In the traditional description of the CF cycle, it was assumed that the muonic atom was always thermalized at the target temperature , and the energy dependence of the reaction rates, if considered at all, was averaged out by integrating over the Maxwellian distribution. This type of analysis using the constant rates has proven very successful in particular for the pure liquid and gaseous deuterium system .
The importance of going beyond the constant rate approach was recognized by the suggestion of epithermal (i.e., non-thermalized) transient phenomena by Kammel  and by Cohen and Leon , but it was not until the complete set of theoretical cross sections [15,16,17,18,19,20,21] became available that an energy dependent analysis could be performed in CF .
It is absolutely essential for our measurement to take into account the energy dependence; or rather, our experiment is designed to take advantage of its sensitivity to the energy dependent cross sections.
Theoretically, collisional processes of muonic atoms present challenging few-body problems. Because the muon mass is comparable to that of nuclei, the system is highly non-adiabatic, i.e., nuclear and muonic motions cannot be separated, as we shall see in detail in Section 2.1.2. The main collisional processes of muonic hydrogen isotope atoms include elastic scattering, muon transfer (also called charge exchange), and hyperfine transitions (spin flip). Table 1.1 lists the main properties of muonic atoms, while Fig. 1.2 compares the various cross sections of collisions with hydrogen isotope nuclei.