KAHIBARO
Table of Contents
The Idea of Criticality
In a fission chain reaction, neutrons released by one fission can cause new fissions in other nuclei. Criticality describes whether this multiplication exactly sustains itself, dies away, or grows.
If each generation of neutrons produces fewer fissions than the previous one, the chain reaction weakens. If each generation produces exactly the same number of effective fission-causing neutrons, the reaction is steady. If each generation produces more, the reaction grows.
The key quantity is the effective multiplication factor, written as $k_{\text{eff}}$. It compares one neutron generation with the next.
$$
k_{\text{eff}} = \frac{\text{number of neutrons in one generation}}{\text{number of neutrons in the previous generation}}
$$
This gives three basic cases.
| Condition | Meaning | Behavior of chain reaction |
|---|---|---|
| $k_{\text{eff}} < 1$ | Subcritical | Reaction dies out |
| $k_{\text{eff}} = 1$ | Critical | Reaction is self-sustaining and steady |
| $k_{\text{eff}} > 1$ | Supercritical | Reaction grows |
A chain reaction is critical when $k_{\text{eff}} = 1$.
Subcritical means $k_{\text{eff}} < 1$.
Supercritical means $k_{\text{eff}} > 1$.
Why Neutrons Are Lost
Not every neutron from a fission event causes another fission. Some neutrons escape from the material. Some are absorbed without causing fission. Because of these losses, sustaining the chain reaction requires the system to be arranged so that enough neutrons remain available.
Criticality is therefore a balance between neutron production and neutron loss. If production exactly matches all losses, the system is critical.
We can express this idea simply as
$$
\text{neutron production} = \text{neutron losses}
$$
when the system is critical.
Criticality is a balance condition. The reaction is steady only when neutron production exactly equals neutron loss.
Neutron Generations
It is helpful to imagine the chain reaction step by step. One fission releases neutrons. Some of those neutrons cause new fissions, producing a second generation. Then the process repeats.
If we let $N_0$ be the number of neutrons in one generation, then after one step the next generation is
$$
N_1 = k_{\text{eff}} N_0
$$
After two steps,
$$
N_2 = k_{\text{eff}}^2 N_0
$$
and after $n$ generations,
$$
N_n = k_{\text{eff}}^n N_0
$$
This shows clearly why the value of $k_{\text{eff}}$ matters so much. If it is less than 1, the neutron population shrinks. If it equals 1, it stays constant. If it is greater than 1, it grows.
Physical Meaning of the Three Cases
A subcritical system cannot maintain a chain reaction by itself. It may still have some fissions if an outside neutron source is present, but once that source is removed, the reaction fades.
A critical system is in a stable self-sustaining state. This is the basic condition desired in a controlled reactor during steady operation.
A supercritical system has a growing neutron population. If the growth is controlled and small, reactor power can be increased. If the growth is too rapid, it becomes dangerous.
In practice, engineers often discuss not just whether $k_{\text{eff}}$ is above or below 1, but how far from 1 it is. Even a small change can matter greatly because the multiplication repeats generation after generation.
Factors That Affect Criticality
Criticality depends on how easily neutrons produce further fissions before being lost. Several physical features influence this.
The amount of fissile material matters because a larger amount gives neutrons more opportunity to interact before escaping. Shape matters because some shapes reduce neutron leakage better than others. A compact shape, such as a sphere, tends to lose fewer neutrons than a long thin shape of the same volume. Material composition matters because some materials absorb neutrons, while others help preserve them. Density matters because closely packed nuclei increase the chance of interaction.
The surrounding environment also matters. If neutrons are reflected back into the fissile material by nearby material, neutron loss is reduced and criticality becomes easier to achieve.
Geometry and Neutron Escape
A central issue in criticality is the competition between volume and surface area. Fissions occur throughout the volume, but neutron escape happens through the surface. As an object becomes larger, its volume grows faster than its surface area. That means larger samples generally lose a smaller fraction of neutrons.
This is one reason why a small sample of fissile material can be subcritical, while a larger sample of the same material may become critical.
Critical Size and Critical Mass
Because neutron leakage decreases as the system becomes larger or more compact, there is often a minimum size at which a self-sustaining chain reaction becomes possible. The corresponding amount of fissile material is called the critical mass. That topic is treated separately, but criticality is the condition that defines it.
In simple terms, below the critical condition the system is subcritical. At the threshold it is critical. Beyond that threshold it can become supercritical.
Bare and Reflected Systems
A bare system has no surrounding material intended to send neutrons back. A reflected system is surrounded by a reflector material that scatters escaping neutrons back inward. Reflection reduces neutron loss, so a reflected assembly can reach criticality with less fissile material than a bare one.
This does not create neutrons. It simply saves neutrons that would otherwise be lost.
A reflector helps criticality by reducing neutron leakage, not by creating additional neutrons.
Criticality in Reactors
In a nuclear reactor, criticality usually means steady operation. The goal is often to keep the reactor at or very near $k_{\text{eff}} = 1$ so that power remains constant. If operators want to raise power, the reactor is made slightly supercritical for a time. If they want to lower power, it is made slightly subcritical.
The important idea is that criticality is not the same as maximum power. It means balanced, self-sustaining behavior.
Prompt and Delayed Effects
The neutrons from fission are not all equally important for control. Most appear almost immediately, while a small fraction are emitted later by certain fission products. These delayed neutrons make control much easier because they slow the time scale of reactor changes. A system can be critical in a controlled way because of them.
A full discussion of delayed neutrons belongs elsewhere, but for understanding criticality it is enough to know that the exact condition $k_{\text{eff}} = 1$ can describe a steady state whose practical control depends strongly on neutron timing.
A Simple Numerical Example
Suppose one neutron generation contains 100 neutrons that are effective in continuing the chain.
If $k_{\text{eff}} = 0.9$, then
$$
N_1 = 0.9 \times 100 = 90
$$
$$
N_2 = 0.9 \times 90 = 81
$$
so the chain reaction fades.
If $k_{\text{eff}} = 1.0$, then
$$
N_1 = 100,\quad N_2 = 100,\quad N_3 = 100
$$
so the reaction remains steady.
If $k_{\text{eff}} = 1.1$, then
$$
N_1 = 110,\quad N_2 = 121,\quad N_3 \approx 133
$$
so the reaction grows.
Distinguishing Criticality from Related Ideas
Criticality is about the balance of a chain reaction. It is not identical to the existence of fission itself, and it is not identical to energy release alone. Fission can occur in a subcritical sample, but it will not sustain itself. Likewise, a critical system is self-sustaining, but not necessarily rapidly increasing in power.
It is also important not to confuse criticality with critical mass. Critical mass is a particular amount of material under specified conditions. Criticality is the state of the neutron balance.
Criticality is a condition of neutron balance.
Critical mass is the amount of fissile material needed to reach that condition under specified circumstances.
Summary
Criticality is the condition that determines whether a fission chain reaction dies out, sustains itself, or grows. The central quantity is the effective multiplication factor $k_{\text{eff}}$. When $k_{\text{eff}} < 1$, the system is subcritical. When $k_{\text{eff}} = 1$, it is critical and self-sustaining. When $k_{\text{eff}} > 1$, it is supercritical and the reaction increases. Criticality depends on neutron production, absorption, and leakage, and is strongly affected by size, shape, density, and surrounding materials.