Rhombohedral crystal systems

Crystal system Rhombohedral Orthorhombic Tetragonal Cubic Tetragonal Tetragonal Amorphous... 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

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°. 

Figure A.1.1. Miller axes applied to crystal systems other than hexagonal (including rhombohedral = trigonal) system.

Only fourteen space lattices, called Bravais lattices, are possible for the seven crystal systems (Fig. 328). Designations are P (primitive), / (body-centered), F (face-centered),34 C pace-centered in one set of laces), and R (rhombohedral) Thus our monoclinic structure P2Jc belongs to the monoclinic crystal system and has a primitive Bravais lattice. 

The trigonal system can be considered as a subdivision of the hexagonal unit. On this basis there would be only six different crystal systems, but conventionally, the trigonal system (also called the rhombohedral system) is retained separately. Figure 2.1 shows two rhombohedral cells within a hexagonal cell. 

The six crystal systems are described in Table 1.1. The hexagonal system is sometimes divided into two, the rhombohedral and hexagonal systems, making a total of seven. 

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. 

Rhombohedral unit cell Figure 1.26) is a special unit cell that is allowed only in a trigonal crystal system. It contains two additional lattice points located at 1/3, 2/3, 2/3 and 2/3, 1/3, 1/3 as shown by the ends of the two vectors inside the unit cell, which results in a total of three lattice points per unit cell. 

The first position in a structure type symbol is occupied by a small letter designating the crystal system of the material c for cubic, t for tetragonal, h for hexagonal, trigonal and rhombohedral, o for orthorhombic, m for monoclinic, and a for triclinic (anorthic). 

Other crystal systems -> Tetragonal, orthorhombic, rhombohedral, monoclinic, and triclinic... 

Crystal System /Habit Hexagonal/rhombohedral crystal system see Figure 2.11. Crystals are rare. Commonly massive, coarse to fine granular, or compact, earthy or chalky, or lamellar or coarsely fibrous. 

Crystal System/Habit Hexagonal/rhombohedral. Crystals are usually broad tabular. Commonly, foliated massive fibrous with fibers separable and elastic. 

---