Nonprimitive unit cells

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],...

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 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). 

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. 

The supercell approach consists of a periodic replica of the defect, which is enclosed in a large nonprimitive unit cell. A pictorial view (in 2-D) of the supercell approach is given in Figure 47, where, by starting from the perfect... 

Taking into account such nonprimitive unit cells, the structure of any crystalline solid can be represented in terms of one of 14 possible basic types called Bravais lattices (Figure 6.14). 

FIGURE 2.18 Left the paradigm of generating a periodic structure by the combination between Base and Reason is noted the possibility of identify inside the lattice of nonprimitive unit cells (NP) and those primitive (P) right a structural realization formed by the cubic lattice with the faces centered with the fullerene base (C60) after Heyes (1999). 

A unit cell of a crystal consists of a volume which contains the array of atoms that are repeated. The smallest possible unit cell that can form the structure is a primitive unit cell although is it often convenient to describe to solid using nonprimitive unit cells that have higher symmetries. A unit cell is described by three lattice vectors that define the directions the cell can be propagated to form the crystal lattice or framework of the structure. The basis of the crystal describes the positions of the atoms within the unit cell. The basis plus the lattice defines the structure. TTae set of all possible translations along the lattice vectors, forms the translation group. 

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