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From the given set of symmetry operations for a space group, say, P-3m1 or Cmma, how can one identify the type of all symmetry operations. For example, how to distinguish the axis of rotation symmetry from the roto-inversion symmetry in the set. Same for screw vs glide operations.

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    $\begingroup$ Are you asking how to read the space group given or how to envision the various kinds of symmetry operations within a particular lattice? $\endgroup$
    – legolizard
    Commented Jun 8 at 12:25
  • $\begingroup$ I want to know the kinds of symmetry operations $\endgroup$
    – AbPhys
    Commented Jun 9 at 8:29
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    $\begingroup$ If you are looking for algorithms to do it, see RWGK's paper: Algorithms for deriving crystallographic space-group information, Acta Cryst. (1999). A55, 383-395. But usually it's more practical to use Tables or online resources. $\endgroup$
    – marcin
    Commented Jun 13 at 9:20

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The simplest answer would be "click on the horizontal arrow at the bottom right". You'll be redirected to another page.

The column on the left lists a triad of coordinates for each Wyckoff site pertaining to the space group. The remaining ones can be calculated by the list opening when you click on x,y,z.

The column on the right lists the symmetry operations constituting the space group. The first one is the Identity, then you have the 3-fol axis rotations (for P-3m1), and so on.

Anyway, if possible, use the International Tables for Crystallography vol. A

The most difficult answer is:

  • for point 1: choose arbitrary values ​​for x,y,z (between 0 and 1) and identify the position in a trigonal lattice
  • for point 2: use the same triad of initial values ​​x,y,z transforming it into -y,x-y,z and identify the position of this new point in the trigonal lattice
  • identify what type of symmetry operation correlates the two points
  • go on in the same way for points 3, 4, ...

Actually, with practice, some easier operations are immediately recognizable, especially for orthorhombic, tetragonal and cubic systems; for example -x,-y,z is a reflection, whereas y,x,-z is a 2-fold axis rotation.

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  • $\begingroup$ Thank you @gryphys. I appreciate if you could help me with these queries as well- How different are 3- or 3+ (in P-3m1 case) or simply 3 rotational axes from each other? The symbol d or tA (in Fddd) or tC (in C2/m) denote translation, and appears glide or screw symmetry when combined with mirror or rotation. So how to find whether d, tA, tC notations indicate glide or screw operation? $\endgroup$
    – AbPhys
    Commented Jun 10 at 7:47
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    $\begingroup$ 3 indicates a simple 3-fold rotation axis; 3+ and 3- (better 3_1 and 3_2) indicate a screw rotation of order 3 that can be right- or left-handed. d indicates a 'diamond’ glide plane. Sincerely, I don't fully understand what you mean with tA and tC. I guess that A and C could indicate the type of lattice and t the translation, But Fddd space group has an F-type lattice (lattice points on the cell corners with one additional point at the center of each of the faces of the cell), not A (base centered). $\endgroup$
    – gryphys
    Commented Jun 11 at 9:00
  • $\begingroup$ Thanks @gryphys If 3+ and 3- indicate a screw rotation, what does this 3+_1 and 3-_2 indicates? I guess latter are screw rotations. If it is so, why to use conflicting symbols that indicates same symmetry operations? Then there is 3bar+ also, I guess that is a rotoinversion symmetry. Ref-: img.chem.ucl.ac.uk/sgp/large/142az2.htm, img.chem.ucl.ac.uk/sgp/large/144az2.htm $\endgroup$
    – AbPhys
    Commented Jul 20 at 8:56
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    $\begingroup$ actually, the notation on the webpage seems to be a mixture; you should refer to the International Tables of Crystallography vol A where the 2 Symmetry operations are written as 3+(0,0,1/3 ) 0,0,z (indicating a counter-clockwise rotation of 120° around the line 0,0,z, combined with a translation of 1/3c) and 3−(0,0,2/3 ) 0,0,z (indicating a clockwise rotation of 120° around the line 0,0,z, combined with a translation of 2/3c). 3_1 and 3_2 indicate right-handed screw rotation of 120° and 240°, respectively $\endgroup$
    – gryphys
    Commented Jul 25 at 14:02

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