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Crystal Class

The crystal class is derived from the point group formed by the symmetry operators of the space group with the translational components set to zero. This is demonstrated in the table below for both the monoclinic space-group P21 and the orthorhombic space-group Pnma referred to previously:

Space
  Group  		   Symmetry  
Operators		   Symmetry  
Elements		     Symmetry  
Operators		   Symmetry  
Elements		   Crystal  
Class
P21 x,y,z 1 → x,y,z 1 2
  -x,1/2+y,-z 21 → -x,y,-z 2  
 
Pnma x,y,z 1 → x,y,z 1 mmm
  1/2-x,1/2+y,1/2+z n → -x,y,z m  
  x,1/2-y,z m → x,-y,z m  
  1/2+x,y,1/2-z a → x,y,-z m  
  -x,-y,-z -1 → -x,-y,-z -1  
  1/2+x,1/2-y,1/2-z 21 → x,-y,-z 2  
  -x,1/2+y,-z 21 → -x,y,-z 2  
  1/2-x,-y,1/2+z 21 → -x,-y,z 2  

The concept of the crystal class is important because the crystal class is related to the Laue class observed in the diffraction experiment. The relationship between these two properties will be discussed in more detail under the subject of space-group determination.

Polar Space Groups

Space groups whose crystal class is polar are themselves polar. Polar space groups have an origin that is not fixed in either x, y, or z. Therefore, structure analysis and refinement of a crystal with this space-group symmetry requires the origin to be fixed arbitrarily. This is conveniently done, either by fixing one or more coordinates of the centre of gravity of a molecule (if the structure is molecular), or by fixing one or more coordinates of a heavy atom in the structure. Examples of polar space groups are P1, where x, y, and z must be fixed, Pc, where x and z must be fixed, and P21, where only y must be fixed.

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