Epithermal Collisions

Let us consider an epithermal collision between $\mu t$ and a DX molecule, with a $\mu t$ lab energy (velocity ). Throughout this appendix, solid state effects such as lattice binding and phonon exchange are neglected unless otherwise specified.

Resonant scattering may occur via the following sequence of processes: (1) the molecular complex (MMC) is formed in a collision , (2) MMC receives a recoil from the impact of $\mu t$, (3) MMC may be (partly) thermalized via collisions with the rest of the target molecules, (4) ro-vibrational states of MMC may change as a result of collisions with other molecules, and (5) back decay occurs leaving molecule DX in either the ground state (elastic channel) or in an excited state (inelastic).

The energy of the $\mu t$ after back decay is important. It can be shown that if the $\mu t$ energy is the same before and after the back decay, the effective formation approximation gives the correct fusion yield (condition (b) above). However, if the $\mu t$ energy changes such that it is removed from the resonant region, the effective formation approximation fails as we shall see. It should be noted that resonant structure has a narrow width for low temperature targets, therefore a small change in the $\mu t$ energy is sufficient to remove it from the resonance.

Even in the completely elastic case (no MMC thermalization, no MMC relaxation, and back decay via the elastic channel), the mean energy of $\mu t$in the lab frame is significantly reduced after back decay, because of recoil of the MMC in process (2) and that of DX in (5). In reality, MMC thermalization, MMC ro-vibrational relaxation, and back decay via inelastic channels, all give contributions to reducing the $\mu t$ energy, hence the $\mu t$ is likely to be removed from the resonance region.