Planar and Filamentary Structures

Although our discussion in the next seven chapters will center on simple tetrahedral structures, in which all electrons form simple two-electron bonds, it is desirable to introduce two other types of structures in these, some of the electrons form two-electron bonds (and are understandable in the same terms used for the tetrahedral solids) and other electrons are accommodated in pure p slates, similar to the 71 states discussed in Chapter 1 for diatomic molecules. For a discussion of the stability of these structures, see Friedel (1978). The two-electron bonds are simpler geometrically in these systems than in the tetrahedral solids and will provide very good problems. (Many of the problems at the ends of the following chapters will explore the theory of the bonds for these simpler systems.) 

Graphite is a simple planar structure in which the carbon atoms are arranged as in Fig. 3-10. The nearest neighbors are separated by 1.42 A (in diamond they are separated by 1.54 A), but the distance between successive planes is much larger (3.4 A), so that to a good approximation we may think of the planes as isolated sets of atoms when we discuss the electronic structure. Successive planes are stacked one above the other in the crystal. (The successive planes are also displaced laterally.) 

The electronic structure in graphite may be understood in terms o(sp hybrids (see Problem 3-2) oriented in the direction of the bond and bond orbitals constructed from these hybrids. The shorter bond length and different composition of the hybrid lead to a covalent energy value, Fj. that is also different in graphite than it is in diamond. 

The bond orbitals for graphite can accommodate three electrons per carbon atom. The remaining electrons go into p states oriented perpendicular to the plane in Fig. 3-10 and are analogous to the a states of diatomic molecules, as discussed in Chapter 1. The tc states are coupled by small matrix elements and broaden into a rather narrow band. There are enough electrons to half-fill this band (because of the two spin states). By filling only the lower half of the band these electrons 

The arrangement of carbon atoms in one plane of the graphite structure. Boron nitride can also form this structure (hexagonal BN) with atoms of B and N alternating as indicated by shaded and open circles. 

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Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

Planar and filamentary structures are of interest in their own right, but understanding their electronic structure and properties seems to depend upon a combination of concepts applicable also to other, more general systems rather than upon a unique set of concepts. We will not undertake a special study of them, therefore, but will carry them along as illustrative examples at the end of each chapter. 

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