A Lawson-inspired Cycle-Closure Criterion for Deuterium--Tritium Muon-Catalyzed Fusion

Abstract

Deuterium--tritium muon-catalyzed fusion is limited by a cycle-closure problem: a negative muon must complete enough catalytic cycles before decay or effective alpha sticking removes it from reuse. We formulate a Lawson-inspired criterion for this single-muon cycle. The effective cycle strength is defined as ${\mathcal{L}}_{\mu }={\Lambda }_c{\tau }_{\mu }$, where ${\Lambda }_c$ is the effective cycle-completion rate and ${\tau }_{\mu }$ is the muon lifetime. Together with the residual effective sticking probability ${\omega }_S^{\mathrm{eff}}$, it gives the mean fusion yield per useful muon, $N_{\mathrm{fus},\mu }={\mathcal{L}}_{\mu }/(1+{\omega }_S^{\mathrm{eff}}{\mathcal{L}}_{\mu })$. Introducing the useful D--T cycle energy $E_{\mathrm{use}}$, the system factor ${\eta }_{\mathrm{sys}}$, and the effective muon cost $E_{\mu }^{\mathrm{cost}}$, the one-muon gain is $G_{\mu }=({\eta }_{\mathrm{sys}}{E}_{\mathrm{use}}/E_{\mu }^{\mathrm{cost}})N_{\mathrm{fus},\mu }$. This leads to the required cycle strength ${\mathcal{L}}_{\mu }^{\mathrm{req}}=G_{\mu }{N}_L/(1-{\omega }_S^{\mathrm{eff}}{G}_{\mu }{N}_L)$, with $N_L=E_{\mu }^{\mathrm{cost}}/({\eta }_{\mathrm{sys}}{E}_{\mathrm{use}})$, and to the conditional sticking boundary ${\omega }_S^{\mathrm{eff}}<1/(G_{\mu }{N}_L)$. The criterion separates rate-limited, sticking-limited, and cost-limited regimes in the $({\omega }_S^{\mathrm{eff}},{\mathcal{L}}_{\mu })$ plane. When representative historical D--T $\mu \mathrm{CF}$ anchors are projected onto this plane, they lie in a high-yield region but remain constrained by the effective-sticking boundary under conventional multi-GeV muon-cost accounting. The framework provides a compact diagnostic for assessing whether future improvements act mainly by increasing the effective cycle-completion rate, reducing residual sticking, or lowering the useful cost of delivered muons.

Comments: 7 pages, 2 figures