The hyperfine structures of the muonic ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atom (${}_{}{}^3{}_{}{}^{}\mathrm{He}{\mu }^{-}{e}^{-}$) and the muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom (${}_{}{}^4{}_{}{}^{}\mathrm{He}{\mu }^{-}{e}^{-}$) are calculated with the use of variational wave functions. Relativistic and radiative corrections are included. The theoretical hyperfine structure interval is $\Delta \nu =4166.8\mathrm{}\pm 0.3$ MHz for the muonic ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atom, and $\Delta \nu =4465.0\mathrm{}\pm 0.3$ MHz for the muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom; these values are in good agreement with the experimental values.

• Received 17 May 1982

DOI:https://doi.org/10.1103/PhysRevA.26.2330