Bravais base-centered

The Seven Systems of Crystals are shown in Figure 2.2. The relationship between the trigonal and rhombohedral systems is shown in Figure B.la. The possibilities of body-centered and base-centered cells give the 14 Bravais Lattices, also shown in Figure 2.2. A face-centered cubic (fee) cell can be represented as a 60° rhombohedron, as shown in Figure B.lb. The fee cell is used because it shows the high symmetry of the cube. 

Within a given crystal system, a supplementary subdivision is necessary to be made, in order to produce the 14 Bravais lattices. In this regard, it is necessary to make a distinction between the following types of Bravais lattices, that is, primitive (P) or simple (S), base-centered (BC), face-centered (FC), and body-centered (BoC) lattices [1-3]. 

Figure 2.14. Models of the 14 Bravais lattices. The various types of Bravais centering are given the symbols P (primitive/simple), F (face-centered), I (body-centered), and C (base-centered). The primitive rhombohedral Bravais lattice is often given its own symbol, R, and corresponds to a primitive unit cell possessing trigonal symmetry.

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

At first glance, the list of Bravais lattices in Table 2-1 appears incomplete. Why not, for example, a base-centered tetragonal lattice The full lines in Fig. 2-4 delineate such a cell, centered on the C face, but we see that the same array of lattice points can be referred to the simple tetragonal cell shown by dashed lines, so that the base-centered arrangement of points is not a new lattice. However, the base-centered cell is a perfectly good unit cell and, if we wish, we may choose to use it rather than the simple cell. Choice of one or the other has certain consequences, which are described later (Problem 4-3). 

Body-, face-, or base-centering translations, if present, must begin and end on atoms of the same kind. For example, if the structure is based on a body-centered Bravais lattice, then it must be possible to go from an A atom, say, to another A atom by the translation ... 

The number of atoms per unit cell in any crystal is partially dependent on its Bravais lattice. For example, the number of atoms per unit cell in a crystal based on a body-centered lattice must be a multiple of 2, since there must be, for any atom in the cell, a corresponding atom of the same kind at a translation of from the first. The number of atoms per cell in a base-centered lattice must also be a multiple of 2, as a result of the base-centering translations. Similarly, the number of atoms per cell in a face-centered lattice must be a multiple of 4. 

Figure 2.3 The Bravais lattices (P, R = primitive cells, F = face centered, I = body centered, and C = base centered). (From B.D. Cullity, Elements of X-ray Diffraction, 1978 by Addison-Wesley Publishing Company. Reprinted with permission of the publisher.)...

Standard ASTM E157-82a has the Bravais lattices designations as following C - primitive cubic B - body-centered cubic F - face-centered cubic T - primitive tetragonal U - body-centered tetragonal R - rhombohedral H - hexagonal O - primitive orthorhombic P - body-centered orthorhombic Q - base-centered orthorhombic S - face-centered orthorhombic M - primitive monoclinic N - centered monoclinic A - triclinic. 

Standard ASTM E157-82ahasthe Bravais lattices designations as following C — primitive cubic B — body-centered cubic F — face-centered cubic T — primitive tetragonal U—body-centered tetragonal R—rhombohedral H — hexagonal O—primitive orthorhombic P — body-centered orthorhombic Q — base-centered orthorhombic S — face-centered orthorhombic M — primitive monoclinic N — centered monoclinic A — triclinic. 

The lattice types are labeled by P (simple or primitive), F (face-centered), I (body-centered) and A B,C) (base-centered). Cartesian coordinates of basic translation vectors written in units of Bravais lattice parameters are given in the third column of Table 2.1. It is seen that the lattice parameters (column 4 in Table 2.1) are defined only by syngony, i. e. are the same for all types of Bravais lattices with the point symmetry F and all the crystal classes F of a given syngony. 

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. 

Monoclinic System It has two Bravais lattices, i.e., primitive (P) and base-centered C, and three point groups 2, m, and 2/m. In detailed study of symmetry, the array of atoms that constitutes the structure of the crystal, a macroscopic mirror plane m, might be a glide plane c, while twofold rotation axis might be a screw axis as 2i. Considering these aspects of possible symmetry, the complete set is given as follows ... 

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. 

It is difficult to get more primitive than the tetragonal or triclinic lattice, and the base-centered monoclinic lattice (Figure 4.12g) rounds out the 14 Bravais lattices. 

It was mentioned that the base-centered hexagon is not considered as one of the Bravais lattices because it adds no new symmetry. There is an important hexagonal structure similar in some respects to the fee lattice called the hexagonal close-packed (hep) lattice. Like the base-centered hexagon, the hep lattice is not considered to be one of the 14 Bravais lattices either. The structure consists of a sandwich of two vertically aligned hexagonal planar arrays with three atoms between them that sit on three of the six vertices as shown in Figure 4.13. 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

Seven different point lattices can be obtained simply by putting points at the corners of the unit cells of the seven crystal systems. However, there are other arrangements of points which fulfill the requirements of a point lattice, namely, that each point have identical surroundings. The French crystallographer Bravais worked on this problem and in 1848 demonstrated that there are fourteen possible point lattices and no more this important result is commemorated by our use of the terms Bravais lattice and point lattice as synonymous. For example, if a point is placed at the center of each cell of a cubic point lattice, the new array of points also forms a point lattice. Similarly, another point lattice can be based on a cubic unit cell having lattice points at each corner and in the center of each face. 

The simplest crystals one can imagine are those formed by placing atoms of the same kind on the points of a Bravais lattice. Not all such crystals exist but, fortunately for metallurgists, many metals crystallize in this simple fashion, and Fig. 2-14 shows two common structures based on the body-centered cubic (BCC) and face-centered cubic (FCC) lattices. The former has two atoms per unit cell and the latter four, as we can find by rewriting Eq. (2-1) in terms of the number of atoms, rather than lattice points, per cell and applying it to the unit cells shown. 

Note that diamond and a metal like copper have quite dissimilar structures, although both are based on a face-centered cubic Bravais lattice. To distinguish between these two, the terms diamond cubic and face-centered cubic are usually used. The industrially important semiconductors, silicon and germanium, have the diamond cubic structure. 

The reverse of these propositions is not true. It would be a mistake to assume, for example, that if the number of atoms per cell is a multiple of 4, then the lattice is necessarily face-centered. The unit cell of the intermediate phase AuBe, for example (Fig. 2-20), contains 8 atoms and yet it is based on a simple cubic Bravais lattice. The atoms are located as follows ... 

The other crystal lattices can be generated by adding to some of the above-defined cells extra high-symmetry points by the so-called centering method. TableB.2 shows the new systems added to the simple crystal lattices (noted s, or P, for primitive) and the numbers of lattice points in each conventional unit cell. The body-centred lattices are noted be or I (for German Innenzentrierte), the face-centred, fc or F, and the side-centred or base-centred lattices are noted C (an extra atom at the Centre of the base). These 14 lattice systems are known as the Bravais lattices (noted here BLs). A representation of their unit cells can be found in the textbook by Kittel [7]. 

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