Chemistry Reference
In-Depth Information
m
A
I
Figure 6.1
An electron moving in a classical orbit of area
A
has magnetic moment
m
=
I
A
due to its orbital motion, where
I
is the current flowing, and the
moment points perpendicular to the loop.
6.3 Magnetic moment of the electron
Consider an electron with charge
e
moving in a classical circular orbit of
area
A
(fig. 6.1). It can be shown that the electron has a magnetic moment
m
due to its motion, given by
=
m
I
A
(6.8)
where
I
is the current flowing in the closed loop, and the vector
A
has
magnitude equal to the loop area and points perpendicular to the loop,
with the current flowing clockwise when looking along the direction of
A
.
The current
I
equals the charge passing any point on the loop per unit time,
and is given by
2
I
=−
e
(6.9)
π
where
is the angular frequency of the electronmotion, and theminus sign
follows from the negative charge of the electron. Substituting in eq. (6.8),
the magnitude of the electron's magnetic moment due to its orbital motion
is then given by
ω
e
2
r
2
m
=−
·
(π
)
π
e
2
m
e
·
(
−
m
e
r
2
=
ω)
(6.10)
m
e
r
2
where
m
e
equals the electron mass and
Q
is its angular momen-
tum. This result remains true in quantum mechanics, where the angular
momentum is quantised, as shown in Appendix B, with orbital quantum
number
l
. If we consider an externally applied field,
B
, directed along the
z
-direction, then the component of angular momentum
Q
z
along the field
direction is given by
=
ω
Q
z
=
l
z
(6.11)
