WOLFRAM|DEMONSTRATIONS PROJECT

Fermi-Dirac Distributions for Free Electrons in Metals

​
elements
lithium
temperature (°K)
0.01
General
:Exp[-5.5615×
7
10
] is too small to represent as a normalized machine number; precision may be lost.
General
:Exp[-5.5615×
6
10
] is too small to represent as a normalized machine number; precision may be lost.
General
:Exp[-5.5615×
7
10
] is too small to represent as a normalized machine number; precision may be lost.
This Demonstration shows the variation in density of free electrons as a function of energy (in eV) for some representative metals at different temperatures
T
. According to the Fermi–Dirac distribution, the number of free electrons per electron volt per cubic meter is given by
ρ(ϵ)=0.682×
28
10
1/2
ϵ
(exp[(ϵ-
ϵ
F
)/(
k
B
T)]+1)
, where
ϵ
F
is the Fermi energy of the metal and
k
B
is the Boltzmann constant.
The dashed orange lines show the density of free electrons as a function of energy. The area under each curve equals the total number of free electrons per cubic meter of the metal.
The blue area shows where electrons have vacated energies in the neighborhood of
ϵ
F
. The red area shows the electrons that have acquired energies greater than
ϵ
F
as the temperature is increased.
For reference, the distribution of energy of free electrons at 0.001 °K is shown with bold purple lines.