Coherent Smith-Purcell $\gamma$-Ray Emission

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.