Symmetry cubic/hexagonal

When the atomic size ratio is near 1.2 some dense (i.e., close-packed) structures become possible in which tetrahedral sub-groups of one kind of atom share their vertices, sides or faces to from a network. This network contains holes into which the other kind of atoms are put. These are known as Laves phases. They have three kinds of symmetry cubic (related to diamond), hexagonal (related to wurtzite), and orthorhombic (a mixture of the other two). The prototype compounds are MgCu2, MgZn2, and MgNi2, respectively. Only the simplest cubic one will be discussed further here. See Laves (1956) or Raynor (1949) for more details. 

In most oxides, the oxygen atoms are present as close-packed layers stacked either to produce cubic symmetry or hexagonal symmetry. Some of the cubic cases have already been discussed. Now some hexagonal cases will be considered. 

J Cubic Hexagonal Tetragonal Lower symmetry J Cubic All other symmetries... 

These systems can be described in terms of their symmetry elements. A triclinic crystal has only a center of symmetry. Monoclinic crystals have a single axis of twofold rotational symmetry. Orthorhombic crystals have three mutually perpendicular axes of twofold symmetry. With tetragonal symmetry, there is a single axis of fourfold symmetry. Cubic crystals are characterized by four threefold axes of symmetry, the <111> axes. There is a single axis of threefold symmetry in the rhombohedral system. The hexagonal system involves a single axis of sixfold symmetry. 

Contrary to this chemical cluster growth is the formation of naked, ligand free clusters in a more physical sense. Considering the cluster growth in Fig. 8 and 9, it is to be recognized that both five-fold symmetry and hexagonal (hep) or cubic (ccp) structures are realized. While in smaller clusters the icosahedral and cuboctahedral symmetry are... 

The object of this experiment is to determine the crystal structure of a solid substance from x-ray powder diffraction patterns. This involves determination of the symmetry classification (cubic, hexagonal, etc.), the type of crystal lattice (simple, body-centered, or face-centered), the dimensions of the unit cell, the number of atoms or ions of each kind in the unit cell, and the position of every atom or ion in the unit cell. Owing to inherent limitations of the powder method, only substances in the cubic system can be easily characterized in this way, and a cubic material will be studied in the present experiment. However, the recent introduction of more accurate experimental techniques and sophisticated computer programs make it possible to refine and determine the structnres of crystals of low syimnetiy from powder diffraction data alone. 

Theoretioaiiy a maximum of 21 tensor eiements of eiastio stiffness for the triclinic crystal (the lowest-symmetry crystal) can be determined with one specimen. However, it is diffioult to assimilate properties relating to stress waves and elasticity for such a low-symmetry orystal. In praotioe, RUS oan determine nine tensor elements for orthotropic symmetry as well as for higher symmetry (isotropic, cubic, hexagonal or tetragonal). 

The structural units of surfactant-containing liquids are self-assembled aggregates, such as spherical or cylindrical micelles, or bilayers. These supramolecular structural units can then further self assemble into ordered phases, with cubic, hexagonal, smectic, or other symmetry. Consequently, the structural and flow properties of such liquids are amazingly rich. Laws of mass action, combined with geometric packing arguments, allow rationalization, if not prediction, of the phase behavior of many surfactant solutions. 

We consider a compound such as (Fe, Zn)S. It exists in two forms, distinguished by cubic symmetry or hexagonal symmetry, and in either form the ratio of Fe to Zn is variable. There is a range of pressures and... 

As noted in Chap. 1, every unit cell can be characterized by six lattice parameters — three edge lengths a, b, and c and three interaxial angles a, j3, and 7. On this basis, there are seven possible combinations of a, h, and c and a, ff and 7 that correspond to seven crystal systems (see Fig. 1.2). In order of decreasing symmetry, they are cubic, hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic, and triclinic. In the remainder of this section, for the sake of simplicity the discussion is restricted to the cubic system for which a — b c and a = / = 7 = 90°. Consequently, this system is characterized by only one parameter, usually denoted by a. 

Fig. 8.1. Form of the matrices % or c j for materials of different symmetries (a) hexagonal, (b) cubic, (c) isotropic. All matrices are symmetric about the leading diagonal and the following symbols are used , zero component , non-zero component — , equal components x, z(sn —Sia) for Sf/ or Cia) for...

Polarity Centro symmetry No. of point groups Cubic Hexagonal Tetragonal Rhombohedral Orthorhombic Monoclinic Triclinic... 

Detailed evaluation of the optical indicatrix can give useful information on the crystal symmetry. Cubic crystals and amorphous solids are optically isotropic trigonal, tetragonal, and hexagonal crystals are uniaxial orthorhombic, monoclinic, and triclinic crystals are biaxial. Extinction directions vary with wavelength in monoclinic and triclinic crystals. 

---