Time-of-flight analysis

  

Figure 8.19: The simulated correlations between the fusion time and the energy at which $d\mu t$ molecular formation takes place for the TOF measurements using 3 T$\cdot l$ layer.

  

Figure 8.20: The simulated time vs. energy for total events in a thick (20 T$\cdot l$) layer, where the correlation is weaker than the thin layer.

In this section, we shall extract two important physics parameters, the rate for the resonant molecular formation and its resonant energies. We show in Fig. 8.19 the results of our Monte Carlo calculations for a 3 T$\cdot l$ DS layer using the nominal input to illustrate the features of our TOF measurements. Plotted are simulated two dimensional scatter-plots of the fusion time and the energy at which $d\mu t$ formation takes place. The direct events (top) refer to a process in which $\mu t$ forms $d\mu t$ directly, before being scattered by D₂ in the DS layer, while the total events (bottom) include also indirect processes where the $\mu t$ first loses energy in collision with another D₂ molecule in the DS layer. The direct process shows a strong correlation between the fusion time and formation energy exhibiting individual resonance structure as shown on the projection on the time axis, whereas for the total events the correlation is obscured by the indirect processes. The role of the indirect processes is more prominent, hence the correlations further weakened, in thick DS layer measurements as shown Fig. 8.20, in which only the total events are plotted (the direct process is nearly independent of layer thickness in these examples). In addition, the thick layer measurements are sensitive to the low energy process as also seen in the US measurement, thus there are relatively large theoretical uncertainties due to solid state effects (notice that a considerable amount of fusion is taking place at very low energies in Fig. 8.20). This is why we focused our measurements and analysis on thin layers. It should be stressed, however, that despite the presence of the indirect process, a significant correlation is preserved between the fusion time and formation energy in the thin layer measurements.

In extracting physical quantities, formation rate and resonance energy, our approach is to perform iterative fits to the data using Monte Carlo calculations with varied input parameters. We varied the formation rate and the resonant energies by scaling

  

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where $S_{\lambda }$ is taken as energy-independent, and S_E, rate-independent. The fusion probability W was kept at Faifman's value during scaling.

We made full use of our accurate absolute fusion yield determination. In each fit (i.e., fit to each Monte Carlo result for a particular physics scaling input), the $\chi ^{2}$ was minimized by varying the overall normalization by the Monte Carlo spectrum by a factor . But since our MC yields are already normalized, taking into account factors such as the number of incident muons, the muon stopping fraction, and the Si solid angle, together with experimental corrections due to nitrogen contamination (if present), $d\mu d$ proton contribution, the energy cut efficiency, and systematic effect in the background subtraction, all of which have been discussed in great detail in this thesis, we expect in an ideal case. Thus constraining to its uncertainty (which is the relative error in our overall normalization), we define our $\chi ^{2}$ as:

where h^exp_i, h^MC_i are experimental and simulated counts in bin i, with , being the respective uncertainties. Thus the likelihood function is maximized, hence $\chi ^{2}$ minimized, when the shape and the yield match between experiment and Monte Carlo.

  

Figure 8.21: Fit of the calculated MC spectrum (upper histogram) to the experimental data (error bars) for 3 T$\cdot l$ in Series A, giving (confidence level 54.6%). The contribution from the direct process is also plotted (lower histogram). Both the data and MC results are normalized to GMU.