49- 



NATURE 



[July io, 1913 



structure the atomic point-system. The latter deter- 

 mines the class of symmetry (which of the thirty-two 

 classes is exhibited), and therefore governs any hemi- 

 hedrism or tetartohedrism, as the development of less 

 than full systematic symmetry used to be called. But 

 it is the space-lattice which governs the crystal 

 system, that is, which determines whether the sym- 

 metry is cubic, tetragonal, rhombic, monoclinic, tri- 

 elinic, trigonal, or hexagonal, and also deter- 

 mines the crystal angles and the disposition of faces 

 in accordance with the law of rational indices, the law 

 which limits the number of possible faces to those which 

 cut off small whole-number relative lengths from the 

 crystal axes. Indeed, it is because only those planes 



a b c o' 



b'lG. t. — Kedorow's types of parallelohedra. 



which contain the points of the space-lattice are pos- 

 sible as crystal faces that the law of rational indices 

 obtains. For any three points of the space-lattice 

 determine a plane in which similar points are 

 analogously regularly repeated, and which is a pos- 

 sible crystal face obeying the law of rational indices. 

 Moreover, those facial planes which are most densely 

 strewn with points are of the greatest crystallographic 

 importance, being what are known as the primary 

 faces, either parallel to the crystal axes or cutting oif 

 unit lengths therefrom, as well as being usually the 

 planes of cleavage. 



As the space-lattice units are all sameways orien- 

 tated, any one atom of the molecular or polymolecular 

 grosser unit might be equally well chosen as the 

 representative point of the lattice, so long as a similar 

 choice were made in every space-lattice unit, and the 

 resulting space-lattice would be. the same whichever 

 atom were so selected. Consequently, the space- 

 lattice is afforded by the similarly (identically) situated 

 atoms of the same chemical element throughout the 

 crystal structure. The combined point-system (one of 

 the 230 possible point-systems) may thus be con- 

 sidered to be built up of as many identical but inter- 

 penetrating space-lattices as there are atoms in the 

 space-lattice grosser unit. These facts are concisely 

 expressed in the definition of crystal structure which 

 was stated as follows by Prof, von Groth at the Cam- 

 bridge meeting of the British Association in 1904 : — 



"A crystal — considered as indefinitely extended — 

 consists of » interpenetrating regular point-systems, 

 each of which is formed from similar atoms ; each of 

 these point-systems is built up from n interpenetrat- 

 ing space-lattices, each of the latter being formed 

 from atoms occupying parallel positions. All the 

 space-lattices of the combined system are geometric- 

 ally identical or are characterised by the same elemen- 

 tary parallelepipedon." 



Having thus arrived at a comprehensive idea of 

 crystal structure on the assumption of each atom and 

 each grosser space-lattice unit being only a point, 

 as far as which we are on safe and assured ground, 

 we may proceed to the consideration of the various 

 ideas advanced concerning the character of the units 

 of structure thus represented by points ; that is, con- 

 cerning: the mode in which the space around the point 

 is more or less filled up. 



The valency theory of Barlow and Pope considers 

 NO. 2280. VOL. 91] 



the atomic point to be expanded into the sphere of 

 the atom's influence, the relative size of which in 

 any one substance is supposed to be proportional to 

 the fundamental valency of the chemical element of 

 which the atom is composed. The spheres are further 

 assumed to be pressed together on crystallisation 

 until thev fill space, becoming thereby deformed 

 into polyhedra. The theory of von Fedorow, on the 

 other hand, considers the grosser or space-lattice 

 units to be parallelohedra ; besides those corresponding" 

 to the fourteen space-lattices there are nine other 

 parallelohedra (making twenty-three in all) composed 

 of simple Sohnckian point-systems compounded of in- 

 terpenetrating space-lattices. All the twenty-three 

 parallelohedra are arranged parallelwise, 

 and fill space without interstices. There 

 are, however, only four types, namely tin- 

 cube, the rhombic dodecahedron (which 

 has a second vertically elongated variety), 

 the cubo-octahedron, and the hexagonal 

 prism, the first three being all of cubic 

 symmetry, and the fourth of obviously 

 e hexagonal symmetry. They are shown, 



including the second variety of the dode- 

 cahedron, in the next screen picture 

 (Fig. 2). He further considers that 

 [ all four may be homogeneously deformed into 

 j analogous parallelohedra of lower orders of sym- 

 metry, without ceasing to fill space when closely 

 packed. Hence, von Fedorow concludes that all 

 crystal structures are of either cubic or hexagonal 

 type, including not only truly cubic and hexagonal 

 crystals, but their deformed derivatives. 



Unlike the atomic polyhedra of Pope and Barlow, 

 these parallelohedra of von Fedorow are either mole- 

 cular or polymolecular, in the latter event being made 

 up of a small number of identically or symmetrically 

 similar subpolvhedra, termed by him " stereohedra," 

 which represent the chemical molecules, just as already 

 explained, when the grosser space-lattice unit is poly- 

 molecular, the stereohedra being arranged to build up 



Fig. 3. — Fedorow's stereohedra. 



the main parallelohedron (the space-lattice unit) on a 

 definite plan, which may involve mirror-image juxta- 

 position. For example, a rhombohedral system of stereo- 

 hedra is shown on the screen (Fig. 3), consisting of 

 two kinds, R and L, one sort being the mirror-image of 

 the other. Each rhombohedron representing the com- 

 bined system is composed of six stereohedra, three of 

 each kind, and a series of points, similarly situated 

 one within each stereohedron R, would constitute a 

 Sohncke point-system, while a "double-system" is 

 obtained by adding a series similarly situated one 

 within each stereohedron L. If a single point were 

 taken to represent analogously each rhombohedral set 

 of six stereohedra, we should have a rhombohedral 

 space-lattice produced. 



The valency theory of Barlow and Pope may or 

 may not in the sequel prove to be correct, and some 



