Closed orbits continuous band

Electrical conduction in metals can be explained in terms of molecular orbitals that spread throughout the solid. We have already seen that, when N atomic orbitals merge together in a molecule, they form N molecular orbitals. The same is true of a metal but, for a metal, N is enormous (about 1023 for 10 g of copper, for example). Instead of the few molecular orbitals with widely spaced energies typical of small molecules, the huge number of molecular orbitals in a metal are so close together in energy that they form a nearly continuous band (Fig. 3.43). 

As Fig. 6.9 illustrates, the orbitals are very close in a metal and form an almost continuous band of levels. In fact, it is impossible to detect the separation between levels. The bands behavior is in many respects similarly to the orbitals of the molecule as shown in Fig. 6.8 If there is little overlap between the electrons, the interaction is weak and the band is narrow. Such is the case for d orbitals (Fig. 6.10), which have pronounced shapes and orientations that are largely retained in the metal. Hence the over-... 

The metallic bond can be seen as a collection of molecular orbitals between a large number of atoms. As Fig. A.6 illustrates, the molecular orbitals are very close and form an almost continuous band of levels. It is impossible to detect that the levels are actually separated from each other. The bands behave in many... 

The electronic properties of solids can be described by various theories which complement each other. For example band theory is suited for the analysis of the effect of a crystal lattice on the energy of the electrons. When the isolated atoms, which are characterized by filled or vacant orbitals, are assembled into a lattice containing ca. 5 x 1022 atoms cm 3, new molecular orbitals form (Bard, 1980). These orbitals are so closely spaced that they form essentially continuous bands the filled bonding orbitals form the valence band (vb) and the vacant antibonding orbitals form the conduction band (cb) (Fig. 10.5). These bands are separated by a forbidden region or band gap of energy Eg (eV). 

Molecular orbital (MO) theory has been used to explain the bonding in metallic crystals, such as pure sodium or pure aluminum. Each MO, instead of dealing with a few atoms in a typical molecule, must cover the entire crystal (might be 1020 or more atoms ). Following the rule that the number of MOs must equal the number of atomic orbitals (AOs) combined, this many MOs must be so close on an energy level diagram that they form a continuous band of energies. Because of this factor, the theory is known as band theory. 

For an assemblage of two identical molecules spaced d nm apart, the HOMO and LUMO energies split into four levels, each split by 2t eV apart ("dimer splitting") [26] here t is akin to the Hiickel69 resonance integral (i of Section 3.15 Indeed, chemists will remember Eq. (8.6.10) from the simple Hiickel molecular orbital theory for aromatic 7r-electron systems. As the number of molecules N increases, the energy levels become spaced more closely, until they form a quasi-continuous band of bandwidth W, where... 

The metals we have discussed so far adopt close-packed structures and this provides extensive overlap of the atomic orbitals so that a wide continuous band is formed. 

Figure 12.35 The band of molecular orbitals in lithium metal. Lithium atoms contain four valence orbitals, one 2s and three 2p(left). When two lithium atoms combine (IJ2), their AOs form eight MOs within a certain range of energy. Four Li atoms (LIa) form 16 MOs. A mole of Li atoms forms 4A/a MOs (A/a = Avogadro s number). The orbital energies are so close together that they form a continuous band. The valence electrons enter the lower energy portion (valence band), while the higher energy portion (conduction band) remains empty. In lithium (and other metals), the valence and conduction bands have no gap between them.

Electronic Properties When molecular orbitals are formed from N atoms, atomic orbital combined to form N molecular orbitals. In solids, N is very large, resulting in a large number of orbitals [40]. The overlap of a large number of orbitals leads to closely spaced molecular orbitals which form a virtually continuous band (Fig. 3) [41]. The overlap of the highest occupied molecular orbitals (HOMO) results in the formation of a valence band and a conduction band is formed from... 

Dopant atoms chemical impurities that are deliberately introduced into the semiconductor lattice to provide control over the conductivity and Fermi level of the solid Doping the introduction of specific chemical impurities into a semiconductor lattice to control the conductivity and the Fermi level of the semiconductor Effective density of states the number of electronic states within ikT of the edge of an energy band, where k is the Boltzmann constant and T is the temperature Energy bands a cluster of orbitals in which the individual molecular orbitals are packed closely together to form an almost continuous distribution of energy levels... 

The Hg atom has a 6s closed electronic shell. It is isoelec-tronic with helium, and is therefore van der Waals bound in the diatomic molecule and in small clusters. For intermediate sized clusters the bands derived from the atomic 6s and 6p orbitals broaden as indicated in fig. 1, but a finite gap A remains until the full 6s band overlaps with the empty 6p band, giving bulk Hg its metallic character. This change in chemical binding has a strong influence, not only on the physical properties of mercury clusters, but also on the properties of expanded Hg, and on Hg layers on solid and liquid surfaces. For a rigid cluster the electronic states are discreet and not continuous as in fig. 1. Also the term band for a bundle of electronic states will be used repeatedly in this paper, although incipient band might be better. As the clusters discussed here are relatively hot, possibly liquid, any discreet structure will be broadened into some form of structured band . 

Band A series of very closely spaced, nearly continuous molecular orbitals that belong to the crystal as a whole. 

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