Fission — chain reaction, critical mass: Question bank

Symbols you will see (all built in the parent note): $k$ = multiplication factor (neutrons this generation ÷ last generation); $ν$ = neutrons emitted per fission ( $\approx 2.5$ ); $R$ = radius of the fissile lump; $M_{c}$ = critical mass; $τ$ = generation time; $Q$ = energy released per fission.

True or false — justify

State true/false AND the reason. A bare truth value scores nothing.

A single

^{235} U

nucleus needs the whole critical mass present before it can fission.

False — one nucleus fissions the instant a neutron is absorbed, at any mass. Critical mass is the threshold for a self-sustaining chain, i.e. for neutron survival across generations, not for one split.

Below critical mass no fission events happen at all.

False — individual fissions still occur; they just don't cascade. With

k < 1

each generation produces fewer neutrons than the last, so the chain fizzles out even though single splits keep firing.

At exactly

k = 1

the number of neutrons stays constant in time.

True —

N_{n} = N_{0} k^{n} = N_{0}

for all

n

when

k = 1

, so the population is steady. This is a reactor's normal operating state.

Doubling the mass of a subcritical lump always makes it supercritical.

False in general — it helps (leakage fraction

\propto 1/ R

drops as

R

grows), but whether

k

crosses 1 depends on shape, density, purity and reflectors. A flat sheet of twice the mass can still leak enough to stay subcritical.

Fission releases energy because it breaks strong nuclear bonds.

False — energy comes from the fragments being more tightly bound (higher BE/A,

\approx 8.5

7.6

MeV). Going from loosely to tightly bound releases the difference; you gain binding, you don't spend it. See Binding Energy per Nucleon Curve.

A sphere is the optimal shape because it holds the most fuel.

False — a sphere has the least surface area per unit volume, minimising leakage (which scales as surface

R^{2}

against production

R^{3}

). It minimises critical mass, not maximises fuel.

Compressing fissile material lowers its critical mass.

True — higher density

ρ

shrinks

R

for the same mass and packs nuclei closer so neutrons hit one sooner;

M_{c} \propto 1/ ρ^{2}

. This is the physics behind implosion assembly.

A moderator speeds up the chain reaction by giving neutrons more energy.

False — a moderator slows neutrons via elastic collisions. Slow (thermal) neutrons have a far larger fission cross-section in

^{235} U

, so slowing them raises the chance of the next fission. See Neutron Cross-section.

Delayed neutrons make a reactor harder to control.

False — they stretch the effective generation time

τ

from

\sim 1 0^{- 8} s

\sim 1 0^{- 1} s

, giving operators seconds (not microseconds) to respond. They are exactly what makes reactors steerable.

Since

Q = Δ m c^{2}

, the fragments together weigh more than the original nucleus.

False — they weigh less; that lost mass

Δ m = m_{before} - m_{after}

became the released energy. See Mass-Energy Equivalence E=mc^2.

Spot the error

Each statement contains one flaw. Name it and correct it.

"Fusion and fission both release energy by climbing toward iron on the BE/A curve."

Correct — and worth internalising. Both move nuclei toward the iron peak: fission brings heavy nuclei down from the right, fusion brings light nuclei up from the left. See Nuclear Fusion.

"With

k = 1.01

, the neutron count grows by 1% of its original value each generation."

Error: growth is multiplicative, not additive. Each generation multiplies by

1.01

, so it grows by 1% of the current count — that's exponential (

N_{n} = N_{0} (1.01)^{n}

), not linear.

"Control rods work by reflecting neutrons back into the core."

Error: control rods absorb neutrons (cadmium/boron) to remove them and hold

k = 1

. Reflecting neutrons back is the job of a reflector/tamper, which lowers critical mass — the opposite purpose.

"Enriching fuel to 3%

^{235} U

makes a reactor able to explode like a bomb."

Error: 3% enrichment cannot go prompt-supercritical. A bomb needs

> 90%

^{235} U

and fast neutrons; reactor-grade fuel physically cannot assemble into a nuclear explosion.

"Because

ν \approx 2.5

neutrons come out per fission,

k

is always about

2.5

Error: $ν$ is production; $k$ is net survival. Most of those

2.5

neutrons leak, get absorbed non-productively, or fail to cause fission.

k

counts only those that trigger the next generation — near

1

in a working reactor.

"Leakage and non-fission absorption both scale with surface area, so both dominate small lumps."

Error: absorption scales with volume ( $R^{3}$ ), like production. Only leakage scales with surface (

R^{2}

). That surface/volume mismatch,

\propto 1/ R

, is why small lumps go subcritical.

Why questions

Explain the mechanism, not just the fact.

Why does critical mass exist as a finite number rather than "any big enough amount works trivially"?

Because the neutron loss fraction from leakage falls as

1/ R

: only at a specific radius does production

\propto R^{3}

exactly balance losses, giving

k = 1

. That balance point is a genuine geometric threshold.

Why does a neutron reflector (tamper) lower the critical mass?

It bounces would-be-escaping neutrons back into the core, cutting the leakage loss. Less leakage means production wins at a smaller radius, so less material is needed to reach

k = 1

Why do reactors deliberately keep

k

1

rather than comfortably above it?

k > 1

means exponential growth of power — runaway. Holding

k = 1

gives constant, controllable output; small deliberate excursions above/below let operators raise or lower power smoothly.

Why is the generation time

τ

so crucial to reactor safety even though it doesn't change whether the reaction sustains?

τ

sets the timescale of change,

N (t) = N_{0} k^{t / τ}

. A tiny

τ

makes any

k > 1

blow up in microseconds; a large

τ

(thanks to delayed neutrons) turns the same

k

into a slow, catchable drift.

Why can't you make a bomb just by piling up enough natural uranium?

Natural uranium is only

\sim 0.7%

^{235} U

; the dominant

^{238} U

absorbs neutrons without fissioning at those energies, so

k

stays below 1. You must enrich to concentrate the fissile isotope.

Why does

^{235} U

prefer slow neutrons when the fission emits fast ones?

The fission cross-section — the effective target size a neutron sees — is much larger for slow (thermal) neutrons. Fast neutrons whizz past too quickly to be captured efficiently, so they must be moderated first. See Neutron Cross-section.

Edge cases

Boundary and degenerate scenarios the topic quietly assumes.

What happens to

k

R \to \infty

(an effectively infinite slab of fuel)?

Leakage fraction

\propto 1/ R \to 0

, so no neutrons escape and

k

approaches its maximum "infinite-medium" value

k_{\infty}

. This is the ideal upper limit; every real, finite lump has

k < k_{\infty}

What happens to

k

R \to 0

(a speck of fissile dust)?

Surface-to-volume

\to \infty

, so nearly every neutron leaks before causing fission and

k \to 0

. That's why fine powder or thin foils are safe from chain reactions regardless of material.

k

is exactly

1

but a fluctuation removes a few neutrons, does the reactor die instantly?

No — at

k = 1

the population is only marginally stable; a dip drops it slightly and it drifts, but there's no exponential collapse. The large

τ

from delayed neutrons gives ample time to nudge

k

back to 1.

Consider the degenerate case

ν = 0

(no neutrons emitted per fission). Is a chain reaction possible?

No — with zero neutrons produced, each fission is a dead end and

k = 0

always. A chain requires

ν > 1

so that after all losses at least one neutron survives to continue.

At the melting/compression limit, can raising density alone turn any subcritical lump critical?

Only until physical limits (solid-density, then nuclear-matter density) are reached, and only if enough total mass is present;

M_{c} \propto 1/ ρ^{2}

falls but a truly tiny mass still can't sustain a chain no matter how compressed.

Recall Quick self-test

The single sentence that resolves most traps here ::: Critical mass is about neutron survival across generations ( $k$ ), governed by geometry and leakage, not about whether one nucleus can split or how much total energy is stored.