Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. II. Dynamical theory

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
V. B. Molodkin ◽  
S. I. Olikhovskii ◽  
S. V. Dmitriev ◽  
V. V. Lizunov
Keyword(s):  
Reciprocal Lattice ◽  
Dynamical Theory ◽  
Bragg Diffraction ◽  
Three Dimensions ◽  
X Ray ◽  
X Ray Scattering ◽  
Diffraction Geometry ◽  
Lattice Space ◽  
Integrated Intensities ◽  
Ray Scattering

The analytical expressions for coherent and diffuse components of the integrated reflection coefficient are considered in the case of Bragg diffraction geometry for single crystals containing randomly distributed microdefects. These expressions are analyzed numerically for the cases when the instrumental integration of the diffracted X-ray intensity is performed on one, two or three dimensions in the reciprocal-lattice space. The influence of dynamical effects, i.e. primary extinction and anomalously weak and strong absorption, on the integrated intensities of X-ray scattering is investigated in relation to the crystal structure imperfections.

Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. I. Semi-dynamical model

2020 ◽  
Vol 76 (1) ◽  
pp. 45-54
Author(s):  
V. B. Molodkin ◽  
S. I. Olikhovskii ◽  
S. V. Dmitriev ◽  
A. I. Nizkova ◽  
V. V. Lizunov
Keyword(s):  
Silicon Crystal ◽  
Azimuthal Angle ◽  
Dynamical Theory ◽  
Bragg Diffraction ◽  
X Ray Diffraction ◽  
X Ray ◽  
X Ray Scattering ◽  
Diffraction Geometry ◽  
Coulomb Type ◽  
Analytical Expressions

The analytical expressions for the coherent and diffuse components of the integrated reflection coefficient are considered in the case of asymmetric Bragg diffraction geometry for a single crystal of arbitrary thickness, which contains randomly distributed Coulomb-type defects. The possibility to choose the combinations of diffraction conditions optimal for characterizing defects of several types by accounting for dynamical effects in the integrated coherent and diffuse scattering intensities, i.e. primary extinction and anomalous absorption, has been analysed based on the statistical dynamical theory of X-ray diffraction by imperfect crystals. The measured integrated reflectivity dependencies of the imperfect silicon crystal on azimuthal angle were fitted to determine the diffraction parameters characterizing defects in the sample using the proposed formulas in semi-dynamical and semi-kinematical approaches.

Crystal truncation rod X-ray scattering: exact dynamical calculation

2008 ◽  
Vol 41 (1) ◽  
pp. 18-26 ◽  
Author(s):  
Václav Holý ◽  
Paul F. Fewster
Keyword(s):  
Reciprocal Lattice ◽  
Dynamical Theory ◽  
Lattice Points ◽  
Space Distribution ◽  
Matrix Formalism ◽  
X Ray ◽  
Transmission Coefficients ◽  
X Ray Scattering ◽  
Crystal Truncation Rod ◽  
Scattering Geometry

A new method is presented for a calculation of the reciprocal-space distribution of X-ray diffracted intensity along a crystal truncation rod. In contrast to usual kinematical or dynamical approaches, the method is correct both in the reciprocal-lattice points and between them. In the method, the crystal is divided into a sequence of very thin slabs parallel to the surface; in contrast to the well known Darwin dynamical theory, the electron density in the slabs is constant along the surface normal. The diffracted intensity is calculated by a matrix formalism based on the Fresnel reflection and transmission coefficients. The method is applicable for any polarization of the primary beam and also in a non-coplanar scattering geometry.

Fundamentals of Kinematical X-Ray Scattering Theory

2014 ◽  
pp. 1-14
Keyword(s):  
Scattering Theory ◽  
Profile Analysis ◽  
Reciprocal Lattice ◽  
Line Profile Analysis ◽  
X Ray ◽  
X Ray Scattering ◽  
Basic Concepts ◽  
Lattice Planes ◽  
The Relationship ◽  
Ray Scattering

The broadening of X-ray line profiles is usually described by the kinematical scattering theory. In this chapter, the basic concepts and equations of the kinematical X-ray scattering are presented in order to better understand the theory of line profile analysis. The correlation between the crystal structure and the diffracted intensity distribution is shown. The scattering angles of the diffracted peak maxima are given by the Ewald construction in the reciprocal space. The correspondence between the reciprocal lattice vectors and the lattice planes is also presented, and the relationship between the scattering angle and the lattice plane spacing is given by Bragg’s law.

Characterizing nanoparticles with a laboratory diffractometer: from small-angle to total X-ray scattering

2014 ◽  
Vol 29 (S1) ◽  
pp. S47-S53 ◽  
Author(s):  
Marco Sommariva ◽  
Milen Gateshki ◽  
Jan-André Gertenbach ◽  
Joerg Bolze ◽  
Uwe König ◽  
...  
Keyword(s):  
Small Angle ◽  
Structural Features ◽  
Bragg Diffraction ◽  
X Ray Diffraction ◽  
X Ray ◽  
X Ray Scattering ◽  
Transmission Electron ◽  
Pdf Analysis ◽  
Comprehensive Picture ◽  
Ray Scattering

X-ray diffraction and scattering on a single multipurpose X-ray platform have been used to probe the structure, composition, and thermal behavior of TiO2 nanoparticles ranging in size from 1 to 10 nm. Ambient and non-ambient Bragg diffraction, small-angle X-ray scattering (SAXS), as well as total scattering and pair-distribution function (PDF) analysis are combined to obtain a comprehensive picture of the samples. At these ultrasmall particle-size dimensions, SAXS and PDF prove powerful in distinguishing the salient features of the materials, in particular the size distribution of the primary particles (SAXS) and the identification of the TiO2 polymorphs (PDF). Structural features determined by X-ray scattering techniques are corroborated by high-resolution transmission electron microscopy. The elemental make-up of the materials has been measured using X-ray fluorescence spectrometry and energy-dispersive X-ray analysis.

Temperature dependence of the thermal diffuse X-ray scattering from sodium single crystals

1990 ◽  
Vol 68 (11) ◽  
pp. 1279-1290
Author(s):  
W. Mayr ◽  
G. Fritsch ◽  
E. Lüscher
Keyword(s):  
Temperature Dependence ◽  
Single Crystals ◽  
Cross Section ◽  
Reciprocal Lattice ◽  
Scattering Cross Section ◽  
X Ray ◽  
Scattering Cross ◽  
X Ray Scattering ◽  
Thermal Diffuse ◽  
Ray Scattering

We report on experimental results for the thermal diffuse X-ray-scattering cross section from Na single crystals. Data are presented for the [100], [110], and [111] directions taken in the temperature range from 38 K to the melting point. In addition we present a numerical calculation of the harmonic diffuse-scattering cross section including all orders of multiphonon contributions using a realistic phonon-dispersion relation. The results of this model are compared with a simpler approximation for the higher order multiphonon terms. The differences between the calculations and the experimental data show a distinct asymmetrical behaviour with respect to the reciprocal lattice points. Owing to this fact and their temperature dependence they can be related to anharmonic scattering. The contributions of the four lowest order terms are derived from the data. The lowest order antisymmetric contribution agrees quite well with available theoretical calculations.

Lattice dynamics, X-ray scattering and one-body potential theory

1970 ◽  
Vol 317 (1530) ◽  
pp. 359-366 ◽  
Keyword(s):  
Integral Equation ◽  
Potential Theory ◽  
Lattice Dynamics ◽  
Reciprocal Lattice ◽  
Dynamical Matrix ◽  
X Ray ◽  
X Ray Scattering ◽  
Body Potential ◽  
Exchange And Correlation ◽  
Ray Scattering

One-body potential theory, which includes the effect of exchange and correlation forces, is used to calculate the change in the electron density due to small displacements of the ions. The final result contains a Dirac density matrix for the perfect crystal, the diagonal element being the exact ground state density ρ 0 ( r ). The basic quantity R ( r ) determining the electronic contribution to the dynamical matrix is such that the gradient of ρ 0 ( r ) is obtained by superposition of R ( r - l ) on each lattice site l . An integral equation is obtained which gives R ( r ) uniquely once the exchange and correlation energy is known. The Fourier transform R k of R ( r ) is given in term s of the Fourier components ρ K n of the charge density, which are known from X-ray scattering, by R K n = i ρKn K n the reciprocal lattice vectors K n . This is the same result as the rigid-ion model at the K n 's, which makes the assumption that this is true for all k . Deviations from rigid ions can be evaluated quantitatively from the integral equation obtained here. Such deviations reflect the role of many-body forces in lattice dynamics and the present theory provides a systematic basis for their calculation.

scholarly journals Observation of correlated X-ray scattering at atomic resolution

2014 ◽  
Vol 369 (1647) ◽  
pp. 20130315 ◽  
Author(s):  
Derek Mendez ◽  
Thomas J. Lane ◽  
Jongmin Sung ◽  
Jonas Sellberg ◽  
Clément Levard ◽  
...  
Keyword(s):  
Disordered Systems ◽  
Rational Drug Design ◽  
Three Dimensions ◽  
Atomic Resolution ◽  
Synchrotron Radiation Beam ◽  
X Rays ◽  
Radiation Beam ◽  
X Ray ◽  
X Ray Scattering ◽  
Ray Scattering

Tools to study disordered systems with local structural order, such as proteins in solution, remain limited. Such understanding is essential for e.g. rational drug design. Correlated X-ray scattering (CXS) has recently attracted new interest as a way to leverage next-generation light sources to study such disordered matter. The CXS experiment measures angular correlations of the intensity caused by the scattering of X-rays from an ensemble of identical particles, with disordered orientation and position. Averaging over 15 496 snapshot images obtained by exposing a sample of silver nanoparticles in solution to a micro-focused synchrotron radiation beam, we report on experimental efforts to obtain CXS signal from an ensemble in three dimensions. A correlation function was measured at wide angles corresponding to atomic resolution that matches theoretical predictions. These preliminary results suggest that other CXS experiments on disordered ensembles—such as proteins in solution—may be feasible in the future.

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