Biomedical Engineering Reference
In-Depth Information
assumed the existence of a spin angular momentum vector
s
, which had similar properties
to
l
above. In order to account for the splitting into two beams, they postulated a spin
quantum number for electrons s of
1
/
2
and so a spin magnetic quantum number
m
s
of
±
1
/
2
. The spin wavefunctions are usually written α (corresponding to the +
1
/
2
spin quantum
number) and β (corresponding to the -
1
/
2
quantum number). So we write, just as for ordinary
angular momentum
1
2
+
1
h
2
1
2
ˆ
s
2
α
=
4π
2
α
1
2
+
1
h
2
1
2
ˆ
s
2
β
=
4π
2
β
(13.35)
1
2
h
2π
α
ˆ
s
z
α
=
1
2
h
2π
β
s
z
β
ˆ
=−
A problem appeared once the splittings were analysed. According to classical theory, the
spin magnetic moment of an electron should be
e
2
m
e
s
−
p
spin
=
(13.36)
whilst to get agreement with the splittings, an extra factor of (almost) 2 was needed. The
solution to the problem was the introduction of the
g
-factor, an experimentally determined
quantity that made Equation (13.36) correct. Thus we write
g
e
e
2
m
e
s
The electronic
g
e
factor is another of physical science's very accurately known constants,
having a value of
g
e
=
p
spin
=−
2.0023193043787
±
(82
×
10
−
12
)
13.10 Total Angular Momentum
When orbital angular momentum and spin angular momentum both exist in the same atom,
the magnetic moments that result from these two angular momenta interact to cause a
splitting of the energy level. The interaction that leads to this splitting is called
spin-
orbit coupling
and it couples the two into a resultant
total angular momentum
. A simple
vector model that is very similar to the model used to describe orbital angular momentum
can describe this. According to this model, the total angular momentum of an electron is
characterized by a quantum number
j
. For any given nonzero value of
l
the possible values
of
j
are given by
j
=
l
±
s
(The use of
j
as an atomic quantum number is not to be confused with the use of j for the
square root of
−
1). The rule is that the quantum number
j
must always be positive, so if
l
=
1 then
j
=
3/2 and 1/2, but if
l
=
0 then we only have
j
=
1/2).
