Crystallographic groups

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1. Feb 8, 2015

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

2. Feb 8, 2015

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

3. Feb 8, 2015

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".

4. Feb 8, 2015

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

5. Feb 9, 2015

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.

6. Feb 9, 2015

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.

7. Feb 9, 2015

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 simmetry 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...

8. Feb 13, 2015

tirrel

Up!

9. Feb 13, 2015

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:

We could discuss that as a formal mathematical definition.