Nonprimitive cells

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes, a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry... 

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,... 

Sometimes the smallest, or primitive, unit cell does not have the full symmetry of the crystal lattice. If so, a larger nonprimitive unit cell that does have the characteristic symmetry is deliberately chosen (Fig. 21.8). Only three types of nonprimitive cells are commonly used in the description of crystals body-centered, face-centered, and side-centered. They are shown in Figure 21.9. 

The fourteen Bravais lattices are described in Table 2-1 and illustrated in Fig. 2-3, where the symbols P, F, /, etc., have the following meanings. We must first distinguish between simple, or primitive, cells (symbol P or R) and nonprimitive cells (any other symbol) primitive cells have only one lattice point per cell while nonprimitive have more than one. A lattice point in the interior of a cell belongs to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight. The number of lattice points per cell is therefore given by... 

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 number of k points required to reach a given accuracy for total energy decreases when the unit cell is larger than the ones considered so far. In fact, the adjective reciprocal before space qualifies the relation of inverse proportionality between direct and reciprocal space (Eq. [3]), so that the bigger a unit cell in real space, the smaller the volume of the corresponding cell in reciprocal space. In those cases where the volume of the first Brillouin zone is small, it is sufficient to solve Schrodinger s equation only at a few k points. To illustrate this point, we consider what happens when we repeat the calculation for magnesium oxide with nonprimitive cells. In particular, we refer to unit cells with volumes of 4, 16, and 64 times the primitive cell... 

Centered Cell n A unit cell which has entities (atoms, molecules, ions) at locations in addition to the ell corners. A nonprimitive cell. 

R = rhombohedral (unit cell can be primitive or nonprimitive see notes to Table 1-11) Principal axis of ration given number n = order e.g., 2 = twofold axis of rotation... 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

It is frequently formd that it is not possible to find a primitive rrrrit cell with edges parallel to crystal axes chosen on the basis of symmetry. In such a case the crystal axes, chosen on the basis of symmetry, are proportiorral to the edges of a unit of stracture that is larger than a primitive unit cell. Such a rrrrit is called a nonprimitive unit cell, and there is more than one lattice point per nonprimitive unit cell. If the nonprimitive unit cell is chosen as small as possible consistent with the symmetry desired, it is found that the extra lattice points (those other than the comer points) lie in the center of the unit cell or at the centers of some or all of the faces of the imit cell. The coordinates of the lattice points in such a case are therefore either integers or half-integers. 

The crystal structure of l2(i). The primitive unit cell, outlined in heavy lines, contains two molecules, identified hy dots at the atomic centers (one-half molecule each at the upper left and lower right corners, and one molecule in the body center). The light lines outline an orthorhomhic nonprimitive unit cell of dimensions Of) = 0.727 nm, feo = 0.479 nm, Co = 0.979 nm. All molecules are in planes parallel to the b and c axes. (Not all molecules in the orthorhombic cell are shown.)... 

It is often convenient to choose a unit cell larger than the primitive unit cell. Nonprimitive unit cells contain extra lattice points, not at the vertices. For example, in three dimensions, nonprimitive unit cells may be of three kinds ... 

These are denoted as F, I, and C, respectively, while primitive cells are denoted as P, and rhombohedral as R. Several symmetry-related copies of the asymmetric unit may be contained in the nonprimitive unit cell, which can generate the entire crystal structure by means of translation in three dimensions. Although primitive unit cells are smaller than nonprimitive unit cells, the nonprimitive unit cell may be preferred if it possesses higher symmetry. In general, the unit cell used is the smallest one with the highest symmetry. 

Bravais lattice Classification of fourteen three-dimensional lattices based on primitive and nonprimitive unit cells. Named after Auguste Bravais, who first used them. 

Figure 17. The molecular structure and temperature dependence of the "Fe Mdssbauer parameters (IS and QS) and half-width of the absorption line (h) and some parameters of the nonprimitive unit cell of the FeNx i(BHd)2 crystal [67],...

FIGURE 4.1. (a) Symmetry in a repeating pattern. One possible unit cell, indicated by broken lines contains one object per unit cell and is described as primitive. Another possible unit cell, indicated by solid lines, contains two objects per unit cell and is said to be centered (and therefore nonprimitive ). Thus, while the simplest unit cell is not rectangular, a larger unit cell with higher symmetry, which in this case is rectangular, can readily be picked out from the arrangement of objects. 

FIGURE 21.18 Two ionic lattices in the fee system. A single (nonprimitive) cubic unit cell of each is shown. 

The lattice points in a nonprimitive unit cell can be extended through space by repeated applications of the unit-cell vectors a, b, c just like those of a primitive cell. We may regard the lattice points associated with a unit cell as being translated one by one or as a group. In either case, equivalent lattice points in adjacent unit cells are separated by one of the vectors a, b, c, wherever these points happen to be located in the cell (Fig. 2-5). 

If nonprimitive lattice cells are used, the vector from the origin to any point in the lattice will now have components which are nonintegral multiples of the unitcell vectors a, b, c. The position of any lattice point in a cell may be given in terms of its coordinates, if the vector from the origin of the unit cell to the given point has components xa, yb, zc, where x, y, and z are fractions, then the coordinates of the point are x y z. Thus, point A in Fig. 2-7, taken as the origin, has coordinates 0 0 0 while points B, C, and D, when referred to cubic axes, have coordinates 0 0 i, and i 0, respectively. Point E has coordinates i 1 and... 

The silver halides, with the exception of one crystal modification of Agl, have cubic crystal structures. Crystal structure data, relevant lattice properties and low temperature dielectric constants, are collected in Table 3. AgF, AgCl, and AgBr all have the NaCl rocksalt structure in which there are four silver halide pairs per nonprimitive unit cell with cation-anion nearest neighbor distances equal to one-half a lattice constant. 

Rhombohedral is the primitive cell. Lattice parameters given are for the nonprimitive hexagonal cell. 

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