scholarly journals Group-Theoretical Approach to Reduced Orientation Spaces for Crystallographic Textures

1995 ◽  
Vol 23 (4) ◽  
pp. 201-219
Author(s):  
V. P. Yashnikov ◽  
H. J. Bunge
Keyword(s):  
Theoretical Approach ◽  
Point Group ◽  
Three Dimensional ◽  
Rotation Group ◽  
Orientation Space ◽  
Proper Point ◽  
Symmetry Properties ◽  
Voronoi Partition ◽  
Invariant Derivation ◽  
A Finite Group

A unified group-theoretical approach to the reduction problem for the orientation space of a crystallographic texture is developed. After preliminary considerations of the three-dimensional rotation group SO(3) the concept of the invariant inner distance function in the group space has been introduced. Left and right group translations, inner auto-morphisms, motions of general form, and inversion transforms in the space SO(3) are analysed. The concept of Dirichlet-Voronoi partition dual to an arbitrary finite set of rotations has been considered. It is shown that the Dirichlet-Voronoi partition, dual to the proper point group for the grain lattice of original orientation, is regular with respect to the group of motions generated by elements of proper point group.It is demonstrated that the true orientation space of a texture (at the absence of specimen symmetry) may be obtained by passing to the topological closure of any Dirichlet-Voronoi domain with the next, identifying crystallographically equivalent rotations belonging to its topological boundary. Thus an invariant derivation for the reduced (true) orientation space is given that does not require using any particular parametrization for the group space SO(3).Symmetry properties of Dirichlet-Voronoi domains are studied in conclusion. It is shown that any domain of such kind admits a finite group of symmetries generated by elements of a proper point group and by an appropriate inversion of the group space SO(3). It is proved that only part of them may be extended onto the true orientation space.

Crystal Structure Analysis of Cu-Zr Alloys by Convergent Beam Diffraction

1981 ◽  
Vol 39 ◽  
pp. 354-355
Author(s):  
A. F. Marshall ◽  
J. W. Steeds ◽  
D. Bouchet ◽  
S. L. Shinde ◽  
R. G. Walmsley
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Intermetallic Phase ◽  
Three Dimensional ◽  
Crystal Structure Analysis ◽  
Ion Milling ◽  
Composition And Structure ◽  
Unit Cells ◽  
Sputter Deposited ◽  
Convergent Beam

Convergent beam electron diffraction is a powerful technique for determining the crystal structure of a material in TEM. In this paper we have applied it to the study of the intermetallic phases in the Cu-rich end of the Cu-Zr system. These phases are highly ordered. Their composition and structure has been previously studied by microprobe and x-ray diffraction with sometimes conflicting results.The crystalline phases were obtained by annealing amorphous sputter-deposited Cu-Zr. Specimens were thinned for TEM by ion milling and observed in a Philips EM 400. Due to the large unit cells involved, a small convergence angle of diffraction was used; however, the three-dimensional lattice and symmetry information of convergent beam microdiffraction patterns is still present. The results are as follows:1) 21 at% Zr in Cu: annealed at 500°C for 5 hours. An intermetallic phase, Cu3.6Zr (21.7% Zr), space group P6/m has been proposed near this composition (2). The major phase of our annealed material was hexagonal with a point group determined as 6/m.

Semi-automated methods for 3-D reconstruction of macromolecular complexes

1995 ◽  
Vol 53 ◽  
pp. 42-43
Author(s):  
B. Carragher ◽  
M. Whittaker
Keyword(s):  
Three Dimensional ◽  
Time Saving ◽  
Macromolecular Complexes ◽  
Symmetry Properties ◽  
Dimensional Reconstruction ◽  
Experienced Operator ◽  
Projection Images ◽  
The Individual ◽  
Natural Symmetry

Techniques for three-dimensional reconstruction of macromolecular complexes from electron micrographs have been successfully used for many years. These include methods which take advantage of the natural symmetry properties of the structure (for example helical or icosahedral) as well as those that use single axis or other tilting geometries to reconstruct from a set of projection images. These techniques have traditionally relied on a very experienced operator to manually perform the often numerous and time consuming steps required to obtain the final reconstruction. While the guidance and oversight of an experienced and critical operator will always be an essential component of these techniques, recent advances in computer technology, microprocessor controlled microscopes and the availability of high quality CCD cameras have provided the means to automate many of the individual steps.During the acquisition of data automation provides benefits not only in terms of convenience and time saving but also in circumstances where manual procedures limit the quality of the final reconstruction.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

scholarly journals Nearly uniform sampling of crystal orientations

2018 ◽  
Vol 51 (4) ◽  
pp. 1162-1173 ◽  
Author(s):  
Romain Quey ◽  
Aurélien Villani ◽  
Claire Maurice
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Uniform Distributions ◽  
Orientation Space ◽  
Unit Quaternions ◽  
Spatial Uniformity ◽  
Unit Hypersphere ◽  
Open Source Software Package ◽  
Diffraction Patterns ◽  
Thomson Problem

A method is presented for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry. The method is based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere – a configuration of high spatial uniformity. Orientations are represented as unit quaternions, which lie on a unit hypersphere in four-dimensional space. Expressions of the electrostatic potential energy and Coulomb's forces are derived by working in the tangent space of orientation space. Using the forces, orientations are evolved in a conventional gradient-descent optimization until equilibrium. The method is highly versatile as it can generate uniform distributions for any number of orientations and any symmetry, and even allows one to prescribe some orientations. For large numbers of orientations, the forces can be computed using only the close neighbourhoods of orientations. Even uniform distributions of as many as 106 orientations, such as those required for dictionary-based indexing of diffraction patterns, can be generated in reasonable computation times. The presented algorithms are implemented and distributed in the free (open-source) software package Neper.

scholarly journals The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

1983 ◽  
Vol 74 ◽  
pp. 213-224
Author(s):  
I.A. Robin ◽  
V.V. Markellos
Keyword(s):  
Recent Work ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Three Dimensional ◽  
General Situation ◽  
Orbital Symmetry ◽  
Resonant Orbits ◽  
Symmetry Properties ◽  
Elliptic Restricted Problem ◽  
Symmetric Periodic Orbits

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

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