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On 08/02/2012 04:37 AM, Careina Edgooms wrote:
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Dear ccp4

I ask a very fundamental question because I have not had formal training in this and I would like to understand. 
How can I obtain the multiplicity (z) from the space group? So for example if the space group is P222 how do I know that there are 4 monomers in the unit cell? Or if it is P422 then there is 8? I am only concerning myself with a primitive lattice for now because I am sure the others are more complicated.
thanks
Careina
If you want to derive this number yourself (instead of looking it up in ITC), do this:

1. Write down all the symmetry operators for the spacegroup.  To save time , I'll use P2 for an example:

(x,y,z)
(-x,y,-z)

2. Keep applying them until you get a closed list of symmetry mates:

(x,y,z) - primitive

(-x,y,-z) - second copy

Now apply second operator to the second copy and you get

(-(-x),y,-(-z)) = (x,y,z)  - but that is the same as the primitive operator, so further application of symmetry will not lead to new copies.

3.  Count the unique symmetry copies you found - in this case 2 of them and you are done.


Other space groups are not really more complicated, the same steps apply, you just have more operators.  Notice that symmetry mates should always be shifted back into the origin unit cell, e.g. in P21 the second operator is

(-x, y+1/2, -z)

which after two applications results in

(-(-x), (y+1/2)+1/2, -(-z)) = (x, y+1, z)

but this is the same as (x,y,z) after you translate it back by (0,-1,0).

Where do you find symmetry operators?  You can derive them yourself from the space group symbol or look them up in ITC.  Once you master this, you will be able to explain to others why there is no P22 space group :)

Cheers,

Ed.