Band formation by orbital overlap

Key point: The overlap of atomic orbitals in solids gives rise to bands of energy levels separated by energy gaps. The overlap of a large number of atomic orbitals in a solid lead to a large number of molecular orbitals that are closely spaced in energy and so form an almost continuous band of energy levels (Fig. 3.61). Bands are separated by band gaps, which are values of the energy for which there is no molecular orbital. The formation of bands can be understood by considering a line of atoms, and supposing that each atom has an s orbital that overlaps the s orbitals on its immediate neighbours (Fig. 3.62). When the line consists of only two atoms, there is a bonding and an antibonding molecular orbital. When a third atom joins them, there are three molecular orbitals. The central orbital of the set is nonbonding and the outer two are at low energy and high energy, respectively. As more atoms are added, each one contributes an atomic orbital, and hence one more molecular orbital is formed. When there are N atoms in the line, there are N molecular orbitals. The orbital of lowest energy has no nodes between neighbouring atoms. The orbital of highest energy has a node between every pair of neighbours. The remaining orbitals have successively 1, 2,...internuclear nodes and a corresponding range of energies between the two extremes. The total width of the band, which remains finite even as N approaches infinity (as shown in Fig. 3.63), depends on the strength of the interaction between neighbouring at oms. The greater the strength of interaction (in broad terms, the greater the degree of overlap between neighbours), the greater the energy separation of the non-node orbital and the all-node orbital. However, whatever the number of atomic orbitals used to form the molecular orbitals, there is only a finite spread of orbital energies (as depicted in Fig. 3.63). It follows that the separation in energy between neighbouring orbitals must approach zero as N approaches infinity, otherwise the range of orbital energies could not be finite. That is, a band consists of a countable number but near-continuum of energy levels. The band just described is built from s orbitals and is called an s band. If there are p orbitals available, a p band can be constructed from their overlap as shown in Fig. 3.64.

Because p orbitals lie higher in energy than s orbitals of the same valence shell, there is often an energy gap between the s band and the p band (Fig. 3.65). However, if the bands span a wide range of energy and the atomic s and p energies are similar (as is often the case), then the two bands overlap. The d band is similarly constructed from the overlap of d orbitals. The formation of bands is not restricted to one type of atomic orbital and bands may be formed in compounds by combinations of different orbital types, for example the d orbitals of a metal atom may overlap the p orbitals of neighbouring O atoms.

Fig. 3.65 (a) The s and p bands of a solid and the gap between them. Whether or not there is in fact a gap depends on the separation of the s and p orbitals of the atoms and the strength of the interaction between them in the solid. (b) If the interaction is strong, the bands are wide and may overlap.

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مواضيع ذات صلة

The Fermi level

Densities of states

The conductivities of inorganic solids

Nonstoichiometry

Intrinsic point defects

Solubility

Thermal stabilities of ionic solids

The Born–Mayer equation

Structure maps

The radius ratio

Ionic radii

Characteristic structures of ionic solids