Bravais primitive

Since the Bravais cell contains lO atoms, it has 3x10 — 3 = 27 normal vibrations, excluding three translational motions of the cell as a whole.l These 27 vibrations can be classified into various symmetry species of the factor group Djd, using a procedure similar to that described in Sec. 1-7 for internal vibrations. First, we calculate the characters of representations corresponding to the entire freedom possessed by the Bravais primitive cell [a r( )] translational motions of the whole cell [ (T)], translatory lattice modes [ f(T )],... 

Ever molecule (or ion) in a Bravais primitive cell can be superimposed on that of the neighboring cell by simple translation. In the case of calcite, the Bravais primitive cell is the same as the crystallographic unit cell. Hotvever, this is not always the case. 

Figure A.2 The five surface Bravais lattices square, primitive rectangular, centered rectangular, hexagonal, and oblique.

The dye molecules are positioned at sites along the linear channels. The length of a site is equal to a number ns times the length of c, so that one dye molecule fits into one site. Thus ns is the number of unit cells that form a site we name the ns-site. The parameter ns depends on the size of the dye molecules and on the length of the primitive unit cell. As an example, a dye with a length of 1.5 nm in zeolite L requires two primitive unit cells, therefore ns = 2 and the sites are called 2-site. The sites form a new (pseudo) Bravais lattice with the primitive vectors a, b, and ns c in favorable cases. 

We consider the case where the sites form a Bravais lattice, which means that the position Rt of an / s-site i can be expressed by the primitive vectors a, b, and ns-c of the hexagonal lattice and the integers na[Pg.21]

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.

If we consider a primitive Bravais lattice with cell edges defined by vectors a, 82, and aj, the corresponding reciprocal lattice is defined by reciprocal vectors bj, b2, and bj so that... 

The complete charge array is built by the juxtaposition of this cell in three dimensions so that to obtain a block of 3 x 3 x 3 cells, the cluster being located in the central cell. In that case the cluster is well centered in an array of475 ions. Practically and for computational purposes, the basic symmetry elements of the space group Pmmm (3 mirror planes perpendicular to 3 rotation axes of order 2 as well as the translations of the primitive orthorhombic Bravais lattice) are applied to a group of ions which corresponds to 1/8 of the unit cell. The procedure ensures that the crystalline symmetry is preserved. 

An example is the (110) plane of III-V semiconductors, such as GaAs(llO). The only nontrivial symmetry operation is a mirror reflection through a line connecting two Ga (or As) nuclei in the COOl] direction, which we labeled as the X axis. The Bravais lattice is orthorhombic primitive (op). In terms of real Fourier components, the possible corrugation functions are... 

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. 

We can now complete our answer to the question, What information is conveyed when we read that the crystal structure of a substance is monodime P2JC7" The structure belongs to the monoclinic crystal system and has a primitive Bravais lattice. It also possesses a two-fold screw axis and a glide plane perpendicular to it. The existence of these two elements of symmetry requires that there also be a center of inversion. The latter is not specifically included in the space group notation as it would be redundant. 

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). 

Figure 16.12. Brillouin zones, with symmetry points marked, of (a) the primitive cubic Bravais lattice and (b) the cubic close-packed or fee Bravais lattice.

One of the most important points, adequately emphasized by Fateley, McDevitt and Bently u) in their excellent paper, is that the spectroscopic unit cell must be chosen, and not the crystallographic unit cell. The spectroscopic unit cell is identical with the Bravais cell the Bravais cell is identical with the crystallographic unit cell in the P (primitive) cells, but the other structures, designated by B, C, I, F.in the crystallographic tables 14) contain 2, 2,3,4.Bravais unit cells. Fateley, McDevitt and Bentley 10 de-... 

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]. 

Among the 14 cells that generate the Bravais lattices (see Figure 1.4), only the P-type cells are considered primitive unit cells. It is possible to generate the other Bravais lattices with primitive unit cells. However, in practice, only unit cells that possess the maximum symmetry are chosen (see Figure 1.4 and Table 1.2) [1-6]. 

Thus, there are seven crystal types for which fourteen different types of lattices (7 primitive, 3 body centred, 2 face centred and 2 end centred) are possible. These fourteen different types of lattices are known as Bravais lattices. 

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

As previously mentioned, the primitive unit cell is the smallest unit of a crystal that reproduces itself by translations. Figure 1-37 illustrates the difference between a primitive and a centered or nonprimitive cell. The primitive cell can be defined by the lines a and c. Alternatively, we could have defined it by the lines a and c. Choosing the cell defined by the lines a" and c" gives us a nonprimitive cell or centered cell, which has twice the volume and two repeat units. Table 1-11 illustrates the symbolism used for the various types of lattices and records the number of repeat units in the cell for a primitive and a nonprimitive lattice. The spectroscopist is concerned with the primitive (Bravais) unit cell in dealing with lattice vibrations. For factor group selection rules, it is necessary to convert the number of molecules per crystallographic unit cell Z to Z, discussed later, which is the number of molecules per primitive cell. For example,... 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The five planar Bravais lattices (a) primitive oblique (parallelogram) (b) primitive rectangular, (c) centered rectangular, (d) primitive square, (e) primitive hexagonal. 

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. 

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