Fission: Chain Reactions, Neutrons, and Reactor Basics

What fission is (and what makes it special)

Nuclear fission is the splitting of a heavy nucleus into two (occasionally three) lighter nuclei, typically after the nucleus absorbs a neutron. The key features that matter for reactors are:

• Energy release (mostly as kinetic energy of the fission fragments, plus some gamma energy).
• Additional neutrons emitted promptly (and a small fraction emitted later as delayed neutrons).
• Two new nuclei (fission products) that are usually neutron-rich and therefore radioactive.

A representative reaction (one of many possible “fission channels”) is:

n + U-235  →  (U-236*)  →  Ba-141 + Kr-92 + 3 n + energy

The exact fragments and the neutron count vary from event to event, but the average number of neutrons per fission for U-235 with thermal neutrons is roughly 2–3. Those extra neutrons are what make a chain reaction possible.

Why energy comes out: linking to binding energy per nucleon

In fission, a very heavy nucleus splits into medium-mass nuclei. Medium-mass nuclei have a higher binding energy per nucleon than very heavy nuclei. That means the final products are, in total, more tightly bound. The difference in total binding energy appears as released energy.

Energy accounting is often expressed using mass-energy equivalence:

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Q = (m_initial − m_final) c^2

Where Q is the energy released. In fission, m_final is smaller because the products have greater total binding energy. Most of Q becomes kinetic energy of the two heavy fragments; this kinetic energy is what heats reactor fuel.

Fissile vs fertile: what can (and cannot) sustain a chain reaction

Fissile materials

A fissile nuclide can undergo fission after absorbing a slow (thermal) neutron and can therefore support a chain reaction in a moderated reactor. Common fissile nuclides include:

• U-235
• Pu-239
• U-233

“Fissile” is a practical reactor term: it emphasizes that thermal neutrons are sufficient to induce fission with a high probability.

Fertile materials

A fertile nuclide is not readily fissioned by thermal neutrons but can absorb a neutron and later transform (through beta decays) into a fissile nuclide. Key fertile nuclides include:

• U-238 (can breed Pu-239)
• Th-232 (can breed U-233)

Fertile materials matter because they influence neutron economy: they can “consume” neutrons by capture, but they can also create new fissile fuel over time.

Neutrons in a reactor: production, slowing down, and losses

Fast vs thermal neutrons

Neutrons born from fission are typically fast (high kinetic energy). Many fissile materials have a higher fission probability for thermal neutrons (slow neutrons in equilibrium with the surrounding material). Therefore, many reactors include a moderator to slow neutrons down efficiently without capturing too many.

Moderation: the step-by-step picture

Moderation is mainly repeated elastic scattering: a neutron collides with nuclei in the moderator and loses energy gradually.

1. Neutron is emitted fast from a fission event.
2. It scatters off light nuclei (commonly hydrogen in water, deuterium in heavy water, or carbon in graphite).
3. Each collision reduces neutron energy; light nuclei are effective because they can take a larger fraction of the neutron’s kinetic energy per collision.
4. After many collisions, the neutron becomes thermal and is more likely to induce fission in fissile fuel (in a thermal reactor design).

Moderators must balance two properties:

• Good slowing-down power (efficient energy loss per collision).
• Low absorption (do not capture too many neutrons).

Neutron losses: why not every neutron continues the chain

Even if fission produces multiple neutrons, not all of them cause new fissions. Neutrons can be lost by:

• Non-fission capture in fuel (capture without fission) or in structural materials, coolant, moderator, impurities.
• Leakage out of the core (especially near boundaries).
• Absorption in control materials (intentionally introduced to regulate the chain reaction).

Criticality: subcritical, critical, supercritical

Reactor behavior is summarized by the effective multiplication factor, usually written k-effective or k_eff. It is defined conceptually as:

k_eff = (number of neutrons in one generation) / (number of neutrons in the previous generation)

• Subcritical: k_eff < 1. The chain reaction dies out.
• Critical: k_eff = 1. The chain reaction is steady (power constant in time, ignoring delayed effects).
• Supercritical: k_eff > 1. The neutron population grows (power increases).

In power reactors, the goal during normal operation is k_eff very close to 1, with control systems adjusting it as fuel composition, temperature, and other conditions change.

A simplified chain-reaction model (generation-by-generation)

To build intuition, use a “neutron generation” model. Let N0 be the number of neutrons that successfully start a generation (available to cause fission). Then:

N1 = k_eff * N0  (next generation neutrons that survive losses and are available again)

After g generations:

Ng = (k_eff)^g * N0

Interpretation:

• If k_eff = 0.95, then after 10 generations: N10 ≈ 0.95^10 ≈ 0.60 of the original (shrinking).
• If k_eff = 1.00, then Ng stays the same generation to generation (steady).
• If k_eff = 1.05, then after 10 generations: N10 ≈ 1.05^10 ≈ 1.63 (growing).

This model hides many details (time dependence, energy spectrum, spatial effects), but it captures the core idea: small changes in k_eff matter a lot because they compound across generations.

What determines k_eff qualitatively?

Think of k_eff as the result of a “neutron budget”:

• Source: neutrons produced per fission (and how many fissions occur).
• Helpful pathways: neutrons that slow down to useful energies and reach fuel.
• Loss pathways: capture without fission and leakage.

Any change that increases losses tends to push k_eff down; any change that increases the fraction of neutrons causing fission tends to push k_eff up.

Geometry and neutron leakage: why size and shape matter

Leakage is strongly tied to geometry because neutrons are born throughout the core and move randomly. A useful qualitative rule is:

• Neutron production scales with volume (more fuel volume → more fissions).
• Neutron leakage scales with surface area (more boundary area → more ways to escape).

As a reactor core gets larger, volume grows faster than surface area, so the surface-to-volume ratio decreases, and leakage becomes less significant. This is one reason why a small assembly of fuel may be subcritical while a larger arrangement of the same material can become critical.

Shape matters too: for a given volume, a sphere minimizes surface area, so it tends to minimize leakage compared with elongated or thin shapes.

Control materials: how reactors regulate k_eff

Control materials are neutron absorbers introduced to adjust the neutron budget. They reduce k_eff by increasing absorption losses. Common absorber elements include boron, cadmium, hafnium, and gadolinium (used in different forms and strategies).

Control rods (step-by-step control logic)

1. Insert rods deeper into the core → more neutrons absorbed → fewer neutrons available to cause fission → k_eff decreases.
2. Withdraw rods → fewer neutrons absorbed → more neutrons survive to cause fission → k_eff increases.
3. Fine adjustments maintain k_eff near 1 as conditions change (fuel burnup, temperature effects, xenon buildup, etc.).

Soluble absorbers and burnable absorbers

• Soluble absorber (e.g., boron dissolved in coolant/moderator in some designs): changes absorption throughout the core volume, providing smooth reactivity control.
• Burnable absorber (e.g., gadolinium-bearing fuel or separate absorber rods): starts with high absorption and gradually depletes, compensating for excess reactivity early in fuel life.

Putting it together: controlled fission as neutron economy

Controlled fission is not just “fissions happen”; it is “enough neutrons from each fission event survive all losses, slow down appropriately (if needed), and cause new fissions so that k_eff stays at 1.” The main levers are:

• Fuel composition (fissile fraction, presence of fertile absorbers).
• Moderation quality (how effectively neutrons are slowed without being captured).
• Core geometry (leakage control).
• Absorbers/control systems (intentional neutron removal).

Conceptual exercises (qualitative k_eff reasoning)

Exercise 1: Interpreting k_eff from a neutron story

A fuel assembly produces on average 2.5 neutrons per fission. Out of these, 0.7 leak out, 0.6 are captured in non-fuel materials, and the rest reach fuel. Of those that reach fuel, only 0.4 per fission actually cause a new fission (the rest are captured without fission).

• Task: Based on this story, is the system subcritical, critical, or supercritical?
• Hint: Count how many neutrons per fission effectively lead to the next generation of fission-causing neutrons. If fewer than 1 fission is induced per previous fission, k_eff < 1.

Exercise 2: Geometry change

You take the same fuel and moderator and rearrange it from a compact shape into a long, thin geometry with much more surface area.

• Task: Predict the direction of change in k_eff and explain using leakage arguments.
• Expected reasoning: More surface area relative to volume increases leakage, reducing the fraction of neutrons that remain to cause fission, so k_eff decreases.

Exercise 3: Moderator swap (qualitative)

A thermal reactor core is modified so that neutrons are less effectively slowed down (moderation becomes weaker), while absorption in the moderator remains low.

• Task: For a fissile fuel that fissions more readily with thermal neutrons, predict what happens to k_eff.
• Expected reasoning: Fewer neutrons reach thermal energies, so fewer induce fission; k_eff tends to decrease.

Exercise 4: Control rod insertion

During operation, control rods are inserted slightly.

• Task: Explain, in neutron-budget terms, why reactor power begins to drop.
• Expected reasoning: Rods increase absorption losses, reducing the number of neutrons available to cause fission in the next generation, so k_eff becomes less than 1 and the neutron population declines.

Exercise 5: “Small change, big effect”

Two cores are identical except one has k_eff = 0.99 and the other has k_eff = 1.01.

• Task: Without doing detailed time calculations, explain why these two cases behave very differently over many generations.
• Hint: Repeated multiplication compounds: (0.99)^g shrinks and (1.01)^g grows as g increases.