Ferroelectric space groups

1986 ◽  
Vol 42 (1) ◽  
pp. 44-47 ◽  
Author(s):  
D. B. Litvin
Keyword(s):  
Electric Dipole ◽  
Space Time ◽  
Discrete Space ◽  
Symmetry Groups ◽  
Space Groups ◽  
Point Groups

The 440 ferroelectric space groups, viz the Heesch-Shubnikov (magnetic) space groups, which are symmetry groups of ferroelectric electric-dipole arrangements in crystals, are derived and tabulated. By considering automorphisms induced by the automorphisms of the discrete space-time group, we show that although ferroelectric, ferromagnetic and ferrocurrent point groups all number 31, the number of ferroelectric space groups differs from 275, which is that of both ferromagnetic and ferrocurrent space groups.

3D Braided Geometry Structures Based on Space Group

2010 ◽  
Vol 152-153 ◽  
pp. 1156-1161 ◽  
Author(s):  
Wen Suo Ma ◽  
Bin Qian Yang ◽  
Xiao Zhong Ren
Keyword(s):  
Group Theory ◽  
Symmetry Groups ◽  
The Novel ◽  
Geometrical Structures ◽  
Braided Group ◽  
Space Group ◽  
Space Groups ◽  
Geometry Structure ◽  
Point Groups ◽  
Space Point

3D braided group theory is dissertated. The analysis procedure is described from the existing braided geometry structure to the braided space group; 3D braided geometrical structures are finally described by means of group theory. Some of novel 3D braided structures are deduced from the braided space groups. By describing the 3D braided materials with braided space point and braided space groups, the braided space groups are not always the same as symmetry groups of crystallographic because novel lattices can be produced and the reflection operation cannot exist in braided space point groups. Braided point and space groups are theoretical basis for deriving the novel braided geometry structure.

Crystallography and Diffraction

2021 ◽  
pp. 220-276
Author(s):  
Kannan M. Krishnan
Keyword(s):  
Rotational Symmetry ◽  
Reciprocal Lattice ◽  
Real Space ◽  
Crystalline Materials ◽  
Space Groups ◽  
Ewald Sphere ◽  
Point Groups ◽  
Diffraction Patterns ◽  
Point Line ◽  
Periodic Arrangement

Crystalline materials have a periodic arrangement of atoms, exhibit long range order, and are described in terms of 14 Bravais lattices, 7 crystal systems, 32 point groups, and 230 space groups, as tabulated in the International Tables for Crystallography. We introduce the nomenclature to describe various features of crystalline materials, and the practically useful concepts of interplanar spacing and zonal equations for interpreting electron diffraction patterns. A crystal is also described as the sum of a lattice and a basis. Practical materials harbor point, line, and planar defects, and their identification and enumeration are important in characterization, for defects significantly affect materials properties. The reciprocal lattice, with a fixed and well-defined relationship to the real lattice from which it is derived, is the key to understanding diffraction. Diffraction is described by Bragg law in real space, and the equivalent Ewald sphere construction and the Laue condition in reciprocal space. Crystallography and diffraction are closely related, as diffraction provides the best methodology to reveal the structure of crystals. The observations of quasi-crystalline materials with five-fold rotational symmetry, inconsistent with lattice translations, has resulted in redefining a crystalline material as “any solid having an essentially discrete diffraction pattern”

2. Symmetry

2016 ◽  
pp. 24-38
Author(s):  
A. M. Glazer
Keyword(s):  
Crystal Structure ◽  
Reflection Point ◽  
Notation System ◽  
Space Group ◽  
Space Groups ◽  
Different Types ◽  
Point Groups

In order to explain what crystals are and how their structures are described, we need to understand the role of symmetry, for this lies at the heart of crystallography. ‘Symmetry’ explains the different types of symmetry: rotational, mirror or reflection, point, chiral, and translation. There are thirty-two point groups and seven crystal systems, according to which symmetries are present. These are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Miller indices, lattices, crystal structure, and space groups are described in more detail. Any normal crystal belongs to one of the 230 space group types. Crystallographers generally use the International Notation system to denote these space groups.

GALILEAN AND DISCRETE SPACE-TIME SYMMETRIES OF ROY-SINGH DETERMINISTIC QUANTUM MECHANICS

1995 ◽  
Vol 10 (32) ◽  
pp. 4641-4650
Author(s):  
ARVIND KUMAR
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Time Reversal ◽  
Space Time ◽  
Discrete Space ◽  
Galilean Space

The recent deterministic quantum theory of Roy and Singh is shown to be covariant with respect to Galilean, space reflection and time reversal transformations.

Transition from a continuous to a discrete space-time scheme

1970 ◽  
Vol 66 (4) ◽  
pp. 645-656 ◽  
Author(s):  
E. A. B. Cole
Keyword(s):  
Space Time ◽  
Discrete Space
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