Chemistry - A Molecular Science

Figure 8.8 extends the discussion to the case of a very large number of orbitals that
would describe the delocalized orbitals in a meta

l. The orbitals in the solid are constructed

in the same way as those in a molecule, but they

apply to a crystal rather than a molecule,

so they are called

crystal orbitals

rather than molecular orbitals. As the number of orbitals

increases, the adjacent orbitals become more

similar, and the resulting energy levels get

closer together. In a piece of metal, the number of atoms and atomic orbitals in each crystal orbital is enormous (on the order of Avogadro’s number), so the energy levels are so close that they can no longer be distinguished. At this point, the energy levels are said to form an

energy band

. We will represent energy bands with rectangles to indicate that

essentially any energy within the rectangle is

accessible to the electron. However, the

crystal orbitals in a band are filled in the

same way as those in a molecule: from lowest

energy up and two electrons per orbital. The o

ccupancy of a band is usually the same as

the occupancy of the atomic orbitals used to

construct it. For example, a band constructed

from filled atomic orbitals will be full, and one

constructed from half-filled atomic orbitals

will be half full. The band occupancy is re

presented by shading the occupied portion of the

band. Thus, Figure 8.9 shows a half-filled band

because only half of the band is shaded.

Electrical conductivity relies on the ability of the electrons to move through the crystal

orbitals that are delocalized over the entire me

tal, but electrons are free to move only if

there are empty orbitals available for them to

move into. Thus, it is the highest energy

electrons, those closest in energy to the empty orbitals that are responsible for electrical conductivity, and it is the energy separation be

tween the highest occupied crystal orbital

and the lowest unoccupied crystal orbital that dictates the conduction properties of the solid. Indeed, the highest occupied crystal orbita

l is such an important characteristic of the

metal that it is given a name, the

Fermi level

.* Crystal orbitals above the Fermi level are

empty and those below it are full. In a partially filled band, such as the one shown in Figure 8.9, the energy separation at the Fermi level is essentially zero, so thermal energy is sufficient to move electrons from filled into em

pty orbitals, where they are mobile and can

conduct electricity. Thus, substances with pa

rtially filled bands like the one shown in

Figure 8.9 are

metallic conductors.

Band Energy BandEnergy

Energy

Number of orbitals in systemNumber

of

orbitals

in

system

21020 40

10

6

2

10

20

40

10

6

Figure 8.8 Energy diagrams for some many-atom systems Energy diagrams for two-, ten-, twenty-, forty-, and 10

6 -orbital

systems. The energy separation in the 10

6 orbital system is so small

that an energy band is formed. The dotted, red line shows the behavior of the maximum and minimum energies in the systems. Note how they level off for systems with large numbers of orbitals. Energy

empty

orbit

als

filled

orbit

als

Fermi

level

Met

allic

conductor

Figure 8.9 Band diagram for a metallic conductor

* The Fermi level is to crystal orbita

ls what the HOMO is to molecular

orbitals.

The band structure in Figure 8.9 applies only to

metals with half-filled atomic orbitals,

but atoms with filled atomic orbitals also form bands. The band structure for an atom with no partially filled bands is show

n in Figure 8.10. The highest energy filled band, which is

filled with valence electrons, is called the

valence band

, while the lowest energy empty

band is called the

conduction band

. The Fermi level is at the top of the valence band. The

Chapter 8 Solid Materials

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