Quadrupole Shielding and Antishielding Factors for Atomic States

The atomic shielding (or antishielding) factor $R$ for the quadrupole hyperfine structure has been evaluated for several excited atomic states. The values of $R$ were determined by means of the perturbed wave functions $v_1^{}{}_{}{}^{\prime }(nl\to l^{\prime })$ as obtained by solving the inhomogeneous Schrödinger equation for each type of excitation of the core electrons by the nuclear quadrupole moment $Q$. For the case of copper, the calculations provide a quantitative explanation of the fact that the values of $Q\left({\mathrm{Cu}}^{65}\right)$ obtained from two different excited configurations ($3d^{9}4s^2$ and $3d^{10}4p$) differ considerably from each other (by ≳40%), when the correction factor $\frac{1}{\mathrm{}(1-R)\mathrm{}}$ is not applied. The calculated values of $R$, namely $R_{3d}=+0.178\mathrm{}$ (shielding) and $R_{4p}=-0.175\mathrm{}$ (antishielding), bring the two corrected values of $Q$ into very good agreement. We thus obtain $Q\left({\mathrm{Cu}}^{65}\right)=(-0.195\mathrm{}\pm 0.004)\times {10}^{-24\mathrm{}}$ ${\mathrm{cm}}^2$. Besides the work on Cu $3d$ and $4p$, the following calculations have also been carried out: (1) $R_{5d}$ for the $5d$ states of Pr and Tm; (2) $R_{2p}$ and $R_{3p}$ for the $2p$ and $3p$ states of Li; (3) $R_{2p}$ for the $2s2p{}_{}{}^3{}_{}{}^{}P$ excited state of beryllium.