Show C5 Not Crystal Point Group

In summary, the conversation discussed the problem of showing that the C5 group is not a crystal point group. It was mentioned that point symmetry must not only fix one of the points, but also be a symmetry of the overall structure. It was then explained that rotating by 1/5 of a rotation counterclockwise about the center point does not send the set of vertices to the set of vertices, while rotating by 1/6 of a rotation does. This means that C5 would imply a pentagonal crystal system, which is not possible, therefore making it not a crystal point group.
  • #1
WolfOfTheSteps
138
0

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.
 
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  • #2
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].
 
  • #3
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?
 
  • #4
Yeah, that's correct.
 
  • #5
Great. Thanks again.
 

1. What is "Show C5 Not Crystal Point Group"?

"Show C5 Not Crystal Point Group" is a mathematical concept used in crystallography to describe the symmetry properties of crystals. It refers to the group of symmetry operations that are possible for a crystal with a C5 point group, but not for a crystal with a crystal point group.

2. What is a C5 point group?

A C5 point group is a type of crystal symmetry in which the crystal has a five-fold rotational axis. This means that if you rotate the crystal by 72 degrees around this axis, it will look the same as it did before the rotation.

3. How is "Show C5 Not Crystal Point Group" useful in crystallography?

"Show C5 Not Crystal Point Group" is useful in crystallography because it helps to classify crystals based on their symmetry properties. This information can be used to predict the physical and chemical properties of crystals, as well as to understand their formation and behavior.

4. Can you provide an example of a crystal with a C5 point group?

Yes, one example of a crystal with a C5 point group is quartz. Quartz crystals have a five-fold rotational axis and exhibit many of the symmetry properties associated with this point group.

5. Are there crystals that do not fall under the "Show C5 Not Crystal Point Group" category?

Yes, there are many crystals that do not fall under the "Show C5 Not Crystal Point Group" category. This concept only applies to crystals with a C5 point group, while there are many other types of symmetry groups that are possible for crystals, such as cubic, hexagonal, and tetragonal point groups.

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