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Symmetry of Crystals, Point Groups and Space Groups
Published in Dong ZhiLi, Fundamentals of Crystallography, Powder X-ray Diffraction, and Transmission Electron Microscopy for Materials Scientists, 2022
The seven crystal systems, combined with the four lattice types, can generate fourteen Bravais lattices, as discussed in Chapter 1. Some centering types are not allowed for some crystal systems because they would lower the symmetry of the unit cell. For example, the one-face-centered cubic lattice does not exist, as the three-fold axes along the diagonal directions do not allow it. Some centering types are unnecessary. For example, the one-face-centered tetragonal can be described using a smaller tetragonal primitive unit cell. It is worthwhile mentioning that, in the trigonal system with a rhombohedral cell, R is used for both the rhombohedral description (primitive cell) and the hexagonal description (triple cell).
Electron States in Solids
Published in John P. Xanthakis, Electronic Conduction, 2020
The crystalline lattice of a crystalline solid has an underlying lattice called a Bravais lattice. The Bravais lattice is defined as the set of points on each of which a molecular unit of the solid “sits” and, if it is repeated according to the translations of equation 2.1, it reproduces the whole crystal
Crystal Structure
Published in Alan Owens, Semiconductor Radiation Detectors, 2019
Combining the central crystal systems with the various possible lattice centerings leads to the Bravais lattices (sometimes known as space lattices). Bravais [12] showed that crystals could be divided into 14 unit cells for which (a) the unit cell is the simplest repeating unit in the crystal; (b) opposite faces of a unit cell are parallel and (c) the edge of the unit cell connects equivalent points. There are 14 unique Bravais lattices, based on the seven basic crystal systems, which are distinct from one another in the translational symmetry they contain. All crystalline materials fit in to one of these arrangements. The 14 three-dimensional lattices, classified by crystal system, are shown in Fig. 3.2.
Modelling of a displacive transformation in two-dimensional system within four-body approximation of Continuous Displacement Cluster Variation Method
Published in Philosophical Magazine, 2018
Our previous study based on the Bragg–Williams (BW) approximation [16] successfully reproduced essential features such as the Gaussian-type distribution of a displaced atom around a Bravais lattice point in the square phase and the disappearance of the polarisation of atomic displacement in the distorted phase upon transformation to the square phase. However, the resultant atomic distribution in the distorted phase is only the d-type, which is referred to as displacive behaviour in the Refs. [17,18], and its atomic distribution is like a Gaussian-type distribution. Other characteristic distributions such as the o-type, which is referred to as order–disorder behaviour in these references, and its double-peaked atomic distribution were by no means possible to realise, irrespective of the atomic interaction energies used. In addition to the type of atomic distribution, the order of phase transformation only leads to the second-order transformation. Such inconveniences may originate from the BW approximation in the entropy term, which is the most primitive approximation in the CVM hierarchy, and this approximation is sometimes misleading with regard to the order of transformation for a replacive-type transformation [19]. Hence, the main objective of the present study is to extend the former study of distorted to square transformation by employing a higher order approximation, namely a four-body approximation of the CDCVM, and to examine whether qualitatively different results are obtained with the four-body approximation.
Superconductivity in the twisted bilayer graphene: emergent mystery in the magic angle, the topological bosons and the Bardeen Cooper Schrieffer – Bose Einstein unconventional crossover
Published in Philosophical Magazine, 2021
Each monolayer of the bilayer graphene (BLG) consists of carbon atoms in the form of a honeycomb lattice (Figure 4(a)). The hexagonal lattice consists of two trigonal sublattices AAA and BBB. The primitive translation vectors e1 and e2 form the rhombic unit cell, and the basis consists of two carbon atoms shown as A and B. The Bravais lattice (consider, e.g. the lattice formed by the A atoms) is triangular with the Bravais lattice spacing , where a is the spacing between neighbouring atoms.
Nanosized oxide phases in oxide-dispersion-strengthened steel PM2000
Published in Philosophical Magazine, 2021
Yuan Wang, Jian Lin, Bocong Liu, Yifeng Chen, Dehui Li, Hui Wang, Yinzhong Shen
YAlO3 has a perovskite-like orthorhombic P Bravais lattice (a = 5.330 Å, b = 7.375 Å, c = 5.180 Å) with space group Pnma [18]. According to the orthorhombic structure’s mechanical stability criteria, YAlO3 has a mechanical stability from 0–20 GPa and 0–1300 K [19–21]. YAlO3 also has another crystal structure as hexagonal P Bravais lattice (a = 3.678 Å, b = 3.678 Å, c = 10.483 Å) [22].