to study of the Structure of Crystals. 137 



The positions of the atoms in the crystal molecule Rve 

 defined by having as many groups of equivalent points 

 associated with each point of the lattice as there are kinds 

 of c ry stall oyraphically different atoms in the unit. This 

 number will be as great as, and may be greater than, the 

 number of different kinds of atoms in the cliemlccd 

 molectde. For instance, in the case of calcite^^ the posi- 

 tions of the carbon atoms must be assigned to at least one 

 set of equivalent points, the positions of the calcium 

 atoms to another set and the oxygen atoms to still 

 another. With two chemical molecules associated with 

 the unit cell, it might be conceivable for the two carbon 

 atoms and the two calcium atoms to be alike and for all 

 six of the oxygen atoms to be crystallographically alike, 

 for four of them to be alike and two different,^ '-^ or that 

 there should be three sets of two like atoms or two sets 

 of three that are alike. There might thus be as many as 

 seven different groups of points associated with each unit 

 of calcite. 



The manner of obtaining, with the aid of the theory of 

 space groups, all of the crystallographically possible ways 

 of arranging the atoms of a compound in the fundamental 

 unit has been illustrated in detail in dealing with calcite. 

 So detailed an application of the theory to the case of 

 magnesium oxide, which follows, is not possible here 

 because of the large number of space groups that must 

 be considered. 



Summary. 



Such details of the theory of space groups as are of 

 importance in the application of this theory to the deter- 

 mination of the structure of crystals are briefly con- 

 sidered. Point groups, space lattices and space groups 

 are illustrated by simple examples. The relations be- 

 tween space groups and crystals is discussed and those 

 modifications in the results of the theory of space groups 

 that are required in order that it may serve as the basis 

 for a general method for the study of the structure of 

 crystals, are indicated. 



Geophysical Laboratory, 



Carnegie Institution of Washington, 

 Washington, D. C. 

 October, 1920. 



^' Ralph W. G. Wyckoff, this Journal, (4) 50, 317, 1920. 

 "Some of these possibilities are actually ruled out by considerations of 

 symmetry. 



