From atoms to ions: power-law scaling in the periodic table of highly charged ions

Abstract We previously identified a power-law relationship between atomic number and atomic weight in neutral atoms, characterized by a Pareto exponent of approximately 1.09, indicative of a scaling behavior close to Zipf’s law. A similar Zipf-like scaling law was subsequently reported for energy splittings in highly charged ions (HCIs). In the present study, we refine this analysis by examining fine-structure splittings between ground-state multiplet levels of 3 d + 2 HCIs across a wide range of atomic numbers, with level energies extracted from spectroscopic data. Our nonlinear regression analysis reveals a common power-law exponent of μ ≈ 1.78 ± 0.10 for the J = 0 → 4 and J = 2 → 4 transitions, differing only in amplitude. This shared scaling exponent indicates a universal power-law regime governing relativistic and electron-correlation contributions within the multiplet. Collectively, these findings reinforce the presence of robust power-law structures across the periodic table, deepening our understanding of relativistic scaling and electron–electron interactions, and providing a refined empirical foundation for spectroscopic predictions and technological applications involving HCIs.