André Felipe F. da Silva

Earlier work by some of the present authors identified power-law statistical regularities across the periodic table: a near-Zipf relationship $W\propto Z^{1.09}$ between atomic weight and atomic number in neutral atoms, and a fine-structure splitting scaling $\Delta E\propto Z^{1.78}$ in the $3d^{2+}$ (Cr-like) isoelectronic sequence of highly charged ions (HCIs). The apparent tension between this superlinear empirical exponent and the linear-in-Z scaling derived analytically by Lyu, Keitel, and Harman for relativistic clock-state transitions in $n{d}^6$ ions is resolved by writing the scaling law in the form $\Delta E=A{(Z-Z^{\ast })}^{\gamma }$, which gives $\gamma =1.03$ and $Z^{\ast }=25.83$ for the $3d^{2+}$ data and restores a near-Zipf interpretation. We extend the thresholded model from the single $3d^{2+}$ sequence to fourteen isoelectronic sequences spanning C-like ($Z\ge 6$) through Mo-like ($Z\ge 42$), drawing on the NIST Atomic Spectra Database, evaluated compilations of strontium, copper, and tungsten ion data, and digitization of GRASP MCDHF Mo-like results. Comparing the fitted threshold $Z^{\ast }$ against the classical Slater shielding ${\sigma }_{\mathrm{Slater}}$ of the valence shell reveals three regimes: (i) Coulomb-LS sequences (C/N/O/Ne-like) with $Z^{\ast }$ fixed at zero by construction; (ii) a mainstream cluster of nine sequences (a mix of spin–orbit fine-structure and, for the $d^2$ Ca-/Sr-like members, Coulomb term separations) for which $Z^{\ast }$ tracks ${\sigma }_{\mathrm{Slater}}$ within $\pm 1.5$ units; (iii) a pair of $n{d}^6{}^{5}D_0$ sequences (Cr-like $3d^6$ and Mo-like $4d^6$) with $Z^{\ast }-{\sigma }_{\mathrm{Slater}}=+7.20$ and $+10.36$, scaling linearly with Z in the jj regime (after resolving a low-Z LS–jj crossover for the Mo-like member). The slope ratio $A_{\mathrm{Mo}}/A_{\mathrm{Cr}}\approx 0.48$ is of the order of the hydrogenoid radial-extent ratio $⟨r^2⟩(3d)/⟨r^2⟩(4d)\approx 0.32$, exceeding it by ∼50% as expected from relativistic $4d$ contraction. A falsifiable extrapolation for the W-like $5d^6{}^{5}D_0$ sequence is offered: $\Delta \in [+7,+11]$, $A_W\in [0.12,0.18]$ eV/Z, with explicit failure modes specified for an independent GRASP MCDHF test.