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Total and effective formation rates

For our analysis, it is convenient to express the formation rates as a function of the $\mu t$ laboratory (lab) energy, as opposed to the target temperature given in Eq.  2.27. Specifying the initial and final states, we then have[*]:

\lambda ^{F}_{d\mu t}(E_{\mu t}^{lab}) =
\sum _{\nu _fS K_f}\lambda ^{FS}_{\nu _f, K_f},
\end{displaymath} (47)

$\displaystyle \lambda ^{FS}_{\nu _f, K_f}$ = $\displaystyle \sum _{K_i} \omega _K (K _i)
\lambda ^{SF}_{\nu _i=0 K_i, \nu _f K_f}$ (48)
$\displaystyle \lambda ^{SF}_{\nu _i K_i, \nu _f K_f}$ = $\displaystyle 2\pi N \gamma (E_{\mu t}^{lab}, \epsilon _{res})
A_{if} \vert<i\vert H'\vert f>\vert^2$ (49)

where $\omega _K(K_i)$ is the initial rotational population distribution, which for equilibrated targets is the standard Boltzmann distribution, and Aif is a coefficient which depends on initial and final quantum numbers. The factor $\gamma (E, \epsilon _{res})$ is the Doppler broadening profile due to target motion and recoil, whose exact form is derived in Ref. [133], but for $E_{\mu t}^{lab} >> kT$ can be approximated [157] as a Gaussian distribution with the width of:

\sigma _D = \sqrt{ \frac{4E_{\mu t}^{lab}kT M_{\mu t}}{M_{D_2}}}.
\end{displaymath} (50)

In Eq. 2.35, a $\delta$ function resonance profile was assumed (the classical Vesman model), but even with that, the formation rate in the lab frame has a distribution with non-zero width due to the above-mentioned Doppler broadening.

At epithermal energies, the transition formation matrix elements become very large, which means both the formation rate and back decay width are large, resulting in a significant probability for $d\mu t$ not fusing but returning to the entrance channel $\mu t +
D_2$. The effective formation rate is a renormalized rate taking into account the fusion probability, as defined in Ref. [134]:

\tilde \lambda ^F _{d\mu t}= \sum _{\nu _f, S, K_f} W_{\nu _f}^{SF} (K_f)
\lambda _{\nu _F}^{SF},
\end{displaymath} (51)

$\displaystyle W_{\nu _f}^{SF} (K_f)$ = $\displaystyle \frac {\tilde \lambda _f}
{\tilde \lambda _f + \Gamma _{\nu _f K_f}^{SF}},$ (52)
$\displaystyle \Gamma _{\nu _f K_f}^{SF}$ = $\displaystyle \sum _ {\nu _i' K_i'} \Gamma ^{SF}_{\nu _f K_f \nu _i' K_i'}.$ (53)

For a high density ($\phi$ more than about 0.1) target such as ours, Faifman assumes complete rotational relaxation of the Kf levels, hence dropping the Kf dependence,
$\displaystyle \tilde \lambda ^F _{d\mu t}$ = $\displaystyle \sum _{\nu _f, S} W_{\nu _f}^{SF}
\lambda _{\nu _f K_f}^{SF},$ (54)
$\displaystyle W^{SF}_{\nu _f}$ = $\displaystyle \frac {\tilde \lambda _f}
{\tilde \lambda _f + \Gamma _{\nu _f}^{SF}},$ (55)
$\displaystyle \Gamma _{\nu _f}^{SF}$ = $\displaystyle \sum _{K_f} \omega _B (K_f)
\Gamma ^{SF}_{\nu _f K_f},$ (56)

where $\omega _B (K_f)$ is the Boltzmann distribution of the Kf states [*].

The effective fusion probability for the molecular complex is defined as [133]:

W^F=\frac {\tilde \lambda ^F_{d\mu t} + \lambda ^{nr} _{d\mu t}}
{\lambda ^F _{d\mu t} + \lambda ^{nr} _{d\mu t}},
\end{displaymath} (57)

where $\lambda ^{nr} _{d\mu t}$ is the rate for non-resonant formation, which does not back-decay. For $\tilde \lambda ^F_{d\mu t} >> \lambda ^{nr}
_{d\mu t}$, WF can be written:

W^F=\frac {\displaystyle \sum _{\nu _f, S, K_f} W^{SF}_{\nu...
...playstyle \sum _{\nu _f, S, K_f}
\lambda ^{SF}_{\nu _f K_f}},
\end{displaymath} (58)

which can be understood as the average of fusion probabilities from each state weighted by the formation rate of that state.

Figure 2.4 shows the molecular formation rates $\lambda
_{d\mu t}^{F}(E)$ and effective fusion probabilities WF for Ki=0(ortho) and Ki=1 (para) cases, calculated by Faifman et al. [70,71,72] for a 3 K target. Also shown are the effective rates $\tilde \lambda ^{F}_{d\mu t} \approx W^F
\lambda _{d\mu t}^F $. Note that $\lambda ^{nr} _{d\mu t}$ less than about 107s-1 is invisible in the scale plotted.

Figure 2.4: Formation rates $\lambda ^F _{d\mu t}$ for $\mu t + D_2
\rightarrow [(d\mu t)dee]^*$ (top) and the fusion probability WF(bottom), calculated by Faifman [70,71,72] for 3 K. Also shown in dashed lines are the effective rate $\tilde \lambda _{d\mu
t}^F \sim \lambda ^F_{d\mu t} W^F$. The rates are normalized to liquid hydrogen density.

next up previous contents
Next: Subthreshold resonances Up: The standard Vesman model Previous: Decay of the molecular