Shielding and Antishielding Effects for Various Ions and Atomic Systems

We report the results of calculations of several shielding and antishielding effects pertaining to the hyperfine structure of various ions and atomic systems. The following shielding (or antishielding) parameters have been considered in the present work: (1) the parameter ${\sigma }_2$ for the rare-earth ions, which measures the reduction of the $A_2{r}^2{P}_2\left(cos\theta \right)$ term of the crystal field at the location of the $4f$ electrons due to the shielding effect of the more external $5s$ and $5p$ electrons; (2) the quadrupole antishielding factor ${\gamma }_{\infty }$ which gives the total quadrupole moment induced in the closed shells of an ion by the nuclear quadrupole moment $Q$; (3) the corresponding atomic shielding factor $R$ which describes the effect of the induced quadrupole moment on the valence electrons; (4) the hexadecapole antishielding factor ${\eta }_{\infty }$, which is analogous to ${\gamma }_{\infty }$, except that it pertains to the induced hexadecapole (16-pole) moment produced by a possible nuclear electric hexadecapole moment. The calculated values of ${\sigma }_2$ for the ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Tm}}^{3+}$ ions are in good agreement with the experimental results. We have obtained improved values of ${\gamma }_{\infty }$ for the following ions: ${\mathrm{Al}}^{3+}$, ${\mathrm{Cs}}^{+}$, ${\mathrm{I}}^{-}$, ${\mathrm{Pr}}^{3+}$, and ${\mathrm{Tm}}^{3+}$. The atomic shielding factor $R$ has been calculated for the $4f$ electrons of ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Tm}}^{3+}$, and was found to be in reasonable agreement with experiment. A discussion is given of the most likely regions of atomic number for which it might be possible to detect a nuclear hexadecapole moment. In addition, an evaluation of the second-order terms in the antishielding factor ${\gamma }_{\infty }$ for the ${\mathrm{Cl}}^{-}$ ion has been carried out. Expressions for ${\gamma }_{\infty }(nl\to l\pm 2)$ for the case of hydrogenic wave functions have been obtained.