Nuclear Reactions & Scattering: Optical Model, Direct/Compound Processes & Resonances

1. Energetics and the Reaction Q-Value

Consider a generic two-body nuclear reaction in which projectile $a$ collides with stationary target $X$, producing ejectile $b$ and residual nucleus $Y$:

$$a + X \to Y + b \quad \text{or} \quad X(a, b)Y$$

The reaction $Q$-value is defined as the difference between initial and final nuclear rest masses:

$$Q \equiv \left[ (m_a + M_X) - (m_b + M_Y) \right] c^2 = T_b + T_Y - T_a$$

where $T_i$ denotes kinetic energy.

• Exoergic (Exothermic) Reaction ($Q > 0$): Nuclear mass is converted into kinetic energy. The reaction can proceed even at zero incident projectile energy ($T_a \to 0$).
• Endoergic (Endothermic) Reaction ($Q < 0$): Incident kinetic energy is converted into mass. The reaction cannot occur unless the projectile energy exceeds a minimum threshold energy $E_{\text{th}}$.

2. Derivation of the Laboratory Threshold Energy

In the center-of-mass (CM) frame, total linear momentum is zero. To create the final rest masses at threshold, the center-of-mass kinetic energy $E_{\text{cm}}$ must at least equal the mass deficit $|Q|$:

$$E_{\text{cm}} = \frac{M_X}{m_a + M_X} T_{\text{lab}} \ge |Q| = -Q$$

Solving for the minimum laboratory kinetic energy of projectile $a$:

$$E_{\text{th}} = |Q| \left( \frac{m_a + M_X}{M_X} \right) = -Q \left( 1 + \frac{m_a}{M_X} \right)$$

The additional energy fraction $\frac{m_a}{M_X}|Q|$ is unavoidable: it represents the center-of-mass kinetic energy that must be conserved to maintain total linear momentum.

1. The Cloudy Crystal Ball Model

When nucleons or composite projectiles collide with nuclei, two competing processes occur:

1. Elastic Scattering: Projectile and target remain in their ground states ($a + X \to a + X$).
2. Absorption / Inelastic Reactions: The projectile is removed from the elastic channel via inelastic scattering, nucleon transfer, or compound nucleus formation.

In 1954, Herman Feshbach, Charles Porter, and Victor Weisskopf introduced the Optical Model, drawing an analogy to a light wave propagating through a semi-transparent, absorbing sphere of glass (a "cloudy crystal ball").

In optics, an absorbing medium is described by a complex index of refraction:

$$\tilde{n} = n + i \kappa$$

The optical wavevector becomes $k = \tilde{n} k_0 = n k_0 + i \kappa k_0$, causing the electromagnetic wave intensity to attenuate exponentially:

$$I(z) \propto e^{-2 \kappa k_0 z}$$

In quantum mechanics, this spatial attenuation is produced by adding a negative imaginary potential to the Schrödinger equation:

$$U(r) = -V(r) - i W(r)$$

2. The Complex Optical Potential Parameterization

Standard phenomenological optical potentials (such as the Becchetti-Greenlees and Koning-Delaroche global parameterizations) take the form:

$$U(r) = -V_R f_{\text{WS}}(r, R_R, a_R) - i W_V f_{\text{WS}}(r, R_V, a_V) + 4 i a_S W_S \frac{d}{dr}f_{\text{WS}}(r, R_S, a_S) + V_{so} \vec{l}\cdot\vec{s} \left(\frac{\hbar}{m_\pi c}\right)^2 \frac{1}{r}\frac{df_{\text{WS}}}{dr}$$

where $f_{\text{WS}}(r, R, a) = [1 + \exp((r-R)/a)]^{-1}$ is the Woods-Saxon function.

• Real Depth ($V_R \approx 45\text{ to }55\text{ MeV}$): Refracts the incoming wave, determining the average phase shift and shape of diffraction oscillations.
• Volume Imaginary Depth ($W_V \approx 5\text{ to }12\text{ MeV}$): Represents absorption throughout the nuclear volume (dominant at high energies $E > 50\text{ MeV}$).
• Surface Imaginary Depth ($W_S \approx 8\text{ to }14\text{ MeV}$): Peaked strictly at the nuclear surface via $\frac{df}{dr}$. At low energies ($E < 20\text{ MeV}$), the Pauli principle prevents collisions in the nuclear interior, restricting reactions to the surface.

1. The Scattering Matrix (S-Matrix) with Absorption

In partial-wave scattering from a complex potential, probability is not conserved within the elastic channel alone. The asymptotic radial wavefunction for partial wave $l$ is:

$$u_l(r) \xrightarrow{r\to\infty} \frac{i}{2} \left[ e^{-i(kr - l\pi/2)} - S_l e^{i(kr - l\pi/2)} \right]$$

where $S_l$ is the complex scattering matrix element ($S$-matrix). Writing $S_l$ in terms of a real phase shift $\delta_l$ and inelastic absorption parameter $\eta_l$:

$$S_l \equiv \eta_l e^{2i\delta_l}, \qquad 0 \le \eta_l \le 1$$

• If there is no absorption ($W = 0$), the flux in the outgoing wave equals the incoming flux: $\eta_l = 1 \implies |S_l| = 1$ (unitary $S$-matrix).
• If absorption occurs ($W > 0$), flux is lost to other channels: $\eta_l < 1 \implies |S_l| < 1$.
• If complete absorption occurs for partial wave $l$ (black disk limit): $\eta_l = 0 \implies S_l = 0$.

2. Elastic, Reaction, and Total Cross Sections

Integrating the probability flux gives the exact partial-wave cross sections:

1. Elastic Scattering Cross Section: $$\sigma_{\text{el}} = \frac{\pi}{k^2} \sum_{l=0}^\infty (2l+1) |1 - S_l|^2 = \frac{\pi}{k^2} \sum_{l=0}^\infty (2l+1) \left[ 1 + \eta_l^2 - 2\eta_l \cos(2\delta_l) \right]$$
2. Reaction (Absorption) Cross Section: $$\sigma_{\text{reac}} = \frac{\pi}{k^2} \sum_{l=0}^\infty (2l+1) \left( 1 - |S_l|^2 \right) = \frac{\pi}{k^2} \sum_{l=0}^\infty (2l+1) (1 - \eta_l^2)$$ The factor $T_l \equiv 1 - |S_l|^2 = 1 - \eta_l^2$ is the transmission coefficient.
3. Total Cross Section: $$\sigma_{\text{tot}} = \sigma_{\text{el}} + \sigma_{\text{reac}} = \frac{2\pi}{k^2} \sum_{l=0}^\infty (2l+1) \left[ 1 - \text{Re}(S_l) \right] = \frac{4\pi}{k}\text{Im}[f(0)]$$ (The Optical Theorem!).

Important Physical Theorem: If $\sigma_{\text{reac}} > 0$ (absorption occurs, $\eta_l < 1$), then $|1 - S_l|^2 > 0$ is unavoidable. Therefore, absorption is always accompanied by elastic scattering (shadow/diffraction scattering). You cannot have absorption without elastic scattering!

1. Characteristics of Direct Nuclear Reactions

When the incident projectile has moderate to high energy ($E \ge 10\text{ MeV/nucleon}$), a collision can proceed via a direct reaction:

• Interaction Time: Extremely brief, equal to the transit time across the nucleus: $$\tau_{\text{direct}} \sim \frac{2 R}{v} \approx \frac{10\text{ fm}}{0.2 c} \approx 1.5 \times 10^{-22}\text{ s}$$
• Peripheral Nature: Direct reactions involve only one or two valence nucleons at the surface of the nucleus; the core nucleons remain undisturbed "spectators".
• Forward-Peaked Angular Distribution: The differential cross section $\frac{d\sigma}{d\Omega}(\theta)$ is strongly peaked at forward laboratory angles ($\theta \approx 0^\circ \text{ to } 30^\circ$).

2. Stripping and Pickup Reactions

1. Deuteron Stripping $(d, p)$ or $(d, n)$: The loosely bound deuteron ($B = 2.22\text{ MeV}$) grazes the target nucleus. The nuclear force captures the neutron into an empty single-particle shell model orbital $(n, l, j)$, while the proton escapes without entering the nucleus.
2. Pickup Reactions $(p, d)$ or $(d, t)$: The incoming proton snatches a valence neutron from an occupied shell model orbital of the target, emerging as a deuteron.

3. Butler Theory and Spectroscopic Factors

In the semi-classical Butler theory, the transferred nucleon carries orbital angular momentum $l$ into the nucleus at impact parameter $R$:

$$\hbar l \approx p_{\text{trans}} R = q R$$

where $\vec{q} = \vec{k}_i - \vec{k}_f$ is the momentum transfer vector:

$$q = \sqrt{k_i^2 + k_f^2 - 2 k_i k_f \cos\theta}$$

The differential cross section is proportional to the square of the spherical Bessel function $j_l(q R)$:

$$\frac{d\sigma}{d\Omega}(\theta) \propto |j_l(q R)|^2$$

Because $j_l(x)$ has its first maximum at a characteristic value $x_{\max} \approx l$:

• $l = 0$: Peak is at $\theta = 0^\circ$ ($q = 0$).
• $l = 1$: Peak is at a small finite angle $\theta_1 > 0^\circ$.
• $l = 2$: Peak is at a larger angle $\theta_2 > \theta_1$.

By simply measuring the angular position of the first maximum in $\frac{d\sigma}{d\Omega}$, experimentalists uniquely determine the orbital angular momentum $l$ of the single-particle state! The absolute cross-section magnitude yields the spectroscopic factor $S$, measuring the degree to which the state matches an unperturbed single-particle shell model state.

1. The Niels Bohr Compound Nucleus Hypothesis

In 1936, Niels Bohr proposed that low-energy nuclear reactions (such as thermal or slow neutron capture) proceed through a two-stage mechanism:

$$a + X \xrightarrow{\text{Stage 1: Formation}} C^* \xrightarrow{\text{Stage 2: Decay}} Y + b$$

1. Stage 1 (Formation): The projectile $a$ enters the target nucleus $X$ and undergoes multiple collisions with nucleons, rapidly sharing its kinetic energy and binding energy among all $A$ nucleons. The system forms a highly excited quasi-equilibrium state called the compound nucleus $C^*$.
2. Stage 2 (Statistical Decay): The compound nucleus has a long lifetime compared to direct transit times: $$\tau_{\text{compound}} \sim 10^{-18} \text{ to } 10^{-15}\text{ s} \quad \left( 10^4 \text{ to } 10^7 \times \tau_{\text{direct}} \right)$$ During this long lifetime, the energy fluctuations statistically concentrate sufficient energy on a single nucleon (or cluster) to allow it to evaporate from the nucleus.

2. Bohr's Independence Hypothesis

Because the lifetime is so long, the compound nucleus loses all memory of its mode of formation, retaining only conserved macroscopic constants of motion: total energy $E^*$, total angular momentum $J$, and parity $\pi$.

Therefore, the cross section factorizes into the formation cross section $\sigma_{\text{form}}(a+X \to C^*)$ multiplied by the decay branching ratio $P_{\text{decay}}(C^* \to Y+b) = \frac{\Gamma_b}{\Gamma_{\text{total}}}$:

$$\sigma(a, b) = \sigma_{\text{form}}(a+X \to C^*) \times \frac{\Gamma_b}{\sum_c \Gamma_c}$$

3. Classic Experimental Test: Ghoshal's Experiment (1950)

S. N. Ghoshal brilliantly tested Bohr's independence hypothesis by forming the same compound nucleus ${}^{64}_{30}\text{Zn}^*$ via two completely different entrance channels at the same excitation energy ($E^* \approx 30\text{ MeV}$):

$$\text{Channel A: } p + {}^{63}_{29}\text{Cu} \to {}^{64}_{30}\text{Zn}^*$$ $$\text{Channel B: } \alpha + {}^{60}_{28}\text{Ni} \to {}^{64}_{30}\text{Zn}^*$$

He measured the cross sections for three different exit channels:

$${}^{64}\text{Zn}^* \to \begin{cases} {}^{63}\text{Zn} + n \\ {}^{62}\text{Cu} + p + n \\ {}^{62}\text{Zn} + 2n \end{cases}$$

Ghoshal found that the ratios of cross sections for the three exit channels were identical for both entrance reactions:

$$\sigma(p, n) : \sigma(p, pn) : \sigma(p, 2n) = \sigma(\alpha, n) : \sigma(\alpha, pn) : \sigma(\alpha, 2n)$$

This landmark result provided irrefutable confirmation of the compound nucleus independence hypothesis.

1. The Statistical Evaporation Model

At high excitation energies ($E^* > 10\text{ MeV}$), individual compound nucleus resonances overlap so densely that discrete quantum states blend into a continuum.

In the continuum regime, particle emission is modeled as statistical evaporation from a boiling droplet at nuclear temperature $T$. The kinetic energy spectrum of evaporated neutrons follows a Maxwellian-type distribution:

$$\frac{d N(E)}{d E} \propto E \exp\left( -\frac{E}{T} \right)$$

where the nuclear temperature $T$ (in MeV) is related to excitation energy $E^*$ by $E^* = a T^2$.

2. Bethe's Nuclear Level Density Formula

Hans Bethe (1936) treated the nucleus as an ensemble of non-interacting Fermi gas quasiparticles. Applying the grand canonical partition function:

$$\rho(E^*) = \frac{1}{\sqrt{48} E^*} \exp\left( 2\sqrt{a E^*} \right)$$

where the level density parameter $a$ is proportional to the single-particle density of states $g(\epsilon_F)$ at the Fermi surface:

$$a = \frac{\pi^2}{6} g(\epsilon_F) \approx \frac{A}{8} \text{ to } \frac{A}{10}\text{ MeV}^{-1}$$

Key physical properties:

• Exponential Explosion: The level density grows exponentially with $\sqrt{E^*}$. In heavy nuclei such as ${}^{238}\text{U}$ at neutron separation energy ($E^* \approx 6.5\text{ MeV}$), $a \approx 25\text{ MeV}^{-1}$, yielding $\rho \sim 10^6\text{ levels/MeV}$ (average level spacing $D = 1/\rho \sim 1\text{ eV}$).
• Shell Effects: Near magic numbers ($Z, N = 20, 28, 50, 82, 126$), the single-particle density $g(\epsilon_F)$ drops sharply, reducing $a$ and causing level densities to be orders of magnitude lower than in mid-shell deformed nuclei.

1. The Single-Level Breit-Wigner Formula

Gregory Breit and Eugene Wigner (1936) derived the dispersion formula for a nuclear reaction proceeding through an isolated compound nucleus resonance state of energy $E_0$, total spin $J$, and total width $\Gamma$:

$$\sigma(a, b) = \frac{\pi}{k^2} g_J \frac{\Gamma_a \Gamma_b}{(E - E_0)^2 + (\Gamma/2)^2}$$

where:

• $k = \frac{\sqrt{2\mu E}}{\hbar}$ is the incident relative wavevector.
• $g_J$ is the statistical spin factor: $$g_J = \frac{2J + 1}{(2s_a + 1)(2I_X + 1)}$$
• $\Gamma_a$ is the partial width for the entrance channel ($a + X$).
• $\Gamma_b$ is the partial width for the exit channel ($Y + b$).
• $\Gamma$ is the total resonance width: $\Gamma = \sum_c \Gamma_c = \Gamma_a + \Gamma_b + \Gamma_\gamma + \dots$
• By the Heisenberg uncertainty relation, the mean lifetime of the compound resonance is: $$\tau = \frac{\hbar}{\Gamma}$$

2. Resonant and Potential Scattering Interference

In the elastic channel ($b = a$), the total scattering amplitude is the coherent quantum sum of the hard-sphere potential scattering amplitude $A_{\text{pot}} = \frac{i}{2k}(1 - e^{2i\delta_{\text{pot}}})$ and the resonant Breit-Wigner amplitude $A_{\text{res}}$:

$$f(\theta) = f_{\text{pot}}(\theta) + f_{\text{res}}(\theta)$$

For an $s$-wave resonance with potential phase shift $\delta_{\text{pot}} = -k R$:

$$\sigma_{\text{el}}(E) = \frac{\pi}{k^2} g_J \left| e^{2i\delta_{\text{pot}}} - 1 + \frac{i\Gamma_n}{(E - E_0) + i\Gamma/2} \right|^2$$ $$\sigma_{\text{el}}(E) = \frac{4\pi}{k^2}\sin^2(k R) + \frac{\pi}{k^2} g_J \frac{\Gamma_n^2 + 2\Gamma_n\Gamma\sin^2(kR) - 4\Gamma_n(E - E_0)\sin(kR)\cos(kR)}{(E - E_0)^2 + (\Gamma/2)^2}$$

The cross-term produces asymmetric Fano-type interference:

• Just below the resonance energy ($E < E_0$), the resonant and potential amplitudes interfere destructively, causing the cross section to dip sharply below the potential baseline.
• Just above the resonance energy ($E > E_0$), the amplitudes interfere constructively, generating a steep peak.