Chemistry Reference
In-Depth Information
the
n
electrons in the atom, with
n
S
=
s
zi
(6.28)
i
=
1
an
d the to
tal spin angular momentum then being of magnitude
√
S
(
S
+
1
)
.
2
Subject to the constraint of the first rule, the electrons are then distrib-
uted among the possible orbital angular momentum states. Each such
state has a quantumnumber,
l
z
, associatedwith its angularmomentum
component along the quantisation direction. The electrons are distrib-
uted among the possible
l
z
states so that
L
=|
l
z
is maximised, with
the
resultan
t total orbital angular momentum then being of magnitude
|
√
L
(
L
+
1
)
.
3
The orbital and spin angular momenta are coupled to each other in an
isolated ion, and we can define a total quantum number
J
S
associated with this total angular momentum. The magnitude of
J
is
given by
=
L
+
J
=|
L
−
S
|
(6.29a)
when the shell is less than half-full (in which case
L
and
S
point in
opposite directions), and
J
=
L
+
S
(6.29b)
when the shell is more than half-full (in which case
L
and
S
point along
the same direction).
Using these rules, the ground state of any isolated ion can be determined,
with the
z
-component of the magnetic moment given by
m
z
=−
g
µ
B
J
z
(6.30)
where
g
is called the Landé splitting factor, and is given by
3
2
+
S
(
S
+
1
)
−
L
(
L
+
1
)
g
=
(6.31)
2
J
(
J
+
1
)
The Landé splitting factor reflects the relative contribution of spin and
orbital motion to the total angular momentum. If we have a pure spin
state, for which
L
=
=
=
0and
J
S
, then eq. (6.31) gives
g
2, as we had
earlier in eq. (6.14), while when
S
=
0, and
J
=
L
,wefind
g
=
1, as we had
in eq. (6.13) for pure orbital angular momentum.
