An assessment of the energy liberated in a nuclear reaction-or in a sequence of reactions-relative to the energy cost of causing that reaction, is fundamental for evaluating the attractiveness of any nuclear energy system. Recall that such a criterion was employed in the energy balance assessment for magnetic and inertial confinement fusion. We now seek to formulate a similar energy balance criterion for a muon-catalyzed d-t system, i. e. a |xdt system.

One unique feature of muon-catalyzed fusion is that muons have to be produced by appropriate accelerators and then directed into a liquid deuterium — tritium mixture. The accelerator beam energy requirements are such that the average energy cost of a muon is of the order of 3000 MeV. Then, since each d-t fusion reaction catalyzed by a muon releases the usual 17.6 MeV of fusion energy, energy breakeven will require each muon to catalyze, on average, several hundred d-t fusion reactions during its short lifetime of 2.2×10"6 s. Consequently, the rates at which the various processes of Fig. 12.2 occur are most important.

To establish a tractable formulation for this energy balance problem, we consider a unit volume of liquid deuterium and tritium into which muons are injected at a rate density Fig.12.3. These muons then initiate and sustain a complex reaction network as suggested in Fig. 12.2. The fusion energy so generated heats the fluid-fuel mixture; this heat is transported with the moving fluid to heat exchangers for subsequent conversion. For the muon-sustained reactions, it is known that an acceptable loss in accuracy results if the reduced reaction network of Fig. 12.4 is used instead of that of Fig. 12.2.

Here, in Eq. (12.15b), the factor Kd. t is to account for the net reaction transfer from pd to pt, also shown in Fig. 12.4. These equations represent a so-called point-kinetics representation in the sense that any spatial effects can be

considered to be minimal.

By analogy to the energy viability analysis of magnetic and inertial systems, we consider a reactor chamber of volume Vr and an operating period x during which muon injection occurs. Then, the total energy supplied is

T T

El = V rFu (0 E^dt = Vr J F^iDdt (12.16)

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where Euc is the average energy cost of producing one muon. The total energy released as a consequence of muon injection can be written as

where now E^s is the average energy deposited in the medium per muon as the high energy muons are slowing down. Note the addition of the muon mean lifetime Тц to the injection period; this is necessary because a muon will continue

to catalyze nuclear reactions during its lifetime after injection has ceased.

Since the reaction energy Qdt is released whenever the (|adt)-fusion compound decays, we write for the fusion rate density in Eq.(12.17)

Here, the ratio ЕЦ5/ЕЦС accounts for the recoverable muon beam energy and is of the order of 1/10. Of greater interest therefore is the first term of Eq.(12.20) where Q* and E,,c are constants and the remaining factor defines the important parameter which represents the average number of d-t fusion events catalyzed by one muon:

i-riM

f^N^ftJdt

ZM = —T—————— • (12-21)

J FM(t)dt

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This parameter fa, called the muon recycle efficiency, is evidently of utmost importance and will be considered in detail next.