131 E. W. G. Wyckoff — Theory of Space Groups 



unit prism will contain the following equivalent positions. 

 The unit of groups having this lattice, with the primitive 

 translations 2tx ; Ty , Tz ; Ty , - Tz , is best considered as a 

 prism, two of whose sides are centered (figure 4a). There 

 are then two crystal molecules in the unit. The coordi- 

 nates of the equivalent positions are those of the 

 equivalent points about a corner of the lattice (the origin) 

 and those of the crystal molecule about a lattice point 

 which is at the center of a side. Thus since the coordi- 

 nates of this second lattice point are 0, Ty , t,, , the equiv- 

 alent positions mthin the unit prism are : 



xyz; xyz; lujz ; xifz ; 



The positions of the equivalent points in the unit of 

 structure can be obtained in a similar manner for any 

 space group. 



The Significance of the Space Groups in the Study of 

 the Structure of Crystals. — If an atom occurs at a 

 general position x, y, z Avithin the unit of structure, then 

 in order that the conditions of symmetry may be fulfilled 

 there must be as many more atoms of the same kind in the 

 unit prism as there are equivalent points. For instance 

 in the case of the holohedry of the cubic system the unit 

 cube of a space group having the simple cubic lattice 

 ( Fc ) as a basis has 48 equivalent positions contained 

 within it; the unit cell of a space group having the 

 face-centered lattice (TcO with four points of the 

 lattice (crystal molecules) associated with it, has 192 

 equivalent positions. In the first of these two cases, if 

 an atom, say an oxygen atom, has a general position 

 {xyz) within the cube, then there must be 47 other oxygen 

 atoms of the same sort within the unit; or in the latter 

 case there would have to be 191 other similar oxygen 

 atoms. 



■In actual practice, for the present, we are dealing with 

 simple compounds having relatively few atoms in the 

 chemical molecule ; X-ray spectrum measurements seem 

 to indicate at the same time that only a small number of 

 chemical molecules are associated with the unit cell. 

 Consequently but few atoms of the same kind (less than 

 the number of general equivalent positions) occur within 

 the unit and, as a result, these atoms must take up special 



