Resonant scattering of $\mu t$

In this appendix, we discuss the influence of resonant scattering in the modelling of the muon catalyzed fusion processes. As we have seen elsewhere in this thesis, resonant formation of $d\mu t$ molecular ions in a loosely bound state of J=v=1 occurs via formation of a metastable muonic molecular complex (denoted MMC hereafter) in a collision of $\mu t$ on a molecule DX,

where X = H,D,T and x = p,d,t. Here we shall denote the total rate for the formation of the complex as . After the formation of the complex, two competing processes take place: stabilization of the complex leading to fusion (mainly via Auger de-excitation of the $d\mu t$ molecule) with the effective fusion rate $\tilde \lambda _f$, and back decay [151] to with the rate . The above rates generally depend on the quantum numbers, such as ro-vibrational, spin, and hyperfine states, of initial and final states, but in the first part of this appendix we consider the averaged rates for simplicity and drop indices for quantum numbers. This simplification does not affect our conclusion.

In conventional analyses of $\mu$CF, the effective formation rate , a renormalized rate taking into account the back decay probability, has been widely used, and is defined (when ignoring the indices for the quantum numbers) as:

(126)

where W is the branching ratio for the fusion channel

(127)

With the recent recognition of the importance of transport properties of muonic atoms, theoretical cross sections for muonic atom scattering have been calculated to high accuracy with sophisticated methods. Despite the vast theoretical efforts in pushing the accuracy of various scattering cross sections, including electronic screening, atomic and molecular structure, and most recently solid state effects, little detailed attention has been paid to the back decay process as a scattering mechanism of muonic atoms, except in the context of spin flip in the $d\mu d$ system [34].

In early work, resonant scattering processes were neglected or treated incorrectly. For example, in the pioneering studies of Markushin [75,27,9,82], who first performed the full three-dimensional Monte Carlo calculations of muonic processes, renormalized effective formation rates were used to account for the back decay processes. On the other hand, in the theoretical analysis of the $\mu$CF kinetics in D/T mixture [238], Somov claimed that $\mu t$ emitted after back decay of the muonic molecular complex has a thermal distribution with a temperature of the surrounding medium. Jeitler et al. followed this in their Monte Carlo analysis of D/T and H/D/T mixtures [74,22]. They claim that this is justified when one assumes complete thermalization (of the translational motion) and the rotational relaxation of the MMC [74].

The purpose of this appendix is to point out the importance of the resonant scattering process resulting via backdecay of the molecular complex, particularly in the atomic beam type of experiments, as performed in this thesis. We will show first that the use of renormalized effective rates as was done in Refs. [75,27,9,82] significantly overestimates the calculated fusion yield, and second that thermalized $\mu t$ emission after back decay is not justified even if one assumes the complete MMC thermalization.