I am trying to understand space groups in crystallography. In Internation tables for crystallography, for a nonsymmorphic space group, they list some symmetry operations. 8 of them are listed under the (0,0,0)+ set and 8 in the (1/2, 1/2, 1/2)+ set. What does this mean? Are there 16 operations in total? How do the sets differ?
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I don't know what space group you're looking at, but this has nothing to do with symmorphic or non-symmorphic space groups.
take a look for example at the space group n° 67 (Cmme). The symmetry operations constituting the group can move the atom at x,y,z in other positions. for example if you consider operation (5) you see that is an inversion (-1) located at $(0,0,0)$ and $(1/4,1/4,0)$. so if you apply "-1" at $(0,0,0)$ it will move the atom at $(-x,-y,-z)$, if you apply "-1" at $(1/4,1/4,0)$, you will move the atom at $(-x+1/2,-y+1/2,-z)$. In Table you see explicitely the 8 coordinates of the operations constituting the $(0,0,0)$+set; the 8 remaining of the $(1/2,1/2,0)$+set are obtained by simply adding $(1/2,1/2,0)$ to the previous 8 ones.
You can see that at the end you have 16 symmetry operations and 16 positions at the general position 16 a
