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Basic Theory on Optical Properties of Solids
Published in Moriaki Wakaki, Optical Materials and Applications, 2017
Moriaki Wakaki, Keiei Kudo (deceased)
The result satisfies the condition indicated above. These groups of symmetry elements express the lattice points of the crystal and are called the point group. Each unit cell is kept invariant for respective symmetry operation in the point group. But each unit cell must also cover the whole crystal by the translational movement. To meet the requirement, the rotation angles in Cn and Sn are limited to the integral multiple of 60° or 90°, which is derived from the properties of the point group. Only 32 point groups are allowed from these results. The 32 point groups give the possible arrangement of the crystal lattice and are often called the crystal class.
Introductory Concepts
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. These include pure translations, which move a point along a vector; screw axis, which rotate a point around an axis while translating parallel to the axis; and glide planes, which reflect a point through a plane while translating it parallel to the plane. There are 230 distinct space groups.
Crystallization
Published in George A. Lane, Solar Heat Storage: Latent Heat Materials, 1983
The crystal symmetry groups have been established by the use of point group theory. With this technique, crystallographers determined all the ways that equivalent points can be arranged around a given point in space, maintaining the symmetry properties of crystals. Considering axes of symmetry, planes of symmetry, and axes of inversion, there are 32 groups. Table 2 shows how these are distributed among the crystal systems.
Ferroelectric, Piezoelectric Mechanism and Applications
Published in Journal of Asian Ceramic Societies, 2022
Arun Singh, Shagun Monga, Neeraj Sharma, K Sreenivas, Ram S. Katiyar
The symmetry elements are used to describe symmetry about a point in space, and employing these elements of symmetry, all crystals can be separated into 32 diverse point groups or classes [3]. These 32-point groups are subcategories of the seven basic crystal systems, the crystal system is a grouping of crystal structures that are categorized according to the axial system used to describe their “lattice”. A crystal’s lattice is a three dimensional network of atoms that are arranged in a symmetrical pattern. Each crystal system consists of a set of three axes in a particular geometrical arrangement. The seven unique crystal systems, listed in order of decreasing symmetry, are: 1. Isometric System, 2. Hexagonal System, 3. Tetragonal System, 4. Rhombohedric (Trigonal) System, 5. Orthorhombic System, 6. Monoclinic System, 7. Triclinic System. Twenty-one out of the 32-point groups lack a center of symmetry, and 20 exhibit piezoelectricity as they become polarized on the application of mechanical stress. The remaining one point group out of the 21 does not exhibit piezoelectricity, even though it lacks a center of symmetry, due to the combination of symmetry elements. Ten out of the 20 piezoelectric point groups display a polarization that has a finite and permanent value, termed as spontaneous polarization, which exists when the applied stress or field is zero, and such dielectric materials are known as polar materials or pyroelectric materials.
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
The point group of a molecule is composed of the set of symmetry operations such as reflections, rotations, inversion, etc. that leave the molecular framework unchanged. These operations form a group in the mathematical sense, with multiplication defined as the composition of symmetry operations. When an object (such as a molecular orbital) is subjected to one of these operations, its behaviour can be described by its character, that is the overlap of the object before and after the operation. The collection of the characters for each operation gives the representation. For combinations of objects, such as a charge distribution defined by a pair of orbitals, the individual characters of the overall object are the product of the individual characters. Taking the representations as a whole, this element-wise multiplication to give a new combined representation is called the direct product (⊗), not to be confused with the outer or direct product used in other areas of mathematics. In order to describe and classify arbitrary representations, the irreducible representations (irreps) of the group form a unique basis: the way in which any particular object transforms (its representation) can be uniquely decomposed into a linear combination of the irreps. Each group has an irrep called the totally-symmetric irrep which has characters of unity. In order for an integral to be non-zero, the combined representation of the integrands must include (have non-zero projection onto) this totally-symmetric irrep. Individual components of the integrand (integrals, amplitudes, etc.) are generally chosen to be symmetry-adapted to a particular irrep, meaning that e.g. the direct product of the orbital representations for a given integral or amplitude tensor element must include the chosen overall irrep in order for that element to be non-zero. This sparsity is the origin of reduced computational cost when using point group symmetry.