Environmental Engineering Reference
In-Depth Information
We shall neglect the spin-orbit coupling of the conduction electrons,
and assume their wavefunctions to be the Bloch functions
ψ
n
k
(
r
)=
u
n
k
(
r
)
e
i
k
·
r
=
ψ
n
k
(
r
−
R
i
)
e
i
k
·
R
i
,
(5
.
7
.
1)
independent of the spin state
σ
.
u
n
k
(
r
)=
u
n
k
(
r
−
R
i
)hastheperiod-
icity of the lattice, and
n
is the band index. The Hamiltonian of the
conduction electrons in
second quantization
is
H
s
=
n
k
σ
ε
n
k
c
n
k
σ
c
n
k
σ
,
(5
.
7
.
2)
where the index
s
is conventionally used for the conduction electrons
even though, as we saw in Section 1.3, they have predominantly
d
char-
acter.
c
n
k
↑
annihilates a spin-up electron in the band-
state (
n
k
), and they are Fermi-operators which satisfy the
anticommu-
tation relations
creates and
c
n
k
↑
c
n
k
σ
,c
n
k
σ
}≡
c
n
k
σ
c
n
k
σ
+
c
n
k
σ
c
n
k
σ
=
δ
nn
δ
kk
δ
σσ
{
(5
.
7
.
3)
c
n
k
σ
,c
n
k
σ
}
{
=
{
c
n
k
σ
,c
n
k
σ
}
=0
.
An exposition of second quantization may be found, for example, in
White (1983). The exchange interaction between a pair of electrons is
−
2
I
s
1
·
s
2
,where
I
is the exchange integral. If
s
1
is the spin of a 4
f
electron at site
i
, then the sum over all the 4
f
electrons at this site gives
4
fel.
−
2
I
s
1
·
s
2
=
−
2
I
S
i
·
s
2
=
−
2
I
(
g
−
1)
J
i
·
s
2
,
where
I
is an average value of the exchange integral for the 4
f
elec-
trons, and states other than those in the ground-state
J
-multiplet are
neglected. The spin-density of the conduction electrons at
r
may be
expressed in second-quantized form so that, for instance,
s
2
z
(
r
)=
nn
c
n
k
↑
c
n
k
↓
.
ψ
n
k
(
r
)
ψ
n
k
(
r
)
2
c
n
k
↓
c
n
k
↑
−
(5
.
7
.
4)
kk
The
sf-exchange interaction
is determined by the following exchange
integral:
d
r
1
d
r
2
ψ
n
k
(
r
1
)
φ
4
f
(
r
2
−
R
i
)
e
2
|
r
1
−
r
2
|
ψ
n
k
(
r
2
)
φ
4
f
(
r
1
−
R
i
)
1
N
I
(
n
k
,n
k
)
e
−i
(
k
−
k
)
·
R
i
,
=
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