Crystal structure primitive cell

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

The first crystal structure to be detennined that had an adjustable position parameter was that of pyrite, FeS2 In this structure the iron atoms are at the comers and the face centres, but the sulphur atoms are further away than in zincblende along a different tln-eefold synnnetry axis for each of the four iron atoms, which makes the unit cell primitive. 

The rhombohedral structure, obtained via the optimization of crystal energy [equation (1)] without the JT term inclusion or setting IVj = 0, is stable with respect to deformations of the crystal. The experimental and calculated structure parameters are listed in Table 2. As the temperature decreases down to about 400 K [5,8] the crystal symmetry changes to monoclinic phase. The major difference between the monoclinic and rhombohedral phases is the presence of JT distortions and a doubled primitive cell. The local values of JT distortions on different Mn3+ ions could be obtained via the projection of oxygen ions coordinates onto the normal local Eg modes of each [Mn06] octahedra (the numbering of Mn3+ is carried out according to Fig. 1). 

There are very many unit cell structures if we consider all the atoms or ions in the crystal. However, if we focus on just one atom or ion, we can reduce the number to just 14 primitive cells. Three of these, the simple cubic, face-centered cubic (fee), and body-centered cubic (bcc) unit cells, are shown in Figure 9-2. The lattice points, represented by small spheres in the drawings, correspond to the centers of the atoms, ions, or molecules occupying the lattice. 

Cesium chloride crystallizes with a structure derived from the simple cubic primitive cell. Ch ions occupy the 8 comer sites with Cs+ in the center of the cell note that this is not a body-centered cubic unit cell since the ion at the center is not the same as those at the comers. Thus there is one CsCl unit per unit cell and the coordination numbers of Cs+ and Ch are both 8. Crystals of CsBr and Csl adopt the CsCl structure, but all other alkali halides crystallize in the NaCl structure. 

The crystal structure of many pure metals adopts the hep of identical spheres, which has a primitive hexagonal lattice in space group D h - Pbi/mmc. There are two atoms in the hexagonal unit cell with coordinates (0,0,0) and(f,i,i). 

Cl-. The lattice type is reduced from face-centered cubic to primitive cubic, and the space group of CsCl is 0 - Pm3m. Figure 10.3.3(a) shows a unit cell in the crystal structure of CsCl. 

Fig. 3.1. (a) Primitive cell (heavy lines) of the wurtzite-structure lattice placed within a hexagonal prism, a and c are the lattice constants, (b) Schematic drawing of surfaces cut from a hexagonal single crystal with different crystallographic orientations (surface planes)... 

The rocksalt crystal structure belongs to the cubic system with space group 0 (Pm3m). It consists of two face-centered cubic (fee) sublattices, which are occupied by one atom species each. The two sublattices are shifted along one half of the diagonal of the primitive unit cell against each other. The rocksalt lattice is sixfold coordinated. 

To derive factor group (space group) selection rules, it is necessary to utilize X-ray data for a molecule from a literature source or from Wyckoffs (54) Crystal Structures. The factor group and site symmetries of the ion, molecule, or atoms must be available, as well as the number of molecules per unit cell reduced to a primitive unit cell. 

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.)... 

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. 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

In all properties studied with pseudopotenlial theory, the first step is the evaluation of the structure factors. For simplicity, let us consider a metallic crystal with a single ion per primitive cell -either a body-centered or face-centered cubic structure. We must specify the ion positions in the presence of a lattice vibration, as we did in Section 9-D for covalent solids. There, however, we were able to work with the linear force equations and could give displacements in complex form. Here the energy must be computed, and that requires terms quadratic in the displacements. It is easier to keep everything straight if we specify displacements as real. Fora lattice vibration of wave number k, we write the displacement of the ion with equilibrium position r, as... 

Figure 9.16 Crystal structure of a-AID3 (a) The unit cell ofa-AIDs, (b) illustration ofthe connectivity ofthe octahedra. Each octahedron shares one corner with one other octahedron, building a distorted primitive Al sublattice, Ref [48].

Figure 2.t. The crystal structure of CeO> (a) unii celt as cep array of cerium atoms The cep layers are parallel to the [Ml] planes of the fe e. unit cell, (b) and (c) the same stniclure redrawn as a primitive cubic array of oxygens. 

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