Crystal truncation rods in kinematical and dynamical x-ray diffraction theories

Abstract

Crystal truncation rods calculated in the kinematical approximation are shown to quantitatively agree with the sum of the diffracted waves obtained in the two-beam dynamical calculations for different reflections along the rod. The choice and the number of these reflections are specified. The agreement extends down to at least ∼ 10 −7 of the peak intensity. For lower intensities, the accuracy of dynamical calculations is limited by truncation of the electron density at a mathematically planar surface, arising from the Fourier series expansion of the crystal polarizability.

Structure ofFe3Si∕GaAs(001)epitaxial films from x-ray crystal truncation rods

“…Since our experiments are performed with an incidence angle ⌽ in = 0.3°, just barely above the critical angle ␣ c = 0.24°, we extend the kinematical approximation by the distorted-wave Born approximation ͑DWBA͒. 36 In this approximation, the incident and scattered plane waves in vacuum are replaced by the waves in the medium having the same mean polarizability as in the system under investigation. Two sets of crystal waves are involved, corresponding to vacuum waves incident on the surface under the incidence angles ⌽ in and ⌽ out , respectively.…”

“…Moreover, dynamical theory, in either its two-beam or multibeam formulation, inaccurately treats the surface layer, 36 which gives rise to an error around 10 −7 -10 −8 of the peak intensity. The error originates from the Fourier expansion of the electron density in a periodic medium that is used in dynamical theory.…”

“…[26][27][28][29][30][31][32][33][34][35] Recently, we have shown 36 that the dynamical calculation can be extended to the whole CTR by summing up the amplitudes of the diffracted waves of the two-beam diffraction problems for all Bragg reflections along the CTR. Dynamical and kinematical scattering intensities quantitatively agree everywhere except in the vicinity of the Bragg peaks, where the kinematical intensity diverges.…”

Many‐Body Interactions in Nanoscale Materials by Angle‐Resolved Photoemission Spectroscopy

“…If we closely examine the photoemission data along a plane containing these necks Figure 6.17b, we can clearly see their signature as vertical rods. (Such structures are exactly analogous to the crystal truncation rods seen in X-ray or electron diffraction from crystal surfaces [22,23].) Identification of these vertical rods is the standard method to distinguish surface from bulk states.…”

Silicon Σ13(501) grain boundary interface structure determined by bicrystal Bragg rod X-ray scattering

“…Some Bragg rods arise from the truncation of the crystal at a plane (surface or interface) and are formed from the tails of Bragg peaks overlapping in reciprocal space in the direction normal to the surface [12]. These rods are known as crystal truncation rods (CTR) [13] and are widely used to determine the atomic arrangement at surfaces.…”