Parametric Mössbauer radiation in the grazing-diffraction geometry: Specular diffraction and superradiant amplification

Parametric x-ray radiation (PXR) is a tunable and versatile x-ray production mechanism that relies on relativistic electrons traversing through a crystal. When the crystal contains Mössbauer nuclei, they may also contribute to the x-ray generation, giving rise to spectrally narrow parametric Mössbauer radiation (PMR) as a promising alternative to undulator-based x-ray sources. It is well known that the x-ray generation strongly depends on the diffraction geometry. Motivated by this, here, we develop a framework for parametric x-ray generation in general diffraction geometries, encompassing the entire x-ray emission range from the conventional grazing extremely asymmetric diffraction (EAD) geometry with a large emission direction to the crystal surface up to the grazing specular diffraction (SD) geometry with emission near parallel to the surface. The framework further incorporates a range of physical effects arising from more realistic modeling of both the Mössbauer crystal and the electron beam. We show that the PMR generation in the grazing SD geometry qualitatively differs from the previously considered grazing EAD case, and explain the physical origins of these differences. We further apply our framework to analyze the geometry dependence of superradiant parametric Mössbauer radiation (SPMR) generated from microbunched electron beams provided by x-ray free-electron lasers. Our results set the stage for the exploration of PXR, PMR, superradiant amplification of parametric x-ray radiation, and SPMR beyond the established diffraction geometries.