Chemistry Reference
In-Depth Information
5
Reducible and Irreducible
Representations
5.1
Introduction
Figure 5.1 shows the fundamental vibrations of the sound board of a guitar. The motions
of the sound board as it vibrates have been resolved using a laser interferometer to give a
contour plot of the distortions of the surface caught at an instant in time. These vibrational
motions are sustained because they correspond to the resonant frequencies of the sound
board. For the listener, the correct resonance of the instrument gives the guitar the mixture
of tones and overtones that distinguishes it from other instruments.
From a symmetry point of view the sound board of the guitar would be classified as
belonging to the point group
C
2v
: there is a
C
2
axis which runs along the line defined by
the finger board (vertical in the images of Figure 5.1), a vertical mirror plane perpendicular
to the board
σ
v
. However, the shape of the distortion
due to the vibrations does not follow all of the symmetry operations in the point group
in the same way. For example, the second vibration in Figure 5.1 involves the left- and
right-hand sides of the sound board moving in opposite directions at a given instant: if the
left-hand side is moving toward the viewer, then the right-hand side will be moving away.
This is why the centre of the sound board, along the line of the
C
2
axis, stays stationary
at all times: it is a node in this vibrational mode. If the distortion were taken into account,
both of the vertical mirror planes would no longer be symmetry elements; but rather than
assign a new point group to the distortion, we will use the irreducible representations of
the point group to describe it.
The characters under the operations of the
C
2v
point group for the Figure 5.1b vibration
are set out in Figure 5.1g. This shows that the
C
2
rotation leaves the distortion of the
sound board apparently unchanged, giving character 1, while for a reflection through either
mirror plane the motion of the vibrating sound board would appear to be reversed. This
is equivalent to multiplying the original distortions by
σ
v
and the plane of the board itself
−
1, and so this is the character
for the
σ
v
and
σ
v
operations. Along with the character of 1 under the identity operation,
