370 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 11, NO. 15 



The summation is to be taken over each atom in the unit cell. The 

 spacings between like planes in a simple cubic lattice (Fig. 1) is pro- 

 portional to 



-==^= (1) 



where h, k, and / are the indices of the plane. 



With the aid of these two expressions, and without any precise 

 knowledge of the "laws" of scattering, it is possible to determine the 

 ratio of the spacings between like planes to be observed from any 

 arrangement of atoms. jP$ 



Every such, structure which possesses complete (holohedral) cubic 

 symmetry must be either a general or a special case of one of the ten 

 space groups O/, ^~^". The ratio of the spacings to be observed from 

 each of these space groupings for the (100), (110) and (111) planes has 

 been calculated.^ Of these ten groups but one, 0;^^ shows the sequence 

 observed for sodium chloride. Any number of conceivable structures 

 for sodium chloride can be developed from this space group by simply 

 assigning different coordinate values to the positions of the sodium and 

 chlorine atoms ; and except for very special values of these coordinates 

 the ratio of the spacing from the three planes under discussion will 

 always be the same as that experimentally observed for sodium 

 chloride.*' For the sake of simplicity of illustration the least compli- 

 cated of the special cases of this space group will be treated. The 

 three special cases containing the fewest number of molecules within 

 the unit cell are:^ 



Four equivalent positions : 



(1)000; Il0;l0h0ll 

 (2)^2-2; OO^OiO;,^00. 

 Twenty-four equivalent positions : 



(3) uOO;u-\- 1, 1,0; u-\- i, 0,1; uH; 



uOO ; \ - u,io ;\-u, 0,\ ;u\\; 



OuO;\,u + \,0; 0,u -\-\,\;\u\; 



OuO;\,i-u,0;0,\-u,\;lu\; 



00u;\,0,u-\-\;0,\,u-\r-2;\\u; 



00u;2,0,-2-u;0,\,\-u;-2 \u. 



^ p. NiGGLi. Geometrische Krystallographie des Discontinuums , p. 492. 



^ In the general case there are 192 equivalent positions within the unit cell of this 

 group so that there may be as many as 192 molecules of sodium chloride within it. 



' These results are taken from a book now being prepared for publication which con- 

 tains an analytical expression of all of the special cases of each of the space groups. 



