June i8, 1891 



NATURE 



159 



by twelve others. Now suppose the spheroids arranged 

 as in Fig. 6, with their axes perpendicular to the plane of 

 the figure ; place the next layer in the black triangular 

 spaces, and complete the pyramid. The three faces of 

 the pyramid will be equal isosceles triangles ; and if the 

 spheroids be oblate, and the axis half the greatest dia- 

 meter, the three angles at the apex of the pyramid will 

 be right angles. The crystal will have cubic symmetry, 

 but the relative condensation in the faces of the cube, 

 octahedron, and dodecahedron, will be as i :o'5774: 07071. 

 The easiest cleavage would therefore be cubic, as in rock 

 salt and galena. 



Again, if the spheroids have their axes and greatest 

 diameters in the ratio of i : \/2, and we place four, as in 

 Fig. 7, with their axes perpendicular to the plane of the 

 figure, then place one upon them in the middle, and then 

 four more upon it, in positions corresponding to those of 

 the first four, we get a cubical arrangement, the centre of 



a spheroid in each angle of a cube, and one in the centre 

 of the cube. Crystals so formed will have cubic sym- 

 metry, but the concentration of molecules will be greatest 

 in the faces of the dodecahedron, and their easiest cleav- 

 age will be, like that of blende, dodecahedral. 



If spheroids of any other dimensions be arranged, as 

 in Figs. I and 2, with their axes perpendicular to the 

 plane of Fig. i, we shall get a crystal with the symmetry 

 of the pyramidal system. If the spheroids be prolate, 

 the fundamental octahedron will be elongated in the 

 direction of the axis, and if sufficiently elongated, the 

 greatest condensation will be in planes perpendicular to 

 the axis, and the easiest cleavage, as in prussiate of 

 potash, in those planes. On the other hand, if the 

 spheroids be sufficiently oblate, the easiest cleavage will 

 be parallel to the axis. 



If spheroids be arranged, as in Fig. 6, with their axes 



Fig. 7. 



perpendicular to the plane of the figure, they will, in 

 general, produce rhombohedral symmetry, with the 

 rhombs acute or obtuse, according to the length or short- 

 ness of the axes of the spheroids. The cubical form 

 already described is only a particular case of the rhombo- 

 hedral. If the ratio between the axes of the spheroids 

 and their greatest diameters be only a little greater, or a 

 little less, than i : 2, the condensation will be greatest in 

 the faces of the rhombohedron, and the easiest cleavage 

 will be rhombohedral, as in calcite. If the spheroids be 

 prolate, the easiest cleavage will be perpendicular to the 

 axis of symmetry, as in beryl and many other crystals. 

 Such crystals have a tendency to assume hexagonal 

 forms — equiangular six-sided prisms and pyramids. To 

 explain this, it may be seen in Fig. 6 that, in placing the 

 next layer upon the spheroids represented in the figure, 

 the three spheroids which touch that marked a may 



NO. I 129, VOL. 44] 



occupy either the three adjacent white triangles or the 

 three black ones. Either position is equally probable. 

 The layer occupying the white triangles is in the position 

 of a twin to that occupying the black triangles. So far 

 as the central parts of the layer are concerned, it will 

 make no difference in which of these ways the molecules 

 are packed. It is only at the edges that the surface- 

 tension will be affected. If the form growing be a 

 rhombohedron, a succession of alternating twins will 

 produce a series of alternating ridges and furrows in the 

 rhombohedral faces, which will give rise to increased 

 surface-tension, which will tend to prevent the twinning. 

 On the other hand, an hexagonal form and its twin, 

 formed in the way indicated, are identical, and we have 

 in this fact a cause tending to the production of hexa- 

 gonal forms. This tendency is increased by the fact 

 that, for a given volume, the total surface of the hexagonal 

 forms is in general less than that of the rhombohedral. 

 Indeed, such forms lend themselves to the formation of 

 almost globular crystals, as is well seen in pyromorphite 

 and mimetite. 



If the spheroids be arranged with their axes in other 

 positions than those we have been discussing, or if the 

 molecules occupy ellipsoidal space?, they will, when 

 packed so that each is touched by twelve others, give 

 figures of less symmetry. The results may be worked 

 out on the lines indicated in the foregoing discussion, 

 and will be found to correspond throughout to the 

 observed facts. 



Bravais long ago proposed various arrangements of 

 molecules to account for crystalline forms, and Sohncke 

 has extended them to further degrees of complication in 

 order to account for additional facts in crystallography. 

 But neither of them has given any reason why the 

 molecules should assume such arrangements. To me it 

 seems that only one arrangement can be spontaneously 

 assumed by the molecules, and that the varieties of crys- 

 talline form depend on the dimensions of the ellipsoids 

 and the orientation of their axes. Curie also has in- 

 dicated that the development of combined forms, as those 

 of cube and octahedron, will depend on the surface-ten- 

 sions in the faces of these forms, but he has not indicated 

 how the surface-tension is connected with the crystalline 

 arrangement, or why the energy of a cubic face should be 

 greater or less than that of an octahedral face. 



We are now in a position to understand the interesting 

 facts brought forward by Prof. Judd in a discourse de- 

 livered at the Royal Institution early this year. How- 

 ever long a crystal has been out of the solution, or vapour, 

 from which it was formed, its surface-tension will remain 

 unaltered, and when it is replaced it will grow exactly as 

 if it had not been removed. Also, if any part be broken 

 off it, the tension of the broken surface will, if it be not a 

 cleavage face, be greater than on a face of the crystal, 

 and in growing, the laws of energy necessarily cause it 

 to grow in such a way as to reduce the potential energy — 

 that is, to replace the broken surface by the regular planes 

 of less surface energy. The formation of "negative 

 crystals" by fusing a portion in the interior of a crystal- 

 line mass, is due to the same principle. Surfaces of least 

 energy will be most easily produced inside as well as 

 outside, and in a crystalline mass of course they will be 

 parallel to the external faces of the crystal. We see the 

 same thing in the action of solvents. Most metals assume 

 a crystaUine texture on cooling from fusion, and when 

 slowly acted on by dilute acids the surfaces of greater ■ 

 energy are most easily attacked, in accordance with the 

 laws of energy, and the undissolved metal is left with 

 surfaces of least energy which aie the faces of crystals. 

 This is easily seen on treating a piece of tin plate, 

 or of galvanized iron, with very dilute aqua regia. In 

 fact, solution is closely connected with surface energy. 

 It is probably the low surface energy of one form of 

 crystals of sulphur which makes them insoluble in carbon 



