Electric Quadrupole Moments of the $D$ States of Alkaline-Earth-Metal Ions

The electric quadrupole moment for the $4d{}^{2}D_{5/2}$ state of ${}^{88}\mathrm{Sr}^{+}$; one of the most important candidates for an optical clock, has been calculated using the relativistic coupled-cluster theory. This is the first application of this theory to determine atomic electric quadrupole moments. The result of the calculation is presented and the important many-body contributions are highlighted. The calculated electric quadrupole moment is $(2.94\pm 0.07)e{a}_0^2$, where $a_0$ is the Bohr radius and $e$ the electronic charge while the measured value is $(2.6\pm 0.3)e{a}_0^2$. This is so far the most accurate determination of the electric quadrupole moment for the above mentioned state. We have also calculated the electric quadrupole moments for the metastable $4d{}^{2}D_{3/2}$ state of ${}^{88}\mathrm{Sr}^{+}$ and for the $3d{}^{2}D_{3/2,5/2}$ and $5d{}^{2}D_{3/2,5/2}$ states of ${}^{43}\mathrm{Ca}^{+}$ and ${}^{138}\mathrm{Ba}^{+}$, respectively.

Figure 1

The diagrams (a) and (c) are subsets of DPC diagrams. Diagram (b) is one of the direct DCP diagram.

Figure 2

Diagram indicating the clock transition in ${}^{88}\mathrm{Sr}^{+}$. Excitation energies of the $4d{}^{2}D_{3/2}$ and $4d{}^{2}D_{5/2}$ levels are given in ${\mathrm{cm}}^{-1}$.