PRINCIPLES OF STRUCTURE 



27 



case of crystals showing a high degree of symmetry (cubic system), the 

 lattice points are identical with the points of intersection of symmetry planes 

 and their distances can therefore be calculated from the distances in the 

 X-ray diagram. In the case of lattices having a lower degree of symmetry, 

 however, the situation of the points in the lattice planes is not determined 

 unambiguously by symmetry elements; they possess certain degrees of 

 freedom. Accordingly, the determination of the structure with the aid of 

 the distances in the X-ray diagram, alone, is not possible; additional mea- 

 surements of the intensity of the interferences are then required. In this 

 case, however, the position of all lattice points in the unit cell can often be 

 only approximately determined. (Niggli, 1929, 1941/42). 



Fig. 25 



Fig. 26 



Crystal lattices. The encircled points belong to the unit cell 

 Fig. 25. Gold, a = 4.07 A, # Au - Fig. 26. Sodium chloride, a = 5.60 A, # Na, O CI 



Figs. 25 and 26 represent two of the best-known lattices, viz. that of the 

 element gold and of the compound sodium chloride. Both lattices are cubic: 

 this means that the dimensions and shape of the unit cell are determined by 

 a single identity period a which is the same in three mutually perpendicular 

 directions. Once the spacing a has been determined by means of X-rays, the 

 volume a^ and, from the known density of the crystalline substance, the 

 weight of the unit cell can be calculated. Dividing this weight by the absolute 

 weight of the atom or molecule in question ( = atomic or, as the case may 

 be, molecular weight/LoscHMiDT's number 0.606 • 10-^), one finds the 

 number of atoms or molecules in the unit cell. 



For example, the elementary cell of gold contains 4 Au atoms, that of 

 sodium chloride 4 Na- and 4 Cl-ions. These points have been encircled in 

 Figs. 25 and 26; the other points marked on the planes of the cube are to 

 be considered as having originated from the encircled ones by a simple 

 translation, thus belonging to a neighbouring unit cell. The lattice type of 

 gold is termed face-centred because the points of intersection of the 

 diagonals of the faces are all occupied by atoms. Numerous elements, such 

 as Ag, Cu, Al, Pb, etc., crystallize in accordance with the same scheme, 

 though with different identity periods. In the NaCl type of lattice, which is 



