Environmental Engineering Reference
In-Depth Information
to leading order.
U
k
(
n, n
) = 0 and, for
n
=
n
,
I
(
n
k
,n
k
)
ε
n
k
−
U
k
(
n, n
)=(
g
−
1)
J
z
.
(5
.
7
.
11
b
)
ε
n
k
The (approximately) diagonal form of (5.7.9) implies that the thermal
expectation values are
c
n
k
σ
c
n
k
σ
=
δ
nn
δ
kk
δ
σσ
f
n
k
σ
,
(5
.
7
.
12
a
)
where
1
e
β
(
ε
n
k
σ
−µ
F
)
+1
f
n
k
σ
=
(5
.
7
.
12
b
)
is the
Fermi-Dirac distribution function
and
µ
F
is the chemical poten-
tial, equal to the Fermi energy
ε
F
in the temperature regime in which
we shall be interested. The moment density is determined by (5.7.4),
and introducing the new Fermi operators and using (5.7.12), we obtain
c
.
el
.
=
µ
B
nn
ψ
n
k
(
r
)
ψ
n
k
(
r
)
c
n
k
↓
c
n
k
↑
c
n
k
↓
µ
z
(
r
)
c
n
k
↑
−
kk
,
=
µ
B
nn
ψ
n
k
(
r
)
ψ
n
k
(
r
)
{
δ
nn
+
U
k
(
n
,n
)
}
(
f
n
k
↑
−
f
n
k
↓
)
k
f
n
k
↓
)
.
+
U
k
(
n, n
)(
f
n
k
↑
−
(5
.
7
.
13)
The uniform, averaged part of this moment density can be obtained
by an integration of eqn (5.7.13) over space, and remembering that the
wavefunctions are orthogonal and normalized, we find the magnetic mo-
ment of the conduction electrons per ion to be
N
n
k
f
n
k
↑
−
f
n
k
↓
.
1
µ
z
c
.
el
.
=
µ
B
(5
.
7
.
14)
We note that, in addition to this uniform polarization of the conduction
electrons, there is a spatially non-uniform component of the polarization
density with the periodicity of the lattice. This non-uniform component
reflects the variation in the electronic density, including the perturba-
tive changes due to the
interband
contributions proportional to
U
k
(
n, n
).
Furthermore, when the spin-orbit coupling of the conduction electrons
is of importance, the interband coupling may induce a positional depen-
dence in the direction of the spin polarization.
In order to obtain order-of-magnitude estimates of the exchange
effects, we introduce a reasonable but somewhat crude approximation
for the exchange integral, which is due to Overhauser (1963) and has
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