Calculation of resonant states of muonic molecular ions pμHe, dμHe, tμHe in the adiabatic hyperspherical approach

The resonant states of muonic molecular ions x mu He-3,He-4 (x = P, d, t) are important for the theory of nuclear collisions at ultra low energies [1]. The positions and the widths of these resonances, as well as G- and gamma -factors, are calculated in the adiabatic hyperspherical approach (AHSA) for total angular momentum J = 0, 1. This approach was used previously for bound states [2],[3],[4] and scattering [5] in the Coulomb three-body systems. The numerical method developed for resonances is based on the reduction of the hyperradial scattering problem to the boundary-value problem on the finite interval with the boundary conditions of R-matrix type [6]. This method coincides essentially with the stabilisation one, but the boundary conditions are chosen in a specific way. The algorithm developed overcomes the difficulties connected with numerous quasicrossings of AHS terms, long range character of nonadiabatic couplings and strong E-dependence near the resonance. It allows to study in details E-dependence of the three-body wave function, its local characteristics ( G- and gamma- factors) and the phase shift 6 in the resonance range. The numerically calculated G-factor strongly depends on E within the resonance width. In contrast to G-factor the numerically obtained gamma -factors do not depend on E noticeably in the resonance range. The results obtained by the method developed for the resonance (d mu He-3)(J=0) were used for the calculation of the rate lambda (0)(f) of the nuclear fusion reaction d + He-3 --> He-4 + p. in paper [1].