Page 235 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

226 Systems of identical particles

8.5 The free-electron gas
The concept of non-interacting fermions may be applied to electrons in a metal.
A metal consists of an ordered three-dimensional array of atoms in which some
of the valence electrons are so weakly bound to their parent atoms that they
form an `electron gas'. These mobile electrons then move in the Coulombic
®eld produced by the array of ionized atoms. In addition, the mobile electrons
repel each other according to Coulomb's law. For a given mobile electron, its
Coulombic interactions with the ions and the other mobile electrons are long-
ranged and are relatively constant over the range of the electron's position.
Consequently, as a ®rst-order approximation, the mobile electrons may be
treated as a gas of identical non-interacting fermions in a constant potential
energy ®eld.
The free-electron gas was ®rst applied to a metal by A. Sommerfeld (1928)
and this application is also known as the Sommerfeld model. Although the
model does not give results that are in quantitative agreement with experi-
ments, it does predict the qualitative behavior of the electronic contribution to
the heat capacity, electrical and thermal conductivity, and thermionic emission.
The reason for the success of this model is that the quantum effects due to the
antisymmetric character of the electronic wave function are very large and
dominate the effects of the Coulombic interactions.
Each of the electrons in the free-electron gas may be regarded as a particle
in a three-dimensional box, as discussed in Section 2.8. Energies may be
de®ned relative to the constant potential energy ®eld due to the electron±ion
and electron±electron interactions in the metallic crystal, so that we may
arbitrarily set this potential energy equal to zero without loss of generality.
Since the mobile electrons are not allowed to leave the metal, the potential
energy outside the metal is in®nite. For simplicity, we assume that the metallic
3
crystal is a cube of volume v with sides of length a, so that v ˆ a .Asgiven
by equations (2.82) and (2.83), the single-particle wave functions and energy
levels are
r 
8 n x ðx n y ðy n z ðz
ˆ sin sin sin (8:55)
v a a a
ø n x ,n y ,n z
h 2 2 2 2
ˆ (n ‡ n ‡ n ) (8:56)
E n x ,n y ,n z 2 x y z
8m e a
where m e is the electronic mass and the quantum numbers n x , n y , n z have
values n x , n y , n z ˆ 1, 2, 3, ... :
We next consider a three-dimensional cartesian space with axes n x , n y , n z .
Each point in this n-space with positive (but non-zero) integer values of n x , n y ,