Novel theoretical approach to the GISAXS issue: the Green function formalism using the q-Eigenwaves propagating through a twofold rough-surfaced medium

To describe the 1D and 2D patterns of the grazing-incidence small-angle X-ray scattering (GISAXS) from a rough fractal surface, the novel integral equations for the amplitudes of reflected and transmitted waves are derived. To be specific, the analytical expression for the 2D total intensity distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{dR_{tot} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$\end{document}dRtotθ,ϕ;θ0dΩ is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{dR_{spec} \left( {\theta ;\theta_{0} } \right)}}{d\Omega }$$\end{document}dRspecθ;θ0dΩ and diffuse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{dR_{dif} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$\end{document}dRdifθ,ϕ;θ0dΩ patterns, respectively. Hereafter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}θ is the scattering meridian angle, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}ϕ is the scattering azimuth angle; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta_{0}$$\end{document}θ0 is the angle of incidence. By using the above analytical expressions, the 1D and 2D GISAXS patterns are numerically calculated. Some new experimental measurements of the specular reflectivity curves R_spec(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta_{0}$$\end{document}θ0) related to the fused quartz and crystal Si(111) samples have been carried out. Based on the theoretical approach developed, a direct least-squared procedure in a χ²-fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.

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 {\cal F}\{ {\cal P} \} in a χ² sense in order to recover the function f_Ctpd(r - r₀) [{\cal P} 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 {\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.

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.

Dynamic Newton-gradient-direction-type algorithm for multilayer structure determination using grazing X-ray specular scattering: numerical simulation and analysis

A new dynamic iterative algorithm code for retrieving macroscopic multilayer structure parameters (the layer thickness and complex refraction index for each layer, the surface roughness and the interface roughness between the layers) from specular scattering angular scan data is proposed. The use of conventional direct methods, particularly the well known Newton algorithm and gradient-direction-type algorithm operating dynamically to minimize the error functional in a least-squares fashion, is explored. Such an approach works well and seems to be effective in solving the inverse problem in the high-resolution X-ray reflectometry (HRXR) method. In order to demonstrate some features of the proposed iterative algorithm, numerical calculations for retrieving three-layer structure parameters are carried out using simulated HRXR angular scan data. The calculations indicate clearly that the dynamic iterative algorithm is convergent and capable of yielding the true solution. It is important that the performance coefficient for successful iterative cycles for the absolute minimization of the HRXR error functional is quite high even if the initial values of the search parameters are chosen rather far from the true values. It is particularly noteworthy that the relative number of successful iterative cycles is of the order of 90-40% when only moderately accurate initial parameter values, varying by +/-10-40% from the true values, are presumed.

Ab initiocrystal structure determination using X-ray fluorescence holography for different noise levels: numerical simulation and analysis

X-ray fluorescence holography (XFH) two-dimensional angular scans with the fluorescing Cu atom of a Cu(3)Au single crystal for different noise levels have been calculated and the structure factors have been numerically restored, supporting the ab initio structure determination method first discussed by Chukhovskii & Poliakov [Acta Cryst. (2003), A59, 109-116]. In the case of resultant XFH scans where noise levels are up to the regular signal values at each angular scan point, the elaborated method is found to work well. With the use of the linear regression algorithm code [Chukhovskii & Poliakov Acta Cryst. (2003), A59, 109-116], the restored structure factors show clearly not just good accuracy of the restoration code procedure but also the efficiency of the structure-determination method that can utilize the XFH data even for high noise levels.

X-ray fluorescence holography: a novel treatment for crystal structure determination

It is shown that it is possible to use a linear regression algorithm direct method to solve crystal structures from X-ray fluorescence holography (XFH) data. It is found that, in contrast to conventional X-ray structure determination methods, which do not always work unambiguously, the sustainable method utilizing the XFH data generally provides the unique phase-retrieval structure solution and is able, in many cases, to replace the above for determining both the absolute values (moduli) and phases of structure factors. The XFH (theta, varphi) scan with a fluorescing Cu atom from a spherical cluster of a Cu(3)Au single crystal, at an energy of 10 keV for the incident unpolarized plane-wave X-radiation, is numerically simulated to test the performance of the method in finding a unique solution for the structure factors involved in the restoration procedure using the linear regression algorithm.

Domino phase-retrieval algorithm for structure determination using electron diffraction and high-resolution transmission electron microscopy patterns

Direct-method formalism to determine atomic structures using electron diffraction data is here aimed at a general solution of the phase-retrieval problem, consequently combining electron diffraction (ED) and high-resolution transmission electron microscopy (HRTEM) patterns in a 'domino' fashion. While there are similarities to conventional (kinematical) direct methods, there remain major differences; in particular, owing to the dynamical effects in the data, the ED structure factors prove to be complex and then the positivity of the reconstructed electron density is no longer a valid constraint for 'dynamical' direct methods. Besides, owing to the dynamical effects, heavy atoms no longer dominantly contribute to the HRTEM images. Thus, the 'dynamical' direct-methods concept is based on the phase-retrieval algorithm utilizing both the dynamical ED and the HRTEM data. The fusion of the traditional direct-method technique, which is described here, allows realization of a full-phase restoration of complex structure factors. A numerical example, using the dynamical ED and HRTEM data for (Ga,In)(2)SnO(5) ceramic, shows that the method is capable of yielding a unique phase-retrieval solution. The clear sense is that the domino transform algorithm proposed works well and represents a valuable method for phasing diffraction patterns in electron structural crystallography using an experiment that is readily performed when the ED and HRTEM data are collected.

Statistical dynamical direct methods. II. The three-phase structure invariant

The triplet distribution used for kinematical diffraction is extended to the complex case appropriate for dynamical transmission electron diffraction. It is demonstrated that this gives good results if the distributions are handled statistically rather than relying upon single triplet relationships. As a consequence, conventional statistical direct methods will yield a reasonable approximation to the effective dynamical potential for thicknesses when kinematical theory is not appropriate. The recovered effective dynamical potential may be similar to the kinematical potential, but does not have to be and in general will not be.