Testing the predicted deformation-driven quenching of the $N=50$ shell closure in neutron-rich Sc–Co nuclei ($21\leqslant Z\leqslant 27$) and the $N=20$ discrepancy in Ti: experimental confirmation and the missing correlations in continuum covariant density functional theory

Deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) with the PC-PK1 functional predicts that the conventional neutron shell closure at $N=50$ disappears or is strongly quenched in the isotopic chains from Sc through Co ($21\leqslant Z\leqslant 27$), driven by quadrupole deformation that mixes the $sd$ and $dg$ orbitals, and simultaneously predicts a robust $N=20$ closure in Ca, Sc, and Ti that conflicts with existing titanium mass data. Neither prediction has been tested: no charge radii, masses, or spectroscopy exist for the relevant $N=50$ nuclei ($^{71}$Sc–$^{77}$Co), and independent spherical mean-field analyses instead find $N=50$ shell-closure signatures persisting in Sc and Ti. The open problem is to determine, experimentally and/or theoretically, whether these shell closures persist or are quenched in this neutron-rich region, and to identify which correlations absent from the current mean-field treatment — neutron–proton pairing, pairing-gradient terms, tensor and exchange (Fock) contributions, and beyond-mean-field configuration mixing — are needed to reconcile the theory with data on charge radii, odd–even staggering, and two-neutron gaps.

Background

A nucleus is said to have a "magic" neutron number $N$ (a shell closure) when a large energy gap separates the last filled neutron orbital from the next empty one; such gaps make the nucleus comparatively rigid and spherical, and they reveal themselves observably as sharp drops in the two-neutron separation energy $S_{2n}$ (the energy released when two neutrons are added), pronounced peaks in the two-neutron gap $\delta_{2n}=S_{2n}(N)-S_{2n}(N+2)$, kinks in the charge radius along an isotopic chain, and small quadrupole deformations $\beta_2$ (a measure of the nucleus's departure from spherical shape). Conventional magic numbers such as $N=20$, 28, and 50 are well established near stability, but they are known to erode far from stability where deformation and intruder orbitals compete with the spherical shell gaps — most famously in the "islands of inversion", where deformed intruder configurations from the shell above become the ground state and the nominal closure loses its observable signatures.

A state-of-the-art tool for mapping such shell evolution across the nuclear chart is the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc): a covariant (relativistic) density functional theory in which the mean field is computed with axial quadrupole deformation, pairing correlations are included via the Bogoliubov quasiparticle formalism, and the scattering of weakly bound neutrons into the continuum of unbound states is treated exactly by expanding in a Woods–Saxon (rather than harmonic-oscillator) basis. With the point-coupling density functional PC-PK1 (in which the interaction is represented by contact terms rather than meson exchange, and which omits explicit tensor-force contributions and Fock exchange terms — the exchange part of the interaction that a Hartree–Fock treatment would include on top of the direct density-dependent contact terms), the DRHBc mass table collaboration has published ground-state properties of essentially all nuclei from the proton to the neutron drip line (the boundaries of nuclear existence).

Zheng, Sun, Shen, and Geng (Chin. Phys. C 48, 014107 (2024)) applied DRHBc+PC-PK1 to all $20\leqslant Z\leqslant 30$ nuclei between the drip lines and traced the $N=20$, 28, and 50 closures through charge radii, $S_{2n}$, $\delta_{2n}$, deformations, and single-particle levels. They found the $N=28$ closure disappearing in the Mn chain (consistent with data), but also two provocative results: (i) the $N=50$ closure disappears or is quenched for $21\leqslant Z\leqslant 27$ — the region between the doubly magic $^{78}$Ni and calcium — predominantly through deformation-driven mixing of the $sd$ and $dg$ neutron orbitals, while it remains robust in Ni, Cu, and Zn; and (ii) the $N=20$ closure appears robust in Ca, Sc, and Ti in the calculated $S_{2n}$, whereas the existing Ti data suggest the contrary. The same study exposed why these predictions cannot yet be trusted as final: the mean-field framework omits neutron–proton pairing and pairing-gradient terms that are known to control the odd–even staggering of charge radii (the alternating radius jump between odd-$N$ and even-$N$ isotopes, which DRHBc fails to reproduce around $N=20$ in Ca, Sc, and Ti); it omits tensor and exchange terms, which elsewhere are established as essential for the neighboring $N=32,34$ subshell closures; and it lacks beyond-mean-field correlations (configuration mixing and symmetry restoration), which are suspected to matter for the experimentally observed charge-radius kink at $N=20$ in Sc (König et al., Phys. Rev. Lett. 131, 102501 (2023)) — where DRHBc does find a prolate second minimum with a large radius, but not as the mean-field ground state. Charge radii are measured for short-lived isotopes by collinear laser spectroscopy (overlapping a laser beam with a fast ion beam and extracting the radius from isotope shifts of optical transitions), and masses to sub-keV precision by Penning-trap and multireflection time-of-flight techniques; both techniques have so far stopped far short of the $N=50$, $Z\leqslant 27$ nuclei.

Problem Statement

Determine, with experimental evidence and/or quantitatively improved theory, whether the conventional neutron shell closures survive in the neutron-rich $20\leqslant Z\leqslant 30$ region, by answering the following three questions, which set the required scope of the problem. (1) Does the $N=50$ shell closure — robust in the Ni, Cu, and Zn chains — disappear or become quenched in the Sc, Ti, V, Cr, Mn, Fe, and Co chains ($21\leqslant Z\leqslant 27$), as DRHBc+PC-PK1 predicts through deformation-driven mixing of the $sd$ and $dg$ neutron orbitals? Decide this with the standard shell-closure observables: kinks in charge radii along each isotopic chain, two-neutron separation energies $S_{2n}$ and two-neutron gaps $\delta_{2n}$ from mass measurements, first $2^+$ excitation energies and $B(E2)$ values (the reduced transition probability for electric-quadrupole excitation, which measures collectivity), and ground-state deformations, with the degree of persistence or quenching judged against the signatures of the established robust closures — the Ni, Cu, and Zn chains and the doubly magic $^{78}$Ni. (2) Does the $N=20$ shell closure persist in the Ti chain, where the existing mass data suggest its absence while DRHBc predicts it to be prominent — i.e., is the discrepancy an artifact of missing correlations in the model or an unrecognized experimental effect? (3) Identify which physics absent from the current mean-field treatment — neutron–proton pairing and pairing-gradient terms, tensor-force and exchange (Fock) contributions, and beyond-mean-field configuration mixing and symmetry restoration — is required to reproduce simultaneously the measured charge radii, their odd–even staggering, and the two-neutron gaps across the Ca–Zn region. These three questions serve one overarching objective: establishing the true evolution of the $N=20$, 28, and 50 closures across this region and the microscopic mechanism (shell-gap quenching versus deformation and correlation effects) that produces it, so that predictions for unreachable neutron-rich nuclei — including drip-line positions — rest on validated physics rather than on a single unconfirmed model.

References

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