End-centered lattice

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

For each of these shapes, there may be choices as to where the lattice points may be placed in the unit cell while preserving the symmetry. If the lattice points are located only at the vertices of the unit cell, we have a prunitiye lattice. For some lattice systems, there may also be lattice points located in the centers of two opposite faces (end-centered lattice), in the center of each face (face-centered lattice), or in the middle of the unit cell (body-centered lattice). There are 14 distinct possibilities, and these are known as the Bravais lattices. The conventional unit cells for these lattices are shown in Fig. 13.5. 

Table 21.2 also lists another standard nomenclature for referring to the types of Bravais lattice. Generally, the letter P is used to indicate a primitive lattice, I is used for body-centered lattices, F for face-centered lattices, and C for end-centered lattices. 

Some of the crystal systems have more than one kind of lattice. Apn/nrtrv lattice or simple lattice (denoted by P) is one in which lattice points occur only at the corners of the unit cell. A unit cell of a primitive lattice contains one basis (one-eighth of the basis at each corner). A body-centered lattice (denoted by I, for German imenzentriert) is one in which there is a lattice point at the center of the unit cell as well as at the corners. A face-centered lattice (denoted by F) is one in which there is a lattice point at the center of each face of the unit cell as well as at the corners. The sodium chloride lattice is a face-centered cubic lattice. A base-centered lattice or end-centered lattice (denoted by C) is one in which there is a lattice point at the center of one pair of opposite faces as well as at the corners. Table 28.1 and Figure 28.2 show the 14 possible lattices, which are called Bravais lattices. 

---

See also in sourсe #XX -- [ , , ]

---