Occupation Probability
According to classical
theory all the electrons in a metal can have the same energy, so that, at 0K,
they all condense into the lowest available energy. But when quantum statistics
is applied we find that these electrons occupy states between the energy values
O and
.
Fermi has shown that the probability of occupancy of a particular quantum state is given as
...(1)Where
Fermi factor or Fermi function. E is
the energy of the given state and
is the Fermi energy at temperature T. The
function (1) is plotted in Fig. for various values of T.
Let us consider the
behavior of Fermi factor at different temperature -
(a) At T= 0K, the
exponential term when
approaches to zero as
And we get
.
When
we get
.
The meaning of
for
is that all the quantum states are occupied
and all the states having
are empty at 0K.
(b) At
, 
Hence the probability
of occupation of Fermi level is fifty percent. The plot for
and
temperatures is also shown in Fig. The value
of
is still practically unity.
For energies above
,
the Fermi distribution becomes identical with that of Boltzman distribution.
The meaning of this is that all the states below
are not filled and hence all the states above 
or
.
(c) At very high
temperatures when
the relation (1.74) does not remain valid and
the entire distribution becomes Maxwell-Boltzmann. This is shown for
temperature
in Fig.
For next topic 'Effective mass' - Click Here
0 Comments
If you have any doubt, please let me know