Epithermal molecular formation

Muonic molecule formation with epithermal (non-thermalized) $\mu t$ atoms has attracted much recent interest because of the very large rate predicted by theory [70]. With the maximum rate approaching 10¹⁰ s^-1, this is nearly two orders of magnitude larger than formation rates with $\mu t$ in thermal equilibrium at room temperature ($\sim 10^8$ s^-1). Figure 1.4 shows the recently calculated epithermal resonant molecular formation rate in $\mu t + D_2$collisions [71,72], normalized to liquid hydrogen density (LHD). Also plotted is the elastic scattering rate of $\mu t$ on the d nucleus [17].

  

Figure 1.4: Resonant molecular formation rate in $\mu t + D_2$ collisions calculated for a 3 K target [71,72]. The rates are normalized to LHD and averaged over F. More detailed structure will be given in Fig. 2.4. Also shown is the $\mu t$ elastic scattering rate on the d nucleus [17].

Epithermal molecular formation, in fact, was first suggested by Cohen and Leon [14], and independently by Kammel [13], to explain the fusion time spectra observed in the PSI experiment with low density gas [64], in which an unexpected prompt peak was initially misinterpreted as due to the hyperfine effect. Later Jeitler et al. [74,22] carried out Monte Carlo studies in the homogeneous mixture of D/T. They obtained qualitative agreement between the experimental data and the simulations assuming strong epithermal resonances, but neither the strength nor the position the resonances could be determined by their analysis, because (a) the neutron time spectra are essentially insensitive to the position of epithermal resonances, and (b) the magnitude of the prompt peak is dependent both on the formation rate strength and the poorly known initial population of epithermal $\mu t$, hence with the lack of knowledge of the latter, the former cannot be extracted reliably.

In order to access these resonances with thermal energies in a conventional target, one would require a temperature of several thousand degrees. The Dubna group has recently developed a target which can operate at up to 800 K, but this is still substantially smaller than the required temperature for the main resonances. Efforts are being made by the PSI/Vienna group to develop an ambitious target system for 2000 K, but the technical difficulties in handling tritium at such a high temperature have so far prevented the realization of such a system.

In this thesis, we will develop an alternative approach [76,77,78,79] for investigating the epithermal molecular formation, as illustrated in Fig. 1.5. Taking advantage of the Ramsauser-Townsend effect, a neutral atomic beam of muonic tritium is obtained [80]. By placing a second interaction layer, separated by a drift distance in vacuum, reaction measurements are possible in an event-by-event fashion. This is in sharp contrast to conventional methods where a muon is stopped in a bulk gas or liquid target in which complex, and inevitably interconnected chains of reactions take place, as seen in Section 1.3.1. The use of the muonic atom beam would thus help us to isolate the process of interest from the rest of the ``mess.'' Furthermore, the time of flight of $\mu t$ across the drift distance provides information on the $\mu t$ kinetic energy, hence the energy dependence of the formation rates can be tested.

  

Figure 1.5: Conceptual drawing of the time-of-flight measurement of resonant molecular formation rates using the $\mu t$ atomic beam.

The Basic processes involved in our experiments are as follows [81,82]:

Production:

Muonic atoms are produced by stopping low-momentum muons ($\sim 27$ MeV/c) in a thin layer ($\sim 0.4$ mm) of hydrogen with a small tritium concentration. The knowledge of the fraction of muons which stops in the hydrogen layer is of great importance for our measurements, and will be discussed in Section 6.3.

Acceleration:

The muon transfer reaction $\mu p + t \rightarrow p + \mu t$, taking place in $\sim 100$ ns [83], is used to create a fast $\mu t$ with a kinetic energy of $\sim 45$ eV, which results from the reduced mass difference in the binding energy.

Extraction:

$\mu t$ loses its energy via scattering mainly with protons, until it reaches around 10 eV, where due to the Ramsauer-Townsend (RT) effect, the $\mu t + p$ elastic cross section drops by a few orders of magnitude, and the rest of the target becomes nearly transparent. Since there is no RT effect in the $\mu p + p$ scattering, $\mu t$ is selectively extracted from the layer with the characteristic energy of $\sim 10$ eV. The understanding of the $\mu t$ emission mechanism is essential to our measurements, and will be discussed in Chapter 7.

Moderation:

The $\mu t$ beam energy can be degraded, when necessary, by placing a moderation layer on top of the production layer. For the epithermal formation rate measurement, where the largest peak lies at around 0.5 eV, the beam energy of $\sim 10$ eV is indeed too high, hence a D₂ overlayer was used as a moderator (not shown in Fig. 1.5). A thin layer ( $\sim 5\; \mu$m) of D₂ is sufficient to slow down the $\mu t$, due to the large elastic cross section (see Fig. 1.2). Fusion events also occur at the upstream (US) moderation layer, which become background to the fusion at the downstream (DS) reaction layer, but because of their mostly prompt time structure, the US events can be separated from the DS signal.

Reaction:

The $\mu t$, after travelling across the drift distance, collides with a target D₂ molecule. Resonant $d\mu t$ formation can take place in two different manners: (a) the direct process, where $d\mu t$ is formed directly before $\mu t$ is scattered, and (b) the indirect process, in which $\mu t$ first loses energy in collision with another D₂ molecule, before molecular formation (see Fig. 1.4 for a comparison of the rates). The process (a) is most interesting because it preserves the correlation between the $\mu t$ time of flight (which is roughly equal to the time between the muon stop and the detection of a fusion product, due to short emission and fusion times) and the energy at which molecular formation takes place. On the other hand, in the process (b), while the correlation is obscured ( e.g., a fast $\mu t$ with a short time-of-flight can slow down significantly in the reaction layer and form $d\mu t$ at much lower energy), it can still give us useful information, such as the magnitude of the resonance. The use of a thin layer, together with a moderation of the $\mu t$ energy mentioned above, reduces the fraction of the indirect process, thereby enhancing our sensitivity.

There are, of course, difficulties and disadvantages involved in our approach. In addition to the obvious technical challenges in dealing with cryogenic and ultra-high vacuum targets with the potential hazard of tritium, there are difficulties and limitations which we did not foresee in advance (which are reflected in the volume of this thesis!). They will be discussed in due course.

Before describing the details of our measurements and the analysis, we shall discuss some theoretical aspects of muon catalyzed fusion in the following chapter, with particular emphasis on the muonic three body problem and molecular formation, in order to highlight the importance of the physics involved in our experiment.