Symmetry of solids/Topic 5

From WolfWikis

Exploiting the normal modes of the translation group

As decomposing into irreps makes things more tractable for hydrogen wave functions or molecular vibrations, let us see if we can exploit the irreps of the translation group in a comparable way.

Let us consider a series of N data points that were sampled at regular intervals, e.g. one point per second. We could call this the sampling frequency: 1 point per second. If all points are the same we have a system with perfect translation symmetry (assuming closure of course).

This is basically the analog of a CO₂ molecule at rest: it also has perfect symmetry. However if we make the molecule vibrate it could stretch asymmetrically or even bend, distorting its symmetry. As said before we can always decompose such motion in terms of the normal modes (read: irreps) of the symmetry group of the molecule.

Can we do the same for our translation symmetry system. The answer is yes.

A simple example, the N=4 case

Let us take a very small translation group T⁴, with only four elements: E,t,t²,t³. It has four irreps and the basic root-of-one to generate them all is ε=exp(2πi/4). This is the imaginary number i. Its quare is i.i= -1 and its third power is i.i.i=-i. Adn of course its four power i⁴=1. We can assign these values to represent the element t and find the other characters by multiplication. Thus the four irreps look like

This table will be familiar to anyone that has looked at the character table for the point group C₄. In fact it is identical, because the elements of the two groups have the same multiplication table. This is a direct result of the closure applied to obtain a translation group.

Another important observation is that two of the irreps are real: Γ₀ and Γ₂. The other two are each other's complex conjugates. We could also have generated Γ₃ from ε^-1, because that also produces -i. This is why Γ₃ is often denoted as Γ_-1 and Γ_-1=Γ*₁.

Decomposing a signal

Suppose we sample a signal four times at regular intervals, say one second and get:

It will be clear that this data set has full translation symmetry on the time axis. But just to make sure, let's decompose it by multiplying each value with the characters in the character table and sum them:

This calculation shows that the data indeed transform as the totally symmetric irrep Γ₀. The other irreps are not represented in the data.

As soon as the data show variation with time we can use the same calculation to decompose them is the 'normal modes' (irreps) of our translation group. For example:

This gives:

We could look at the final number as an amplitude with which the particular 'mode' (irrep) is present in the data. In this case some of the amplitude of the totally symmetric irrep was 'stolen' by the irrep Γ₂. In addition we note that the remaining magnitude of the totally symmetric one amounts to four (i.e. N) times the average over the data set. Using the amplitudes we can now decompose our data into two superposed signals:

Discrete Fourier Transforms

The above decomposition is an example of a DFT: the Discrete Fourier Transform, albeit a rather special one because the complex irreps are still empty so that the result is real. In general that is not the case and to work with DFT's we must examine that a little more closely.

This is no luxury as the algorithm that Cooley and Tuckey bequeathed upon mankind in the 1960's to do it fast: the Fast Fourier Transform FFT has made this type of decomposition a ubiquitous feature of data processing and many scientific measurements rely upon it.