The neutron shell gap at the magic number N=50 decreases along the N=50 isotone chain from Z=38 (88Sr, 5.37 MeV in the two-neutron separation-energy gap) down to Z=32 (82Ge, 3.15 MeV), and the doubly magic nucleus 78Ni (Z=28, N=50) sits at the extrapolated end of this trend. Yet the actual magnitude of the gap at 78Ni remains undetermined: empirical extrapolations spread from roughly 3.4 to 4.3 MeV depending on whether one-neutron or two-neutron separation energies are used, shell-model effective single-particle gaps are near 5 MeV, and the masses of the Ni isotopes that would settle the question empirically have never been measured. The problem is to determine the N=50 gap at 78Ni with quantified uncertainty, to decide whether the decrease continues to Z=28 or is reinforced there by the simultaneous closure of the Z=28 proton shell, and to identify the dominant microscopic mechanism — tensor-force monopole shell evolution, configuration-mixing and quadrupole correlations, or mutually enhanced magicity — including through the particle–hole excitation systematics in 82Ge, 84Se and 86Kr that first anchored the trend.

Background

Nuclear shell structure: protons and neutrons in a nucleus fill quantized single-particle orbitals grouped into shells, and the energy gaps between shells produce the experimentally robust magic numbers (2, 8, 20, 28, 50, 82...) of protons or neutrons at which nuclei are especially tightly bound. The size of the gap at a closed neutron shell — the spacing between the last orbital below the closure and the first orbital above it — governs excitation spectra, deformation tendencies, and nucleosynthesis reaction rates. For neutrons, the gap at N=50 separates the filled g9/2 orbital below the closure from the d5/2 orbital above it. Far from the valley of stability these gaps are not constant: as protons are removed from an N=50 isotone (a nucleus with exactly 50 neutrons), the proton–neutron tensor interaction shifts the relevant orbital energies, a phenomenon known as shell evolution, so the N=50 gap measured at, e.g., 88Sr (Z=38) need not persist at 78Ni (Z=28, N=50), a flagship doubly magic candidate — a nucleus in which the proton and neutron shells are closed simultaneously — far from stability.

Empirically, shell gaps are extracted from masses. The two-neutron separation energy is the energy required to remove two neutrons from a nucleus. Working with mass excesses — the mass excess ME of an atom is the difference between its atomic mass and its mass number, expressed in energy units — the empirical two-neutron gap is Δ2n(Z)=ME(Z,48)−2 ME(Z,50)+ME(Z,52)\Delta_{2n}(Z)=\mathrm{ME}(Z,48)-2\,\mathrm{ME}(Z,50)+\mathrm{ME}(Z,52), where ME(Z,N)\mathrm{ME}(Z,N) denotes the mass excess of the nucleus with ZZ protons and NN neutrons; this combination cancels odd–even pairing effects. Alternatively, the gap is inferred from the drop of one-neutron separation energies across the closure, or computed as a spacing of effective single-particle energies (ESPE — orbital energies obtained by averaging the residual nucleon–nucleon interaction over the occupied orbits) in shell-model calculations. These definitions differ by correlation and pairing energies, and reconciling them is itself part of the physics.

Hwang, Hamilton and Ramayya (2012) analyzed the excited 0+ states of 82Ge (Z=32), 84Se (Z=34) and 86Kr (Z=36) as two-particle–two-hole (2p-2h) neutron excitations across N=50 — two neutrons promoted from the g9/2 shell below N=50 to the orbitals above — and used their excitation energies as evidence that the N=50 gap decreases as Z falls from 40 to 32; a linear extrapolation of the data for Z=30–36 gave an estimated gap of 3.67 MeV at 78Ni. In the same period, Porquet and Sorlin (2012) extrapolated one-neutron-based gaps to 3.44 MeV at Z=28, noted that the two-neutron method instead suggests an increasing gap toward 78Ni (which they attributed to correlation energies), and raised the possible role of the doubly magic effect — now usually called mutually enhanced magicity, the empirically observed extra binding reinforcement when proton and neutron shells close simultaneously, as discussed in the review by Sorlin and Porquet (2008) — which could strengthen the gap at Z=28. Hwang et al. further described a Hartree–Fock–Bogoliubov (HFB — mean-field nuclear structure theory including pairing) calculation with the tensor monopole term that reproduced the empirical N=50 gaps for 82Ge, 84Se and 88Sr but missed 86Kr by 0.45 MeV, and listed open issues: the possible proton admixture in the neutron 2p-2h 0+ states (for which they proposed a (t,α) proton pick-up measurement on 84Se as a test of the 2p-2h interpretation), the incompleteness of a purely tensor description of the gap evolution, the unquantified uncertainty of the linear extrapolation to 78Ni, the missing double-magic enhancement in the theory, and the need to confirm the 80Zn (Z=30) anchor value (3.69 MeV) that then rested on one-neutron separation energies.

The question that crystallized from this work is still the live one: how strong is the N=50 closure at 78Ni, does the erosion seen from Z=38 to Z=32 continue all the way to Z=28, and which microscopic effect — tensor monopole shell evolution, configuration mixing and quadrupole correlations, or mutually enhanced magicity — dominates the evolution?

Problem Statement

Determine the strength of the N=50 neutron shell closure at the doubly magic nucleus 78Ni (Z=28) and the mechanism of its evolution along the N=50 isotone chain from Z=38 down to Z=28. Concretely, an answer should: (a) fix the empirical two-neutron N=50 gap at Z=28, Δ2n(28)=ME(28,48)−2 ME(28,50)+ME(28,52)\Delta_{2n}(28)=\mathrm{ME}(28,48)-2\,\mathrm{ME}(28,50)+\mathrm{ME}(28,52) (the quantity used for the Z=38 and Z=32 anchors), with quantified uncertainty — e.g., from high-precision masses of the Ni isotopes at N=48–52 (whose direct measurement, replacing the current linear extrapolations from Z=30–36, is the suggested decisive route), or from equivalent spectroscopic constraints on neutron particle–hole excitations across N=50 — thereby settling the present method-dependent spread between the one-neutron extrapolations (3.44–3.67 MeV) and the two-neutron extrapolation (about 4.2–4.3 MeV), and reconciling these with the near 5 MeV shell-model effective-single-particle estimates; (b) establish whether the decrease of the gap observed from Z=38 (5.37 MeV in the two-neutron separation-energy gap) to Z=32 (3.15 MeV) continues to Z=28 or is reinforced by the simultaneous closure of the Z=28 proton shell (the mutually enhanced magicity or doubly magic shell-gap effect); and (c) identify which microscopic mechanism dominates the evolution — tensor-force monopole shell evolution, configuration mixing and quadrupole correlations (including the deformed intruder configurations found near 78Ni), or doubly-magic enhancement. All three of (a), (b) and (c) are required for a complete answer. In (a), a value of Δ2n(28)\Delta_{2n}(28) obtained by extrapolation or from theory counts as fixing the gap, provided the method and its uncertainty are justified, since the direct masses of 78Ni and its N=52 neighbour needed for the two-neutron formula are not yet measured. Because (b) compares that same two-neutron quantity at Z=28 and at Z=32, its direction follows from the value fixed in (a) and need not be established independently of (a); (c) must instead be supported by its own evidence and not merely by restating (b)'s reinforcement clause.

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