Lattice parallelogram

FIGURE 5.35 The 14 B [,t, iis lattices. P denotes primitive I, body-centered F, face-centered C with a lattice point on two opposite faces and R, rhomltohedral (a rhomb is an oblique equilateral parallelogram). 

Figure 6.8 summarizes the most important properties of the reciprocal lattice. It is important that the base vectors of the surface lattice form the smallest parallelogram from which the lattice may be constructed through translations. Figure 6.9 shows the five possible surface lattices and their corresponding reciprocal lattices, which are equivalent to the shape of the respective LEED patterns. The unit cells of both the real and the reciprocal lattices are indicated. Note that the actual dimensions of the reciprocal unit cell are irrelevant only the shape is important. 

The unit cells for the two-dimensional lattices are parallelograms with their corners at equivalent positions in the array (i.e., the corners of a unit cell are lattice points). In Figure 1.17, we show a square array with several different unit cells depicted. All of these, if repeated, would reproduce the array it is conventional to choose the smallest cell that fully represents the symmetry of the structure. Both unit cells (la) and (lb) are the same size but clearly (la) shows that it is a square array, and this would be the conventional choice. Figure 1.18 demonstrates the same principles but for a centred rectangular array, where (a) would be the conventional choice because it includes information on the centring the smaller unit cell (b) loses this information. It is always possible to define a non-centred oblique unit cell, but doing so may lose information about the symmetry of the lattice. 

The five planar Bravais lattices (a) primitive oblique (parallelogram) (b) primitive rectangular, (c) centered rectangular, (d) primitive square, (e) primitive hexagonal. 

Figure 8-27 shows three planar networks based on the same plane lattice. Two and only two lines intersect in each point of all three networks. Accordingly, the parallelograms of all three networks have the same area. All of them are unit cells, in fact, primitive cells. Each of these parallelograms is determined by two sides a and b, and the angle y between them. These are called the cell parameters. 

Doslic135 proved that there is a one-to-one correspondence between the number of Kekule structures in a benzenoid parallelogram and the number of all square-lattice paths from (0,0) to (n,m) with steps (1,0) and (0,1). As illustrative examples we give all Kekule structures of anthanthrene in Figure 7 and the corresponding square-lattice paths in Figure 8. 

Therefore, both approaches, the Pascal recurrence algorithm and the approach based on counting the square-lattice paths are similar. However, the Pascal recurrence algorithm is applicable to a wider range of benzenoids and not only to benzenoid parallelograms. 

In addition to different types of crystal system there are also different types of lattice within those crystal systems, which correspond to specific arrangements of the atoms/ions within them. As discussed earlier, the two-dimensional system with the simplest sort of lattice which contains only one lattice point, is termed primitive. Similarly, for each three-dimensional crystal system there is always a primitive unit cell which consists of atoms located at the comers of the particular parallelepiped (i.e. a solid figure with faces which are parallelograms). For example. Figure 1.5 shows primitive lattices (symbol P) for both tetragonal and hexagonal systems. 

The closed but boundless network of hexagons embedded as an all-hexagon toroidal polyhex can be represented by an infinite planar lattice, on which the set of hexagons forms a parallelogram which repeats itself endlessly in two dimensions (i.e., it is doubly periodic). Figure 14 shows an example of a torus with nine hexagons (A-I). 

The general plane lattice (a) shown in Figure 8-27 is called a parallelogram lattice. The other four plane lattices of Figure 8-27 are special cases of the general lattice. The rectangular lattice (b) has a primitive cell with unequal sides. The so-called diamond lattice (c) has a unit cell with equal sides. A... 

The periodically repeating unit is called a motif. In Fig. 1.8(a), the contents of one of the parallelograms can be considered to be the motif. It is important to distinguish clearly between the terms lattice, motif and structure ... 

By choosing two non-collinear translations a and b in Fig. 1.8(a), we describe the lattice by the translation vectors r = wa H- t b,M and v being integers. We call this coordinate system the lattice base. The parallelogram (a, b) is the cell (unit cell). Analogously, the base a, b, c of a three-dimensional lattice is defined by three non-coplanar translations. The cell is hence a parallelepiped. The coordinates x, y, z of a point inside this cell are referred to this non-unitary coordinate system. The set of all the points equivalent by translation to the point Xpyp Zj is given by... 

The parallelogram formed by the lattice vectors, the shaded region in Figure 12.3, defines the unit cell. In two dimensions the unit cells must tile, or fit together in space, in such a way that they completely cover the area of the lattice with no gaps. In three dimensions the unit cells must stack together to fill all space. 

A FIGURE 12.3 A crystalline lattice in two dimensions. An infinite array of lattice points is generated by adding together the lattice vectors a and b. The unit cell is a parallelogram defined by the lattice vectors. 

You may wonder why the hexagonal unit cell is not shaped like a hexagon. Remember that the unit cell is by definition a parallelogram whose size and shape are defined by the lattice vectors a and b. 

In two dimensions the unit cell is a parallelogram whose size and shape are defined by two lattice vectors (a and b). There are four primitive lattices, lattices where the lattice points are located only at the corners of the unit cell square, hexagonal, rectangular, and oblique. In three dimensions the unit cell is a parallelepiped whose size and shape are defined by three lattice vectors (a, b and c), and there are seven primitive lattices cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic, and triclinic. Placing an additional lattice point at the center of a cubic unit cell leads to a body-centered cubic lattice, while placing an additional point at the center of each face of the unit cell leads to a face-centered cubic iattice. 

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