Environmental Engineering Reference
In-Depth Information
For discussing the rare earths, we may restrict ourselves to the case
of electrons localized around the lattice sites in a crystal. Further, we
define
r
e
=
R
j
+
r
,with
r
now being the relative position of the electron
belonging to the
j
th atom at the position
R
j
. Equation (4.1.6) may
then be written
(
Q
p
+
Q
s
)
e
−i
κ
·
R
j
,
H
int
(
κ
)=8
πµ
B
µ
n
·
(4
.
1
.
7
a
)
introducing
i
hκ
κ
×
p
e
−i
κ
·
r
e
−i
κ
·
r
.
Q
p
=
;
Q
s
=
κ
×
s
×
κ
(4
.
1
.
7
b
)
In order to calculate the matrix element
<i|
Q
p,s
| f>
,thefactor
exp(
−i
κ
·
r
) is expanded in spherical Bessel functions
j
n
(
ρ
), and with
ρ
=
κr
and cos
θ
=
κ
·
r
/ρ
,
e
−i
κ
·
r
=
∞
i
)
n
j
n
(
ρ
)
P
n
(cos
θ
)
(2
n
+1)(
−
(4
.
1
.
8)
n
=0
j
0
(
ρ
)
−
3
ij
1
(
ρ
)cos
θ
=
j
0
(
ρ
)
−
i
κ
·
r
{
j
0
(
ρ
)+
j
2
(
ρ
)
}
,
using
j
n
(
ρ
)=
ρ
/
(2
n
+ 1). The truncation of the
series is valid for small values of
ρ
,where
{
j
n−
1
(
ρ
)+
j
n
+1
(
ρ
)
}
j
n
(
ρ
)=(
ρ
n
/
(2
n
+ 1)!!)
ρ
2
/
(4
n
+6)+
{
1
−
···}
.
κ
×
p
commutes with exp(
We note that, although
κ
·
r
), it does not
commute with the individual terms in (4.1.8). Introducing this expan-
sion in the expression for
Q
p
, we find
−
i
i
,
hκ
j
0
(
ρ
)
p
+
1
(
κ
·
r
)
p
+
Q
p
=
κ
×
h
{
j
0
(
ρ
)+
j
2
(
ρ
)
}
···
which can be rearranged to read
Q
p
=
2
{
}
κ
×
l
×
κ
+
Q
p
.
j
0
(
ρ
)+
j
2
(
ρ
)
(4
.
1
.
9
a
)
We have defined
i
hκ
j
0
(
ρ
)
p
+
,
(4
.
1
.
9
b
)
1
2
h
{
Q
p
=
κ
×
(
κ
·
r
)
p
+(
κ
·
p
)
r
}
j
0
(
ρ
)+
j
2
(
ρ
)
}{
+
···
where the orbital momentum
h
l
=
r
×
p
and
h
l
=
h
l
+
c
r
×
A
e
,and
used
h
l
×
κ
(
r
×
p
)
(
κ
·
r
)
p
−
(
κ
·
p
)
r
}
κ
×
=
−
κ
×{
κ
×
}
=
κ
×{
,
where [
l
,j
n
(
ρ
)] =
0
and [
κ
×
r
,
κ
·
p
]=
0
.
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