Inverse-Scattering Framework in Nuclear Physics

• Inverse-scattering framework is a set of mathematical and computational methods for deducing interaction potentials from scattered wavefield data in nuclear physics.
• It employs iterative inversion techniques and basis function expansions to match calculated and target S-matrix elements with minimal model assumptions.
• Its applications include extracting nonlocal interactions, parity-dependent forces, and dynamic polarization effects from empirical scattering observables.

The inverse-scattering framework refers to the set of mathematical, computational, and physical methodologies for deducing the properties of an interaction potential or scatterer from measurements of scattered wavefields, typically encoded in the scattering matrix (S-matrix), phase shifts, or related observables. In nuclear physics, this framework is employed to reconstruct the local interaction potential between two colliding nuclei or nucleons starting from the experimentally determined or theoretically computed S-matrix elements, instead of the traditional "direct" approach which solves the scattering problem for a given potential. This reversal enables extraction of model-independent information about nuclear forces, nonlocality, parity dependence, energy-dependence, and dynamical effects from data.

1. Foundational Elements of the Inverse-Scattering Problem

The direct scattering problem in nuclear physics starts with a local interaction potential $V(r)$ between two particles and computes the S-matrix (scattering matrix) by solving the Schrödinger equation: $$\left( -\frac{\hbar^2}{2m} \frac{d^2}{dr^2} + V(r) + \frac{l(l+1)\hbar^2}{2mr^2} \right) u_l(r) = E u_l(r)$$ The S-matrix $S_l(k)$ encodes all elastic scattering observables for each orbital angular momentum $l$ and is related to the phase shift via: $S_l(k) = \exp(2i\,\delta_l)$ where $k$ is the center-of-mass wave number. The asymptotic form of the radial wave function is: $$u_l(r) \to I_l(r) - S_l(k) O_l(r) \quad \text{as } r\to\infty$$ In the inverse problem, one aims to reconstruct $V(r)$ so that the solution of the Schrödinger equation reproduces a given target S-matrix or set of phase shifts, ideally matching empirical data.

2. Role and Manipulation of the S-matrix

The S-matrix is central to both forward and inverse scattering formulations. For local and l-independent interactions, the S-matrix is diagonal, and the problem reduces to determining $V(r)$ such that, for each partial wave,

$$S^{\mathrm{calc}}_l(k; V) \approx S^{\mathrm{target}}_l(k)$$

For coupled channels (e.g., when spin is included), the S-matrix is non-diagonal, and the inverse problem generalizes to finding a set of local (possibly matrix-valued) potentials.

The inverse mapping depends critically on the functional sensitivity of $S_l(k)$ to changes in $V(r)$. For small variations, to leading order: $$\Delta S_l = \frac{i}{2k} \int_0^\infty \left[u_l(r)\right]^2 \Delta V(r)\, dr$$ This expression underlies iterative schemes for potential reconstruction.

3. Inversion Methodologies

Inverse-scattering techniques can be classified according to the type of S-matrix data available:

• Fixed-l inversion: S-matrix is known for all energies for a fixed $l$. The classic Gel'fand–Levitan–Marchenko equations are applicable.
• Fixed-E inversion: S-matrix for all $l$ at a fixed energy is available; suitable when many partial waves contribute.
• Mixed-case or energy-dependent inversion: Combines data at multiple energies, especially when only a few $l$ are accessible at low energy.
• Iterative-Perturbative (IP) inversion: The most widely applied modern scheme. Starting from a reference potential $V_{\mathrm{SR}}(r)$, the algorithm iteratively updates the potential as:

$$V_{\text{new}}(r) = V_{\text{curr}}(r) + \sum_{i} d_i v_i(r)$$

where $v_i(r)$ are a set of basis functions (e.g., Gaussians, Bessel functions, splines) and $d_i$ are amplitudes solving an over-determined system,

$\mathbf{A} = \mathbf{M} \mathbf{X}$

with $\mathbf{A}$ containing mismatches between calculated and target S-matrix changes, $\mathbf{M}$ encoding overlap integrals $M_{ij} = \int_0^\infty [u_l(r)]^2 v_j(r) dr$, and $\mathbf{X}$ the unknown amplitudes. Techniques like singular value decomposition (SVD) are employed to ensure robustness. The process is iterated until $S_l(k; V_{\mathrm{new}})$ is acceptably close to $S_l^{\mathrm{target}}(k)$.

4. Physical Significance and Applications

The inverse-scattering framework provides a model-independent route to extracting nuclear potentials directly from scattering observables, circumventing the need for assumed parametric forms (such as Woods–Saxon). Applications and physical insights include:

• Model-agnostic extraction: No prior assumption on potential shape, facilitating unbiased tests of nuclear interaction models.
• Local equivalence: Enables construction of local potentials equivalent in S-matrix to genuinely non-local forces arising from exchange or antisymmetrization.
• Parity dependence: The existence of parity-dependent potentials, especially in light-ion systems, is revealed and quantified by inversion of empirical S-matrices.
• Dynamic Polarization Potentials (DPPs): In cases including inelastic channels or projectile breakup, DPPs—local, energy-dependent modifications to the bare potential—can be reconstructed from S-matrix analyses using coupled-channel calculations.
• Two-step nuclear elastic scattering phenomenology: Empirical angular distributions are first fit to extract S-matrix elements, then inversion yields the potential profile, illuminating phenomena such as surface transparency or reaction-induced polarization.
• Coupled-channel inversion: For projectiles with spin, the formalism accommodates spin-orbit and tensor couplings, yielding a richer set of potential functions.

5. Central Equations in the Framework

Below is a summary table of key analytical relationships used in the inverse-scattering framework:

Purpose	Formula	Notes
Asymptotic radial behavior	$u_l(r) \to I_l(r) - S_l(k) O_l(r)$	$I_l, O_l$: incoming/outgoing (Coulomb/Bessel) fns
S-matrix from phase shift	$S_l(k) = \exp(2i\delta_l)$	$\delta_l$: empirical or theoretical phase shift
S-matrix linear response	$$\Delta S_l = \frac{i}{2k} \int [u_l(r)]^2 \Delta V(r)\, dr$$	Foundation of IP inversion
Potential update	$$V_{\text{new}}(r) = V_{\text{curr}}(r) + \sum_{i} d_i v_i(r)$$	$v_i(r)$: basis functions, $d_i$: amplitudes
Matrix inversion system	$\mathbf{A} = \mathbf{M} \mathbf{X}$	$\mathbf{A}$: S-matrix mismatch vector

These equations translate the comparison between target and calculated S-matrix elements into updates in the potential, facilitating iterative convergence.

6. Limitations and Practical Considerations

The power of the framework is balanced by practical constraints:

• Completeness of input: For a unique reconstruction, S-matrix elements must be known over a sufficient range of $l$ and energies.
• Uncertainties and noise: Input phase shifts derived from experimental data can carry significant uncertainty, potentially destabilizing the inversion if not suitably regularized.
• Non-locality and l-dependence: While local, l-independent potentials are the target, real nuclear forces are non-local and may exhibit explicit l-dependence, meaning inverted potentials are only "local equivalent" to the full interaction.
• Resource demands: The IP inversion and SVD-based solution of large, overdetermined systems can be computationally intensive, especially for coupled-channel or high-dimensional problems.

7. Impact and Theoretical Ramifications

By providing an independent route to nuclear potentials, the inverse scattering framework has led to verification and refinement of existing optical model analyses, discovery of parity dependence and non-local effects (such as the Perey effect), and the construction of dynamic polarization potentials fundamental to understanding nuclear reaction mechanisms. Insights from inversion have directly influenced theoretical development in the structure and reaction subfields, including the modeling of nucleon-nucleus and light-ion systems, and the explicit inclusion of channel coupling and collective excitation effects. The framework exemplifies the interplay between empirical data, mathematical inversion, and physical interpretation central to modern nuclear many-body theory.

In summary, the inverse-scattering framework in nuclear physics constitutes a mathematically rigorous, physically grounded scheme to reconstruct interaction potentials directly from scattering observables, enabling detailed, model-independent insights into nuclear structure and dynamics and serving as both an analytical tool and a benchmark for theory and phenomenology (Mackintosh, 2012).