The complete analysis of this complex chain of processes is beyond the scope of this appendix, and we deal with two limiting cases of (b), i.e., no rotational relaxation at all, and complete rotational relaxation (no vibrational relaxation assumed in either case). We develop in the following expressions for the double differential rates for resonant scattering which depend on both outgoing and incoming energies in the lab frame . Our formulation is analogous to the resonant formation rate calculations by Faifman et al. . We explicitly treat the back-decayed energy , paying attention to the energy balance. Thermal motion of the MMC is included to give the Doppler broadening  of the distribution.
In the first limit of no relaxation at all in the MMC states, the rate for
the resonant scattering rate
can be written:
In the other limit that MMC rotational relaxation is
complete, the following substitution should occur:
Because the expression given in Eq. B.13 is rather
complicated, we can alternatively take advantage of the already calculated
and fusion probability WF to
write, in the limit of no MMC rotational relaxation, that
Note in Eq. B.18, we made an approximation by factorizing the state dependence, and this is not rigorously accurate when the state dependence of each factor is large, hence the approach given here should be taken as a first approximation. On the other hand, the first expression given in Eq. B.13 does not rely on the factorization approximation hence it is more accurate, though more complicated.
In order to numerically evaluate these expressions, we need some several hundred matrix elements for MMC transitions. It should be stressed, however, these have been already calculated, for example by Faifman et al. for calculations of , thus with their assistance we can readily estimate the energy distribution of back-decayed (within the approximation that the MMC is translationally thermalized, and is rotationally relaxed or not relaxed at all).
We note that in case of the elastic scattering, with translationally thermalized MMC, the energy of back-decayed is given by the resonance energy divided between and DX (convoluted with the Doppler broadening profile). Hence the claims by Somov and Jeitler [238,74,22] of thermalized after back decay is not physically justified. Jeitler also considered the limit of no MMC rotational relaxation at all, but the simple expression given for the energy distribution for that case ( , Eq. 4.72 in Ref. ) is not accurate, as the back decay into (the resonant excitation of DX), and the Doppler broadening due to MMC thermal motion are neglected.