What distinguishes the 32 crystallographic point groups?

[tirrel]

Hello, some weeks ago I was having a first look at the world of crystals:

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

Now I forgot the bit that I've understood but before trying to study the topic again I would like to ask an other simple question: " What makes the 32 crystallographic point groups "different"? " (the question applies also to space groups).
I've found only just the enumeration of these groups and some kind of construction. But from the mathematical point of view I think that there are isomorphic point groups among these 32 (it can also be confirmed from wikipedia page, under the voice "abstract group" on the table of point gropus). Therefore group isomorphism it seems to me it's not the criterion of classification.
What is this criterion then?

Tirrel

 

[M Quack]

They are different because physics is not mathematics.

There are a few isomorphic groups within the 32, but physically they are different. E.g. a mirror plane is not the same thing as a 2-fold rotation axis. Space inversion is not the same thing as a 2-fold rotation or a mirror plane, etc.

The classification, given in the first table of the Wiki page, follows the "required symmetries of point group" in column 3. These lead to macroscopically observable symmetries of real world crystals. This classification is much older than our microscopic understanding of crystals. The relation between microscopic and macroscopic symmetries is called "Neumann's principle", after Franz Ernst Neumann

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

 

[Stephen Tashi]

I think what physicists call "groups" are often what mathematicians call "group actions". http://en.wikipedia.org/wiki/Group_action So it isn't surprising to a mathematician that two different "group actions" are associated with the same "group".

 

[M Quack]

Looking at the Wiki page, that sounds about right. To typical physicists (especially the experimental type, like me) the difference is a bit lost...

For the physical properties, wave functions, etc. the representations of the symmetry group are important.

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

 

[tirrel]

Thanks a lot for the reply. Very interesting! So the remaining point would be two understand how two different group actions (on the same space) are discriminated. Never came across in my studies of such a criterion ) I'm very curious.

 

[M Quack]

Take the 3 most simple examples:

C_i (-1)
C_2 (2)
C_s (m)

The all contain the identity and one other symmetry element, and are therefore all isomorphic to the cyclic group Z_2.

For C_i the extra symmetry is space inversion. The compatible space groups are triclinic, with no restrictions on the lattice parameters and angles.

For C_2 this is a 2-fold rotation
For C_s this is a mirror plane (=2-fold rotation followed by space inversion).

In these two cases the compatible space groups are monoclinic, i.e. 2 lattice angles have to be 90 deg.

 

[tirrel]

I see that two isomorphic groups can be accounted as different point groups. But this was the starting point... Actually what is then the answer? Two space groups are "different"... when? When there is a one to one correspondence (which should also be an isomorphism) beween its symmetry elements that has also the proprierty of sending axes of rotations into axes of rotations (of the same order), mirror planes into mirror planes, glide planes into glide planes? Could this be the definition? But I do not find it anywhere...

 

[tirrel]

Up!

 

[Stephen Tashi]

What properties of the crystallographic groups make them useful? If we would answer that question, perhaps we'd understand the formal definition. Is it just the idea of a group action that maps an infinite crystal structure onto itself in some physically plausible way. (i.e. as a rigid motion? or perhaps as a crystal forming around a "seed" that can have various initial orientations?)

On the "talk" page for the Wikipedia article on "Space group" https://en.wikipedia.org/wiki/Space_group, someone proposed the following introduction to the article:

In mathematics and physics, a space group is the symmetry group of a configuration in space, usually in three dimensions. A symmetry group of R3 belongs to a space group iff the subgroup of all translations in that group is generated by 3 linearly independent translations and the symmetry group has only finitely many cosets of the subgroup of all translations in it. Two symmetry groups that belong to a space group belong to the same space group iff there exists a linear transformation R such that the function that assigns to each transformation T in the first symmetry group R-1TR is a bijection from the first symmetry group to the second symmetry group. Note that R-1TR doesn't necessarily have to be an isometry for all isometries T; it just has to be an isometry for all symmetry operations T of the first symmetry group. In total there are 219 such symmetry groups. For each space group, either all structures that belong to that space group are chiral or none of them are. Some authors consider chiral copies of a space group to be distinct, that is, there define a space group pretty much the same way except for replacing the criterion "there exists a linear transformation R such that" with "there exists a non-inverting linear transformation R such that". 11 of the members of the first definition of a space group can be split into 2 members of the second definition leaving a total of 230 space groups according to the second definition. Those 11 space groups are called chiral space groups. Although all structures that belong to a chiral space group are chiral structures, not all chiral structures that belong to a space group belong to a chiral space group. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002))." to better explain what a space group is.

Can anyone find sources for any of the information I added? Blackbombchu (talk) 01:17, 24 November 2014 (UTC)

We could discuss that as a formal mathematical definition.