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 − r₀) = h · u(r − r₀) due to a Coulomb-type point defect (Ctpd) located at a point r₀ within a crystal [h is the diffraction vector and u(r − r₀) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function in a χ² sense in order to recover the function f_Ctpd(r − r₀) [ is the parameter vector that characterizes the 3D function f_Ctpd(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 to tend towards the global minimum even if the vector in the search is initially chosen rather a long way from the true one.