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 $\mathcal{L}_μ=Λ_cτ_μ$, where $Λ_c$ is the effective cycle-completion rate and $τ_μ$ is the muon lifetime. Together with the residual effective sticking probability $ω_S^{\rm eff}$, it gives the mean fusion yield per useful muon, $N_{\rm fus,μ}=\mathcal{L}_μ/(1+ω_S^{\rm eff}\mathcal{L}_μ)$. Introducing the useful D--T cycle energy $E_{\rm use}$, the system factor $η_{\rm sys}$, and the effective muon cost $E_μ^{\rm cost}$, the one-muon gain is $G_μ=(η_{\rm sys}E_{\rm use}/E_μ^{\rm cost})N_{\rm fus,μ}$. This leads to the required cycle strength $\mathcal{L}_μ^{\rm req}=G_μN_L/(1-ω_S^{\rm eff}G_μN_L)$, with $N_L=E_μ^{\rm cost}/(η_{\rm sys}E_{\rm use})$, and to the conditional sticking boundary $ω_S^{\rm eff}<1/(G_μN_L)$. The criterion separates rate-limited, sticking-limited, and cost-limited regimes in the $(ω_S^{\rm eff},\mathcal{L}_μ)$ plane. When representative historical D--T $μ{\rm CF}$ 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.