Lattice, primitive

Let us imagine an infinite crystal, e.g., a system that exhibits the translational symmetry of charge distribution (nuclei and electrons). The translational q mmetry will be fully determined by three (linearly independent) basis vectors ai, 02 and 03 having the property that ai beginning at any atom, extends to identically located atom of the crystal. The lengths of the basis vectors ai, 02 and 03 are called the lattice constants along the three periodicity axes.  

There are many such basis sets possible. Any basis vectors choice is acceptable from the point of view of mathematics. For economic reasons we choose one of the possible vector sets that give the least volume parallelepiped with sides a, 02 and 03. This parallelepiped (arbitrarily shifted in space, Fig. 9.1) represents our choice of the unit cell, which together with its content (motif) is to be translationally repeated.  

An example of a jigsaw puzzle shows that other choices are possible as well. A particular choice may result from its convenience. This freedom will be used on p. 438. 

Electronic Motion in the Mean Field Periodic Systems 

The points indicated by all the translation vectors ( lattice vectors ) are called the crystallographic lattice or the primitive lattice or simpty the lattice. 

---

Fig. 1. Two-dimensional honeycomb lattice of graphene primitive lattice vectors R and R2 are depicted outlining primitive unit cell.

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

Let us give a brief summary of the LSGF method. We will consider a system of N atoms somehow distributed on the underlying primitive lattice. We start with the notion that if we choose an unperturbed reference system which has an ideal periodicity by placing eciuivalent effective scatterers on the same underlying lattice, its Hamiltonian may be calculated in the reciprocal space. Corresponding unperturbed Green s... 

What this means is that the primitive lattice is composed of points at the corners of the lattice, whereas the inversion lattice hcis an additional point at the center of the lattice, i.e.- "body-centered". Face-centered has points in the middle of each face of the lattice in addition to those at the corners of the lattice. 

The linking pattern of two zeolites is shown in Fig. 16.24. They have the /I-cage as one of their building blocks, that is, a truncated octahedron, a polyhedron with 24 vertices and 14 faces. In the synthetic zeolite A (Linde A) the /3-cages form a cubic primitive lattice, and are joined by cubes. j3-Cages distributed in the same manner as the atoms in diamond and linked by hexagonal prisms make up the structure of faujasite (zeolite X). 

Figure 9.40 The primitive lattice of sites and bonds and bonds in the problem of sites the sites of different types are as shown as black and white.

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.

As shown in Table E.l, there is only one centered lattice, oc. It is easy to show that for monoclinic, orthorhombic, and hexagonal cases, the centered lattice reduces to primitive lattices with halved unit cells. 

Another idea to be clear about is that a centered cell or centered lattice need not necessarily be used. An equivalent primitive lattice can always be chosen, as illustrated by the relationship between parts (d) and (e) in Figure 11.3. 

If, as in Figure 11.14, we make the sixfold axes of all 2D nets coincide, we obtain a primitive lattice that retains all the symmetry present in the 2D lattice p6. We call this the primitive, hexagonal lattice. However, we can also choose the stacking pattern shown in Figure 11.15a, where we place the origin of the cell in the nth layer over the point 1, i in the (n - l)th layer. The result of this stacking scheme is seen in elevation in Figure 11.156. It has several important properties. 

Plate 7 shows alanine in hypothetical unit cells of two space groups. A triclinic unit cell (Plate 7a) is designated PI, being a primitive lattice with only a onefold axis of symmetry (that is, with no symmetry). P2 j (Plate 7b) describes a primitive unit cell possessing a twofold screw axis parallel to c, which points toward you as you view Plate 7. Notice that along any 2l screw axis, successive alanines are rotated 180" and translated one-half the axis length. A cell in space group T>212121 possesses three perpendicular twofold screw axes. 

The reciprocal lattice will then be a simple cubic lattice with primitive lattice... 

In case of non-primitive lattices with different atoms in the elementary cell, the sub-lattices can vibrate against each other (optical modes, see Figure 1.10). A vibration with a frequency iv 0 becomes possible even for k = 0. The opposite movement of neighboring atoms evokes large dipole moments allowing a coupling to electromagnetic waves. 

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. 

---