  Efficient excitation of nuclei via exchange of a real or virtual photon has a
fundamental importance for nuclear science and technology development. Here, we
present a new mechanism of nuclear excitation based on the capture of a free
muon into the atomic orbits (NE$\mu$C). The cross section of such a new process
is evaluated using the Feshbach projection operator formalism and compared to
other known excitation phenomena, i.e. photo-excitation and nuclear excitation
by electron capture (NEEC), showing up to ten orders of magnitude increase in
cross section. NE$\mu$C is particularly interesting for MeV excitations that
become accessible thanks to the stronger binding of muons to the nucleus. The
binding energies of muonic atoms have been calculated introducing a state of
the art modification to the Flexible Atomic Code (FAC). An analysis of an
experimental scenarios in the context of modern muon production facilities
shows that the effect can be detectable for selected isotopes. The total
probability of NE$\mu$C is predicted to be $ P \approx 10^{-6}$ per incident
muon in a beam-based scenario. Given the high transition energy provided by
muons, NE$\mu$C can have important consequences for isomer feeding and
particle-induced fission.
