---
title: Chain Reactions and Reactor Physics
module: Nuclear Fission
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  A self-sustaining chain reaction is a fixed point of neutron bookkeeping: the
  multiplication factor k counts the neutrons in one generation per neutron in the last,
  and criticality is k = 1. The four-factor formula tracks a neutron through fast fission,
  resonance escape, thermal utilization, and reproduction; moderation slows fission
  neutrons to the thermal energies where the fission cross section is largest; and the
  small delayed-neutron fraction sets the timescale that makes a reactor controllable.
  Breeding converts fertile U-238 and Th-232 into new fissile fuel.
topics: [Nuclear Fission]
sources:
  - book: Krane
    ref: "Ch. 13 — Nuclear Fission; §13.4 Controlled Fission Reactions, §13.5 Fission Reactors, §13.6 Radioactive Fission Products"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion; §6-4 Nuclear Fission"
draft: false
---

A single fission of ${}^{235}\mathrm{U}$ releases about $2.4$ neutrons. If on average
exactly one of them induces a further fission, the reaction sustains itself at a constant
rate; if more than one does, the rate grows; if fewer, it dies away. The whole of reactor
physics is the bookkeeping that decides which of these three outcomes occurs, and the
engineering that holds a core at the balance point.[^krane-control]

## The multiplication factor and criticality

Follow the neutron population from one fission generation to the next. The **multiplication
factor** is the ratio

$$
k = \frac{\text{number of neutrons in one generation}}{\text{number in the preceding generation}}.
$$

If $N_0$ neutrons start the first generation, after $n$ generations the population is
$N_0 k^{n}$. Three regimes follow from the value of $k$:

- **Subcritical**, $k < 1$: the population decays geometrically and the chain cannot
  sustain itself without an external source.
- **Critical**, $k = 1$: the population is stationary, one fission neutron surviving to
  induce exactly one further fission. A power reactor runs here.
- **Supercritical**, $k > 1$: the population grows geometrically; a controlled excursion
  raises the power level, an uncontrolled one is a runaway.

The fractional departure from criticality is the **reactivity**

$$
\rho = \frac{k - 1}{k},
$$

zero at criticality, positive above, negative below. If a generation lasts a mean time
$\ell$, the population obeys $\d N/\d t = (k-1)N/\ell$, so

$$
N(t) = N_0\,e^{(k-1)t/\ell} = N_0\,e^{t/T}, \qquad T = \frac{\ell}{k-1},
$$

where $T$ is the **reactor period**, the time for the power to change by a factor $e$. The
period is the operational handle on a reactor: a long period means a slow, controllable
change; a short one means a fast excursion.

$$
% caption: Neutron population versus time for the three regimes. Below criticality the
% chain dies, at criticality it holds steady, and above criticality it grows exponentially
% with a reactor period set by how far k exceeds one.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {time};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {neutron population};
  % supercritical: growing (bezier approx of exp)
  \draw[acc, very thick]
    (0,1.2) .. controls (3.0,1.7) and (4.8,2.6) .. (6.6,4.0);
  \node[acc, anchor=south west, font=\scriptsize] at (5.3,3.4) {$k > 1$};
  % critical: flat
  \draw[black, very thick] (0,1.2) -- (7.6,1.2);
  \node[black!70, anchor=south, font=\scriptsize] at (6.9,1.25) {$k = 1$};
  % subcritical: decaying
  \draw[black, thick, dashed]
    (0,1.2) .. controls (2.0,0.6) and (4.0,0.35) .. (7.4,0.2);
  \node[black, anchor=north, font=\scriptsize] at (5.6,0.35) {$k < 1$};
  \fill[black] (0,1.2) circle (1.6pt);
\end{tikzpicture}
$$

## The four-factor formula

In an idealized infinite medium (no leakage) the multiplication factor is built from four
probabilities that a neutron encounters as it cycles through one generation. The
**four-factor formula** is

$$
k_\infty = \eta\,\varepsilon\,p\,f.
$$

Reading the neutron's life cycle from a thermal fission outward:

- **Reproduction factor $\eta$**: the number of fast neutrons produced per thermal neutron
  absorbed in the fuel. Not every absorption causes fission, so
  $\eta = \bar\nu\,\Sigma_f/\Sigma_a$ over the fuel, where $\Sigma_f$ and $\Sigma_a$ are the
  macroscopic fission and absorption cross sections. For ${}^{235}\mathrm{U}$,
  $\eta \approx 2.07$; for natural uranium, diluted by ${}^{238}\mathrm{U}$ capture,
  $\eta \approx 1.34$.
- **Fast fission factor $\varepsilon$**: a small bonus above unity, $\varepsilon \approx
  1.03$, from the few extra fissions that fast neutrons induce in ${}^{238}\mathrm{U}$
  before they slow down.
- **Resonance escape probability $p$**: the fraction of neutrons that slow through the
  ${}^{238}\mathrm{U}$ capture resonances between roughly $6\ \mathrm{eV}$ and
  $200\ \mathrm{eV}$ without being captured, typically $p \approx 0.9$.
- **Thermal utilization factor $f$**: the fraction of thermal neutrons absorbed in the fuel
  rather than in the moderator, cladding, or structure, $f = \Sigma_a^{\text{fuel}}/
  \Sigma_a^{\text{total}} \approx 0.9$.

$$
% caption: The four-factor neutron life cycle. A thermal fission produces fast neutrons
% (reproduction eta), which gain a few fast fissions (epsilon), slow past the U-238
% resonances (escape probability p), and are absorbed in the fuel (thermal utilization f);
% the product of the four factors is the infinite-medium multiplication factor.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \tikzset{stg/.style={draw=black, very thick, minimum width=2.4cm, minimum height=0.85cm, align=center, font=\scriptsize}}
  \node[stg] (fis) at (0,2.4) {thermal\\split};
  \node[stg] (fast) at (3.6,2.4) {reproduction\\(eta)};
  \node[stg] (res) at (7.2,2.4) {fast split\\(epsilon)};
  \node[stg] (util) at (7.2,0) {resonance\\escape (p)};
  \node[stg] (repro) at (3.6,0) {thermal\\use (f)};
  \node[stg] (back) at (0,0) {new\\split};
  \draw[->, black, thick] (fis) -- (fast);
  \draw[->, black, thick] (fast) -- (res);
  \draw[->, black, thick] (res) -- (util);
  \draw[->, black, thick] (util) -- (repro);
  \draw[->, black, thick] (repro) -- (back);
  \draw[->, black, thick] (back) -- (fis);
\end{tikzpicture}
$$

A real, finite core also leaks neutrons from its surface. Two non-leakage probabilities,
$P_{\text{FNL}}$ for fast neutrons and $P_{\text{TNL}}$ for thermal, extend the result to
the **six-factor formula**

$$
k_{\text{eff}} = \eta\,\varepsilon\,p\,f\,P_{\text{FNL}}\,P_{\text{TNL}}.
$$

Leakage scales with the surface-to-volume ratio, so a core must exceed a **critical size**:
below it, too many neutrons escape and $k_{\text{eff}} < 1$ regardless of composition. The
critical size for a bare sphere follows from balancing production against diffusion loss and
defines the critical mass of the fuel.[^krane-fourfactor]

## Moderation and the thermal advantage

Fission neutrons are born fast, near $2\ \mathrm{MeV}$, but the fission cross section of
${}^{235}\mathrm{U}$ is far larger at thermal energies, $\sigma_f \approx 585\ \mathrm{b}$ at
$0.025\ \mathrm{eV}$ against $\sim 1\ \mathrm{b}$ at $\mathrm{MeV}$ energies, following a
$1/v$ law at low energy. A thermal reactor deliberately slows its neutrons in a
**moderator** to reach this large cross section, threading them past the ${}^{238}\mathrm{U}$
resonances on the way down.

$$
% caption: Fission cross section of uranium-235 versus neutron energy. The cross section
% follows a 1/v rise toward low energy and is hundreds of times larger at thermal energy
% than in the fast region where fission neutrons are born, which is why moderation helps.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {neutron energy (log)};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {cross section (log)};
  % x ticks
  \foreach \x/\lab in {0.8/{0.01 eV},3.0/{1 eV},5.2/{1 keV},7.2/{1 MeV}} \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab};
  % 1/v thermal region
  \draw[acc, very thick]
    (0.5,3.8)
    .. controls (1.4,3.0) and (2.0,2.55) .. (2.7,2.3);
  % resonance region (jagged) drawn as small spikes
  \draw[very thick] (2.7,2.3)
    -- (3.0,2.8) -- (3.15,2.2) -- (3.4,3.1) -- (3.55,2.1)
    -- (3.8,2.7) -- (3.95,2.0) -- (4.2,2.5) -- (4.4,1.85);
  % falling toward fast
  \draw[very thick]
    (4.4,1.85) .. controls (5.4,1.3) and (6.4,0.9) .. (7.6,0.6);
  \node[acc, anchor=west, font=\scriptsize] at (0.7,3.5) {$\frac{1}{v}$};
  \node[black, anchor=south, font=\scriptsize] at (3.5,3.15) {resonances};
  \node[black, anchor=west, font=\scriptsize] at (6.0,0.95) {fast region};
  \draw[black, dashed] (0.8,0) -- (0.8,3.7);
  \node[acc, anchor=south east, font=\scriptsize] at (1.7,3.65) {thermal};
\end{tikzpicture}
$$

Moderation works by elastic scattering: a neutron transfers energy to a light nucleus at
rest. The average loss of the logarithm of the energy per collision, the **logarithmic
energy decrement**, is independent of energy and depends only on the moderator mass number,

$$
\xi = 1 + \frac{(A-1)^2}{2A}\ln\frac{A-1}{A+1}.
$$

For hydrogen $\xi = 1$; for carbon $\xi = 0.158$. The number of collisions to slow a
$2\ \mathrm{MeV}$ neutron to thermal energy $0.025\ \mathrm{eV}$ is
$n = \xi^{-1}\ln(E_0/E_{\text{th}})$, about $18$ in hydrogen but $115$ in carbon. A good
moderator combines a large $\xi$, a large scattering cross section, and a small absorption
cross section; the figure of merit is the **moderating ratio** $\xi\Sigma_s/\Sigma_a$.

| Moderator | $\xi$ | Collisions to thermal | Moderating ratio |
| --- | --- | --- | --- |
| Ordinary water | $0.92$ | $19$ | $62$ |
| Heavy water | $0.51$ | $35$ | $4800$ |
| Graphite | $0.158$ | $115$ | $170$ |

Ordinary water thermalizes in the fewest collisions but its hydrogen absorbs neutrons, so a
light-water reactor needs uranium enriched to a few percent ${}^{235}\mathrm{U}$. Heavy
water and graphite absorb far less, which is why reactors built on them can run on natural
uranium.[^krane-moderation]

## Delayed neutrons and control

The prompt-neutron generation time in a thermal reactor is short, $\ell \approx 10^{-4}\
\mathrm{s}$. If the population responded on that timescale, a reactivity of $\rho = 0.001$
would give a period $T = \ell/\rho \approx 0.1\ \mathrm{s}$, far too fast for mechanical
control. The **delayed neutrons** rescue the situation. Although they are only a fraction
$\beta \approx 0.0065$ of the total, they arrive with the seconds-long half-lives of their
beta-decaying precursors, and this stretches the effective generation time enormously. The
mean generation time weighted over prompt and delayed neutrons is

$$
\langle\ell\rangle = (1-\beta)\ell + \sum_i \beta_i\,\tau_i \approx 0.08\ \mathrm{s},
$$

with the precursor mean lives $\tau_i$ averaging about $13\ \mathrm{s}$. A small positive
reactivity now produces a period of tens of seconds, slow enough to control with rods.

The critical distinction is between reactivity below and above $\beta$. As long as
$\rho < \beta$ the chain can be sustained only with the help of the delayed neutrons, and
the reactor responds on the slow delayed timescale. If $\rho$ reaches $\beta$ the reactor is
**prompt critical**: the prompt neutrons alone sustain the chain, the delayed neutrons no
longer matter, and the period collapses to the prompt value. Reactivity is therefore
measured in units of $\beta$, the **dollar**, and safe operation keeps the reactivity well
below one dollar.

> **Definition (Prompt criticality).** A reactor is prompt critical when its reactivity
> equals the delayed-neutron fraction, $\rho = \beta$, so that the prompt neutrons alone
> give $k = 1$. Below this threshold ($\rho < \beta$) the effective generation time is set
> by the delayed precursors and the reactor is controllable; at or above it the period
> drops to the prompt generation time and control by mechanical rods is lost.

Control acts on the reactivity through several mechanisms:

- **Control rods** of strong thermal absorbers (boron, cadmium, hafnium) inserted into the
  core to lower $f$ and hence $k_{\text{eff}}$.
- **Soluble neutron poison**, boric acid dissolved in the coolant, for slow bulk
  adjustment.
- **Reactivity feedback**, the change in $k$ with temperature. A negative temperature
  coefficient, from Doppler broadening of the ${}^{238}\mathrm{U}$ resonances (which lowers
  $p$ as fuel heats) and from moderator density changes, is a passive safety feature: a
  power rise reduces reactivity and self-limits.
- **Fission-product poisoning**, above all ${}^{135}\mathrm{Xe}$, whose thermal absorption
  cross section of $2.6\times 10^{6}\ \mathrm{b}$ builds up after a power change and shifts
  the reactivity on a timescale of hours.

## Reactor types and the core

A thermal reactor core arranges fuel, moderator, coolant, and control into a lattice that
achieves $k_{\text{eff}} = 1$ at operating power. The fuel is uranium (as metal or oxide) in
clad rods; the moderator surrounds them; the coolant removes the fission heat; and control
rods trim the reactivity.

$$
% caption: Schematic thermal reactor core. Fuel rods sit in a lattice within the moderator,
% the coolant flows past them to carry off heat, and control rods drop between the fuel to
% absorb neutrons and lower the multiplication factor.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % vessel
  \draw[black, very thick] (0.4,0.3) rectangle (7.4,4.2);
  \node[black, anchor=south west, font=\scriptsize] at (0.45,4.25) {moderator + coolant};
  % fuel rods
  \foreach \x in {1.2,2.2,3.2,4.2,5.2,6.2} {
    \draw[acc, very thick, fill=acc!14] (\x,0.7) rectangle (\x+0.35,3.2);
  }
  \node[acc, anchor=north, font=\scriptsize] at (1.375,0.65) {fuel};
  % control rods dropping in (between fuel)
  \foreach \x in {1.75,3.75,5.75} {
    \draw[black, very thick, fill=black!14] (\x,2.4) rectangle (\x+0.3,4.6);
  }
  \node[black!70, anchor=south, font=\scriptsize] at (3.9,4.6) {control rods};
  % coolant flow arrows
  \draw[->, black, thick] (0.75,0.5) -- (0.75,3.9);
  \draw[->, black, thick] (6.9,0.5) -- (6.9,3.9);
  \node[black, anchor=west, font=\scriptsize] at (6.95,2.1) {coolant};
\end{tikzpicture}
$$

The major designs differ in the moderator and coolant they choose:

- **Pressurized and boiling water reactors** use ordinary water as both moderator and
  coolant and burn uranium enriched to $3$–$5\%$ ${}^{235}\mathrm{U}$. Water's strong
  negative void and temperature coefficients make them self-stabilizing.
- **Heavy-water reactors** (CANDU) moderate with $\mathrm{D_2O}$ and run on natural
  uranium, at the cost of a large moderator inventory.
- **Graphite-moderated reactors** (Magnox, RBMK) use a graphite stack with gas or water
  cooling; their large cores tolerate natural or slightly enriched fuel.
- **Fast reactors** omit the moderator entirely, sustaining the chain on fast neutrons in
  fuel rich in ${}^{239}\mathrm{Pu}$, and use a liquid-metal coolant that does not
  moderate.

## Breeding and the fuel cycle

The dominant isotope of natural uranium, ${}^{238}\mathrm{U}$, is not fissile but
**fertile**: neutron capture followed by two beta decays converts it into fissile
${}^{239}\mathrm{Pu}$,

$$
{}^{238}\mathrm{U} + n \to {}^{239}\mathrm{U}
\xrightarrow[23\ \mathrm{min}]{\beta^-} {}^{239}\mathrm{Np}
\xrightarrow[2.4\ \mathrm{d}]{\beta^-} {}^{239}\mathrm{Pu}.
$$

The parallel thorium cycle turns fertile ${}^{232}\mathrm{Th}$ into fissile
${}^{233}\mathrm{U}$ through ${}^{233}\mathrm{Th}$ and ${}^{233}\mathrm{Pa}$. The efficiency
of fuel regeneration is the **conversion ratio**

$$
C = \frac{\text{fissile nuclei produced}}{\text{fissile nuclei consumed}}.
$$

A reactor with $C > 1$ is a **breeder**: it makes more fuel than it burns. Breeding requires
that each fissile absorption yield more than two neutrons on average, one to sustain the
chain and one to convert a fertile nucleus, with a margin for losses. The condition is
$\eta > 2$. The reproduction factor $\eta$ of ${}^{239}\mathrm{Pu}$ is only about $2.1$ for
thermal neutrons, too tight to breed, but rises to about $2.9$ for fast neutrons, which is
why practical breeders are fast reactors.

$$
% caption: The fissile and fertile fuel cycle. Fertile uranium-238 captures a neutron and
% beta-decays twice to fissile plutonium-239; the parallel path converts thorium-232 to
% uranium-233. A breeder produces more fissile fuel than it consumes.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \tikzset{nuc/.style={draw=acc, very thick, fill=acc!8, minimum width=1.7cm, minimum height=0.75cm, align=center, font=\scriptsize}}
  \tikzset{fert/.style={draw=black, very thick, fill=black!8, minimum width=1.7cm, minimum height=0.75cm, align=center, font=\scriptsize}}
  \node[fert] (u8) at (0,2.4) {U-238\\fertile};
  \node[nuc] (pu) at (3.4,2.4) {Pu-239\\fuel};
  \node[fert] (th) at (0,0.4) {Th-232\\fertile};
  \node[nuc] (u3) at (3.4,0.4) {U-233\\fuel};
  \draw[->, black, thick] (u8) -- node[above, black, font=\scriptsize]{capture} (pu);
  \draw[->, black, thick] (th) -- node[above, black, font=\scriptsize]{capture} (u3);
  \node[nuc, draw=black, fill=black!8] (fis) at (6.6,1.4) {energy +\\neutrons};
  \draw[->, black, thick] (pu) -- (fis);
  \draw[->, black, thick] (u3) -- (fis);
  \draw[->, black, thick] (fis) .. controls (6.6,3.4) and (2.0,3.6) .. (u8);
  \node[black, anchor=south, font=\scriptsize] at (3.6,3.4) {extra neutron breeds fuel};
\end{tikzpicture}
$$

> **Result.** A reactor is a critical assembly, $k_{\text{eff}} = \eta\varepsilon p f\,
> P_{\text{FNL}}P_{\text{TNL}} = 1$, held there by control of $f$ and $p$. Moderation moves
> fission neutrons to the thermal energies where $\sigma_f$ is largest; the delayed-neutron
> fraction $\beta \approx 0.0065$ stretches the response time from milliseconds to seconds
> and makes control possible, provided the reactivity stays below one dollar; and breeding
> on the $\eta > 2$ margin of fast ${}^{239}\mathrm{Pu}$ converts fertile ${}^{238}\mathrm{U}$
> and ${}^{232}\mathrm{Th}$ into new fuel.

The same neutron-slowing and cross-section physics that governs a reactor also governs how
radiation deposits energy in matter and how it is detected, and the neutron-rich fission
products are the source of the reactor's radioactive inventory. The complementary route to
nuclear energy, fusing light nuclei rather than splitting heavy ones, is taken up in the
fusion module.

[^krane-control]: **Krane**, _Introductory Nuclear Physics_, §13.4 (Controlled Fission
Reactions). The multiplication factor, criticality, and the neutron balance of a chain
reaction. Reactor-physics parameters are compiled by the IAEA Nuclear Data Services,
[https://www-nds.iaea.org/](https://www-nds.iaea.org/).

[^krane-fourfactor]: **Krane**, §13.4–13.5. The four-factor formula $k_\infty =
\eta\varepsilon p f$, the non-leakage extension to $k_{\text{eff}}$, and the resulting
critical size and mass. Reproduction factors and cross sections from the NNDC,
[https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).

[^krane-moderation]: **Krane**, §13.5 (Fission Reactors), and **Wong**, _Introductory
Nuclear Physics_, §6-4. The logarithmic energy decrement $\xi$, the collision count to
thermalize, the moderating ratio, and the thermal ${}^{235}\mathrm{U}$ fission cross
section of $585\ \mathrm{b}$ at $0.025\ \mathrm{eV}$.

[^krane-breeding]: **Krane**, §13.5. The delayed-neutron control margin and the dollar unit,
the ${}^{238}\mathrm{U}\to{}^{239}\mathrm{Pu}$ and ${}^{232}\mathrm{Th}\to{}^{233}\mathrm{U}$
breeding chains, the conversion ratio, and the $\eta > 2$ breeding condition met by fast
${}^{239}\mathrm{Pu}$.