Treatment of the $t\mu +{\mathrm{D}}_2$ reaction by the methods of quantum reactive scattering

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 $\mathrm{dt}\mu$ muonic molecular ion, in which fusion takes place very rapidly. We have calculated reaction probabilities for the resonances that occur in $t\mu +{\mathrm{D}}_2$ scattering for incident kinetic energies less than 0.6 eV and total angular momentum $J_{\mathrm{tot}}=0.\mathrm{}$ 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 $\mathrm{dt}\mu$ part of the PES was scaled to allow it to exhibit the correct binding energy of the crucial $\mathrm{}(J,v)=(1,1)\mathrm{}$ 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 $\mathrm{dt}\mu$ 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, ${\Gamma }_e^{J_{\mathrm{tot}}}.$ Once these are known for all significant $J_{\mathrm{tot}},$ the rate of formation of $\mathrm{dt}\mu$ 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.