Chemical Forums
Specialty Chemistry Forums => Materials and Nanochemistry forum => Topic started by: oaksoft on January 30, 2011, 02:13:42 PM
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A cubic unit cell is described as having minimum symmetry of four 3-fold rotation axes.
I know what 3-fold symmetry is and I can see the axes on a cubic unit cell.
What I'm unsure about is the precise meaning of "minimum symmetry" in the context above.
I want to use that definition to understand how to deduce the minimum symmetry of other unit cell types such as tetragonal and orthorhombic.
Obviously I can simply look them up but I'm trying to understand how to derive them myself.
So.....does anyone know what is meant by minimum symmetry?
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Cubic groups can also show 2-fold, 4-fold axes and planes
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"Minimum symmetry" covers the symmetry elements that are present in every space groups of a given crystal system (cubic for example).
In the case of cubic systems, they all have four 3-fold rotation axes along the cube diagonals. These can define a cubic space group.
But for example not all cubic space groups have 4-fold rotation axes. So you can't use 4-fold axes to define cubic space groups, as it would be too restrictive.
If you look at the space groups list (http://en.wikipedia.org/wiki/Space_group, large table near the end of the page) for the cubic crystal system, you can see that all and every cubic space group has 3-fold axes (examples: P23, P432, Pm-3m,...).
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"Minimum symmetry" covers the symmetry elements that are present in every space groups of a given crystal system (cubic for example).
In the case of cubic systems, they all have four 3-fold rotation axes along the cube diagonals. These can define a cubic space group.
But for example not all cubic space groups have 4-fold rotation axes. So you can't use 4-fold axes to define cubic space groups, as it would be too restrictive.
If you look at the space groups list (http://en.wikipedia.org/wiki/Space_group, large table near the end of the page) for the cubic crystal system, you can see that all and every cubic space group has 3-fold axes (examples: P23, P432, Pm-3m,...).
I'm afraid we've not had anywhere near enough lectures on this stuff for me to be able to understand that table yet but your bit about a cubic unit cell not always having 4 fold symmetry helps a bit (although I can't imagine an example where it wouldn't).
Thanks for helping.
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You can have a look at this website http://www.unioviedo.es/qcg/d-MolSym/.
There, you can find this picture (and more) of a cubic symmetry without 4-fold rotation axes:
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fwww.unioviedo.es%2Fqcg%2Fd-MolSym%2Fmol-cube-t.png&hash=14faeee01f825e8fa751f6b9ddd27b10cf8eb436)
In green are 2-fold rotation axes and in yellow are 3-fold rotation axes. No other symmetry elements.