bei48482_FM

324 Chapter Nine

and the total specific heat of the metal should be

cV 3 R R R

at high temperatures where a classical analysis is valid. Actually, of course, the Dulong-

Petit value of 3Rholds at high temperatures, from which we conclude that the free

electrons do not in fact contribute to the specific heat. Why not?

If we reflect on the characters of the entities involved in the specific heat of a metal,

the answer begins to emerge. Both the harmonic oscillators of Einstein’s model and the

phonons of Debye’s model are bosons and obey Bose-Einstein statistics, which place

no upper limit on the occupancy of a particular quantum state. Electrons, however,

are fermions and obey Fermi-Dirac statistics, which means that no more than one elec-

tron can occupy each quantum state. Although both systems of bosons and systems of

fermions approach Maxwell-Boltzmann statistics with average energies^12 kT per

degree of freedom at “high” temperatures, how high is high enough for classical be-

havior is not necessarily the same for the two kinds of systems in a metal.

According to Eq. (9.29), the distribution function that gives the average occupancy

of a state of energy in a system of fermions is

fFD() (9.29)

What we also need is an expression for g()d, the number of quantum states avail-

able to electrons with energies between and  d.

We can use exactly the same reasoning to find g()dthat we used to find the num-

ber of standing waves in a cavity with the wavelength in Sec. 9.5. The correspon-

dence is exact because there are two possible spin states, ms^12 and ms^12 (“up”

and “down”), for electrons, just as there are two independent directions of polariza-

tion for otherwise identical standing waves.

We found earlier that the number of standing waves in a cubical cavity Lon a side is

g(j)djj^2 dj (9.33)

where j  2 L. In the case of an electron, is its de Broglie wavelength of hp.

Electrons in a metal have nonrelativistic velocities, so p 2 mand

j dj  d

Using these expressions for jand djin Eq. (9.33) gives

g()d d

As in the case of standing waves in a cavity the exact shape of the metal sample does

not matter, so we can substitute its volume Vfor L^3 to give

g() d d (9.55)

(^8)  (^2) Vm^3 ^2

h^3
Number of
electron states
(^8)  (^2) L^3 m^3 ^2

h^3
2 m


L

h
2 L 2 m

h
2 Lp

h
2 L

1

e(F)kT 1
Average occupancy
per state
9

2
3

2
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