Physical Chemistry Third Edition

28.3 The Electronic Structure of Crystalline Solids 1175

5 electrons

3 d

4 s 5 electrons

3 d

(a)

Fermi surface Copper

1 electron

4.46 electrons

3 d

4 s 5 electrons

3 d

(b)

Fermi surface

0.54 electron

Fermi surface

0.54 electron

Nickel at 0 K

4.73 electrons

3 d

4 s 4.73 electrons

3 d

(c)

0.27 hole

Nickel above the

curie temperature

0.54 hole

Figure 28.14 The 4sand 3dBands of Nickel and Copper.The vertical axis in each

diagram represents the electronic energy. From N. B. Hannay,Solid-State Chemistry,

Prentice-Hall, Englewood Cliffs, NJ, 1967, p. 38.

The Free-Electron Theory

This simple theory is based on the assumption that the mobile electrons in a solid can

be represented as a gas of noninteracting fermions. The orbitals for the electrons are

free-particle wave functions like those of Eq. (15.3-41):

ψeik·rei(kxx+kyy+kzz) (28.3-2)

The vectorkis called thewave vectorand the vectorris the position vector of the

electron. The scalar product (dot product)k•ris equal tokxx+kyy+kzz, as described

in Appendix B.

Consider a cubic region with dimensionsLbyLbyLthat is part of a very large

crystal. We imposeperiodic boundary conditions, which means that

ψ(x+L,y,z)ψ(x,y,z) (28.3-3)

with similar equations foryandz. To satisfy this condition,

kx 2 πnx/L, ky 2 πny/L, kz 2 πnz/L (28.3-4)