Applications of dynamical theory of X-ray diffraction by perfect crystals to reciprocal space mapping

The dynamical theory of X-ray diffraction in a crystal modulated by a surface acoustic wave (SAW) is developed for spatially restricted beams. It is shown that this approach is applicable to X-ray reciprocal space mapping. Rayleigh's surface-wave model is used to describe ultrasonic excitation. Based on the recurrent relations, a numerical simulation of the dynamical diffraction in a crystal modulated by a SAW is performed. Within the framework of the triple-axis diffraction scheme, the effect of the instrumental function on X-ray diffraction data is studied.

Download Full-text

BINoculars: data reduction and analysis software for two-dimensional detectors in surface X-ray diffraction

Journal of Applied Crystallography ◽

10.1107/s1600576715009607 ◽

2015 ◽

Vol 48 (4) ◽

pp. 1324-1329 ◽

Cited By ~ 25

Author(s):

Sander Roobol ◽

Willem Onderwaater ◽

Jakub Drnec ◽

Roberto Felici ◽

Joost Frenken

Keyword(s):

Data Reduction ◽

Modular Design ◽

Reciprocal Lattice ◽

Measurement Techniques ◽

Three Dimensional ◽

Two Dimensional ◽

X Ray Diffraction ◽

Data Set ◽

High Resolution Data ◽

X Ray

BINocularsis a tool for data reduction and analysis of large sets of surface diffraction data that have been acquired with a two-dimensional X-ray detector. The intensity of each pixel of a two-dimensional detector is projected onto a three-dimensional grid in reciprocal-lattice coordinates using a binning algorithm. This allows for fast acquisition and processing of high-resolution data sets and results in a significant reduction of the size of the data set. The subsequent analysis then proceeds in reciprocal space. It has evolved from the specific needs of the ID03 beamline at the ESRF, but it has a modular design and can be easily adjusted and extended to work with data from other beamlines or from other measurement techniques. This paper covers the design and the underlying methods employed in this software package and explains howBINocularscan be used to improve the workflow of surface X-ray diffraction measurements and analysis.

Download Full-text

Determination by high-resolution X-ray diffraction of shape, size and lateral separation of buried empty channels in silicon-on-nothing architectures

Journal of Applied Crystallography ◽

10.1107/s0021889807003603 ◽

2007 ◽

Vol 40 (2) ◽

pp. 338-343 ◽

Cited By ~ 1

Author(s):

Marco Servidori

Keyword(s):

High Resolution ◽

Lattice Point ◽

Reciprocal Lattice ◽

Reciprocal Lattice Vector ◽

X Ray Diffraction ◽

Lattice Vector ◽

Reciprocal Lattice Point ◽

One Dimensional ◽

X Ray ◽

Planar Array

High-resolution multi-crystal X-ray diffraction was employed to characterize silicon-on-nothing samples made by a one-dimensional periodic planar array of buried empty channels. When the channels are normal to the scattering plane, under the constraint of lattice continuity from the perfect substrate to the surface, this periodic array gives rise to a well defined Fraunhofer diffraction in a scan crossing a selected reciprocal lattice point and normal to the reciprocal lattice vector (transverse or ω scan). In a longitudinal scan (ω/2θ scan crossing the reciprocal lattice point and parallel to the reciprocal lattice vector) interference fringes are observed. By analysis of the ω scan and numerical fit of the ω/2θ scan, the period of the buried empty channels and their shape, size and lateral gap were easily determined, thanks to the high-resolution optics used for the measurements.

Download Full-text

Structural Characterization of Reactive Ion Etched Semiconductor Nanostructures Using X-Ray Reciprocal Space Mapping

MRS Proceedings ◽

10.1557/proc-406-457 ◽

1995 ◽

Vol 406 ◽

Author(s):

G. Bauer ◽

A. A. Darhuber ◽

V. Holy

Keyword(s):

Quantum Dot ◽

Reciprocal Lattice ◽

Space Mapping ◽

Reciprocal Space ◽

Lattice Points ◽

Scanning Electron Micrographs ◽

X Ray Diffraction ◽

Reciprocal Space Mapping ◽

Entire Volume ◽

X Ray

AbstractWe have studied GaAs/AlAs periodic quantum dot arrays using high resolution x-ray diffraction (reciprocal space mapping) around the (004) and (113) reciprocal lattice points. From the distribution of the diffracted intensities we deduced the average strain status of the dots. From the numerical simulations it is evident that random elastic strain fields are present, which extend through almost the entire volume of the quantum dot. The simulations of the x-ray measurements revealed that the crystalline part of the dots is considerably smaller as scanning electron micrographs would indicate.

Download Full-text

High Resolution X-ray Diffraction from Epitaxial Gallium Nitride Films

MRS Proceedings ◽

10.1557/proc-449-483 ◽

1996 ◽

Vol 449 ◽

Cited By ~ 2

Author(s):

T. Lafford ◽

N. Loxley ◽

BK Tanner

Keyword(s):

Gallium Nitride ◽

High Resolution ◽

Reciprocal Lattice ◽

Wide Distribution ◽

Reciprocal Lattice Vector ◽

Mosaic Structure ◽

X Ray Diffraction ◽

Lattice Vector ◽

Instrument Function ◽

X Ray

ABSTRACTThe width of double axis X-ray rocking curves of epitaxial GaN layers is shown to be critically dependent on the width of the detector aperture. We show that triple axis diffraction measurements using a crystal analyser before the detector enables the instrument function to be defined and the tilt and dilation distributions separated. All GaN samples examined showed a mosaic structure of misoriented sub-grains with little dilation within the mosaic blocks. In reciprocal space maps this was revealed as a wide distribution of intensity in a direction perpendicular to the reciprocal lattice vector.

Download Full-text

X-ray diffraction in superabsorbing crystals: absorption intrinsic width

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273319009732 ◽

2019 ◽

Vol 75 (5) ◽

pp. 772-776

Author(s):

A. N. C. Lima ◽

M. A. R. Miranda ◽

J. M. Sasaki

Keyword(s):

Diffraction Theory ◽

Dynamical Theory ◽

X Rays ◽

X Ray Diffraction ◽

Linear Absorption ◽

New Techniques ◽

Diffraction Profile ◽

X Ray ◽

Kinematical Theory ◽

Special Case

The several mathematical formulations of X-ray diffraction theory facilitate its understanding and use as a materials characterization technique, since one can opt for the simplest formulation that adequately describes the case being studied. As synchrotrons advance, new techniques are developed and there is a need for simple formulations to describe them. One of these techniques is soft resonant X-ray diffraction, in which the X-rays suffer large attenuation due to absorption. In this work, an expression is derived for the X-ray diffraction profiles of reflections where the linear absorption is far greater than primary extinction; in other words, the crystal is superabsorbing. The case is considered of a parallel plate crystal, for which the diffraction profile of the superabsorbing crystal is computed as a function of crystal size normal to the diffraction planes. For thin crystals or those with negligible absorption, the diffraction profile of a superabsorbing crystal coincides with the result of the kinematical theory. For thick crystals, the absorption intrinsic profile is obtained, described by a Lorentzian function and characterized by the absorption intrinsic width. This absorption intrinsic width is proportional to the linear absorption coefficient and its expression is similar to that for the Darwin width, while the absorption intrinsic profile is a special case of the Laue dynamical theory, and it is similar to the Ornstein–Zernike Lorentzian. The formulation of X-ray diffraction of superabsorbing crystals is simple and provides new perspectives for the soft resonant X-ray diffraction technique.

Download Full-text

Structural Characterization of Reactive Ion Etched Semiconductor Nanostructures Using X-Ray Reciprocal Space Mapping

MRS Proceedings ◽

10.1557/proc-405-359 ◽

1995 ◽

Vol 405 ◽

Cited By ~ 2

Author(s):

G. Bauer ◽

A. A. Darhuber ◽

V. Holy

Keyword(s):

Quantum Dot ◽

Reciprocal Lattice ◽

Space Mapping ◽

Reciprocal Space ◽

Lattice Points ◽

Scanning Electron Micrographs ◽

X Ray Diffraction ◽

Reciprocal Space Mapping ◽

Entire Volume ◽

X Ray

AbstractWe have studied GaAs/AlAs periodic quantum dot arrays using high resolution x-ray diffraction (reciprocal space mapping) around the (004) and (113) reciprocal lattice points. From the distribution of the diffracted intensities we deduced the average strain status of the dots. From the numerical simulations it is evident that random elastic strain fields are present, which extend through almost the entire volume of the quantum dot. The simulations of the x-ray measurements revealed that the crystalline part of the dots is considerably smaller as scanning electron micrographs would indicate.

Download Full-text

Two-dimensional incommensurately modulated structure of (Sr0.13Ca0.87)2CoSi2O7 crystals

Acta Crystallographica Section B Structural Science ◽

10.1107/s0108768100006716 ◽

2000 ◽

Vol 56 (5) ◽

pp. 811-821 ◽

Cited By ~ 13

Author(s):

B. Bagautdinov ◽

K. Hagiya ◽

K. Kusaka ◽

M. Ohmasa ◽

K. Iishi

Keyword(s):

Reciprocal Lattice ◽

Basic Structure ◽

Partial Ordering ◽

Partial Substitution ◽

Two Dimensional ◽

X Ray Diffraction ◽

Modulated Structure ◽

X Ray ◽

Oxygen Coordination ◽

Incommensurate Structure

The incommensurate structure of (Sr0.13Ca0.87)2CoSi2O7 at room temperature has been determined from single-crystal X-ray diffraction data. The compound has a non-centrosymmetric tetragonal basic cell of a = 7.8743 (4) and c = 5.0417 (2) Å with the space group P4¯21 m. The refinements of the basic structure converged to R = 0.038 for 757 main reflections. The two-dimensional incommensurate structure is characterized by the wavevectors q 1 = 0.286 (3)(a* + b*) and q 2 = 0.286 (3)(−a* + b*), where a*, b* are the reciprocal lattice vectors of the basic structure. With the (3 + 2)-dimensional superspace group P p4mg P4¯21 m , the refinements converged to R = 0.071 for 1697 observed reflections (757 main and 940 satellite reflections). The structure is described in terms of displacement of the atoms, rotation, distortion of CoO4 and SiO4 tetrahedra, and the partial ordering of the Sr and Ca atoms accompanied with the modulation. Correlated evolution of these features throughout the crystal gives rise to various oxygen coordination around Ca/Sr. Comparison of the derived modulated structure to that of Ca2CoSi2O7 clarified that the partial substitution of Ca by large alkaline-earth atoms such as Sr should decrease the distortion of the polyhedra around the cations.

Download Full-text

Tubulin structure at intermediate resolution

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100140634 ◽

1995 ◽

Vol 53 ◽

pp. 852-853

Author(s):

K. H. Downing ◽

S. G. Wolf ◽

E. Nogales

Keyword(s):

High Resolution ◽

Structural Data ◽

Therapeutic Drug ◽

Zinc Ions ◽

Two Dimensional ◽

X Ray Diffraction ◽

X Ray ◽

Negative Stain ◽

Functional Mechanisms ◽

Tubulin Structure

Microtubules are involved in a host of critical cell activities, many of which involve transport of organelles through the cell. Different sets of microtubules appear to form during the cell cycle for different functions. Knowledge of the structure of tubulin will be necessary in order to understand the various functional mechanisms of microtubule assemble, disassembly, and interaction with other molecules, but tubulin has so far resisted crystallization for x-ray diffraction studies. Fortuitously, in the presence of zinc ions, tubulin also forms two-dimensional, crystalline sheets that are ideally suited for study by electron microscopy. We have refined procedures for forming the sheets and preparing them for EM, and have been able to obtain high-resolution structural data that sheds light on the formation and stabilization of microtubules, and even the interaction with a therapeutic drug.Tubulin sheets had been extensively studied in negative stain, demonstrating that the same protofilament structure was formed in the sheets and microtubules. For high resolution studies, we have found that the sheets embedded in either glucose or tannin diffract to around 3 Å.

Download Full-text

Effects of Higher-Order Laue Zone reflections on structure images

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100148083 ◽

1993 ◽

Vol 51 ◽

pp. 450-451

Author(s):

H. S. Kim ◽

S. S. Sheinin

Keyword(s):

Wave Vector ◽

Theoretical Calculations ◽

Reciprocal Lattice ◽

Image Plane ◽

Bloch Wave ◽

Dynamical Theory ◽

Higher Order ◽

Lattice Image ◽

Lattice Vector ◽

Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

Download Full-text