Bravais body-centered

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. 

FIGURE 5.33 The 14 Bravais lattices. P denotes primitive I, body-centered F, face-centered C, with a lattice point on two opposite faces and R, rhombohedral (a rhomb is an oblique equilateral parallelogram). 

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]. 

First symbol refers to the Bravais lattice P = primitive lattice C = centered lattice F = face-centered lattice I = body-centered lattice... 

Designate space groups by a combination of unit cell type and point group symbol, modified to include screw axes and glide planes (Hermann-Mauguin) 230 space groups are possible. Use italic type for conventional types of unit cells (or Bravais lattices) P, primitive I, body-centered A, A-face-centered B, B-face-centered C, C-face-centered P, all faces centered and R, rhombohedral. 

The 14 Bravais lattices are enumerated in Table 9-4 as the following types primitive (P, R), side-centered (C), face-centered (F), and body-centered (7). The numbering of the Bravais lattices in Table 9-4 corresponds to that in Figure 9-20. The lattice parameters are also enumerated in the table. In addition, the distribution of lattice types among the crystal systems is shown. 

The actual infinite lattices are obtained by parallel translations of the Bravais lattices as unit cells. Some Bravais cells are also primitive cells, others are not. For example, the body-centered cube is a unit cell but not a primitive cell. The primitive cell in this case is an oblique parallelepiped constructed by using as edges the three directed... 

Within a given crystal system, there are in some cases several different types of crystal lattice, depending upon the type of minimum-size unit cell that corresponds to a choice of axes appropriate to the given crystal system. This unit cell may be primitive P or in certain cases body-centered I, face-centered F, or end-centered A, B, or Q depending on which pair of end faces of the unit cell is centered. The lattices are designated as primitive, body-centered, face-centered, or end-centered depending on whether the smallest possible unit cell that corresponds to the appropriate type of axes is primitive, body-centered, face-centered, or end-centered. There are in all 14 types of lattice, known as Bravais lattices. In the cubic system there are three primitive, body-centered, and face-centered these are shown in Fig. 2. 

Both are body-centered Bravais lattices and for both the site symmetry of the origin is identical with the short space group symbol. The body-center position is of the lowest multiplicity (two-fold) and highest symmetry, and thus is considered as the origin in the lA/mmm space group. However, in the tetragonal lattice, a = b c. Hence, the body center position is not an inversion center. It possesses four-fold rotational symmetry (the axis is parallel to c) with a perpendicular mirror plane and two additional perpendicular mirror planes that contain the rotation axis. 

The body-centered position in the cubic Bravais lattice (a = fa = c) is an inversion center and for this reason is taken as the origin in space group Im 3 m. The position possesses three-fold rotoinversion symmetry with three perpendicular mirror planes. 

Figure 3.11. The cubic cesium-chloride unit cell is not a body-centered cubic Bravais lattice since there are two nonequivalent lattice points.

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.

FIGURE 3.15 The types of unit cells that form the basis for the allowable lattices of all crystals (known as the Bravais lattices). There are 15 unique lattices (see International Tables, Volume I, for further descriptions). All primitive (/ ) cells may be considered to contain a single lattice point (one-eighth of a point contributed by each of those at the corners of the cell), face-centered (C) and body-centered (/) cells contain two full points, and face-centered (F) cells contain four complete lattice points. 

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. 

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 ... 

What is the Bravais lattice of CsCl Figure 2-18(a) shows that the unit cell contains two atoms, ions really, since this compound is completely ionized even in the solid state a caesium ion at 0 0 0 and a chlorine ion at The Bravais lattice is obviously not face-centered, but we note that the body-centering translation i i i connects two atoms. However, these are unlike atoms and the lattice is therefore not body-centered. It is, by elimination, simple cubic. If one wishes, one may think of both ions, the caesium at 0 0 0 and the chlorine at as being... 

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. 

The reader may have noticed in the previous examples that some of the information given was not used in the calculations. In (a), for example, the cell was said to contain only one atom, but the shape of the cell was not specified in (b) and (c), the cells were described as orthorhombic and in (d) as cubic, but this information did not enter into the structure-factor calculations. This illustrates the important point that the structure factor is independent of the shape and size of the unit cell. For example, any body-centered cell will have missing reflections for those planes which have ft + k + 1) equal to an odd number, whether the cell is cubic, tetragonal, or orthorhombic. The rules we have derived in the above examples are therefore of wider applicability than would at first appear and demonstrate the close connection between the Bravais lattice of a substance and its diffraction pattern. They are summarized in Table 4-1. These rules are subject to... 

Powder patterns of cubic substances can usually be distinguished at a glance from those of noncubic substances, since the latter patterns normally contain many more lines. In addition, the Bravais lattice can usually be identified by inspection there is an almost regular sequence of lines in simple cubic and body-centered cubic patterns, but the former contains almost twice as many lines, while a face-centered cubic pattern is characterized by a pair of lines, followed by a single line, followed by a pair, another single line, etc. 

Before considering the ordering transformation in AuCu, which is rather complex, we might examine the behavior of j5-brass. This alloy is stable at room temperature over a composition range of about 46 to almost 50 atomic percent zinc, and so may be represented fairly closely by the formula CuZn. At high temperatures its structure is, statistically, body-centered cubic, with the copper and zinc atoms distributed at random. Below a critical temperature of about 460°C, ordering occurs the cell corners are then occupied only by copper atoms and the cell centers only by zinc atoms, as indicated in Fig. 13-6. The ordered alloy therefore has the CsCl structure and its Bravais lattice is simple cubic. Other alloys which have the same ordered structure are CuBe, CuPd, and FeCo. 

Only the seven primitive unit cell are described above. If the lattice points on the faces of the unit cell (named as A-, B- or C-centered unit cell) or in the center of the unit cell (body-centered, viz. f-centered unit cell) are taken into account, a total of 14 so called Bravais lattices are obtained. The treatment of the Bravais lattices is out of the scope of this Chapter and the reader is encouraged to check Refs. [10]— [12] for further details. 

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. 

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.)...

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