Environmental Engineering Reference
In-Depth Information
independent form, in terms of the state vectors
LSM
L
M
S
>
=
m
l
m
s
|
<m
l
m
s
|
LSM
L
M
S
>
|
m
l
m
s
>.
(1
.
2
.
15)
The exchange and Coulomb interactions are suciently large that the
magnetic properties at all accessible temperatures are determined by the
S
and
L
states of lowest energy. These are found from Hund's rules;
S
is
maximized and, subject to this maximum
S
value,
L
is also maximized.
This results in the values for the trivalent ions shown in Table 1.1.
Tab l e 1 . 1 .
Properties of the tripositive rare earth ions.
4
f
n
Ion
+++
1)
2
J
(
J
+1) ∆(
K
)
LSJ g
(
g
−
0 a
000—
1
2
5
2
6
7
1
e
3
0.18
3150
4
5
2
r
5
1
4
0.80
3100
3
2
9
2
8
11
3
d
6
1.84
2750
3
5
4
m 6
2
4
3.20
2300
5
2
5
2
2
7
5
Sm 5
4.46
1450
6 u
330—
0
7
2
7
2
7
d
0
2
15.75
3
2
8
b
3
3
6
10.50
2900
5
2
15
2
4
3
9
y
5
7.08
4750
5
4
10
Ho
6
2
8
4.50
7500
3
2
15
2
6
5
11
Er
6
2.55
9350
7
6
11950
a
12
Tm
5
1
6
1.17
1
2
7
2
8
7
13
Yb
3
0.32
14800
4 u
000—
a
The lowest excited state in Tm is
3
F
4
at 8490K.
It is a consequence of the
Wigner-Eckart theorem
that the spin-
orbit term in (1.2.13) can be written
H
so
=
±
ζ
(
LS
)
S
·
L
,
(1
.
2
.
16)
where
rR
4
f
(
r
)
dv
π
m
2
c
2
S
ζ
(
LS
)=
dr
dr,
(1
.
2
.
17)
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