bei48482_FM

9.9 FREE ELECTRONS IN A METAL

No more than one electron per quantum state

The classical, Einstein, and Debye theories of specific heats of solids apply with equal

degrees of success to both metals and nonmetals, which is strange because they ignore

the presence of free electrons in metals.

As discussed in Chap. 10, in a typical metal each atom contributes one electron to

the common “electron gas,” so in 1 kilomole of the metal there are N 0 free electrons.

If these electrons behave like the molecules of an ideal gas, each would have ^32 kT of

kinetic energy on the average. The metal would then have

Ee N 0 kT RT

of internal energy per kilomole due to the electrons. The molar specific heat due to

the electrons should therefore be

cVe

V

 R

3



2

Ee



T

3



2

3



2

Statistical Mechanics 323

A

lthough Einstein’s formula predicts that cV→ 0 as T→ 0, as observed, the precise manner

of this approach does not agree too well with the data. The inadequacy of Eq. (9.54) at

low temperatures led Peter Debye to look at the problem in a different way in 1912. In Ein-

stein’s model, each atom is regarded as vibrating independently of its neighbors at a fixed fre-

quency. Debye went to the opposite extreme and considered a solid as a continuous elastic

body. Instead of residing in the vibrations of individual atoms, the internal energy of a solid

according to the new model resides in elastic standing waves.

The elastic waves in a solid are of two kinds, longitudinal and transverse, and range in fre-

quency from 0 to a mximum (^) m. (The interatomic spacing in a solid sets a lower limit to the
possible wavelengths and hence an upper limit to the frequencies.) Debye assumed that the total
number of different standing waves in a kilomole of a solid is equal to its 3N 0 degrees of free-
dom. These waves, like em waves, have energies quantized in units of h. A quantum of acoustic
energy in a solid is called a phonon, and it travels with the speed of sound since sound waves
are elastic in nature. The concept of phonons is quite general and has applications other than
in connection with specific heats.
Debye finally asserted that a phonon gas has the same statistical behavior as a photon gas or
a system of harmonic oscillators in thermal equilibrium, so that the average energy per stand-
ing wave is the same as in Eq. (9.52). The resulting formula for cVreproduces the observed
curves of cVversus Tquite well at all temperatures.
Peter Debye, who was Dutch, did original work in many aspects of both physics and chem-
istry, at first in Germany and later at Cornell University. Although Heisenberg, a colleague for a
time, thought him lazy (“I could frequently see him walking around in his garden and watering
the roses even during duty hours of the Institute”), he published nearly 250 papers and received
the Nobel Prize in chemistry in 1936.
The Debye Theory
bei48482_Ch09.qxd 1/22/02 8:46 PM Page 323