Chemistry Reference
In-Depth Information
(iv) A free
3
F ion without LS-coupling in a magnetic field H
The magnetic moments operators corresponding to the quantum states
characterized by uncoupled L and S are
m
L
¼b
e
L
b
e
S
;
m
S
¼
2
ð
10
:
115
Þ
Then, the potential energy of the magnetic dipole in the uniform
magnetic field H
¼
k H is
H
1
¼ðm
L
þ m
S
Þ
H
¼ b
e
H
ð
L
z
þ
2S
z
Þ
ð
10
:
116
Þ
with
L
z
c ¼
M
L
c;
S
z
c ¼
M
S
c; c ¼ cð
M
L
;
M
S
Þ
ð
10
:
117
Þ
so that the Zeeman energy splitting of the
ð
2L
þ
1
Þð
2S
þ
1
Þ
-sublevels in
presence of the field F will be
ð
L
M
L
L
;
S
M
S
S
Þ
D
E
ð
M
L
;
M
S
Þ¼b
e
H
ð
M
L
þ
2M
S
Þ
ð
10
:
118
Þ
Therearealtogether
ð
2L
þ
1
Þð
2S
þ
1
Þ¼
7
3
¼
21energylevels, seven
of which are still degenerate even inpresence of the field (degeneracies 2,
2, 3, 3, 3, 2, 2), as shown schematically in Figure 10.2.
(v) An LS-coupled
3
F ion in a magnetic field H
In this case:
8
<
L
¼
3
;
S
¼
1
;
J
¼
4
;
3
;
2
3
2
5
J
ð
J
þ
1
Þ
g
e
¼
ð
10
:
119
Þ
:
5
4
;
13
12
;
2
3
J
¼
4
;
g
e
¼
J
¼
3
;
g
e
¼
J
¼
2
;
g
e
¼
There are 9
þ
7
þ
5
¼
21 levels altogether, as before, but now in the
presence of a field H any degeneracy is removed.
