Bravais face-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. 

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. 

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... 

The 14 possible Bravais lattices for crystals of a monoatomic molecule. The full designation shown here bears a numerical prefix—for example, 23Ffor face-centered cubic. When space groups are generated from the Bravais lattices, then this numerical prefix is dropped (e.g., the 23P cubic Bravais lattice reappears simply as P), because the other numbers or letters that follow the P will identify the space group uniquely. 

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. 

In positronium forming materials without optical phonons much longer Ps slowing-down times should be found. This may be tested by AMOC measurements in solid rare gases, crystallizing in the face centered cubic (fee) structure which, being a Bravais lattice, does not have optical phonon branches. 

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. 

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.

We can imagine several models for the onset of simple-cubic order below Tc. The molecules at the corner and three face-centered sites must be made inequivalent. This could be accomplished, for example, by displacements away from fee Bravais lattice sites, qua-drupolar distortions into a football" shape, or development of orientational order. The first two mechanisms correspond, respectively, to finite Ytm or V2m spherical-harmonic components, resulting in terms in the structure factor proportional to Ji(gR) or j2iqR). However, maxima in j and 72 occur at small arguments (corresponding to O 0.59 and 0.93 A in the units of Fig. 1 if / 3.52 A), so these two mechanisms would predict detectable 100 and 200 intensities which are not ob-... 

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. 

Any of the fourteen Bravais lattices may be referred to a primitive unit cell. For example, the face-centered cubic lattice shown in Fig. 2-7 may be referred to the primitive cell indicated by dashed lines. The latter cell is rhombohedral, its axial angle a is 60°, and each of its axes is l/ /2 times the length of the axes of the cubic cell. Each cubic cell has four lattice points associated with it, each rhombohedral cell has one, and the former has, correspondingly, four times the volume of the latter. Nevertheless, it is usually more convenient to use the cubic cell rather than the rhombohedral one because the former immediately suggests the cubic symmetry which the lattice actually possesses. Similarly, the other centered nonprimitive cells listed in Table 2-1 are preferred to the primitive cells possible in their respective lattices. 

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. 

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... 

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. 

Some rather complex crystals can be built on a cubic lattice. For example, the ferrites, which are magnetic and are used as memory cores in digital computers, have the formula MO Fe203, where M is a divalent metal ion like Mn, Ni, Fe, Co, etc. Their structure is related to that of the mineral spinel. The Bravais lattice of the ferrites is face-centered cubic, and the unit cell contains 8 molecules or a... 

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 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 ... 

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. 

At this point, we know that the unit cell of CdTe is cubic and that it contains 4 molecules of CdTe, i.e., 4 atoms of cadmium and 4 atoms of tellurium. We must now consider possible arrangements of these atoms in the unit cell. First we examine the indices listed in Table 10-5 for evidence of the Bravais lattice. Since the indices of the observed lines are all unmixed, the Bravais lattice must be face-centered. (Not all possible sets of unmixed indices are present, however 200,420,... 

The ordered alloy thus produces diffraction lines for all values of hkl, and its diffraction pattern therefore resembles that of a simple cubic substance. In other words, there has been a change of Bravais lattice on ordering the Bravais lattice of the disordered alloy is face-centered cubic and that of the ordered alloy simple cubic. 

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. 

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