bei48482_FM

The quantity depends on the properties of the particular system and may be a func-

tion of T. Its value is determined by the normalization condition that the sum over all

energy states of n() g()f() be equal to the total number of particles in the system.

If the number of particles is not fixed, as in the case of a photon gas, then from the

way is defined in deriving Eqs. (9.26) and (9.27), 0, e1.

The 1 term in the denominator of Eq. (9.26) expresses the increased likelihood

of multiple occupancy of an energy state by bosons compared with the likelihood

for distinguishable particles such as molecules. The 1 term in the denominator of

Eq. (9.27) is a consequence of the uncertainty principle: No matter what the values

of ,, and T,f() can never exceed 1. In both cases, when  kTthe functions

f() approach that of Maxwell-Boltzmann statistics, Eq. (9.2). Figure 9.5 is a com-

parison of the three distribution functions. Clearly fBE( ) for bosons is always greater

at a given ratio of kTthan it is for molecules, and fFD( ) for fermions is always

smaller.

From Eq. (9.27) we see that fFD() ^12 for an energy of

Fermi energy FkT (9.28)

This energy, called the Fermi energy,is a very important quantity in a system of

fermions, such as the electron gas in a metal. In terms of (^) Fthe Fermi-Dirac distribution
function becomes
Fermi-Dirac fFD() (9.29)
To appreciate the significance of the Fermi energy, let us consider a system of fermi-
ons at T0 and investigate the occupancy of states whose energies are less than F
and greater than F. What we find is this:
T0, F: fFD() 1
T0,  F: fFD() 0
1

e^  1
1

e(F)kT 1
1

0  1
1

e
 1
1

e(F)kT 1
1

e(F)kT 1
308 Chapter Nine
Figure 9.5A comparison of the
three distribution functions for
the same value of . The Bose-
Einstein function is always higher
than the Maxwell-Boltzmann one,
which is a pure exponential, and
the Fermi-Dirac function is al-
ways lower. The functions give
the probability of occupancy of a
state of energy  at the absolute
temperature T.
1.2
1.0
0.8
0.6
0.4
0.2
0 kT 2 kT 3 kT 4 kT
Maxwell-Boltzmann
Fermi-Dirac
Bose-Einstein
Distribution function
f(e
)
Energy, e
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