Biomedical Engineering Reference
In-Depth Information
13.7.3 The d-orbitals
=
Similar comments apply to the
l
2 orbitals. There are five in all, four of which come in
complex pairs. We combine together the corresponding values of
m
l
and
−
m
l
just as for
the p-orbitals to give real equivalents. Table 13.4 records the 3d orbitals.
Table 13.4
The 3d orbitals
n
,
l
,
m
l
Symbol Normalized wavefunction
Z
a
0
3
/
2
1
exp
3
ρ
2
3 cos
2
(θ )
1
81
√
6
π
ρ
3, 2, 0
3d
zz
−
−
√
2
81
√
π
Z
a
0
3
/
2
ρ
2
sin
(θ )
cos
(θ )
cos
(φ)
exp
3
ρ
3, 2,
±
1
xz
−
√
2
81
√
π
Z
a
0
3
/
2
ρ
2
sin
(θ )
cos
(θ )
sin
(φ)
exp
3
ρ
3d
yz
−
Z
a
0
3
/
2
ρ
2
sin
2
(θ )
cos
(
2
φ)
exp
3
1
81
√
2
π
ρ
3, 2,
±
2
x
2−
y
2
−
Z
a
0
3
/
2
ρ
2
sin
2
(θ )
sin
(
2
φ)
exp
3
1
81
√
2
π
ρ
3d
xy
−
Once again these are usually represented as contour diagrams (Figures 13.7 and 13.8).
13.8 The Stern-Gerlach Experiment
If we suspend a compass needle in the Earth's magnetic field, it aligns itself along the
magnetic field lines. Acompass needle is an example of a magnetic dipole, and the strength
of the interaction between the compass needle and this external magnetic field is determined
by the
magnetic dipole moment
p
m
. This is a vector quantity pointing by convention from
the south pole of the needle to the north pole. The interaction between the dipole and the
field is determined by the magnetic potential energy
U
m
=−
p
m
.
B
(
r
)
(13.32)
where the position vector
r
is the position of the dipole and
B
the magnetic induction. If
B
is uniform and so does not depend on
r
, the gradient is zero and so the force is also zero,
in accord with experiment.
Around 1820, Oersted discovered experimentally that electric currents could exert
forces similar to those exerted by permanent magnets, for example on a compass needle.
Figure 13.9 shows a simple current loop located in the
xz
plane and carrying steady current
I
. If the loop is flat, the dipole is perpendicular to the plane and if we consider points on
the axis far away from the loop it turns out that the magnitude of
p
m
is
p
m
=
IA
where
A
is the area of the loop (which need not be circular).
