Crystals

by A. E. H. Tutton

Fig. 59.—Triclinic Space-Lattice.

But the fact that the structure is that of a space-lattice also causes the crystal to obey the law of rational indices. To enable us to see how this comes about it is only necessary to regard a space-lattice. In Fig. 59 is represented the general form of space-lattice, that which corresponds to triclinic symmetry. It is obviously built up of parallelepipeda, the edges of which are proportional to the lengths of the three triclinic axes, and their mutual inclinations are those of the latter. As we may take our representative point anywhere in the molecule, so long as the position chosen is the same for all the molecules of the assemblage, we may imagine the points occupying the centres of the parallelepipeda instead of the corners if we choose, for that would only be equivalent to moving the whole space-lattice slightly parallel to itself. Hence, each cell may be regarded as the habitat of the chemical molecule.

Now the faces of the crystal parallel to each two of the three sets of parallel lines forming the space-lattice will be the three pairs of axial-plane faces, and any fourth face inclined to them must be got by removing parallelepipedal blocks in step-wise fashion, precisely like bricks, as already shown in Fig. 12 (page [28]) in Chapter III., in order to illustrate the step by step removal of Haüy’s unit blocks. It will readily be seen that if one more cell be removed from each row than from the row below it, the line of contact touching the projecting corner of the last block of each row will be inclined more steeply than if two more cells were removed from each row. Moreover, the angle varies considerably between the two cases, and if three blocks are removed at a time the angle gets very small indeed. Hence, there cannot be many such planes possible, and we see at once why the indices of the faces developed on a crystal are composed of low whole numbers and why the forms are so relatively few in number. Owing to the minuteness of a chemical molecule, all the irregularities of such a surface are submicroscopic, and the general effect to the eye is that of a smooth plane surface.

The space-lattice arrangement of the molecules in the crystal structure thus causes the crystal to follow the law of rational indices, by limiting and restricting the number of possible facial forms which can be developed. It also determines which one of the seven systems of symmetry or styles of crystal architecture the crystal shall adopt. It does not determine the details of the architecture, however, that is, to which of the thirty-two classes it shall conform, this not being the function of the molecular arrangement but of the atomic arrangement that is, of the arrangement of the cluster of atoms which form the molecule, and this leads us to the next step in the unravelling of the internal structure of crystals.

The credit of this next stage of further progress is due to Sohncke, whose long labours resulted in the discrimination and description of sixty-five “Regular Point-Systems,” homogeneous assemblages of points symmetrically and identically arranged about axes of symmetry, which are sometimes screw axes, that is, axes about which the points are spirally distributed. Sohncke’s point-systems express the number of ways in which symmetrical repetition can occur. Moreover, the points may always be grouped in sets or clusters, the centres of gravity of which form a Bravais space-lattice.

This latter fact is of great interest, for it means that Sohncke’s points may represent the chemical atoms, and that the stereometric arrangement of the atoms in the molecule is that which produces the point-system and determines the crystal class, while the whole cluster of atoms forming the molecule furnishes, as above stated, the representative point of the space-lattice.

This, however, is not the whole story, for the sixty-five Sohnckian regular point-systems only account for twenty-one of the thirty-two crystal classes, the remaining eleven being those of lower than full holohedral systematic symmetry, and which are characterised by showing complementary right and left-handed forms. In other words, they exhibit two varieties, on one of which faces of low symmetry are developed on the right, while on the other symmetrically complementary faces are developed on the left; that is, these little faces modify on the right and left respectively the solid angles formed by those faces of the crystal which are common to both the holohedral class of the system and to the lower symmetry class in question. In some cases, moreover, these two complementary forms are known to exist alone, without the presence of faces common to both the holohedral class and the class of lower symmetry. The two varieties of the crystals are the mirror images of each other, being related as a right-hand glove is to a left-hand one.

Further, the crystals of these eleven classes very frequently exhibit the power of rotating the plane of polarised light to the right or to the left, and complementarily in the cases of the two varieties of any one substance, corresponding to the complementariness of the two crystal forms. The converse is even more absolute, for no optically active crystal has yet been discovered which does not belong to one or other of these eleven classes of lower than holohedral symmetry.

The final step of accounting for the structure of these highly interesting eleven classes of crystals was taken simultaneously by a German, Schönflies, a Russian, von Fedorow, and an Englishman, Barlow, who quite independently and by totally different lines of reasoning and of geometrical illustration showed that they were entirely accounted for by the introduction of a new element of symmetry, that of mirror-image repetition, or “enantiomorphous similarity” as distinguished from “identical similarity.” These three investigators all united in finally concluding that when the definition of symmetrical repetition is thus broadened to include enantiomorphous similarity, 165 further point-systems are admitted, and the whole 230 point-systems then account for the whole of the thirty-two classes of crystals.

Schönflies’ simple definition of the nature of the structure is that every molecule is surrounded by the rest collectively in like manner, when likeness may be either identity or mirror-image resemblance. Von Fedorow finds the extra 165 types to be comprised in “double systems” consisting of two “analogous systems” which are the mirror images of each other. Barlow proceeds to find in how many ways the two mirror-image forms can be combined together, there being in general three distinct modes of duplication, including the insertion of one inside the other. He also shows that all homologous points in a structure of the type of one of these additional 165 point-systems together form one of the 65 Sohnckian point-systems, the structure being capable of the same rotations or translations, technically known as “coincidence movements” (movements which bring the structure to exhibit the same appearance as at first), as those which are characteristic of that point-system.