All Lectures Crystallography J. Crystal Mathematics 1. Introduction We are all familiar wit h the subdivision of the natural world into th e plant, animal, and mineral kingdoms. Of these the realm of mineral s is the study and resear ch area of mineralogy. A min eralogist uses the word mineral, however, in a much more restricted sense than prevails with common usage. If we examine a rock rather closely, we notiee, for example, tha t a sandstone is composed of individual quartz grains. When we examine a specimen of granite, we ean recog­ inze feldspar, dark mica, and perhaps also hornblende, in addition to quartz. From a specimen of certain ores, galena, ehalcopyrite, and sphalerite all shine forth brilliantly. Any such individual rock constituent is a min eral. Mineralogy encompasses the study of these min erals and the manner in which they oceur , and includ es the properties of all rock materials, which, in the sense used here, embra ce ore and salt deposits, as weIl as ot her useful min eral deposits. Minerals occur as eonstituents of rocks not only in assemblage , but are found also as separate individuals. Frequ ently these have the opportunity to develop without restrietion in eavities or in surrounding media which could yield to their growth. Under these conditions they often develop into polyhedra l form s called crystals. Th ese have from time immemorial attracted the attention of thoughtful natural scientists as weIl as laym en. Earlier all natural bodies with planar external boundaries were called crystals. For exa mple, A. G. WERNER, the patriarch of min eralogy, considered that basalt columns were crys tals, because of their somewhat polyh edral shape. Today the concept of a crys tal and a mineral is restrieted to a homoqeneous body, that is, one composed of only one substance. Basalt columns, which are composed of feldspar, augite, and ot her minerals, are not crys tals in the modern sense. In addition we kn ow now that the regular outer form or morphology of a crys tal is neither the unique nor truly diagnostic eharacteristic of the crystalline state . It is the regularity of the internal strueture which determines the unique characteristics of crystalline matter. As this kn owledge developed during th e last century, it was further recognized that almost all solid bodies are erystalline. Since the earlier studies of crystals involved almost exclusively natural produets, minerals, the study of crystals developed as a bran ch of mineralogy. The regular internal structure of crystals determines their extern al geometrie form and is also manifest by other phenomena. If , for example, we strike a piece of galena or rock salt, it always breaks alon g plan es parallel to the surfaces of a eube. The mineral hornblend e on the other hand breaks or cleaves into four­ sided prismatic forms with prism angles of 124 and 56°. A property common to many min erals is exemplified by kyanite, which also occurs as prismatic crystals. Kyanite can be scra tched with a steel needle in the dir ection of elongation of the prism, but not perpendi cular to it . Another interesting mineral is cordierite , a magnesium aluminium silicate named after the French mineralogist CORDIER. If a cube is cut from a cordierite crystal in the proper orientation and viewed in each of the three perpendicular direetions corresponding to the cub e edges, 1· 4 Crystal Mathematics it will be noticed that the color is different in each direction, namely blue, lilac, and yellow. Directional dependence of color has been established for many other minerals also. If in addition we investigate the heat conductivity of quartz, we can ascertain that it is about 40% greater in the direction parallel to a prism edge, than perpendicular to it. This difference can be demonstrated very nicely also in the case of gypsum. If a cleavage fragment is coated with wax or paraffin and then probed with a hot needle, the wax melts around the source of the heat clearly in the form of an ellipse, not a circle. All of these observations lead to the conclusion that crystals are bodies for which many properties, such as external shape, cleavage, color, hardness, and heat conductivity, are dependent upon direction. This directional dependency of geometrical and physical properties is called anisotropism (Greek. iso, equal; tropos, direction; an , neg .), It is characteristic of every kind of crystal. Orystals of very high symmetry show no directional dependence for certain physical properties, such as the transmission of light, and are said to be isotropic. With respect to other properties, such as tensile strength, the same crystal may be anisotropie. Orystals are, therefore, first of all homogeneous substances, and secondly they are anisotropic. Very long ago the recognition of the directional dependence of properties led to attempts at an explanation. The idea finally evolved that crystals were constructed of tiny building blocks. Thus the Hollander, OHR. HUYGENS in 1678 explained the cleavability, the directional variation in hardness, and the double refraction of calcite, in terms of an arrangement and orderly grouping together of very small and invisible flattened ellipsoidal units. In Sweden TORBERN BERGMANN (1773) and in France RENEE JUST HAÜY (1782) introduced the concept of "integrating molecules", of building blocks, whose shapes should correspond to the shapes of cleavage fragments of crystals. As early as 1824, SEEBER, in Freiburg i. Br., formulated a concept of the arrangement of points in space quite similar to that which prevails today. So we see that the fundamental property of a crystal, its directional nonequivalence, has led from the very beginning directly to a concept of orderly arranged building units. For some time mathematicians, mineralogists, and physicists have been concerned with theoretical investigation of such regular repetitive arrangements. As early as 1891 the mineralogist FEDOROV in Petersburg and independently the mathematician SCHÖNFLIESS in Koenigsberg proved that there are only 230 symmetrically different ways of arranging points in space. These regular spatial arrangements are called space groups. Since 1912 it has been possible to investigate these space groups experimentally, thanks to the discovery by MAX VON LAUE of X-ray interference by crystals. This discovery has been of the greatest signif­ icance to crystal science. It has been learned since that the "centers of gravity" of atoms, or ions and molecules, actually confrom to regular point group arrange­ ments. Their spatial relationships can be measured. The interatomic distances are of the order of magnitude of 1 Angstrom (10-8 cm = 1/100,000,000 cm). Discussion of the methods of measurement will be reserved for a later chapter. Weshall next familiarize ourselves with space lattice relationships. A disordered array of points also shows different relationships in different directions, but only as long as we consider a region of few points within the total array. If we integrate some property over long distances in the disordered point system, we will encounter like conditions in all directions. The situation is quite different in ordered point systems. It can be rather easily demonstrated that there is no arrangement of points in space whose regular and periodie spacings do not vary with direction. This would appear to be inconsistent with the concept Introduction 5 of isotropism. Isotropism arises only statistically as the result of integration of som e property over long distan ces in a point array. The separation of lattice particles from one another is so small, that over large distances we obtain only average values with the usual methods of measuring properties. For example, a gas in which particles are constantly in motion is isotropie, as is a fluid or a glass, which is a fluid solidified as a result of supereooling. Such incompletely ordered matter in the solid state is said to be amorphous because it possesses no tend ency toward development of a char­ acteristic form. The most importan t char­ acteristic of a crys tal, on the other hand, is its regular atomic arrangement in space. Thus the directional depend ence of crys­ Fig. 1. P lana r lattice talline properties can be understood. A planar net or lattice is shown in Fig. 1. Along any particular set of parallel lattice lines , points are distributed periodically. In different directions the spacing of points differs. The arrangement is anisotropie. What can we say about the homogeneity of such lattices ? As far as th e small portion in Fig. 1 is concerned, the structure is discontinuous, not hom ogeneous. a b ~I' II ÜW11:1ili1,'11 I 1IIIIIIIIIIIIIII IIt " 1111 11 1111111 LllJlJWU c d e Fig. 2a - e. Different arra ngements of rod-shap ed particles showing the transition from amorp hous (a) to nemati c (b) (nerna, Gr., threa d) and smectitic (c, d) (smektein, Gr., salve) mesoph ascs and to a crystal (e) A point array becomes homogeneous if we consider larger expanses of an unlimited, extended array: that is to say, when each point possesses an infinitely large nu mber of corresponding or equivalent points in the extended arrangement (B ARLow,1888). We can mark off a repetition of points in a straight line giving rows. We can project these linear patterns within a plane creating planar point nets, and finally we can stack and build these together in space pr odu cing a space lattice. The only pre-requisite for real homogeneity is that the structure be regular. The structural units are repeated periodically. Space lattices conform, 6 Crystal Mathematics therefore, not only to the property of anisotropism, but also to that of genuine homogeneity. Crystals possess a corresponding atomic arrangement. An amorphous body also possesses a sort of homogeneity.