Case Study 17.2 — The Optical Model: Treating the Nucleus as a Cloudy Crystal Ball

The Puzzle of Neutron Total Cross Sections

In the late 1940s, with the availability of monoenergetic neutron beams from reactors and the first particle accelerators, experimentalists began measuring total cross sections for neutrons on various nuclei as a function of energy. The data revealed a striking pattern: superimposed on a smooth, slowly oscillating background were sharp, narrow resonances at low energies (below about 1 MeV for heavy nuclei). The narrow resonances could be understood as individual excited states of the compound nucleus (Chapter 18). But the smooth background — showing broad oscillations with maxima and minima as a function of both energy and target mass number $A$ — was puzzling. It could not be explained by either the compound nucleus model or simple potential scattering.

The key observation was that the oscillations had a systematic dependence on $A^{1/3}$: the maxima occurred when the nuclear radius $R \propto A^{1/3}$ satisfied a condition for constructive interference of the nucleon wave inside the nuclear potential. This was the clue that the nucleus could be treated as a refracting and absorbing sphere — a "cloudy crystal ball."

The Feshbach-Porter-Weisskopf Model (1954)

Herman Feshbach, Charles Porter, and Victor Weisskopf at MIT proposed that the nucleus could be described by a complex potential well:

$$V(r) = \begin{cases} -(V_0 + iW_0), & r < R \\ 0, & r > R \end{cases}$$

for the simplest (square-well) version. The real depth $V_0 \approx 42\,\text{MeV}$ corresponds to the mean-field potential known from nuclear structure. The imaginary depth $W_0$ was the new ingredient: it accounts for the loss of flux from the elastic channel into compound-nucleus formation and other inelastic processes.

The physics is transparent in the optical analogy. A plane wave (the neutron beam) encounters a sphere with a complex index of refraction. Part of the wave is reflected (elastic scattering), part is refracted and transmitted (contributing to the forward scattering amplitude), and part is absorbed (removed from the beam by reactions). The interference between reflected and refracted-transmitted waves produces the characteristic oscillating pattern in the total cross section.

Size Resonances: The Signature of Nuclear Transparency

Inside the nucleus, the neutron's kinetic energy is increased by $V_0$, so its internal wavelength is shorter than the external wavelength:

$$K = \frac{1}{\hbar}\sqrt{2m(E + V_0)}, \qquad k = \frac{1}{\hbar}\sqrt{2mE}$$

The ratio $K/k = \sqrt{(E + V_0)/E}$ is the nuclear "index of refraction." When the internal wavelength satisfies:

$$2KR \approx (n + 1/2)\pi, \qquad n = 0, 1, 2, \ldots$$

constructive interference inside the potential well creates a shape resonance (also called a size resonance or single-particle resonance). At these energies, the neutron wave function has a large amplitude inside the nucleus, increasing both the scattering and absorption cross sections.

For a nucleus with $A = 120$ ($R \approx 6.16\,\text{fm}$) and $V_0 = 42\,\text{MeV}$, the condition $2KR = n\pi$ gives size resonances at approximately:

These broad resonances ($\Gamma \sim 1$--$5\,\text{MeV}$) were observed experimentally in the 1950s and provided the first direct evidence that nucleons move in a mean-field potential inside the nucleus — the same potential responsible for the nuclear shell structure (Chapter 6).

The Transition to the Woods-Saxon Optical Potential

The square-well potential, while pedagogically valuable, is too crude for quantitative comparison with data. The sharp edge at $r = R$ produces unphysical oscillations in the calculated cross sections. The modern optical model replaces the square well with a Woods-Saxon form factor:

$$f(r) = \frac{1}{1 + \exp\!\left(\frac{r - R}{a}\right)}$$

with diffuseness $a \approx 0.6\,\text{fm}$, smoothly transitioning from the interior ($r \ll R$) to the exterior ($r \gg R$). Additional refinements include:

1. Surface-peaked imaginary term: At low energies ($E < 50\,\text{MeV}$), absorption occurs preferentially at the nuclear surface (due to Pauli blocking in the interior), described by a term proportional to $df/dr$.

2. Spin-orbit term: The spin-orbit interaction $V_{SO}\,\boldsymbol{\ell}\cdot\boldsymbol{s}\,(1/r)\,df/dr$ is essential for reproducing the polarization (spin asymmetry) of scattered nucleons.

3. Energy dependence: The real potential depth decreases approximately linearly with energy: $V_0(E) \approx V_0(0) - \alpha E$, with $\alpha \approx 0.3$--$0.4$. This reflects the nonlocality of the nuclear mean field.

How the Parameters Are Determined

The optical model parameters are determined by fitting calculated observables to experimental data. The fitting procedure — which can be done for a specific nucleus ("local" fit) or for a broad range of nuclei ("global" fit) — involves:

1. Start with initial guesses based on systematic trends (e.g., $V_0 \approx 50\,\text{MeV}$, $R = 1.25A^{1/3}\,\text{fm}$).
2. Solve the radial Schrodinger equation for each partial wave $l$ to obtain the S-matrix elements $S_l$.
3. Compute the elastic angular distribution from the partial-wave sum and compare to measured data.
4. Iterate using a least-squares minimization algorithm ($\chi^2$ fitting) to optimize the parameters.

The different parameters are constrained by different observables: - $V_0$ (real depth): The overall magnitude and the positions of diffraction minima - $R_V$, $a_V$ (real geometry): The angular positions and shape of the diffraction pattern - $W_D$ (imaginary surface): The depth of the diffraction minima (more absorption = shallower minima) - $V_{SO}$ (spin-orbit): The analyzing power (left-right asymmetry with polarized beams)

A local fit to a single nucleus at a single energy typically achieves $\chi^2/N \sim 1$--$5$ (where $N$ is the number of data points), indicating agreement within the experimental uncertainties. Global fits across hundreds of nuclei and energies sacrifice some precision but provide predictive power for unmeasured systems.

Global Optical Model Parametrizations

A major practical advance was the development of global optical potentials — parametrizations that describe nucleon-nucleus scattering for all stable nuclei across a broad energy range with a single set of parameters (whose values depend smoothly on $A$, $Z$, and $E$).

The most widely used global potential is the Koning-Delaroche (KD) parametrization (2003), fitted to elastic scattering angular distributions and total cross section data for protons and neutrons on nuclei with $24 \leq A \leq 209$ at energies from 1 keV to 200 MeV. The KD potential has approximately 50 parameters, but these are all determined by the global fit — no free parameters remain when applied to a specific nucleus.

Typical KD predictions: - Elastic angular distributions: agreement with data to $\sim 10$--$20\%$ across all angles - Total cross sections: agreement to $\sim 5\%$ - Reaction cross sections: agreement to $\sim 10$--$15\%$ - Analyzing powers (spin observables): qualitative agreement, quantitative accuracy of $\sim 20$--$30\%$

For applications in nuclear technology (reactor physics, shielding), medical physics (proton therapy), and nuclear astrophysics (neutron capture rates), the optical model — often through global parametrizations — provides the essential input for reaction calculations.

Experimental Evidence: Elastic Scattering Angular Distributions

The most stringent test of the optical model is the elastic scattering angular distribution $d\sigma/d\Omega(\theta)$. For medium-energy protons ($E \sim 30$--$60\,\text{MeV}$) on medium and heavy nuclei, the data show a characteristic pattern:

1. Forward peak: Dominated by Coulomb scattering at small angles, transitioning to nuclear scattering at the Coulomb-nuclear interference angle.

2. Diffraction oscillations: Regular minima and maxima, resembling Fraunhofer diffraction from an opaque disk. The angular spacing is $\Delta\theta \approx \pi/(kR)$.

3. Exponential falloff: At large angles, the cross section falls approximately exponentially, reflecting the penetration of the nuclear surface (the diffuseness $a$).

4. Ratio to Rutherford: The ratio $\sigma/\sigma_{\text{Ruth}}$ starts at unity (small angles), oscillates through the Coulomb-nuclear interference region, drops well below unity (nuclear absorption), and shows the diffraction pattern at large angles.

These features are reproduced by the optical model with remarkable fidelity. The positions of diffraction minima constrain $R$; the depth of the minima constrains $W$ (more absorption fills in the minima); and the large-angle falloff constrains $a$.

Limitations and Extensions

The standard optical model has important limitations:

• No internal excitations: The target is treated as inert. Inelastic scattering to specific excited states requires coupled-channels calculations that extend the optical model by including explicit coupling to collective excitations (rotations, vibrations).

• No exchange effects: For proton scattering, the projectile is identical to the target nucleons. Proper treatment requires antisymmetrization (knockout exchange), which modifies the effective potential.

• Unstable nuclei: Global parametrizations like KD are fitted to stable nuclei. For exotic nuclei far from stability — now accessible at radioactive beam facilities — the standard parametrizations may fail, particularly for nuclei with unusual density distributions (halo nuclei, neutron skins).

Despite these limitations, the optical model remains the workhorse of nuclear reaction theory. Every distorted-wave Born approximation (DWBA) calculation for transfer reactions (Chapter 19), every coupled-channels calculation for inelastic scattering, and every Hauser-Feshbach statistical model calculation for compound-nucleus decay (Chapter 18) starts with optical model wave functions.

A Quantitative Example: 30 MeV Protons on Calcium-40

To make the optical model concrete, consider 30 MeV proton elastic scattering from ${}^{40}\text{Ca}$, one of the most thoroughly studied systems. Representative Koning-Delaroche parameters at this energy:

The corresponding radii are $R_V = 1.24 \times 40^{1/3} = 4.24\,\text{fm}$, $R_D = 4.31\,\text{fm}$, and $R_{SO} = 3.76\,\text{fm}$. The real potential at $r = 0$ is $-47.2\,\text{MeV}$; the imaginary potential peaks at $r \approx R_D$ with a maximum value of about $-2.1\,\text{MeV}$ (the surface-peaked derivative form).

The internal wave number is $K = \sqrt{2m_p(E + V_0)}/\hbar = \sqrt{2 \times 938.3 \times 77.2}/(\hbar c) \approx 1.93\,\text{fm}^{-1}$, giving an internal wavelength $\lambda_{\text{int}} = 2\pi/K \approx 3.3\,\text{fm}$. The product $KR \approx 8.2$ tells us that several internal wavelengths fit inside the nucleus, producing the observed diffraction pattern.

The predicted angular distribution shows: a Coulomb peak at forward angles ($\theta < 10°$), Coulomb-nuclear interference oscillations near $\theta \sim 15°$--$25°$, and a Fraunhofer diffraction pattern with minima near $\theta \sim 30°$, $55°$, and $80°$. The measured data from experiments at Indiana University (1980s) and iThemba LABS (South Africa, 2000s) agree with the Koning-Delaroche prediction to within 10--20% across the entire angular range — a remarkable achievement for a potential with no free parameters adjusted to this specific reaction.

The Optical Model in Nuclear Technology

The optical model is not merely an academic exercise — it is a critical input for practical applications:

1. Reactor physics: Neutron transport calculations require cross sections for elastic scattering, inelastic scattering, and absorption for all materials in the reactor. The optical model provides these cross sections for energies above the resolved resonance region, typically $E > 100\,\text{keV}$. The evaluated nuclear data files (ENDF/B-VIII.0, JEFF-3.3, JENDL-5) rely heavily on optical model calculations.

2. Radiation shielding: Designing shielding for reactors, accelerators, and space missions requires knowing the angular distribution of scattered neutrons (not just the total cross section). The optical model's prediction of the elastic angular distribution — forward-peaked with a diffraction tail — directly determines the shielding effectiveness.

3. Medical physics: Proton therapy for cancer treatment requires accurate cross sections for proton-nucleus scattering at 60--250 MeV. The optical model predicts the nuclear scattering contribution that broadens the proton beam laterally and removes protons from the primary beam. The Koning-Delaroche potential is used directly in Monte Carlo treatment planning codes like GEANT4 and FLUKA.

4. Nuclear astrophysics: Neutron capture rates in the r-process and s-process of nucleosynthesis require optical model cross sections for thousands of unstable nuclei where no experimental data exist. Global potentials provide the only available predictions.

Discussion Questions

1. The optical model's imaginary potential is surface-peaked at low energies but volume-dominated at high energies. Explain this transition in terms of the Pauli exclusion principle and available phase space for nucleon-nucleon collisions inside the nucleus.

2. The real potential depth decreases with energy ($V_0(E) \approx 50 - 0.3E$ MeV). This energy dependence reflects the nonlocality of the nuclear mean field. Explain qualitatively why a nonlocal potential (which depends on both $\mathbf{r}$ and $\mathbf{r}'$) is equivalent to an energy-dependent local potential.

3. For proton scattering, there is a Coulomb-nuclear interference region where the Rutherford and nuclear scattering amplitudes are comparable. At what angle does this transition occur (approximately) for 40 MeV protons on ${}^{90}\text{Zr}$? (Hint: equate the Rutherford and nuclear amplitudes using the black-disk model.)

4. The optical model for neutrons has no Coulomb potential. How does this simplify the analysis of neutron scattering data compared to proton data? What experimental challenges does neutron scattering present that proton scattering does not?

5. The optical model has about 10 adjustable parameters for a given nucleus and energy. Some physicists have criticized it as "epicyclic" — having enough parameters to fit anything. How would you respond to this criticism? (Hint: consider the global parametrizations, the dispersion relation constraint, and the connection to nuclear structure.)

6. At FRIB, one can measure elastic scattering of short-lived isotopes ($t_{1/2} \sim 1\,\text{ms}$) by bombarding a hydrogen target in inverse kinematics. Discuss the challenges of extracting optical model parameters from such measurements when: (a) the beam intensity may be only $\sim 100$ particles per second, (b) the angular range is limited by kinematic focusing, and (c) no previous scattering data exist for the isotope.