Brillouin zones definition

The Online Dictionary of Crystallography is maintained by the IUCr Commission for Crystallographic Nomenclature. This forum allows members of the community to comment on current definitions or propose new ones.

Brillouin zones definition

Postby gchapuis » Tue May 17, 2011 6:30 pm

Referring to the definition of the Brillouin zones on the Online dictionary ( Mike Glazer submitted the following message to André Authier

I have been looking at the IUCr Online Dictionary and I would like to suggest a change to the section on Brillouin Zones. The point is that it is not correct to DEFINE the BZ as a Wigner-Seitz construction, as the whole idea of the BZ is to contain all the allowed quantum wave states. Thus it is really just a unit cell in reciprocal space and can therefore be constructed in an infinite variety of ways. The Wigner-Seitz construction is normally used because it always gives a primitive unit cell (i.e. the smallest volume) that when isolated shows the full symmetry of the reciprocal lattice from which it is constructed. However, this is fine for high symmetry cases but really not useful nor recommended for low symmetry cases. For example the shape of the Wigner-Seitz cell for monoclinic lattices is complicated and depends on the particular mix of lattice parameters and beta angle. In such a circumstance it is much easier and more reliable to use a conventional parallepiped cell to describe the wave states. Triclinic is even more horrible. Even in tetragonal symmetry the shape depends on the c/a ratio. This has been pointed out many years ago for example in the classic book by Bradley and Cracknell. The Bilbao Crystallographic Server also for instance in the section dealing with BZ’s doesn’t bother with Wigner-Seitz constructions for the low-symmetry cases.

If you agree, can we alter this section?


Here is André's answer

Dear Mike,

Thank you for your message and your remarks. You are of course right and the definition could do with some improvement. However,

- Kittel does define de B.Z. as the Wigner-Seitz cell

- The solid-state textbooks that I have at hand (Altman, Blakemore) do indeed say that the primitive cell of the reciprocal lattice is sufficient, but that it is not convenient and that the centred primitive cell is usually more convenient.

- After all, the Brillouin Zone (as it is defined) is what one is led to automatically when looking for the solutions of the propagation of a wave in a bounded solid

- It is the definition given by Brillouin in his original paper (see attached)

Personally, I would keep the definition as it is, adding some comments along the lines you have written. If you want to be complete, you could say that this construction is of course much older than solid-state physics, it is due to Dirichlet-Voronoi.


With my best wishes,


You are kindly invited to give your opinion of the definition given by the Online dictionnary
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Re: Brillouin zones definition

Postby glazer » Sun May 22, 2011 11:15 am

I agree that the original definition By Brillouin was to use the Wigner-Seitz construction, but I do not think he was considering the reality of trying to do this for low-symmetry cases, which is much too complicated for mortals. Once one realizes that the whole point of having a Brillouin zone is simply to have a region of reciprocal space that contains all the allowed wave states, then one is simply talking about a unit cell in reciprocal space.

The Wigner-Seitz construction is just a way of always getting a primitive unit cell, whether in real or reciprocal space. It has two advantages over the normal parallelepiped type unit cell that we use to describe crystal structures.

1. It is always primitive and therefore the smallest volume in reciprocal space for the purposes of counting wave states
2. Once isolated and viewed on its own one easily see the full symmetry of the lattice from which it is taken. So for instance if we do a W-S construction on an F-centred cubic lattice we get a shape in which we can immediately see the four 3-fold axes that tell you it is cubic. A parallelepiped primitive cell taken from such a lattice does not easily show the four 3-fold axes.

It is the symmetry advantage No 2 that is thrown away when you consider a low-symmetry lattice e.g. monoclinic and especially triclinic. If you do a W-S construction of a triclinic lattice you get a most complex shape (in fact I dont think I have ever seen anyone create one), the details of which depend on the ratios of the axial lengths and angles. Yet this cell is primitive and contains the same wave states as a conventional primitive parallepiped unit cell, which is much easier to deal with.
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Re: Brillouin zones definition

Postby tedjanssen » Sun May 22, 2011 3:23 pm

Dear contributors to the discussion,

in principle a Brillouin zone is just the unit cell of the reciprocal lattice. As such an infinite number of choices may be made. Characterising quantum states in a crystal is done with wave vectors form the reciprocal unit cell. The Wigner-Seitz is indeed often quite complicated, but it has one very important property. The borders of this cell are the places where gaps occur in spectral curves, because the points of these borders are connected to equivalent points via a reciprocal lattice vector. Of course, in calculating dispersion curves it is not very practical to let the q-vectors run over a
Wigner-Seitz cell. Instead, one uses a paralelopepid or something similar. If computer power is not an important factor, one may do this even with centered lattices. However, the physical properties are more related to the W-S presentation.

Strongly related to this argument is the fact that for irreducible representations of space groups, indicated by a star of wave vectors and an irreducible representation of the point group, projective (or double-valued) representations of the point groups ony occur if the star of wave vectors is ON THE BOUNDARY OF the BZ, at least if one uses as definition of BZ the unit cell of the Wigner-Seitz construction. This has to do with the degeneracy of the bands.

Finally, the symmetry of the BZ as WS cell reflects the point group symmetry of the crystal, and of its physical properties.

Therefore, I am in favour of keeping in the definition of Brillouin zone the Wigner-Seitz construction. For other choices
one may use the term 'unit cell of the reciprocal lattice'.

Best regards,


Re: Brillouin zones definition

Postby glazer » Wed May 25, 2011 4:45 pm

By the way the diagram showing the 2nd zone in the dictionary page is completely wrong. It doesn't even have the right symmetry. :?:
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Re: Brillouin zones definition

Postby glazer » Wed May 25, 2011 5:24 pm

Sorry to keep posting on this before others have had a chance to say anything but I found something interesting in the book by Brillouin 1946 Wave Propagation in Periodic Structures. He says
"We must now formulate a rule for choosing the area to which the wave-vectors are to be confined. We might start at the origin of the basis system and confine all wave-vectors to the first elementary cell of the reciprocal lattice i.e to the parallogram bounded by (0,0), (0,1), (1,0) and (1,1). There are two disadvantages to this.

1. In the first place this method singles out the directions contained in the angle (here he refers to the acute angle between b1 and b2) as preferred directions of propagation for the waves.

2. Further, it does not require the use of the longest possible wavelength for a given disturbance and thus is inconsistent with the conventions set up for the one-dimensional case.

His solution to this is the Wigner-Seitz construction. But with regard to his first disadvantage (1) he has not realized something, namely that he need not put the origin at the corner of the elementary reciprocal cell but in the middle. It was the need to have the origin at the centre of the cell that first led him to adopt the W-S cell. As for the second point (2) he is arguing to adopt conventions used for the 1-dimensional case. But why? As I have already pointed out before when you go to lower symmetry this whole argument for the W-S approach becomes less meaningful.

By the way Brillouin in his book has a nice diagram showing the W-S construction for 1st 2nd and 3rd zones (Figure 27.4) and this is correct.
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Re: Brillouin zones definition

Postby glazer » Thu May 26, 2011 9:17 am

In answer to Ted, my objection to the dictionary wording is with one one word. The word is "is" i.e. The first Brillouin zone "is" the Wigner-Seitz cell of the reciprocal lattice. I would like to suggest a modification that I think would solve the problem.

"The (first) Brillouin zone is a unit cell in reciprocal space that is usually described by the Wigner-Seitz cell of the reciprocal lattice, especially for high-symmetry lattices and it is used to contain all the allowed quantum wave states of the crystal. This construction has the benefit of always giving a shape that has the smallest volume that at the same time displays most conveniently the point group symmetry of the lattice. However, it may be that for low-symmetry lattices a more conventional parallepiped unit cell may be preferred"

I am not sure if we need the word "first" in this.

Regarding Ted's point about the physical meaning of the edges of the BZ, this still applies in fact to conventional cells too, as this is simply a matter of description. There is a nice example given in a book by Cracknell on group theory dealing with Umklapp processes. I dont know how to upload images to this site so I shall try to describe it.
He starts with a square 2-d lattice and uses a conventional W-S construction. He has two wave-vectors that combine together to produce a third wave-vector that lies outside the W-S cell. In just about all solid state texts the argument then is that this wave-vector is :D fact describes a real wave travelling in the opposite direction (by subtracting a reciprocal lattice vector) and then this is responsible for thermal resistivity (Umklapp). Cracknell then shows another unit cell, also primitive with the reciprocal lattice point at the centre, so containing all the same wave states, but this time that third k-vector now lies inside the unit cell. So what has happened to Umklapp? The crystal still has the same thermal resistivity! You might like to think about this and raise the question about what Umklapp really means. (one thing to bear in mind is that if you take components of this wave-vector you then find a component still lies outside the new unit cell directed along one of the cell axes that coincides with one of the axes of the W-S cell, and so it has a component that is reflected back).

Isn't solid state physics fun?

By the way if you take a look at a web page of mine you will see another example of a mistake that occurs in almost all text books related to the misuse of translational symmetry where waves are concerned. :D
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Re: Brillouin zones definition

Postby tedjanssen » Thu May 26, 2011 12:16 pm

1. I agree that one can start the definition with 'a Brillouin zone is a unit cell of the reciprocal lattice', but I don't like to continue with 'usually' and then giving the WS construction. I would say that the BZ is a unit cell obtained with the WS construction. It is not essential that the cell is centered around the origin. Any reciprocal lattice vector would do, but I don't think that has any advantage.

2. In my opinion the history of the concept, and especially what Brillouin thought precisely, is not so important here. We want to have a practical definition for a notion that we can continue giving the name of Brillouin, to honour his contribution.

3. Actually, BZ boundaries, in the traditional sense, may be seen in de Haas-van Alphen measurements: they are the Fermi surface for a band with filling 2.

4. In fact, it is no problem at all to visualize a BZ using a simple computer program. However, in making calculations of dispersion curves or bands it is not practical to go to the BZ boundary. But, in that case, one cannot use the unit cell either, because you want to restrict the range of wave vectors to a fundamental region inside the BZ.

5. I am not so sure whether the concept of first, second etc. BZ is very useful. For high symmetry cells it may have some relevance, but I do not see applications of the concept in general. That implies that I think it is not useful to talk about a 'first' BZ either.

6. I agree completely that the concept 'Umklapp process' is often used in a highly mysterious manner.

7. I have never seen a representation of the difference between acoustic and optic waves as in the figure Mike showed, but it is immediately clear that it is not correct. The waves are given by the displacement vector in the unit cell, and a plane wave vector. And the displacement vectors are different if the masses are different. The movies are indeed very instructive.


Re: Brillouin zones definition

Postby tedjanssen » Thu May 26, 2011 3:32 pm

glazer wrote:By the way the diagram showing the 2nd zone in the dictionary page is completely wrong. It doesn't even have the right symmetry. :?:

That is right. Even for a cubic case it is difficult.

If one looks at the definition in the dictionary, the first BZ is defined by means of a WS construction looking only at first neighbours, and the n-th BZ using n-th neighbours. That does not work for a orthorhombic or even less symmetric lattice, because there the perpendicular planes through the midpoints of lines connecting to first neighbours, do not delimit a finite region. This illustrates why it is doubtful to use notions like n-th BZ.

Re: Brillouin zones definition

Postby glazer » Thu May 26, 2011 4:24 pm


This is what is said in Bradley and Cracknell's Mathematical Theory of Symmetry in Solids, a classic book on crystal group theory. After a discussion on the use of different cells as BZ "We therefore suggest that, while the Wigner-Seitz unit cell of reciprocal space is retained for defining the Brillioun zones of most crystals it should be replaced by the use of the primitive unit cell of reciprocal space for monoclinic and triclinic spce groups."

Now the reason that I am pushing this is that from a pedagogical point of view by saying that the BZ "is" a Wigner-Seitz cell is misleading, because it infers something special about the W-S cell. AS I have pointed out earlier for low-symmetry crystals it is an unnecessary and even confusing concept. It is more important for students learning this stuff to understand that the BZ is a region that contains all the allowed wave states, and that how you define it is arbitrary, in much the same way as in real space we can describe a crystal structure by choosing an infinite number of unit cell descriptions.

Let me give you an example of the kind of difficulty one can get into by not understanding this point and fixating on the W-S construction. Many years ago there was a lot of interest in a material called SN polymer, as a possible one-dimensional conductor. Several papers appeared on the electronic band structure, many of them conflicting. I published a paper with a Japanese colleague on the band structure which I believe was correct. We did this by not using the W-S cell (which was difficult to visualise as it was monoclinic) but by doing the calculation using a conventional cell. Because at the time the referees thought that one had to use a W-S construction we were forced to transform our results to the W-S cell. We believed that it was the confusion over the W-S constructions that caused the other authors to get their band structure wrong.

On the question of 2nd and third BZ,I agree that there is not a lot of point to them but it seems to be something that all the books like to go into. It becomes clear once one uses a reduced zone scheme (which after all reflects the translational symmetry of the crystal) the 2nd,3rd etc zones simply refer to excited states of the ground states seen in the 1st zone. Once again,as you rightly point out, these higher zones are difficult to get right even for high-symmetry crystals, Imagine how difficult it would be for monoclinic or triclinic crystals! So here again you see that by insisting on "defining" the BZ through the W-S construction we are are propagating problems that are not necessary. :D
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Re: Brillouin zones definition

Postby tedjanssen » Fri May 27, 2011 8:20 am

Reaction to Mike's post:
I don't think pedagogical arguments are sufficient reason for a change of the definition of a notion that is deeply rooted in the physical literature. Moreover, in principle, the common definition of BZ is not difficult. It sometimes becomes difficult to visualise and to specify the vectors inside the BZ. But then one can use the notion of unit cell.

In my opinion, the BZ is not just a unit cell. It has special properties (e.g. that additional degeneracies occur at the BZ boundary only) and it has a long history. Moreover, because the frequencies f(q) of an excitation is a plane in 4D, one never plots these frequencies above the BZ. One chooses a line and plots the frequencies along that line. Then it is not important what is the relation between this line and the BZ as long as begin and end points are clear.

Therefore, I propose to mention in the dictionary three notions.

Brillouin Zone A special choice of the unit cell of a reciprocal lattice, defined as the set of vectors closer to the origin than to any other
reciprocal lattice point.

Note 1: For symmetric lattices (square and hexagonal in 2D and cubic in 3D) one uses sometimes the notion of n-th BZ. The first BZ is just the BZ as defined above. The n-th BZ is th set of points that are 1) outside the (n-1)-th BZ and 2) closer to the origin than to any n-th neighbour of the origin. [Since this notion is sometimes used in the literature, is should be mentioned in a dictionary, even if one has doubts about its usefulness.]

Note 2: Points on the boundary of the BZ should be chosen such that no pair of such points is connected by a reciprocal lattice vector.

Note 3: Putting a copy of the BZ in each reciprocal lattice point, the reciprocal space is covered without overlaps or gaps: it is a tiling.

Reciprocal Unit Cell Any unit cell of the reciprocal lattice.

Note: A special choice of the RUC is the BZ.

Reciprocal Fundamenta Region A minimal connected set such that any point in reciprocal space may be reached from a point in the RFU by combinations of operations from the point group and reciprocal lattice vectors. 'Minimal' here means that no 2 points of the RFU are connected by such a combination.

The use of the dictionary is to explain the use of certain terms. In the literature by BZ usually is meant what is said here in the definition, but it makes clear that it is by no means the only choice.

Re: Brillouin zones definition

Postby glazer » Fri May 27, 2011 1:15 pm

I see that we dont quite agree. That's what makes science fun even for old folk like you and me!

I am concerned though that it is just you and I who are discussing this and I think it would be useful to hear from others, e.g. the Bilbao group. Perhaps Arroyo should be invited?

1. Going back to the original dictionary definition.
The (first) Brillouin zone IS the Wigner-Seitz cell of the reciprocal lattice

To define the BZ as a unit cell in reciprocal space is like saying
A crystal structure IS the (parallelepiped) unit cell in direct space.

Note my emphasis on the little word IS. This is clearly wrong and wrong use of language. The point is, even if we accept the Wigner-Seitz cell, the Brillouin zone is a collection of wave states that are repeated by translational symmetry. In direct space we would say that a crystal structure is a collection of atoms that are repeated by translational symmetry.

2. The fact that textbooks for many years have defined the BZ in terms of the W-S cell is no reason to maintain it in the face of a better way to do it. My example of the wrong diagram for the diatomic chain is an example. It has been in all the books for the last 60 years! The fact that it was in all these books and also in the internet, does not make it correct. Unfortunately book writers like Kittel and others make mistakes, just like we do.
The idea of using a conventional cell for low-symmetry crystals has been around for a very long time too. Bradley and Cracknell for instance were proposing this over 40 years ago!

3. Brillouin himself only chose the W-S cell to describe the wave-vectors out of convenience, and certainly this is the case when dealing with high-symmetry crystals. However, much of this advantage is lost if one goes to triclinic symmetry (Brillouin as far as I know did not consider this). The result is something complicated and not unique since it's shape depends on the values of the cell parameters. So why should we saddle ourselves with an unnecessary complication?
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Re: Brillouin zones definition

Postby tedjanssen » Fri May 27, 2011 2:59 pm

Indeed, Mike, a forum should not be restricted to 2 people.

But, in the meantime, I can answer your 3 points.

1. I have proposed another phrasing. I agree that it is wrong to say 'The BZ is the unit cell...'. Instead I have proposed 'The BZ is a particular choice of the unit cell of the reciprocal lattice.' The phrase 'A crystal structure is the (parallelepiped) unit cell' is also clearly wrong. But its structure and meaning has nothing to do with what I proposed.

I am not convinced that it is right to say 'The BZ is a collection of wave states..' No, it is a set of reciprocal space vectors, that can be used to characterize representations of the translation group, and consequently to characterize states. But the vectors of the BZ are not states.

2. I agree that a given definition can be changed, even if it is very old. The point is that the BZ is very useful. Characterizing representations of the translation group can also be done using any other unit cell of the reciprocal lattice. But for a space group, and for characterizing states in a crystal, it is a very useful notion. If there are practical problems one has always the freedom of taking another unit cell for the reciprocal lattice. But that is not a reason to change the definition.

3. I do not agree that calling any unit cell a BZ solves unnecessary complications. As you stressed yourself, it is not important why Brillouin chose his definition, but it works perfectly, albeit that sometimes visualisation is difficult. But that will not be different for other choices. So in what sense is choosing the BZ is something complicated and not unique? It is very unique, and only if you want to make pictures, it may become complicated. But try to imagine a Fermi surface in a simple triclinic crystal. It is not a simple figure, whatever you do.

The longer I think of it, the better I like Brillouin's choice.

Re: Brillouin zones definition

Postby glazer » Fri May 27, 2011 3:54 pm

I think that your suggestion
The BZ is a particular choice of the unit cell of the reciprocal lattice.
is close to what I have in mind, so maybe we can get there! I would prefer to say

The BZ is described normally by a particular choice of the unit cell of the reciprocal lattice.

and then one can go on to explain that this particular choice is the W-S cell, although it may be in some circumstances easier to adopt a conventional parallepiped unit cell, especially when the crystal symmetry is triclinic or monoclinic.

Something like this would I think satisfy my main objects to the way the BZ is often explained. What do you think?

Certainly Brillouin was very clever but I am not sure that he foresaw the problems of using his suggestion at low symmetry.

And yes, we definitely need other opinions.
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Re: Brillouin zones definition

Postby AndreAuthier » Tue May 31, 2011 10:19 am

I agree with Ted's suggestion. I suggest adding that the Wigner-Seitz cell is nothing but the Dirichlet domain/Voronoi cell in reciprocal space. The Dirichlet domain was introduced (1840) in relation to the properties of ternary quadratic forms. It is well-known (Gauss, Seeber) that the square of the distance between two lattice points can be expressed in terms of a ternary quadratic form.

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Re: Brillouin zones definition

Postby glazer » Tue May 31, 2011 4:11 pm

For sure the W-S cell is the same as the Dirichlet Domain or Voronoi cell. But I still maintain that it is not correct to say

"The Brillouin zone IS the W-S cell"

Reading Brillouin's book you see that he started from the point of view of a conventional unit cell in reciprocal space and decided to adopt the Wigner-Seitz construction for convenience as it had certain advantages. But what he did not discuss was whether these advantages were worth having when dealing with low-symmetry cases. I believe that the correct way to word this is along the following lines

"The Brillouin Zone is defined by a unit cell in reciprocal space. It is conventional to use the Wigner-Seitz construction to describe this unit cell, especially for high-symmetry cases. It may be preferable, however, to adopt a conventional reciprocal unit cell when dealing with low-symmetry lattices, such as in monoclinic or triclinic systems"

This I think is a clear statement and fits with your own arguments in favour of the W-S construction but at the same time agrees with the points made earlier by Bradley and Cracknell. It also agrees with the approach that Brillouin himself made to the subject. It also permits the user to adopt a W-S construction or conventional cell according to his/her wishes.

I really think it would be helpful to have the opinion on this by people like Arroyo in Bilbao.
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Re: Brillouin zones definition

Postby MichaelWidom » Fri Jul 01, 2011 4:05 pm

From a pedagogical point of view when teaching introductory solid state physics it is handy to have a specific definition (e.g. the Wigner-Seitz cell of reciprocal space) rather than a more general one (e.g. arbitrary primitive cell). The WS choice is especially useful because of its relationship to the diffraction condition. It is true, though, that the existing definition is seriously flawed because it misdefines the WS cell. The WS cell is the region enclosed by the set of ALL perpendicular bisectors, not just the nearest neighbors, as has already been pointed out by Janssen in this discussion.
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Re: Brillouin zones definition

Postby glazer » Mon Jul 04, 2011 8:35 am

Michael Widom's point is correct and it is true that for teaching purposes it is useful to have one definition. However, what has concerned me is that because of concentration on the W-S construction to do this, when in fact you can use the conventional unit cell , primitive or not, to define the eigenvectors, this can mislead students. This has the effect that research scientists remain wedded to the W-S construction even when it is much easier and less liable to error to use the conventional cell, especially in low symmetry cases. Because the text books confine themselves mainly to cubic lattices (or at least orthogonal lattices) even experienced solid state scientists can fall into the trap in thinking that everything is cubic. There was enough trouble like this when high Tc superconductors appeared and physicists brought up on Kittel and other textbooks suddenly discovered that they had to relearn crystal symmetry.
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