Hexagonal system space-lattice

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. 

Although the bond-orientational metrics defined above have proven useful for identifying numerous space-filling crystalline morphologies43 like face-centered cubic, body-centered cubic, simple cubic, and hexagonally close-packed lattices, they are inadequate for detecting order in systems that organize... 

There is only one space lattice in the rhombohedral crystal system. This crystal is sometimes called hexagonal R or trigonal R, so don t confnse it with the other two similarly-named crystal systems. The rhombohedral crystal has nniform lattice parameters in all directions and has equivalent interaxial angles, bnt the angles are nonorthogonal and are less than 120°. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

William B. Pearson (1921-2005) developed a shorthand system for denoting alloy and intermetallic structure types (Pearson, 1967). It is now widely used for ionic and covalent solids, as well. The Pearson symbol consists of a small letter that denotes the crystal system, followed by a capital letter to identify the space lattice. To these a number is added that is equal to the number of atoms in the unit cell. Thus, the Pearson symbol for wurtzite (hexagonal, space group PS mc), which has four atoms in the unit ceU, is hPA. Similarly, the symbol for sodium chloride (cubic, space group Fm3m), with eight atoms in the unit cell, is cF8. 

Fig. 2.15. Primitive rhombohedral cell (space group / 32) in hexagonal coordinate system. The hexagonal cell has lattice points at 0,0,0 2/3,1/3,1/3 l/3,2/3,2/3. The threefold axes are indicated by triangles. The twofold axes, whose directions are indicated by broad arrows, occur in pairs separated by Cjj/2 the numbers give the fractional Zh coordinates...

Bravais proved that if the contents of a unit cell have symmetry, the number of distinct types of space lattices becomes fourteen. These are the only lattices that can fill all space and are commonly termed the 14 Bravais lattices. Since there are seven crystal systems, it might be thought that by combining the seven crystal systems with the idea of a primitive lattice a total of seven distinct Bravais lattices would be obtained. However, it turns out that the trigonal and hexagonal lattices so constructed are equivalent, and therefore only six lattices can be formed in this way. These lattices, which are given the label P, define the primitive unit cells (or P-cells) in each case. 

There are seven possible space lattices which entirely describe both inorganic and organic crystalline materials. These are called the seven crystal systems (i.e., cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, and triclinic). 

Bravais in 1849 showed that there are only 14 ways that identical points can be arranged in space subject to the condition that each point has the same number of neighbors at the same distances and in the same directions.Moritz Ludwig Frankenheim, in an extension of this study, showed that this number, 14, could also be used to describe the total number of distinct three-dimensional crystal lattices.These are referred to as the 14 Bravais lattices (Figure 4.9), and they represent combinations of the seven crystal systems and the four lattice centering types (P, C, F, I). Rhombohedral and hexagonal lattices are primitive, but the letter R is used for the former. 

---