  We investigate the Smith-Purcell emission produced by electron- or
ion-beam-driven coherent excitation of nuclei arranged in periodic crystal
lattices. The excitation and subsequent radiative decay of the nuclei can leave
the target in the initial ground state after $\gamma$-ray emission, thus
generating a coherent superposition of the far-field photon amplitude emanating
from different nuclei that results in sharp angular patterns at spectrally
narrow nuclear transition energies. We focus on Fe-57 as an example of
two-level nuclear lossy system giving rise to Smith-Purcell emission at
14.4\,keV with a characteristic delay of 1.2\,ns relative to the excitation
time. These properties enable a clean separation from faster and spectrally
broader emission mechanisms, such as bremsstrahlung. Besides its fundamental
interest, our study holds potential for the design of high-energy, narrow-band,
highly-directive photon sources, as well as a means to store energy in the form
of nuclear excitations.
