Phonon Modeling
In order to completely understand the phonon modes, here are five animations of the operations from the chart below, which is called the Character Table. Before showing any of the animations, let's investigate the Character Table first. In the top row of the table, the first sign of Td shows the symmetry of the tetrahedral structure, and after that are the five symmetries Td structure has as we discussed earlier. The integers before each symmetry is the number of symmetry elements that specific symmetry a Td structure has. For example, there will be a total of eight C3-axes for a Td structure, four along each of the tetrahedral axes, and each counted twice for the clockwise and counter-clockwise rotation. The right column of the table is the five irreducible representations which will be animated later. The integers in the middle are the characters used. Each of the symmetries starts from a vector chosen for convenience. It is important to notice that in these animations only two of the C3's from the same axis are shown. After that, these three vectors produced from the C3 operations are included in each of the three C2-operations, that is equivalent to eight C3 operations.

Table Characters in Td Group
Td e 8C2 3C2 6S4 6σd
A1 1 1 1 1 1
A2 1 1 1 -1 -1
E 2 -1 2 0 0
T1 3 0 -1 1 -1
T2 3 0 -1 -1 1


A1
The A1 is the most basic operation because the characters of all symmetries are one. In the beginning of the A1, an arbitrary vector is rotated 120° from the C3 twice around the purple axis of rotation (one represents a clockwise and one represents a counter-clockwise rotation). Only these two will be completed, and the C2 will do the other six. The C2 is then completed; and it is followed by the six σs, which are mirrored over the plane dropped down. Lastly, six S4 are finished. After that, all the red vectors on the same atoms are summed up to form a green vector. This final illustration shows how each atom moves, and it is obvious that it makes the bonds stretched or compressed. This mode is considered a breathing mode.
Td e 8C2 3C2 6S4 6σd
A1 1 1 1 1 1
[ Download full size video (avi/73.8MB) ]

A2
The A2 is a bit different from the A1 because of the negatives on the S4's and the σ's. In the animation, this is represented by a vector that goes to the opposite direction of the vector produced by the symmetry operation. In this A2 animation, there are twelve atoms because if there is no movement for the center five atoms. Other than these, the rest of this operation is the same as the A1 shown above. Once again, the symmetries start from an arbitrary red vector that is conveniently chosen, and will rotate around the axis of rotation.
Td e 8C2 3C2 6S4 6σd
A2 1 1 1 -1 -1
[ Download full size video (avi/69.4MB) ]

E
The E is different from any other operation because there is no S4's or σ's. There is also a character of two in the identity column. Therefore, near the end of the animation, another red vector will be added, so there will be a total of two for the arbitrary vector. The same is also true for the C2. There will be another red vector added from the rotation of the arbitrary red vector so there will be a total of two. The C3 is also pointed in the opposite direction. A shadow map is added to help understand the final direction of the vectors.
Td e 8C2 3C2 6S4 6σd
E 2 -1 2 0 0
[ Download full size video (avi/31.4MB) ]

T1
The T1 is different because of the three for the identity. Near the end, two red vectors will be added to the arbitrary red vector for a total of three. There is also no C3's rotation. Otherwise, the symmetry operations are similar to above. At the end of this animation, a shadow map is used where the small "x" is the axis of rotation, and each of the three shadow vectors are perpendicular to the axis of rotation.
Td e 8C2 3C2 6S4 6σd
T1 3 0 -1 1 -1
[ Download full size video (avi/73.8MB) ]

T2
The T2 is just like T1 except the negatives on the S4 and σ's are switched to the opposite direction. Otherwise, the rest of the symmetries are self-explanatory. A shadow map is formed here to help understand the direction of the final green vectors.

Td e 8C2 3C2 6S4 6σd
T2 3 0 -1 -1 1


[ Download full size video (avi/76.9MB) ]