Symbols, centered unit cells

For example, considering the crystal structure of copper, which has cubic face-centered lattice Figure 6.2) and a total of 4 atoms in the unit cell, its Pearson s symbol is cF4. On the other hand, if the material has Pearson s symbol oI32, this means that its crystal structure is orthorhombic, and one body-centered unit cell contains a total of 32 atoms. 

More systematic (but not always unambiguous) is the designation by Pearson symbols their use is recommended by IUPAC (International Union of Pure and Applied Chemistry). A Pearson symbol consists of a lower case letter for the crystal system (cf. the abbreviations in Table 3.1, p. 24), an upper case letter for the kind of centering of the lattice (cf. Fig. 2.6, p. 8) and the number of atoms in the unit cell. Example sulfur-< F128 is orthorhombic, face centered and has 128 atoms per unit cell (a-sulfur). 

P = 98 92(6)°. The center of inversion is indicated by a dot in the center of the unit cell, and the two two-fold screw axes are perpendicular to the plane of the paper and are marked with the symbol i Two glide planes perpendicular to the screw axes in the. xy plane (parallel with the plane of the paper) are not indicated but are found at distances of one-fourth and three-fourths unit cell depth. Note that a, b, and c do not correspond exactly to x, y, and a because one of the three angles of a monoclinic structure is unequal to 90°. The fluorine atoms have been omitted for clarity. [Modified from Hadj-Baghcri, N. Strickland, D. S., Wilson, S. R. Shapley, J. R. J. Organomet. Chem. 1991,410, 231-239 Courtesy of S. R. Wilson and C. L. Stern.]... 

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. 

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,... 

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. 

Figure 9-23. The diamond structure after Shubnikov and Koptsik [33], (a) A unit cell the edges of the cube are the a, b, and c axes (b) Two face-centered cubic sublattices displaced along the body diagonal of the cube (c) Projection of some symmetry elements of the Fd im space group onto a horizontal plane. The vertical screw axes 4 and 43 are marked by appropriate symbols. Used with permission.

One has to take into account, however, that the unit cell which is relevant for spectroscopy is the primitive (or Wigner-Seitz) unit cell. It is a parallelepiped from which the entire lattice may be generated by applying multiples of elementary translations. Face- and body-centered cells are multiple unit cells. The content of such a cell has to be divided by a factor m to obtain the content of a primitive unit cell. This factor m is implicitly given by the international symbol for a space group P and R denote primitive cells (m = 1), face-centered cells are denoted A, B, C (m = 2), and F m = 4), and body-centered cells are represented by I m = 2). Examples are described by Turrell (1972). 

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.

Since every unit cell in the crystal lattice is identical to all others, it is said that the lattice can be primitive or centered. We already mentioned (Eq. 1.1) that a crystallographic lattice is based on three non-coplanar translations (vectors), thus the presence of lattice centering introduces additional translations that are different from the three basis translations. Properties of various lattices are summarized in Table 1.13 along with the international symbols adopted to differentiate between different lattice types. In a base-centered lattice, there are three different possibilities to select a pair of opposite faces, which is also reflected in Table 1.13. 

Centering of the lattice Lattice points per unit cell International symbol Lattice translation(s) due to centering... 

FIGURE 6.8 The lattice of asymmetric units, represented by the symbol for a question ( ), is the same in both (a) and (b). The corresponding reciprocal lattice is accordingly the same for both (c) and (d). If the unit cell is chosen as in (a) to be a primitive unit cell, then every lattice point in (c) is occupied. If, however, the real unit cell is chosen to be centered as in (h), then half of the reciprocal lattice points, indexed according to this centered cell, are systematically absent, and a checkerboard pattern of diffraction intensities is observed. 

The first letter denotes the crystal system triclinic (a), monoclinic (m), orthorhombic (o), tetragonal (t), hexagonal (h) and cubic (c). Trigonal (rhombohedral) system is denoted by combination hR. The second letter of Pearson s symbol denotes lattice type primitive (P), edge-(base-) centered (C), body-centered (I) or face-centered (F). The following number denotes number of atoms in the crystal unit cell. 

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