We calculate the hyperfine structure of the ground state of the muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom by the variational approach, employing an improved set of the Hylleraas-type basis functions to deal with the nonrelativistic three-body wave functions of the muon, the electron, and the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ nucleus. The convergence of the series is very fast, thus eliminating the problem with the propagation of the truncation errors of the computer machine. Using the recoil correction, ${\delta }_{\mathrm{rec}}$ =Δ${\nu }_F$(3α/π)($m_e$/$m_{\mu }$) ln($m_{\mu }$/$m_e$), we arrived at the value Δν=4464.87±0.05 MHz for the total hyperfine-structure splitting of the ground state of the muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom including relativistic and other radiative corrections. But if we follow the treatment of Borie [Z. Phys. A 291, 107 (1979)] for the recoil corrections by treating the (μ${\mathrm{-}}^4$He${)}^{+}$ system as a pseudonucleus, we will arrive instead at the value Δν=4464.00±0.05 MHz. The former seems to be in better agreement with the experimental result of 4465.004±0.029 MHz.

• Received 24 March 1989

DOI:https://doi.org/10.1103/PhysRevA.40.5520