Most minerals have structures that are lower symmetry than cubic. As a consequence, symmetrical anions such as SiO42- and SO42- often find themselves bound in local environments that have lower symmetries than the full symmetry of the tetrahedral anionic unit. In such an environment, the tetrahedral unit will vibrate with energies different from those it would have in a totally symmetric environment. We must learn to deal with the vibrational energies of these units in low symmetry structures.
First, we will consider an isolated SO42- unit that has textbook perfect tetrahedral symmetry.
a) Verify that it belongs to the point group Td.
We will first determine the number of fundamental vibrational modes it has when isolated from the rest of the world. Then we will “place” the group into a series of sites of progressively lower symmetry and calculate how the number and symmetry of vibrational modes should change. Finally, we will compare these predictions with some actual data obtained on a suite of selected sulfate minerals.
b) Consider the isolated SO42-
group: Show that the application of the symmetry operations of Td
upon the five atoms in the SO42- generates
the reducible representation:
| E | C3 | C2 | S4 | sd |
| 15 | 0 | -1 | -1 | 3 |
Reduce the representation above into its component symmetry species. If you are finished with this part, check your answer with the TA before proceeding with the rest of this problem or at
http://minerals.gps.caltech.edu/GE214/TA/index.htm
d) The character tables for Td can be used to determine which of these representations correspond to pure translation and pure rotation. Rotations are represented by Rz for rotation about the (molecular) z-axis, etc. Translations are designated in the same column by the cartesian coordinates x, y, and z. Specifically, the translations in the x, y, and z directions are degenerate and comprise a basis for the representation of Td.
Subtract out the two representations of translations and rotations from the symmetries of the genuine vibrations. Remember, a T state has 3 components and an E state has 2.
e) Draw a picture of the SO42- ion and indicate the direction of motion of the atoms in the A1 vibration (which is also called n1, the symmetric stretch).
f) From the character tables determine which modes are infrared active. There will be the modes whose symmetry is the same as one of the three coordinates, x, y, and z. Note that some authors call a T mode by the letter F (e.g. T1 becomes F1.)
g) Next allow one and only one of the oxygens to interact strongly with a cation so as to destroy the effective Td symmetry.
h) Determine the point group of this new unit. Next, determine the number of components that are generated in the lower symmetry by the splitting of the original bands and the symmetry of each of these individual components. How many bands are infrared active? You can either work it out completely if you wish, or you may use the correlation tables such as those in Wilson EB, Decius JC, and Cross PC (1955) Molecular Vibrations. McGraw-Hill. [that is on page 340 in my 1980 Dover paperback reprint]. This table tells you what happens to vibrations of different symmetry when you decrease the symmetry of the vibrating unit. A copy of this table for the Td group follows
Determine again the manner in which
the original tetrahedral bands split.
j) Below, you will find three spectra which show the infrared absorption spectrum in the spectral region which corresponds to n4
(pictured below) one of the T2 modes in Td symmetry.
Based on the results of the analysis that you have just
completed, explain why the three spectra differ as they do. Look
up the structure of these minerals to see if you can correlate
the infrared results with the X-ray structural data.
Spectral data are from Omori K, Kerr PF (1964) Science Report. Tohoku University, 9,1.