2022/12/18 by Mamchuev, Murat O., Chukhovskii, Felix N.
#35A08 #35C15 #35L40 #35Q70 #35Q92 #45F05 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Primary 35F35 #Secondary 35F40
paper · doi:10.48550/arxiv.2212.09133
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.