Nuclear Magnetic Moments of Francium-207–213 from Precision Hyperfine Comparisons

We report a fourfold improvement in the determination of nuclear magnetic moments for neutron-deficient francium isotopes 207–213, reducing the uncertainties from 2% for most isotopes to 0.5%. These are found by comparing our high-precision calculations of hyperfine structure constants for the ground states with experimental values. In particular, we show the importance of a careful modeling of the Bohr-Weisskopf effect, which arises due to the finite nuclear magnetization distribution. This effect is particularly large in Fr and until now has not been modeled with sufficiently high accuracy. An improved understanding of the nuclear magnetic moments and Bohr-Weisskopf effect are crucial for benchmarking the atomic theory required in precision tests of the standard model, in particular atomic parity violation studies, that are underway in francium.

Figure 1

Radial dependence of the hyperfine operator [Eq. (1)], with magnetization distribution as modeled for Fr by: a pointlike nucleus, a ball of constant magnetization, and the single-particle model.

Figure 2

Calculated ratios of the $7s$ to $7p_{1/2}$ hyperfine constants for ${}^{207–213}\mathrm{Fr}$ using the ball and single-particle (SP) nuclear magnetization models, and comparison with experiment [18]. The odd-even staggering is due to the addition of neutrons; the slight negative slope is due to the changing nuclear radius. The dashed blue line shows the calculated (SP) ratios corrected by the factor $\xi ({}^{133}\mathrm{Cs})$; see text for details.