Towards a solution of the inverse X-ray diffraction tomography challenge: theory and iterative algorithm for recovering the 3D displacement field function of Coulomb-type point defects in a crystal

Abstract

The theoretical framework and a joint quasi-Newton–Levenberg–Marquardt–simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field functionfCtpd(r−r0) =h · u(r−r0) due to a Coulomb-type point defect (Ctpd) located at a pointr0within a crystal [his the diffraction vector andu(r−r0) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function {\cal F}\{ {\cal P} \} in a χ2sense in order to recover the functionfCtpd(r−r0) [{\cal P} is the parameter vector that characterizes the 3D functionfCtpd(r−r0) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi–Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function {\cal F} \{ {\cal P} \} to tend towards the global minimum even if the vector {\cal P} in the search is initially chosen rather a long way from the true one.