The neutron multiplication factor describes how the neutron population changes from one generation to the next inside
a reactor core. It is one of the central ideas in reactor physics because it tells you immediately whether the chain
reaction dies away, remains steady, or grows.
If one generation contains \(N_g\) neutrons and the next contains \(N_{g+1}\), then the effective multiplication factor
is
Effective multiplication factor.
\[
\begin{aligned}
k_{\text{eff}} &= \frac{N_{g+1}}{N_g}
\end{aligned}
\]
This simple ratio leads directly to the three standard reactor states:
| Condition |
State |
Meaning |
| \(k_{\text{eff}} < 1\) |
Subcritical |
The neutron population decreases from generation to generation |
| \(k_{\text{eff}} = 1\) |
Critical |
The neutron population remains steady |
| \(k_{\text{eff}} > 1\) |
Supercritical |
The neutron population grows from generation to generation |
Simple model
A compact educational model treats neutron reproduction, absorption, and leakage in a single product:
Simple multiplication model.
\[
\begin{aligned}
k_{\text{eff}} &= \eta \cdot a \cdot P_{nl}
\end{aligned}
\]
Here \(\eta\) is the reproduction factor, \(a\) is a simplified effective absorption factor, and \(P_{nl}\) is the
non-leakage factor, meaning the fraction of neutrons that remain in the core rather than leaking out. This form is useful
when you want a quick estimate without resolving the full neutron economy in detail.
Four-factor formula
In an infinite medium, a more refined model uses the classic four-factor formula:
Infinite-medium multiplication factor.
\[
\begin{aligned}
k_{\infty} &= \eta \cdot \varepsilon \cdot p \cdot f
\end{aligned}
\]
The factors have the following meanings:
| Factor |
Name |
Interpretation |
| \(\eta\) |
Reproduction factor |
Useful neutrons produced per neutron absorbed in fuel |
| \(\varepsilon\) |
Fast fission factor |
Extra neutrons contributed by fast fission |
| \(p\) |
Resonance escape probability |
Chance a neutron avoids resonance capture while slowing down |
| \(f\) |
Thermal utilization factor |
Fraction of thermal neutrons absorbed in fuel |
A real core is not infinite, so leakage must still be included. The effective multiplication factor is then
Including leakage.
\[
\begin{aligned}
k_{\text{eff}} &= k_{\infty} \cdot P_{nl}
\end{aligned}
\]
This is the form used in the calculator’s four-factor mode.
Population growth over generations
Once \(k_{\text{eff}}\) is known, the neutron population after \(g\) generations follows directly:
Generation growth law.
\[
\begin{aligned}
N_g &= N_0 \cdot k_{\text{eff}}^{\,g}
\end{aligned}
\]
This formula explains why even a small deviation of \(k_{\text{eff}}\) from 1 matters so much. If \(k_{\text{eff}} = 0.98\),
the chain reaction gradually fades. If \(k_{\text{eff}} = 1.02\), the growth is modest at first but compounds with each
generation. If \(k_{\text{eff}}\) is much larger than 1, the population grows rapidly.
Sample calculation
In the simple example from the prompt, take
\(\eta = 2.1\),
\(a = 1.0\),
and
\(P_{nl} = 0.95\).
Then
Step 1. Compute the effective multiplication factor.
\[
\begin{aligned}
k_{\text{eff}} &= \eta \cdot a \cdot P_{nl} \\
&= 2.1 \cdot 1.0 \cdot 0.95 \\
&= 1.995
\end{aligned}
\]
Since \(1.995 > 1\), the system is supercritical.
Step 2. Compute the population after several generations.
\[
\begin{aligned}
N_g &= N_0 \cdot 1.995^{\,g}
\end{aligned}
\]
If \(N_0 = 1\), then the neutron population nearly doubles each generation. That is why a \(k_{\text{eff}}\) value close
to 2 implies very rapid growth.
Reactivity
A related quantity often used in reactor physics is the reactivity
Reactivity estimate.
\[
\begin{aligned}
\rho &= \frac{k_{\text{eff}} - 1}{k_{\text{eff}}}
\end{aligned}
\]
This gives another way to quantify how far the reactor is from the critical state. A positive \(\rho\) corresponds to
a supercritical system, while a negative \(\rho\) corresponds to a subcritical one.
Physical meaning
The multiplication factor summarizes the balance between neutron production and neutron loss. Production is increased by
favorable fission behavior and efficient fuel absorption. Loss is increased by leakage and parasitic capture. Reactor
control, moderation, fuel composition, and geometry all influence these terms.
That is why the multiplication factor is one of the first quantities analyzed in any reactor design problem. It immediately
links the microscopic neutron processes to the macroscopic question of whether the core can sustain a chain reaction.
Advanced note
At university level, neutron multiplication is extended beyond the simple and four-factor models into six-factor formulas,
neutron diffusion theory, transport theory, and time-dependent point kinetics. Those methods include spatial leakage in a
more realistic way and describe how the neutron field evolves throughout the reactor. This calculator deliberately stays
at the educational level so the main reactor-physics idea remains clear: the chain reaction depends on whether
\(k_{\text{eff}}\) is below, equal to, or above 1.
| Concept |
Main relation |
Meaning |
| Simple model |
\(k_{\text{eff}} = \eta \cdot a \cdot P_{nl}\) |
Quick estimate including leakage and effective absorption |
| Four-factor model |
\(k_{\infty} = \eta \cdot \varepsilon \cdot p \cdot f\) |
Infinite-medium multiplication factor |
| Leakage correction |
\(k_{\text{eff}} = k_{\infty} \cdot P_{nl}\) |
Finite-core effective multiplication factor |
| Criticality rule |
\(k_{\text{eff}} \lessgtr 1\) |
Subcritical, critical, or supercritical behavior |
| Population law |
\(N_g = N_0 \cdot k_{\text{eff}}^{\,g}\) |
Generation-by-generation neutron growth or decay |
| Reactivity |
\(\rho = (k_{\text{eff}} - 1)/k_{\text{eff}}\) |
Distance from the critical state |