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

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 function f(Ctpd)(r - r(0)) = h . u(r - r(0)) due to a Coulomb-type point defect (Ctpd) located at a point r0 within a crystal [h is the diffraction vector and u(r - r(0)) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function F{P} in a chi(2) sense in order to recover the function f(Ctpd)(r - r(0)) [P is the parameter vector that characterizes the 3D function f(Ctpd)(r - r(0)) 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 F{P} to tend towards the global minimum even if the vector P in the search is initially chosen rather a long way from the true one.