Structure of Neutral Mesonic Atoms Formed in Liquid Helium

Abstract

Wave functions and binding energies are calculated for some states of the $\alpha {\pi }^{-}{e}^{-}$, $\alpha K^{-}{e}^{-}$, and $\alpha \overline{p}{e}^{-}$ atoms in which the electron is in a $1s$ orbit and in which the meson or the antiproton is in a circular or a nearly circular orbit with large principal quantum number $n$. The results which are obtained in the present paper provide a basis for an estimate, described elsewhere, of the probability that atoms in some of these states can exist for an anomalously long time after being formed in liquid helium. Two methods are employed to calculate the wave functions and binding energies. A variational method which employs hydrogenic functions is used for states in which the meson is in a circular orbit. A more elaborate method, which makes use of the Born-Oppenheimer approximation and which is applicable to states with meson orbits which are not necessarily circular, is used for circular and nearly circular orbits of a few levels of the $\alpha {\pi }^{-}{e}^{-}$ and $\alpha K^{-}{e}^{-}$ atoms. It is found, in particular, that an $E4\mathrm{}$ or higher multipole transition is required for an Auger electron to be ejected by a $K^{-}$ meson in a circular orbit of the $\alpha K^{-}{e}^{-}$ atom with $n=27, 28, or 29$. It is found that for levels of the $\alpha K^{-}{e}^{-}$ atom with $n=27, 28, or 29$ the magnitude of the energy difference between two states with the same value of $n$, but with orbital angular momenta ($n-2\mathrm{}$) and ($n-1\mathrm{}$), is roughly 0.5 eV. It is also found that, in some instances, replacing the absolute square of the meson wave function with a $\delta$ function does not significantly affect the accuracy of a calculation.