Nonrelativistic hyperfine splitting in muonic helium by adiabatic perturbation theory

Huang and Hughes have recently discussed the hyperfine splitting $\Delta \nu$ of muonic helium (${\alpha }^{++}{\mu }^{-}{e}^{-}$) using a variational approach. In this paper, the Born-Oppenheimer approximation is used to simplify the evaluation of $\Delta \nu$ in the nonrelativistic limit. The first-order perturbed wave function of the electron is obtained in closed form by slightly modifying the method used by Dalgarno and Lynn. The result $\Delta \nu =4450\mathrm{}$ MHz, is quite close to the published result of Huang and Hughes 4455.2±1 MHz, which required a very large Hylleraas expansion as well as considerable extrapolation.