Comprehensive Recovery of Point Defect Displacement Field Function in Crystals by Computer X-ray Diffraction Microtomography

In the case of the point defect in a crystal, the inverse Radon's problem in X-ray diffraction microtomography has been solved. As is known, the crystal-lattice defect displacement field function f(r) = hu(r) determines phases - (+/- h)-structure factors incorporated into the Takagi-Taupin equations and provides the 2D image patterns by diffracted and transmitted waves propagating through a crystal (h is the diffraction vector and u(r) is the displacement field crystal-lattice-defects vector). Beyond the semi-kinematical approach for obtaining the analytical problem solution, the difference-equations-scheme of the Takagi-Taupin equations that, in turn, yield numerically controlled-accuracy problem solutions has been first applied and tested. Addressing the inverse Radon's problem solution, the chi(2)-target function optimization method using the Nelder-Mead algorithm has been employed and tested in an example of recovering the Coulomb-type point defect structure in a crystal Si(111). As has been shown in the cases of the 2D noise-free fractional and integrated image patterns, based on the Takagi-Taupin solutions in the semi-kinematical and difference-scheme approaches, both procedures provide the chi(2)-target function global minimum, even if the starting-values of the point-defect vector P-1 is chosen rather far away from the reference up to 40% in relative units. In the cases of the 2D Poisson-noise image patterns with noise levels up to 5%, the figures-of-merit values of the optimization procedures by the Nelder-Mead algorithm turn out to be high enough; the lucky trials number is 85%; and in contrast, for the statistically denoised 2D image patterns, they reach 0.1%.