	—
$k$	effective multiplication factor	—	1	—
$\nu$	neutrons released per fission (≈2.4 for U-235)	—	1	—
$p$	probability a neutron causes another fission	—	1	—
$n$	generation number (one generation ≈ 10 ns – 0.1 s)	—	1	—

Discovery

Leó Szilárd · Enrico Fermi · 1942

Szilárd conceived the chain reaction in 1933 while crossing a London street — annoyed by Rutherford calling atomic energy 'moonshine' — and patented it before fission was even discovered. On 2 December 1942, under the stands of a Chicago squash court, Fermi's team slid the control rods out of Chicago Pile-1 inch by inch until k reached 1.0006. The world's first self-sustaining chain reaction ran for 28 minutes. The coded message to Washington: 'The Italian navigator has landed in the new world.'

Real-world applications

Power reactors hold $k = 1$ exactly; delayed neutrons (0.65% arriving seconds late) stretch the response time from microseconds to seconds, making control rods fast enough.
Control rods (boron, cadmium) and dissolved boric acid trim p; pulling them raises k a hair above 1 to raise power, then back to 1.
Criticality safety: fissile solutions are stored in tall thin 'geometrically safe' tanks that leak too many neutrons to ever go critical.
Natural reactor at Oklo, Gabon: 1.7 billion years ago groundwater moderated uranium seams into hours-long critical pulses.
Chernobyl (1986): a positive feedback drove k well above prompt-critical; power rose $\sim 100\times in 4$ seconds — why reactor design obsesses over keeping feedback negative.

Common misconceptions

“A reactor can explode like a bomb.” — It can't: 3–5% enriched fuel can never reach prompt supercriticality of bomb magnitude; accidents are steam/chemical explosions.
“All neutrons matter equally.” — Reactors live on the 0.65% of delayed neutrons; without them control would require microsecond reflexes.
“k > 1 means instant boom.” — $k = 1.001$ on delayed neutrons is a gentle power ramp; danger starts only past prompt-critical $(k \approx 1.0065)$ .

Experimental verification

Chicago Pile-1 (1942) measured the predicted exponential pile period; modern reactors verify k to five decimal places daily via inverse kinetics; pulsed reactors like Godiva confirm prompt-burst growth rates exactly matching

N_{0}e^((k-1)t/\tau )

.

Derivation

Each fission releases $\nu$ neutrons $(\nu \approx 2.42$ for thermal fission of U-235).
Each neutron causes a new fission with probability p (it may instead escape, or be captured by U-238, control rods, or structure).
Expected neutrons in the next generation per current neutron: $k = \nu p$ .
After n generations: $N_n = N_{0}k^{n}$ — geometric growth or decay.
With generation time $\tau$ , continuous form $N(t) = N_{0}e^((k-1)t/\tau )$ . $For \tau \sim 10^{-8} s$ in a bomb core, $k = 2$ gives 80 doublings in under a microsecond.

Limiting cases

k < 1⟶

N \to 0

Subcritical: every burst dies away geometrically — the safe state.

k = 1

⟶ N constantCritical: steady power, the operating point of every reactor.

k >

1 + \beta

(prompt critical)⟶ explosive growthGrowth on prompt neutrons alone, microsecond timescales — the weapon regime.

What if…

What

if \nu

were 1.1 instead of 2.4?

After unavoidable losses, k could barely exceed 1 even in pure U-235 — no bombs, marginal reactors. Civilization's nuclear age hinges on uranium being a generous neutron emitter.

What if there were no delayed neutrons?

Reactors would respond in microseconds instead of seconds; mechanical control rods would be hopeless and fission power likely impractical.

1

From one neutron to a gram of fissioned uranium

Given ·

k = 2

, generation time

\tau = 10\,\mathrm{ns}

, energy per fission

= 200\,\mathrm{MeV} = 3.2\times 10^{-11} J

Find · Generations and time to fission

10^{21}

nuclei

(\sim 0.4\,\mathrm{g}

of U-235)

Steps

Need $2^{n} \approx 10^{21} \to n = 21 \ln 10 / \ln 2 \approx 70$ generations.
Time $= 70 \times 10\,\mathrm{ns} = 0.7 \mu s$ .
Energy $= 10^{21} \times 3.2\times 10^{-11} J = 3.2\times 10^{10} J \approx 7.6$ tons of TNT.

Result ·

\sim 70

generations, under a microsecond — geometric growth is why fission weapons need no fuse, only assembly.

2

Reactor power ramp on delayed neutrons

Given ·

k = 1.0005

, effective generation time (with delayed neutrons

) \tau \approx 0.1\,\mathrm{s}

Find · Time to double the reactor power

Steps

$N(t) = N_{0}e^((k-1)t/\tau )$ → doubling when $(k-1)t/\tau = \ln 2$ .
$t = 0.693 \times 0.1 / 0.0005 \approx 139\,\mathrm{s}$ .

Result · Power doubles

in \sim 2.3

minutes — leisurely enough for human operators, thanks entirely to delayed neutrons.

Nuclear Binding Energy→Binding Energy per Nucleon→Q-Value of Nuclear Reaction→Radioactive Decay Law→Semi-Empirical Mass Formula→