Muonic-hydrogen molecular bound states, quasibound states, and resonances in the Born-Oppenheimer approximation.

Abstract

The Born-Oppenheimer approximation is used as an exploratory tool to study bound states, quasibound states, and scattering resonances in muon (\ensuremath{\mu})--hydrogen (x)--hydrogen (y) molecular ions. Our purpose is to comment on the existence and nature of the narrow states reported in three-body calculations, for L=0 and 1, at approximately 55 eV above threshold and the family of states in the same partial waves reported about 1.9 keV above threshold. We first discuss the motivation for study of excited states beyond the well-known and well-studied bound states. Then we reproduce the energies and other properties of these well-known states to show that, despite the relatively large muon mass, the Born-Oppenheimer approximation gives a good, semiquantitative description containing all the essential physics. Born-Oppenheimer calculations of the s- and p-wave scattering of d-(d\ensuremath{\mu}), d-(t\ensuremath{\mu}), and t-(t\ensuremath{\mu}) are compared with the accurate three-body results, again with general success. The places of disagreement are understood in terms of the differences in location of slightly bound (or unbound) states in the Born-Oppenheimer approximation compared to the accurate three-body calculations.The analytic properties of the function ${\mathit{k}}^{2\mathit{L}+1}$cot\ensuremath{\delta} are used to illustrate the locating of bound states and resonance poles in the complex k or E plane from the scattering data and to deduce the expected widths of resonances of a given L value. The prominent L=3 resonance at 22 eV (\ensuremath{\Gamma}=1.4 eV) in d-(t\ensuremath{\mu}) scattering provides a benchmark for the unsuccessful search within the Born-Oppenheimer approximation for the claimed narrow s-wave d-t-\ensuremath{\mu} resonance at 54.35 eV (\ensuremath{\Gamma}=0.74 eV). Although absolute proof is obviously lacking (because of the approximate calculation), the Born-Oppenheimer results for the s-wave gerade scattering are entirely reasonable; the absence of a centrifugal barrier makes it implausible that there could be resonances at 50--60 eV with narrow widths. We argue that the threshold at 48 eV of the t+(d\ensuremath{\mu}) channel is unlikely to have a major effect. The family of states at 1.9 keV and above are of a different nature. They are, as is already known, molecular states based on the 3d ${\mathrm{\ensuremath{\sigma}}}_{\mathit{g}}$ ``electronic'' potential-energy curve, which is asymptotically the energy of the x(y\ensuremath{\mu}; n=2) system. Born-Oppenheimer calculations of binding energies, and properties depending dominantly on the wave functions in the classically allowed region, agree reasonably well with the accurate three-body computations for the d-t-\ensuremath{\mu} system. Expected disagreement occurs in the probability density ${\mathrm{\ensuremath{\rho}}}_{\mathit{N}}$(0) for the nuclei to be at vanishing internuclear separation (relevant for the fusion rate), in which configuration mixing is more important in the classically forbidden region. Nonetheless, the reported values of ${\mathrm{\ensuremath{\rho}}}_{\mathit{N}}$(0), when compared to the average probability density in the classically allowed region of nuclear motion computed here, appear excessively large. The likelihood that this family of continuum resonances would play a significant role in molecular formation in the muonic cascade is discussed briefly.