Symmetries of a Diamond Unit Cell - Point Group Confusion

  • #1

Main Question or Discussion Point

Dear All,

I've been recently reading the very clear text of Burns and Glazer entitled Space Groups for Solid State Scientists in the context of my thesis which requires understanding of symmetries of crystals, more specifically symmetries of (approximate triply periodic minimal surfaces) ATPMSs.

Currently I am stuck at a point on the symmetries of diamond unit cells, and their ATPMS counterparts are defined. These structures are classified to be under m \bar{3} m, which seem to be involving four fold rotational symmetries along the normals of the cell surfaces as well as centrosymmetry. But neither the diamond crystal nor its ATPMS version encapsulates these symmetries, at least to my understanding.

What am I missing here? Thank you very much for the comments.
Kumar
 

Answers and Replies

  • #2
7
3
I believe diamond's space group is Fd3m that is F4sub1/d -3 2/m, not Fm3m like NaCl-type

This is a non-symmorphic space group: it has fourfold screw axes and diamond glides: symmetry elements that combine rotation with partial translation. For some properties like optical ones that does not matter and so this group resorts under point group m-3m. The actual space group may not have any position that has that point group symmetry because screw axes tend to cross each other rather than intersect. The inversion center may also not be where you'd expect it but somewhere between the crossing axes in open space rather than at the origin or the intersection of the axes. Look at the international tables
 
  • #3
I believe diamond's space group is Fd3m that is F4sub1/d -3 2/m, not Fm3m like NaCl-type

This is a non-symmorphic space group: it has fourfold screw axes and diamond glides: symmetry elements that combine rotation with partial translation. For some properties like optical ones that does not matter and so this group resorts under point group m-3m. The actual space group may not have any position that has that point group symmetry because screw axes tend to cross each other rather than intersect. The inversion center may also not be where you'd expect it but somewhere between the crossing axes in open space rather than at the origin or the intersection of the axes. Look at the international tables
Dear ngonyama,

Thank you very much for your detailed answer. I understand that the positions of the centers of transformations may alter. However, one aspect of your comment confuses me: You mention that "properties like optical ones that does not matter and so this group resorts under point group m-3m". Now, to my understanding, one should distinguish purely geometric properties from the physical properties, e.g. optical, thermal, mechanical. According to Neumann-Curie principle the latter should encapsulate at least a point symmetry group of the former. What do you think?

Along with this same discussion, I see most mechanics books tend to use crystal symmetry with physical property symmetry interchangeably, which creates great ambiguity.
 
  • #4
7
3
The difference between a mirror operation and a glide operation is that the latter is followed by a translation of half a unit cell, i.e. a few Angstroms at most. If you want to describe, say the refractive index and your wavelength is 500nm or 5000A (green light) do you really think that tiny shift matters? You will observe the rotational point group symmetry, not the space group one.

If you use X-rays with wavelengths in the Angstrom range that becomes a very different matter. It also matters if you consider band structures, particularly at the edges of the Brillouin zone. It is phenomenon known as 'bands-sticking-together'. The non-symmorphic symmetry causes the local representations in points other than Γ not to be fully reducible.
 

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