Physical Chemistry Third Edition

28.3 The Electronic Structure of Crystalline Solids 1177

A typical metal has a density of mobile electrons approximately equal to 10^28 m−^3 ,

corresponding to a value of the Fermi level equal to several electron volts.

EXAMPLE28.7

The density of copper is 8960 kg m−^3. Find the density of mobile electrons, assuming one

mobile electron from each atom and find the zero-temperature value of the Fermi level in

joules and in electron volts.

Solution

N

(8960 kg m−^3 )(6. 022 × 1023 mol−^1 )

0 .063546 kg mol−^1

 8. 49 × 1028 m−^3

εF0(5. 842 × 10 −^38 Jm^2 )(8. 49 × 1028 m−^3 )^2 /^3  1. 129 × 10 −^18 J 7 .04 eV

In the free-electron theory the Fermi level for nonzero temperature is approximately^9

μεF≈εF0

(

1 −

(πkBT)^2

12 ε^2 F0

)

(28.3-12)

whereεF0is the Fermi level at 0 K.

The energy per unit volume of the free-electron gas at 0 K is

U 0 

∫εF0

0

εg(ε)dε

1

5 π^2

(

2 m

h ̄^2

) 3 / 2

ε^5 F0/^2 

3 NεF0

5

(28.3-13)

wheremis the electron mass. IfT<<εF/kB, the energy at a nonzero temperature is

approximately^10

Uel≈U 0 +

Nπ^2 k^2 BT^2

4 εF

(28.3-14)

The heat capacity per unit volume is

Cel≈

(

∂U

∂T

)

V



Nπ^2 k^2 BT

2 εF

(28.3-15)

If the electron gas obeyed classical mechanics the heat capacity would be 3kB/2 per

electron so that

Cel

π^2 kBT

3 εF

Cclass (28.3-16)

The quantum mechanical electron gas is sometimes called thedegenerate electron gas

because its heat capacity is “degenerated” from the classical value by the factor given

in Eq. (28.3-16). This meaning of the word “degenerate” is different from the usage in

previous chapters, where it applied to the number of states in an energy level.

(^9) J. S. Blakemore,op. cit., p. 176 (note 4).
(^10) J. S. Blakemore,op. cit., p. 176 (note 4).