Periodicity in Three Dimensions—Graphite

Graphite is an important commercial material. Some of its uses derive from the fact that it is a very good conductor of electricity—nearly as good as metals. The most stable crystalline form of graphite (Bernal graphite) is depicted in Fig. 15-30a. It comprises 

Chapter 15 Molecular Orbital Theory of Periodic Systems 

Points of special interest in the RFBZ are labeled F, M, and K. The point F corresponds to the k, k2) values (0,0). This is a unique point in the FBZ, so it produces one nondegenerate Bloch sum for each basis function in the unit cell. The point M 

Since there are two carbon atoms per unit cell and four valence AOs per carbon, we have eight basis functions per unit cell for a minimal valence basis set calculation. Therefore, we expect eight bands. Since the system is planar we can distinguish two n AOs and six a AOs. We know that these two symmetry types will not interact for any k, so we can predict or interpret the a and n band behaviors independently. 

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If electron-pair, or covalent, bonding is periodic in two or three dimensions, crystals result. The most important case is the carbon-carbon bond. If it is extended periodically in two-dimensions the result is graphite in three-dimensions it is diamond. Other elements that form electron-pair bonds are Si, Ge, and a-Sn. Some binary compounds are A1P (isoelectronic with Si),... 

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