MMC recoil and thermalization

In this section, we shall take a more detailed look at each step in the back decay in order to estimate the energy of $\mu t$ after back decay.

When the muonic molecular complex is formed in the collision, it will receive recoil velocity . For the case of $\mu t + D_2$ collisions at the main resonance of eV, cm$\mu$s^-1. The final velocity of MMC is important since it affects the lab energy of back-decayed $\mu t$. Two kinematic extremes are the complete thermalization of MMC before it decays, and no thermalization at all. For either case, the maximum possible energy of back-decayed $\mu t$ (for a low temperature target) is obtained when ``elastic scattering'' (i.e. no excitation of the target D₂ molecule) takes place. However, even in elastic scattering, $\mu t$ is decelerated in the lab frame due to the recoil of MMC and D₂ as discussed above.

Cross sections for the interaction of has been calculated by Padial et al. [154,239]. Extrapolating a figure given in Ref. [239] to epithermal energies, the elastic scattering cross section appears to have a value of order cm^-2. The average time between elastic collision , where n is the D₂ number density, can be compared to the MMC life time, .

In the case of eV, which corresponds to the largest resonance for molecular formation in $\mu t + D_2$, the MMC recoil velocity is cms^-1, hence at a molecular density of 1.4 N₀/2 ( cm^-3, the atomic density of liquid hydrogen) corresponding to a solid at 3 K, s. Comparing this to  [156,152], we have

suggesting a fair number of collisions occur before the MMC decays.

The average number of collisions needed to slow the MMC of E_MMC^initto E_MMC^fin, can be roughly estimated, by generalizing the formula for neutron thermalization [240]:

where

(135)

For eV in $\mu t + D_2$, the MMC recoil energy is eV, and we have for E_MMC^fin=0.4 meV. Obviously, the formula B.10 gives simply a crude estimate, but together with Eq. B.9, one can assume that the MMC slows down significantly before back decay occurs.