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I am doing an introductory course to Crystallography and I have found in the following web page: http://pd.chem.ucl.ac.uk/pdnn/symm3/sgmonoc.htm a guide for the elaboration of the diagrams of the spatial groups.

In this tutorial, they say the following: "The symmetry operator for the c-glide plane is of the form x,T-y,1/2+z, while the symmetry operator for the two-one screw axis is of the form T′-x,1/2+y,T′′-z. It is possible to show (using matrices) that the only values for T, T′, and T′′, which permit the symmetry operators to form a closed group, are 1/2, 0, and 1/2, respectively"

They could tell me how to calculate the values of T, T' and T'' using the matrices.

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  • $\begingroup$ Why, that's simple. Apply each operator twice and see where this gets you. $\endgroup$ – Ivan Neretin Aug 16 at 15:11
  • $\begingroup$ Sorry, but I don't understand you $\endgroup$ – aprendiendo-a-programar Aug 16 at 15:23
  • $\begingroup$ They say the symmetry operator has a certain form. What does that mean, if anything? I'll tell you what. It means that the operator grabs any point (x,y,z) and relocates it to certain other coordinates. Now I suggest looking at what happens if you apply the operator twice. $\endgroup$ – Ivan Neretin Aug 16 at 15:40
  • $\begingroup$ But, I would like to know how to obtain the values of T, T', T'' through the use of matrices. $\endgroup$ – aprendiendo-a-programar Aug 16 at 15:41
  • $\begingroup$ Take a huge whiteboard marker and write "Matrices" on your chair where you will be sitting while doing this. I think that will count as the use of matrices. Seriously, what is the point of using a specific instrument you don't really need? $\endgroup$ – Ivan Neretin Aug 16 at 15:49
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You are missing an important part in the quote (the first sentence):

Fixing the point of inversion at the origin implies that one of the symmetry operators is -x,-y,-z. What are the symmetry operators for the glide plane and screw axis? (The answer is clearly listed, but suppose it was not given here!) The symmetry operator for the c-glide plane is of the form x,T-y,1/2+z, while the symmetry operator for the two-one screw axis is of the form T′-x,1/2+y,T′′-z. It is possible to show (using matrices) that the only values for T, T′, and T′′, which permit the symmetry operators to form a closed group, are 1/2, 0, and 1/2, respectively. This produces the symmetry operators listed on the right-hand side of the space-group diagram shown above.

Once you fix the origin, T, T' and T'' are indeed fixed. Otherwise, they are not. So to show that they are, you have to combine a glide plane or screw axis with the inversion operation for the chosen origin, and then make an argument.

These arguments usually go as in this example: Two symmetry operations combined give a pure translation. This translation has to be compatible with the crystal lattice, otherwise you have a contradiction.

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  • $\begingroup$ And, this can be done mathematically. In other words, without knowing these arguments, I can deduct them. $\endgroup$ – aprendiendo-a-programar Aug 17 at 14:15
  • $\begingroup$ For example, in the space group Pnma. Can I know from matrices that the helical binary axis is at 1/4? Or that the "a" plane is too? $\endgroup$ – aprendiendo-a-programar Aug 17 at 14:18
  • $\begingroup$ Or, for example, why does a diagonal plane (type "n") appear in the spatial group Ccc2? $\endgroup$ – aprendiendo-a-programar Aug 17 at 16:08

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