Cycling measurements in D/T mixture

In conventional experiments [61,62,63,64,65,,67,68,69], the molecular formation rates are determined from the measurements of the cycling rate $\lambda _c$, which is $1/\tau _c$, where $\tau _c$ is the mean time between the 14 MeV neutrons from fusions catalyzed by the same muon.

In this method, the cycling rates, measured at various target conditions (such as the tritium concentration c_t, temperature T, and the density $\phi$), are fitted to an expression describing the kinetics model. Molecular formation rates $\lambda ^F _{d\mu t}$, together with quantities such as the $d\rightarrow t$ transfer rate $\lambda _{dt}$ and q_1s, are the fitting parameters. A simplified expression for $\lambda _c$ reads [67]:

 

$$\frac{1}{\lambda _c} \approx \frac{c_d q_{1s}}{c_t \lambda _{dt}} + \frac{3/4}{c_t \lambda _{1... ...1/4 + 3/4 \chi _1}{c_d \lambda ^0 _{d\mu t} {\chi _2}}, \;\;\;\;\;\; {\rm with}$$

$$\chi _1 = \frac{c_t \lambda _{10}}{c_t \lambda _{10} + c_d \lambda ^1_{d\mu t}},$$ (13)

$$\chi _2 = \frac{\lambda ^f _{t\mu t}}{c_t \phi \lambda _{t\mu t} + \lambda ^f _{t\mu t}},$$

where $\chi _1$ is the fraction of $\mu t$, $F=1\rightarrow 0$ transition with the spin flip rate $\lambda _{10}$, and $\chi _2$ is a correction factor taking into account the $t\mu t$ fusion time, with $\lambda _{t\mu t}$ and $\lambda ^f _{t\mu t}$ being $t\mu t$ formation and fusion rates, respectively (note $t\mu t$ formation is non-resonant, hence assumed to be independent of F). An improved formula would include a further correction term due to the muon recycling in $d\mu d$ and $t\mu t$ branches without producing the 14 MeV neutrons, which depends on parameters like $d\mu d$formation and spin flip rates.

The molecular formation rate depends not only on the $\mu t$ hyperfine state F, but also on the target molecular species. Thus, $\lambda ^F _{d\mu t}$ consists of different components

$$c_d\lambda ^F_{d\mu t} = c_{D_2} \lambda ^F _{d\mu t - d} +... ..._d \lambda ^{nr} _{d\mu t} \;\;\;\;\;\; \mbox{\rm for F=0,1},$$ (14)

which have to be disentangled in the fit. Often, assumptions have to be made in actual fits with guidance from theory. For example, at low temperatures all the terms in $\lambda ^F _{d\mu t}$ except $\lambda ^0_{d\mu t-d}$ are assumed to be zero [67,22]. We shall come back to the implications of these assumptions later in this thesis.

As Cohen points out [4], further complication in the conventional method comes from the lack of precise characterization of the parameter q_1s (the probability that the muon reaches the ground state of $d\mu$ before transferring to a triton). Its behavior suggested by the cycling fit is in large disagreement with theoretical expectations of its c_t and $\phi$ dependence. Independent measurements of q^pd_1s for pd transfer are emerging [37], but no direct information is yet available for the dt case.

Furthermore, the recent suggestion by Froelich and Wallenius of a resonant side-path in D/T cycling [44,49] (see Section 1.2.4) could have significant impact in the extraction of the $d\mu t$ formation rate from the cycling rate.

Finally, it should be noted that in these analyses using single rates, the energy dependences of the processes are averaged out.