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As an example, Let's consider a material $Ba_{2}YMoO_{6}$,(ref:PRB 81,224409), the space group of this material is Fm3m, the crystal structure is shown below

enter image description here

(https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.224409).

If I want to know the crystal field symmetry (i.e. the point group of the crystal field) of Mo (which is surrounded by 6 oxygen and $MoO6$ forms a octahedron), how should I do?

First step: I look up the space group information of Fm3m (No.225), and find the site symmetry of Mo atom in Wyckoff Positions of this space group, and we can find the site symmetry is m-3m, i.e. $O_{h}$ group. Is this mean the crystal field of Mo is $O_{h}$ symmetry?

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  • $\begingroup$ I am not a crystallographer, but I know that in 3D there are only 32 different crystallographic point groups (en.wikipedia.org/wiki/Crystallographic_point_group), so all you need to do is eliminate the wrong ones. You know that your structure accepts the $O_h$ symmetry, so see which of the point groups have $O_h$ as a sub-group. Then try to restrict the range of possibilities further. Do you have reflection planes? Inversion points? What about the behaviour under time-inversion (i.e. are the magnetic point groups relevant)? $\endgroup$ – Cryo Aug 4 at 21:34

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