Geodesic
Direct and inverse problems of studying the properties of multilayer nanostructures based on a two-dimensional model of $X$-ray reflection and scattering
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 6, pp. 977-987
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A mathematical model of $X$-ray reflection and scattering by multilayered nanostructures in the quasi-optical approximation is proposed. $X$-ray propagation and the electric field distribution inside the multilayered structure are considered with allowance for refraction, which is taken into account via the second derivative with respect to the depth of the structure. This model is used to demonstrate the possibility of solving inverse problems in order to determine the characteristics of irregularities not only over the depth (as in the one-dimensional problem) but also over the length of the structure. An approximate combinatorial method for system decomposition and composition is proposed for solving the inverse problems. 
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R. V. Khachaturov. Direct and inverse problems of studying the properties of multilayer nanostructures based on a two-dimensional model of $X$-ray reflection and scattering. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 6, pp. 977-987. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_6_a8/

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