Valence crystals diamond

The binding of valence crystals can also be explained from the standpoint of energy bands. In Fig. XXIX-12 we show energy bands for diamond, a typical crystal held by homopolar bonds. We see that the... 

Valence crystals are formed by combinations of atoms of the lighter elements in the middle column of the periodic table, such as diamond and silicon carbide. These crystals conform to valence rules, and the interatomic bonds are due to the sharing of electron pairs. Valence crystals are characterized by very great hardness, poor cleavage, and poor electrical conductivity. 

Figure Al.3.22. Spatial distributions or charge densities for carbon and silicon crystals in the diamond structure. The density is only for the valence electrons the core electrons are omitted. This charge density is from an ab initio pseudopotential calculation [27].

In the course of the PP calculations of these quantities for Si [10] and Ge [11], a characteristic local pattern which reflects position, shape and size of a specific atom in the crystal is observed on the contour map of the valence electron A(r)-function. The atom is one of the two atoms in the unit cell of diamond structure. It seems as... 

It is interesting that all of these crystals except diamond are boron compounds. Note also, that most of them consist exclusively of relatively small atoms. The exception is ReB2. Since Re has a large number of valence electrons the general rule is followed that high hardness is associated with high VED (valence electron density). 

Diamond is the prototype of all alijihatic compounds. One would expect on its surface free valences which are capable of surface compound formation. The surface compounds on diamond should differ somewhat in character as compared to the surface compounds on aromatic graphite or microcrystalline carbon. Apart from singly bonded carbon atoms on the edges and corners of diamond crystals... 

Fig. 1.10. Crystal structure of silicon, (a) The bonding of Si atoms. Each Si atom has four valence electrons, which form four. vp orbitals, directed to the four corners of a regular tetrahedron. Each of the. sp orbitals is bonded with an. sp orbital of another Si atom, (b) The structure of the Si crystal, the so-called diamond structure.

The formation of a Si crystal is shown in Fig. 1.10. Aside from the core, each Si atom has four valence electrons two 3s electrons and two 3p electrons. To form a Si crystal, one of the 3s electrons is excited to the 3p orbital. The four valence electrons form four sp hybrid orbitals, each points to a vertex of a tetrahedron, as shown in Fig. 1.10. Thpse four sp orbitals are unpaired, that is, each orbital is occupied by one electron. Since the electron has spin, each orbital can be occupied by two electrons with opposite spins. To satisfy this, each of the directional sp orbitals is bonded with an sp orbital of a neighboring Si atom to form electron pairs, or a valence bond. Such a valence bonding of all Si atoms in a crystal form a structure shown in (b) of Fig. 1.10, the so-called diamond structure. As seen, it is a cubic crystal. Because all those tetrahedral orbitals are fully occupied, there is no free electron. Thus, similar to diamond, silicon is not a metal. 

Typical values of transfer coefficients a and ji thus obtained are listed in Table 4 for single crystal and polycrystalline thin-film electrodes [69] and for a HTHP diamond single crystal [77], We see for Ce3+/ 41 system (as well as for Fe(CN)63 /4 and quinone/hydroquinone systems [104]), that, on the whole, the transfer coefficients are small and their sum is less than 1. We recall that an ideal semiconductor electrode must demonstrate a rectification effect in particular, a reaction proceeding via the valence band has transfer coefficients a = 0, / =l a + / = 1 [6], Actually, the ideal behavior is rarely the case even with single crystal semiconductor materials fabricated by advanced technologies. Departure from the ideal semiconductor behavior is likely because the interfacial potential drop is located in part in the Helmholtz layer (due e.g. to a high density of surface states), or because the surface states participate in the reaction. As a result, the transfer coefficients a and ji take values intermediate between those characteristic of a semiconductor (0 or 1) and a metal ( 0.5). 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

Crystal orbitals are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums - one for each of the s- and p-atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. 

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