Cubic system, classes

In metallic solids, the reticular positions are occupied by cations immersed in a cloud of delocalized valence electrons. In sohd Na, for instance (figure 1.2D), the electron cloud or electron gas is composed of electrons of the s sublevel of the third shell (cf table 1.2). Note that the type of bond does not limit the crystal structure of the solid within particular systems. For example, all solids shown in figure 1.2 belong to the cubic system, and two of them (NaCl, Ne) belong to the same structural class Fm3m). 

As the cubic system was often found to be an important structural class for good superconductors, another myth was generated that suggested one should focus on compounds having a cubic-type crystalline structure, or a structure possessing high symmetry. This myth was also abandoned when lower symmetry systems were found... 

Cubic (sometimes called isometric, or lesser al). All crystals having four secondary threefold axes have three mutually perpendicular directions all equivalent to each other. The unit cell is thus a cube, the secondary threefold axes being the cube diagonals. The five classed of the cubic system are 23, m3 (= 2/m3), 43m, 43 (= 432), and (= 4/m 3 2/m). Examples are shown in Fig 37. 

Fig. 37. Cubic system. (See also Figs. 16-22.) a. Unit cell type, b and c. Two habits of pyrites, FeSs. Class m3, d. Tetrahedrite, Cu3SbSa. Class 43m. e. Spinel, MgAls04. Class m3m. /. Almandine (Garnet), Fe8Al2(Si04)3. Class mZm.

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

Let us now refer to the cubic system. For all the space groups belonging to the Laue class m3 the 24 equivalent reflections are ... 

Smith and Snyder compared T n and M20 performances, analyzing a set of compounds belonging to triclinic, orthorhombic and cubic systems. They emphasized the superiority of T n with respect to M20 because the latter is (a) defined for exactly 20 lines, (b) strongly dependent on the crystal class and the space group. But Werner ° noted that the increasing value of M20 with symmetry is not a disadvantage, since a cubic indexing of a powder pattern is more probable than a triclinic one. 

Figure 27.16 Tetrahedral class of the cubic system, (a) Truncated cube, (b) Tetrahedron developed from the cube.

Cubic system uses the S5mibol T (from tetrahedron) for the 3A 4A class and O (from octahedron) for the 3A" 4A 6A class. For the rest of the classes Tj will be the 3AMA" 3nC class, T the 3A44A 6P class, and Oj the 3AW6A26P3rtC class, Table 2.3. 

There is very important to establish correctly the indices in order to be identified the simple forms. For example, in the cubic system in each class we have at least two forms of the same number of faces which are equally inclined in relation with two axes and different towards the third one. 

Many secondary phenomena and principles, such as holosymmetry and holohedry, derive from the principles elucidated above but are not dealt with here. Attention is merely drawn to one common mistake It is because an ordinary perfect cube shape has a tetrad axis relating its four vertical sides that it is often thought that the cubic system contains a tetrad, or rather three of them, as a cube can be seen to be the same when looked at from any one of three orthogonal directions. However, this refers only to the external symmetry of crystal faces. Of the given cubic classes only two contain a tetrad, one (class 43m) contains an inversion tetrad and the other two classes contain only a combination of a triad and a diad (and hence not a tetrad). In those classes without a tetrad, a crystal cannot be simply rotated 90° to obtain an identical atomic arrangement even if the external crystal faces might suggest this. 

The cubic system has the most symmetry of all as can be seen in Figure 4.10g. It has three mutually orthogonal axes of fourfold symmetry along the (100) directions, four axes of threefold symmetry along the (111) directions, and six axes of twofold symmetry along the (110) directions. Mirror planes consist of the 200 and 110 families. The Schoenflies symbol for the full symmetry of this class is Oh, the O standing for octahedral. The full international symbol is, A/mSl/m indicating mirror planes perpendicular to the fourfold axes as well as to the twofold axes and a center of inversion on the threefold axes. (The 111 family of planes are not mirror planes.) However, this symbol is usually abbreviated as m3m (in some text, this is written as m3m). 

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

The information obtainable from the Laue symmetry is meagre it consists simply in the distinction, between crystal classes, and then only in the more symmetrical systems—cubic, tetragonal, hexagonal, and trigonal (see Table VI). But it is useful in cases in which morphological features do not give clear evidence on this point. 

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