Chapter 8 Solid Materials
8.6BAND THEORY OF SIMPLE METALS
The lattice sites of simple metals are occupiedby metal atoms that are held together by
metallic bonds
. The strength of metallic bonds can vary substantially, so their properties
vary over a wide range. For example, melting points of simple metals range from -39
oC
for mercury, a liquid at room temperature, to 3410
oC for tungsten, while densities vary
from about 0.5 g·cm
-3 for lithium to 22.6 g·cm
-3 for osmium. In the classical picture of
metallic bonding, metal atoms lose all or some
of their valence electrons to form positively
charged ions and a ‘sea of electrons’ that is delocalized over the entire metal. The metal cations are immersed in that sea, and the electrostatic force exerted between the cations and the surrounding electrons holds the positively charged metal ions in place, just as the bonding electrons hold the two nuclei to
gether in a covalent bond.
The molecular orbital description of the
delocalized electrons in metallic bonds
provides a more complete picture of metallic bonding as well as an explanation for the electrical conductivity of metals. Th
e explanation, which is called
band theory
, applies
the concepts presented in the molecular orbital discussion in Chapter 6 to a very large number of orbitals. In molecules, the number of
atomic orbitals that are used to construct
the MO’s is relatively small because the number of atoms involved is small, but recall that there are ~10
6 unit cells/mm in a solid, so the number
of orbitals involved in a metal is
enormous. Two effects of increasing the number
of orbitals can be seen by comparing a
system with two s orbitals with one of ten s orbitals as shown in Figure 8.7.
B A
Z Y X
DE
1
DE3DE
2Energy
Figure 8.7 Energy levels in two- and ten-orbital systems Only the two orbitals at lowest energy and the one at highest energy of the ten-orbital system are shown.Increasing the number of orbitals increases their energy spread
(Δ
E^2
EΔ
). 1
Orbital X has nine bonding interactions
, while orbital A has only one. Each
bonding interaction lowers the energy of the orbital, so X is at lower energy. Orbital Z has nine antibonding interactions, while orbital B has only one. Thus, Z is at a higher energy than B. We conclude that increasing the number of orbitals increases the energy
difference between the highest and
lowest energy orbitals;
i.e.
, the energy spread.
Increasing the number of orbitals reduces the energy separation beween adjacent orbitals
(Δ
E^3
<
EΔ
). Orbital A is completely bonding, but orbital B 1
is completely antibonding. They are very different types of orbitals, so their energies of very different. Orbital X has nine bonding interactions, while orbital Y has eight bonding interacti
ons and one antibonding interaction.
Orbitals X and Y are both strongly bonding, so they are close in energy.
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