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Space-Group Frequencies

It has already been shown that certain types of symmetry element are preferred for molecular packing: screw axes and glide planes allow closer packing of molecules than rotation axes and mirror planes, respectively. You should therefore not be surprised by the fact that although there are 230 3-dimensional space groups, in practice the symmetry of the known crystal structures is not evenly distributed among them. The following numbers were extracted in April 1998 from the Cambridge Structural Database (CSD), which contains crystallographic data on organic and organometallic compounds, i.e. substances of a molecular nature. (There are other crystallographic databases covering inorganic compounds, metals and alloys, but these together with the CSD will be discussed in a later section of this course.)

The table below shows the distribution by crystal system of the 177,291 entries examined in the CSD:

Crystal SystemFrequency
Orthorhombic           20.98%  

This table clearly demonstrates that molecular crystals have low symmetry crystal structures, i.e. over 95% of the structures belong to either the triclinic, monoclinic, or orthorhombic crystal systems. For this reason, the remainder of this section will concentrate on these three crystal systems. The second reason for studying space groups from the low-symmetry crystal systems is that they have fewer symmetry operators, which makes them simpler to understand than the high-symmetry space groups. You may be surprised by the latter statement since the high-symmetry rock-salt structure encountered in the diffraction section looks pretty simple: the structure of NaCl is simple, but its space-group symmetry (Fm-3m) certainly is not!

The next table shows the 10 most frequently observed space groups found in the CSD as released in April 1998:

Symbol No. Frequency
P21/c 14 36.0%  
P-1   2 19.9%  
P212121 19 9.2%  
C2/c 15 7.3%  
P21   3 5.8%  
Pbca 61 3.9%  
Pnma 62 1.6%  
Pna21 33 1.6%  
Cc   9 1.0%  
P1   1 1.0%  

This second table shows that among the low-symmetry space groups, molecular compounds form crystal structures with certain preferred symmetries: the first three space groups in the list account for over 50% of the known crystal structures in the database. In addition to concentrating on low-symmetry space groups, this section on space-group symmetry will therefore explicity go into further depth on these "top" three space groups. This is very convenient from a teaching perspective since an understanding of each these three space groups will provide you with an example from each of the three low-symmetry crystal systems.

The final table is a statistical proof of the point made earlier concerning the preference of screw axes over rotation axes for molecular structures. The major difference between the four orthorhombic spacer groups listed below is that, in going from P222 to P212121, a twofold rotation axis is successively replaced by a two-one screw axis:

Symbol No. Frequency
P222 16 18  
P2221 17 39  
P21212 18 847  
P212121 19 16256  

As with any statistical study, care must be taken in interpreting the numbers. The above numbers were all based on a search of the Cambridge Structural Database. A different picture would be obtained from a statistical study of, say, the crystal structures of inorganic compounds. In the latter example, cubic space groups will have a much higher frequency. However, to illustrate the point that care must be taken, I would suggest that a statistical study of the cubic space groups might lead one to believe that space-group number 230, symbol Ia-3d, is very popular. It certainly occurs frequently in the database, but this is due to the presence of one type of crystal structure, namely the structure of the mineral garnet for which many isostructural compounds have been synthesised.

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© Copyright 1995-2006.  Birkbeck College, University of London. Author(s): Jeremy Karl Cockcroft