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Recently I'm working on a compound with space group P63/m. The top view of its structure is shown below (where only atoms of z=1/4 are shown). Top view of the structure

From the list of space groups (Wiki: List of Space Groups), I found that the space group P63/m has a subgroup of $C_{6h}$, which means that the system should have $C_{6}$ rotation symmetry. However, by checking the previously mentioned structure, I cannot find such a $C_{6}$ rotation axis.

Is there anyone who can help me to find out the "missing" $C_{6}$ rotation axis? Or did I misunderstand the symmetry of this system?

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  • $\begingroup$ Have you tried rotating the crystal around to put the $a$ or $b$ axis perpendicular to the plane of observation? Maybe one of those directions has the $C_6$ in your structure. $\endgroup$
    – KF Gauss
    Dec 11 '19 at 7:52
  • $\begingroup$ I didn't consider the axis along other direction because the lattice constant $c$ can be changed independently from $a$ and $b$. Thus if there exists a 6-fold rotation axis which is not along the $z$ direction, the rotation would mix the $z$-coordinate with $x$ and $y$. Such a rotation should not be preserved if we arbitrarily change the $c$ value, I think. $\endgroup$
    – Siqi Wu
    Dec 11 '19 at 8:12
  • $\begingroup$ your logic doesn't make sense to me, axes are arbitrary. I would try looking at those other axes instead of ignoring them. $\endgroup$
    – KF Gauss
    Dec 11 '19 at 8:21
  • $\begingroup$ Let me put it this way: the point group is $C_{6h}$, which means there is a mirror plane perpendicular to the rotation axis. And in this system, the only presented mirror plane is the $xy$-plane, thus the corresponding 6-fold axis should be along the $z$ direction. $\endgroup$
    – Siqi Wu
    Dec 11 '19 at 9:06
  • $\begingroup$ Which material is this? Can you link to a cif or crystallography database entry? $\endgroup$
    – KF Gauss
    Dec 11 '19 at 9:18
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You don't have simple (proper) 6-fold rotation axes, but (improper) 6-fold screw axes, that is a rotation followed by a translattion along the c-axis. In your picture only atoms at z=1/4 are shown; so you're missing the translation and you cannot see the screw axis. enlarge the z- range and you will see the 6-fold symmetry; look at the TlMo3Se3 below, the 6-fold axis is evident. enter image description here

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The answer is that space group P6$_{3}$/m does have a corresponding point group of $C_{6h}$, and it does not have the symmetry of $C_6$.

This is not contradictory because P6$_{3}$/m is a non-symmorphic space group. A non-symmorphic group has at least one element that contain non-interger translation. If we put all pure translations into a group T, we can see that T is a subgroup of the space group G. And we can divide G by T to get a factor group F, which is a subgroup of G.

For a symmorphic group, the factor group F is nothing but its corresponding point group P, while for a non-symmorphic group, its point group is obtained by omitting the translation part of the factor group. In this case the point group is isomorphic to F, but it is not a subgroup of G.

Therefore, despite that $C_6$ is an element of point group $C_{6h}$, since $C_{6h}$ is not a subgroup of P6$_3$/m, $C_6$ is not an element of space group P6$_3$/m. The corresponding element in the factor group is the screw operation {$C_6$|(0,0,1/2)}.

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