Condensed matter effects

An important advance was the observation of a striking condensed matter effect in solid D₂ [77,166], where we measured an unexpectedly high $d\mu d$ formation rate. This stimulated the theoretical efforts in $\mu$CF to be extended to solid state physics. In addition to the study of thermalization processes [167], investigation of molecular formation in solids has been started by several authors [168,167,169,170].

Condensed matter effects, important mainly at collision energies comparable to or below the Debye temperature of hydrogen ( $k\Theta _B \approx 10$ meV), include the reduced mass effect which shifts the resonance energies in the lab frame, and phonon assisted resonant molecular formation. The latter is similar to the three-body collision mentioned in the previous section, but in this case it is one or more phonons which carry away the excess energy.

Fukushima made the first and so far only calculation of $d\mu t$ formation in solid hydrogen [168], but he considered only metallic hydrogen targets at extremely high pressure in order to avoid the quantum crystal nature (involving large non-harmonic lattice vibrations) of solid hydrogen at zero pressure. Therefore its applicability to our experiments is questionable even at an order of magnitude level. Menshikov and Filchenkov claimed that our measured $d\mu d$ formation rate [77,166] could be explained by the phonon assisted resonant formation alone, but again because of the lack of proper treatment of the phonon spectrum, the reliability of their calculation is rather unclear. Adamczak has recently reported the first realistic calculations of resonant molecular formation in a solid for the $d\mu d$case [170], extending his work on muonic atom thermalization in solid targets [167]. Unfortunately, there are no realistic calculations for $d\mu t$ formation in the solid state available to date. We further note that the role of condensed matter effects in the ro-vibrational relaxation of the molecular complex (perhaps involving rotons or vibrons), which affects the fusion probability W^F, is an open question.

Finally, let us note that the use of renormalized effective rates $\tilde \lambda ^F_{d\mu t}$(Eq. 2.37-2.42), which takes into account the fusion probability, needs very careful consideration as to its applicability. In the case of epithermal molecular formation, the use of the renormalized rates in the Monte Carlo calculations, as was done in Refs. [27,82] would result in a significant overestimate of the fusion yield [171]. We shall give a detailed discussion on the use of effective rates in Appendix B (Section B.1).

I shall come back to some of the theoretical details when we discuss our results. In the chapters that follow, we shall see what we can contribute to the understanding of the rich physics described in this chapter, involving resonant molecular formation as well as the muonic few body problem. We shall start in the next chapter with a description of our experimental apparatus, which makes these measurements possible.