Why Isn't C5 a Crystal Point Group?

[WolfOfTheSteps]

Homework Statement

"Show that the C5 group is not a crystal point group."

2. Relevant information

1) "There exists another type of symmetry operation, called point symmetry, which leaves a point in the structure invariant"

2) "In crystallography, the angle of rotation cannot be arbitrary but can only take the following fractions of 2*pi: THETA= 2*pi/n where n = 1,2,3,4,6"

The Attempt at a Solution

So, the problem states that C5 is a group, mathematically, but just not a crystal point group. But obviously, C5 is also a point symmetry, since the point at the rotation axis is invariant. So the only thing I can think of is saying "by definition," because of the undemonstrated statement given by 2) above.

I have no idea how to proceed. I mean, it's a group. It's a point symmetry. If that's all I know, it should be a point group. Why isn't it a crystal point group? My book never explains what technical meaning modifying a phrase by "crystal" would yield.

Any hints would be greatly appreciated.

Thanks.

 

[AKG]

A point symmetry must not only fix one of the points, but must also be a symmetry of the overall structure. So consider a hexagon with a point at its center - this gives 7 points. Then rotating by 1/5 of a rotation CCW about the center point fixes the center point, but doesn't send the set of vertices to the set of vertices. On the other hand, if you rotate by 1/6 of a rotation CCW about the center, then the center is fixed, and the vertices get sent to the vertices (in particular, each vertex gets sent to the "next" one that's adjacent to it in the CCW direction) [CCW = counter-clockwise].

 

[WolfOfTheSteps]

Thanks AKG,

I think I see what you are saying. Would it also be correct to say that C5 would imply a pentagonal crystal system, which is not possible?

 

[AKG]

Yeah, that's correct.

 

[WolfOfTheSteps]

Great. Thanks again.