ABSTRACT
Since the second term on the left-hand side of Eq.(40.2), ( ) ,ee jU r depends on the
functions of all the other electrons, a self-consistent procedure, similar to that outlined before for the Hartree equations (36.52) should, in general, be used. It is usual to rewrite Eq.(40.2) in the compact form
( ) ( ) ( )rrrU me
GGG= εψψ =⎥⎦ ⎤⎢⎣
⎡ +∇− 2 2
2 (40.3)
where ( )rU G represents the last two terms on the left-hand side of Eq.(40.2), that are assumed to be dependent on the position rG of a given electron only. In other words, the N-electron system represented by Eq.(40.1) can be treated as N one-electron systems described by Eq.(40.3). We will consider a metal to consist of positive ions located on lattice sites, bathed in a sea of conduction electrons, and take a particular example where N valence electrons move in a structure containing N monovalent ion cores. Then the second term in Eqs.(40.2), which describes the positive interaction between equal amounts of uniform electronic charge, and the third term, which represents the negative interaction between the uniform ionic charge and an equal amount of electronic charge, will cancel. We can therefore drop ( )rU G in Eq.(40.3), in the zeroth order of approximation, to obtain the oneelectron, free-electron Schrödinger equation as
( ) ( )rr me
GG= εψψ =∇− 2 2
2 (40.4)
Although this free-electron approximation is a drastic assumption, Eq.(40.4) provides an excellent model for the physical properties of metals. The plane wave solutions of Eq.(40.4), which read
( ) rkier GGG ⋅= V 1ψ (40.5)
are normalized with respect to the volume V of the crystal and correspond to energy eigenvalues given by
em
k 2
22==ε (40.6) The momentum eigenvalues of the free electrons result from
( ) ( )rke i
rp rki G G
==GG G G ψψ =⎥⎦ ⎤⎢⎣
⎡∇= ⋅ V 1ˆ
in the form
kp G
=G = (40.7)
If we assume the volume V as a finite parallelepiped of sides , and ,Pa Qb Sc GG G we
obtain the free-electron gas model where, since the interaction between electrons is
neglected, we only need to describe the motion of one electron at a time inside an impenetrable rectangular box. Although the physical situation requires that the wave function ( )rGψ vanishes at the walls of the box, we shall use instead the periodic boundary condition which requires that the solutions should be periodic over the distances Pa, Qb and Sc, respectively, and hence
( ) ( ) ( ) ( )rcSrbQraPr GGGGGGG ψψψψ =+=+=+ Substituting Eq.(40.5), it follows that
2 , 2 , 2Pk a p Qk b q Sk c sπ π π⋅ = ⋅ = ⋅ =G G G GG G where p, q and s are arbitrary integers. Comparing these equations with the definitions of the reciprocal base set (39.10), it is clear that k
G may be considered to be a vector in
of the form -spacek G
*p q sk a b P Q S
∗ ∗= + + cG GG G
Note that, for a vector in k G
-space, the components p/P, q/Q and s/S can be fractions of an integer. If they are integers, say / , / and ,h p P k q Q l s/S= = = G becomes a reciprocal lattice vector, as defined by Eq.(39.8) and denoted by
k .hklK
G In the unit cell
volume ( )∗∗∗∗ ×⋅= cba GGG0V of -space, defined as in Eq.(39.3) which gives for the direct lattice, Eq.(39.3), there are PQS
k G
0V k G
-states of the free electron, so that the number of states per unit volume of k
G -space is
( ) ( ) ( ) ( )
V V V 2 2
PQSPQS PQS a b c 3π π∗ ∗ ∗ ∗= = =⋅ ×GG G (40.8)
where V is the volume of the crystal. Since the states in k G
-space are uniformly distributed, the number of states with wave vector up to an arbitrary value of ,kk =G that is, with energy less than is ,2/22 emk==ε
( ) 2/3
6 V
6 V
2 V
3 4 ⎟⎠
⎞⎜⎝ ⎛==⎟⎠
⎞⎜⎝ ⎛
= ε
πππ π emkk i.e.
