Finite-nuclear-size effect in hydrogenlike ions with relativistic nuclear structure

The finite-nuclear-size (FNS) effect has a large contribution to atomic spectral properties, especially for heavy nuclei. By adopting the microscopic nuclear charge-density distributions obtained from the relativistic continuum Hartree-Bogoliubov (RCHB) theory, we systematically investigate the FNS corrections to atomic energy levels and bound-electron $g$ factors of hydrogenlike ions with nuclear charge up to 118. The comparison of the present numerical calculations with the predictions from empirical nuclear charge models, the nonrelativistic Skyrme-Hartree-Fock calculations, and the results based on experimental charge densities indicate that both the nuclear charge radius and the detailed shape of the charge-density distribution play important roles in determining the FNS corrections. The variation of FNS corrections to energy levels and $g$ factors with respect to the nuclear charge are investigated for the several lowest bound states of hydrogenlike ions. It is shown that they both increase by orders of magnitude with increasing nuclear charge, while the ratio between them has a relatively weak dependence on the nuclear charge. The FNS corrections to the $s_{1/2}$ and $p_{1/2}$ bound-state energies from the RCHB calculations are generally in good agreement with the analytical estimations by Shabaev [V. M. Shabaev, J. Phys. B 26, 1103 (1993)] based on the homogeneously charged sphere nuclear model, with the discrepancy indicating the distinct contribution of microscopic nuclear structure to the FNS effects.

Figure 1

Nuclear charge-density distributions ${\rho }_{\mathrm{c}}\left(r\right)$ of (a) ${}^{40}\mathrm{Ca}$ and (c) ${}^{208}\mathrm{Pb}$ obtained from the RCHB calculations (red dashed line), in comparison with several empirical charge distribution models such as a homogeneously charged sphere (purple short-dashed line), a Gaussian charge distribution (blue dash-dotted line), and the two-parameter Fermi model (magenta dash–double-dotted line), as well as the Fourier-Bessel analysis of experimental scattering data [91] (black solid line). (b) and (d) Corresponding electrostatic potentials $V\left(r\right)$ compared with the point-nucleus Coulomb potential (green short-dash–dotted line). See the text for details.

Figure 2

(a) FNS corrections to the energy levels $\mathrm{\Delta }{E}_{\mathrm{FNS}}$ and (b) bound-electron $g$ factors $\mathrm{\Delta }{g}_{\mathrm{FNS}}$ in the $n{s}_{1/2}$ and $n{p}_{1/2}$ states with $n\le 4$ for hydrogenlike ions with $8\le Z\le 118$.

Figure 3

Difference of the ratio $D=\frac{\mathrm{\Delta }{g}_{\mathrm{FNS}}}{\mathrm{\Delta }{E}_{\mathrm{FNS}}}$ between the approximate formula of Eq. (40) and the present RCHB numerical calculations for $n{s}_{1/2}$ and $n{p}_{1/2}$ states with $n\le 4$. The inset shows the specific ratio for the ground state evaluated by Eq. (40) (red solid line) and the corresponding RCHB numerical results (black square scatter).

Figure 4

Comparison of the present RCHB calculations and the two approximate formulas for $\mathrm{\Delta }{E}_{\mathrm{FNS}}$ proposed by Shabaev [43] and Deck et al. [44] for the $1s_{1/2}, 2s_{1/2}$, and $2p_{1/2}$ states of hydrogenlike ions with $8\le Z\le 96$. (a) Direct comparison and (b) absolute differences of the RCHB numerical calculations and the estimation of Deck et al. [44] with respect to Shabaev's [43] approximation. The squares and circles for the RCHB results indicate the positive and negative absolute differences of $\delta E=\mathrm{\Delta }{E}_{\mathrm{RCHB}}-\mathrm{\Delta }{E}_{\mathrm{Shabaev}}$, respectively. The dashed lines represent the positive absolute difference of $\delta E=\mathrm{\Delta }{E}_{\mathrm{Deck}}-\mathrm{\Delta }{E}_{\mathrm{Shabaev}}$. The four vertical dotted lines denote the nuclei with magic proton numbers of 20, 28, 50, and 82.