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

TL;DR

This paper introduces a fractional calculus approach to the Takagi-Taupin equations for X-ray diffraction, deriving solutions using Green functions and allowing for adjustable fractional order based on experimental data.

Contribution

It extends the classical Takagi-Taupin equations to fractional derivatives, providing a new theoretical framework for modeling X-ray diffraction in distorted crystals.

Findings

Derived integral form of fractional Takagi-Taupin equations using Green functions.

Obtained solutions for inhomogeneous incident X-ray beams.

Showed the fractional order parameter can be tuned from experimental data.

Abstract

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 $\alpha\in(0,1]$ 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, $\alpha=1.$ In turn, notice that the value of order $\alpha$ may be adjusted from the experimental X-ray diffraction data.