From WolfWikis

Basis functions and Brillouin zones

As interesting and fundamental as FFT's are, (you can do a lot with them!), let us return to solids. If we want to describe their electronic states or their vibrational ones, we need to find proper functions to stick into Schrödinger's equation.

We have seen before that Bloch functions like ψ= exp(ikx) will transform as the irreps of our translation group, but the number of values of the spatial frequency (aka wave vector) k is limited to N, the size of the group. For example for T⁸ the 'allowed' values of k are 0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8 and 7/8. What happens if k= 11/8 e.g.?

It is tempting to just subtract 8 and equate this case with 3/8, but the functions ψ₃= exp(i3x/8) and ψ₁₁= exp(i11x/8) are not the same functions. Both run around on the unit circle of the complex plane but one goes quite a bit faster than the other!

However what they do have in common is that upon application of the translations of the group they get multiplied with the same coefficients:

Tψ₃= Texp(i3x/8)=exp(i3[x+1]/8)= ε³.exp(i3x/8)=ε³.ψ₃

Tψ₁₁= Texp(i11x/8)=exp(i11[x+1]/8)= ε¹¹.exp(i11x/8)=ε³.ψ₁₁

(ε¹¹=ε³ if ε⁸=1!)

In other words ψ₃ and ψ₁₁ are both basis functions of the same irrep as they transform the same. Although different functions they are each other's alias from a symmetry point of view.

This phenomenon is quite general. If we add integer value Q to our k-vector we always find a different function but it does belong to the same irrep. As they belong to the same irrep they do mix in the quantum mechanical sense and a proper wave function Ψ to try would be a linear combination of them all:

Ψ= Σc_Q.exp(i(k+Q)x) for all integer Q's

We might note that Q could also be negative integers.

Brillouin zones

If we let the spatial frequency increase in steps of 1/8 we get an endless repetition of irreps:

The zone indicated in pink is known as the first Brillouin zone. As you can see the Г₄ Nyqvist irrep is indicated on both edges of the zone. Although k= -1/2 and k=+1/2 represent different wave functions they do mix because they represent the same irrep. From a symmetry point of view the choice of what is the Brillouin zone is arbitrary. It simply denotes the point on the complex circle we have taken as zero angle. Physically however the choice presented here is the most sensible.

The yellow part of our 1D reciprocal space is known as the second Brillouin zone, the white part outside its borders the third etc.

Aliasing

The zone structure of reciprocal space has important consequences for FFT treatments of experimental data. If we study a fast phenomenon with a sampling frequency that is not sufficient to keep up with the phenomenon we get what is known as aliasing. In the sampled data the phenomenon will appear to happen at the wrong speed. E.g. if the real frequency is 11/8 in the above example, we will see it move at a frequency of 3/8, its equivalent inside the first Brillouin zone.

This effect was well known to early movie-makers who tried to show propellers of starting planes. The propeller first seems to turn right, then jump up and down (at the Nyqvist frequency) then move backwards then stand still when its speed has reached Г₀ in the second zone, then move forward again etc.

Aliasing dictates that sound recordings need to be sampled with pretty high sampling frequencies, otherwise those components of the recorded sound that are above the Nyqvist frequency (half the sampling frequency) will get aliased and the sound will distort in often rather unpleasant ways.

Time reversal

The experience of the folks in Hollywood illustrates an important point about the irreps above Nyqvist. We had first numbered them with positive integers but later renamed them with negative ones to better represent the conjugate symmetry. Here they resurface as true negative frequencies, as in: things moving backwards in time. This is why people sometimes introduce the concept time reversal.

However, we should be careful not to throw time and space in one basket, when talking about time reversal. If we invert the spatial frequency k to -k, this is equivalent of a spatial inversion center operation i, not time reversal.

Of course for objects traveling at constant speed there is a direct link (their velocity) between temporal and spatial frequency. Just think of what the (time) frequency is, that you pass lamp posts at a spatial frequency 100ft apart while traveling at 50 mph along the highway.

For magnetic structures however time reversal is something else than spatial inversion. Time reversal means that all spins flip (because all currents go the other way) without changing the spatial structure. In other words time reversal R takes t => -t, but i takes k=(x,y,z) to -k=(-x,-y,-z). It is the addition of the element R that expands the space groups to the magnetic double groups.

Resolution and contrast harmonics

Another way of looking at the frequencies of the DFT is to consider them as a resolution. This is easy to see if your data are represented as an image, each point a pixel with a grey scale. The Nyqvist irrep always contrasts one point with its closest neighbour: +-+-+-+-+ etc. This is the highest resolution your data set allows. If you want more, you need more pixels, data points, i.e. a higher sampling rate.

Conversely if we look at the irrep Γ₁ it represents the lowest resolution. If your data form a picture it contrasts the left half of your picture with the right one as it superposes a single cosine/sine wave over the whole image. We could also call that the first harmonic, Γ₂ being the second etc.

Γ₀ represents the average over the whole image, i.e. its background level. Symbolically we can show the contrast patterns as:

This picture also invites us to look at the harmonics in a slightly different way. They each have their own 'repeat' unit, say the distance d between one blackest point and the next. For Γ₁ that would be d=32 steps in the above figure, for Γ₂ d=16 and for Γ_Nyqvist it is d=2 steps. In crystallography people often use this 'd-spacing' terminology.