X-Ray Diffraction Tomography Recovery of the 3D Displacement-Field Function of the Coulomb-Type Point Defect in a Crystal

F. N. Chukhovskii,V. V. Volkov,P. V. Konarev

doi:10.1038/s41598-019-50833-6

Abstract

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 [bar{{bf{2}}}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.

Highlights

The first X-ray diffraction tomography (XRDT) experiments have been successfully performed on the synchrotron radiation sources ESRF11 and SPring-812
It is important to find out approximate analytical solutions of the dynamical Takagi–Taupin equations, which would allow describing some features of the X-ray diffraction topograms related to particular regions in the neighborhood of single defects in crystals
One derives the solution of the dynamical Takagi–Taupin equations in the semi-kinematical approximation, which seems to be effective for recovering the 3D displacement-field function using the inclined XRDT 2D projections data

Summary

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 important to find out approximate analytical solutions of the dynamical Takagi–Taupin equations, which would allow describing some features of the X-ray diffraction topograms related to particular regions in the neighborhood of single defects in crystals As is expected, this is a key factor for solving the inverse XRDT problem, in particular, for the 3D recovery of the displacement-field function near around single defects. An approximate analytical solution of the dynamical Takagi–Taupin equations is found out that seems to represent by itself a principal point to the inverse XRDT problem Such the semi-kinematical approximation for the diffracted wave Eh(r) is used for recovering the 3D displacement-field function f (r − r0) = h ⋅ u(r − r0) in the case of the Coulomb-type point defect (r0 is the radius-vector of a point defect in a crystal).

Results and Discussion

Γh πΛ λ

Target Function

Final Target function CP

Conclusions

Additional Information