Space Group Duplicates: Is it Possible for Monoatomic Crystals?

[g_mogni]

Hello,

I was building some test crystal structures with VESTA software and I noticed when generating the POSCAR output files to be read by VASP that for a simple monoatomic basis such as in elemental silicon crystals some distinct space groups produce the same exact POSCAR files (i.e. same primitive vectors and basis atom positions in the unit cell). My question therefore is: is it possible for a simple monoatomic crystal structure such as those of the pure elements to belong to several degenerate (i.e. indistinguishable) space groups (for example space groups 210 and 227)? In this case how can we tell the difference and know for example that at ambient conditions crystalline silicon exists in space group 227 and not 210 given that they appear to have the same structure?

Thanks and sorry if this is a silly question,

Gabriele

 

[M Quack]

#210 is a subgroup of #227 that is obtained by removing the center of inversion.
In principle you can represent the Si crystal structure in each and any subgroup of #227 Fd-3m as long as you position the Si atoms at the correct positions and fix the lattice parameters "accidentally" at 90 deg and a=b=c (not necessary for #210 as it is cubic).
By doing that you lose information, as the lower space groups do not contain all the symmetries that exist in the Si crystal.
You can see for example that in #227 the point symmetry of the Si position 8a is given as -432, whereas in #210 it is only 23. - even though the positions are fixed and there are no additional degrees of freedom in the lattice parameters.
So #210 correctly describes the positions of the Si atoms, but does not list all existing symmetries.
You can see that if you put an atom at the general positions (xyz), then in #227 you need 192 atoms, and in #210 you need 96 atoms to maintain all space group symmetries.

http://it.iucr.org/Ab/ch7o1v0001/sgtable7o1o210/
http://it.iucr.org/Ab/ch7o1v0001/sgtable7o1o227/

 

[g_mogni]

OK so I suppose that for a complex molecular crystal the difference between 210 and 227 will be apparent in the positions of the basis atoms, but not so for a simple monoatomic crystal like silicon...

So if I understand correctly for the purpose of computing the Gibbs Free energy of silicon under pressure the structures of 210 and 227 don't make any difference, and further considerations are required to distinguish between the two, right? Does this mean that simply minimising the Gibbs free energy to determine relative phase stability at a given pressure is not guaranteed to yield a unique answer for the space group?

 

[M Quack]

OK so I suppose that for a complex molecular crystal the difference between 210 and 227 will be apparent in the positions of the basis atoms, but not so for a simple monoatomic crystal like silicon...

Yes.

]
So if I understand correctly for the purpose of computing the Gibbs Free energy of silicon under pressure the structures of 210 and 227 don't make any difference, and further considerations are required to distinguish between the two, right? Does this mean that simply minimising the Gibbs free energy to determine relative phase stability at a given pressure is not guaranteed to yield a unique answer for the space group?

I'm not an expert in this. It is possible that by going to the lower symmetry space group you allow degrees for freedom that are not present in the higher symmetry, and that increases the entropy... If you find a lower energy for #210 than #227 that would probably mean that #227 is unstable under pressure.
In crystallography, typically if one assumes that the correct space group is the highest symmetry compatible with the data - e.g. if you have a position that refines to a small value with large error bars in one space group, and is exactly zero by symmetry in another, higher symmetry space group.

 

[g_mogni]

uhm the thing is that in DFT calculations (including entropy calculations using phonon vibrational frequencies) all the information about the space group is enclosed in the structure file like POSCAR file in the case of VASP. So if two space groups have exactly the same structure file as it is the case for 227 and 210 in crystalline silicon then I don't see how the DFT code can distinguish between the two, which means that they will inevitably end up having the same Gibbs free energy at any pressure. This makes me wonder if thermodynamics is simply incapable of differentiating between such degenerate space groups? In that case determining the precise space group of the stable phase of a crystal (such as determining that the stable phase of silicon at ambient conditions is 227 and not 210) might require careful further crystallographic analysis (including experimental observation) and not just a simple thermodynamical calculation... unfortunately I don't think that this kind of analysis lies within my range of expertise :(

Gabriele

 

[M Quack]

It is conceivable that the phonons break the space group symmetry. In that case you would get two domains, e.g. left- and right handed. In this case that appears unlikely, though.
If you impose a symmetric arrangement of atoms, then it is not surprising that the DFT result is also symmetric.
What you would probably have to do (no idea if that is feasible, though) is to manually introduce a small deviation from symmetry, compatible with one of the subgroups of #227 (#210 may not be the best example) and see if that lowers the Gibbs energy - in that case the high symmetry structure would be unstable.
In any case, this is outside my narrow field of expertise, and somebody else might be able to supply a more precise answer.