Page 237 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

228 Systems of identical particles
8ðv
E tot ˆ (2m e ) 3=2 E F 5=2
5h 3
which may be simpli®ed to
3
E tot ˆ NE F (8:62)
5
The average energy E per electron is, then
E tot 3
E ˆ ˆ E F (8:63)
N 5
Equations (8.57) and (8.58) are valid only for values of E suf®ciently large
and for energy levels suf®ciently close together that E can be treated as a
3
continuous variable. For a metallic crystal of volume 1 cm , the lowest energy
level is about 10 ÿ14 eV and the spacing between levels is likewise of the order
of 10 ÿ14 eV. Since metals typically possess about 10 22 to 10 23 free electrons
3
per cm , the Fermi energy E F is about 1.5 to 8 eVand the average energy E per
electron is about 1 to 5 eV. Thus, for all practical purposes, the energy of the
lowest level may be taken as zero and the energy values may be treated as
continuous.
The smooth surface of the spherical octant in n-space which de®nes the
Fermi energy cuts through some of the unit cubic cells that represent single-
particle states. The replacement of what should be a ragged surface by a
smooth surface results in a negligible difference because the density of single-
particle states near the Fermi energy E F is so large that E is essentially
continuous. At the Fermi energy E F , the density of single-particle states is
1=3
2ðvm e 3N
ù(E F ) ˆ (8:64)
h 2 ðv
which typically is about 10 22 to 10 23 states per eV. Thus, near the Fermi energy
E F , a differential energy range dE of 10 ÿ10 eV contains about 10 11 to 10 12
doubly occupied single-particle states.
Since the potential energy of the electrons in the free-electron gas is assumed
to be zero, all the energy of the mobile electrons is kinetic. The electron
velocity u F at the Fermi level E F is given by
1 2 (8:65)
F
2 m e u ˆ E F
and the average electron velocity u is given by
1 2 3 (8:66)
m e u ˆ E ˆ E F
2 5
8
ÿ1
For electrons in a metal, these velocities are on the order of 10 cm s .
The Fermi temperature T F is de®ned by the relation
E F ˆ k B T F (8:67)