In order to emphasize some essential concepts involving multi-stage low temperature reactions, we first consider the following general reaction process involving arbitrary а-type and b-type particles:

cl + b —> ^ c + d. (12.7)

Here, the formation of the intermediate (ab) comes about as a result of a binary collision process and this intermediate is subsequently transformed by a decay process. Thus, though the same straight arrow symbol is used for the formation and decay processes in the reactions of Eq.(12.7), the two constitute distinctly different physical phenomena.

We introduce a reaction parameter каь for the formation of the intermediate species (ab) in reaction (12.7) so that the rate density of formation of the new species (ab) of density Nab is given by

R+ab = KabNaNb ■ (12.8)

Thus, if this formalism is applied to the d-t fusion process, we can equate = <cv>dt.

The decay of (ab) in reaction (12.7) is equivalent to the process of radioactive decay of an (ab) particle of density Nat>- Thus, the decay rate density can be characterized by a mean lifetime 1/А, аь for the intermediate, giving therefore

R-ab = XabNab • (12-9)

By inspection of the reactions in Eq.(12.7), the rate equations for the various particles a, b, ab, c and d in terms of their densities Na, Nb, Nab, Nc and Nd are given by the following:

dNa

dt

dNb

dt

With the imposition of initial conditions for each of the reacting species, the dynamical description of reaction (12.7) is thus fully specified. Note, however, that these equations are nonlinear and also display some redundancy.

The above characterization will now be used to provide a reduced dynamic description of the muon-catalyzed d-t fusion process. As an initial simplified case, if muon decay and its parasitic loss by alpha capture are ignored, then a typical muon-catalyzed chain would be written, with x in the reactions of Eqs. (12.3) replaced by p, as

(12.13)

where is the muon decay constant. The reaction shown here represents the

dominant processes involved in muon-catalyzed d-t fusion. However, a number of additional reactions are possible as we display in Fig. 12.2.