28 FUNDAMENTALS OF SUBMICROSCOPIC MORPh6lOGY I 



found in several binary compounds (NaF, KCl, PbS, etc.) with different 

 values of a, two of such face-centred cubic lattices overlap. 



If the atoms of a crystal lattice are not represented by distinct points, 

 but by spheres touching each other, their space requirement related to the 

 volume of the unit cell can be calculated. It is then found that of all possible 

 crystal lattices the cubic face-centred lattice of Fig. 25 has the closest possible 

 packing. The volume of the spheres amounts to 0.74 of the total space 

 available. There is another possibiHty of closest packing where the arrange- 

 ment of the spheres is hexagonal (hexagonal space-centred lattice). The 

 ratio of the axis is a:c = 1.63:1 and the space required exactly the same 

 as in the cubic closest packing (0.74). In other types of close packing the 

 space requirement is always smaller than 0.74. For instance, in the space- 

 centred cubic lattice the spheres fill only 0.68 of the volume of the unit cell. 



Primary valency lattice. Next to the geometrical relations between the 

 points in the crystal lattices, the forces which keep the atoms together 

 are of primary importance. The purely geom.etrical consideration of 

 the lattice is quite independent of this. As soon, however, as one is 

 interested in the reason why certain distances in a lattice are great and 

 others small, this question must be considered. In fact, the lattice 

 forces are of a varied nature. Actually, in the examples given, the forces 

 are different. In Fig. 25 similar atoms, in Fig. 26 oppositely charged 

 ions attract each other. In both cases primary valencies act as lattice 

 forces which can join together uncharged as well as oppositely 

 charged particles. In the first case one speaks of a homopolar lattice, in 

 the second of a heteropolar or ion lattice. 



The morphological similarity of these two types of lattice is due to 

 the fact that in both cases the construction of the lattice is founded 

 on the rules of the theory of co-ordination. According to Werner's 

 chemistry of complexes, each atom is surrounded by a fixed number 

 of neighbouring particles, either 4, 6, 8 or 1 2, depending on volume 

 conditions. This theory, based originally on the composition of salts 

 containing crystal water [e.g., Ca(H20)gCl2] and other complex salts, 

 has also proved useful in the elucidation of crystal structures of other 

 compounds and of the elements. In fact, in Fig. 25 each Au-atom at the 

 corners of the cube is surrounded by 1 2 neighbouring atoms and in 

 Fig. 26 each Na-ion by 6 Cl-ions or, vice versa, each Cl-ion by 6 Na- 

 ions. 



The theory of co-ordination has led to another fundamental re- 

 cognition which has become of the greatest importance to the sub- 



