Properties and hyperfine structure of helium-muonic atoms

Basic geometrical and dynamical properties of the ${}^3{\mathrm{He}}^{2+}{\mu }^{-}{e}^{-}$ and ${}^4{\mathrm{He}}^{2+}{\mu }^{-}{e}^{-}$ helium-muonic atoms are determined from high-precision, variational calculations. Only the bound ground states with $L=0\mathrm{}$ are discussed. The estimated hyperfine splitting between ${}^{1/2}S$ and ${}^{3/2}S$ states in the ${}^3{\mathrm{He}}^{2+}{\mu }^{-}{e}^{-}$ atom is $\Delta \nu =4166.389\mathrm{}\hspace{0.5em}\mathrm{MHz}.$ The analogous result for the hyperfine splitting between ${}^{0}S$ and ${}^{1}S$ states in the ${}^4{\mathrm{He}}^{2+}{\mu }^{-}{e}^{-}$ atom is $\Delta \nu =4464.555\mathrm{}\hspace{0.5em}\mathrm{MHz}.$ Both obtained figures agree very well with the known experimental values. The evaluated hyperfine splitting for the ${}^6{\mathrm{Li}}^{3+}{\mu }^{-}{e}^{-}$ ion is $\approx 29311.4\hspace{0.5em}\mathrm{MHz}$ and for the ${}^7{\mathrm{Li}}^{3+}{\mu }^{-}{e}^{-}$ ion is $\approx 36790.8\hspace{0.5em}\mathrm{MHz}.$