Correlation trends in the hyperfine structures of ${}^{210,212}\mathrm{Fr}$

We demonstrate the importance of electron correlation effects in the hyperfine structure constants of many low-lying states in ${}^{210}\mathrm{Fr}$ and ${}^{212}\mathrm{Fr}$. This is achieved by calculating the magnetic dipole and electric quadrupole hyperfine structure constants using the Dirac-Fock approximation, second-order many-body perturbation theory, and the coupled-cluster method in the singles and doubles approximation in the relativistic framework. By combining our recommended theoretical results with the corresponding experimental values, improved nuclear magnetic dipole and electric quadrupole moments of the above isotopes are determined. In the present work, it is observed that there are large discrepancies between the hyperfine structure constants of the $7D_{5/2}$ state obtained from the experimental and theoretical studies, whereas good agreement is found for the other $D_{5/2}$ states. Our estimated hyperfine constants for the $8P,6D,10S$, and $11S$ states could be very useful as benchmarks for the measurement of these quantities.

Figure 1

Goldstone diagrammatic representation of $O,O{T}_1,O{S}_{1v}$, and $O{S}_{2v}$ terms corresponding to the DF and correlation contributions to the hyperfine structure constants. Lines with double arrows or a single arrow pointing upwards denote valence $\left(v\right)$ and virtual particle $\left(p\right)$ orbitals, respectively, while lines with a downward arrow stand for the occupied orbitals $\left(a\right)$.

Figure 2

Demonstration of correlation trends in the evaluation of $A_{\text{hf}}/g_I$ values (in MHz) in a Fr atom through a few important CCSD terms. Results among the considered (a) $S$, (b) $P$, (c) $D_{3/2}$, and (d) $D_{5/2}$ states are compared from the DF, $O{T}_1,O{S}_{1v}$ (after subtracting the DF value), and $O{S}_{2v}$ terms along with their Hermitian conjugate terms in a sequence in each arbitrary unit distance.

Figure 3

Comparison of correlation trends in the evaluation of $B_{\text{hf}}/Q$ values (in MHz) of the considered states of Fr through a few important CCSD terms. Results among the considered (a) $P_{3/2}$, (b) $D_{3/2}$, and (c) $D_{5/2}$ states are compared from the DF, $O{T}_1,O{S}_{1v}$ (after subtracting the DF value), and $O{S}_{2v}$ terms along with their Hermitian conjugate terms in a sequence in each arbitrary unit distance.