Crystal Class 
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 spacegroup P2_{1} and the orthorhombic spacegroup Pnma referred to previously:
Space Group 
Symmetry Operators 
Symmetry Elements 
Symmetry Operators 
Symmetry Elements 
Crystal Class 


P2_{1}  x,y,z  1  →  x,y,z  1  2 
x,1/2+y,z  2_{1}  →  x,y,z  2  
Pnma  x,y,z  1  →  x,y,z  1  mmm 
1/2x,1/2+y,1/2+z  n  →  x,y,z  m  
x,1/2y,z  m  →  x,y,z  m  
1/2+x,y,1/2z  a  →  x,y,z  m  
x,y,z  1  →  x,y,z  1  
1/2+x,1/2y,1/2z  2_{1}  →  x,y,z  2  
x,1/2+y,z  2_{1}  →  x,y,z  2  
1/2x,y,1/2+z  2_{1}  →  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 spacegroup 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 spacegroup 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 P2_{1}, where only y must be fixed.
© Copyright 19952006. Birkbeck College, University of London.  Author(s): Jeremy Karl Cockcroft 