An optical method of studying the diffraction from imperfect crystals II. Crystals with dislocations

1957 ◽  
Vol 239 (1217) ◽  
pp. 192-201 ◽  
Keyword(s):  
Intensity Distribution ◽  
Optical Method ◽  
Reciprocal Lattice ◽  
Lattice Points ◽  
Dimensional Problem ◽  
Single Edge ◽  
General Complex ◽  
Diffraction Patterns ◽  
Special Points ◽  
The Ideal

Lipson’s optical diffractometer has been used to determine the diffraction patterns of gratings representing crystals with dislocations. The optical method lends itself readily to the solution of the two-dimensional problem of diffraction by a single edge dislocation. The intensity distribution near the ideal reciprocal lattice points is, in general, complex, but for certain special points it is relatively simple and in agreement with that deduced by a new theory of Suzuki. An earlier theory of Wilson’s fails to explain the observed intensity distribution. Diffuse scattering, not predicted by Suzuki’s theory, occurs in the form of streaks joining the reciprocal lattice points. The diffraction has also been studied from gratings consisting of photographs of dislocated bubble rafts, and from a grating whose diffraction pattern is related to that of a screw dislocation. A brief discussion is given of the expected diffraction patterns from crystals with various arrays of dislocations.

Fine Structure in the Selected Area Electron Diffraction Patterns of Beidellite

1975 ◽  
Vol 33 ◽  
pp. 72-73
Author(s):  
N. Güven ◽  
R.W. Pease
Keyword(s):  
Fine Structure ◽  
Electron Diffraction ◽  
Reciprocal Lattice ◽  
Linear Chain ◽  
Lattice Points ◽  
Selected Area Electron Diffraction ◽  
General Terms ◽  
Regular Network ◽  
Diffraction Patterns ◽  
Pass Through

Selected area electron diffraction (SAD) patterns of beidellite exhibit fine structure in the form of nonradial streaks and extra spots between the normal Laue spots. The streaks form a regular network as shown in Figure 1A andvery clearly after a long exposure, in Fig. IB. These streaks do not pass through the origin and they are not symmetrical with respect to the reciprocal lattice points. Therefore they cannot be caused by finite crystallite size. The distribution of the streaks suggests a strong anisotropy in the beidellite structure as they are restricted to the directions parallel to [11], [11], and [02]. However, there are no streaks along the actual [11], [11] and [02] directions. In general terms, these linear streaks are explained by the presence of ‘continuous sheets’ or ‘walls’ of intensity in reciprocal space. These intensity 'walls' are associated with a linear chain of scatterers in the crystal in the direction perpendicular to the intensity sheets. Such linear scatterers may be produced by small shifts of certain atoms due to thermal motion, isomorphic substitutions, distortions, or other lattice imperfections.

scholarly journals Lattice row distance

2014 ◽  
Vol 70 (a1) ◽  
pp. C1454-C1454
Author(s):  
Hejing Wang ◽  
Ting Li ◽  
Ling Wang ◽  
Zhao Zhou ◽  
Lei Yuan
Keyword(s):  
Reciprocal Lattice ◽  
Early Time ◽  
Basic Feature ◽  
Lattice Points ◽  
Lattice Parameters ◽  
Parallel Lines ◽  
Basic Characteristics ◽  
Diffraction Patterns ◽  
Distance Formulae

Lattice and diffraction are two relating aspects of a crystal. The former reflects the nature of a crystal and the latter describes the basic feature of a crystal. A lattice possesses points and rows two basic characteristics. Great attention has been paid to the points and their distances and directions (angles) they form since the early time of crystallography. Starting from lattice points people have already revealed and found so many regulations in crystals and made great progresses in crystallography. What about the lattice rows? Starting from the geometric relations of reciprocal lattice, we propose six general formulae [1] to describe the relationships between the lattice row distance, the Miller indices h, k, l and the lattice parameters for all crystal systems along any direction. This, like the lattice points, establishes the foundation of the row-indexing, row-refinement of lattice parameters and row-determination of incidence direction theoretically. It is a new method from the lattice row distance to the Miller indices, to the lattice parameters or to the incidence direction. Five steps are optimized for the procedure of "Row-indexing" or "Row-refinement". For example, the procedure of row-indexing is described as 1) measurement of row distance; 2) calculation of row distance; 3) comparison of the measured with the calculated row distances; 4) indexing, and 5) check according to the crystallographic regulations. In respect to diffraction patterns, a series of diffraction spots (points) comprise row(s) and arrange into a series of parallel "lines". When diffraction is strong, diffraction spots are isolated and sharp. However, when diffraction is weak, those spots are obscure or gloomy and often distorted into elongation, asymmetry, deformation, etc. This leads to the outstanding of the rowing "lines" relatively and hence, the row-distance formulae are able to be utilized to structure analysis for those "linear diffraction patterns".

X-Ray Line Profile Analysis for Single Crystals

2014 ◽  
pp. 242-270
Keyword(s):  
Single Crystals ◽  
Intensity Distribution ◽  
Profile Analysis ◽  
Reciprocal Lattice ◽  
Lattice Points ◽  
Polycrystalline Materials ◽  
Line Profile Analysis ◽  
Line Profiles ◽  
X Ray ◽  
Lattice Misorientation

The features of the dislocation structure in plastically deformed single crystals can be determined from diffraction line broadening. Both the measuring and the evaluation procedures of X-ray line profiles are somewhat different from the methods used for polycrystalline materials. In this chapter, these procedures are overviewed, and their effectiveness is illustrated by representative examples. It is shown that the intensity distribution in the vicinity of the reciprocal lattice points can be mapped by rocking the single crystal about appropriate axes. From the detected intensity distribution, the density, the slip systems, and the arrangement of dislocations, as well as the lattice misorientation can be determined. The average misorientation obtained from rocking curve measurement can be related to the density of geometrically necessary dislocations. It is also shown that the inhomogeneous distribution of dislocations in plastically deformed single crystals usually results in asymmetric line profiles. The evaluation of these peaks enables the determination of the long-range internal stresses besides the dislocation densities in the dislocation cell walls and interiors.

Determination of dispersion and polarization of thermal plane waves in crystals

1971 ◽  
Vol 134 (3-4) ◽  
Author(s):  
P. L. La Fleur
Keyword(s):  
Intensity Distribution ◽  
Phonon Scattering ◽  
Reciprocal Lattice ◽  
Plane Waves ◽  
Lattice Points ◽  
X Ray Diffraction ◽  
Diffraction Intensity ◽  
Reciprocal Lattice Point ◽  
X Ray

AbstractThe dispersion of the thermal plane waves (phonons) in crystals can be determined from the x-ray diffraction intensity distribution around a reciprocal lattice point. In the method presented here no higher-order phonon-scattering corrections are necessary. It is shown furthermore that polarizations and dispersion of the phonons can be determined from the intensity distributions around six properly chosen reciprocal lattice points.

An optical method of studying the diffraction from imperfect crystals III. Layer structures with stacking faults

1958 ◽  
Vol 248 (1253) ◽  
pp. 183-198 ◽  
Keyword(s):  
Intensity Distribution ◽  
Optical Method ◽  
Reciprocal Lattice ◽  
Stacking Faults ◽  
Hexagonal Structure ◽  
Optical Diffraction ◽  
Layer Structures ◽  
The Face ◽  
Cubic Structure

The reciprocal lattice intensity distribution has been determined quantitatively using a photomultiplier to measure the optical diffraction intensities. The systems examined included the simple-cubic structure with 'wollastonite-type’ stacking faults, the close-packed-hexagonal structure with growth faults, and the face-centred-cubic structure with de­formation faults on one set and on two sets of {111} planes. It is shown that the Paterson (1952) analysis of the diffraction from deformation faulted f. c. c. crystals can be extended to intersecting faulted {111} planes, provided that the faulting parameter, α , is not greater than about 0·1. The main limitation of the optical method concerns the restriction in the number of layers (10 3 ) which can be conveniently represented in one grating. This restriction gives rise to weak fluctuations in the observed intensity distribution and to an uncertainty of up to 0·03 in the determination of α from this distribution.

A new method of complex structure analysis by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 628-629
Author(s):  
V.V. Rybin ◽  
E.V. Voronina
Keyword(s):  
Complex Structure ◽  
Reciprocal Lattice ◽  
Stacking Faults ◽  
Plastic Strains ◽  
Base Position ◽  
Structural Regularity ◽  
Orientation Matrix ◽  
Computerized Processing ◽  
Fully Automatic ◽  
Diffraction Patterns

Recently, it has become essential to develop a helpful method of the complete crystallographic identification of fine fragmented crystals. This was maainly due to the investigation into structural regularity of large plastic strains. The method should be practicable for determining crystallographic orientation (CO) of elastically stressed micro areas of the order of several micron fractions in size and filled with λ>1010 cm-2 density dislocations or stacking faults. The method must provide the misorientation vectors of the adjacent fragments when the angle ω changes from 0 to 180° with the accuracy of 0,3°. The problem is that the actual electron diffraction patterns obtained from fine fragmented crystals are the superpositions of reflections from various fragments, though more than one or two reflections from a fragment are hardly possible. Finally, the method should afford fully automatic computerized processing of the experimental results.The proposed method meets all the above requirements. It implies the construction for a certain base position of the crystal the orientation matrix (0M) A, which gives a single intercorrelation between the coordinates of the unity vector in the reference coordinate system (RCS) and those of the same vector in the crystal reciprocal lattice base : .

A program for simulation of changes in diffraction patterns with crystal rotations

1990 ◽  
Vol 48 (1) ◽  
pp. 544-545
Author(s):  
F.C. Mijlhoff ◽  
H.W. Zandbergenl
Keyword(s):  
Electron Microscope ◽  
Reciprocal Lattice ◽  
Extensive Training ◽  
Diffraction Mode ◽  
Diffraction Patterns ◽  
Electron Microscopist

Orientation of crystals for HREM is done in diffraction mode. To do this efficiently thorough knowledge of the electron microscope and the reciprocal lattice of the investigated material is essential. With respect to the electron microscope extensive training is required to obtain the ability to tilt a crystal in the desired orientation. Familiarity with the reciprocal lattice of the investigated materials has to be obtained by tilt experiments on a relatively large number of crystals in the electron microscope. Even for experienced electron microscopists this can be very time consuming.In order to be able to practice tilt experiments without using the electron microscope, a program to simulate the electron microscope in diffraction mode was developed. The inexperienced electron microscopist may use the program to practice tilting of crystals. The experienced microscopist can use the program to familiarize with the reciprocal lattice of materials, which have not been studied by him before.

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.

scholarly journals Comparison of EMC and CM methods for orienting diffraction images in single-particle imaging experiments

IUCrJ ◽  
2021 ◽  
Vol 8 (6) ◽  
Author(s):  
Miklós Tegze ◽  
Gábor Bortel
Keyword(s):  
Intensity Distribution ◽  
Single Particle ◽  
Free Electron ◽  
Free Electron Laser ◽  
Biological Molecules ◽  
Crucial Step ◽  
X Ray ◽  
Diffraction Patterns ◽  
Particle Imaging ◽  
Single Particle Imaging

In single-particle imaging (SPI) experiments, diffraction patterns of identical particles are recorded. The particles are injected into the X-ray free-electron laser (XFEL) beam in random orientations. The crucial step of the data processing of SPI is finding the orientations of the recorded diffraction patterns in reciprocal space and reconstructing the 3D intensity distribution. Here, two orientation methods are compared: the expansion maximization compression (EMC) algorithm and the correlation maximization (CM) algorithm. To investigate the efficiency, reliability and accuracy of the methods at various XFEL pulse fluences, simulated diffraction patterns of biological molecules are used.

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