Why R32 Space Group is so named? I mean, if R stands for Rhombohedral and 3 for Threefold Axis, what 2 stands for?

I don't see any Twofold Axis in the cell:



What does it means 1/2, 2/3 and 1/6 on the arrows?

R32 have two cells.

This one: http://img.chem.ucl.ac.uk/sgp/large/155bz1.htm

And this other: http://img.chem.ucl.ac.uk/sgp/large/155az1.htm

Are they equivalent?


I have traced some lines to mark the screw-axes and the twofold-axis, but i can't see any point that follow the simmetry operator and the element of simmetry. Maybe i don't understand how it works.

Which is the height of the green line? And, which are the points of the screw-axis of the yellow line?enter image description here


The 2-fold axes are parallel to the "paper plane ", i.e. perpendicular to the three-fold. They are depicted as arrows, the half-arrows indicate screw-axes. The numbers next to it show the "height" of the axis, i.e. where the plane parallel to the paper-plane cuts the unit cell: 1/2 would be in the middle, 1/3 would be a third from the bottom/top, etc. (obviously one only gives relative numbers here, as the absolute unit cell length varies depending on the real-life crystal you are looking at).

Also have a look here for the symbols: http://img.chem.ucl.ac.uk/sgp/misc/guide.htm

Otherwise, the two links you provided are equivalent, as you can see from the pictogram in the lower lefthand corner - the first link is in the "tri-/hexagonal", the second link in the "rhombohedral" setting. The symmetry for both is obviously the same (otherwise they would not be the same space group), it is just a different way of depicting the spacegroup by changing the point of origin.

See here how they relate to each other: https://www.uwgb.edu/dutchs/symmetry/rhombo-lattice.htm

This one should also be quite enlightening - but I am not quite sure whether it might be behind a paywall: http://it.iucr.org/Ab/ch7o1v0001/sgtable7o1o155/ (Alternatively, have a look at the hardcopy of the International Tables of Crystallography, Volume A and look up spacegroup 155 - any decent crystallography lab or library should have it).


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