Fission — chain reaction, critical mass, reactors (thermal vs fast): Visual walkthrough

Step 1 — Follow ONE neutron

WHAT. Imagine a single free neutron — a tiny electrically-neutral ball — drifting inside a block of $^{235} U$ . A neutron is a nuclear particle with no charge, so nothing pushes it away from the positively-charged nuclei; it drifts until it physically runs into one.

WHY start here. The entire chain reaction is just this one story, repeated. If we understand the fate of one neutron, multiplying by "how many neutrons there are" gives us everything.

PICTURE. In the figure, the magenta dot is our neutron. It travels in a straight line (the dashed arrow) until it meets a violet uranium nucleus. Three things can happen next, shown as three branching arrows — and only ONE of them keeps the story going.

Step 2 — Count the offspring: define $ν$ and $k$ term by term

WHAT. When our neutron does cause a fission (fate 1), that fission releases a fresh batch of neutrons. Let us name that number:

$ν = average number of neutrons released per fission \approx 2.5$

$ν$ (Greek "nu") — read it as "the litter size": how many baby neutrons each fission produces. For $^{235} U$ , $ν \approx 2.5$ .

WHY a new symbol. Because $2.5$ neutrons are born, but they are not all useful. We need a second number that counts only the ones that go on to cause the next fission.

$k = \frac{neutrons causing fission in the NEXT generation}{neutrons that caused fission in THIS generation}$

The top counts survivors — babies that actually grow up to split a nucleus.
The bottom counts their parents.
So $k$ is simply "how many of your children reach adulthood, on average."

PICTURE. One parent fission (orange star) fires off $ν = 2.5$ neutrons (magenta arrows). Some leak, some are absorbed, and only a fraction — shown surviving to the right — cause the next fission. That surviving fraction, per parent, is $k$ .

Step 3 — Turn survival into probabilities

WHAT. Of the $ν$ neutrons born, let a fraction $ℓ$ (ell) leak out and a fraction $a$ get absorbed-without-fission. The rest cause fission. We write $k$ as $ν$ times the fraction that survives:

$k = ν \times stayed inside P_{no leak} \times split a nucleus P_{fission ∣ stayed}$

$ν$ — how many we started with.
$P_{no leak}$ — probability a neutron does not escape through the surface (a number between 0 and 1).
$P_{fission ∣ stayed}$ — given it stayed, the probability it hits fuel and fissions rather than being uselessly absorbed.

WHY split it this way. Because the two losses have completely different causes. Leakage depends on geometry (how big the lump is) — that's Step 4. Absorption depends on chemistry / materials (control rods, U-238) — that's how reactors tune $k$ . Separating them lets us control each independently.

PICTURE. A funnel: $ν$ neutrons pour in the top; the leak fraction spills out one side, the absorbed fraction out the other, and only the surviving stream at the bottom counts toward $k$ .

Step 4 — Why SIZE decides leakage (surface vs volume)

WHAT. Fissions happen everywhere inside the lump, so neutron production scales with the volume. Leakage happens only at the skin, so neutron loss scales with the surface area. For a sphere of radius $r$ :

$production \propto V = \frac{4}{3} π r^{3}, leakage \propto S = 4 π r^{2}$

$\frac{leakage}{production} \propto \frac{S}{V} = \frac{4 π r ^{2}}{\frac{4}{3} π r ^{3}} = \frac{3}{r}$

$V \propto r^{3}$ — triple the radius, the volume (and production) grows $\times 27$ .
$S \propto r^{2}$ — the same tripling grows surface (and leakage) only $\times 9$ .
$\frac{3}{r}$ — the punchline: the bigger the lump, the smaller the relative leakage.

WHY this is the whole secret of critical mass. Because $P_{no leak}$ from Step 3 grows as $r$ grows. A tiny lump leaks so much that $k < 1$ no matter what. Grow it, leakage shrinks, and at one special radius $k$ climbs to exactly 1.

PICTURE. Two spheres — a small one (mostly-surface, lots of red escaping arrows) and a large one (mostly-interior, few escaping arrows). The $1/ r$ curve underneath shows leakage-fraction falling as the sphere grows.

Step 5 — The generation-by-generation avalanche

WHAT. Start with $N_{0}$ neutrons in "generation 0." Each generation multiplies the count by $k$ :

$N_{1} = k N_{0}, N_{2} = k N_{1} = k^{2} N_{0}, \dots, N_{n} = N_{0} k^{n}$

$N_{0}$ — how many neutrons you begin with.
$k$ — the survival-to-fission multiplier from Steps 2–3.
$n$ — the generation number (a fission generation lasts only about $1 0^{- 3}$ s in a thermal reactor).
$k^{n}$ — $k$ multiplied by itself $n$ times; this is where the avalanche lives.

WHY a power, not a product. Because each generation feeds the next: the output of one round is the input to the next. Repeated multiplication by the same factor is exactly what an exponential $k^{n}$ means. We use a power (not, say, addition) precisely because the process is self-similar — every generation is a scaled copy of the last.

PICTURE. Three side-by-side "family trees" for $k < 1$ , $k = 1$ , $k > 1$ . Watch the number of active branches per row shrink, hold steady, and explode.

Step 6 — The three regimes (cover EVERY case)

WHAT. The single formula $N_{n} = N_{0} k^{n}$ hides three totally different futures, depending only on whether $k$ is below, equal to, or above 1.

k < 1 \Rightarrow N_{n} \to 0 (dies), k = 1 \Rightarrow N_{n} = N_{0} (steady), k > 1 \Rightarrow N_{n} \to \infty (explodes)

$k < 1$ : each generation is smaller — the avalanche fizzles. This is a subcritical lump, or a reactor with control rods pushed in.
$k = 1$ : every generation is a perfect replacement — steady power. This is the reactor's critical operating point.
$k > 1$ : each generation grows — power surges or, if uncontrolled, a bomb.

WHY the boundary is razor-sharp. Because raising a number to a high power is unforgiving. With $n \approx 1000$ generations, $k = 0.999$ gives $k^{n} \approx 0.37$ (fading) while $k = 1.001$ gives $k^{n} \approx 2.7$ (nearly tripling). A change in $k$ of just $0.002$ flips the whole system — which is exactly why reactors need fast, precise control.

PICTURE. Three curves of $N_{n}$ versus generation $n$ : a decaying magenta curve ( $k < 1$ ), a flat violet line ( $k = 1$ ), and a soaring orange curve ( $k > 1$ ).

Step 7 — Tuning $k$ back to 1: what a reactor actually does

WHAT. A power reactor is a machine that holds $k = 1$ . Since Step 4's geometry is fixed once the fuel is loaded, the reactor tunes the absorption term of Step 3 using control rods.

$k = ν P_{no leak} P_{fission ∣ stayed}, rods in \Rightarrow P_{fission} ↓\Rightarrow k ↓$

Push rods in: they soak up neutrons → fewer survive to fission → $k$ drops below 1 → power falls.
Pull rods out: fewer neutrons absorbed → $k$ rises above 1 → power climbs.
Hold rods just so: $k = 1$ exactly → steady output.

WHY it is controllable at all. A small fraction of neutrons are delayed — emitted seconds after fission, not instantly. Those extra seconds stretch the generation time enough for mechanical rods to respond. Without them, the $1 0^{- 3}$ s generation would be far too fast to steer.

PICTURE. A dial labelled $k$ with a needle resting on 1; control rods slide in and out, dragging the needle down and up.

The one-picture summary

Everything on this page collapses into a single flow: one neutron → $ν$ babies → losses to leakage (geometry) and absorption (materials) → survivors define $k$ → $k^{n}$ decides the whole avalanche.

Recall Feynman retelling — say it in plain words

Imagine one neutron rattling around inside a chunk of uranium. It slams into a nucleus and splits it — out fly about two and a half new neutrons. But not all of them find work: some shoot out the sides of the chunk, some get eaten by control rods or the wrong kind of uranium. Whatever survives to split another nucleus, counted per parent, is the number $k$ . If a small chunk lets too many escape out its big surface, $k$ dips below 1 and the whole thing fizzles. Make the chunk bigger and its skin becomes relatively smaller, so fewer escape and $k$ rises. At the perfect size, $k = 1$ — every fission exactly replaces itself, forever: that's a reactor. Nudge $k$ above 1 with control rods pulled out, and because each generation multiplies by $k$ and there are thousands of generations per second, the count runs away as $k^{n}$ — an explosion of energy. The reactor operator's whole job is to keep that one number sitting exactly on 1.

Recall Test yourself

A lump has $ν = 2.5$ , and 60% of born neutrons are lost to leakage plus absorption. Is it super-, sub-, or critical? ::: Survivors $= 2.5 \times 0.40 = 1.0$ , so $k = 1$ — critical. If $k = 1.002$ and there are 1000 generations, roughly what factor does the neutron count grow by? ::: $1.00 2^{1000} \approx 7.4$ — over sevenfold, showing how tiny $k$ shifts explode. Why does a sphere give the smallest critical mass? ::: It has the least surface area per unit volume, so it leaks the least — least leakage means $k$ reaches 1 at the smallest mass.

See also: Nuclear Fusion · Radioactive Decay & Half-life · Einstein Mass-Energy Equivalence · Nuclear Reactor Safety & Waste.