A literal interpretation of the parameter is that it represents the average number of d-t fusions a muon can catalyze during its lifetime. Equation (12.21) could be evaluated if the muon injection rate density F^(t) is known as a function of time together with the time dependent concentration N^dt(t); the former is specified by accelerator operation but the latter can only be determined from a solution of the system of dynamical particle balances given by Eqs.(12.15) in combination with the energy dynamics in the reaction domain. To avoid having mathematical complexity obscure the physical features of the problem, we will assume steady-state operation. That is, we assume a constant fuel temperature

allowing therefore a constant reaction rate parameter and also take a constant injection rate, F^(t) = F^° during the operating time x of interest. Hence, after an initial transient during start-up, the density of the NMdt(t) molecular ions reaches a constant value, N0^, characteristic of steady-state. Then the muon recycle

Note that the reactor operating time x will invariably be much in excess of the muon mean lifetime so that x + хд = x.

By maintaining constant fuel and muon densities by appropriate feed rates, a constant N^d, implies that all the other intermediate particle densities will also be constant in time; that is, we will have in Eqs.(12.15)

dN fi _ dN і _ dN d _ dN ju _ dN ^ _ dN fjdt

dt dt dt dt dt dt Under these conditions, the system of linear algebraic equations for N^, N^, and N^dt, Eq.(12.15a) and Eqs.(12.15d) to (12.15f), can be solved to yield an explicit ratio N’^ / F^° as required for Eq.( 12.22). This gives

NU_

Вц)

and may be interpreted as a muon "residence unavailability" penalty. In compact form the muon recycling efficiency therefore reduces to

‘ 1 Л

a + Bnj

For the case of muon catalysis at liquid hydrogen conditions, Table 12.1, the muon recycle efficiency is calculated to be

Ад-34. (12.26)

That is, on average, one muon catalyzes some 34 d-t fusions during its mean lifetime of 2.2×1 O’6 s.

The energy multiplication assessment follows similarly from a substitution of Eq.(12.21) into Eq.(12.20); we take Qdt = 17.6 MeV and use an estimate of Ец ~ 3000 MeV to find

This provides a useful estimate of the energy viability of a muon catalyzed fusion system which, as is evident, is too low by perhaps a factor of thirty.

Some additional considerations may, however, be introduced which suggest possible increases in ME. Clearly Qdt is a constant and Ецс could not be further reduced except for possible altemative-and highly speculative-methods of muon production. Considerable research has in recent years been undertaken to determine if Хц, the average number of d-t fusions catalyzed by one muon, could be increased. Experiments have revealed that at specific temperatures and at significantly elevated pressures, values of 160 are possible-corresponding to an increase in the energy multiplication of Eq.(12.27) by a factor of about 3.5. However, while a definite dependence on medium temperature and composition has been established, it is not yet evident that an according optimization can be sufficient for energy viability.

There exists, however, another approach which can be summarized by the following. Supposing the fusion domain is surrounded by a neutron multiplication and breeding blanket domain, the function of which is to multiply the neutron by (n, xn) reactions and to breed both tritium and fissile fuel for companion fission reactors. (Fusion-fission hybrids and similar integrated systems will be discussed in greater detail in Chs.14 and 15.) The fission energy eventually thus generated might be considered a benefit. For example, if a fission

(12.28)

to yield a significant positive energy balance. Such a hybrid system, based on muon-catalyzed fusion, would have to be evaluated on the same basis as other fusion based hybrid concepts and will be further discussed. A potential advantage of this system could be the relative simplicity of the reaction chamber; however, accelerators which produce sufficient numbers of collimated muons at an acceptable average energy cost per muon represent a significant design uncertainty.