Growth of the Coulomb-induced enhancement of nuclear deformation and outward shift of the neutron drip line in heavy nuclei (Z > 60)

A 2025 study by Hagihara, Nakatsukasa, and Hinohara compared paired Skyrme Hartree-Fock-Bogoliubov (HFB) calculations of even-even nuclei with and without the Coulomb interaction for proton numbers $2 \le Z \le 60$, finding that, as a general trend, the Coulomb interaction enhances the quadrupole deformation $|\beta_2|$ and shifts the neutron drip line toward larger neutron numbers, with 29 nuclei bound only when the Coulomb interaction is included. The open problem is to extend this comparison to the heavier nuclear regions — rare-earth, actinide, and super-heavy nuclei, $Z > 60$ — to determine whether these Coulomb effects grow with proton number, and to establish the robustness of the conclusions against the Coulomb-exchange treatment (exact exchange versus the Slater approximation).

Background

The chart of nuclides maps all bound nuclear species by proton number $Z$ and neutron number $N$. Its boundary on the neutron-rich side is the neutron drip line: the locus beyond which adding two neutrons leaves the nucleus unbound, conventionally identified through the two-neutron separation energy $S_{2n}(Z,N)=E(Z,N-2)-E(Z,N)$ (positive for a bound nucleus, with $E$ the total energy). Because nuclei are self-bound (no external potential holds the nucleons together), the standard tool for chart-wide predictions is nuclear energy density functional (EDF) theory, in which each nucleon moves in an average potential generated by all the nucleons; the Skyrme functionals are a widely used family of such EDFs (the source study uses the parameterization SLy4). Superfluid pairing correlations are incorporated by solving the Hartree-Fock-Bogoliubov (HFB) equations, a mean-field framework that mixes particle and hole degrees of freedom into quasiparticles. Nuclear shapes are characterized by the axial quadrupole deformation parameter $\beta_2$, which measures the elongation (prolate or oblate) of the mass distribution; classically the liquid-drop surface energy favors spherical shapes, and ground-state deformation arises from quantum shell effects. Some nuclei exhibit shape coexistence: two or more potential-energy minima of different deformation are near-degenerate, and then a single mean-field configuration may not describe the ground state — a superposition of shapes, for instance via generator-coordinate-method configuration mixing (building collective wave functions as superpositions of mean-field states of different deformation), is required.

The Coulomb repulsion among protons is the primary breaker of the neutron-proton (isospin) symmetry of the nuclear force. Its effects are usually discussed for proton-rich stability, alpha decay, and fission. Hagihara, Nakatsukasa, and Hinohara (2025) performed a systematic paired comparison — HFB calculations with and without the Coulomb EDF for every even-even nucleus with $2 \le Z \le 60$ — and found: (i) whether a nucleus is deformed at all is decided by shell effects and is unchanged by the Coulomb interaction, but the Coulomb interaction enhances the magnitude $|\beta_2|$ as a general trend; (ii) in shape-coexistence regions such as the Zr isotopes with $N=60$–$70$, the Coulomb interaction reorders the near-degenerate potential-energy minima, producing jumps in the ground-state deformation; (iii) near the neutron drip line the Coulomb interaction provides additional binding — 29 nuclei are bound only when it is included (an effect the authors find strongest for $Z \gtrsim 40$), so the neutron drip line shifts outward to larger $N$, which is unexpected for a repulsive interaction acting only among protons; (iv) the mechanism is quantum mechanical: the Coulomb-induced increase of the nuclear radius lowers canonical neutron single-particle energies near the particle-emission threshold (demonstrated for $^{170}$Sn), raising $S_{2n}$, which is well approximated by $-2\lambda_n$ with $\lambda_n$ the neutron Fermi energy.

The study treated the Coulomb exchange energy in the Slater approximation, a local-density approximation replacing the non-local exchange kernel by a functional of the proton density. Adjacent literature quantifies exchange treatments in related frameworks: Gu, Liang, Long, Van Giai, and Meng compared the exact Coulomb exchange term with local-density approximations in relativistic Hartree-Fock(-Bogoliubov) theory for semi-magic isotopic chains running from the proton to the neutron drip line, finding the exact exchange energies reproduced within about 5%; Naito, Roca-Maza, Colò, and Liang constructed a generalized-gradient-approximation Coulomb exchange functional for self-consistent Skyrme Hartree-Fock. Neither work addresses the Coulomb effect on deformation or drip-line location, and neither provides the quantitative exact-versus-Slater check conjectured to be negligible in the source study.

Problem Statement

The conclusions of Hagihara, Nakatsukasa, and Hinohara — that the Coulomb interaction enhances the quadrupole deformation and shifts the neutron drip line outward — are established only for even-even nuclei with $2 \le Z \le 60$, within a single EDF calculation using the Slater approximation for the Coulomb exchange. The problem is to determine how the Coulomb interaction reshapes deformation and the neutron drip line across the heavier nuclear chart, and how large these electromagnetic effects become, by:

(1) Performing the systematic paired HFB comparison (calculations with and without the Coulomb interaction, with stated EDF, pairing, and basis conventions) for even-even nuclei in the rare-earth, actinide, and super-heavy regions ($Z > 60$), and establishing whether the Coulomb-induced enhancement of $|\beta_2|$ and the outward shift of the neutron drip line (in the $S_{2n}>0$ sense) grow with proton number $Z$ — as expected since the Coulomb energy increases with $Z$ — and by how much, including the count and location of nuclei that are bound only when the Coulomb interaction is included.

(2) Verifying the robustness of these conclusions with respect to the Coulomb-exchange treatment by repeating the comparison with the exact Coulomb exchange term in place of the Slater approximation, and quantifying the resulting sensitivity of the deformation and drip-line results.

In interpreting the deformation results in shape-coexistence regions where the Coulomb interaction reorders near-degenerate potential-energy minima (as in the Zr isotopes with $N=60$–$70$), the source study's methodological caveat should be kept in mind: for such cases with approximately degenerate multiple potential minima, a superposition of different shapes is often required, and a single mean-field configuration may not give a reliable ground state. This caveat is retained as background relevant to the heavier regions, not as an additional completion requirement.

The problem is complete when the growth, or absence of growth, of both Coulomb effects with $Z$ is established for the even-even nuclei of the heavier regions — read from the $Z$-dependence of the Coulomb-induced change in $|\beta_2|$ and of the two-neutron drip-line position — together with the exchange-treatment robustness.

References

1. Kenta Hagihara, Takashi Nakatsukasa, Nobuo Hinohara, 'Effect of the Coulomb Interaction on Nuclear Deformation and Drip Lines', Particles 8(3), 72 (2025). DOI: 10.3390/particles8030072, https://doi.org/10.3390/particles8030072; arXiv:2505.01211, https://arxiv.org/abs/2505.01211
2. Huai-Qiang Gu, Haozhao Liang, Wen Hui Long, Nguyen Van Giai, Jie Meng, 'The Slater approximation for Coulomb exchange effects in nuclear covariant density functional theory', Physical Review C 87, 041301(R) (2013). DOI: 10.1103/PhysRevC.87.041301, https://doi.org/10.1103/PhysRevC.87.041301; arXiv:1210.3162, https://arxiv.org/abs/1210.3162
3. Tomoya Naito, Xavier Roca-Maza, Gianluca Colò, Haozhao Liang, 'Coulomb exchange functional with generalized gradient approximation for self-consistent Skyrme Hartree-Fock calculations', Physical Review C 99, 024309 (2019). DOI: 10.1103/PhysRevC.99.024309, https://doi.org/10.1103/PhysRevC.99.024309; arXiv:1810.02500, https://arxiv.org/abs/1810.02500