X-Ray Diffraction Tomography Recovery of the 3D Displacement-Field Function of the Coulomb-Type Point Defect in a Crystal

A successive approach to the solution of the inverse problem of the X-ray diffraction tomography (XRDT) is proposed. It is based on the semi-kinematical solution of the dynamical Takagi–Taupin equations for the σ-polarized diffracted wave amplitude. Theoretically, the case of the Coulomb-type point defect in a single crystal Si(111) under the exact conditions of the symmetric Laue diffraction for a set of the tilted X-ray topography 2D-images (2D projections) is considered provided that the plane-parallel sample is rotated around the diffraction vector [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{{\bf{2}}}$$\end{document}2¯20]. The iterative simulated annealing (SA) and quasi-Newton gradient descent (qNGD) algorithm codes are used for a recovery of the 3D displacement-field function of the Coulomb-type point defect. The computer recovery data of the 3D displacement-field function related to the one XRDT 2D projection are presented. It is proved that the semi-kinematical approach to the solution of the dynamical Takagi–Taupin equations is effective for recovering the 3D displacement-field function even for the one XRDT 2D projection.

It is obvious, that the 3D computer image recovery based on any experimental XRDT data is connected equally, if not to a greater extent, with the same difficulties as the interpretation of the XRDT 2D projections on the X-ray diffraction topograms. They are due to the complicated contrast mechanisms of the image formation related to various regions around defects in crystals [1][2][3] . In this respect, it is important to find out approximate analytical solutions of the dynamical Takagi-Taupin equations, which would allow describing some features of the X-ray diffraction topograms related to particular regions in the neighborhood of single defects in crystals. As is expected, this is a key factor for solving the inverse XRDT problem, in particular, for the 3D recovery of the displacement-field function near around single defects.
In the present study, an approximate analytical solution of the dynamical Takagi-Taupin equations is found out that seems to represent by itself a principal point to the inverse XRDT problem. Such the semi-kinematical approximation for the diffracted wave E h (r) is used for recovering the 3D displacement-field function − = ⋅ − h u f r r r r ( ) ( ) 0 0 in the case of the Coulomb-type point defect (r 0 is the radius-vector of a point defect in a crystal). By applying iterative simulated annealing (SA) 16 and quasi-Newton gradient descent (qNGD) algorithm codes 17,18 , we will show that in the case of the symmetric Laue diffraction of the X-ray characteristic MoK α1 -radiation with the diffraction vector h = [220] from a single crystal Si(111) the 3D displacement-field point defect function − = ⋅ − f r r h u r r ( ) ( ) 0 0 can be recovered. Note that the first attempt to recover the 3D displacement-field point defect function was made in 19 using the so-called simultaneous algebraic reconstruction technique (SART) algorithm (cf. 13 ). Unfortunately, the SART solutions turn out to converge only for a limited number of voxels (no more than 5 × 5 × 5 = 125 voxels) near around a point defect in a crystal.

Results and Discussion
The 2D diffraction imaging of a point defect. Semi-kinematical approximation. In this section, one derives the solution of the dynamical Takagi-Taupin equations in the semi-kinematical approximation, which seems to be effective for recovering the 3D displacement-field function using the inclined XRDT 2D projections data.
As is well-known 1-3 , a direct image of defect in a crystal is due to the interbranch scattering of the X-ray Bloch waves in the strongly distorted region in the immediate vicinity of a defect, which can be interpreted as the kinematical scattering of Bloch waves propagating in a perfect crystal far from the defect central region. This assertion was confirmed by numerous numerical calculations (see, e.g. 2,3,10 ).
For simplicity, in order to avoid cumbersome formulas and calculations, we restrict ourselves to the case of propagation of a σ-polarized X-ray wave-field in a non-absorbing (thin) crystal "on average" oriented in the exact Bragg position, , where k 0 is the wave vector of the monochromatic X-ray plane wave incident onto a crystalline sample. In this case, the propagation of the total X-ray wave-field in a distorted crystal under conditions of the symmetric two-wave Laue diffraction is set in the form of the dynamical Takagi-Taupin equations (cf. 8,9 ) E s s E s s  Original coordinate system (X, S) for the inclined X-ray diffraction geometry; 0Y-axis is perpendicular to the (X, S)-plane. Φ is the crystal rotation angle around the diffraction vector h parallel to the 0X-axis. The detector plane is perpendicular to the unit vector s h = k h| /k along the diffracted wave propagation.
Note that the boundary conditions (2) are automatically followed from the modified Eq. (1) with the added term − δ + s s ( ) h h 0 satisfies the inhomogeneous differential hyperbolic-type second-order equation in the partial derivatives over the dimensionless variables S 0 and S h (S 0 = s 0 /Λ, where C is the X-ray polarization factor equal to 1 for σ-polarization and cos 2θ B for π-polarization. Further, we will choose a plane-parallel single crystal Si(111) sample with diffraction vector h = [220]; the wavelength λ of incident characteristic X-ray radiation equal to 0.071 nm of the characteristic X-ray Mo K α1 -radiation, photon energy of 17.48 keV. Correspondingly, the susceptibility coefficients Re( within a perfect crystal and the kernel function 0 are the corresponding diffracted wave amplitude and the Green (point source) function in a perfect crystal; 0 is the zero-order Bessel function of the first kind.
Further, we will use the image peculiarity of the XRDT 2D projections that are directly linked with strongly distorted regions near around a single defect core. As is shown in 1-3 , in such the regions the X-ray diffraction scattering is, in general, kinematical owing to the interbranch scattering of the Bloch X-ray waves. Physically, it does mean that the so-called direct defect image contrast on the XRDT 2D projections is due to the diffracted wave propagation through the strongly distorted crystal region along the wave vector k h . It immediately follows that the direct defect image on the XRDT 2D projections is formed due to the kinematical scattering.
The above assertion is equivalent to building the first-order perturbation theory solution of the dynamical Takagi-Taupin Eq. (1) if the corresponding zero-order approximation for the X-ray transmitted amplitude in a perfect crystal is used.
Thus, after some obvious straightforward manipulations with the integral Eq. (6), one obtains the following expression www.nature.com/scientificreports www.nature.com/scientificreports/ for the diffracted wave amplitude in the scope of the semi-kinematical approach, where the dynamical transmitted wave amplitude E S P ( , ) within a perfect crystal is defined by ain vicinity of the defect core, which represents by itself the pure phase object. That is why the formula (8) can be treated as the semi-kinematical approximate solution of the dynamical Takagi-Taupin Eq. (1).
Further, the theoretical formula (8) rewritten as where Ф is the rotation angle around the diffraction vector h (see Fig. 1). Note that the dimensionless X, Y, Z coordinates are linked with the intrinsic Cartesian coordinates in the crystal. The formulas (10), (11) for the 3D displacement-field function − f r r ( ) 0 are the basic theoretical expressions for solving the inverse problem XRDT under consideration. Finally, one has to determine the error-functional (the target function) and generate its optimization procedure. The latter has to apply to the target function in the standard form as follows Recovery of the 3D displacement-field function. The SA and qNGD algorithm codes. In this Section, we apply the iterative SA 16 and qNGD 17,18 algorithm codes, adapted to solving the inverse XRDT problem.
SA algorithm: setting the displacement-field function in analytical form. As is known, the SA algorithm 16 is applied to minimize nonlinear target χ 2 -functions. Essentially, it is one of the efficient methods for solving inverse problems with a large number of variables and combinatorial nature of iterative calculations. Starting with a model specified as the initial model and varying pseudo-randomly its parameters, the SA algorithm works until the current model fits best the data set 'observed' that means the minimum of the target function is achieved. The advantage of the SA algorithm seems to overcome local minima of the target function, which are the main obstacles for other nonlinear optimization methods, e.g., for the qNGD methods 17,18 . In our case, the position of a point defect is set by a radius vector r 0 = nT/2, where n is the internal normal to the input crystal surface, Z = 0. Note, the crystal thickness T of a single crystal Si(111) is chosen such that the X-ray absorption in a sample can be neglected. In the first study stage, one needs to checkup that the SA algorithm code can be effective in the case of one XRDT 2D projection when only the term with Φ = 0 is taken into account in formula (12).
Correspondingly, formula (12) reduces to where I X T Y T { ( ), ( ); 0} h true , is the true X-ray topography 2D projection simulated according formula (10) with the true displacement-field function (11). The intensity I X T Y T { ( ), ( ); 0} h calc , is calculated according to formula (10) with trial displacement-field functions according to an iterative procedure of the target function optimization.
For simplicity, without the loss of generality, the scaling coefficient G can be set to unity in formula (11). Besides, to estimate the convergence of the iterative optimization procedure, one uses the error parameter CP, which is defined as The recovery data of the SA algorithm code application for the 3D displacement-field function − f r r ( ) sol 0 are listed in Table 1. As it follows from Table 1, in the case of the one descending index p i for i = 1, the solution for the 3D displacement-field function SA, qNGD algorithm codes: numerical displacement-field function assignment. From the practical viewpoint, it is of great interest to perform the 3D displacement-field function − f r r ( ) 0 recovering with its numerical assignment. In contrast to the case above described, we consider the 3D function Based on formulas (10), (11), the true XRDT 2D projection for a single crystal Si(111) sample calculated on the spatial grid {15, 15, 15} is displayed in Fig. 2. As is easily seen from Fig. 2, the XRDT 2D projection is symmetric towards the coordinate Y and shifted from the center along the X coordinate by T/2 · tan θ B . The recovery data of the 3D function  Table 2, in the case of the spatial grid {15, 15, 15} the SA algorithm code application provides the target function value to be reduced by more than six orders of magnitude, whereas the error parameter CP increases from 0.0658 to 0.118. This may occur because of the solution ambiguity of the inverse XRDT problem under consideration.
We compare the above3D displacement-field function recovery results with the corresponding ones by using the nonlinear qNGD algorithm based on the Levenberg-Marquardt scheme 20 . Further, one utilizes the open access NL2SNO program code as an implementation of the qNGD algorithm (see 18 for details). The corresponding optimization results of the target function (13) obtained by the qNGD algorithm are listed in Table 2. It is   www.nature.com/scientificreports www.nature.com/scientificreports/ The upper row (Fig. 3a,b) gives cross-sections of the true 3D function

Conclusions
The theoretical semi-kinematical approach has been developed that describes the X-ray diffraction propagation in the vicinity of the point defect core in a crystal. To solve the inverse XRDT problem the iterative SA and qNGD algorithm codes have been applied.
In the case of the Coulomb-type point defect the recovery of the 3D displacement-field function − f r r ( ) 0 has been obtained. With certain limitations on a class of the searched functions − f r r ( ) 0 specified in both the analytical and numerical forms, the iterative SA and qNGD algorithm codes in use work well even for the one XRDT 2D projection.
At this point, it should be noted once more that the present semi-kinematical approach embraces the inverse XRDT problem even if alternative algorithm codes have to be applied. On the other hand, a convergence of the optimization procedure for the target function based on the SA and qNGD algorithm codes might be improved by exploiting a number of the true XRDT 2D projections and, besides, some modifications of the SA and qNGD algorithm codes in use. The question of whether the elaborated theoretical approach overall is effective or whether it should be given up in favor of other approaches, which should be applied instead, remains a good topic for future work.  Table 2. Initial and final recovery data of the 3D displacement-field function by the qNGD and SA algorithms. The 3D displacement-field function is determined in the numerical form.