Decay of the molecular complex

Once the [($d\mu t$)dee] complex is formed, it can either stabilize itself, which leads to fusion, or go through a back decay [151] into the entrance channel $\mu t + D_2$:

 

$${[(d\mu t)^S _{J\nu }dee]_{\nu _f K_f}} \mbox{\hspace*{3cm}} \mbox{} \mbox{} \mbox{} \mbox{}$$

$$\longrightarrow (\mu t)_F + [D_2]_{\nu _i' K_i'} (\Gamma ^{SF}_{\nu _f K_f \nu _i' K_i'})$$ (45)

$$\longrightarrow {\rm stabilization} (\tilde \lambda _f)$$ (46)

The back decay width $\Gamma ^{SF}_{\nu _f K_f \nu _i' K_i'}$ is determined by the same matrix element as the formation, though it should be noted that, in general, $\nu _i' K_i' \neq \nu _i K_i$ (see footnote).

The rate for stabilization, often called the effective fusion rate $\tilde \lambda _f$, includes contributions from: (1) fusion from the $(d\mu t)_{11}$ state, (2) radiative decay of the $(d\mu t)_{11}$ into a lower state, (3) Auger deexcitation of $(d\mu t)_{11}$, and (4) collisional deexciation of the complex due to interaction with the surrounding environment. Because of the centrifugal barrier, fusion from J=1 states is relatively slow (< 10⁸ s^-1; see Ref. [3] for a recent summary), and Lane estimates the radiative decay of $d\mu t$ to be even slower ($\leq 10^6$ s^-1 [151]). Hence $\tilde \lambda _f$ is dominated by the Auger deexcitation rate, at least at low densities. This rate has been calculated by several authors with an increasing degrees of sophistication, most recently by Armour et al. [152], who took into account the molecular nature of the host complex, and estimated the rates for $(J,v)= (1,1)\rightarrow (0,1)$ to be in the range (6.9-10.3) $\times 10^{11} \; {\rm s}^{-1}$, depending on the model used.

The collisional rotational de-exciation of the complex was first calculated by Ostrovskii and Ustimov [153], and later by Padial, Cohen and Walker [154]. The latter, who claim better accuracy but still neglect the ro-vibrational transitions in the target (i.e., the surrounding) molecule, found the rates substantially smaller than the former (by a factor of two to ten depending on the transition).

To our knowledge, there is no accurate calculation of collisional vibrational quenching, except a rough estimate by Lane of $\sim 10^7 \; {\rm s}^{-1}$ at room temperature [151,155], which is much slower than other processes. However, he claims this rate increases drastically with target temperature (for equilibrated targets) and increasing $\nu$ [155], rising to the order of 10¹⁰ s^-1at 2000 K, therefore it may compete with Auger decay for the molecular formation at epithermal energies with high $\nu _f$ (see footnote).

In Faifman's calculations used in our analysis [71,72], the rotational relaxation rate of $\lambda _{K _f K _f'} = 10^{13} \phi$s^-1, with $\phi$ being the target density in units of LHD (liquid hydrogen density), and the effective fusion rate of $\tilde \lambda _f = 1.27\times 10^{12}$s^-1 [156], were used. The latter is the sum of the dipole E1 Auger transition from the (J,v)=(1,1) state to (0,1), (2,0), or (0,0) with each rate being 11.4, 1.3 and 0.02 ( $\times 10^{11}$ s^-1), respectively.