[en] Highlights: • Cluster-layered and aperiodical multilayer Fe/Cr system is investigated • The depth-profile of element (z) from X-ray reflectivity data is obtained. • The regularization method is used for solving an inverse problem for low contrast systems. • The efficiency of the method is confirmed by model calculations. - Abstract: A new modelless method of determining the element concentration profile of metallic multilayer nanoheterostructures for low-contrast systems is applied to study periodical and aperiodical multilayer Fe/Cr system by X-ray reflectivity. As a special case, we discuss a cluster-layered Fe/Cr film with Kondo-like behavior of the resistance. The method does not require any a priori information on the structure of the multilayer nanostructures, such as position and width of interfaces, as well as, their shape. The method is based on solution of the Fredholm integral equation of the first kind, which relates the reflection coefficient and the concentration profile of the chemical elements of the sample. The ill-posed inverse problem of determination of the element concentration profile is solved by the regularization method. Efficiency of the method is confirmed by model calculations.

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[en] Highlights: •A depth-resolved method for investigation of the local atomic structure is proposed. •We combined data of X-ray reflectivity and angle-depended EXAFS. •The regularization method and Levenberg–Marquardt (L–M) algorithm are used. •Mathematical algorithm comprises three process steps. •This general mathematical scheme cannot be interrupted. - Abstract: A depth-resolved method for the investigation of the local atomic structure by combining data of X-ray reflectivity and angle-resolved EXAFS is proposed. The solution of the problem can be divided into three stages: 1) determination of the element concentration profile with the depth z from X-ray reflectivity data, 2) determination of the X-ray fluorescence emission spectrum of the element i absorption coefficient ${\mu }_i^a$(z,E) as a function of depth and photon energy E using the angle-resolved EXAFS data $I_i^f\left(E,{\vartheta }_l\right),$ 3) determination of partial correlation functions $g_{\mathit{ij}}\left(z,r\right)$as a function of depth from ${\mu }_i\left(z,E\right)$. All stages of the proposed method are demonstrated on a model example of a multilayer nanoheterostructure Cr/Fe/Cr/Al₂O₃. Three partial pair correlation functions are obtained. A modified Levenberg-Marquardt algorithm and a regularization method are applied.

[en] Highlights: • A depth-resolved method for investigation of the local atomic structure is proposed. • We combine data of the X-ray reflectivity and the angular resolved EXAFS. • The regularization method is used. • A mathematical algorithm comprises three process steps. • Depth-resolved atomic structure for the cluster-layered Cr/Fe model was obtained. - Abstract: An experimental method for studying the local atomic structure with a depth resolution in low-contrast multilayered thin films is proposed. We consider combination of the X-ray reflectivity and the extended X-ray absorption fine structure spectroscopy with angular resolution. To obtain the structural characteristics, the most advanced methods for solving inverse ill-posed problems are used, namely, the Tikhonov regularization method in the case of linear integral equations and the Levenberg-Marquardt algorithm for nonlinear problems. The proposed mathematical algorithms do not require a priori information about the studied system, such as the width and shape of the interface, the width and depth of the bands of the individual elements, or detailed information about the atomic structure. A software for the method has been developed. It allows us to obtain the most complete information about the local atomic structure of interface and surface. To evaluate the capabilities of the method, we used the system of cluster-layered nanostructures Al₂O₃/Cr/Fe-Cr/Cr/Fe-Cr/Cr as a model. A depth resolution of 1–2 Å was reached.