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 [Formula: see text] is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular [Formula: see text] and diffuse [Formula: see text] patterns, respectively. Hereafter, [Formula: see text] is the scattering meridian angle, [Formula: see text] is the scattering azimuth angle; [Formula: see text] 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 Rspec([Formula: see text]) 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.

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 [[Formula: see text]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.

Grazing-incidence small-angle X-ray scattering in a twofold rough-interface medium: a new theoretical approach using the q-eigenwave formalism.

Based on the rigorous Green function formalism to describe the grazing-incidence small-angle X-ray scattering (GISAXS) problem, a system of two linked integral equations is derived with respect to amplitudes of the reflected and transmitted plane q-eigenwaves (eigenstate functions) propagating through two homogeneous media separated from each other by a rough surface interface. To build up the coupled solutions of these basic equations beyond the perturbation theory constraint 2kσθ0 < 1, a simple iteration procedure is proposed as opposed to the self-consistent wave approach [Chukhovskii (2011). Acta Cryst. A67, 200-209; Chukhovski (2012). Acta Cryst. A68, 505-512]. Using the first-order iteration, analytical expressions for the averaged specular and non-specular scattering intensity distributions have been obtained. These expressions are further analysed in terms of the GISAXS parameters {k, θ, θ0} and surface finish ones {σ, l, h}, where θ and θ0 are the scattering and incidence angles of the X-rays, respectively, σ is the root-mean-square roughness, l is the correlation length, h is the fractal surface model index, k = 2π/λ, and λ is the X-ray wavelength. A direct way to determine the surface finish parameters from the experimental specular and diffuse scattering indicatrix scan data is discussed for an example of GISAXS measurements from rough surfaces of α-quartz and CdTe samples.

X-ray specular scattering from statistically rough surfaces: a novel theoretical approach based on the Green function formalism.

The Green function formalism is applied to the problem of grazing-incidence small-angle X-ray scattering from statistically rough surfaces. Kirchhoff's integral equation is used to describe the X-ray wavefield propagation through a single rough surface separating vacuum and medium. Taking into account multiple diffuse X-ray scattering effects, the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained using the Gaussian statistical model of rough surfaces in terms of the two-point height-height correlation function. In the limiting cases when the correlation length xi is equal to zero or infinity, analytical formulae for the reflection R(coh)(θ) and transmission T(coh)(θ) coefficients of the specular wave are obtained. It is important that in the case xi --> infinity they coincide with the corresponding reflection R(DW)(θ) and transmission T(DW)(θ) coefficients related to the conventional Debye-Waller approximation for describing the grazing X-ray scattering from a rough surface. In the case of finite values of correlation length \xi the reflection |R(coh)(θ)|(2) and transmission |T(coh)(θ)|(2) scans are numerically calculated.

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 initio crystal 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.