Understanding Point Groups in Crystals: Definition and Key Concepts

[tirrel]

Hello, I've got difficulties in understanding what is the point group a o crystal. I read that it is the subset of symmetry operations leaving at least one point of the lattice fixed. But I do not understand:
1) This point must be the same for all the members of the point group?
2) if it must be the same, then the definition is point dependent?
3) if it must not be the same, then how can we say that composition of two elements of the point group is still a member of the point group?

Thanks a lot for any help,

Tirrel

 

[mfb]

1) This point must be the same for all the members of the point group?

It doesn't matter, the point of a crystal is to have a repeating structure.

 

[M Quack]

The space group contains symmetry operations that can contain combinations of rotations, translations and inversion. For example, a mirror plane is a 2-fold rotation combined with inversion. A glide plane combines a mirror plane with a translation, a screw axis combines a rotation with a translation, etc.
The corresponding point group is made up of the same symmetry operations with all translations removed, leaving only rotation and inversion - for a glide plane you keep the mirror operation, for a screw axis you keep the rotation.

All rotations, mirrors, etc of the point group a carried out about the origin.

If your space group does not contain glide planes or screw axes, then it is called symmorphic. In that case there is a point, usualy chosen at the origin, that has the full point group symmetry. This is the case e.g. for the rock salt structure.
If the space group is non-symmorphic (i.e. contains screw axes or glide planes), then there is no such point. This is the case e.g. for the Diamond crystal structure.

 

[tirrel]

So if I understand well:

1) the symmetry operations of the point group should always be visualized as keeping the origin fexed and the origin does not have any actual meaning with relation to the crystal;

2) it can happen that the point group has operations that are not symmetry of the crystal (whatever atom or point in space we identify withi the origin), but become actual symmetries only when combined with translations? And this is the case for non symmorphic groups? Did I understand right?

In order to compare with mfb answer, he points out that in a Bravais lattice all points are equivalent. This means that all point groups of a Bravais lattice are symmorphic?

 

[M Quack]

1) yes, you can see it that way. The point group is the symmetry at the origin of the reciprocal lattice. It is also the symmetry of macroscopic properties which do not "see" the microscopic lattice translations. The space groups are ordered by point symmetry. In the Schoenflies notation that is particularly obvious.

2) yes.

All lattice points are equivalent, i.e. all points that are obtained by translation from the origin by multiples of lattice vectors. But that is not really related to the point group.

In complicated crystal structures you have a "basis", i.e. several atoms within the unit cell. In general these will have lower symmetry than the point group. Symmetry operations may then "throw" that position onto another equivalent, but different point. Such positions are classified by symmetry and are tabulated as "Wyckoff positions". If you look at the International Tables for Crystallography you will find them listed for each space group, along with information about the local symmetries, etc.

 

[tirrel]

Ah.. thanks a lot! I've got other question though... Sorry but I've always had some difficulties in understanding crystallography:

0) when I put a bases the atoms in the bases are considered different from a symmetry point of view or do I have also to declare the type of atom in the bases?

1) In the sentence "Symmetry operations may then "throw" that position onto another equivalent, but different point" you mean "equivalent" in the sense of the underlying bravais lattice? And "different" in the sense of the crystal structure (there can be no atom in that point, or an atom of different type)?

2) can there be a crystal, a Bravais lattice with a bases, whose positions are described equivalenty by an other different Bravais lattice (different= with different point group) with an associated different bases?

3) the Wyckoff positions are a proprierty of the Bravais lattice (identified by its space group), not of the bases and not of the Bravais point group right?

4) which is the difference between a "crystal point group" and a "Bravais point group"? I understand that adding a bases to a Bravais lattice can lower the symmetry (can make it higher? Actually if we identify all atoms (see question 0) there are more places in which the original Bravais lattice can be sent) but I see that there are 230 different crystallographic space groups. Does it mean that whatever bases I put into a Bravais lattice, with whatever identification of atoms, I will always will end up with one of these 230 space groups?

I hope that the number of questions will not diverge! :)

 

[M Quack]

Ah.. thanks a lot! I've got other question though... Sorry but I've always had some difficulties in understanding crystallography:

0) when I put a bases the atoms in the bases are considered different from a symmetry point of view or do I have also to declare the type of atom in the bases?

The space group says nothing about the basis. In order to fully define the crystal structure you need to specify the space group, the lattice parameters, and the basis.

On the other hand, the basis has to be compatible with the space group.

1) In the sentence "Symmetry operations may then "throw" that position onto another equivalent, but different point" you mean "equivalent" in the sense of the underlying bravais lattice? And "different" in the sense of the crystal structure (there can be no atom in that point, or an atom of different type)?

Equivalent in the sense that the two positions have the same local point symmetry, and that they are related by crystal symmetry. If in a given space group you have an atom at one position, then symmetry requires that you have one at the other.

2) can there be a crystal, a Bravais lattice with a bases, whose positions are described equivalenty by an other different Bravais lattice (different= with different point group) with an associated different bases?

I am not sure if I understand the question. In general, you can always describe a high-symmetry crystal structure as a special case of a low-symmetry structure. This is sometimes useful when you have phase transitions that change the symmetry from high to low. In that case the low symmetry point group has to be a subgroup of the high symmetry group.

3) the Wyckoff positions are a proprierty of the Bravais lattice (identified by its space group), not of the bases and not of the Bravais point group right?

The space group is based on the Bravais lattice, but to determine the space group you need extra information. There are 14 3D Bravais lattices, but 230 3D space groups.

Within each space group there is a list of Wyckhoff positions, usually listed from high local symmetry to low local symmetry.

Which of these Wyckoff positions is occupied by what atom (or any atom at all) is defined by the basis.

4) which is the difference between a "crystal point group" and a "Bravais point group"? I understand that adding a bases to a Bravais lattice can lower the symmetry (can make it higher? Actually if we identify all atoms (see question 0) there are more places in which the original Bravais lattice can be sent) but I see that there are 230 different crystallographic space groups. Does it mean that whatever bases I put into a Bravais lattice, with whatever identification of atoms, I will always will end up with one of these 230 space groups?

The basis can lower the symmetry, it can never raise it. If your basis is not compatible with the symmetry of the Bravais lattice, then you have to choose a lower symmetry Bravais lattice and space group. Again, this sometimes happens when you have phase transitions, e.g. in magnetic systems or ferroelectrics. The "fake" higher symmetry of the "original" Bravais lattice is called an accidental symmetry, and in general there is no good reason why this symmetry should exist - in most cases, if you look carefully you find that the symmetry of the Bravais lattice is broken and only conforms with the symmetry of the basis - no free symmetries.

I hope that the number of questions will not diverge! :)

\

I sure hope so, too. But if it helps you understand crystals, keep them coming :-)

 

[tirrel]

Unluckily, I'm still more confused. Let's begin with answer 0.
I guess, reading your answer, that you consider the space group as a property defined by the crystal (Bravais lattice + bases) and not of only the Bravais lattice , otherwise I do not understand the compatibility condition you are referring to in the answer.
But I thought that the expression "space group" referred to the symmetry elements of the Bravais lattice and the expression "crystallographic space group" was the expression for the symmetry properties of the whole crystal. Is there such a distinction?

 

[M Quack]

In nature, the Bravais lattice and the basis define the space group. But if you start out from a symmetry point of view with the space group, then the basis has to match the space group. In the end it is a bit of a chicken-and-egg question: Which one comes first?

 

[tirrel]

Ok, I'm trying to put pieces together. I understand from your answers that a space group is a proprierty of the crystal and not only of its Bravais lattice. It is the group of symmetry operations of all the crystal, considering also the bases and the different type of atoms. Am I right? In this case may I ask how the symmetry operations consider the different type of atoms (for example in NaCl)?

So after understanding this I go back to your message:

"In complicated crystal structures you have a "basis", i.e. several atoms within the unit cell. In general these will have lower symmetry than the point group. Symmetry operations may then "throw" that position onto another equivalent, but different point. Such positions are classified by symmetry and are tabulated as "Wyckoff positions". If you look at the International Tables for Crystallography you will find them listed for each space group, along with information about the local symmetries, etc."

What do you mean here by saying "In general these will have lower symmetry than the point group". In this sentence "these" stands for the crystal structure or stands for "atoms"? In the first case I do no understand what does it mean, in the second case I do not understand what a symmetry of an "atom" is.

 

[M Quack]

NaCl crystallizes in the space group Fm-3m, #225. This is a non-symmorphic space group.

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=225

Na sits at Wyckhoff position 4a = (0,0,0)
Cl sits at Wyckhoff position 4b =(1/2, 1.2, 1/2)

This fully defines the basis.

Actually, these two positions are completely equivalent, and you might as well put Na on 4b and Cl on 4a.

"4" means that there are 4 equivalent positions within the cubic unit cell. This is because it is a FCC lattice.
The 4 positions are (x,y,z), (x,y,z)+(1/2,1/2,0), (x,y,z)+(0,1/2,1/2), (x,y,z)+(1/2,0,1/2), where (x,y,z) is (0,0,0) for 4a and
(x,y,z)=(1/2,1/2,1/2) for 4b. "a" and "b" are just labels. Note that these positions are fully determined by symmetry, whereas the positions 24e, for example, have a free parameter "x", i.e. the atoms can be shifted in this direction without affecting the symmetry.

Both sites 4a and 4b have local symmetry m-3m, i.e. the full point group symmetry of the space group.

 

[tirrel]

mmm... actually for me it would be better to clarify my last questions before looking at a definite example...

 

[DrDu]

But I thought that the expression "space group" referred to the symmetry elements of the Bravais lattice and the expression "crystallographic space group" was the expression for the symmetry properties of the whole crystal. Is there such a distinction?

I don't think so.
However, the symmetry of the Bravais lattices (as opposed to the whole crystal including the basis) due to rotations and inversions gives rise to the 7 crystal systems (tricline, monocline etc.) while the corresponding group of the whole crystal are the point groups.

 

[tirrel]

Thanks DrDu. Let's refer to Aschroft Mermin, table 7.1, for notations, at least in this thread. I find the nomenclature is quite confusing in this field...

 

[M Quack]

"Space group" and "crystallographic space group" are the same thing, at least as long as you are talking about crystals.

As DrDu pointed out, there are 7 crystal systems that give rise to the 14 Bravais lattices.

http://en.wikipedia.org/wiki/Crystal_system

On the other hand, there are 32 crystallographic point groups.

Each crystal system is compatible with a limited number of crystallographic point groups. This is because the crystal system already defines certain symmetries. The Bravais lattices combine these basic symmetries with certain basic translational symmetries, e.g. SC, BCC and FCC for cubic.

Each combination of Bravais lattice and point group then gives rise to a number of space groups, depending on how the point group symmetries are combined with translations.

The higher the symmetry, the more choices there are. That's why there are only 2 triclinic space groups, but 36 cubic space groups.

 

[DrDu]

Maybe this relatively short article is helpful:
http://www.crystallography.fr/pages_perso/Nespolo/en/spacegroups.php

 

[M Quack]

Well, he certainly defines every single last detail... If defining all these names that are rarely used in practice is helpful in understanding the underlying concept is another question.

For the record: My use of "space group" corresponds to Nespolo's "space group type".

 

[DrDu]

I have seen this kind of classification sometimes, although I must confess that I never really bothered to understand the details. However, there are quite some subtleties which have been cleared up by the mathematicians. For example, the following paragraph seems to answer a question we had recently in a different thread:
https://www.physicsforums.com/threads/bravais-lattice-with-one-angle-equal-90.771412/#post-4857015

"Commonly, a space group is associated with the same Bravais class of its lattice. In some cases, however, the lattice may accidentally correspond to a higher Bravais class ( e.g. the case of a monoclinic crystal whose lattice has β = 90°)."

Also the non recognizance of the difference between crystal families, lattice systems and crystal systems is a constant source of confusion.

 

[M Quack]

Point taken. One can rarely argue with the mathematicians hyper-precise point of view.

Note that the monoclinic Bravais lattice already has alpha=gamma=90deg, such that an accidental beta=90 would take it to orthorhombic.

A triclinic lattice with just one angle at 90deg, however, is still triclinic.

 

[tirrel]

Thinking about this concept:

"The basis can lower the symmetry, it can never raise it."

If I take a continuous bases filling the whole cell, the symmetry raises a lot :)

 

[M Quack]

Well, that means that you have badly chosen your unit cell - in fact there is no unit cell. The system is just completely homogenous.

 

[tirrel]

Thanks, a question you didn't answer before is if the space group of a Bravais lattice is symmorphic? By this I mean it is the direct sum of the point group and the translationa vectors? It seems to me that it is like that, reading Ashcroft and Mermin...

 

[M Quack]

Thanks, a question you didn't answer before is if the space group of a Bravais lattice is symmorphic? By this I mean it is the direct sum of the point group and the translationa vectors? It seems to me that it is like that, reading Ashcroft and Mermin...

Yes. If you give the origin of the lattice the full point group symmetry and then apply the translations of the Bravais lattice the result is a symmorphic space group (or space group type, according to Nespolo's nomenclature). Furthermore, in his nomeclature the point group determines the geometric crystal class.

This then gives you 73 different space group types, one for each arithmetic crystal class.

http://reference.iucr.org/dictionary/Arithmetic_crystal_class

 

[tirrel]

Thanks a lot, hope step by step to understand crystals I have still to put togethere a lot of points :)