Chapter 18 — Compound Nucleus Reactions and Resonances

"The capture of the bombarding particle by the target nucleus leads to a close combination of all the nuclear particles into a compound system from which the mode of disintegration is largely independent of the mode of formation." — Niels Bohr, "Neutron Capture and Nuclear Constitution" (1936)

Chapter Overview

In Chapter 17, we developed the general framework for nuclear reactions: kinematics, Q-values, cross sections, and the partial-wave expansion. We introduced the optical model as a way to describe average scattering from a nuclear potential. But the optical model, by design, averages over the rich structure that appears in measured cross sections — sharp peaks, sometimes hundreds of barns tall and fractions of an eV wide, that rise and fall as the beam energy is changed by tiny amounts.

These peaks are resonances, and they are the signature of a remarkable phenomenon: the formation of a compound nucleus. When a projectile is absorbed by a target, the kinetic energy of the projectile is rapidly shared among all nucleons in the combined system. This compound nucleus is a highly excited, long-lived (by nuclear standards) intermediate state that has completely "forgotten" how it was formed. It decays statistically, with branching ratios that depend only on the quantum numbers of the compound system — not on which particle went in.

This chapter derives the theory of compound nuclear reactions:

• Section 18.1 develops the physical picture of the compound nucleus and states the Bohr independence hypothesis — the threshold concept of this chapter.
• Section 18.2 derives the Breit-Wigner resonance formula from the S-matrix, connecting the resonance shape to the underlying physics of quasi-bound states.
• Section 18.3 examines resonance parameters — widths, spacings, and the statistical spin factor — with real nuclear data.
• Section 18.4 develops the nuclear level density using the Fermi gas model, a key input to statistical reaction theory.
• Section 18.5 introduces the Hauser-Feshbach statistical model, which averages over many resonances to predict compound nuclear cross sections.
• Section 18.6 treats neutron capture cross sections in detail: the $1/v$ law, resonance structure, and the resonance integral.
• Section 18.7 connects compound nuclear reactions to astrophysics — the $s$-process and $r$-process of nucleosynthesis.
• Section 18.8 describes experimental methods for measuring resonance parameters.

By the end of this chapter, you will understand why a plot of the neutron cross section of ${}^{238}$U looks like a forest of skyscrapers, why reactor designers must worry about "resonance escape," and why the same physics that governs a nuclear reactor also builds the elements heavier than iron in stellar explosions.

🏃 Fast Track: If you are primarily interested in applications, read Sections 18.1 (physical picture), 18.2.4 (Breit-Wigner result), 18.6 (neutron capture), and 18.7 (astrophysics). Return to the derivation later.

🔬 Deep Dive: The full Breit-Wigner derivation (Section 18.2) and the level density derivation (Section 18.4) reward careful study — they connect to the S-matrix formalism of scattering theory and to the nuclear partition function.

18.1 The Compound Nucleus Model

18.1.1 Bohr's Physical Picture

In 1936, Niels Bohr proposed a model of nuclear reactions that was, in its own way, as revolutionary as his earlier atomic model. Bohr argued that when a projectile enters a nucleus, it does not simply interact with one or two nucleons and re-emerge. Instead, the projectile collides with nucleons in the nuclear surface, each of which collides with other nucleons, and within a time on the order of the nuclear transit time ($\sim 10^{-22}$ s), the projectile's energy is shared among all nucleons in the combined system. The result is a compound nucleus — a highly excited system in thermodynamic equilibrium (or something close to it), with no memory of how it was formed.

Bohr's analogy was vivid: imagine a billiard ball entering a box filled with other billiard balls. After many collisions, the energy of the incoming ball is distributed among all the balls. Eventually, by statistical fluctuation, enough energy concentrates on one ball (or a cluster of balls) near the edge for it to escape. The identity and energy of the escaping ball carry no information about the original incoming ball — only about the total energy available.

The nuclear analogue: a neutron with kinetic energy $T$ enters a target nucleus ${}^A_Z X$ and forms a compound nucleus ${}^{A+1}_Z X^*$ at excitation energy:

$$E^* = S_n + T_{\text{cm}}$$

where $S_n$ is the neutron separation energy of the compound nucleus and $T_{\text{cm}}$ is the center-of-mass kinetic energy. For a neutron on a heavy target, $T_{\text{cm}} \approx T \cdot A/(A+1)$.

The compound nucleus then decays — by neutron emission, gamma-ray emission, fission, or other channels — with probabilities that depend on $E^*$, the angular momentum $J$, and the parity $\pi$ of the compound state, but not on the entrance channel.

18.1.2 The Bohr Independence Hypothesis

The central claim of the compound nucleus model is the Bohr independence hypothesis:

Bohr Independence Hypothesis: The cross section for the reaction $a + A \to C^* \to b + B$ factorizes as:

$$\sigma_{a \to b} = \sigma_{\text{form}}(a + A \to C^*) \times P_{\text{decay}}(C^* \to b + B)$$

The formation cross section depends only on the entrance channel $(a, A)$, and the decay probability depends only on the properties of the compound nucleus $C^*$ — not on how $C^*$ was formed.

This is the threshold concept of this chapter. It is physically remarkable: the compound nucleus is a system of $A+1$ strongly interacting nucleons at high excitation, and the energy is so thoroughly shared that all information about the entrance channel is erased.

Evidence. The most direct test of the independence hypothesis is to form the same compound nucleus through different entrance channels and check whether the decay probabilities are the same. A classic example was provided by Ghoshal (1950), who formed the compound nucleus ${}^{64}$Zn$^*$ at the same excitation energy via two different reactions:

$$p + {}^{63}\text{Cu} \to {}^{64}\text{Zn}^* \qquad \text{and} \qquad \alpha + {}^{60}\text{Ni} \to {}^{64}\text{Zn}^*$$

By adjusting the beam energies so that both reactions populated ${}^{64}$Zn$^*$ at the same excitation energy, Ghoshal measured the relative yields of the exit channels ($n$, $pn$, $2n$). After accounting for the different formation cross sections, the decay branching ratios agreed — confirming the independence hypothesis.

💡 The Compound Nucleus Forgets: This is worth pausing over. In everyday experience, the way you arrive at a state usually matters — a room heated by a fireplace feels different from one heated by a heat pump, even at the same temperature, because the heat distribution differs. In the compound nucleus, the "thermalization" is so complete that the analogy to a thermodynamic system is almost exact. The compound nucleus is a tiny droplet of nuclear matter in thermal equilibrium.

18.1.3 The Mean Free Path Argument

Why does energy sharing happen so quickly? Bohr's argument rests on the short mean free path of a nucleon inside nuclear matter. Consider a neutron entering a nucleus of radius $R \approx 1.2 A^{1/3}$ fm. The nucleon-nucleon cross section at typical nuclear energies ($\sim 10$–$40$ MeV in the Fermi gas) is $\sigma_{NN} \approx 3$–$5$ fm$^2$ in free space. Inside the nucleus, Pauli blocking reduces the effective cross section (because many final states are already occupied), but the mean free path remains short:

$$\lambda_{\text{mfp}} = \frac{1}{n_{\text{nuc}} \, \sigma_{NN}^{\text{eff}}} \approx \frac{1}{0.17 \, \text{fm}^{-3} \times 2 \, \text{fm}^2} \approx 3 \, \text{fm}$$

This is comparable to the nuclear radius for medium and heavy nuclei ($R \sim 5$–$7$ fm). Therefore, the incoming nucleon suffers at least one or two collisions before traversing the nucleus, and the secondary nucleons suffer their own collisions, creating a rapid cascade of energy sharing. After just a few collision times ($\sim 3 \times 10^{-23}$ s each), the available energy is distributed among many nucleons, and the system has "thermalized."

For heavy nuclei such as ${}^{238}$U, the incoming neutron collides multiple times before crossing the nuclear volume, and the mean free path argument strongly supports the compound nucleus picture. For very light nuclei (e.g., $A < 20$), the nuclear radius becomes comparable to the mean free path, the number of collisions is small, and the compound nucleus model is less reliable — direct reactions (Chapter 19) become significant even at low energies.

18.1.4 Time Scales and the Validity of the Model

The compound nucleus model rests on a separation of time scales:

1. Transit time $\tau_{\text{transit}} \sim R/v \sim 10^{-22}$ s — the time for the projectile to cross the nucleus. For a 1 MeV neutron ($v \approx 1.4 \times 10^7$ m/s) on a heavy nucleus ($R \approx 7$ fm): $\tau_{\text{transit}} \approx 5 \times 10^{-22}$ s.
2. Thermalization time $\tau_{\text{therm}} \sim$ a few $\times 10^{-22}$ s — the time for the projectile's energy to be shared among all nucleons via nucleon-nucleon collisions. Typically 2–5 collision times.
3. Compound nucleus lifetime $\tau_C \sim \hbar/\Gamma \sim 10^{-19}$ to $10^{-14}$ s — the time before the compound nucleus decays (where $\Gamma$ is the total width, typically eV to keV).

The model is valid when $\tau_C \gg \tau_{\text{therm}}$ — that is, when the compound system lives long enough for the energy to be fully shared before it decays. This hierarchy is impressive: the lifetime of a typical heavy-nucleus resonance ($\tau_C \sim 10^{-14}$ s) exceeds the thermalization time ($\sim 10^{-21}$ s) by a factor of $10^7$. The compound nucleus lives long enough for the energy to be redistributed $\sim 10^7$ times over — amply justifying the statistical picture.

The model breaks down when:

• The excitation energy is very high, so that the level density becomes so large and the widths so broad that $\Gamma \gtrsim D$ (overlapping resonances) — here, the Hauser-Feshbach statistical model (Section 18.5) replaces the single-resonance picture, but the compound nucleus concept remains valid.
• The projectile energy is very high (say, above $\sim 50$–$100$ MeV per nucleon), so that the projectile's de Broglie wavelength becomes short enough to resolve individual nucleons and the projectile can "punch through" the nucleus — that regime is direct reactions (Chapter 19).
• The target is very light (small $A$), so that few nucleon-nucleon collisions occur during transit.

At low to moderate energies (say, neutron energies from thermal to a few MeV), the compound nucleus model dominates for medium and heavy nuclei. At higher energies, direct reactions become increasingly important. In practice, the optical model describes the average (direct + compound) and the Hauser-Feshbach model describes the compound nuclear part, while the difference gives the direct component.

18.1.5 Compound Elastic Scattering

An important subtlety: the compound nucleus can decay back into the entrance channel. The reaction $a + A \to C^* \to a + A$ is compound elastic scattering — the projectile is absorbed, a compound nucleus forms, and then a particle identical to the projectile is re-emitted, leaving the target in its ground state. This is physically distinct from shape elastic (potential) scattering, where the projectile bounces off the nuclear surface without being absorbed.

Experimentally, compound elastic and shape elastic scattering are indistinguishable event by event (both produce the same final state: $a + A$). However, they have different angular distributions: shape elastic scattering produces a smooth, forward-peaked pattern (diffractive), while compound elastic scattering produces a symmetric $1/\sin\theta$ pattern (because the compound nucleus decays isotropically in its rest frame). The two amplitudes interfere, and the full elastic cross section near a resonance shows the characteristic asymmetric pattern discussed in Section 18.2.4.

18.1.6 A Roadmap of Compound Nuclear Physics

The compound nucleus picture leads naturally to several questions that structure the rest of this chapter:

• What is the cross section near a single resonance? $\to$ Breit-Wigner formula (Section 18.2)
• What determines the resonance parameters? $\to$ Level densities, widths (Sections 18.3–18.4)
• What is the cross section when there are many overlapping resonances? $\to$ Hauser-Feshbach model (Section 18.5)
• What does this look like for neutrons? $\to$ Neutron capture, $1/v$ law, resonance integral (Section 18.6)
• Why does it matter? $\to$ Reactors, stellar nucleosynthesis (Sections 18.6–18.7)

18.2 The Breit-Wigner Resonance Formula

18.2.1 Physical Origin of Resonances

Consider a neutron impinging on a nucleus. The neutron sees a potential well (the nuclear potential, attractive) surrounded by a centrifugal barrier (for $\ell > 0$) and, for charged particles, a Coulomb barrier. At certain energies, the neutron's de Broglie wavelength matches the "cavity" of the nuclear potential well in just the right way to form a quasi-bound state — a state that would be truly bound if the barrier were infinitely high, but that can decay by tunneling back through the barrier.

These quasi-bound states are the resonances of the compound nucleus. They correspond to excited states of the compound nucleus ${}^{A+1}X^*$ that lie above the threshold for particle emission. Each resonance has:

• A resonance energy $E_R$ — the center-of-mass energy at which the cross section peaks.
• A total width $\Gamma$ — related to the lifetime of the state by $\tau = \hbar / \Gamma$.
• Partial widths $\Gamma_a, \Gamma_b, \ldots$ for each open decay channel — these satisfy $\Gamma = \sum_c \Gamma_c$.
• Definite quantum numbers: spin $J$ and parity $\pi$.

📊 Typical Numbers: For low-energy neutron resonances in heavy nuclei (e.g., ${}^{238}$U), $E_R \sim 1$–$1000$ eV, $\Gamma \sim 0.02$–$1$ eV, and the neutron partial width $\Gamma_n \sim 10^{-3}$–$10^{-1}$ eV. The radiation width $\Gamma_\gamma \sim 25$ meV is remarkably constant across resonances (because it is an average over many gamma-ray transitions to lower-lying states). The lifetime is $\tau = \hbar/\Gamma \sim 10^{-16}$–$10^{-14}$ s — an eternity compared to the nuclear transit time.

18.2.2 The S-Matrix and Resonance Poles

We derive the Breit-Wigner formula from the partial-wave S-matrix introduced in Chapter 17. Recall that for a single partial wave $\ell$, the elastic scattering cross section is:

$$\sigma_\ell^{\text{el}} = \frac{\pi}{k^2}(2\ell + 1)|1 - S_\ell|^2$$

where $S_\ell = e^{2i\delta_\ell}$ is the S-matrix element and $\delta_\ell$ is the phase shift. For a resonance, we need to allow for inelastic channels (absorption), so $|S_\ell| \leq 1$.

Near a resonance, the S-matrix element for partial wave $\ell$ has a pole in the complex energy plane. The most general single-pole form consistent with unitarity is:

$$S_\ell(E) = S_\ell^{(\text{bg})}(E) \cdot \frac{E - E_R - i\Gamma/2}{E - E_R + i\Gamma/2}$$

where $S_\ell^{(\text{bg})}$ is a slowly varying background (potential scattering) contribution, $E_R$ is the resonance energy, and $\Gamma$ is the total width. The pole is at $E = E_R - i\Gamma/2$ in the lower half of the complex energy plane, corresponding to a decaying state.

18.2.3 Derivation of the Breit-Wigner Formula

We now derive the cross section for the reaction $a + A \to C^* \to b + B$ proceeding through a single isolated resonance with spin $J$ and parity $\pi$ in partial wave $\ell$.

Step 1: Decompose the S-matrix.

Write $S_\ell = S_\ell^{(\text{pot})} + S_\ell^{(\text{res})}$, separating the potential scattering (background) from the resonance contribution. The resonance part of the scattering amplitude for the transition from channel $a$ to channel $b$ is:

$$f_{a \to b}^{(\text{res})} \propto \frac{\gamma_a \gamma_b}{E - E_R + i\Gamma/2}$$

where $\gamma_a$ and $\gamma_b$ are the reduced-width amplitudes for the entrance and exit channels, and $\Gamma_a = 2P_\ell(E)\gamma_a^2$, $\Gamma_b = 2P_{\ell'}(E)\gamma_b^2$ with $P_\ell$ being the barrier penetrability.

Step 2: Square the amplitude to get the cross section.

For the reaction (non-elastic) cross section through a single resonance, the potential scattering does not interfere with the reaction amplitude (it only contributes to elastic scattering). The reaction cross section is:

$$\sigma_{a \to b}(E) = \frac{\pi}{k^2} g_J \frac{\Gamma_a(E) \, \Gamma_b(E)}{(E - E_R)^2 + (\Gamma/2)^2}$$

where we have introduced the statistical spin factor $g_J$.

Step 3: The statistical spin factor.

When the projectile (spin $i$) hits the target (spin $I$), the number of magnetic substates in the entrance channel is $(2i+1)(2I+1)$. The compound nucleus has spin $J$, with $2J+1$ substates. The fraction of entrance-channel states that can couple to the compound state $J$ is:

$$g_J = \frac{2J + 1}{(2i+1)(2I+1)}$$

This factor accounts for the quantum-mechanical probability that the entrance channel has the correct angular momentum to form the resonance.

Step 4: The Breit-Wigner resonance formula.

Assembling the pieces, and writing $\lambdabar = 1/k$ so that $\pi/k^2 = \pi\lambdabar^2$:

$$\boxed{\sigma_{a \to b}(E) = \pi\lambdabar^2 \, g_J \, \frac{\Gamma_a \, \Gamma_b}{(E - E_R)^2 + (\Gamma/2)^2}}$$

This is the Breit-Wigner single-level resonance formula, first derived by Gregory Breit and Eugene Wigner in 1936 — the same year as Bohr's compound nucleus paper. It is one of the most important formulas in nuclear physics.

💡 Reading the Formula: - The factor $\pi\lambdabar^2$ sets the scale — the geometrical cross section associated with the de Broglie wavelength of the projectile. For thermal neutrons ($E = 0.025$ eV), $\lambdabar \approx 4.5 \times 10^{-10}$ cm, so $\pi\lambdabar^2 \approx 6.4 \times 10^{5}$ barns. This is why thermal neutron cross sections can be enormous. - The factor $g_J$ selects the angular momentum channel. - The factor $\Gamma_a \Gamma_b / [(E-E_R)^2 + (\Gamma/2)^2]$ is a Lorentzian — the same line shape that appears in atomic spectroscopy, electrical resonance circuits, and mechanical oscillators. It peaks at $E = E_R$ with peak value $\Gamma_a\Gamma_b/(\Gamma/2)^2$ and has full width at half maximum (FWHM) equal to $\Gamma$.

18.2.4 Special Cases

Elastic scattering ($b = a$):

$$\sigma_{\text{el}}^{(\text{res})}(E) = \pi\lambdabar^2 \, g_J \, \frac{\Gamma_a^2}{(E - E_R)^2 + (\Gamma/2)^2}$$

Note that this is only the resonance part. The full elastic cross section includes interference with potential scattering, which produces the characteristic asymmetric resonance shapes visible in high-resolution data.

Total cross section: Summing over all exit channels $b$, using $\sum_b \Gamma_b = \Gamma$:

$$\sigma_{\text{tot}}^{(\text{res})}(E) = \pi\lambdabar^2 \, g_J \, \frac{\Gamma_a \, \Gamma}{(E - E_R)^2 + (\Gamma/2)^2}$$

At the peak ($E = E_R$):

$$\sigma_{\text{peak}} = \pi\lambdabar_R^2 \, g_J \, \frac{4\Gamma_a \Gamma_b}{\Gamma^2} = 4\pi\lambdabar_R^2 \, g_J \, \frac{\Gamma_a \Gamma_b}{\Gamma^2}$$

where $\lambdabar_R = \lambdabar(E_R)$. For elastic scattering when only one channel is open ($\Gamma_a = \Gamma$), the maximum cross section is the unitary limit:

$$\sigma_{\text{peak}}^{\text{el}} = 4\pi\lambdabar_R^2 \, g_J$$

18.2.5 Energy Dependence of Partial Widths

The partial widths are not strictly constant — they depend on energy through the barrier penetrability:

$$\Gamma_n(E) = 2P_\ell(E) \gamma_n^2$$

where $\gamma_n^2$ is the reduced width (an intrinsic property of the nuclear state) and $P_\ell(E)$ is the penetrability of the centrifugal (and, for charged particles, Coulomb) barrier. For s-wave neutrons ($\ell = 0$), $P_0 \propto k \propto \sqrt{E}$, so:

$$\Gamma_n^{(\ell=0)}(E) \propto \sqrt{E}$$

This energy dependence is critical for understanding the $1/v$ law (Section 18.6). For the radiation width $\Gamma_\gamma$, the energy dependence is negligible over the width of a resonance because gamma emission couples to the enormous density of states below the neutron separation energy.

18.2.6 Connection to the Time-Energy Uncertainty Relation

The resonance width $\Gamma$ and the compound nucleus lifetime $\tau$ are related by:

$$\tau = \frac{\hbar}{\Gamma}$$

This is a direct consequence of the energy-time uncertainty relation $\Delta E \cdot \Delta t \gtrsim \hbar$. A narrow resonance ($\Gamma$ small) corresponds to a long-lived state; a broad resonance ($\Gamma$ large) to a short-lived state.

🔄 Check Your Understanding: The first resonance of ${}^{238}$U + n is at $E_R = 6.67$ eV with $\Gamma = 24.9$ meV. (a) Compute the lifetime. (b) How many times does the neutron traverse the nucleus during this time? [Answer: (a) $\tau = 2.6 \times 10^{-14}$ s; (b) Using $R \sim 7.4$ fm and $v \sim 3.6 \times 10^4$ m/s (for 6.67 eV neutron in CM), the transit time is $\sim 4 \times 10^{-22}$ s. The neutron "rattles around" $\sim 6.5 \times 10^7$ times — fully justifying the statistical picture.]

18.2.7 Worked Example: The First Resonance of ${}^{238}$U + n

Let us apply the Breit-Wigner formula to the most important resonance in reactor physics.

Given: The first s-wave resonance of n + ${}^{238}$U: - $E_R = 6.674$ eV, $J = 1/2^+$, $\ell = 0$ - $\Gamma_n = 1.493$ meV, $\Gamma_\gamma = 23.00$ meV - Target: ${}^{238}$U, $I = 0^+$; neutron spin $i = 1/2$

Step 1: Total width and lifetime.

$$\Gamma = \Gamma_n + \Gamma_\gamma = 1.493 + 23.00 = 24.49 \text{ meV}$$

$$\tau = \frac{\hbar}{\Gamma} = \frac{6.582 \times 10^{-16} \text{ eV}\cdot\text{s}}{24.49 \times 10^{-3} \text{ eV}} = 2.69 \times 10^{-14} \text{ s}$$

Step 2: Statistical spin factor.

$$g_J = \frac{2J+1}{(2i+1)(2I+1)} = \frac{2(1/2)+1}{(2(1/2)+1)(2(0)+1)} = \frac{2}{2 \times 1} = 1.0$$

Since the target has $I = 0$, there is only one possible $J$ value for s-wave neutrons: $J = 1/2$. All the entrance-channel flux can couple to this resonance.

Step 3: De Broglie wavelength.

The reduced wavelength at the resonance energy, in the center-of-mass frame:

$$E_{\text{cm}} = E_R \frac{A}{A+1} = 6.674 \times \frac{238}{239} = 6.646 \text{ eV}$$

$$\lambdabar = \frac{\hbar}{\sqrt{2 m_n E_{\text{cm}}}} = \frac{\hbar c}{\sqrt{2 m_n c^2 E_{\text{cm}}}}$$

Using $\hbar c = 197.3$ MeV$\cdot$fm and $m_n c^2 = 939.565$ MeV:

$$\lambdabar = \frac{197.3 \text{ MeV}\cdot\text{fm}}{\sqrt{2 \times 939.565 \times 6.646 \times 10^{-6}}} = \frac{197.3}{\sqrt{0.01249}} = \frac{197.3}{0.1118} = 1764 \text{ fm}$$

$$\pi\lambdabar^2 = \pi (1764)^2 = 9.78 \times 10^6 \text{ fm}^2 = 9.78 \times 10^4 \text{ b}$$

Step 4: Peak capture cross section.

$$\sigma_\gamma(E_R) = \pi\lambdabar_R^2 \, g_J \, \frac{4\Gamma_n \Gamma_\gamma}{\Gamma^2} = 9.78 \times 10^4 \times 1.0 \times \frac{4 \times 1.493 \times 23.00}{(24.49)^2}$$

$$= 9.78 \times 10^4 \times \frac{137.4}{599.8} = 9.78 \times 10^4 \times 0.2290 = 2.24 \times 10^4 \text{ b}$$

The peak capture cross section is approximately 22,400 barns — more than $10^4$ times the geometrical nuclear cross section. This enormous enhancement is entirely due to the quantum-mechanical resonance and the large de Broglie wavelength at low energies.

Step 5: Capture-to-total ratio at the peak.

$$\frac{\Gamma_\gamma}{\Gamma} = \frac{23.00}{24.49} = 0.939$$

At this resonance, 93.9% of compound nuclei that form will decay by gamma emission (radiative capture), and only 6.1% will re-emit the neutron. The capture cross section dominates because $\Gamma_\gamma \gg \Gamma_n$.

18.2.8 Multi-Level and Multi-Channel Extensions

The single-level Breit-Wigner formula is exact only for an isolated resonance — one where the tails of neighboring resonances are negligible. When resonances overlap (either because they are closely spaced or broad), interference between levels must be accounted for.

The multi-level Breit-Wigner (MLBW) formula sums the amplitudes (not the cross sections) of nearby resonances:

$$\sigma_{a \to b}(E) = \pi\lambdabar^2 g_J \left|\sum_\lambda \frac{\sqrt{\Gamma_{\lambda,a}\Gamma_{\lambda,b}}}{E_\lambda - E - i\Gamma_\lambda/2}\right|^2$$

where the sum runs over resonances $\lambda$. This correctly captures the interference between overlapping resonances, which produces the asymmetric and complicated line shapes observed in high-resolution cross section data.

For the most rigorous treatment, the R-matrix formalism (Wigner and Eisenbud, 1947; Lane and Thomas, 1958) provides a general, unitary parameterization of the scattering matrix in terms of resonance parameters. The R-matrix is:

$$R_{cc'}(E) = \sum_\lambda \frac{\gamma_{\lambda c} \gamma_{\lambda c'}}{E_\lambda - E}$$

where $\gamma_{\lambda c}$ are reduced-width amplitudes and $E_\lambda$ are the R-matrix level energies. The collision matrix (and hence all cross sections) is obtained from $R$ by the Lane-Thomas transformation. The R-matrix formalism is the theoretical foundation of all modern nuclear data evaluation — codes like SAMMY, REFIT, and EDA use R-matrix theory to fit cross section data.

⚠️ Notation Warning: Be careful to distinguish between the single-level Breit-Wigner formula (SLBW), the multi-level Breit-Wigner (MLBW), and the Reich-Moore approximation (RM, a practical variant of the R-matrix used in most evaluated nuclear data files). For well-separated resonances in non-fissile nuclei, SLBW is adequate. For fissile nuclei (where the fission channel introduces multi-channel interference), the Reich-Moore formalism is essential.

18.3 Resonance Parameters and Their Systematics

18.3.1 Partial Widths and Branching Ratios

The total width is the sum of all partial widths:

$$\Gamma = \Gamma_n + \Gamma_\gamma + \Gamma_f + \Gamma_p + \Gamma_\alpha + \cdots$$

where the subscripts denote neutron emission, gamma-ray emission, fission, proton emission, alpha emission, etc. For neutron-induced reactions on medium to heavy nuclei below the fission barrier:

• $\Gamma_n$ (neutron width): Varies strongly from resonance to resonance, reflecting the overlap of the resonance wavefunction with the continuum. Typical: $10^{-3}$–$10^{-1}$ eV.
• $\Gamma_\gamma$ (radiation width): Remarkably constant from resonance to resonance, typically $\sim 25$–$30$ meV for heavy nuclei. This constancy arises because $\Gamma_\gamma$ is a sum over thousands of individual gamma transitions to lower-lying states, and by the central limit theorem, the sum fluctuates little.
• $\Gamma_f$ (fission width): Present only for fissile/fissionable nuclei. Varies from resonance to resonance.

The branching ratio for channel $c$ is simply:

$$b_c = \frac{\Gamma_c}{\Gamma}$$

For neutron capture (radiative capture), $b_\gamma = \Gamma_\gamma / \Gamma$.

18.3.2 Level Spacing and the Distribution of Resonance Parameters

Level spacing $D$: The average energy spacing between resonances of the same spin and parity. For s-wave ($\ell = 0$) neutron resonances:

Note the enormous range: from tens of keV for light nuclei to fractions of an eV for actinides. The trend is that heavier nuclei with higher excitation energies have smaller spacings (higher level densities). The even-even target ${}^{238}$U has a larger spacing than odd-$A$ ${}^{235}$U because the neutron separation energy places the compound state at a lower effective excitation above the yrast line.

Porter-Thomas distribution: The reduced neutron widths $\gamma_n^2$ follow a chi-squared distribution with one degree of freedom (the Porter-Thomas distribution):

$$P(x) = \frac{1}{\sqrt{2\pi x}} e^{-x/2}$$

where $x = \Gamma_n^0 / \overline{\Gamma_n^0}$ and $\Gamma_n^0 = \Gamma_n / \sqrt{E_R}$ is the reduced neutron width (corrected for the $\sqrt{E}$ penetrability factor). This distribution, predicted by random matrix theory, has been confirmed experimentally for many nuclei and is one of the great successes of statistical nuclear physics.

💡 Why Does Random Matrix Theory Work? The compound nuclear states are superpositions of an enormous number of shell-model configurations. The matrix elements coupling these configurations behave, to a good approximation, as random Gaussian variables. The eigenvalue statistics of random matrices then predict the level spacing distribution (Wigner distribution) and the width distribution (Porter-Thomas). This is a deep connection between nuclear physics and quantum chaos.

18.3.3 Wigner Distribution of Level Spacings

The distribution of nearest-neighbor level spacings $s = D/\overline{D}$ (normalized to the mean spacing) follows the Wigner surmise:

$$P(s) = \frac{\pi s}{2} \exp\left(-\frac{\pi s^2}{4}\right)$$

This distribution exhibits level repulsion — the probability of finding two levels at the same energy ($s = 0$) vanishes. This is in stark contrast to a random (Poisson) distribution $P(s) = e^{-s}$, where there is no repulsion. The Wigner distribution has been verified to high precision for nuclear resonances and is another signature of quantum chaos in the compound nucleus.

18.4 Nuclear Level Densities

18.4.1 Why Level Densities Matter

The density of excited states $\rho(E^*)$ — the number of levels per unit energy — is a fundamental input to compound nuclear reaction theory. At low excitation energies, individual states can be resolved and their properties measured. But above a few MeV of excitation (depending on the nucleus), the level density becomes so high that individual states overlap and only statistical descriptions are meaningful. The transition from resolved resonances to the statistical regime is one of the most important conceptual boundaries in nuclear reaction physics.

18.4.2 The Fermi Gas Model

The simplest and most widely used model for nuclear level densities is the Fermi gas model, which treats the nucleus as a gas of non-interacting fermions confined in a potential well.

Derivation. Consider $A$ nucleons (fermions) in equally spaced single-particle levels with spacing $\epsilon_0$ near the Fermi energy. At excitation energy $E^*$, the number of ways to distribute the excitation among the nucleons is related to the thermodynamic entropy $S$:

$$\rho(E^*) \propto e^{S(E^*)}$$

For a degenerate Fermi gas, the entropy at temperature $T$ is:

$$S = 2aT$$

where $a$ is the level density parameter, related to the single-particle level density at the Fermi energy $g(\epsilon_F)$ by:

$$a = \frac{\pi^2}{6} g(\epsilon_F)$$

The excitation energy is related to the temperature by:

$$E^* = aT^2$$

Solving for $T = \sqrt{E^*/a}$ and substituting into $S = 2aT = 2\sqrt{aE^*}$, we obtain:

$$\rho(E^*) \propto \exp\left(2\sqrt{aE^*}\right)$$

A more careful calculation, accounting for the proper thermodynamic treatment (using the saddle-point approximation for the inverse Laplace transform of the partition function), gives the Bethe formula (1936):

$$\boxed{\rho(E^*) = \frac{\sqrt{\pi}}{12} \frac{\exp\left(2\sqrt{aE^*}\right)}{a^{1/4} (E^*)^{5/4}}}$$

This is sometimes written as:

$$\rho(E^*) = \frac{\sqrt{\pi}}{12} \frac{\exp(2\sqrt{aU})}{a^{1/4} U^{5/4}}$$

where $U = E^* - \Delta$ incorporates a pairing correction $\Delta$ that accounts for the energy gap in even-even nuclei ($\Delta \approx 12/\sqrt{A}$ MeV for even-even, $\approx 0$ for odd-$A$).

18.4.3 The Level Density Parameter

The level density parameter $a$ is the single most important parameter in the Fermi gas model. A simple estimate from the independent-particle model gives:

$$a \approx \frac{A}{K} \quad \text{MeV}^{-1}$$

where empirically $K \approx 7$–$9$ MeV. A common first approximation is:

$$a \approx \frac{A}{8} \quad \text{MeV}^{-1}$$

More refined estimates by Ignatyuk and collaborators (1975) include shell corrections:

$$a(E^*) = \tilde{a}\left[1 + \frac{\delta W}{E^*}\left(1 - e^{-\gamma E^*}\right)\right]$$

where $\tilde{a} \approx A/8$ is the asymptotic value, $\delta W$ is the shell correction energy (from the SEMF minus the measured mass), and $\gamma \approx 0.04$–$0.06$ MeV$^{-1}$ controls the damping of shell effects with excitation energy. Shell effects wash out at high excitation — by $E^* \sim 30$–$50$ MeV, the level density approaches the smooth Fermi gas prediction regardless of magic numbers.

18.4.4 Comparison to Experiment

The level density can be tested against measured quantities:

1. Resolved resonance spacings at the neutron separation energy $S_n$: From the observed average spacing $D_0$ of s-wave resonances, $\rho(S_n) \approx 1/D_0$ (after correcting for the spin distribution).

2. Discrete levels at low excitation energy: Counting known levels from nuclear data compilations.

3. Evaporation spectra: The energy spectrum of particles emitted from a highly excited compound nucleus is approximately a Maxwellian $\propto E_p \exp(-E_p / T)$, and the "nuclear temperature" $T = \sqrt{E^*/a}$ can be extracted.

For ${}^{239}$U (the compound nucleus of n + ${}^{238}$U), the observed $D_0 \approx 20.3$ eV at $S_n = 4.806$ MeV gives $\rho(S_n) \approx 3.2 \times 10^4$ MeV$^{-1}$ (after spin correction), in good agreement with the Fermi gas model using $a \approx 238/8 \approx 29.7$ MeV$^{-1}$.

⚠️ Common Pitfall: The level density increases exponentially with excitation energy. At the neutron separation energy ($E^* \sim 5$–$8$ MeV for heavy nuclei), there are thousands to millions of levels per MeV. Students often underestimate how rapidly the level density grows. For ${}^{239}$U at $S_n$: $\rho \sim 3 \times 10^4$ MeV$^{-1}$ — that is one level every 30 eV.

18.4.5 The Spin Distribution

The total level density $\rho(E^*)$ counts levels of all spins. The density of levels with a specific spin $J$ is given by:

$$\rho(E^*, J) = \rho(E^*) \cdot f(J)$$

where the spin distribution factor is:

$$f(J) = \frac{2J+1}{2\sigma^2} \exp\left[-\frac{(J+1/2)^2}{2\sigma^2}\right]$$

and $\sigma$ is the spin cutoff parameter, related to the nuclear moment of inertia and temperature:

$$\sigma^2 \approx 0.0888 \, a^{1/2} \, A^{2/3} \, T$$

with $T = \sqrt{E^*/a}$. For heavy nuclei at the neutron separation energy, $\sigma \approx 5$–$7$, meaning that levels with spins $J \approx 3$–$8$ are the most common.

This spin distribution is essential for connecting the observed s-wave level spacing $D_0$ (which selects a specific $J = I \pm 1/2$) to the total level density $\rho(S_n)$:

$$\frac{1}{D_0} = \rho(S_n, J) = \rho(S_n) \cdot f(J)$$

The correction factor $f(J)$ is typically 0.03–0.10, meaning that the observed spacing is 10–30 times larger than the total spacing $1/\rho$.

18.4.6 Level Densities and the Compound Nucleus

The physical connection between level densities and the compound nucleus model is deep: the compound nucleus "forgets" how it was formed precisely because there are so many states available. At $S_n$ for a heavy nucleus, there are $\sim 10^4$ levels per MeV. The incoming neutron couples to a superposition of these levels, and the resulting interference pattern evolves on a time scale $\sim \hbar D^{-1}$. After enough time (many recurrence times), the system has explored a large fraction of the accessible phase space, and the memory of the initial state is effectively erased.

This connection also explains why the compound nucleus model works better for heavier nuclei (higher level density at $S_n$) and at higher excitation energies (exponentially growing level density). For light nuclei or near thresholds, the level density may be too low for the statistical picture to apply, and the cross section retains information about the specific structure of the resonance states.

18.5 The Hauser-Feshbach Statistical Model

18.5.1 Beyond Single Resonances

The Breit-Wigner formula (Section 18.2) describes the cross section near an isolated resonance. But in many practical situations — especially for medium-to-heavy nuclei at excitation energies above a few MeV — the resonances overlap, and a statistical description is more appropriate.

The Hauser-Feshbach model (1952) provides this statistical description. It computes the compound nuclear cross section by averaging over the contributions of many resonances, using the level density to determine the statistical weights.

18.5.2 The Hauser-Feshbach Formula

For the reaction $a + A \to C^* \to b + B$, the angle-integrated cross section in the Hauser-Feshbach model is:

$$\sigma_{a \to b} = \sum_{J,\pi} \sigma_{\text{CN}}^{J\pi}(a) \cdot \frac{T_b^{J\pi}}{\sum_c T_c^{J\pi}}$$

where:

• $\sigma_{\text{CN}}^{J\pi}(a) = \frac{\pi\lambdabar^2}{(2i+1)(2I+1)} (2J+1) T_a^{J\pi}$ is the compound nucleus formation cross section for spin $J$ and parity $\pi$.
• $T_c^{J\pi}$ are the transmission coefficients for channel $c$ (obtained from the optical model).
• The sum in the denominator runs over all open channels (neutron, proton, alpha, gamma, fission, etc.).

The transmission coefficients $T_c$ are related to the average widths and spacings by:

$$T_c = \frac{2\pi \overline{\Gamma_c}}{D}$$

This is the optical model-statistical model connection: the optical model provides the transmission coefficients (which encode the average probability of absorption into the compound nucleus), and the statistical model distributes the subsequent decay among all available exit channels.

18.5.3 Width Fluctuation Corrections

The Hauser-Feshbach formula as written assumes that widths take their average values. In reality, widths fluctuate (recall the Porter-Thomas distribution). When the number of open channels is small, the fluctuations can enhance the elastic channel relative to the inelastic channels — an effect known as the elastic enhancement or Ericson fluctuations.

The width fluctuation correction factor $W$ modifies the Hauser-Feshbach formula:

$$\sigma_{a \to b} \to \sigma_{a \to b} \times W_{ab}$$

For elastic scattering ($a = b$), $W \approx 2$–$3$ when few channels are open; for reactions ($a \neq b$), $W \leq 1$. The Moldauer (1975) and HRTW (Hofmann, Richert, Tepel, Weidenmuller, 1975) models provide the standard prescriptions.

18.6 Neutron Capture Cross Sections

18.6.1 The $1/v$ Law

One of the most important results in nuclear physics is the $1/v$ dependence of neutron absorption cross sections at low energies. We derive this from the Breit-Wigner formula.

Consider a single resonance at energy $E_R$ with $\Gamma_n \ll \Gamma_\gamma$, and assume we are far below the resonance ($E \ll E_R$). Then the denominator of the Breit-Wigner formula is approximately $(E_R)^2 + (\Gamma/2)^2 \approx E_R^2$ (since typically $\Gamma \ll E_R$). The neutron width has the energy dependence $\Gamma_n(E) \propto \sqrt{E}$ (for s-wave). Therefore:

$$\sigma_{\gamma}(E) = \pi\lambdabar^2 g_J \frac{\Gamma_n(E) \Gamma_\gamma}{E_R^2 + (\Gamma/2)^2} \propto \frac{1}{E} \cdot \sqrt{E} = \frac{1}{\sqrt{E}} \propto \frac{1}{v}$$

since $\lambdabar^2 \propto 1/E$ and $\Gamma_n \propto \sqrt{E}$. Explicitly:

$$\sigma_\gamma(E) = \sigma_\gamma(E_0) \sqrt{\frac{E_0}{E}} = \sigma_\gamma(E_0) \frac{v_0}{v}$$

where $E_0 = 0.0253$ eV is the thermal energy (at 293.6 K) and $v_0 = 2200$ m/s is the thermal neutron speed.

💡 Why $1/v$? Physically, a slower neutron spends more time near the nucleus, increasing the probability of capture. The $1/v$ law holds for any process where the cross section is dominated by well-separated resonances at energies above the thermal region.

For ${}^{238}$U, the thermal neutron capture cross section is $\sigma_\gamma(E_0) = 2.68$ b, and the $1/v$ law holds from thermal energies up to $\sim 1$ eV, where the first resolved resonance at 6.67 eV begins to influence the cross section.

18.6.2 Resonance Structure in Neutron Cross Sections

Above the $1/v$ region, the cross section displays a rich resonance structure. The total neutron cross section of ${}^{238}$U is one of the most studied and visually striking examples in nuclear physics. Between 1 eV and 10 keV, there are hundreds of resolved resonances, with peak cross sections ranging from tens to tens of thousands of barns, separated by valleys where the cross section drops to a few barns.

Each peak corresponds to an excited state of the compound nucleus ${}^{239}$U$^*$. The resonance parameters — $E_R$, $\Gamma_n$, $\Gamma_\gamma$, $J^\pi$ — are tabulated in evaluated nuclear data libraries such as ENDF/B-VIII.0 (United States), JEFF-3.3 (Europe), and JENDL-5 (Japan). These libraries are maintained by large international collaborations and are essential infrastructure for reactor physics, criticality safety, and astrophysical modeling.

The cross section data are organized into three energy regions:

1. Resolved resonance region (RRR): From thermal to a few keV (the exact boundary depends on the nucleus). Individual resonances are fitted with R-matrix parameters. For ${}^{238}$U, the RRR extends to 20 keV in ENDF/B-VIII.0, encompassing $\sim 200$ s-wave and p-wave resonances.

2. Unresolved resonance region (URR): From the upper boundary of the RRR to $\sim 100$–$300$ keV. Resonances still exist but cannot be individually resolved. The cross section is described by average resonance parameters ($\overline{\Gamma_n}$, $\overline{\Gamma_\gamma}$, $D$) and their statistical distributions. Monte Carlo sampling of resonance "ladders" provides the fluctuating cross section needed for self-shielding calculations.

3. Continuum region: Above the URR, the resonances are so densely packed that the cross section is smooth. The optical model and Hauser-Feshbach theory apply directly.

📊 Important Cross Section Values for ${}^{238}$U:

Energy	Cross section type	Value
0.0253 eV (thermal)	$\sigma_\gamma$	2.68 b
0.0253 eV (thermal)	$\sigma_{\text{tot}}$	12.0 b
6.67 eV (first resonance peak)	$\sigma_{\text{tot}}$	$\sim 23{,}000$ b
100 eV–10 keV (average)	$\sigma_\gamma$	$\sim 1$–$10$ b (between resonances)
1 MeV	$\sigma_{\text{tot}}$	$\sim 7$ b
1 MeV	$\sigma_f$ (fission)	$\sim 0.5$ b (above fission threshold)

18.6.3 The Resonance Integral

In reactor physics, the resonance integral quantifies the total capture probability for neutrons slowing down through the resonance region in a moderator:

$$I_\gamma = \int_{E_{\text{Cd}}}^{\infty} \sigma_\gamma(E) \frac{dE}{E}$$

where $E_{\text{Cd}} \approx 0.5$ eV is the cadmium cutoff energy (cadmium absorbs thermal neutrons strongly but is transparent to epithermal neutrons). The $dE/E$ weighting reflects the $1/E$ energy spectrum of neutrons slowing down in a moderator.

For ${}^{238}$U, the resonance integral is $I_\gamma = 275$ b. This is a crucial quantity for the resonance escape probability $p$ in the four-factor formula for reactor criticality (Chapter 26):

$$p = \exp\left(-\frac{N_{238} I_\gamma}{\xi \Sigma_s}\right)$$

where $N_{238}$ is the ${}^{238}$U atom density, $\xi$ is the average logarithmic energy decrement per collision, and $\Sigma_s$ is the macroscopic scattering cross section of the moderator. The resonance escape probability is one of the most sensitive parameters in reactor design — it determines how many neutrons survive slowing down without being captured by ${}^{238}$U, and hence whether the reactor can sustain a chain reaction.

18.6.4 Self-Shielding and the Narrow Resonance Approximation

In a real reactor, the effective resonance integral is reduced by self-shielding: neutrons near a resonance energy are strongly absorbed in the outer layers of a fuel rod, so neutrons deeper inside see a depleted flux at that energy. The effective resonance integral depends on the fuel geometry and temperature.

Temperature dependence (Doppler broadening): At finite temperature, the thermal motion of the ${}^{238}$U atoms broadens the effective resonance shape (the laboratory-frame cross section is a convolution of the Breit-Wigner shape with the thermal velocity distribution). This broadening lowers the peak cross section but increases the width, with the net effect of increasing the effective resonance integral. This Doppler broadening is the basis of the Doppler reactivity coefficient — a negative temperature coefficient of reactivity that is one of the inherent safety features of thermal reactors. As the fuel heats up, the resonance absorption of ${}^{238}$U increases, reducing reactivity. We will develop this quantitatively in Chapter 26.

18.7 Astrophysical Importance: From Resonances to Stars

18.7.1 Neutron Capture in Stars

The compound nucleus model is not just reactor physics — it is astrophysics. All elements heavier than iron are produced primarily by neutron capture reactions in stellar environments. There are two main processes:

The $s$-process (slow neutron capture): In asymptotic giant branch (AGB) stars, neutrons are produced by ${}^{13}$C($\alpha$, n)${}^{16}$O and ${}^{22}$Ne($\alpha$, n)${}^{25}$Mg reactions. These neutrons are captured by seed nuclei (primarily iron-peak elements from a previous supernova), building up heavier elements one neutron at a time. The neutron density is low enough ($n_n \sim 10^7$–$10^{11}$ cm$^{-3}$) that beta-unstable nuclei have time to decay before capturing another neutron. The $s$-process path therefore follows the valley of stability.

The rate of neutron capture per target nucleus is:

$$\lambda_{\text{cap}} = n_n \langle \sigma v \rangle$$

where $\langle \sigma v \rangle$ is the Maxwellian-averaged cross section (MACS) at the stellar temperature, typically $kT \approx 8$–$90$ keV for the $s$-process. The MACS is:

$$\langle \sigma v \rangle = \frac{2}{\sqrt{\pi}} \frac{1}{(kT)^2} \int_0^\infty \sigma(E) \, E \, e^{-E/kT} \, dE$$

The individual resonances in the cross section directly determine $\langle \sigma v \rangle$, and hence the $s$-process abundances. The $s$-process abundance of a nuclide on the $s$-process path is approximately inversely proportional to its MACS: $\sigma N_s \approx$ const (the "local approximation"). Elements with small neutron capture cross sections (magic neutron numbers $N = 50, 82, 126$) accumulate in higher abundances — these are the $s$-process peaks at $A \approx 88$ (Sr, Y, Zr), $A \approx 138$ (Ba, La, Ce), and $A \approx 208$ (${}^{208}$Pb).

🔗 Connection: The magic numbers that produce small neutron capture cross sections are the same magic numbers explained by the nuclear shell model (Chapter 6). The nuclear structure of individual nuclei directly determines the chemical composition of the universe.

18.7.2 The $r$-Process

The $r$-process (rapid neutron capture) occurs in environments with extreme neutron densities ($n_n \sim 10^{20}$–$10^{28}$ cm$^{-3}$): neutron star mergers and possibly core-collapse supernovae. Under these conditions, neutron capture is much faster than beta decay, and the capture path runs far into the neutron-rich side of the chart of nuclides. The path is determined by the competition between neutron capture (governed by compound nuclear cross sections and level densities) and photodisintegration (the reverse reaction).

The detection of gravitational waves and electromagnetic radiation from the neutron star merger GW170817 in August 2017 provided the first direct evidence that $r$-process nucleosynthesis occurs in neutron star mergers. The infrared "kilonova" signal was powered by the radioactive decay of freshly synthesized $r$-process elements — particularly the lanthanides and actinides with their high opacities.

📊 By the Numbers: About half of all elements heavier than iron in the Solar System were produced by the $s$-process, and about half by the $r$-process. A few proton-rich nuclei are produced by the $p$-process (photodisintegration) and $\nu p$-process. The compound nucleus model — Breit-Wigner resonances, level densities, Hauser-Feshbach calculations — is the theoretical engine that connects nuclear physics to the observed abundances.

18.7.3 The Role of Level Densities in Astrophysics

For the $r$-process, most of the relevant nuclei are far from stability and have never been studied experimentally. Their neutron capture cross sections must be calculated using the Hauser-Feshbach model, with level densities from the Fermi gas model or more sophisticated microscopic calculations (Hartree-Fock-Bogoliubov with combinatorial level densities). The uncertainties in these level densities propagate directly into uncertainties in the predicted $r$-process abundances — a major frontier in nuclear astrophysics.

Radioactive beam facilities such as FRIB (Facility for Rare Isotope Beams, Michigan State University, operational since 2022) are designed to produce and study these exotic nuclei, providing the nuclear data needed to reduce these uncertainties.

18.8 Experimental Methods for Resonance Measurements

18.8.1 Time-of-Flight (TOF) Technique

The workhorse method for measuring neutron cross sections and resonance parameters is the time-of-flight technique. A pulsed neutron source produces bursts of neutrons with a broad energy spectrum. The neutrons travel a known distance $L$ (the flight path, typically 10–400 m) to the sample and detector. The neutron energy is determined from the measured time of flight $t$:

$$E = \frac{1}{2} m_n v^2 = \frac{m_n L^2}{2 t^2} = \frac{72.3 \; [\text{eV}]}{(L[\text{m}]/t[\mu\text{s}])^{-2}}$$

or equivalently:

$$E [\text{eV}] = \left(\frac{72.3 \, L[\text{m}]}{t[\mu\text{s}]}\right)^2$$

The energy resolution is:

$$\frac{\Delta E}{E} = 2\sqrt{\left(\frac{\Delta t}{t}\right)^2 + \left(\frac{\Delta L}{L}\right)^2}$$

where $\Delta t$ is the pulse width (or the timing resolution) and $\Delta L$ accounts for the finite sample thickness and moderator geometry. Longer flight paths give better resolution but lower counting rates (intensity drops as $1/L^2$).

Major TOF facilities include ORELA (Oak Ridge), GELINA (Geel, Belgium), and n_TOF at CERN (with flight paths of 20 m, 185 m, and 230 m).

18.8.2 Transmission Measurements

In a transmission measurement, a collimated neutron beam passes through a sample of known thickness, and the transmitted fraction is measured as a function of energy (via TOF). The transmission is:

$$T(E) = e^{-n \sigma_{\text{tot}}(E) \ell}$$

where $n$ is the atom density and $\ell$ is the sample thickness. At a resonance, the cross section is large, the transmission dips sharply, and the resonance energy and total width can be extracted.

For the first resonance of ${}^{238}$U at 6.67 eV, the peak cross section is $\sigma_{\text{tot}} \approx 23{,}000$ b. Even a thin sample (0.1 mm) produces nearly complete absorption ("black resonance"), making careful treatment of sample thickness effects essential.

18.8.3 Capture Measurements

Neutron capture cross sections are measured by detecting the prompt gamma rays emitted when the compound nucleus de-excites. Modern detectors — such as large liquid-scintillator tanks (total absorption calorimeters) or arrays of $\text{C}_6\text{D}_6$ scintillation detectors with the pulse height weighting technique — achieve near-complete detection of the gamma cascade.

The measured capture yield $Y(E)$ is related to the capture cross section:

$$Y(E) = \frac{\sigma_\gamma(E)}{\sigma_{\text{tot}}(E)} \left[1 - e^{-n\sigma_{\text{tot}}(E)\ell}\right]$$

In the thin-sample limit, $Y \approx n\sigma_\gamma(E)\ell$. From the yield, the capture area $A_\gamma = g_J \Gamma_n \Gamma_\gamma / \Gamma$ of each resonance is extracted.

18.8.4 R-Matrix Analysis

The raw data from transmission and capture measurements are analyzed using R-matrix theory — the formal framework for parameterizing nuclear cross sections in terms of resonance parameters. The standard code for this analysis is SAMMY (Oak Ridge National Laboratory), which fits the multi-level R-matrix formula to the measured data, accounting for Doppler broadening, resolution broadening, self-shielding, multiple scattering, and other experimental effects.

The resulting resonance parameters are compiled in the Atlas of Neutron Resonances (Mughabghab, 2018), which contains evaluated parameters for thousands of resonances across the chart of nuclides. This atlas is one of the foundational references of nuclear data science.

18.9 Summary: The Compound Nucleus in Perspective

The compound nucleus model is one of the great unifying ideas of nuclear physics. From Bohr's 1936 insight that the nucleus can "forget" how it was formed, we have developed:

1. The Breit-Wigner formula — the mathematical description of resonance cross sections, derived from the S-matrix pole structure and parameterized by resonance energies, widths, and spins.

2. Nuclear level densities — the Fermi gas model provides the exponential growth of states with excitation energy, quantified by the level density parameter $a \approx A/8$ MeV$^{-1}$.

3. The Hauser-Feshbach model — the statistical extension of the compound nucleus model to the regime of overlapping resonances, connecting the optical model transmission coefficients to decay branching ratios.

4. Neutron capture cross sections — the $1/v$ law at low energies, the resonance structure at intermediate energies, and the resonance integral that governs neutron absorption in reactors.

5. Astrophysical nucleosynthesis — the same resonance parameters that govern reactor physics also determine the abundances of elements heavier than iron in the universe.

The compound nucleus sits at the center of a web of connections: to quantum scattering theory (Chapter 17), to nuclear structure (Chapter 6), to fission (Chapter 20), to reactor physics (Chapter 26), and to stellar nucleosynthesis (Chapter 22–24). The Breit-Wigner formula is to nuclear reactions what the Coulomb scattering formula is to nuclear structure — the starting point from which everything else follows.

In Chapter 19, we will study the opposite limit: direct reactions, where the projectile interacts with one or a few nucleons and exits quickly, without forming a compound nucleus. The contrast between compound and direct mechanisms — statistical versus selective, slow versus fast, all-nucleons versus few-nucleons — is one of the central themes of nuclear reaction physics.

🔄 Looking Back: The nuclear shell model (Chapter 6) predicts magic numbers where the level spacing is large and the neutron capture cross section is small. The compound nucleus model explains why the cross section is small: fewer resonances means a smaller capture probability. Magic-number nuclei at $N = 50, 82, 126$ are the bottlenecks of the $s$-process, and the resulting abundance peaks at $A \approx 88, 138, 208$ are direct observational evidence of nuclear shell structure imprinted on the composition of the universe.

🔭 Looking Ahead: In Chapter 20, we will see that the compound nucleus model also describes fission — but with a twist. The fission width $\Gamma_f$ depends on the structure of the fission barrier, which varies from resonance to resonance. The competition between neutron emission and fission ($\Gamma_n / \Gamma_f$) determines whether a nucleus is fissile or fissionable — a distinction with profound consequences for both reactors and weapons.