The dynamical scattering theory is developed for the Laue diffraction of the Mössbauer rays and x-rays, whose angular distribution is comparable with the diffraction angular range. Both the Rayleigh and the resonant nuclear scattering are taken into account. We consider typical case when incident radiation first passes through an entrance slit and afterwards diffracts at the crystal planes within the Borrmann triangle. In calculations of the wave function for γphotons, refracted or diffracted in such strongly absorbing crystal, we apply the saddle-point method. The distribution of their intensities over the basis of the Borrmann triangle is analyzed. In the spherical wave approximation of Kato, when aperture of the incident beam much exceeds the diffraction interval, the derived formulae well correlate with the familiar equations of the diffraction theory of X-rays.
I. INTRODUCTIONThe diffraction of x-rays, synchrotron radiation, Mössbauer rays and neutrons is widely used for analysis of crystal structure. In this way such unique phenomena were discovered as the pendellösung effect and the anomalous transmission of γ-photons and neutrons through a perfect crystal in the Laue (transmission) geometry. In the x-ray optics the latter effect is frequently referred to as the Borrmann effect [1-3]. The explanation of these phenomena has been given by the dynamical scattering theory [1-3]. In the two-wave case the incident plane x-ray wave generates inside the crystal two couples of waves, both of which are coherent superpositions of the transmitted and reflected waves. One such couple has nodes at the scattering atoms and is therefore anomalously weakly absorbed, whereas another, having antinodes, is strongly absorbed.The dynamical scattering theory has been extended to the case of elastic diffraction of Mössbauer plane waves by Afanas'ev and Kagan [4]. They predicted that it can be realized complete suppression of γ-quanta by Mössbauer nuclei in perfect crystals, that was confirmed in numerous experiments.Multiple scattering of x-ray photons by crystals is always described by the Maxwell equations [1-3]. In the same quasi-classical manner Afanas'ev and Kagan [4] treated the resonant scattering of Mössbauer radiation by a crystal. A quantum approach for the inelastic diffraction of γ-radiation in crystals, exposed to alternating external fields, has been presented in [5].In typical Laue-diffraction experiments the incident γ-quanta are first collimated by a slit, lying on the crystal surface and being parallel to the reflecting planes (see Fig.1). And after that the radiation flows within the angular region, which forms a so-called Borrmann triangle (fan). The intensity distribution of the transmitted and reflected beams over the basis of the Borrmann triangle is analyzed with the aid of one more slit, which is also parallel to the reflecting crystal planes.Standard plane-wave dynamical theory is not able to describe this situation. Therefore Kato [6] considered the Laue diffraction of x-ray spherical waves, treating th...
