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Geometry of X-ray Diffraction
Published in Dong ZhiLi, Fundamentals of Crystallography, Powder X-ray Diffraction, and Transmission Electron Microscopy for Materials Scientists, 2022
In X-ray diffraction, Bragg´s Law can be represented through Ewald sphere construction and the vector form. In the Ewald sphere construction, the wave number 1/λ is the radius of the sphere. The origin of vector k⇀0 is the center of the Ewald sphere, and the tail of the vector k⇀0 is the origin of the reciprocal lattice O∗. Based on the definition of the reciprocal lattice, once the direct lattice parameters are known, the reciprocal lattice parameters can be obtained. When we know the incident X-ray beam direction with respect to the direct lattice of the crystalline sample, the incident beam direction with respect to the reciprocal lattice is determined, as the orientations between the direct lattice and reciprocal lattice is coupled by the definition.
Experimental Considerations of 2D Graphene
Published in Andre U. Sokolnikov, Graphene for Defense and Security, 2017
Transmission electron microscopy (TEM) provides methods of graphene surface characterization. Different diffraction patterns give an image of the reciprocal lattice of graphene crystal structure16. Diffraction spots from graphene from the reciprocal lattice atoms in case of a single layer look like rods (Fig. 8.10 a) and b)). The appearance of rods comes from the two-dimensionality of graphene which may be visualized as a three-dimensional structure stretched in one direction. Thus, diffraction points of one of the 3D become a line. The Ewald sphere intersects the diffraction rods (Fig. 8.10 c)). The Ewald sphere is a geometrical figure used in crystallography investigated by X-rays, electron or neutron bombardment. The Ewald sphere allows finding wave vectors of the incident diffracted beams, diffraction angles if reflection angle is known. It also permits building of the reciprocal lattice (Fig. 8.11).
E
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[solid-state] The spherical configuration with respect to electromagnetic or particle wave (Note: de Broglie wavelength) interrogation of a lattice structure. The vector length resulting from crystalline interference with respect to the reciprocal lattice point outlines a spherical shape described by Paul Peter Ewald (1888–1985), a physicist from Germany. The graphical representation by means of the Ewald sphere applies to electron diffraction, neutron diffraction as well as X-ray crystallography and is a variation of Bragg’s law for lattice diffraction defined by drawing a sphere with radius reciprocal wavelength (λ): R=1/λ, originating from the “real crystal,” where the circumference intersects with the reciprocal lattice for the transmitted beam. The distance between the planes of the crystal is defined by the cord (dhkl) for the location of the reciprocal lattice point on the sphere with respect to the direction of the interrogating beam as 1/dhkl (see Figures E.75 and E.76).
Evaluation of anisotropic small-angle neutron scattering data from metastable β-Ti alloy
Published in Philosophical Magazine, 2018
Pavel Strunz, Jana Šmilauerová, Miloš Janeček, Josef Stráský, Petr Harcuba, Jiří Pospíšil, Jozef Veselý, Peter Lindner, Lukas Karge
When the abovementioned modifications are applied, the general structure of the modelling of the SANS pattern with NOC program is the following: - creation of the 3D model of particle system with a local size distribution (Monte Carlo calculation of the size and distance distributions with the mean size and the mean distance corresponding to the mean size and the mean distance of the global size distribution – these are two of the input parameters of the model),- Fourier transform of the 3D real-space model to the 3D cross section in the reciprocal space,- directional averaging,- implementation of the global size distribution (log normal, its width is another input parameter) by scaling the 3D cross-section model in the reciprocal space,- inclusion of the orientation distribution (its width is another input parameter of the model) of particles in the reciprocal space by performing series of sections of the 3D cross-section model along the Ewald sphere surface (the appropriate number of sections in the reciprocal space usually equals to 133 = 2197, which is a compromise between the angular step between the sections and the computing time [13]),- in case of more sample orientations, cuts through the 3D cross-section model corresponding to different sample orientations are performed, resulting in simulated 2D patterns,- multiple-scattering corrections and instrumental smearing corrections are carried out for the 2D modelled SANS patterns.
Stealth and equiluminous materials for scattering cancellation and wave diffusion
Published in Waves in Random and Complex Media, 2021
S. Kuznetsova, J. P. Groby, L. M. Garcia-Raffi, V. Romero-García
To interpret the scattering produced by these distributions, we first discuss how the scattering is directly interpreted in the reciprocal space using the von Laue formulation. Let us consider that the system is excited by an incident plane wave the wavevector of which is , with the unitary vector along the x direction and . We choose in this particular example to analyze the Bragg scattering in the periodic case. This wavevector is represented in Figure 1(b,c,g,h) for the periodic [random] case pointing one of the points of the reciprocal space. The von Laue formulation of the wave diffraction [65] stipulates that the difference between the vector of the scattered wave, , and that of the incident wave, must be a vector of the reciprocal space, i.e. , for constructive interference to occur. Assuming elastic scattering, , only the vectors pointing non zero values of along the Ewald sphere [65] can lead to scattered waves for 3D problems. This sphere of radius is centered at the origin of in the reciprocal space. More precisely, all the possible scattered waves are given by the Ewald sphere. The scattered wavevectors, , are then given by the vector connecting the center of the sphere and the points along this sphere having a non-null value of . The scattering is finally activated along the direction given by these vectors . This discussion is valid for any dimension, the Ewald sphere is a circumference in 2D of radius centered in , and is given by the limits of a segment of length centered in in 1D.
Iron nitride, α″-Fe16N2, around <100> interstitial type dislocation loops in neutron-irradiated iron
Published in Philosophical Magazine, 2021
T. Yoshiie, K. Inoue, K. Yoshida, T. Toyama, Y. Satoh, Y. Nagai
The observed image changes of α″-Fe16N2 with +/− g (Figure 3) and +/− s can be explained as follows. The illustration of an electron diffraction pattern and an Ewald sphere with a tilted beam approximately along the zone axis [111] of iron is shown in Figure 10. As α″-Fe16N2 is at three {100} planes of iron, then depending on the direction of the three cFeN axes, three extra spots should exist near the iron spots. For example, near (), there are three () spots of α″-Fe16N2. They have the reciprocal lattice vectors of (, 0, ) with , with , and with . The spot overlaps with of iron because the lattice constants (aFeN and bFeN) of α″-Fe16N2 are twice that of iron aFe. Essentially, the {111} planes of α″-Fe16N2 incline 2.5° relative to the iron {111} planes. In Figure 10, the ratio aFeN/cFeN of α″-Fe16N2 ( = 0.909) is exaggerated to 0.5; otherwise, the spots of α″-Fe16N2 are too close to the iron spots. The α″-Fe16N2 spots have streaks from the thin precipitates [17] as shown in lines. The direction of the streaks also tilts 35.3° from the Fe {111} reciprocal lattice plane. When the projection of the centre of the Ewald sphere is at ‘A’ for WBDF-TEM using of iron, the sphere intersects the streak of diffraction spot of α″-Fe16N2 on (100) plane as indicated ‘P’ and does not intersect the other two streaks. Therefore, only α″-Fe16N2 on the (100) plane is bright under WBDF conditions. Conversely, when the projected centre is at ‘B’, the sphere intersects the streak of α″-Fe16N2 on the (001) plane, as indicated by ‘Q’, and only the α″-Fe16N2 on the (001) plane are observed. Note that similar to the alternation of the diffraction vectors, this behaviour also occurs by changing the s sign.