X-ray standing waves on the surface of the crystal with the uniform strain gradient

Abstract

The results of theoretical investigation of X-ray Standing Waves (XSW) on the surface of the crystal with the Uniform Strain Gradient (USG) are presented. Bent crystals are the typical examples of the crystals with the USG. The proposed theory is based on the Takagi-Taupin theory of X-ray propagation in a weakly deformed crystal. This theory is applied for calculation of the angular dependencies of reflection coefficient and XSW on the surface of the crystal with USG for the specific case of Si(400) thick crystal, CuKa radiation. The behaviour of the XSW with the different position of adsorbed atoms (different values of the phase shift) are analyzed

Theory and applications of x-ray standing waves in real crystals

“…Different calculation approach was used in the paper [91] (see also ([92])) for the analysis of x-ray standing wave field on the surface of the crystals with USG (deformation field was taken in the same form hu(r) = B • z 2 , where B is a constant). According to this approach the amplitudes of the x-ray field E 0 (r) and E h (r) were obtained from the direct integration of the TT equations in the form (13).…”

“…implanted crystals, heterostructures) are often curved. Below we will present results obtained in [89,90,91].…”

“…Theory of x-ray standing wave behavior in crystals with USG and as a special case bent crystals was for the first time developed in papers [89,90,91]. Later similar theoretical approach for the analysis of the crystals with USG was performed in [92].…”

“…In the paper [91] the angular dependencies of x-ray reflectivity and intensity of x-ray standing waves (99) on the surface of the crystal with USG were calculated for the case of (400) symmetrical Bragg diffraction, CuK α , σ−polarized radiation in a thick crystal of silicon. Results of these calculations are presented on Fig.…”

“…Therefore, although for the perfect crystals applications of the standing wave method are limited to thin subsurface layers in the case of the crystals with USG this method can be extended to investigation of thin buried layers (monolayers) of impurity atoms even located at the depths z ≥ L ex . Different calculation approach was used in the paper [91] (see also ([92])) for the analysis of x-ray standing wave field on the surface of the crystals with USG (deformation field was taken in the same form hu(r) = B · z 2 , where B is a constant). According to this approach the amplitudes of the x-ray field E 0 (r) and E h (r) were obtained from the direct integration of the TT equations in the form (13).…”