---
title: The Compound Nucleus and Resonance Reactions
module: Nuclear Reactions
moduleNumber: 8
lessonNumber: 2
order: 802
summary: >
  Low-energy reactions proceed through a long-lived intermediate state whose decay
  forgets how it formed. Bohr's independence hypothesis factorizes the cross section
  into a formation step and a branching ratio, an isolated level gives the single-level
  Breit-Wigner line shape with total width Γ tied to the lifetime by Γτ = ħ, and at
  high excitation overlapping levels merge into a statistical continuum described by
  evaporation spectra and the Hauser-Feshbach average.
topics: [Nuclear Reactions]
sources:
  - book: Krane
    ref: "Ch. 11 — Nuclear Reactions; §11.9 Compound-Nucleus Reactions, §11.10 The Optical Model (statistical treatment)"
  - book: Wong
    ref: "Ch. 8 — Nuclear Reactions; §8-4 Compound Nucleus, §8-5 Resonances"
draft: false
---

A projectile with a few MeV of energy incident on a medium or heavy nucleus rarely
scatters off cleanly. It is absorbed, its energy shared among all the nucleons until
no single one carries enough to escape, and the excited nucleus persists far longer
than the time a nucleon needs to cross it. This intermediate object is the **compound
nucleus**, and its formation and subsequent decay are the dominant mechanism of
low-energy nuclear reactions.[^krane-cn]

## Bohr's two-stage picture

Bohr (1936) proposed that a compound-nucleus reaction separates into two independent
stages,

$$
a + X \longrightarrow C^\ast \longrightarrow Y + b,
$$

with a genuine intermediate state $C^\ast$ formed at excitation energy

$$
E^\ast = E_{\text{cm}} + S_a,
$$

the center-of-mass kinetic energy plus the separation energy $S_a$ released when the
projectile binds into the compound system. The lifetime of $C^\ast$ is
$\sim 10^{-16}\,\mathrm{s}$ to $10^{-18}\,\mathrm{s}$, several orders of magnitude
longer than the transit time $\sim 2R/v \sim 10^{-22}\,\mathrm{s}$ of a nucleon across
the nucleus. During this interval the excitation is thermalized among all nucleons and
the memory of the entrance channel is lost.

$$
% caption: A compound-nucleus reaction proceeds in two stages: the projectile a is
% absorbed by X to form the excited compound C, which lives long enough to forget its
% origin and then decays into one of several open channels.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % entrance
  \node[black!70] (a) at (0,0.8) {$a$};
  \node[circle, draw=black, fill=black!6, minimum size=12pt, inner sep=1pt] (X) at (0,-0.2) {$X$};
  \draw[->, black] (0.35,0.65) -- (1.5,0.15);
  % compound blob
  \draw[acc, thick, fill=acc!12] (2.6,0.3) circle (0.75);
  \node[acc] at (2.6,0.3) {$C$};
  \node[acc, font=\scriptsize, anchor=north] at (2.6,-0.6) {compound};
  % exit channels
  \draw[->, black] (3.35,0.55) -- (4.7,1.2);
  \draw[->, black] (3.35,0.3) -- (4.7,0.3);
  \draw[->, black] (3.35,0.05) -- (4.7,-0.6);
  \node[black!70, anchor=west] at (4.75,1.2) {$Y + b$};
  \node[black!70, anchor=west] at (4.75,0.3) {$Y_2 + b_2$};
  \node[black!70, anchor=west] at (4.75,-0.6) {$X + a$};
  \node[black, font=\scriptsize, anchor=south] at (2.6,1.15) {form};
\end{tikzpicture}
$$

> **Hypothesis (Independence).** The decay of the compound nucleus is independent of
> its mode of formation. The cross section for $a + X \to Y + b$ then factorizes into
> the formation cross section and the branching ratio for the exit channel,
> $$
> \sigma_{ab} = \sigma_C(a)\,\frac{\Gamma_b}{\Gamma},
> $$
> where $\sigma_C(a)$ is the cross section to form $C^\ast$ through channel $a$,
> $\Gamma_b$ is the partial width for decay into channel $b$, and $\Gamma = \sum_c \Gamma_c$
> is the total width.

The branching ratio $\Gamma_b/\Gamma$ depends only on the properties of $C^\ast$ at
excitation $E^\ast$, not on how the state was reached. Ghoshal's 1950 experiment tested
this directly by forming the same compound nucleus $^{64}\mathrm{Zn}^\ast$ two ways,
$p + {}^{63}\mathrm{Cu}$ and $\alpha + {}^{60}\mathrm{Ni}$, at matched excitation. The
measured cross sections for the exit channels $(\text{,}n)$, $(\text{,}2n)$, and
$(\text{,}pn)$ tracked in the same ratios from both entrances, as the factorization
requires.

$$
% caption: Ghoshal's independence test: the same compound nucleus reached from two
% entrance channels decays with matching branching ratios into the neutron, two-neutron,
% and proton-neutron exit channels.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % two entrances
  \node[black!70] (e1) at (0,1.4) {$p + {}^{63}\mathrm{Cu}$};
  \node[black!70] (e2) at (0,-1.0) {${}^{4}\mathrm{He} + {}^{60}\mathrm{Ni}$};
  % compound
  \draw[acc, thick, fill=acc!12] (3.2,0.2) circle (0.72);
  \node[acc] at (3.2,0.35) {$^{64}$Zn};
  \node[acc, font=\scriptsize] at (3.2,-0.05) {compound};
  \draw[->, black] (1.0,1.3) -- (2.55,0.55);
  \draw[->, black] (1.0,-0.9) -- (2.55,-0.1);
  % exits
  \draw[->, black] (3.9,0.55) -- (5.3,1.2);
  \draw[->, black] (3.9,0.2) -- (5.3,0.2);
  \draw[->, black] (3.9,-0.15) -- (5.3,-0.8);
  \node[black!70, anchor=west] at (5.35,1.2) {$n$ channel};
  \node[black!70, anchor=west] at (5.35,0.2) {$2n$ channel};
  \node[black!70, anchor=west] at (5.35,-0.8) {$pn$ channel};
\end{tikzpicture}
$$

## Widths and lifetimes

An excited state of mean life $\tau$ has an energy uncertainty $\Gamma$ related by the
time-energy relation

$$
\Gamma\,\tau = \hbar, \qquad \Gamma = \frac{\hbar}{\tau}.
$$

A state that can decay through several channels has a decay rate that is the sum of the
rates for each channel, and since each rate is $\Gamma_c/\hbar$, the widths add:

$$
\frac{1}{\tau} = \sum_c \frac{1}{\tau_c} \;\Longrightarrow\; \Gamma = \sum_c \Gamma_c.
$$

Each **partial width** $\Gamma_c$ measures the coupling of the level to channel $c$; the
ratio $\Gamma_c/\Gamma$ is the probability that the compound nucleus, once formed, decays
into that channel. A compound state at $E^\ast = 8\,\mathrm{MeV}$ with $\tau = 10^{-16}\,\mathrm{s}$
has

$$
\Gamma = \frac{\hbar}{\tau} = \frac{6.58\times10^{-16}\,\mathrm{eV\,s}}{10^{-16}\,\mathrm{s}}
\approx 6.6\,\mathrm{eV},
$$

narrow compared with the MeV scale of the excitation. Neutron resonances in heavy nuclei
have widths from a fraction of an eV up to keV, and the spacing $D$ between adjacent levels
of the same spin and parity ranges from eV in heavy nuclei to keV in light ones.

## The single-level Breit-Wigner formula

Near an isolated level at energy $E_R$ the cross section takes a universal resonant shape.
Treat the entrance channel as a partial wave whose amplitude acquires the resonant
denominator $E - E_R + i\Gamma/2$, characteristic of a decaying state with complex energy
$E_R - i\Gamma/2$. The cross section for $a + X \to C^\ast \to Y + b$ through a single level
of spin $J$ is the **single-level Breit-Wigner formula**,

$$
\sigma_{ab}(E) = \pi\,\bar\lambda^2\,g\,
\frac{\Gamma_a\,\Gamma_b}{(E - E_R)^2 + (\Gamma/2)^2},
$$

where $\bar\lambda = \hbar/p$ is the reduced de Broglie wavelength of the projectile in the
center-of-mass frame, $\Gamma_a$ and $\Gamma_b$ are the entrance and exit partial widths,
and

$$
g = \frac{2J + 1}{(2s_a + 1)(2I + 1)}
$$

is the statistical spin factor built from the compound-state spin $J$, the projectile spin
$s_a$, and the target spin $I$. The line shape is a Lorentzian: the denominator drops to
half its peak value when $|E - E_R| = \Gamma/2$, so the **full width at half maximum equals
the total width $\Gamma$**, and the width read off a measured resonance directly gives the
compound-state lifetime $\tau = \hbar/\Gamma$.[^krane-bw]

$$
% caption: The single-level Breit-Wigner cross section is a Lorentzian peak centered at the
% resonance energy; its full width at half maximum equals the total width, which fixes the
% compound-state lifetime through the time-energy relation.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {energy};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {cross section};
  % Lorentzian-like bell via bezier, peak at x=4, height 3.8
  \draw[acc, very thick]
    (0.4,0.18)
    .. controls (2.2,0.35) and (3.0,0.9) .. (3.4,1.9)
    .. controls (3.7,2.7) and (3.85,3.8) .. (4.0,3.8)
    .. controls (4.15,3.8) and (4.3,2.7) .. (4.6,1.9)
    .. controls (5.0,0.9) and (5.8,0.35) .. (7.6,0.18);
  % peak marker
  \draw[black, dashed] (4.0,0) -- (4.0,3.8);
  \node[anchor=north, black!70, font=\scriptsize] at (4.0,-0.03) {$E_R$};
  % half maximum line
  \draw[black, dashed] (3.15,1.9) -- (4.85,1.9);
  \node[anchor=east, black, font=\scriptsize] at (0.0,1.9) {half max};
  % width arrows at half height
  \draw[<->, black] (3.4,1.9) -- (4.6,1.9);
  \node[anchor=south, black!70, font=\scriptsize] at (4.0,1.95) {width};
  \node[acc, anchor=south west, font=\scriptsize] at (4.15,3.5) {peak};
\end{tikzpicture}
$$

At the peak, $E = E_R$, the cross section reaches

$$
\sigma_{\max} = 4\pi\,\bar\lambda^2\,g\,\frac{\Gamma_a\,\Gamma_b}{\Gamma^2}.
$$

For an elastic-resonance ($b = a$) or a channel with $\Gamma_a \approx \Gamma_b \approx \Gamma/2$,
the peak approaches $\pi\bar\lambda^2 g$, the unitarity limit for a single partial wave. Slow
neutrons, with their large $\bar\lambda$, can therefore reach cross sections of thousands of
barns on resonance even though the geometric size of the nucleus is a fraction of a barn.

> **Worked example (a neutron resonance).** The $^{113}\mathrm{Cd}(n,\gamma)$ reaction has a
> strong level at $E_R = 0.178\,\mathrm{eV}$ with total width $\Gamma = 0.113\,\mathrm{eV}$,
> almost all of it radiative ($\Gamma_\gamma \approx \Gamma$), and neutron width
> $\Gamma_n \approx 4\times10^{-4}\,\mathrm{eV}$. For a thermal neutron with
> $\bar\lambda \approx 4.8\times10^{-11}\,\mathrm{m}$ and $g \approx 1$, the peak capture cross
> section is of order
> $$
> \sigma_{\max} \approx 4\pi\,\bar\lambda^2\,\frac{\Gamma_n}{\Gamma}
> \sim 2.5\times10^{-24}\,\mathrm{m^2} \approx 2.5\times10^{4}\,\mathrm{barn},
> $$
> which is why cadmium is a strong thermal-neutron absorber and a standard control-rod
> material.

Away from the peak but still at low energy, expanding the neutron width $\Gamma_n \propto v$
(the s-wave penetrability scales with the neutron speed) and taking $E \ll E_R$ reproduces
the $1/v$ absorption law: the capture cross section rises as the neutron slows, because the
Breit-Wigner tail of a nearby positive-energy or bound level dominates.

## From isolated levels to a statistical continuum

The level density $\rho(E^\ast)$ of a nucleus climbs steeply with excitation. A Fermi-gas
model gives

$$
\rho(E^\ast) \propto \exp\!\bigl(2\sqrt{a\,E^\ast}\,\bigr),
$$

with a level-density parameter $a \approx A/8\ \mathrm{MeV^{-1}}$. As the levels crowd
together their average spacing $D = 1/\rho$ shrinks. Resonances stay **isolated and
resolvable** while $\Gamma \ll D$; once $\Gamma \gtrsim D$ they **overlap** and the cross
section becomes a smooth function on which individual levels can no longer be picked out.
The reaction then enters the statistical regime, where only energy-averaged cross sections
are meaningful.

$$
% caption: As excitation rises the level spacing shrinks while the widths grow; sharp
% isolated resonances at low energy merge into a smooth continuum once the width exceeds
% the spacing.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {energy};
  \draw[->, black] (0,0) -- (0,3.8) node[above, black!70] {cross section};
  % isolated resonances at low energy (sharp, well separated)
  \draw[acc, very thick]
    (0.3,0.25)
    .. controls (0.7,0.25) and (0.85,2.6) .. (1.05,2.6)
    .. controls (1.25,2.6) and (1.4,0.3) .. (1.9,0.28)
    .. controls (2.2,0.28) and (2.35,2.1) .. (2.55,2.1)
    .. controls (2.75,2.1) and (2.9,0.3) .. (3.4,0.3)
    .. controls (3.7,0.3) and (3.85,2.9) .. (4.05,2.9)
    .. controls (4.25,2.9) and (4.4,0.35) .. (4.9,0.4);
  % overlapping region merging to continuum
  \draw[acc, very thick]
    (4.9,0.4)
    .. controls (5.4,0.9) and (5.7,1.5) .. (6.1,1.5)
    .. controls (6.4,1.5) and (6.55,1.15) .. (6.85,1.35)
    .. controls (7.2,1.55) and (7.4,1.25) .. (7.7,1.45)
    .. controls (8.1,1.65) and (8.5,1.5) .. (9.0,1.55);
  \draw[black, dashed] (4.9,0) -- (4.9,3.4);
  \node[anchor=south, black, font=\scriptsize] at (2.4,3.0) {isolated levels};
  \node[anchor=south, black, font=\scriptsize] at (7.2,1.9) {overlapping continuum};
\end{tikzpicture}
$$

## Evaporation spectra

A highly excited compound nucleus behaves like a heated liquid drop: it de-excites by
"boiling off" nucleons one at a time. The energy spectrum of the evaporated particles
follows from the statistical factor $\rho_f(E^\ast - E)$, the level density of the residual
nucleus after a particle of kinetic energy $E$ leaves, multiplied by the phase-space factor
$E\,\sigma_{\text{inv}}(E)$ for the inverse capture. Approximating the residual level density
near the top of the excitation by an exponential with **nuclear temperature** $T$, defined
through

$$
\frac{1}{T} = \frac{\d \ln\rho}{\d E^\ast},
$$

the emitted-neutron spectrum takes the Maxwellian evaporation form

$$
N(E)\,\d E \propto E\,\exp\!\Bigl(-\frac{E}{T}\Bigr)\,\d E.
$$

The spectrum rises linearly from threshold, peaks at $E = T$, and falls exponentially, so
the slope of $\ln[N(E)/E]$ against $E$ measures the nuclear temperature directly. For a
Fermi gas $T = \sqrt{E^\ast/a}$, giving $T \sim 1$–$2\,\mathrm{MeV}$ at excitations of
$10$–$20\,\mathrm{MeV}$. Evaporated particles emerge nearly isotropically in the
center-of-mass frame, a hallmark of compound decay that distinguishes it from the
forward-peaked direct reactions of the next lesson.

$$
% caption: The evaporation spectrum of neutrons from a compound nucleus rises linearly from
% zero, peaks at the nuclear temperature, and falls off exponentially; the high-energy slope
% measures the temperature.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {emitted energy};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {yield};
  % Maxwellian E exp(-E/T): rise then fall, peak around x=1.6
  \draw[acc, very thick]
    (0.2,0.05)
    .. controls (0.9,1.2) and (1.3,3.4) .. (1.7,3.55)
    .. controls (2.4,3.5) and (3.2,2.1) .. (4.2,1.15)
    .. controls (5.2,0.6) and (6.4,0.25) .. (7.6,0.12);
  \draw[black, dashed] (1.7,0) -- (1.7,3.55);
  \node[anchor=north, black!70, font=\scriptsize] at (1.7,-0.03) {peak};
  \node[black, anchor=west, font=\scriptsize] at (4.2,1.25) {exponential tail};
  \node[black, anchor=south west, font=\scriptsize] at (0.35,0.6) {linear rise};
\end{tikzpicture}
$$

## The Hauser-Feshbach average

When many levels overlap, the observable is the energy-averaged cross section over the
resonances in an interval. Averaging the Breit-Wigner form and using the independence
hypothesis, the compound cross section for $a \to b$ becomes the **Hauser-Feshbach formula**,

$$
\langle\sigma_{ab}\rangle = \pi\,\bar\lambda^2\,g\,
\frac{T_a\,T_b}{\sum_c T_c}\,W_{ab},
$$

where the transmission coefficients $T_c = 2\pi\langle\Gamma_c\rangle/D$ replace the individual
partial widths, the sum runs over all open channels, and $W_{ab}$ is a width-fluctuation
correction of order unity that accounts for correlations between entrance and exit widths.
This factorized structure — a formation factor $T_a$, a branching factor $T_b/\sum_c T_c$,
and the statistical weight — is the direct energy-averaged descendant of Bohr's two-stage
picture, and it predicts cross sections and angular distributions for compound reactions from
the transmission coefficients supplied by the optical model.

The compound-nucleus mechanism accounts for the resonance structure of low-energy cross
sections, the isotropic and Maxwellian character of evaporation products, and the statistical
averages at high excitation. It fails, however, for the fast, forward-peaked reactions that
proceed before equilibration, which require the optical model and direct-reaction theory of
the [next lesson](/nuclear-physics/nuclear-reactions/direct-reactions-optical-model).

[^krane-cn]: **Krane**, _Introductory Nuclear Physics_, §11.9 — the compound-nucleus mechanism, the two-stage reaction and lifetime scales, and Ghoshal's test of the independence hypothesis. Resonance parameters (energies, widths, spins) are compiled by the NNDC, [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/), and evaluated by the IAEA Nuclear Data Services, [https://www-nds.iaea.org/](https://www-nds.iaea.org/).

[^krane-bw]: **Krane**, §11.9–11.10, and **Wong**, _Introductory Nuclear Physics_, §8-4–8-5 — the single-level Breit-Wigner formula, the statistical spin factor $g$, partial and total widths with $\Gamma\tau = \hbar$, the level-density and evaporation treatment, and the Hauser-Feshbach energy-averaged cross section.
</content>