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Crystal Structure
Published in Alan Owens, Semiconductor Radiation Detectors, 2019
While Pearson symbols categorize crystal structure into particular patterns and are easy to use and conceptually simple, not every crystal structure is uniquely defined. The space group designation, also known as the International or Hermann-Mauguin [19] system, is a mathematical description of the symmetry inherent in a crystal’s structure and is also represented by a set of numbers and symbols. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, belonging to one of the basic crystal systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering and the point group symmetry operations of reflection and rotation improper rotation. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries. The International Union of Crystallography publishes comprehensive tables [20] of all space groups and assigns each a unique number. For example, rock salt is given the number “225” (in Hermann-Mauguin notation it is designated “Fm3m”). The relationships among the basic crystal systems, the Bravais lattices and the point and space groups are shown in Fig. 3.3.
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.
Crystalline Polymers
Published in Timothy P. Lodge, Paul C. Hiemenz, Polymer Chemistry, 2020
Timothy P. Lodge, Paul C. Hiemenz
The unit cell contains the smallest number of atoms, in the appropriate spatial relationships, necessary to enable prediction of the full structure of a macroscopic single crystal by repetitive close stacking of unit cells. A schematic unit cell is illustrated in Figure 13.3a. The lengths of the three sides are designated a, b, and c, and the corresponding angles are α, β, and γ. A single crystal can thus be generated by filling space with unit cells, so that the structure repeats exactly every a Å as one moves along the a direction, etc. However, the precise location of the various atoms within the unit cell requires further information than is contained in the three lengths and three angles. There are seven crystal classes (cubic, trigonal, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic), which are defined by different constraints on the values (a, b, c) and (α, β, γ). These are specified in Table 13.1. Within these classes there are further subdivisions, for example, a cubic unit cell could be face-centered, body-centered, or primitive. When these possibilities are included it turns out that there are 14 distinct structures, called Bravais lattices, within these seven classes. Finally, within a certain Bravais lattice there can be many different ways in which the atoms are arranged in detail. These arrangements are described by sets of symmetry operations that leave the structure unchanged, such as rotation about an axis by an angle of 60°, 90°, or 180°, or reflection through a plane, etc. The set of symmetry operations that applies to a particular crystal specifies a space group; in total there are 230 different space groups. The determination of the space group and the full unit cell structure of a polymer crystal, including bond angles, bond lengths, and interchain distances, is the first goal of polymer crystallography.
A new stable porous Pr-organic framework constructed by multi-iodine-substituted aromatic polycarboxylates: Synthesis, characterization, and selective adsorption of dyes
Published in Journal of Coordination Chemistry, 2019
Ying Sun, Feng Ying Bai, Xue Min Wang, Yu Wang, Li Xian Sun, Yong Heng Xing
The data was obtained after X-ray diffraction analysis showed that 1 was a cubic crystal system, and the space group was P2(1)3. Nine coordination bonds are formed around the central Pr ion, three of which are occupied by three oxygen atoms (O1) and six oxygen atoms from three H3TIBTC ligands (O2, O3) (Figure 1a). In each structural unit, H3TIBTC is connected in the form of three building blocks with μ3-η1η1η1η1η1η1, in which two oxygen atoms on each carboxyl group are coordinated in a bidentate chelation coordination mode (Figure 1b). The DMF molecule acts as a terminal ligand to prevent the extension of the structural unit in other directions. Each central Pr ion forms a building block [Pr(COO)3] with three carboxyl groups from the same ligand. Each building block connects three ligands, while each ligand connects three building blocks. Three porous structures are formed by alternating N atoms in the ligand. In this structure, Pr-O bonds are divided into two groups. One is the coordination of three equivalent oxygen atoms (O1) in the three coordinated DMF molecules, and the Ln-O bond length is 2.534 Å. The other is the six oxygen atoms (O2, O3) of the three H3TIBTC ligands have a bond length of 2.574 Å and 2.644 Å. All of these bond lengths are similar to those reported in the related literature, which indicates that the bond lengths involved for 1 are all within a reasonable range [38]. Considering the complexity of the structure, we tried to resolve the topology of the framework by treating the central metal and ligand as nodes. The simplified structure of the coordination compound is shown in Figure 1(c). Each Pr(COO)3 is reduced to a 3-connected node, and each ligand is also reduced to a 3-connected node. The entire structure is shaped into a 3,3-connected srs topology (Figure S5c in Supplementary Material).
Investigations on the enhancement of thermomagnetic properties in Fe2.4Ga0.6O4
Published in Phase Transitions, 2023
K. Rećko, M. Orzechowska, W. Olszewski, A. Beskrovnyy, M. Biernacka, U. Klekotka, A. Miaskowski, K. Szymański
A space group is formed when combining a point symmetry group with a set of lattice translation vectors. The space group Fdm requires four transition vectors: (1) – corresponding to the vertex translation and three related to the centering of unit cells on the face (the first numerical row in the second column of Table 1). According to Wyckoff notation, atomic coordinates, as a result of the symmetry operation, guarantee the invariance of the sublattice. The atomic coordinates together with selective transition vectors allow tracking variants of the spin order in sublattices with mixed occupancy and variable magnetic modes. Analyzing the occupancy of the A-sublattice by gallium cations, one can simulate all possible configurations of ion arrangements in the spinel structure (Figure 3(a–f)). Considering the equivalent combinations of two of the four translation vectors (row 2 of Table 1), it is possible to analyze the distribution of cations in the B sublattice, in which only iron ions are found. Depending on the implemented translation vectors, for example, (1) and (2) (Figure 3(a)) or (1) and (3) (Figure 3(b)) the magnetic ordering propagates differently at (16d) positions. Two of the six configurations favor the formation of homogeneous chains of Fe2+ ions along the diagonal of the base separated by non-magnetic gallium chains and correspondingly perpendicular chains formed by clearly weaker magnetic moments of Fe3+ (see Figure 3(b,e)). It is easy to predict that, under the action of an alternating magnetic field, such systems will require longer relaxation times and will be associated with slower superparamagnetic fluctuations. The other four configurations will be related with shorter relaxation times and faster superparamagnetic fluctuations. Obviously, the configuration distribution can change regardless of the increasing gallium content in the layout or the cation distribution in the sublattices in favor of random configurations. The other source leading to the creation of non-equivalent configurations is non-stoichiometric sample composition.