⚛️ nuclear theory

Non-locality function of microscopic optical potentials from Skyrme-based nuclear structure models at low energies

This study demonstrates that the non-locality of the absorptive optical potential in neutron elastic scattering cannot be described by a single universal constant, but instead requires a nucleus-, energy-, and radially-dependent function derived from self-consistent Skyrme-based particle-vibration coupling calculations.

Original authors: Do Quang Tam, N. Hoang Tung, N. Hoang Phuc, T. V. Nhan Hao

Published 2026-09-11

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CC BY 4.0

Original authors: Do Quang Tam, N. Hoang Tung, N. Hoang Phuc, T. V. Nhan Hao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of nuclear physics, scientists often rely on a tool called the optical potential to understand how particles like neutrons bounce off atomic nuclei. Think of an atomic nucleus not as a solid ball, but as a cloud of protons and neutrons that can absorb some of the energy of an incoming particle while scattering the rest. To predict exactly how this happens, physicists use a mathematical map that describes the forces at play. For decades, a standard assumption has guided these maps: the idea that the "fuzziness" or non-locality of this force is a fixed, unchanging number, much like a universal constant. This assumption, known as the Perey-Buck value, suggests that the range over which a nucleus can "feel" a passing neutron is the same everywhere, regardless of whether the neutron hits the center of the atom or its edge, or whether the neutron is moving slowly or quickly.

However, this long-held belief treats the complex, living interior of an atom as if it were a static, uniform object. In reality, atomic nuclei are dynamic systems where particles interact in intricate ways, influenced by the specific shape of the nucleus and the speed of the incoming particle. If the standard map is too simple, it risks missing crucial details about how nuclear reactions occur. This is particularly important for modern research into exotic, unstable atoms, where understanding these subtle interactions is key to unlocking the secrets of matter. A new study challenges the idea of a single, universal rule for this non-locality, proposing instead that the behavior of the nuclear force is far more nuanced and dependent on the specific conditions of the collision.

A team of researchers set out to test this assumption by building a highly detailed, microscopic model of how neutrons scatter off four different atomic nuclei: oxygen-16, calcium-40, calcium-48, and lead-208. They focused on low-energy collisions, where the incoming neutrons move at speeds up to 30 million electron volts. Rather than relying on a pre-set, fixed number for the non-locality of the force, the team used a sophisticated computer framework that simulates the nucleus from the ground up. They started with a basic description of the nucleus and then added layers of complexity, accounting for how the nucleus vibrates and how the incoming neutron couples with these vibrations. This approach allowed them to calculate the "absorptive" part of the force—the part that represents the nucleus soaking up energy from the neutron—without using any arbitrary adjustments or guesswork.

The researchers examined the results at two distinct locations within each nucleus: the deep interior, or volume, and the outer edge, or surface. They measured how the strength of the absorptive force changed as the distance between the interacting points varied. When they analyzed the data, they found that the force did not behave according to a single, universal rule. Instead, the "range" of the non-locality, which describes how far the influence of the interaction extends, changed depending on where the neutron was, how fast it was moving, and which nucleus it was hitting. For the lightest nucleus they studied, oxygen-16, the non-locality at the surface was significantly different from that in the center. As the energy of the neutron increased, the non-locality at the surface shrank, while the value in the center remained relatively steady.

The findings became even more striking when looking at the heavier nuclei, particularly lead-208. In this heavy atom, the researchers discovered that at low energies, the non-locality in the deep interior was actually stronger and more extended than at the surface. This was a reversal of the pattern seen in lighter atoms and contradicted previous models that suggested the interior had no significant absorptive strength at all. As the energy of the neutron increased, this relationship flipped, and the surface became the dominant region for this extended interaction. The study showed that the value of this non-locality range was not a fixed constant but a variable that shifted between approximately 1.01 and 2.03 femtometers (a unit of length used for atomic scales). This variation was consistent across all the nuclei they tested, proving that the force is sensitive to the specific mass of the target and the energy of the collision.

These results directly challenge the decades-old practice of using a single, fixed number to describe the non-locality of nuclear forces. The study demonstrates that the widely accepted value of 0.85 femtometers, which was derived from fitting data on lead-208 at just two specific energies, cannot capture the full complexity of the interaction. That single number fails to account for the fact that the non-locality changes as you move from the center of the nucleus to its edge, and as the energy of the incoming particle changes. The researchers found that the non-locality is not a static feature but a dynamic one, shaped by the collective vibrations of the nucleus and the specific quantum states involved in the collision.

The team confirmed that their detailed calculations could accurately reproduce experimental data for how neutrons scatter, validating the reliability of their model. By solving the equations exactly, without simplifying the complex interactions into a local approximation, they showed that the microscopic details of the nucleus matter. The study concludes that to achieve precision in nuclear reaction calculations, especially for unstable or exotic nuclei, scientists must move away from universal constants. Instead, they need descriptions of the nuclear force that are specific to the nucleus in question, the energy of the collision, and the exact location within the nucleus where the interaction occurs. This shift from a one-size-fits-all approach to a tailored, detailed understanding represents a necessary step forward for accurately modeling the behavior of matter at the smallest scales.

Technical Summary: Non-locality Function of Microscopic Optical Potentials from Skyrme-based Nuclear Structure Models at Low Energies

Problem Statement
In nuclear reaction theory, the non-locality parameter $\beta$ is central to modeling nucleon-nucleus interactions within the optical potential framework. Historically, the phenomenological Perey-Buck potential has utilized a single, universal, energy-independent constant ( $\beta = 0.85$ fm) to describe the non-locality of the imaginary part of the potential via a Gaussian kernel. However, recent studies suggest that non-locality effects cannot be fully captured by this simplified correction factor, particularly for modern applications involving exotic nuclei and inverse kinematics. While previous microscopic calculations using Gogny interactions or limited Skyrme-based models have provided insights, they have often been restricted to medium-mass nuclei or specific surface regions, leaving a gap in understanding the radial, energy, and target-mass dependence of non-locality across a broad range of nuclei, including the nuclear interior.

Methodology
The authors investigate the non-locality of the imaginary part of the microscopic optical potential ( $W$ ) for neutron elastic scattering on four target nuclei: $^{16}\text{O}$ , $^{40}\text{Ca}$ , $^{48}\text{Ca}$ , and $^{208}\text{Pb}$ , at incident energies up to 30 MeV.

The theoretical framework is built upon a fully self-consistent particle-vibration coupling (PVC) approach using the Skyrme-Hartree-Fock (SHF) plus Random Phase Approximation (RPA) formalism with the SLy5 interaction. Key methodological features include:

Self-Consistency: The mean field, RPA excited states, and the residual interaction are all derived from the same effective SLy5 force, without ad hoc adjustable parameters.
Model Space: The calculations include all natural-parity RPA excited states with multipolarities $L = 0^+, 1^-, 2^+, 3^-, 4^+, 5^-$ and excitation energies below 50 MeV.
Potential Definition: The optical potential is defined as $V_{\text{opt}} = V_{\text{HF}} + \Delta\Sigma$ , where the non-locality arises exclusively from the dynamical part of the nucleon self-energy, $\Delta\Sigma$ . The absorptive potential is defined as $W(R, s, E) \equiv \text{Im}\Delta\Sigma$ , where $R$ is the center-of-mass coordinate and $s$ is the non-locality coordinate.
Extraction of $\beta$ : The dependence of $W$ on the non-locality coordinate $s$ at fixed radial positions $R$ and energies $E$ is fitted to a Gaussian form: $W(R, s, E) = W(R, 0, E) \exp[-s^2/\beta^2(R, E)]$ . This allows the extraction of a radius- and energy-dependent non-locality function $\beta(R, E)$ at representative volume ( $R_v$ ) and surface ( $R_s$ ) radii.
Validation: The quality of the Gaussian fit is assessed using the coefficient of determination ( $R^2$ ), and the underlying potential's validity is confirmed by comparing calculated differential cross-sections with experimental data for $^{208}\text{Pb}$ at 14.5 MeV using the exact non-local Schrödinger equation solver (DWBA98).

Key Results
The study reveals that the non-locality parameter $\beta$ is not a universal constant but a function $\beta(R, E)$ with values ranging from 1.01 to 2.03 fm. The results exhibit distinct systematic trends:

Radial Dependence: There is a clear distinction between volume ( $\beta_v$ ) and surface ( $\beta_s$ ) non-locality.
- For light nuclei ( $^{16}\text{O}$ ), a pronounced separation exists, with $\beta_s$ significantly larger than $\beta_v$ across the energy range.
- For medium-mass nuclei ( $^{40}\text{Ca}$ , $^{48}\text{Ca}$ ), the difference $\Delta\beta = \beta_s - \beta_v$ is smaller, with $^{40}\text{Ca}$ showing near-degeneracy at higher energies.
- For $^{48}\text{Ca}$ , a crossover occurs where $\beta_v$ exceeds $\beta_s$ at very low energies ( $E < 7$ MeV), a feature attributed to neutron excess and isovector modes.
- For heavy nuclei ( $^{208}\text{Pb}$ ), $\beta_v$ is significantly larger than $\beta_s$ at low energies, with a crossover occurring around 22 MeV.
Energy Dependence: Generally, both $\beta_v$ and $\beta_s$ decrease as incident energy increases. This is interpreted as the diminishing relative importance of low-lying collective excitations at higher energies, where the coherent coupling to vibrations (the source of long-range non-locality) is diluted by the opening of more particle-hole configurations. The variation is most dramatic for $\beta_v$ in $^{208}\text{Pb}$ , decreasing by approximately 32.5% from its peak.
Comparison with Previous Work:
- The results differ from the seminal work of Bernard and Van Giai (Ref. [34]), who found no absorptive strength in the nuclear interior of $^{208}\text{Pb}$ and could not extract a volume $\beta_v$ . The current study, utilizing the SLy5 interaction and a complete natural-parity model space, finds significant interior absorption, enabling the first extraction of $\beta_v$ for $^{208}\text{Pb$ in a Skyrme-based PVC framework.
- The extracted surface values ( $\beta_s$ ) are generally larger than those in Ref. [34] but follow a similar decreasing trend with energy.
- The Gaussian approximation for $W(R, s)$ is found to be highly accurate ( $R^2 \geq 0.97$ ), particularly in the surface region, validating the functional form used for extraction.

Significance and Claims
The paper claims that the traditional Perey-Buck assumption of a single universal constant ( $\beta = 0.85$ fm) is insufficient to capture the complex radial, energy, and target-mass dependence of the absorptive optical potential. The study demonstrates that:

A nucleus-dependent, energy-dependent, and radially-resolved description of the absorptive potential $W$ is required for precision nuclear reaction calculations.
The non-locality structure is governed by different mechanisms in the nuclear interior (dominated by mean-field correlations and Pauli blocking) versus the surface (dominated by particle-vibration coupling).
The fully self-consistent Skyrme-based approach successfully reveals non-locality features in the nuclear interior that were previously inaccessible in similar models, particularly for heavy nuclei like $^{208}\text{Pb}$ .

The authors conclude that while their approach is constrained by the use of zero-range Skyrme interactions, the extracted $\beta(R, E)$ functions provide meaningful insights and a necessary step toward developing next-generation optical potentials applicable across the nuclear chart.

Technical Summary: Non-locality Function of Microscopic Optical Potentials from Skyrme-based Nuclear Structure Models at Low Energies

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