ANALYSIS OF CRYSTAL-STRUCTURE BY X-RAYS 397 



principle and a knowledge of their specific gravities without 

 making a single X-ray observation. 



Bragg has applied these arguments to the elucidation of 

 the crystal-structure of ice. Quite independently of X-ray 

 analysis, he arrives at an arrangement which agrees admirably 

 with experimental results obtained previously by other workers. 

 The reasoning is so beautiful that it is well worth giving in 

 some detail. The hypothesis is that each positive ion is 

 surrounded symmetrically by negative ions and vice versa ; 

 and, in view of the low density of ice, let the number of neigh- 

 bours be in each case as small as possible. The crystal is to 

 be hexagonal and have the right density. 



A simple calculation shows that the structure must be 

 such that each ion of one sign is closer to its neighbours of the 

 other sign than the fluorspar arrangement allows, and the low 

 density of ice suggests that the most economical arrangement 

 should be tried. Now there are two arrangements of like 

 atoms in which each atom is at the centre of gravity of its 

 four neighbours. One is exemplified by diamond, and in this 

 the number of atoms could not be fewer, and the other is a 

 hexagonal arrangement obtained from the other by parallel 

 shifts of the tetrahedral planes, without finally altering the 

 value of the universal distance between two neighbours. Zinc 

 oxide is a case of this latter structure. Replace now each 

 carbon atom of the hexagonal structure by oxygen and insert 

 a hydrogen between each pair of oxygens. Each oxygen is 

 now surrounded by four hydrogens and each hydrogen by two 

 oxygens ; and, moreover, the number of neighbours has been 

 reduced to a minimum. 



The dimensions of this structure can be obtained directly 

 by comparison with diamond. Each point of the lattice 

 now represents a molecular weight of 18 instead of 12, and yet 

 the specific gravity has been lowered from 3-52 to 0-9165. 

 Hence, if the linear dimensions are increased in the ratio 

 I '-p, we have 



^3 18 X 3-52 



P = ^-^^ :. p = 1 79. 



^ 12 X 0-9165 ^ 



The distance between the centres of two carbon atoms 

 in diamond is 1-54 A.U. Hence the distance between the 

 centres of two oxygen atoms in ice is 1-54 x 1-79 = 2-76 A.U. 

 The distance between two consecutive basal planes will be now 

 3-67 A.U., and the distance between two atoms in the same plane 

 4-52 A.U. These figures agree exactly with the observations 

 of Dennison {Science, September 24, 1920 ; Phys. Rev., June 

 1 921). As would be expected, the structure is extremely 



