Towards to solution of the fractional Takagi–Taupin equations. The Green function method

Developing the comprehensive theory of the X-ray diffraction by distorted crystals remains to be topical of the mathematical physics. Up to now, the X-ray diffraction theory grounded on the Takagi–Taupin equations with the first-order partial derivatives over the two coordinates within the X-ray scattering plane. In the work, the theoretical approach based on the first-order fractional Takagi–Taupin equations with the ‘quasi-time variable’ of the order α∈ (0,1] along the crystal depth has been suggested and the corresponding X-ray Cauchy issue is formulated. Accordingly, using the Green function method in the scope of the Cauchy issue, the fractional Takagi–Taupin equations in the integral form have been derived. In the case of the inhomogeneous incident X-ray beam, the solution of the Cauchy issue of the X-ray diffraction by perfect crystal has been obtained and compared with the corresponding one based on the solution of the conventional Takagi–Taupin equations, α =1. In turn, notice that the value of order α may be adjusted from the experimental X-ray diffraction data.