IB Physics HLTopic 5 — The Atomic & Nuclear WorldPaper 1 & 2keeping it self-sustaining~15 min read
Chain Reactions
A single fission is a one-off. What makes nuclear power — and nuclear weapons — possible is that each fission throws out spare neutrons that can trigger more fissions. Get the balance just right and the reaction keeps itself going at a steady rate; tip it too far and it runs away out of control. The whole art of a reactor is keeping that balance perfectly on the knife-edge. Let’s see how.
📚 What you need to know
A chain reaction is when neutrons from one fission go on to cause further fissions
Each U-235 fission releases 2 or 3 neutrons on average
The multiplication factork is the average number of neutrons from each fission that cause another fission
k < 1: reaction dies out (subcritical)
k = 1: steady, self-sustaining reaction (critical) — what a reactor wants
k > 1: reaction grows rapidly (supercritical)
The critical mass is the minimum fuel mass for a sustained chain reaction
Neutrons can be lost by escaping the surface or being absorbed without fission
The multiplication factor k
Each fission produces 2 or 3 neutrons, but not all of them go on to cause another fission. Some escape from the surface of the fuel, and some are absorbed by nuclei without causing a split. What matters is the average number of neutrons from each fission that successfully trigger another fission. We call this the multiplication factor, k.
Multiplication factork = neutrons causing next fission ÷ neutrons from previous fission
Everything hinges on the value of k. It’s the single number that decides whether the reaction fades away, holds steady, or explodes in growth.
The value of k decides everything: below 1 the reaction shrinks, at exactly 1 it holds steady, above 1 it multiplies out of control.
The three states explained
Reading k tells you immediately what the reaction will do:
Subcritical (k < 1): fewer and fewer fissions each generation, so the reaction dies away. Too much fuel is being lost or too many neutrons are escaping.
Critical (k = 1): exactly one neutron per fission causes the next fission, so the rate stays constant. This is the sweet spot a reactor is designed to sit at.
Supercritical (k > 1): more fissions every generation, so the reaction — and the power — grows rapidly. Useful briefly for raising power, dangerous if uncontrolled.
Here’s the mental picture I use: think of k as “how many children each fission has.” If every fission has fewer than one child, the family line dies out. Exactly one child — the population stays the same forever. More than one child each — the population explodes. A reactor is kept at exactly one child per fission, generation after generation.
k < 1 shrinks
add fuel / reduce loss
k = 1 steady
still more neutrons
k > 1 grows
What makes neutrons get lost?
For a chain reaction to sustain, enough neutrons must survive to cause the next round of fissions. Two things steal them away:
Escape through the surface: in a small piece of fuel, many neutrons simply fly out before hitting a nucleus. A bigger mass has proportionally less surface, so fewer escape — this is why critical mass exists.
Absorption without fission: some neutrons are captured by nuclei (including non-fuel material) without causing a split.
The critical mass is the minimum mass of fuel where enough neutrons are captured rather than lost, so that k reaches 1 and the reaction sustains itself.
WE 1
In a reactor, each fission produces on average 2.5 neutrons. If the reactor is running steadily (critical), how many of those neutrons, on average, go on to cause another fission? What must happen to the rest?
Step 1 — what “critical” means
Steady running means k = 1 exactly.
Step 2 — neutrons causing next fission
By definition k = 1 means exactly 1 neutron per fission triggers the next one.
Step 3 — the rest
The other 2.5 − 1 = 1.5 neutrons must be lost.
1 causes fission; 1.5 are absorbed or escapeThis is exactly the job of the control rods — they absorb the “extra” 1.5 neutrons so that precisely one is left to keep k at 1. Take the rods out a little and fewer are absorbed, nudging k above 1.
WE 2
A chain reaction starts with 1000 fissions in the first generation and has a multiplication factor k = 1.05. How many fissions occur in the 10th generation? Comment on what this shows.
Step 1 — growth is a power of k
After n generations, count = start × k(n−1)Step 2 — substitute= 1000 × 1.05⁹ = 1000 × 1.551≈ 1550 fissions in the 10th generationEven a tiny k = 1.05 — only 5% growth per step — leads to steady multiplication. That’s why reactors must control k with extreme precision: small changes compound fast.
⚛ Reading a chain reaction
Find k: neutrons causing next fission ÷ neutrons this generation.
k < 1? Subcritical — reaction dies out.
k = 1? Critical — steady, self-sustaining.
k > 1? Supercritical — reaction grows.
Growth over n generations: multiply by k(n−1).
💡 Top tips
k = 1 is what a reactor is held at — steady power.
Neutrons are lost by escaping or being absorbed without fission.
Critical mass exists because bigger masses lose fewer neutrons to the surface.
Control rods absorb the extra neutrons to keep k = 1.
Small changes in k compound quickly over generations.
⚠ Common mistakes
Thinking all the 2–3 neutrons cause fission — most are lost
Saying a reactor runs at k > 1 — steady running is k = 1
Confusing critical (steady) with dangerous — it’s the safe operating state
Forgetting neutrons can escape the surface, not just be absorbed
Mixing up multiplication factor with number of neutrons per fission
Quick recap: A chain reaction sustains when neutrons from one fission trigger the next. The multiplication factor k decides the outcome: k < 1 dies out (subcritical), k = 1 stays steady (critical — what a reactor wants), k > 1 grows (supercritical). Neutrons are lost by escaping or absorption, and the critical mass is the minimum fuel needed to keep k at 1.
Knowing that a reactor must sit exactly at k = 1 is one thing — actually holding it there, safely, for years on end is an engineering marvel. It takes a moderator to slow the neutrons, control rods to soak up the spares, and a coolant to carry the heat away. Let’s put it all together. Next page: How a Nuclear Reactor Works.
Chain reactions and k still confusing?
Book a free meeting and we’ll drill the multiplication factor, the three critical states, and the neutron-economy reasoning examiners test.
