Microscopic mechanism of the Fayans pairing for the enhancement of charge radii

Abstract: The Fayans energy density functional (EDF), and in particular its pairing sector, have been claimed to be able to reproduce the experimental data of charge radii in many instances. A particularly intriguing case is that of the $ \mathrm{Ca} $ isotopes between $ A = 40 $ and $ 48 $, where charge radii exhibit a "bell shape". In our work, we examine the microscopic origin of this behaviour. We prepare in total $ 25 $ paramerizations of the Fayans-like pairing interaction, that are equivalent in fulfilling the same criteria for the reproduction of empirical pairing gaps. We find that both the density and the density-gradient dependence of the pairing interaction are important to reproduce the well-known enhancement of charge radii in the open-shell nuclei, leading to the "bell shape" behaviour of $ \mathrm{Ca} $ isotopes. In particular, this originates from the repulsive nature of the rearrangement potential, and cannot simply be mocked up by a refit of the pairing strength. At the same time, we notice some drawbacks of the Fayans standard EDFs, that may call for investigating a more general form of it.

• The paper presents a microscopic analysis of Fayans pairing that explains enhanced nuclear charge radii trends, particularly the bell shape observed in Ca isotopes.
• It systematically explores 25 pairing parametrizations within the HFB framework to accurately reproduce empirical pairing gaps and charge radii data across isotopic chains.
• The study confirms that density-gradient and rearrangement effects are critical for charge radius predictions, guiding future refinements in nuclear EDF models.

Microscopic Mechanism of the Fayans Pairing for the Enhancement of Nuclear Charge Radii

Background and Motivation

Charge radii represent a fundamental benchmark for nuclear structure models, directly reflecting shell structure, binding, and pairing effects. A central challenge has been the accurate reproduction of anomalous isotopic trends, particularly the "bell shape" in charge radii observed in Ca isotopes between $^{40}$Ca and $^{48}$Ca. Conventional mean-field approaches using standard EDFs, such as Skyrme and Gogny functionals, systematically fail to capture the subtle invariance and enhancement in charge radii, often overestimating differences between closed-shell nuclei. By contrast, the Fayans EDF, distinguished by its sophisticated density and gradient dependencies, has exhibited unique success in modeling these features.

Theoretical Framework

Within nuclear DFT, the Fayans functional is constructed as a sum of kinetic, particle-hole (p-h), and particle-particle (p-p) contributions. The p-h channel incorporates volume and surface terms fitted to ab-initio EOS and empirical radii, as well as spin-orbit and Coulomb terms, with notable flexibility in density dependence. The focus of this study is the p-p (pairing) channel, specifically in the zero-range FaNDF0 parametrization, which uniquely includes density, surface, and gradient terms characterized by parameters $f$ (overall strength), $h_o$ (density dependence), and $h_p$ (gradient dependence).

The functional derivative of Ep-p yields a pairing mean-field and also a rearrangement term affecting the mean-field potential, with the latter being nonzero for non-volume pairing (i.e., $h_o$, $h_p \neq 0$). The rearrangement term is central to the action of the Fayans pairing, driving modifications in charge distributions.

Parametrization and Numerical Approach

A systematic exploration of 25 Fayans-like pairing parametrizations was carried out, varying $h_o$ and $h_p$ across a grid. Each set was constrained to reproduce empirical neutron pairing gaps, particularly in $^{44}$Ca, using the HFB formalism with a robust energy cutoff and spherical symmetry. This allowed the identification of parameter sets simultaneously satisfying constraints for Ca, Sn, and Pb isotopes and preserving doubly-magic character in $^{48}$0Ca, $^{48}$1Sn, and $^{48}$2Pb.

Results and Analysis

Pairing Gaps

Selected sets provided accurate pairing gaps for key nuclei, reflecting a strong dependence of the required strength $^{48}$3 on $^{48}$4 and $^{48}$5. Volume-dominated pairings ($^{48}$6) underestimated charge radii differences, while gradient-rich pairings ($^{48}$7) required increased strength and yielded non-negligible gaps even in closed shells. Notably, higher-order pairing parametrizations compromised gap stability in heavier isotopes.

Charge Radii Trends

Critically, only the Fayans EDFs with substantial gradient dependence ($^{48}$8) reproduced the experimental bell shape in Ca charge radii, yielding near-equal radii in $^{48}$9Ca and $f$0Ca and a pronounced enhancement in $f$1Ca, matching measured values within an order of magnitude better than most EDFs. These parameter sets, however, tended to overshoot radii trends in Sn and Pb isotopic chains, indicating trade-offs in global performance.

Role of the Rearrangement Potential

Detailed calculations with and without the rearrangement potential affirmed its pivotal role: the repulsive contribution pushes proton density outwards, enhancing charge radii. Systematic refitting of pairing strength in absence of the rearrangement potential could not mimic its effect quantitatively. The rearrangement term modifies single-particle energies and increases orbital radii, but occupation probabilities are largely unchanged. Both $f$2 and $f$3 are critical in shaping the spatial profile of the potential and its magnitude.

Comparative Assessment

Results extended to other EDFs (Fy(Ar), SkM*) validate the mechanism: while different pairings may suit different EDF frameworks, the qualitative action of the gradient and rearrangement terms remains robust. Nevertheless, no single pairing form within FaNDF0 could reconcile all isotopic chains simultaneously, highlighting possible deficiencies in the p-h functional, and motivating advanced optimization or generalized pairing forms.

Implications and Future Directions

The results underscore the necessity of incorporating density-gradient-dependent pairing interactions in EDFs to capture subtle isotopic trends in charge radii, especially in Ca isotopes. The rearrangement potential represents a unique and non-trivial mechanism, not reducible to simple strength refitting, driving spatial redistribution of nuclear charge.

This analysis suggests that future advances will require coordinated optimization of both p-h and p-p sectors, perhaps leveraging Bayesian methods for parameter fitting across multiple isotopic chains. Extensions to generalized isovector and finite-range pairing interactions are warranted, as indicated by ongoing refinements in the literature (2606.21491). The improved theoretical fidelity directly impacts interpretations of empirical charge distributions, magic numbers, and structural evolution across the nuclear chart.

Conclusion

This study elucidates the microscopic origin and mechanism of enhanced charge radii in Ca isotopes within the Fayans EDF framework, highlighting the essential role of density and gradient dependencies in the pairing functional and the indispensable rearrangement potential. While the Fayans pairing succeeds in uniquely reproducing the bell shape in Ca radii, limitations in global parameterization point toward the need for further theoretical refinement. These findings inform future work on nuclear structure modeling, with practical and theoretical implications for the development of advanced EDFs and the interpretation of precision nuclear measurements.