Atomic Aftereffects and the Line Shape of Muonic X Rays

The muonic x-ray energy, by virtue of the electron screening, depends on the depletion of the electron states by Auger transitions and on the time needed for the refilling of the electron vacancies. The effect of these processes on the line shape of the muonic x rays is studied here. Equations for the electron $1s$-level population and for the muonic-x-ray center of gravity are derived and applied to representative examples of medium- and high-$Z$ muonic atoms. It is shown that the atomic aftereffects (i.e., refilling of the $1s$ state) result in almost full screening in wide range of atomic numbers. The calculated energy shift with respect to the full screening is only 1-3 eV. The uncertainty, related to the uncertain width of the electron $1s$ vacancy is discussed and estimated.