Biomedical Engineering Reference
In-Depth Information
use the factor of
1
/
2
in their definition. Quadrupole moments are gauge invariant provided
the electric dipole moment and the charge are both zero.
Figure 1.11 shows an octahedrally symmetrical array of point charges. Each point charge
has magnitude
Q
, apart from the central charge that has magnitude
−
6
Q
in order to make
the system neutral. The distance between each axial point charge and the central one is
a
.
Q
a
Q
Q
-6
Q
Q
Q
Q
Figure 1.11
Octahedral charge distribution
If I choose to direct the Cartesian axes along the symmetry axes, then the second moment
matrix is
Qa
2
⎛
⎞
200
020
002
⎝
⎠
q
e
=
whilst the quadrupole moment matrix is zero.
If I now reduce the symmetry of the charge distribution by placing charges 2
Q
along the
vertical axis (taken for the sake of argument as the
x
-axis) and
−
8
Q
at the centre (to keep
the electrical balance), the second moment matrix becomes
Qa
2
⎛
⎞
400
020
002
⎝
⎠
q
e
=
whilst the quadrupole moment matrix is now
Qa
2
⎛
⎞
200
0
⎝
⎠
−
e
=
10
00
−
1
The electric quadrupole moment measures deviations from spherical symmetry. It is zero
when the charge distribution has spherical symmetry.
It always has zero trace (because of the definition), but it is not always diagonal.
Nevertheless, it can always be made diagonal by a rotation of the coordinate axes.
