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We investigate the Smith-Purcell emission produced by electron- orion-beam-driven coherent excitation of nuclei arranged in periodic crystallattices. The excitation and subsequent radiative decay of the nuclei can leavethe target in the initial ground state after $\gamma$-ray emission, thusgenerating a coherent superposition of the far-field photon amplitude emanatingfrom different nuclei that results in sharp angular patterns at spectrallynarrow nuclear transition energies. We focus on Fe-57 as an example oftwo-level nuclear lossy system giving rise to Smith-Purcell emission at14.4\,keV with a characteristic delay of 1.2\,ns relative to the excitationtime. These properties enable a clean separation from faster and spectrallybroader emission mechanisms, such as bremsstrahlung. Besides its fundamentalinterest, 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 formof nuclear excitations.

Coherent Smith-Purcell $\gamma$-Ray Emission