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All of them or only a subset?

This is a famous and fundamental result in solid state physics.

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4 Answers 4

According to Fizicheskaya Entsiklopediya (Physical Encyclopedia, in Russian, http://www.femto.com.ua/articles/part_2/3634.html ), no real crystals had been found for 4 space groups (Pcc2 and three others) as of the encyclopedia's publication in 1988-1999.

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Taking that at face value, we still have the problem that just because we haven't found it doesn't mean that such a crystal doesn't exist somewhere. It is hard to prove a negative, that it is impossible to place objects (molecules) of the required symmetry on the appropriate Bravais lattice to get those missing space group. –  Jon Custer Jul 23 at 13:30

If there were only a few that had not been seen, I suspect that Ashcroft and Mermin would have gleefully pointed that out. However, that is not proof.

On the other hand, I see no way to prove that a given space group definitively could not exist in nature, given the essentially infinite number of molecules that could form crystals. We still spend lots of synchrotron beam time determining crystal structures of proteins - who is to say the next one would not have the magic unseen space group symmetry?

At best, one might say that, given the number of elements identified so far is less than 230, the STP equilibrium elemental crystals do not cover all of the space groups.

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The elements only cover 19 spacegroups (according to ElementData from Wolfram). You have to look at minerals to see a wide range of spacegroups represented. –  alemi Jul 22 at 22:37
    
@alemi - yup, lots of bcc and fcc metals in the elements. The huge number of minerals, not to mention organic molecules make up the rest. I would suspect that if there were a "unseen" space group, a good synthetic chemist could design and make something to have the right symmetry. –  Jon Custer Jul 22 at 22:45

I would be surprised if there wasn't an example of each.

For instance, you can search the American Mineralogist Crystal Structure Database by space group. Click on Cell Parameters and Symmetry and select whichever you like. I'll leave it to you to go through all 230 of them.

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I too chose not to brute force it but thought I'd approach from a slightly different direction. –  Jon Custer Jul 22 at 22:26
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PCC2 (mentioned by akhmeteli as not being observed as of 1999) is not in the dropdown menu. Also "The list box does not contain all possible space groups, but only those represented in the database." Interpret that how you will. –  Chris White Jul 23 at 1:00

Probably all of them and more besides. The 230 does not include quasicrystals, while nature does.

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