A nuclear chain reaction depends on whether each generation of neutrons produces fewer, the same number, or more neutrons
in the next generation. The key quantity is the multiplication factor, usually written as \(k\). It tells
you the average number of neutrons in one generation relative to the previous generation.
If \(k<1\), the neutron population dies away and the system is subcritical. If \(k=1\), the neutron
population stays steady and the system is critical. If \(k>1\), the neutron population grows and the
system is supercritical.
Simple neutron balance
In the simplest model, you can estimate the chain reaction by combining a neutron production factor with a leakage loss.
If \(L\) is the leakage fraction, then the surviving non-leakage probability is
Non-leakage probability.
\[
\begin{aligned}
P_{\mathrm{nl}} &= 1 - L.
\end{aligned}
\]
If \(\eta\) is an effective neutron reproduction factor, then a compact estimate is
\[
\begin{aligned}
k &= \eta (1-L) = \eta P_{\mathrm{nl}}.
\end{aligned}
\]
This model is very useful for fast intuition. For example, if \(\eta = 2.0\) and the leakage is \(10\%\), then
\[
\begin{aligned}
k &= 2.0 \cdot 0.9 = 1.8.
\end{aligned}
\]
Since \(k>1\), the system is supercritical in the basic sense.
Factor form of the multiplication factor
A slightly richer reactor-physics picture uses several neutron-economy factors. In this simplified preview, the
multiplication factor without leakage is
Infinite-medium factor.
\[
\begin{aligned}
k_{\infty} &= \eta \varepsilon p f.
\end{aligned}
\]
Here:
| Factor |
Name |
Interpretation |
| \(\eta\) |
Reproduction factor |
Average useful neutrons produced per relevant absorption |
| \(\varepsilon\) |
Fast fission factor |
Extra multiplication from fast-fission effects |
| \(p\) |
Resonance escape probability |
Chance a neutron avoids resonance capture while slowing down |
| \(f\) |
Thermal utilization factor |
Fraction of thermal absorptions that occur in the fuel |
Once leakage is included, the effective multiplication factor becomes
\[
\begin{aligned}
k_{\mathrm{eff}} &= k_{\infty} P_{\mathrm{nl}} = \eta \varepsilon p f (1-L).
\end{aligned}
\]
This is the quantity used to decide whether the system is subcritical, critical, or supercritical.
Generation growth
If one starts from an initial neutron population \(N_0\), then after \(g\) generations the mean neutron population is
Generation-to-generation multiplication.
\[
\begin{aligned}
N_g &= N_0 k^g.
\end{aligned}
\]
This equation explains why values of \(k\) only slightly above or below 1 can matter enormously after many generations.
If \(k=0.98\), the chain fades away. If \(k=1.02\), the chain grows, and the increase compounds from one generation to
the next.
Delayed neutrons and prompt criticality
Not all neutrons appear immediately. A small fraction, called the delayed neutron fraction \(\beta\),
arrives later from the decay of fission products. Even though \(\beta\) is small, delayed neutrons are extremely important
because they make reactor control far more manageable.
A useful prompt threshold is
Prompt threshold.
\[
\begin{aligned}
k_{\mathrm{prompt\ threshold}} &= \frac{1}{1-\beta}.
\end{aligned}
\]
This divides the supercritical regime into two important parts:
| Condition |
Regime |
Meaning |
| \(k<1\) |
Subcritical |
Neutron population shrinks |
| \(k=1\) |
Critical |
Neutron population stays steady |
\(1
| Delayed supercritical |
Growth depends on delayed neutrons |
|
| \(k \ge 1/(1-\beta)\) |
Prompt critical or prompt supercritical |
Growth can proceed on the prompt-neutron timescale |
Reactivity
Another common quantity is the reactivity
\[
\begin{aligned}
\rho &= \frac{k-1}{k}.
\end{aligned}
\]
This is a convenient measure of how far the system is from criticality. A related normalized measure divides the
reactivity by \(\beta\), giving reactivity in dollars. In simplified language, one dollar corresponds to
the delayed-neutron threshold scale.
Sample calculation
Suppose a simple system has \(\eta = 2.0\) and leakage \(L = 10\%\). Then
\[
\begin{aligned}
P_{\mathrm{nl}} &= 1 - 0.10 = 0.90
\end{aligned}
\]
and therefore
\[
\begin{aligned}
k &= \eta P_{\mathrm{nl}} \\
&= 2.0 \cdot 0.90 \\
&= 1.8.
\end{aligned}
\]
So the system is clearly supercritical in the basic sense. If the delayed-neutron fraction is small, such a large value
is also far above the prompt threshold in this simplified educational model.
Physical meaning
The multiplication factor is a compact way of summarizing neutron economy. Every process that helps neutrons survive and
cause more fissions pushes \(k\) upward. Every process that removes neutrons, such as leakage or non-fuel absorption,
pushes \(k\) downward. Control systems, moderator behavior, geometry, and fuel composition all act by changing the
underlying factors that determine \(k\).
Advanced note
At university level, chain-reaction analysis is refined using the full six-factor formula, time-dependent point-kinetics
equations, prompt-neutron lifetime, delayed-neutron precursor groups, and spatial diffusion theory. Those refinements are
essential for real reactor design and transient analysis. This calculator intentionally stays at the educational preview
level so that the central idea remains clear: criticality is governed by whether the neutron multiplication factor
is below, equal to, or above 1.
| Concept |
Main relation |
Meaning |
| Non-leakage probability |
\(P_{\mathrm{nl}} = 1-L\) |
Fraction of neutrons that remain in the system |
| Infinite-medium factor |
\(k_{\infty} = \eta \varepsilon p f\) |
Multiplication without leakage |
| Effective factor |
\(k_{\mathrm{eff}} = k_{\infty} P_{\mathrm{nl}}\) |
Actual multiplication factor including leakage |
| Criticality condition |
\(k_{\mathrm{eff}} = 1\) |
Steady chain reaction |
| Generation growth |
\(N_g = N_0 k^g\) |
Mean neutron population after \(g\) generations |
| Prompt threshold |
\(1/(1-\beta)\) |
Approximate threshold for prompt criticality in this preview |
| Reactivity |
\(\rho = (k-1)/k\) |
Distance from criticality |