the Spinel Group of Crystals. 309 



depend on whether the blacks and whites have the same or 

 different significance. 



As regards the (111) planes, the spacing in the case of the 

 diamond (loc. eit. p. 100) is the distance from one plane 

 containing blacks only to the next that contains blacks only; 

 the planes containing only whites lie between those con- 

 taining blacks and divide the spacings of the latter in the 

 ratio 1 to 3. Whether or not the blacks and whites represent 

 the same things again makes no difference in the spacings. 

 It makes a difference in the intensities, of course : there is 

 no second order spectrum in the diamond because the blacks 

 and whites represent the same carbon atoms and there is 

 perfect interference (loc. cit. p. 103). In zinc-blende they do 

 not, and the second (111) spectrum is only partially destroyed 

 (loc. cit. p. 98). In the present case the blacks and whites 

 do not represent the same thing as regards the (111) planes, 

 and so the second (111) spectrum remains as in the case of 

 zinc-blende ; but we are coming to this point presently. 



Thus the replacement of the carbon atoms by oxygen 

 tetrahedra does not interfere with the spacings of the (100) 

 (110) and (111) planes, and we still have them of the same 

 description as those of the diamond, which is in agreement 

 with our experimental results. The placing of the oxygen 

 atoms has been completed satisfactorily. The size of the 

 tetrahedron remains undetermined as yet. 



Let us next consider the iron atoms. We have three, of 

 which chemical considerations would distinguish one as 

 divalent from the other two as trivalent. 



Symmetry considerations are in agreement with such a 

 division. It is not possible to place three atoms round a 

 point so as to have all the symmetries required. We may 

 increase the three to six, provided that we so place each iron 

 atom that it is shared equally by two oxygen tetrahedra, and 

 that each oxygen tetrahedron has shares in four iron atoms. 

 It does not seem possible to do this ; nor do extensions of the 

 number to nine, twelve, and so on seem to offer a solution. 

 In any case, such a disposition would make no difference 

 between the divalent and trivalent atoms. In the spinel 

 MgAl 2 4 the magnesium atoms must surely be placed in 

 some different way to the aluminium or else we should, in 

 securing the trigonal symmetries, be obliged to assume that 

 Mg and Al behave alike to the X-rays, which would be 

 contrary to all our experience. 



Let us, therefore, take the divalent and trivalent irons 

 separately. The most simple and obvious place for the single 



