128 B. W. G. WtjcUff— Theory of Space Groups 



fact that if in a particular case the number of molecules 

 associated with the unit of structure^ and the symmetry 

 are known, all of the possible atomic arrangements can 

 be written down and considered in the light of further 

 X-ray measurements. It can then be told, with a given 

 amount of experimental data, whether the particular 

 structure under examination can or can not be uniquely 

 defined. 



The Theory of Space Groups.^ 



The geometrical theory of space groups can be 

 developed in the following way. If all of the n opera- 

 tions of symmetry that are characteristic of some one of 

 the thirty-two classes of crystal symmetry are made to 

 operate upon a point in space, n points will result which 

 are all crystallographically equivalent. The n equivalent 

 points arising from the operations of symmetry of one 

 of the crystal classes, or these operations themselves, 

 can be taken to define one of the thirty-two point groups. 

 To take a simple example: the holohedry of the mono- 

 clinic system possesses two elements of symmetry, a 180° 

 axis which will be taken to coincide with the Z-axis and a 

 plane of symmetry at right angles to this axis (the XY 

 plane). If these two elements of symmetry are caused 

 to operate upon any point in space, three other crystallo- 

 graphically equivalent points will result. The four 

 symmetry operations are (using Schonflies'^^^ notation) : 



1, A(7r), Sii, A(7r)Sb where 



1 (the identity) may be thought of as a rotation of 2 tt, 

 A (tt) is a rotation of angle tt, 



Sh is a mirroring against the horizontal (XY) plane and \X\q product 

 A (7r)Sii is the combination of the rotation A(7r) and the mirroring 

 Su. 1 and A(7r) correspond to the operation of the 180° axis, S^ 

 and A(7r)Sii are mirrorings of 1 and A,, in the XY-plar\e. In 

 figure 1» where Z is the 180° axis and XY the mirroring plane, 



* This, of course, can be told from X-ray spectroscopy. 



^In descriptive crystallography we are accustomed to consider the ele- 

 ments of symmetry — axes, planes and centers — to be of chief importance. 

 For studying the internal structure of crystals, at least, it is much better to 

 think of particular grouping of points (or atoms) as characterizing the dif- 

 ferent classes of symmetry. The elements of symmetry of these various 

 arrangements then become interesting, but for most purposes unnecessary 

 and complicating, details. Those who are well versed in the more customary 

 crystallography are asked to read the following discussion from this other 

 point of view, forgetting for the time being all associations with planes, axes 

 and the like, except as they may be introduced into the argument. 



"A. Schonflies, op. cit. 



