Towards to Theory of the X-ray Diffraction Tomography of Crystals with Nano-Sized Defects

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Abstract

X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and their subsequent 3D-reconstruction of structural defects in crystals. The most frequent objects of research are linear and helical dislocations in a crystal, for which plane wave diffraction images are the most informative, since they do not contain additional interference artifacts unrelated to the images of the defects themselves. In this work the results of modeling and analysis of 2D plane wave diffraction images of a nano-dimensional Coulomb-type defect in a Si(111) thin crystal are presented based on the construction of numerical solutions of the dynamic Takagi-Taupin equations. An adapted physical expression for the elastic displacement field of the point defect, which excludes singularity at the defect location in the crystal, is used. A criterion for evaluating the accuracy of numerical solutions of the Takagi-Taupin equations is proposed and used in calculations. It is shown that in the case of the Coulomb-type defect elastic displacement field, out of the two difference algorithms for solving the Takagi-Taupin equations used in their numerical solution, only the algorithm for solving the Takagi-Taupin equations where the displacement field function enters in exponential form is acceptable in terms of the required accuracy-duration of the calculations.

About the authors

V. A. Grigorev

Shubnikov Institute of Crystallography, FSRC “Crystallography and Photonics” of the RAS

Author for correspondence.
Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

P. V. Konarev

Shubnikov Institute of Crystallography, FSRC “Crystallography and Photonics” of the RAS

Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

F. N. Chukhovskii

Shubnikov Institute of Crystallography, FSRC “Crystallography and Photonics” of the RAS

Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

V. V. Volkov

Shubnikov Institute of Crystallography, FSRC “Crystallography and Photonics” of the RAS

Email: vasiliy.grigorev.1996@mail.ru
Russian Federation, Moscow

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Supplementary files

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1. JATS XML
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2. Fig. 1. Difference grid for calculating the amplitudes of transmitted and diffracted waves in a crystal. T - thickness of the crystal, p - grid spacing

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3. Fig. 2. Schematic representation of the crystal

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4. Fig. 3. Images of the defect calculated using Takagi-Taupin type 1 (a) and type 2 (b) equations. The intensity value is indicated on the color scale

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5. Fig. 4. Distribution of the number n of points in the Y = 0 computational grid plane by divergence values. The divergence values are grouped into 255 ranges. a) type 1, b) type 2. Grid spacing p = 0.054 µm

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6. Fig. 5. Topogram of the defect (a) and distribution by divergence values (b) for type 2 equations and grid step p = 0.0027 µm

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