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10 Quantum Statistics
At higher temperatures, more and more fermions will be excited into higherenergy single-particle states, while at the same time, more of the low-lying singleparticle states will become vacant, and the energy of the lowest-lying vacant singleparticle state will be reduced. If we wish to add a new fermion to the system without
increasing the entropy, we will have to require this new fermion to go into a vacant
single-particle state that lies considerably below F , while once again cooling the
gas slightly in order to avoid an increase in the number of microstates. In such a
case, U = μ, but now μ is much smaller than F . If we continue this line of
reasoning, we see that μ will take the value 0 for a temperature just below the Fermi
temperature (when even the single-particle ground state is unlikely to be occupied).
After this point, μ becomes negative, and eventually, when T T F , μ behaves in
the same manner that we have seen in Chap. 5 for the ideal classical gas.
10.4.3 Interlude: Zero-Temperature Fermi Distribution
Let us now examine briefly the limiting, but very important, case in which T = 0
for Fermi–Dirac particles. To do this, we shall begin with the expression
n
FD
k =
1
e β(( k −μ) + 1
,
and take the limit that T goes to zero to obtain
lim
T →0
n
FD
k = lim
β→∞
n
FD
k = lim
β→∞
1
e β(( k −μ) + 1
.
We must consider two separate cases for this limit, depending upon whether k is
greater than or less than μ: now, if k > μ, the result is
lim
β→∞
1
e β(( k −μ) + 1
= 0 ,
while if k < μ, the result is
lim
β→∞
1
e β(( k −μ) + 1
= 1 .
Finally, when k = μ, we see that n k = 0.5. Hence, for absolute zero temperature,
the graph of n k for Fermi–Dirac particles is a discontinuous function with the form
shown in Fig. 10.3.
All states with energy k less than F ≡ μ(0) are occupied, and all states with
energy k > μ(0) are unoccupied: F is called the Fermi energy, and its value on an
energy scale is referred to as the Fermi level.
10 Quantum Statistics
At higher temperatures, more and more fermions will be excited into higherenergy single-particle states, while at the same time, more of the low-lying singleparticle states will become vacant, and the energy of the lowest-lying vacant singleparticle state will be reduced. If we wish to add a new fermion to the system without
increasing the entropy, we will have to require this new fermion to go into a vacant
single-particle state that lies considerably below F , while once again cooling the
gas slightly in order to avoid an increase in the number of microstates. In such a
case, U = μ, but now μ is much smaller than F . If we continue this line of
reasoning, we see that μ will take the value 0 for a temperature just below the Fermi
temperature (when even the single-particle ground state is unlikely to be occupied).
After this point, μ becomes negative, and eventually, when T T F , μ behaves in
the same manner that we have seen in Chap. 5 for the ideal classical gas.
10.4.3 Interlude: Zero-Temperature Fermi Distribution
Let us now examine briefly the limiting, but very important, case in which T = 0
for Fermi–Dirac particles. To do this, we shall begin with the expression
n
FD
k =
1
e β(( k −μ) + 1
,
and take the limit that T goes to zero to obtain
lim
T →0
n
FD
k = lim
β→∞
n
FD
k = lim
β→∞
1
e β(( k −μ) + 1
.
We must consider two separate cases for this limit, depending upon whether k is
greater than or less than μ: now, if k > μ, the result is
lim
β→∞
1
e β(( k −μ) + 1
= 0 ,
while if k < μ, the result is
lim
β→∞
1
e β(( k −μ) + 1
= 1 .
Finally, when k = μ, we see that n k = 0.5. Hence, for absolute zero temperature,
the graph of n k for Fermi–Dirac particles is a discontinuous function with the form
shown in Fig. 10.3.
All states with energy k less than F ≡ μ(0) are occupied, and all states with
energy k > μ(0) are unoccupied: F is called the Fermi energy, and its value on an
energy scale is referred to as the Fermi level.