Classification of Transuranium Elements in Terms of `Winding' Numbers in the Bohr-Sommerfeld Model

TL;DR

The paper reexamines the Bohr-Sommerfeld semiclassical model for hydrogen-like ions with $Z$ up to 137, using the Sommerfeld fine-structure formula to trace relativistic elliptical orbits as a function of nuclear charge. By introducing the winding number $N_{ m{winding}} = 2\left(\dfrac{1}{n_{\ m{varphi}}} - 1\right)$ and analyzing open rosettes, the authors classify Coulomb-field strength topologically and reveal the emergence of self-intersecting orbits (e.g., a 'double necklace' for Oganesson) in the high-$Z$ regime. They present extensive data and animations (via a complementary Mathematica notebook) for $92 \le Z \le 137$, showing a progression from single-winding to multi-loop rosettes and proposing a tentative classification (Strong to Ultra-Ultra Strong) aligned with the number of orbital loops. While the semiclassical approach cannot replace Dirac/QED methods, this work offers a conceptual bridge linking early quantum theory to modern superheavy element physics and serves as a pedagogical tool for understanding relativistic orbital topology in extreme Coulomb fields.

Abstract

We revisit the Bohr-Sommerfeld atomic model to explore hydrogen-like ions of Uranium ($Z=92$), Oganesson ($Z=118$), and hypothetical superheavy elements beyond. Although superseded by the Dirac equation and modern quantum electrodynamics, the semiclassical approach offers a historically and pedagogically valuable perspective. Using the Sommerfeld fine structure formula and computer algebra methods, we demonstrate the appearance of self-intersecting orbits in super strong Coulomb fields, beginning with Oganesson and hypothetical elements up to $Z\le137$. These orbits can be classified by their `winding' numbers, providing a simple topological description of Coulomb field strength in this framework. Our results highlight a conceptual bridge between early quantum theory and modern superheavy element physics.

Figures (5)

• Figure 1: Kepler's Classical Elliptical Curve Counterclockwise Motion in Relativistic Uranium Hydrogen-like Ion U$^{91+}.$ The 'winding' number is one.
• Figure 2: Kepler's 'Elliptical' Curve Clockwise Motion with Self-intersection in Hypothetical Oganesson Hydrogen-like Ion Og$^{117+}.$ The 'winding' number is two.
• Figure 3: Kepler's 'Elliptical' Orbit with Self-intersection, $n_{{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }}=n_{r}=1,$ in Hypothetical Oganesson Hydrogen-like Ion Og$^{117+}$ in comparison with the ground state, when $n_{{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }}=1$ and $n_{r}=0.$ The 'winding' number is two.
• Figure 4: Kepler's 'Elliptical' Curve Counterclockwise Motion in Hypothetical Unbihexium Relativistic Hydrogen-like Ion Ubh$^{125+}.$ The 'winding' number is three.
• Figure 5: Kepler's 'Elliptical' Curve Clockwise Motion in Hypothetical Untriquadium Relativistic Hydrogen-like Ion Utq$^{133+}.$ The 'winding' number is seven.