156 



NATURE 



[June i8, 189, 



CRYSTALLIZA TION^ 



''pHERE is something very fascinating about crystals. 

 •'■ It is not merely the. intrinsic beauty of their forms, 

 their picturesque grouping, and the play of light upon 

 their faces, but there is a feeling of wonder at the power 

 of Nature, which causes substances, in passing from the 

 fluid to the solid state, to assume regular shapes bounded 

 by plane faces, each substance with its own set of forms, 

 and its faces arranged with characteristic symmetry : 

 some, like alum, in perfect octahedra ; others, like blue 

 vitriol, in shapes which are regularly oblique. It is this 

 power of Nature which is the subject of this discourse. 

 I hope to show that crystalline forms, with all their regu- 

 larity and symmetry, are the outcome of the accepted 

 ■principles of mechanics. I shall invoke no peculiar 

 force, but only such as we are already familiar with in 

 other facts of Nature. I shall call in only the same force 

 that produces the rise of a liquid in a capillary tube and 

 the surface-tension at the boundary of two substances 

 which do not mix. Whether this force be different from 

 gravity I need not stop to inquire, for any attractive force 

 which for small masses, such as we suppose the molecules 

 of matter to be, is only sensible at insensible distances is 

 sufficient for my purpose. 



We know that the external forms of crystals are inti- 

 mately connected with their internal structure. This is be- 

 trayed by the cleavages with which in mica and selenite 

 everybody is familiar, and which extend to the minutest 

 parts, as is seen in the tiny rhombs which form the dust 

 of crushed calcite. It is better marked by the optical 

 properties, single and double refraction, and the effects 

 of crystals on polarized light. These familiar facts lead 

 up to the thought that it is really the internal structure 

 which determines the external form. As a starting-point 

 for considering that structure, I assume that crystalline 

 matter is made up of molecules, and that, whereas in the 

 fluid state the molecules move about amongst themselves, 

 in the solid state they have little freedom. They are 

 always within the range of each other's influence, and 

 do not change their relative places. Nevertheless, these 

 molecules are in constant and very rapid motion. Not 

 only will they communicate heat to colder bodies in con- 

 tact with them, but they are always radiating, which 

 means producing waves in the ether at the rate of many 

 billions in a second. We are sure that they have a great 

 deal of energy, and, if they cannot move far, they must 

 have very rapid vibratory motions. It is reasonable to 

 suppose that the parts of each molecule swing, back- 

 wards and forwards, through, or about, the centre of 

 mass of the molecule. The average distances to which 

 the parts swing will determine the average dimensions of 

 the molecule, the average space it occupies. 



Dalton fancied he had proved that the atoms of the 

 chemical elements must be spherical, because there was 

 no assignable cause why they should be longer in one 

 dimension than another. I rather invert his argument. 

 I see no reason why the excursions of the parts of a 

 molecule from the centre of mass should be equal in all 

 directions, and therefore assume, as the most general 

 case, that these excursions are unequal in different direc- 

 tions. And, since the movements must be symmetrical 

 with reference to the centre of mass of the molecule, they 

 will in general be included within an ellipsoid, of which 

 the centre is the centre of mass. 



ix^^^^^ ^ ^^^' P^^'haps. guard against a misconception. 

 We chemists are familiar with the notion of complex 

 molecules ; and most of us figure to ourselves a mole- 

 cule of common salt as consisting of an atom of sodium 

 and one of chlorine held together by some sort of force, 

 and It may be imagined that these atoms are the parts of 



Frid^^S^-^^^Sr^C^'lJl'L^^y-^-- of Great Britain on 

 NO. I 129; VOL. 44I 



the molecules which I have in mind. That, however, is 

 not my notion. I am paradoxical enough to disbelieve 

 altogether in the existence of either sodium or chlorine 

 in common salt. Were my audience a less philosophical 

 one I could imagine I heard the retort from many a lip : 

 " Why, you can get sodium and chlorine out of it, and 

 you can make it out of sodium and chlorine ! " But no,, 

 you cannot get either sodium or chlorine out of common 

 salt without first adding something which seems to me 

 of the essence of the matter. You can get neither sodium 

 nor chlorine from it without adding energy ; nor can you 

 make it out of these elements without subtracting energy. 

 My point is that energy is of the essence of the molecule. 

 Each kind of molecule has its own motion ; and in this 

 I think most physicist^ will agree with me. Chemists 

 will agree with me in thinking that all the molecules of 

 the same element, or compound, are alike in mass, and 

 in the space they occupy at a given temperature and 

 pressure. The only remaining assumption I make is 

 that the form of the ellipsoid— the relative lengths of its 

 axes— is on the average the same for all the molecules of 

 the same substance. This implies that the distances of 

 the excursions of the parts of the molecule depend on 

 its constitution, and are, on the average, the same in 

 similarly constituted molecules under similar circum- 

 stances. 



I have come to the end of my postulates. I hope they 

 are such as you will readily concede. I want you to 

 conceive of each molecule as having its parts in extremely 

 rapid vibration, so that it occupies a larger space than it 

 would occupy if its parts were at rest; and that the 

 excursions of the parts about the centre of mass are on 

 the average, at a given temperature and pressure, com- 

 prised within a certain ellipsoid ; that the dimensions of 

 this ellipsoid are the same for all molecules of the same 

 chemical constitution, but different for molecules of 

 different kinds. 



We have now to consider how these molecules will 

 pack themselves on passing from the fluid state, in which 

 they can and do rhove about amongst themselves, into 

 the solid state, in which they have no sensible freedom. 

 If they attract one another, according to any law, and 

 for my purpose gravity will sufiice, then the laws of energy 

 require that for stable equilibrium the potential energy 

 of the system shall be a minimum. This is the same, in 

 the case we are considering, as saying that the molecules 

 shall be packed in such a way that the distances between 

 their centres of mass shall on the whole be the least 

 possible ; or, that as many of them as possible shall be 

 packed into unit space. In order to see how this packing 

 will take place, it will be easiest to consider first the 

 particular case in which the axes of the eUipsoids are all 

 equal — that is, when the ellipsoids happen to be spheres. 

 The problem is then reduced to finding how to pack the 

 greatest number of equal spherical balls into a given 

 space. It is easy to reduce this to the problem of finding 

 how the spheres can be arranged so that each one shall 

 be touched by as many others as possible. In this way 

 the cornered spaces between the balls, the unoccupied 

 room, is reduced to a minimum. You can stack balls so 

 that each is touched by twelve others, but not by more. 

 At first sight it seems as if this might be done in two 

 ways. 



In the first place we may start with a square of balls,, 

 as in Fig. i, where each is touched by four others. We 

 may then place another (shaded in the figure) so as to 

 rest on four, and place four more in adjacent holes to 

 touch it, as indicated by the dotted circles. Above these four 

 more may be placed in the openings abc d,soa.s to touch 

 it — making twelve in all. If the pile be completed, we shall 

 get a four-sided pyramid, of which each side is an equi- 

 lateral triangle, as represented in Fig. 2. It will be seen 

 that, in these triangular faces, each ball (except, of 

 course, those forming the edges) is touched by six others. 



