Small Effects in the Theory of Muonic Atoms. I. Exact Calculation of Eigenvalues for Atoms with Deformed Nuclei

Exact numerical solutions of the coupled Dirac equations for a muon moving in the electrostatic field of a permanently deformed nucleus were obtained and compared with the solutions obtained by the perturbation technique previously employed. Expressions are given for the boundary and continuity conditions upon the exact coupled solutions. For ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ as represented by a reasonable charge distribution, the exact energy eigenvalues were shifted by the following amounts from the perturbation values: $1s$ bindiǹg increased by 7.46 keV, $2p$ increased by various amounts between 1.94 and 6.91 keV, and $3d$ increased by less than 1 keV. The effect upon $2p$ eigenvalues cannot be duplicated by the perturbation technique by merely changing the parameters of the charge distribution.