Popular Science Monthly/Volume 29/May 1886/The Problem of Crystallization
| THE PROBLEM OF CRYSTALLIZATION. |
By ALFRED EINHORN, Ph.D., M. E.
CRYSTALS are symmetrical forms bounded by plane surfaces. A surface is said to be plane or level when its nature is such as is exemplified in a sheet of water extending over dimensions very small when compared to the radius of the earth. Crystals occur abundantly; they are generally diminutive and frequently microscopic in size, and therefore readily escape ordinary observation. Quite different in this respect are many forms caused by the rougher forces active in Nature, and analogous to crystals in the regularity of the shapes they assume. They are not unfrequently noted for their unique and startling appearance, as is instanced in the five-sided columns of basalt, known in some volcanic regions, and distinguished for their weird forms and the awe and superstition they give rise to among the inhabitants. Also many erosion figures resulting from the disintegrating action of water and air upon rocks. Many examples of this category may be seen in the scenic displays of unexcelled grandeur afforded by our far West. Not to these, but to a more commonplace phenomenon, I will now direct the attention of the reader, inasmuch as it is, mechanically speaking, related to and will serve to elucidate the subject under consideration. I have reference to a heap of particles of more or less uniform size, arranging themselves under the influence of the pull or gravity of the earth, with the provision that their magnitude should be very small relative to that of the whole heap. Thus, a grain or gravel heap is an excellent example of the phenomenon I refer to, and it is a very remarkable circumstance that different heaps have the same slope, provided the character of the material and the support upon which they rest remain the same. The slope (the inclination of the sides of the heap with the horizon) is dependent upon the magnitude and shape of the particles, and also upon the nature of the support; the whole system being subject to the gravity of the earth, they assume certain definite relative positions which determine the magnitude of the slope. In order to insure the same slope, the particles need not necessarily be perfectly alike, but the average size and shape of a limited number of them, chosen at random, should be uniform throughout. It is clear that the nature of the support must influence the slope of the heap, for, resting on a polished surface like a plate of glass, the slope is less than when supported on a rough surface, as a wooden floor. Generally, in a heap of gravel the slope is different from that of a heap of grain, inasmuch as the dimensions and shape of the grain-particles differ materially from those of the gravel-particles. Bearing in mind that the magnitude of each of the particles is very small, when compared with that of the heap, and therefore their number very large, we have then considered a state of aggregation of particles, assuming certain definite outward forms, these being dependent upon known causes, which we can readily modify at will, so as to produce forms with stated slopes. Mechanically, this may be said to be entirely analogous to the problem of crystallization. There also we have states of aggregation of particles occurring in definite regular shapes of infinite variety, depending upon the nature of the substance and the nature of the force active between the ultimate particles, and the problem of crystallization is solved when the nature of the ultimate particles and of the force which holds them in their relative positions in the crystal has become known to us.
These are the actual questions under consideration, and before proceeding with their further discussion we cite some instances of crystallization of substances, rendered familiar to us, either through their utility in the arts and industries, or the recognized value they have by reason of their rarity and beauty. In Fig. 1 a crystal of diamond is
| Fig. 1. | Fig. 2. | Fig. 3. |
represented; the beauty and value of this gem are greatly enhanced by the cutting process; the remarkable property of cleavage, which all crystals possess to a greater or less extent, is well developed in the diamond, and skillfully utilized in its cutting. The form shown in the figure occurs at the Cape, and has a yellow tinge; the bluish-white Brazilian diamond is preferred. A crystal of hematite (iron-ore) is shown in Fig. 2; it occurs in the Island of Elba, has an iron-black color and metallic luster, while its powder is reddish-brown like ordinary iron-rust. Fig. 3 is a crystal of calcite remarkable for its optical property of double refraction and its ready cleavability in certain directions; in substance it is the same as ordinary marble; in fact, the latter consists of microscopic crystals of calcite. In Fig. 4 we have a crystal of garnet, not unfrequently seen in the mica-slates of New York. A crystal of sulphur from Girgenti, Sicily, is shown in Fig. 5; that locality abounds in fine transparent crystals of this substance. Fig. 6 represents a cube
| Fig. 4. | Fig. 5. | Fig. 6. |
of native silver as found in Konigsberg, Norway; and, finally (Fig. 7), a crystal of cassiterite (tin-ore) from Cornwall, in England, which has also been discovered in this country in the Black Hills, Dakota Territory. There are seven systems of crystallization, differing in the relative magnitudes and directions of certain lines of symmetry, termed the axes of the crystal. In the first, second, and third systems, these lines bear the same inclination to one another, but their magnitudes are respectively equal in the first system (see Fig. 8) (here A A', B B',
| Fig. 7. | Fig. 8. | Fig. 9. |
C C,' are the three axes equal in magnitude and inclined at right angles to one another), equal in two of them in the second or dimetric system (Fig. 9) (here A A' equals BB', but C C' is different from these), and unequal in all three axes in the third or trimetric system (Fig. 10) (here the axes A A', B B', and C C, are all of unequal magnitudes, but their mutual inclinations in this as well as in the second system are equal). In the three oblique systems the axes are partly or altogether obliquely inclined to one another, while their magnitudes are unequal. Fig. 11 is a crystal of the monoclinic system, and Fig. 12 of the triclinic system. The names of the different oblique systems indicate the mutual inclinations of the axes. Fig. 13 represents a crystal of the hexagonal system, which is allied in symmetry to the dimetric system; but there are four lines of symmetry, of which the three A A', B B', and C C, lying in the
| Fig. 10. | Fig. 11. | Fig. 12. |
same horizon, are equal in their mutual inclination and magnitude, while the fourth axis, D D', is at right angles to these but different in magnitude.
The reader will now have formed a tolerably correct idea of a crystal, and when it is borne in mind that crystallization is a widely diffused and essential property of matter, and also that the solution of this question has engaged some of the ablest minds of the century, the high purpose and importance of this investigation will perhaps become evident to him.
Now, the invariability of certain relations existing between the axes and the planes bounding crystal forms are geometrically similar, and are effects produced by causes similar, 
Fig. 13. to those which occasion the constancy of the slopes in heaps of the same material. In the heap of gravel considered above, the horizon was chosen as the reference plane—in the crystal the planes containing the lines of symmetry are selected as reference planes, whereby to gauge the inclination of the bounding surfaces. From our considerations of the heap of gravel, the reader will perceive the intimate connection between outward form and internal structure, and is in a measure prepared to follow deductions made from the one upon the other. Already in the remote infancy of mineralogy assumptions as to the internal structure of crystals were made to explain the axial relations alluded to. The assumption that the internal structure of a crystal is similar to, and in a measure identical with, the internal structure of a cannon-ball pile, is sufficient to explain the axial relations observed in the first, second, and third systems of crystallization. In the first system the ultimate particles of the crystal are symbolized by the sphere, while in the second and third systems they are figures of oval form. The cannon-ball pile arrangement, or, as it is termed, the tetrad configuration, is represented in Fig. 14 (perspective of vertical circles of contact of the spheres); it derives this name from the fact that its type consists of four equal mutually touching spheres (Fig. 15). If in such an arrangement of particles sections are made in certain directions, we obtain the faces of the several crystal forms. In this manner the octahedral face (Fig. 16), the cubical face (Fig. 17), and the dodecahedral face (Fig. 18), have been obtained. In an octahedron, or in a cube, or in a dodecahedron, represented respectively in Figs. 8, 6, 4, and respectively composed of layers as indicated in Figs. 16, 17, 18, the ultimate particles have the same common arrangement, that is, the tetrad grouping. These forms, as has been shown above, all occur in nature; but as yet the most powerful microscope has been unable to dissolve a crystal face into its ultimate particles. Still, they are not insensibly small; their dimensions are shown to lie between certain limits, ascertained by combined computation and observation, and it is highly satisfactory that physicists have approximately obtained the same results in this direction, although the methods chosen were different. And it is the fact that we are dealing with invisibly small particles which renders the problem under consideration one of peculiar difficulty and interest. Instead of the tetrad configuration, there is a second grouping of particles, which would also serve to explain the observed axial relations of crystals. It is deduced from Fig. 19, by placing the layer of spheres marked a centrally over the layer marked b. But this grouping can not exist permanently in Nature; it is, as I have elsewhere shown, in a mechanical state similar to that of an exceedingly thin coin placed on its edge—the slightest effort, tending to upset the coin, would do so—it is what is termed a position of unstable equilibrium, and therefore can not exist permanently; the tetrad configuration, on the contrary, is in stable equilibrium.
We have thus already almost involuntarily introduced force as a factor in our considerations, and the deductions already made from outward form upon internal structure must necessarily also embrace considerations of the forces that the ultimate particles are subject to; and again, in order to bring the subject within the natural sphere of conception of the human mind, we will analyze the force transitions and the force law in a cannon-ball pyramid, subject to the gravity of the earth, preparatory to proceeding with the more remote and recondite subject of crystallization. In Fig. 14 it is clear that the weight of the top ball is distributed among the lower three, in the three direction lines joining the centers of the top and three lower balls respectively. On examination of a pyramid composed of a larger number of balls, we observe that every ball of the pyramid bears the weight only of those balls that are arranged in three lines parallel to upper edges of the pyramid respectively, and meeting in the center of the ball. Thus, in Figs. 15 a and 15 b are represented in plan the four layers of a pyramid of twenty balls. The ball a, of the lowest layer, can only receive the weights of the balls b1 b1 b1 of the second layer, transmitted in direction-lines parallel respectively to the three upper edges of the pyramid (Fig. 14), namely, D A, D A0, and D A3. The ball a
Fig. 14. can not receive the weight of any other ball of the pyramid; it can not receive the weight of the topmost ball, d, inasmuch as the weight of this ball is transmitted only in the lines DA, DA, and DA3, the three upper edges of the pyramid; nor can it receive the weight of the ball C of the third layer, for that is only transmitted in three lines, of which two, C A, and C A3, can be seen in the figure. By a simple application of the physical principle known as the parallelogram of forces, we arrive at the deduction that all balls equidistant from the vertex of the pyramid are solicited by the same force; or, in other words, that every ball of the pyramid is repulsed from the vertex with a force proportional to its distance from the vertex, as a direct con-sequence of this stress distribution. At the vertex itself the repulsion is zero. The weight of the pyramid is uniformly distributed over its base; a result which can readily be verified by experiment, and is also a verification of the stated force law. Now, an exactly analogous action occurs among the invisibly small particles of a crystal. In the pyramid of balls, it is the pull of the earth upon each ball which is active; in the crystal it is the mutual attraction of the particles.
Fig. 15.
In the pyramid of balls, there are only three stress direction-lines respectively parallel to the upper edges of the pyramid, inasmuch as the pull of the earth acts only vertically downward, hence there is no weight transmission in the three horizontal direction-lines parallel to the basal edges respectively; in the crystal, however, there are six stress direction-lines, inasmuch as the mutual forces between the ultimate particles of the crystal act in all the directions joining the centers of the particles respectively. That the stress can only be transmitted in six direction-lines is evident from the following consideration: In a pyramid of four balls (Fig. 15 b) we have evidently the
Fig. 16. six stress direction-lines joining the centers of the balls respectively. In case of a larger number of particles in contact, it is clear that in an octahedral face (Fig. 16) the stress can only be transmitted in three direction-lines, A A1, B B1, and C C1, for there is no contact between the particles which would allow the stress to be transmitted in any other direction; in the cubical face (Fig. 17) there are but two stress direction-lines, D D1, and E E1, and in the dodecahedral face (Fig. 18) there is but one stress direction-line, F F1; and generally on any particle in the tetrad configuration the stress can only be transmitted in six direction-lines, respectively parallel to the six edges of the pyramid. All this applies not only to the first or monometric system 
Fig. 17. of crystallization, in which the ultimate particles are symbolized by the spherical form, but also to the dimetric and trimetric systems in which the ultimate particles are symbolized by an oval form. But this analogy between the pyramid of balls and crystals holds not only for the stress distribution, but extends also to the law of the forces active between the ultimate particles. In order to satisfy the equilibrium condition, the physical doctrine demands a unique law of force for a stated stress distribution, and elsewhere I have shown this law to be Every particle is attracted to the center of the crystal with a force proportional to its distance from the center; while the law for the ball pyramid is Every particle is repulsed from the vertex of the pyramid with a force proportional to its distance from the vertex.
The proper mathematical interpretation of the stated force law in crystals shows its perfect identity with the Newtonian law of gravitation, 
Fig. 18. according to which every particle of the universe attracts every other particle, with a force proportional to the product of the masses, and inversely as the square of the distance. Thus, the symmetry, beauty, and definiteness displayed in the infinite variety of crystal forms have necessarily impressed themselves upon the observing mind, ever since the remote period of the dawn of the natural sciences, as the silent carriers of a law of profound influence upon the nature of substances. That this law, in obedience to which the planets are swept through space, should also regulate the position of the tiny crystal molecule, is a striking instance of the truism, in accordance with which the essences of things are not affected by their magnitude, and without which the human mind could not conceive the interaction of the forces of Nature.
The stated law also governs the interaction of electrical masses. Now, the only reason why it applies to the ultimate particles of a crystal 
Fig. 19. is their tetrad arrangement. Hence the tetrad grouping of the ultimate particles, and therefore crystallization, is caused by an agent which acts like electricity. Very probably it is electricity itself, as is evidenced by the electrical properties of certain crystal forms, which appear to establish an intimate causal connection between the structure of crystals and this agent. This is illustrated in many so-called hemimorphic forms: these are forms in which opposite ends of a crystal, instead of being bounded by faces of the same form, are bounded by faces belonging to different forms. This phenomenon occurs in crystals of tourmaline, topaz, and calamine. The ends which show this peculiarity alternately exhibit positive and negative electricity—the one kind when the mineral is heating, and the other while it is cooling. The experiments of Faraday and Tyndall also indicate this causal connection. Thus the problem of crystallization may be said to have arrived at the stage of a partial solution, and the manner in which the result has been obtained clearly shows why an agent like electricity is the cause of crystallization; it also shows a perfect definite relation existing between the intensity of this agent and the crystal form. When it is considered that difference in crystal form is, as a rule, associated with difference in chemical composition, it is easy to conceive how profoundly important this relation is in the chemism of substances. The intimate causal connection between electricity and chemical affinity is well accepted.
The law of the periodicity of the elements, discovered by the Russian chemist, Mendeljief; the investigations of Kekule on the aromatic compounds, which throw a strong light upon their structure; the law of Dulong and Petit, as to the constancy of the relation between the heat and atomic weight of the elements—all these give just grounds for the remark that, when brought into proper connection with the stated law of crystallization, an epoch may result in our knowledge of atoms.
