◾ The Reciprocal Lattice

2014 ◽  
pp. 318-361
Keyword(s):  
Reciprocal Lattice

Transmission Electron Diffraction Analysis by Computer

1970 ◽  
Vol 28 ◽  
pp. 40-41
Author(s):  
R.A. Ploc ◽  
G.H. Keech
Keyword(s):  
Electron Diffraction ◽  
Input Data ◽  
Reciprocal Lattice ◽  
Computer Programme ◽  
Transmission Electron Diffraction ◽  
Usual Procedure ◽  
Transmission Electron ◽  
Complex Shapes ◽  
Computer Input ◽  
Diffraction Effects

An unambiguous analysis of transmission electron diffraction effects requires two samplings of the reciprocal lattice (RL). However, extracting definitive information from the patterns is difficult even for a general orthorhombic case. The usual procedure has been to deduce the approximate variables controlling the formation of the patterns from qualitative observations. Our present purpose is to illustrate two applications of a computer programme written for the analysis of transmission, selected area diffraction (SAD) patterns; the studies of RL spot shapes and epitaxy.When a specimen contains fine structure the RL spots become complex shapes with extensions in one or more directions. If the number and directions of these extensions can be estimated from an SAD pattern the exact spot shape can be determined by a series of refinements of the computer input data.

Analysis and lattice imaging of a transitional event on garnets

1978 ◽  
Vol 36 (1) ◽  
pp. 270-271
Author(s):  
J.Y. Laval
Keyword(s):  
Yttrium Iron Garnet ◽  
Reciprocal Lattice ◽  
Transitional Zone ◽  
Lattice Imaging ◽  
Yttrium Iron ◽  
X Ray ◽  
Ionic Process ◽  
The Matrix ◽  
High Resolution Technique ◽  
Iron Garnet

The exsolution of magnetite from a substituted Yttrium Iron Garnet, containing an iron excess may lead to a transitional event. This event is characterized hy the formation of a transitional zone at the center of which the magnetite nucleates (Fig.1). Since there is a contrast between the matrix and these zones and since selected area diffraction does not show any difference between those zones and the matrix in the reciprocal lattice, it is of interest to analyze the structure of the transitional zones.By using simultaneously different techniques in electron microscopy, (oscillating crystal method microdiffraction and X-ray microanalysis)one may resolve the ionic process corresponding to the transitional event and image this event subsequently by high resolution technique.

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 : .

Observation of faint contrast at planar faults in alloys

1978 ◽  
Vol 36 (1) ◽  
pp. 340-341
Author(s):  
K. Kuroda ◽  
Y. Tomokiyo ◽  
T. Kumano ◽  
T. Eguchi
Keyword(s):  
Reciprocal Lattice ◽  
Beam Theory ◽  
Al Alloys ◽  
Small Displacement ◽  
Electron Microscopic ◽  
Lattice Vector ◽  
Fringe Contrast ◽  
Planar Fault ◽  
Lattice Displacement ◽  
The Many

The contrast in electron microscopic images of planar faults in a crystal is characterized by a phase factor , where is the reciprocal lattice vector of the operating reflection, and the lattice displacement due to the fault under consideration. Within the two-beam theory a planar fault with an integer value of is invisible, but a detectable contrast is expected when the many-beam dynamical effect is not negligibly small. A weak fringe contrast is also expected when differs slightly from an integer owing to an additional small displacement of the lattice across the fault. These faint contrasts are termed as many-beam contrasts in the former case, and as ε fringe contrasts in the latter. In the present work stacking faults in Cu-Al alloys and antiphase boundaries (APB) in CuZn, FeCo and Fe-Al alloys were observed under such conditions as mentioned above, and the results were compared with the image profiles of the faults calculated in the systematic ten-beam approximation.

Kinematical and Dynamical CBED Pattern Models: Increasing the Accuracy of the Unit-Cell Parameter Measurements Based on CBED Patterns

1990 ◽  
Vol 48 (2) ◽  
pp. 540-541
Author(s):  
I.N. Yadhikov ◽  
S.K. Maksimov
Keyword(s):  
Reciprocal Lattice ◽  
Unit Cell ◽  
Reciprocal Lattice Vector ◽  
Unit Cell Parameters ◽  
Lattice Vector ◽  
Cell Parameters ◽  
Accuracy Of Measurements ◽  
The Difference ◽  
Image Position ◽  
Convergent Beam

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

SPECTRA: A program for processing electron images of crystals

1992 ◽  
Vol 50 (1) ◽  
pp. 132-133
Author(s):  
M.F. Schmid ◽  
R. Dargahi ◽  
M. W. Tam
Keyword(s):  
Lattice Distortion ◽  
Data Transfer ◽  
Reciprocal Lattice ◽  
Data Entry ◽  
Image Data ◽  
Distortion Correction ◽  
Specimen Preparation ◽  
Electron Image ◽  
Computer Controlled ◽  
Program Modules

Electron crystallography is an emerging field for structure determination as evidenced by a number of membrane proteins that have been solved to near-atomic resolution. Advances in specimen preparation and in data acquisition with a 400kV microscope by computer controlled spot scanning mean that our ability to record electron image data will outstrip our capacity to analyze it. The computed fourier transform of these images must be processed in order to provide a direct measurement of amplitudes and phases needed for 3-D reconstruction.In anticipation of this processing bottleneck, we have written a program that incorporates a menu-and mouse-driven procedure for auto-indexing and refining the reciprocal lattice parameters in the computed transform from an image of a crystal. It is linked to subsequent steps of image processing by a system of data bases and spawned child processes; data transfer between different program modules no longer requires manual data entry. The progress of the reciprocal lattice refinement is monitored visually and quantitatively. If desired, the processing is carried through the lattice distortion correction (unbending) steps automatically.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

Fast calibration of CBED patterns for quantitative analysis

1993 ◽  
Vol 51 ◽  
pp. 674-675
Author(s):  
M.A. Gribelyuk ◽  
M. Rühle
Keyword(s):  
Reciprocal Lattice ◽  
Accurate Determination ◽  
Incident Beam ◽  
Crystal Thickness ◽  
Structure Model ◽  
Beam Direction ◽  
Transmitted Beam ◽  
Reciprocal Vector ◽  
On Line

A new method is suggested for the accurate determination of the incident beam direction K, crystal thickness t and the coordinates of the basic reciprocal lattice vectors V1 and V2 (Fig. 1) of the ZOLZ plans in pixels of the digitized 2-D CBED pattern. For a given structure model and some estimated values Vest and Kest of some point O in the CBED pattern a set of line scans AkBk is chosen so that all the scans are located within CBED disks.The points on line scans AkBk are conjugate to those on A0B0 since they are shifted by the reciprocal vector gk with respect to each other. As many conjugate scans are considered as CBED disks fall into the energy filtered region of the experimental pattern. Electron intensities of the transmitted beam I0 and diffracted beams Igk for all points on conjugate scans are found as a function of crystal thickness t on the basis of the full dynamical calculation.

Effects of Higher-Order Laue Zone reflections on structure images

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.

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.

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