Quadrupole Antishielding Factors, and the Nuclear Quadrupole Moments of Several Alkali Isotopes

Abstract

The atomic antishielding factor $R$ for the quadrupole hyperfine structure has been evaluated for the lowest three excited $\mathrm{np}$ states for each of the alkali atoms, i.e., for a total of fifteen atomic states. The values of $R$ were determined by means of the perturbed wave functions ${v}_{1}^{\ensuremath{'}}(nl\ensuremath{\rightarrow}{l}^{\ensuremath{'}})$, as obtained by solving the inhomogeneous Schr\"odinger equation for each type of excitation of the core electrons by the nuclear quadrupole moment $Q$. In order to obtain reliable values of $R$, we have obtained accurate valence wave functions $v(\mathrm{np})$ for the $\mathrm{np}$ excited states from the requirement that the calculated energy eigenvalues ${E}_{0}(\mathrm{np})$ should reproduce the observed energy levels. The wave function $v({n}_{0}p)$ for the lowest valence $\mathrm{np}$ state ($n={n}_{0}$, e.g., ${n}_{0}=3$ for Na) was checked for each alkali atom by calculating the oscillator strength ${f}_{{n}_{0}s\ensuremath{\rightarrow}{n}_{0}p}$ for the lowest ${n}_{0}s\ensuremath{\rightarrow}\mathrm{np}$ transition. The calculated values of ${f}_{{n}_{0}s\ensuremath{\rightarrow}{n}_{0}p}$ were found to be in good agreement with experiment. As an example of the values of $R(\mathrm{np})$ obtained, we have found for Rb: $R(5p)=\ensuremath{-}0.257$; $R(6p)=\ensuremath{-}0.207$; and $R(7p)=\ensuremath{-}0.190$. The resulting correction factors $\mathcal{C}=\frac{1}{(1\ensuremath{-}R)}$ are $\mathcal{C}(5p)=0.796$, $\mathcal{C}(6p)=0.829$, and $\mathcal{C}(7p)=0.840$. At the end of the paper we give the corrected values of $Q$ for twelve alkali isotopes, for which experimental data are available, namely, for the following nuclei: ${\mathrm{Na}}^{23}$, ${\mathrm{K}}^{39}$, ${\mathrm{K}}^{40}$, ${\mathrm{K}}^{41}$, ${\mathrm{Rb}}^{85}$, ${\mathrm{Rb}}^{87}$, ${\mathrm{Cs}}^{131}$, ${\mathrm{Cs}}^{132}$, ${\mathrm{Cs}}^{133}$, ${\mathrm{Cs}}^{134}$, ${\mathrm{Cs}}^{135}$, and ${\mathrm{Cs}}^{137}$. We have also given a brief summary of the present experimental evidence supporting the existence of the quadrupole shielding and antishielding effects for atomic states.