August ii. 1910] 



NATURE 



liig 



leaving one of the 230 types of homogeneous point-systems 

 (Kig. 2). Imagine, next, that each point of the system 

 expands uniformly in all directions until it touches its 

 neighbours ; a system of spheres packed together in contact 

 is thus obtained (Fig. 3), and, on examination, it is found 

 that no way exists of packing these equal spheres more 

 closely together than the one thus derived. The system 

 is therefore termed the cubic closest-packed assemblage of 

 equal spheres, and, being derived in the manner described, 

 still retains the high symmetry of the cube ; the fragment 

 shown, in fact, outlines a cube. Three directions at right 

 angles in it, those which are parallel to the three cube 

 edges, are seen to be identical in kind ; this identity in kind 

 in the three rectangular directions a, 6, and c is con- 

 veniently expressed by the ratio a:b : c = i : i : i. 



On removing spheres from one corner of the cubic 

 closest-packed assemblage of equal spheres a close 

 triangularly arranged layer is disclosed, and, by similarly 

 treating each corner of the fragment of assemblage, the 

 cube outline gives place to one of octahedral form. The 

 assemblage is now seen to be built up by the superposition 

 of the disclosed triangularly arranged layers, the hollows 

 in one layer serving to accommodate the projecting parts 

 of the spheres in adiacfiit layers. When this operation is 



performed it is perceived, however, that two ways of stack- 

 ing the layers homogeneously are possible. The first of 

 these, in which the fourth layer lies immediately over the 

 first, the fifth over the second, and so on, yields the cubic 

 closest-packed assemblage. The alternative mode of stack- 

 ing, in which the third layer lies immediately over the first, 

 the fourth over the second, and so on, exhibits the same 

 closeness of packing as the first, but possesses the symmetry 

 of the hexagonal crystal system ; it is accordingly termed 

 the hexagonal closest-packed a,ssemblage of equal spheres 

 (•'ig- 4)- E.xamination of the hexagonal assemblage shows 

 that the horizontal directions, in the planes of the layers, 

 are not identical in kind with vertical directions perpen- 

 dicular to the planes of the layers. Corresponding 

 dimensions in these two directions, a and c, are in the 

 ratio of 



a:c = i: ^/(i)= i : 0-8165. 



The final step in the treatment of the closest-packed 

 assemblages of equal spheres consists in converting them 

 into the corresponding assemblages of cells fitting together 

 without interstices which have been already mentioned ; it 

 may be carried out in tjiese, and in all other cases, bv 

 causing the component spheres to expand uniformly in ail 

 directions until expansion is checked by contact with the 

 expanding parts of neighbouring spheres. The cubic 



NO. 2128, VOL. 84] 



closest-packed assemblage then becomes a stack of twelve- 

 sided polyhedra, rhombic dodecahedra, which are so fitted 

 together as to fill space without interstices. It is now 

 seen that the even rate of expansion from each point of the 

 original point-system which gives rise to the closely packed 

 stack of rhombic dodecahedra symbolises an even radiation 

 in all directions of the forces of which the atom is the 

 centre of emanation. On applying the same operation of 

 expansion to the spheres present in hexagonal closest- 

 packing, each becomes converted into a dodecahedron, 

 although of symmetry different from that of the rhombic 

 dodecahedron. In each of the two cases the system exhibits 

 the important property that, with a given density of dis- 

 tribution of the centres, a maximum distance prevails 

 between nearest centres ; these two systems thus represent 

 the equilibrium arrangements of the postulated forces of 

 repulsion exerted between near centres, the repulsions 

 between more distant ones being neglected. 



It will be sufficiently evident from what has been said 

 that the function of the spherical surfaces in the closest- 

 packed assemblages of spheres, as representing crystal 

 structures, is merely a geometrical one ; these surfaces are 

 employed only as so much scaffolding by the aid of which 

 inav be derived arrangements exhibiting a maximum 



iiumber of equal distances between neighbouring centres, 

 and no physical distinction is to be made between portions 

 of space lying within the spheres and portions forming part 

 of the interstices between them. Insistence on this point 

 is necessary, because many investigators have made use, 

 quite illegitimately, of spheres for the representation of 

 atomic domains, piling the spheres together in what they 

 have termed open packing ; this term seems to imply that 

 some physical difference can subsist between the portions 

 of space lying within the spheres and those lying without. 

 The one kind of space is apparently regarded as susceptible 

 to atomic influence in some sense not exhibited by the 

 other. To state th's view in any definite manner probably 

 suflices to demonstrate its superficiality ; the question of 

 ascertaining what proportion of the total space is available 

 for atomic occupation by the use of assemblages of spheres 

 does not arise, because the spheres used are solely the 

 geometrical instruments for producing equality amongst the 

 atomic distances, and so determining the prevailing 

 equilibrium conditions. 



So far as the inquiry has been carried, it would seem 

 that the elements should crystallise either in the cubic or 

 the hexagonal system, and that in the latter case corre- 

 sponding dimensions in the horizontal and vertical direc- 

 tions should be in the ratio of a:c=i : 0-8165. The facts 

 are summarised in Table I. 



