Standard crystallographic notation confusion

[Jezza]

I'm revising for my condensed matter exam, and I've never understood the point group notation, in particular of the 32 crystallographic point groups, so let me try and explain what I understand of it and point out where my confusion lies. Please point out any other misunderstandings I have.

1. We can have up to 3 symbols, which I suspect is because we're in 3D space. I'd love a more complete explanation of this if you've got one. What I don't understand is why we ever need three. As far as I can tell, the first two symbols will unambiguously imply the third. Is this just a matter of convention? (The only exception here is when 2/m 2/m 2/m is sometimes written as mmm which is ambiguous with mm2 if you only give the first two symbols.) I'll assume it is convention in the following.
2. When there's only one symbol, it just means there's some rotation/rotoinversion axis which can arbitrarily be defined as the z axis.
3. If there is more than one symbol and they're all of order-2 then
   1. A third order-2 axis is generated by two order-2 axes, so there must be 3 symbols, and
   2. The condition of closure of a point group means all three axes must be orthogonal, and convention is to put x, y, z axes in order.
4. If there is a higher (n^th) order axis then convention dictates this symbol goes first, and following this:
   1. For odd n there are n equivalent axes which are referred to by a second symbol.
   2. For even n, the second symbol will represent n/2 equivalent directions (the secondary directions). The operators along the secondary directions generate another order-2 rotation or mirror plane along the n/2 directions bisecting the secondary directions. The third symbol represents these tertiary directions. The bisection is required by the closure of the group.
5. For multiple higher order axis, things get complicated, and I don't fully understand how the notation is consistent in these cases. For the two cases, 23 and m3, where the notation might imply only one higher order axis, I guess the fact that the higher order comes second indicates that there are multiple of them? The Wikipedia page implies that these crystallographic point groups are always those of a cubic crystal system. I can't quite see myself to an explanation of this fact. Accepting that, however, I appreciate that the three symbols refer to the equivalent x, y and z axes, the body diagonals and the face diagonals respectively.

 

[Jezza]

As an update, I've found a different crystallography book (Space groups for solid state scientists by Glazer and Burns - highly recommended) which explains things a lot more clearly than the one I was reading which, as you can probably tell, I found extremely confusing.

 

[M Quack]

As you have probably found out by now, there are several notations for crystallographic point and space groups. The Hermann-Mauguin notation you are referring to is trying to be systematic, but somehow fails gloriously.

The reason that there are up to 3 symbols is that there are up to 3 different "types" of symmetry axis. You are probably correct that this is related to the 3 dimensions of space. Unfortunately, what is the same type or a different type depends on the point group.

For cubic groups, for example, the types are the (100), (111) and (110) axes and their equivalents.

For tetragonal, they are (001), (100) and (110), because (100) and (010) are equivalent because of the 4-fold rotation about (001).

For orthorhombic, they are (001), (100) and (010)

etc. You get the picture.

Therefore to read the symbol, you first have to look at it and guess (guess correctly!) to which class it belongs. There are hints: If the second symbol is 3 or -3, it is cubic. if not and the first is 4 or -4, it is tetragonal, and so on.

Once you have figured that out, the symbols tell you what symmetries these axes have. In most cases the abbreviated version is used: When one symmetry implies the other, then the implied one is not given. Therefore 2/m is in many cases listed as just 2 - but not when there is a difference between 2, 2/m and just m, e.g. monoclinic.