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The isotope $ {}^{99} \rm{Mo} $, the generator of $ {}^{99m} \rm{Tc} $ usedfor diagnostic imaging, is supplied by extracting from fission fragments ofhighly enriched uranium in reactors. However, a reactor-free production methodof $ {}^{99} \rm{Mo} $ is searched over the world from the point of view ofnuclear proliferation. Recently, $ {}^{99} \rm{Mo} $ production through a muoncapture reaction was proposed and it was found that about $ 50 \, \% $ of ${}^{100} \rm{Mo} $ turned into $ {}^{99} \rm{Mo} $ through $ {}^{100} \rm{Mo}\left( \mu^-, n \right) $ reaction [arXiv:1908.08166]. However, the detailedphysical process of the muon capture reaction is not completely understood. We,therefore, study the muon capture reaction of $ ^{100} \rm{Mo} $ by atheoretical approach. We used the proton-neutron QRPA to calculate the muoncapture rate. The muon wave function is calculated with considering theelectronic distribution of the atom and the nuclear charge distribution. Theparticle evaporation process from the daughter nucleus is calculated by astatistical model. From the model calculation, about $ 38 \, \% $ of $ {}^{100}\rm{Mo} $ is converted to $ {}^{99} \rm{Mo} $ through the muon capturereaction, which is in a reasonable agreement with the experimental data. It isrevealed that negative parity states, especially $ 1^- $ state, play animportant role in $ {}^{100} \rm{Mo} \left( \mu^-, n \right) {}^{99} \rm{Nb} $.The feasibility of $ {}^{99} \rm{Mo} $ production by the muon capture reactionis also discussed. Isotope production by the muon capture reaction stronglydepends on the nuclear structure.

Theoretical study of $ \rm{Nb} $ isotope productions by muon capture reaction on $ {}^{100} \rm{Mo} $