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*Mathematical Analysis*

The Fermi function is

** f(E) = 1 / [ 1 + exp( (E-EF)/kT ) ] **(1)

whose physical meaning is the pobability of electron occupancy for an energy state at energy E. It is plotted as a function of Energy in the left hand side of this applet.

For E - E_{F} >> kT (say, for E > E_{F} + 4
kT)

** f(E) ~ exp( -(E-E_{F})/kT )**.

This means that the probability of occupancy approaches zero exponentially with increasing E for states at energy E > EF + 4kT. This is the so-called Boltzman approaximation (that is, f(E) approximated by a simple exponential term).

For E - E_{F} << - kT (say, for E < E_{F} -
4kT),

** f(E) ~ 1 - exp( (E-E_{F})/kT )**.

This means that the probability of occupancy approaches unity with decreasing
energy for E < E_{F} - 4kT.

Note that the maximum value of f(E) is unity.

For E_{F} - 4kT < E < E_{F} + 4kT, no simple exponential
function approaximation is possible and the full expression, formula (1),
must be used for f(E).

Note that f(E) = 1/2 at E = E_{F}.