Chemical Forums
Chemistry Forums for Students => Undergraduate General Chemistry Forum => Topic started by: souls_at_zero on April 15, 2010, 08:58:32 AM

Hi all, I'm sorry to be making my first post a question but here goes anyway.
I'm just trying to get my head around crystallography and its conventions. I'm not liking it at all! I really hate to feel stupid, and this particular problem is driving me crazy. It's probably so simple but without a good explanation, staring at diagrams the past hour has just left me confused. I've also come across conflicting ideas, which really doesn't help >:(
Anyway I'll jump right into it  I've got an example of a space group diagram:
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fimg693.imageshack.us%2Fimg693%2F7643%2F032cz1.gif&hash=89f8e9540c0ab3b78b353388b7ddbfb6)
I can see that there are what I believe to be symmetry operators (red circles in the box), with the object of origin in the bottom left corner (the empty circle with a positive sign, bottom left inside the box). To the right of the space group diagram are the coordinates to those symmetry operators, with the origin coordinates in blue.
What I am really not getting is the correlation to the points and those coordinates. I completely understand that the origin would be x, y, z. However, my attempts at getting the other coordinates seem to never work. I get why the origin is being repeated as it is. Let me take the next object above the origin, which I believe to be a translation of the origin in a glide plane (shown as the dotted line) parallel to the z axis.
What I would write down as its coordinates would be: x , 1/2 + y, 1/2 + z
Why?  because there has been no movement along the x axis.
 there has been a shift half way up the unit cell along the y axis (and therefore I think it is 1/2 + y)
 I wont lie, I'm saying it's 1/2 + z because it shows that on the diagram.
Looking at the supplied coordinates, I can see that this set of coordinates do not fit in, and I'm quite obviously wrong. I think that if I can't understand this, the more complicated things get the further frustration will build. So please, if anyone can educate me as to how to derive coordinates from these points, and explain where I'm going wrong  you'd make my day (well night, as it's nearly 1am and I've made no progress in my work!).
Thanks a lot.

Red circles are point objects related by different symmetry elements  you find explanation in International Tables for Crystallography A or in crystallography texbooks.
eg
2. x, y, z (y bar, z bar)
this is a rotation by a twofold axis (represented by arrow) parallel to a direction  this cause the object under plane of paper () and below the twofold axis on the scheme moves over plane (+) over the twofold axis on the scheme after rotation preserving the same x  see left down corner

Let me take the next object above the origin, which I believe to be a translation of the origin in a glide plane (shown as the dotted line) parallel to the z axis.
What I would write down as its coordinates would be: x , 1/2 + y, 1/2 + z
Why?  because there has been no movement along the x axis.
 there has been a shift half way up the unit cell along the y axis (and therefore I think it is 1/2 + y)
 I wont lie, I'm saying it's 1/2 + z because it shows that on the diagram.
The c glide plane you mention will transform the coordinates x, y, z into x, 1/2  y, 1/2 + z :
 as you said, there is no change along the a axis. So the x value remains unchanged.
 along the b axis, you have a mirror that transforms y into 1/2y. It is different from a translation by 1/2 along the b axis which would give you 1/2+y. It is where you made your mistake.
 along the c axis, the glide plane will translate the point from z to 1/2+z. It is a translation by 1/2 along the c axis, and not a mirror image.

Let me take the next object above the origin, which I believe to be a translation of the origin in a glide plane (shown as the dotted line) parallel to the z axis.
What I would write down as its coordinates would be: x , 1/2 + y, 1/2 + z
Why?  because there has been no movement along the x axis.
 there has been a shift half way up the unit cell along the y axis (and therefore I think it is 1/2 + y)
 I wont lie, I'm saying it's 1/2 + z because it shows that on the diagram.
The c glide plane you mention will transform the coordinates x, y, z into x, 1/2  y, 1/2 + z :
 as you said, there is no change along the a axis. So the x value remains unchanged.
 along the b axis, you have a mirror that transforms y into 1/2y. It is different from a translation by 1/2 along the b axis which would give you 1/2+y. It is where you made your mistake.
 along the c axis, the glide plane will translate the point from z to 1/2+z. It is a translation by 1/2 along the c axis, and not a mirror image.
a c glide on y is a mirror on y followed by translation by 1/2 on c.
x,y,z > .c. > x,y,z + 1/2

a c glide on y is a mirror on y followed by translation by 1/2 on c.
x,y,z > .c. > x,y,z + 1/2
That would be exact if the c glide was on y=0. But in this case it is on y=1/4, as shown on the picture.
The position x,y, z + 1/2 does not appear on the list of positions given which are:
1 x, y, z
2 x, y, z
3 x, 1/2y, 1/2+z
4 x, 1/2+y, 1/2z