Time evolution calculation of muon catalysed fusion: Emission of recycling muons from a two-layer hydrogen film

Highlights: • Muon catalyzed fusion can be used to moderate negative muons. • Numerical calculation reveals the emission rate of the recycling muons in the muon catalyzed fusion cycle. • Tritium concentration in a two-layer solid hydrogen target is optimized to maximize the emission of the recycling muons. Recycling muon having approximately 10 keV kinetic energy produced in a muon catalysed fusion ($\mu$CF) cycle is expected to be a source of negative muon beam with high-spatial/time coherence. We report a time evolution calculation of $\mu$CF cycle with a particular focus on the emission of recycling muons from a hydrogen film target. Rate equations for the $\mu$CF reactions are solved using the 4th-order Runge-Kutta method. Considering a two-layer solid hydrogen film target composed of 1 mm hydrogen-deuterium layer and 2 µm deuterium-tritium layer, we have numerically simulated the time evolution of the muonic states and their populations in these layers. The muon atomic and molecular processes in the layers are solved simultaneously by introducing a probability of muonic deuterium atoms traveling between the layers via the Ramsauer-Townsend effect. The use of T${}_2$ amplifies the recycling muon emission. A weak dependence of the emission fraction of the recycling muons on the T${}_2$ concentration is predicted, which can be utilized to minimize the usage of tritium.