Towards the Theory of X-ray Diffraction Tomography of Crystals with Nanosized Defects

Abstract

X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and the 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 nanodimensional 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.

APPENDIX

Let us derive an expression for the divergence of the vector $\left\langle {{{I}_{0}},{{I}_{h}}} \right\rangle$ for the nonabsorbing crystal from the Takagi–Taupin equations. To do this, we multiply the first and second Eqs. of system (1) by complex conjugates $D_{0}^{*}$ and $D_{h}^{*}$:

$$\begin{gathered} - \frac{{2i}}{k}\frac{{\partial {{D}_{0}}}}{{\partial {{s}_{0}}}}D_{0}^{*} = {{\chi }_{{\bar {h}}}}{{D}_{h}}D_{0}^{*}{\text{exp}}\left( { - i{\mathbf{hu}}} \right), \\ - \frac{{2i}}{k}\frac{{\partial {{D}_{h}}}}{{\partial {{s}_{h}}}}D_{h}^{*} = {{\chi }_{h}}{{D}_{0}}D_{h}^{*}{\text{exp}}\left( {i{\mathbf{hu}}} \right). \\ \end{gathered} $$

(7)

Function arguments are omitted for clarity and compactness of expressions. Since we do not take absorption into account, the imaginary parts ${{\chi }_{h}}$ and ${{\chi }_{{\bar {h}}}}$ are equal to zero; that is, they will not change during complex conjugation. In this case, subtracting Eqs. (7) from the complex conjugates we obtain

$$\begin{gathered} \frac{{2i}}{k}\left( {\frac{{\partial {{D}_{0}}}}{{\partial {{s}_{0}}}}D_{0}^{*} + \frac{{\partial D_{0}^{*}}}{{\partial {{s}_{0}}}}{{D}_{0}}} \right) \\ = C{{\chi }_{{\bar {h}}}}\left( {D_{h}^{*}{{D}_{0}}{\text{exp}}\left( {i{\mathbf{hu}}} \right) - {{D}_{h}}D_{0}^{*}{\text{exp}}\left( { - i{\mathbf{hu}}} \right)} \right), \\ \frac{{2i}}{k}\left( {\frac{{\partial {{D}_{h}}}}{{\partial {{s}_{h}}}}D_{h}^{*} + \frac{{\partial D_{h}^{*}}}{{\partial {{s}_{h}}}}{{D}_{h}}} \right) \\ = C{{\chi }_{h}}\left( {D_{0}^{*}{{D}_{h}}{\text{exp}}\left( { - i{\mathbf{hu}}} \right) - {{D}_{0}}D_{h}^{*}{\text{exp}}\left( {i{\mathbf{hu}}} \right)} \right). \\ \end{gathered} $$

(8)

Considering that ${{\chi }_{{\bar {h}}}} = {{\chi }_{h}}$, as well as the obvious relationship:

$$\frac{{\partial {{D}_{i}}}}{{\partial {{s}_{i}}}}D_{i}^{*} + \frac{{\partial D_{i}^{*}}}{{\partial {{s}_{i}}}}{{D}_{i}} = \frac{{\partial (D_{i}^{*}{{D}_{i}})}}{{\partial {{s}_{i}}}} = \frac{{\partial {{I}_{i}}}}{{\partial {{s}_{i}}}},\,\,\,\,~i = 0,h;$$

adding Eqs. (8) we get:

$$\frac{{2i}}{k}\left( {\frac{{\partial {{I}_{0}}}}{{\partial {{s}_{0}}}} + \frac{{\partial {{I}_{h}}}}{{\partial {{s}_{h}}}}} \right) = 0$$

or

$${\text{div}}\left( {{{I}_{0}},~{{I}_{h}}} \right) = \frac{{\partial {{I}_{0}}\left( {{{s}_{0}},~{{s}_{h}}} \right)}}{{\partial {{s}_{0}}}} + \frac{{\partial {{I}_{h}}\left( {{{s}_{0}},~{{s}_{h}}} \right)}}{{\partial {{s}_{h}}}} = 0.$$

Note that the expression is valid for both a defective and an ideal crystal, as well as in the case of an inaccurate Bragg condition (α ≠ 0), but only for the nonabsorbing crystal. In the case of an absorbing crystal, divergence, generally speaking, depends on the coordinates (s₀, s_h).