In terms of the Fermi energy, the density of states can now be written as, 3N 263/28¹/2 g(E) = Using this simplified expression, show that the average energy of an electron in a free-electron gas is given by, 3 (E) EF
In terms of the Fermi energy, the density of states can now be written as, 3N 263/28¹/2 g(E) = Using this simplified expression, show that the average energy of an electron in a free-electron gas is given by, 3 (E) EF
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Transcribed Image Text:For 3D free electron gas, the density of states counts the number of degenerate electron states dn per
energy interval de around a given energy E as
g(E)
=
dn
dE
3
(2m₂)²V 1
-E2
2π²ħ³
At absolute zero temperature, N electrons can fill up all low-lying energy levels (following Pauli
exclusion principle) up to a given energy level E called Fermi energy.
In terms of the Fermi energy, the density of states can now be written as,
3N
263/2
2E
g(E)
-E¹/2
Using this simplified expression, show that the average energy of an electron in a free-electron gas is
given by,
3
(E) = { Ef
1 EF
Clue: interpret g(E)/N as probability density for E, and thus (E) = √³ Eg(E)dE
NJO
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