ANALYSIS OF CRYSTAL-STRUCTURE BY X-RAYS 405 



structure of tartaric acid, but to go into the whole question 

 in detail would unduly prolong this article. However, the 

 broad outlines may be stated very simply. 



It is clear that a single space-lattice constructed of unsym- 

 metrical units can show no symmetry by itself. A crj^stal 

 built up on such a basis would simply obey the fundamental law 

 of cr^'stallography, the Law of Rational Indices, but nothing 

 else. Calcium thiosulphate, CaSgOg, 6H2O, is supposed to be 

 a case in point. The addition of any element of symmetry 

 to a structure composed of unsymmetrical units necessarily 

 involves the presence of another lattice, interpenetrating the 

 first, and bearing to it the relation implied by that element of 

 symmetry. For instance, as we have seen in the case of tartaric 

 acid, the presence of a dyad axis means that there are, in the 

 complete structure, two. interpenetrating simple lattices, either 

 of which can be obtained from the other by a displacement 

 and a rotation through 180°. Similarly, the presence of a 

 plane of symmetry involves the existence of two interpenetra- 

 ting simple lattices, either of which can be obtained from the 

 other by a displacement and a reflection across the plane of 

 symmetry. And so, by the logical extension of this principle, 

 we arrive at the conclusion that, for a crystal composed of 

 unsymmetrical units, the number of molecules per unit cell 

 is equal to the number of elements of symmetry shown by the 

 crystal. 



But suppose now the units are not unsymmetrical, that, 

 in fact, they each possess definite symmetry of their own. 

 Nature then makes use of it in order to reduce the number of 

 molecules required to build up the unit cell. Indeed, it seems 

 to be a general principle that Nature never uses more than the 

 minimum number of molecules necessary to show the com- 

 plete symmetry of the structure. For instance, if the molecules 

 possessed each a dyad axis, this circumstance would be made 

 full use of in a crystal possessing a dyad axis (that is, the 

 dyad axis of the molecule would coincide with the dyad axis 

 of the crystal), and if that dyad axis were the only element of 

 symmetrj^ shown by the crystal, then clearly no more than one 

 molecule per cell would be necessary. Thus we see that, in 

 general, the number of molecules per cell is only an expression 

 of the relation between the symmetry of the complete lattice 

 and the symmetry of the molecules constituting the lattice. 

 It is simply the number of elements of symmetry shown by the 

 crystal divided by the number of elements of symmetry shown 

 by the molecules. 



