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:

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

where

 

$$\lambda ^{FS}_{\nu _f, K_f} \sum _{K_i} \omega _K (K _i) \lambda ^{SF}_{\nu _i=0 K_i, \nu _f K_f}$$ = (48)

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

where $\omega _K(K_i)$ is the initial rotational population distribution, which for equilibrated targets is the standard Boltzmann distribution, and A_if 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:

$$\sigma _D = \sqrt{ \frac{4E_{\mu t}^{lab}kT M_{\mu t}}{M_{D_2}}}.$$ (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]:

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

where

  

$$W_{\nu _f}^{SF} (K_f) \frac {\tilde \lambda _f} {\tilde \lambda _f + \Gamma _{\nu _f K_f}^{SF}},$$ = (52)

$$\Gamma _{\nu _f K_f}^{SF} \sum _ {\nu _i' K_i'} \Gamma ^{SF}_{\nu _f K_f \nu _i' K_i'}.$$ = (53)

For a high density ($\phi$ more than about 0.1) target such as ours, Faifman assumes complete rotational relaxation of the K_f levels, hence dropping the K_f dependence,

   

$$\tilde \lambda ^F _{d\mu t} \sum _{\nu _f, S} W_{\nu _f}^{SF} \lambda _{\nu _f K_f}^{SF},$$ = (54)

$$W^{SF}_{\nu _f} \frac {\tilde \lambda _f} {\tilde \lambda _f + \Gamma _{\nu _f}^{SF}},$$ = (55)

$$\Gamma _{\nu _f}^{SF} \sum _{K_f} \omega _B (K_f) \Gamma ^{SF}_{\nu _f K_f},$$ = (56)

where $\omega _B (K_f)$ is the Boltzmann distribution of the K_f states .

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

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

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

$$W^F=\frac {\displaystyle \sum _{\nu _f, S, K_f} W^{SF}_{\nu... ...playstyle \sum _{\nu _f, S, K_f} \lambda ^{SF}_{\nu _f K_f}},$$ (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 W^F for K_i=0(ortho) and K_i=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 W^F(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.