Quantum Bridge in Superheavy Element Physics

• This topic defines the progression from early quantization rules to modern computational and relativistic models used to describe electron dynamics in superheavy elements.
• It emphasizes the integration of semiclassical approaches, quantum electrodynamics, and many-body relativistic theories to address challenges in atomic structure at extreme Coulomb fields.
• The discussion incorporates nuclear shell effects and nucleosynthesis pathways, providing insights into the stability and synthesis of superheavy elements within an interdisciplinary framework.

The conceptual bridge between early quantum theory and modern superheavy element physics is defined by the evolution of mathematical frameworks, experimental paradigms, and theoretical models that connect the first quantum descriptions of atomic structure to the present-day understanding of supercritical nuclear systems, electron correlation, relativistic effects, and nucleosynthetic environments. This interdisciplinary nexus encompasses semiclassical approaches, quantum electrodynamics, many-body atomic and nuclear models, and computational advances, all grounded in foundational quantum principles.

1. Early Quantum Models and Limits to Atomic Structure

Early quantum theory, instantiated in the Bohr and Bohr-Sommerfeld models, introduced quantization rules for electron orbits. These approaches posited discrete energy levels and orbital angular momentum, leading to concepts such as electron shells and the periodic table. As in Mills’ formula $y = 15(p – 0.9375·x)$ and the empirical reasoning behind uranium's atomic weight ceiling (Kragh, 2012), pioneers speculated on the upper bounds of Z through numerological and periodic trends.

The introduction of relativistic corrections—in Sommerfeld’s fine structure formula

$$1 \pm \frac{E}{mc^2} = \left[1 + \alpha^2\left(\frac{Z}{n_r + k}\right)^2\right]^{-1/2}$$

highlighted the theoretical boundary at $Z \lesssim 137$ where electron velocities would reach $c$, and the stability of the atomic K-shell was called into question. These early quantum calculations established the conditions for electron binding in strong Coulomb fields and set the conceptual groundwork for future models.

2. Semiclassical Topological Descriptors: Winding Numbers

Even as the Dirac equation and quantum electrodynamics (QED) superseded semiclassical models for quantitative accuracy, the Bohr-Sommerfeld approach retains pedagogical value, especially when extended to superheavy ions such as uranium ($Z=92$) and oganesson ($Z=118$) (Suslov, 15 Oct 2025). In extreme Coulomb fields, semiclassical orbits exhibit self-intersecting trajectories classified by winding numbers:

$$N_\text{winding} = 2 \left( \frac{1}{\Delta} - 1 \right)$$

where $\Delta$ is the per-revolution angular advance. This topological metric captures the transition from simple rosette orbits to elaborate multi-loop structures as Z increases—providing an intuitive visualization of electron dynamics in superheavy atoms. Despite the limitations for precise calculations, this approach serves as a bridge between early quantization rules and the complex orbital behavior necessary for understanding atomic structure at the edge of the periodic table.

3. Quantum Electrodynamics and Supercritical Fields

The extension to quantum electrodynamics foregrounds phenomena absent in non-relativistic quantum mechanics—most notably QED vacuum decay, predicted for supercritical systems where $Z \alpha > 1$. Actinide collisions at center-of-mass energies above 1 GeV engender super-strong Coulomb fields and, for contact times $T_\text{coll} \gtrsim 2$ zs, allow the empty electron states to “dive” into the Dirac sea and produce spontaneous $e^+e^−$ pairs (Simenel et al., 2011). This is theoretically formalized via the BV variational principle:

$$S_{BV} = \mathrm{Tr}\,[D(t_1)B(t_1)] - \int_{t_0}^{t_1} dt\, \mathrm{Tr}\left( B\,\frac{\partial D}{\partial t} - i D [H,B] \right)$$

Such events are a rigorous test of QED in ultra-strong fields—a direct realization of concepts suggested by Dirac and extended in the superheavy regime.

4. Relativistic Many-Body Theory and Electronic Structure

Modern atomic and nuclear structure calculations for superheavy elements require the Dirac-Coulomb(-Breit) Hamiltonian:

$H_{DCB} = \Lambda_+ [H_D + H_C + H_B] \Lambda_+$

as in relativistic CI and MBPT treatments (Lackenby et al., 2019, Tupitsyn et al., 2021, Smits et al., 2023). These handle:

• Extreme relativistic contraction and expansion of $s$, $p_{1/2}$, and $d$ orbitals,
• Large spin-orbit splitting, invalidating Hund’s rule for open $\text{6d}$ shells (e.g., Sg, Bh, Hs, Mt),
• Correlation effects in the electron cloud, often computed with Dirac-Fock-Sturm (DFS) orbitals, configuration interaction with perturbation theory (CIPT), and multiconfiguration Dirac-Fock (MCDF) methods.

Observable properties—energy levels, ionization potentials, isotope shifts, transition rates—are dramatically impacted, deviating from monotonic periodic trends. The inclusion of QED corrections, such as the Lamb shift and vacuum polarization via operators

$h^\text{QED} = h^\text{VP} + h^\text{SE}$

further refines predictions, making modern theory both a computational and conceptual extension of early quantum models.

5. Nuclear Shell Effects and the Island of Stability

Advancements in nuclear models—Skyrme energy density functional (SEDF), macroscopic-microscopic Nilsson-Strutinsky, and FRDM—are built upon the quantum shell model (Shi et al., 2014, Ramirez et al., 2014). Closed nucleon shells yield magic numbers (e.g., $Z=114, 120, 126$ and $N=184$) and extra binding energy, leading to the theorized “island of stability.” Direct mapping via Penning-trap mass spectrometry and metrics such as

$$S_{2n}(N,Z) = EB(N,Z) − EB(N−2,Z) \ \delta_{2n}(N,Z) = S_{2n}(N,Z) − S_{2n}(N+2,Z)$$

quantifies how shell effects extend the longevity and robustness of SHE, in accord with quantum mechanical predictions.

6. Alternative Synthesis and Nucleosynthesis Pathways

Multi-nucleon transfer in actinide collisions, especially inverse quasifission, represents a modern quantum approach to synthesizing neutron-rich transfermiums closer to the island of stability (Simenel et al., 2011). Additionally, astrophysical scenarios such as neutron star mergers (NSMs) create environments for robust $r$-process nucleosynthesis (Holmbeck et al., 2023); theoretical computations suggest that kilonovae may yield superheavy elements ($Z \geq 104$) with mass fractions:

$X_{Z \geq 104} \approx 3 \times 10^{-2}$

and heating rates in the ejecta described by:

$\dot{Q}(t) = \sum_i \dot{q}_{i}(t)\,Y_i$

These processes depend intrinsically on quantum nuclear structure inputs: shell closures, fission barrier heights, and decay properties. The implications span laboratory synthesis and cosmic element formation.

7. Philosophical, Epistemic, and Interdisciplinary Perspectives

The criteria for element “discovery”—existence thresholds ($t \ge 10^{-14}$ s), reproducibility, chemical identification—are both grounded in quantum theory and subject to sociological negotiation (Kragh, 2017). The cross-disciplinary boundaries between nuclear physics and chemistry have fostered debates about naming rights, elemental definitions, and the ontological status of transient nuclei. The exceptional conditions required to form, detect, and classify SHE reflect both conceptual advances and continued challenges inherent to the quantum-mechanical description of matter.

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In conclusion, the conceptual bridge from early quantum theory to superheavy element physics is reflected in the progression from speculative quantization models, through the mathematical architectures of QED and many-body relativistic theory, to experimentally grounded approaches in nuclear and atomic physics. Semiclassical descriptors, quantum shell effects, relativistic and QED corrections, nucleosynthetic modeling, and philosophical considerations together define the landscape of modern superheavy element science, grounded in the continuous extension of quantum principles to regimes of previously unimaginable Coulomb strength and nucleon number.