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 $\omega_S^{\rm eff}$, it gives the mean fusion yield per useful muon, $N_{\rm fus,\mu}=\mathcal{L}_\mu/(1+\omega_S^{\rm eff}\mathcal{L}_\mu)$. Introducing the useful D--T cycle energy $E_{\rm use}$, the system factor $\eta_{\rm sys}$, and the effective muon cost $E_\mu^{\rm cost}$, the one-muon gain is $G_\mu=(\eta_{\rm sys}E_{\rm use}/E_\mu^{\rm cost})N_{\rm fus,\mu}$. This leads to the required cycle strength $\mathcal{L}_\mu^{\rm req}=G_\mu N_L/(1-\omega_S^{\rm eff}G_\mu N_L)$, with $N_L=E_\mu^{\rm cost}/(\eta_{\rm sys}E_{\rm use})$, and to the conditional sticking boundary $\omega_S^{\rm eff}<1/(G_\mu N_L)$. The criterion separates rate-limited, sticking-limited, and cost-limited regimes in the $(\omega_S^{\rm eff},\mathcal{L}_\mu)$ plane. When representative historical D--T $\mu{\rm CF}$ 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.