2P→1STransitions in Muonic Tl, Pb, and Bi

Abstract

The mixing of states of a muonic atom with the nucleus in an excited state into the states with the nucleus in the ground state, for natural Tl, Pb, and Bi (due to the multipole interactions between the nucleus and the muon), is calculated. This effect fails to explain the ratio of the number of $2{P}_{\frac{1}{2}}\ensuremath{\rightarrow}1{S}_{\frac{1}{2}}$ to $2{P}_{\frac{3}{2}}\ensuremath{\rightarrow}1{S}_{\frac{1}{2}}$ transitions (expected to be 0.5 in the absence of mixing) as observed by Frati and Rainwater. In Bi, the calculation shows that the mixing is negligible, but the observed ratio was 0.75\ifmmode\pm\else\textpm\fi{}0.05; in Tl (where the ratio was 0.97\ifmmode\pm\else\textpm\fi{}0.09) the off-diagonal matrix elements of the Hamiltonian, required for this effect to be the sole cause of the change of ratio, differs from theory by two standard deviations. For Pb the observed ratio agrees with theory and the calculated mixing is, indeed, negligible. If we assume that some unknown effect is acting in Bi and is of the same order of magnitude in Tl, then the difference between the ratio in Tl from that in Bi is explained by the above mixing. This assumption is suggested by the fact that Tl is one proton below and Bi is one proton above a magic number closed shell (82). We suppose that this unknown effect is absent in Pb since it is a magic number nucleus (82 protons). It is proven that nonresonant effects, due to spin-independent operators, cannot affect the radiative-transition sum rules. This is applied to nonresonant hyperfine mixing and a hypothetical nuclear-Auger effect.