Crystal system, symmetric

Symmetry restrictions for a number of crystal systems are summarized in Table B.l. The local symmetry restrictions for a site on a symmetry axis are the same as those for the crystal system defined by such an axis, and may thus be higher than those of the site. This is a result of the implicit mmm symmetry of a symmetric second-rank tensor property. For instance, for a site located on a mirror plane, the symmetry restrictions are those of the monoclinic crystal system. 

Figure 1.37. Three possibilities to select the crystallographic basis in hexagonal and trigonal crystal systems and the family of (1120) crystallographic planes in the hexagonal crystal system. Indices are shown in the unit cell based on the vectors b and c. Three additional symmetrically related families of planes have indices (1120), (1210) and (2110) in the same basis and we leave their identification to the reader.

Figure 5.6. Schematic representations of the fractions of the volume of the sphere (r = 1/X) in the reciprocal space in which the list of hkl triplets should be generated in six powder Laue classes to ensure that all symmetrically independent points in the reciprocal lattice have been included in the calculation of Bragg angles using a proper form of Eq. 5.2. The monoclinic crystal system is shown in the alternative setting, i.e. with the unique two-fold axis parallel to c instead of the standard setting, where the two-fold axis is parallel to b. ...

In other words, Np ss is the number of symmetrically independent points in the reciprocal lattice limited by a sphere with the diameter d N (= 1/t/v) as established by Eq. 5.3 after substituting the Bragg angle, 0, of the iV observed Bragg peak for Qhu. Additional restrictions are imposed on Nposs in high symmetry crystal systems when reciprocal lattice points are not related by symmetry but when they have identical reciprocal vector lengths due to specific unit cell shape (e.g. h05 and h34 in the cubic, or 05/ and 34/ in the... 

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

II, A). Crystallographic data show that barbiturates generally have low-symmetric crystal systems, i.e., monoclinic or sometimes triclinic (Table V). 

The assignation of axes of reference in relation to the rotational symmetry of the crystal systems defines six lattices that, by definition, are primitive or P-lattices. To determine if new lattices can be formed from these P-lattices, one must determine if more points can be added so that the lattice condition is still maintained, and whether this addition of points alters the crystal system. For example, if one starts with a simple cubic primitive lattice and adds other lattice points in such a way that a lattice still exists, it must happen that the resulting new lattice still possesses cubic symmetry. Since the lattice condition must be maintained when new points are added, the points must be added to hightly symmetric positions of the P-lattice. These types of positions are (a) a single point at the body center of each unit cell, (b) a point at the center of each independent face of the unit cell, (c) a point at the center of one face of the unit cell, and (d) the special centering positions in the trigonal system that give a rhombohedral lattice. 

This makes cyki a symmetric tensor, which possesses at most 21 irrdepertdent coordinates as in the case of crystals of the triclirric crystal system. Apart from the symmetry reqttired by the interchangeability of indices all material properties must also reflect the crystal symmetry, i.e. the point group symmetry. This reduces further the number of irrdeperrderrt material tensor coordinates. The elastic properties of cubic crystals are ttrriquely determined by a total of three elastic rtraterial corrstartts while for isotropic bodies only two are required and srrfficierrt. 

In principle, these can be represented by six 6 by 6 matrices. By this procedure the total of 729 tensor coordinates is reduced to 216 elements of these matrices. Due to the interchange symmetries only 56 of these are independent in the least symmetric case, the triclinic crystal system. Since the representation by these matrices is exceedingly bulky and contains a lot of redundancies they are usually not written out in detail. A table listing the nonzero independent constants is given by Nelson (1979) where, in addition, the structure of all the matrices of material constants discussed in Sects. 6.4.1, 6.4.2,6.4.3, and 6.4.4 for all crystal classes may be found. 

The Avrami equationhas been extended to various crystallization models by computer simulation of the process and using a random probe to estimate the degree of overlap between adjacent crystallites. Essentially, the basic concept used was that of Evans in his use of Poisson s solution of the expansion of raindrops on the surface of a pond. Originally the model was limited to expansion of symmetrical entities, such as spheres in three dimensions, circles in two dimensions, and rods in one, for which n = 2,2, and 1, respectively. This has been verified by computer simulation of these systems. However, the method can be extended to consider other systems, more characteristic of crystallizing systems. The effect of (a) mixed nucleation, ib) volume shrinkage, (c) variable density of crystallinity without a crystallite, and (random nucleation were considered. AH these models approximated to the Avrami equation except for (c), which produced markedly fractional but different n values from 3, 2, or I. The value varied according to the time dependence chosen for the density. It was concluded that this was a powerful technique to assess viability of various models chosen to account for the observed value of the exponent, n. 

Let us demonstrate the procedure of finding the matrix of a symmetrical transformation (4.77) by the example of the rhombohedral crystal system where there is only one lattice type (R). The basic translation vectors of the initial lattice are the following ... 

In the triclinic crystal system an arbitrary matrix with integer elements defines a symmetrical transformation (any transformation seems to be symmetrical because of the low point symmetry of the lattice). 

In the monoclinic crystal system there are two lattices simple (P) and base-centered (A) each of which is defined by five parameters. Therefore the matrices of symmetrical transformations are determined by five integers. 

In the hexagonal crystal system there is only one lattice type (P), but the basic translation vectors may be oriented in two different ways relative to the basic translation vectors of the initial lattice either parallel to them or rotated through an angle of 7t/6 about the -axis. Therefore, two types of symmetrical transformation are possible in this case (with two parameters for each). 

In the tetragonal crystal system there are two types of Bravais lattice P and I). AU their symmetrical transformations may be obtained from the symmetrical transformations for orthorhombic lattices if one sets ni = ri2 and takes into account that base-centered and face-centered orthorhombic lattices become simple and body-centered tetragonal ones, respectively. 

For the symmetrical transformation (4.77) the transformation (4.80) is also symmetrical as it does not change the point symmetry of the reciprocal lattice. The symmetrical transformation is compatible with the change of the reciprocal-lattice type in the limits of the same crystal system too. The vectors bj defining the small BriUouin zone are very important in the theory of special points [86]. Let f K) be the function with a point symmetry F to be integrated over the initial BriUouin zone where the wavevec-tor Kjvaries. Usually, th( point-symmetry group F either coincides with the crystal class F of the crystal (if F contains the inversion 7) or F = F x C (otherwise) [86]. The function f K) may be expanded in Fourier series over symmetrized plane waves... 

The crystal symmetry defines the basic shape of the unit cell. There are seven shapes, which correspond to the ways in which symmetrical solid objects can pack. They can be distinguished by their symmetry properties, i.e. whieh symmetry elements they are based on. The resulting unit-cell dimensions can be classified depending on whether their edges are all different or two or three of them are equal in length, and whether their angles are equal or not, or exactly 90° or 120°. These seven crystal systems are hsted alongside other data, which we refer to later, in Table 10.1. 

Finally, let us introduce another simplification. The matrices of the Cy and tensors are symmetric with respect to their main diagonal, which means that the largest number of independent elasticity or compliance moduli is 6 h- (36 - 6)/2 = 21. This is exactly the case for the least symmetrical crystals belonging to the triclinic crystal system. In the opposite extreme case for isotropic media, the number of independent moduli is reduced to two (but never to a single one). All other crystal systems can be arranged by the number of independent elasticity constants in the following series ... 

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