176 R. W. G. Wyckoff — Determination of the 



The probable space groups corresponding to a few 

 crystals have already been determined by showing that 

 the structures assigned to them by the X-ray studies can 

 be deduced from certain particular space groups. 3 It is 

 the intention of the present paper, however, to show, by 

 taking the cubic crystals as the simplest examples, how 

 the space groups of many crystals can be uniquely deter- 

 mined in advance of a complete elucidation of their struc- 

 tures and to state criteria which serve to distinguish 

 between the various cubic space groups where such a 

 distinction is possible. Not only is such a knowledge of 

 the space group of a crystal an ultimate aim of formal 

 crystallography but it may be of great value in the 

 problem of crystal structure study itself. 



In the crystals whose structures have been determined 

 all, or nearly all, of the atoms of which they are composed 

 have been found to occupy positions within the unit cells 

 whose coordinate values are limited and denned by sym- 

 metry considerations (the corners, center, centers of the 

 edges and of the faces of a unit cube are such positions). 

 Such very special structures can usually be deduced from 

 more than one space group. Most cubic crystals, how- 

 ever, have one or more of their constituent atoms in posi- 

 tions so general that the symmetry requirements permit 

 their x, y and z coordinates to have any values. Physical 

 data concerning the mechanism of the scattering of 

 X-rays by atoms are not yet sufficient for the complete 

 determination of the structure of any crystal having 

 one or more atoms in these general positions. A knowl- 

 edge of the space group to which such a crystal should be 

 assigned serves to determine the manner of arrangement 

 of its atoms in many cases, even though the distances 

 between some of these atoms cannot now be established 

 with accuracy. 



X-ray criteria for distinguishing between the different 

 space groups are most simply and satisfactorily deduced 

 for those crystals having some atoms of appreciable 

 scattering power in general positions. For this reason 

 and also because a knowledge of the space group of such 

 crystals is valuable to the crystal analyst, the discussion 

 which follows will be limited to cubic crystals having one 

 or more atoms in general positions. Similar criteria 



3 A. Johnsen, Physikal. Z., 16, 269, 1915. 



