Abstract
We have applied the methods of quantum reactive scattering to the key resonant reaction in the muon catalyzed fusion (MCF) cycle that leads to the formation of a $\mathrm{dt}\ensuremath{\mu}$ muonic molecular ion, in which fusion takes place very rapidly. We have calculated reaction probabilities for the resonances that occur in $t\ensuremath{\mu}+{\mathrm{D}}_{2}$ scattering for incident kinetic energies less than 0.6 eV and total angular momentum ${J}_{\mathrm{tot}}=0.$ To reduce the six-body problem to a three-body problem, the motions of the electrons were treated in the Born-Oppenheimer (BO) approximation while those of the muon were treated with a sophisticated adiabatic approximation. The resulting three-body potential energy surface (PES) was represented by a pairwise additive approximation. The $\mathrm{dt}\ensuremath{\mu}$ part of the PES was scaled to allow it to exhibit the correct binding energy of the crucial $(J,v)=(1,1)$ state. Scattering calculations were carried out using a hyperspherical formulation, and the positions of the resonances were found to occur at energies of a few meV greater than if $\mathrm{dt}\ensuremath{\mu}$ is assumed to be a point particle. A comparison of the resonances with the Breit-Wigner formula allowed us to calculate partial widths for back decay, ${\ensuremath{\Gamma}}_{e}^{{J}_{\mathrm{tot}}}.$ Once these are known for all significant ${J}_{\mathrm{tot}},$ the rate of formation of $\mathrm{dt}\ensuremath{\mu}$ can be determined. This rate, next to the sticking fraction, is the most important parameter in determining the rate of the entire MCF cycle. We have also carried out a calculation whereby the muon was treated in a BO formalism and have found significant differences in the final results, demonstrating the importance of treating the muon as accurately as possible. This work represents a successful ab initio calculation of this reaction.
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