Bohr-Weisskopf effect: From hydrogenlike-ion experiments to heavy-atom calculations of the hyperfine structure

We study the influence of electron screening on the Bohr-Weisskopf (BW) effect in many-electron atoms. The BW effect gives the finite-nucleus magnetization contribution to the hyperfine structure. Relativistic atomic many-body calculations are performed for $s$ and $p_{1/2}$ states of several systems of interest for studies of atomic parity violation and time-reversal-violating electric dipole moments—Rb, Cs, Fr, ${\mathrm{Ba}}^{+}, {\mathrm{Ra}}^{+}$, and Tl. For $s$ states, electron screening effects are small, and the relative BW correction for hydrogenlike ions and neutral atoms is approximately the same. We relate the ground-state BW effect in H-like ions, which may be cleanly extracted from experiments, to the BW effect in $s$ and $p_{1/2}$ states of neutral and near neutral atoms through an electronic screening factor. This allows the BW effect extracted from measurements with H-like ions to be used, with screening factors, in atomic calculations without recourse to modeled nuclear structure input. It opens the way for unprecedented accuracy in accounting for the BW effect in heavy atoms. The efficacy of this approach is demonstrated using available experimental data for H-like and neutral ${}^{203}\mathrm{Tl}$ and ${}^{205}\mathrm{Tl}$.

Figure 1

Effect of relative changes in the nuclear magnetic radius for ${}^{133}\mathrm{Cs}$ on the $6s$ Bohr-Weisskopf effect (left), the hydrogenlike $\epsilon \left(1s\right)/\epsilon \left(2p_{1/2}\right)$ ratio (middle), and the $6s$ screening factor (right). The Bohr-Weisskopf effect depends strongly on the nuclear model and on the nuclear magnetic radius, while the screening factors and ${\epsilon }_s/{\epsilon }_p$ ratios do not. The scale of the dependence is similar for other atoms. Here, ${\epsilon }_0, x_0$, and ${\eta }_0$ correspond to nuclear single-particle model values found with magnetic radius $r_m=r_c=\sqrt{5/3}{r}_{\mathrm{rms}}$. The nuclear spin and parity for ${}^{133}\mathrm{Cs}$ are $I^{\pi }=7/2^{+}$.

Figure 2

Magnetization radius (in units of the reference radius $r_c=\sqrt{5/3}{r}_{\mathrm{rms}}$) for ${}^{203,205}\mathrm{Tl}$, as constrained by measurements of the H-like Bohr-Weisskopf effect (Table 2) and the neutral-atom differential hyperfine anomaly (Table 3). Solid black lines are from experimental BW effect for ${{}^{203,205}\mathrm{Tl}}^{80+}$. Diagonal colored lines relate $r_m$ for ${}^{203}\mathrm{Tl}$ and ${}^{205}\mathrm{Tl}$ deduced from experimental hyperfine anomaly for $6p_{1/2}$ (magenta) and $7s_{1/2}$ (blue) states. Uncertainties are indicated by shaded regions.