The existence of superheavy nuclei is owed entirely to quantum shell effects: beyond roughly Z = 104 the liquid-drop fission barrier vanishes, and only nuclear shell closures stabilize nuclei up to oganesson (Z = 118). Theory has predicted since the 1960s a spherical 'island of stability' around a neutron shell closure at N = 184 paired with a spherical proton shell closure, but the proton magic number — Z = 114, Z = 120, or Z = 126 — remains undetermined: macroscopic-microscopic and self-consistent nuclear models disagree, and no experimental signature exists. Because no nucleus with Z ≥ 119 has ever been synthesized, the region around the predicted spherical closure cannot be observed directly. The problem, posed in the 2015 review by Hamilton, Hofmann and Oganessian, asks where the next spherical proton shell closure beyond Z = 82 lies and whether elements 119 and 120 can be produced so that it can be observed, given that the most sensitive search to date (50Ti + 249Bk at TASCA/GSI) reached a one-event cross-section sensitivity of 65 femtobarns without observing a single atom. A decade later the question is intact: searches at GSI, RIKEN, and Berkeley have tightened limits and demonstrated the titanium-beam route, decay spectroscopy of lighter nuclei hints against Z = 114, and new dedicated facilities (RIKEN's SRILAC/GARIS-III, the FLNR 'SHE Factory', the Berkeley 88-Inch Cyclotron) are attacking the problem, but neither the proton magic number nor a single atom of element 119 or 120 has been secured.

Background

Nuclear shell structure and the island of stability. In the nuclear shell model, protons and neutrons occupy quantized single-particle energy levels, and large gaps between levels at 'magic numbers' (for neutrons: 2, 8, 20, 28, 50, 82, 126) produce extra binding and stability; 208Pb, with Z = 82 protons and N = 126 neutrons, is the heaviest nucleus with both spherical closures filled. Beyond roughly Z = 104 — the 'superheavy elements' (SHE) — the classical liquid-drop fission barrier would be zero and nuclei would fission within about 10⁻¹⁹ s; their millisecond-to-second lifetimes rest on quantum 'shell corrections', the difference between the quantized single-particle energy sum and its smooth average, which create a small but decisive barrier against fission. Macroscopic-microscopic models (a liquid-drop macroscopic part plus shell corrections from a phenomenological single-particle potential) predicted deformed shell gaps at Z = 100/102/108 and N = 152/162 — where the known deformed superheavy nuclei cluster — and, for spherical shapes, shell gaps at N = 184 together with Z = 114, 120, or 126: the predicted 'island of stability' of spherical, relatively long-lived nuclei. Self-consistent density functional theory (DFT, built on Skyrme, Gogny, or covariant energy density functionals) broadly confirms the neutron closure at N = 184 but splits on the proton closure among Z = 114, 120, and 126, and no theoretical consensus has emerged (Kuzmina et al. 2012; Nazarewicz 2018).

How superheavy elements are made. Two reaction families produced all elements up to 118: 'cold fusion' of 208Pb/209Bi targets with beams from Cr to Zn (GSI Darmstadt and RIKEN; elements 107-113) and 'hot fusion' of actinide targets with neutron-rich 48Ca beams (JINR Dubna; elements 114-118). The key observable is the evaporation-residue (ER) cross section: the probability that two nuclei fuse, that the excited compound nucleus survives fission long enough to evaporate neutrons, and that the residue reaches the detector. Residues are separated in-flight in gas-filled recoil separators (TASCA at GSI, GARIS at RIKEN, DGFRS at JINR, BGS at Berkeley) or velocity filters (SHIP at GSI) and identified by position-correlated alpha-decay chains — sequences of alpha particles (helium nuclei) of characteristic energies, terminating in spontaneous fission (SF) or a known daughter. Cross sections fall steeply with atomic number: from roughly 1 pb (10⁻³⁶ cm²) for oganesson to the femtobarn level (1 fb = 10⁻³⁹ cm²) expected for Z = 119 and 120; the 'one-event sensitivity' of a search is the cross section at which the experiment would on average register one atom.

Status at the source (2015). Elements 113-118 had just been established through cross-bombardments (producing the same nuclide in different beam-target combinations), with oganesson (Z = 118) marking the exhaustion of the 48Ca route: both the 48Ca projectile and suitable actinide targets were running out, so reaching Z = 119/120 requires heavier projectiles — 50Ti, 51V, or 54Cr — on targets such as 249Bk or 249Cf. The review by Hamilton, Hofmann and Oganessian argued from two-neutron separation energies (the energy required to remove two neutrons, a standard probe of shell structure) and decay half-life systematics that reinforced shell gaps govern the survivability of superheavy nuclei, and posed the open questions: the location of the next spherical proton shell closure beyond Z = 82 is undetermined (little evidence for Z = 114, with recent calculations favoring Z ≥ 120); the search for element 119 in the 249Bk + 50Ti reaction at TASCA had yielded only an upper limit of 70 fb for the production cross section; and dedicated high-intensity facilities — notably the 'SHE Factory' under construction at the Flerov Laboratory of Nuclear Reactions (FLNR) of JINR — would be needed to overcome the minuscule production probabilities. Kuzmina et al. (2012) had further shown that the production cross sections of new superheavy nuclei depend decisively on where the proton closure actually is, so the two questions — where the closure lies, and whether elements 119/120 can be made — are physically coupled.

Problem Statement

Where is the next spherical proton shell closure beyond Z = 82 located — at Z = 114, Z = 120, or Z = 126 — in the region of the predicted spherical neutron shell closure at N = 184, and can it be established decisively, most directly through the synthesis and decay characterization of the still-unproduced elements 119 and 120? The problem as posed by the source has two coupled parts: (i) the proton magic number that centers the island of stability is fixed neither by theory (macroscopic-microscopic and self-consistent models disagree) nor by experiment (no signature exists); (ii) no nucleus beyond oganesson (Z = 118) has ever been produced — the most sensitive search, 50Ti + 249Bk at TASCA, reached a one-event sensitivity of 65 fb without observing a decay chain, and the predicted cross sections for all accessible projectile-target combinations are at the femtobarn level or below, so that reaching Z = 119/120 taxes the beam intensities, target technology, and separators of the new dedicated facilities. The coupling is physical: the production cross sections of the new elements depend decisively on where the proton closure actually is, and only nuclei with Z ≥ 119 can directly probe single-particle structure near the predicted spherical closure. A complete answer establishes the location of the next spherical proton shell closure with certainty sufficient to discriminate Z = 114 against Z = 120 against Z = 126 — whether by theory alone, where independent model classes converge and are benchmarked against the known nuclei; by production and unambiguous identification of element 119 and/or 120 together with the decay or mass systematics that fix the single-particle structure; or by a combination of these; producing and identifying elements 119 and/or 120 is the principal experimental route to this determination rather than a separate deliverable. The determination must hold for the spherical N ≈ 184 region: evidence restricted to the deformed nuclei (Z ≤ 118, N ≤ 178) currently known constrains the answer but does not by itself settle it.

References

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