Physical Chemistry Third Edition

1176 28 The Structure of Solids, Liquids, and Polymers

wherenx,ny, andnzare integers. The situation is similar to that of a particle in a

three-dimensional box, except that these integers are not required to be positive.

The kinetic energy of the electron is given by

εk

h ̄^2 k^2

2 m



h ̄^2

2 m

(k^2 x+k^2 y+k^2 z)

h ̄^2

2 m

k^2 (28.3-5)

wheremis the electron mass andk^2 k^2 x+k^2 y+k^2 z. We letnequal to

√

n^2 x+n^2 y+n^2 z

and consider a small increment inn, denoted bydn. The number of sets of integers in

the rangednis similar to that in Eq. (28.2-14):

Number of sets in the range (n,n+dn) 4 πn^2 dn (28.3-6)

This number is 8 times as large as that given by Eq. (28.2-14) to account for the

occurrence of negative integers. There are two possible spin states for each electron so

we double this expression to get the number of states. From Eq. (28.3-4) for a cubical

crystal of dimensionL:

Number of states in the rangedkg(k)dk

L^3 k^2

π^2

dk (28.3-7)

Using Eq. (28.3-5) and the chain rule (see Appendix B),

Number of states indε

L^3

√

2 m^3 /^2

π^2 h ̄^3

ε^1 /^2 dε (28.3-8a)

(

Number of states in

dεper unit volume

)

g(ε)dε

√

2 m^3 /^2

π^2 h ̄^3

ε^1 /^2 dε (28.3-8b)

This degeneracy is depicted in Figure 28.15. At 0 K, electrons will occupy all of the

states from zero energy up to the Fermi level. This occupation is shown by the shaded

area in Figure 28.15. At nonzero temperatures the occupation of states is given by

Eq. (28.3-1), and the occupation of the levels is given byg(ε)f(ε), represented by the

curve in the figure.

kBT

T 50

T. 0

(^) F(0)
Figure 28.15 The Degeneracy of
Energy Levels in the Free-Electron
Theory.From N. B. Hannay,Solid-State
Chemistry, Prentice-Hall, Englewood
Cliffs, NJ, 1967, p. 26.
We denote the number of mobile electrons per unit volume byN:

N 

∫∞

0

g(ε)f(ε)dε (28.3-9)

At 0 K, each of the states with energy less than the Fermi level is occupied by one

electron, and all of the states above the Fermi level are vacant. At 0 K the upper limit

of the integral can be changed from infinity to the Fermi level andf(ε) can be replaced

by unity:

N 

√

2 m^3 /^2

π^2 h ̄^3

∫μ 0

0

ε^1 /^2 dε

1

3 π^2

(

2 mμ 0

h ̄^2

) 3 / 2

(28.3-10)

The Fermi level at 0 K is

μ 0 εF0(3π^2 N)^2 /^3

h ̄^2

2 m

(5. 842 × 10 −^38 Jm^2 )N^2 /^3 (28.3-11)