Hexagonal lattice, basis length

Fig. 11. Crystal structure of graphite. The unit cell is shaded in green, (a) Top view of the surface layer. The hexagonal surface lattice is defined by two unit vectors u and v in the xy-plane with a length of 246 pm and an angle of 120° forming a honeycomb web of hexagonal rings. The basis of the lattice consists of two carbon atoms a, (white) and /3 (red) with a distance of 142 pm. (b) Perspective view, showing the layered structure. The distance between layers is 2.36 times the next-neighbor distance of atoms within one layer, and the bond between layers is weak. The a-atoms (white) are directly above an a-atom in the layer directly underneath at a distance of 334.8 pm the /3-atoms (red) are over hollow sites (h). The unit vector w is parallel to the z-axis with a length of 669.6pm.

The exceptional large value of forbidden gap width seems to be any doubtful. Perhaps it means that not all of the calculations results obtained for small clusters can be used for prediction of crystal structures properties. Must be noted that calculated bonds lengths in squares and hexagons and period of crystal lattice (1.607 A, 1.452 A, 5.894 A) differ from the values (1.503 A, 1.380 A, 5.545 A), obtained in [8], However one can not expect the better accordance from the results of clusters calculations performed moreover in the frames of semi-empirical PM3-basis. 

As noted in Chap. 1, every unit cell can be characterized by six lattice parameters — three edge lengths a, b, and c and three interaxial angles a, j3, and 7. On this basis, there are seven possible combinations of a, h, and c and a, ff and 7 that correspond to seven crystal systems (see Fig. 1.2). In order of decreasing symmetry, they are cubic, hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic, and triclinic. In the remainder of this section, for the sake of simplicity the discussion is restricted to the cubic system for which a — b c and a = / = 7 = 90°. Consequently, this system is characterized by only one parameter, usually denoted by a. 

Primitive three-dimensional lattices have been classified into seven crystalline systems triclinic, monoclinic, orthorombic, tetragonal, cubic, trigonal, and hexagonal. They are different in the relative lengths of the basis vectors as well as in the angles they form. An additional seven nonprimitive lattices, belonging to the same crystalline systems, are added to the seven primitive lattices, which thus completes the set of all conceivable lattices in ordinary space. These 14 different types of lattices are known as Bravais lattices (Figure 3). 

Graphene is a one-atom-thick planar layer of graphite, where the atoms are packed in a hexagonal ( honeycomb ) crystal lattice. The carbon atoms are sp -bonded with a bond length of 1.42 A. The crystal lattice has two atoms per unit cell, A and B, and it is rotationally symmetric for rotations of 120° around any lattice point. One can view the honeycomb lattice as a triangular Bravais lattice with a basis of two atoms per unit cell. Fig. 2a. 

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