The Compound Nucleus and Resonance Reactions

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.¹

Bohr's two-stage picture

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

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

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.

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.

Widths and lifetimes

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

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 ${\Gamma }_c/\hslash$, the widths add:

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

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,

where $\bar{\lambda }=\hslash /p$ 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

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 $\tau =\hslash /\Gamma$.²

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

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.

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

with a level-density parameter $a\approx A/8\mathrm{Me}{\mathrm{V}}^{-1}$. 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.

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

the emitted-neutron spectrum takes the Maxwellian evaporation form

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.

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,

where the transmission coefficients $T_c=2\pi ⟨{\Gamma }_c⟩/D$ 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.