Chain Reactions and Reactor Physics

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.¹

The multiplication factor and criticality

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

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

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

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.

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

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 $\epsilon$: a small bonus above unity, $\epsilon \approx 1.03$, 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, $f={\Sigma }_a^{\text{fuel}}/{\Sigma }_a^{\text{total}}\approx 0.9$.

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

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.²

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.

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,

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 $\xi {\Sigma }_s/{\Sigma }_a$.

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.³

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

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.

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.

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 ${\mathrm{D}}_2\mathrm{O}$ 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 ${}^{239}\mathrm{Pu}$,

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

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.

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.