Physics and Crystallography.

G. V. Wulff.

The years 1919 and 1920 were anniversary years for the physics of crystals. Two hundred and fifty years ago, in 1669, the Danish scholar, a member of the Florentine Academy of Experimental Sciences—Academia del Cimento—Nicolas Steno (Nikolaus Steno) published a book entitled De solido intra solidum naturaliter contento, i.e., “On a solid body naturally contained within another solid body.” A year later there appeared a book by another Danish scholar, Erasmus Bartholinus (Erasmus Bartholinus), under the title Experimenta crystalli Islandici disdiaclastici, quibus mira et insolita refractio detegitur, i.e., “Experiments with Iceland spar, in which a wonderful and unusual refraction is revealed.” With regard to the second work there can be no doubt that it laid the beginning of our knowledge of double refraction in crystals. This was indeed the first physical phenomenon, among all those characteristic of crystals, to attract attention. The observation of Erasmus Bartholinus served as material for the classical works of H. Huygens and formed the basis of all crystal optics. Therefore, it would seem that Erasmus Bartholinus should be regarded as the founder of crystal physics, especially since, with regard to Steno’s work, the objection will undoubtedly follow that, although it did form the basis of a new science, that science was not the physics of crystals, but crystallography. This objection has a certain basis and rests on the generally accepted view of the significance of Steno’s discovery—the so-called law of constancy of crystal angles. However, this view contains a misunderstanding, since upon closer examination it turns out that precisely this law must be regarded as the fundamental and initial law of the whole physics of crystals, and therefore the founder of this branch of physics must be considered to be precisely Nicolas Steno, while Erasmus Bartholinus discovered only one of the many phenomena characteristic of crystals. We, however, have no thought of diminishing the significance of the discovery of double refraction, which has now become one of the most studied phenomena and one of the most important properties of crystals, serving for their determination in mineralogy and petrography. Thus,

Thus, without in the least diminishing the merits of Erasmus Bartholinus, we shall devote our essay to the law discovered by Steno, endeavor to disclose its content, indicate its most important consequences, and show that this law is, in fact, the fundamental law of crystallophysics.

I. The concept of a medium. As is known, physics is concerned with the study of natural phenomena, whose variety is infinite.

Let us consider two universally known physical phenomena—the propagation of sound and the propagation of magnetic forces. Let us take an alarm clock, make it ring, and, placing it on the plate of an air pump, cover it with a bell jar and begin to pump out the air. As the air is pumped out, the sound of the bell becomes weaker and, finally, ceases altogether to be heard, although the hammer of the bell continues to strike the bell of the alarm clock with its former force. If a magnetic needle is placed under the bell jar of the pump, it will begin to turn on its support when a magnet is brought near the glass bell jar of the pump, and this action of the magnet on the needle does not cease even after the air has been pumped out. On the contrary, the needle becomes still more mobile, encountering no resistance from the air to its rotation.

What conclusion do we draw from our experiment? Apparently, that sound phenomena cannot occur in airless space, in a vacuum, whereas magnetic ones can. Light, too, can propagate in a vacuum, since we see both the alarm clock and the magnetic needle through the emptiness surrounding them. If we were to place an electroscope under the bell jar of the air pump, its leaves would diverge from an electrified body brought near the bell jar, which indicates that electric forces too can propagate in a vacuum.

However, the thought of the modern physicist is by no means reconciled to such a conclusion. We believe that every phenomenon can occur only in a medium, that in a vacuum no phenomenon is possible, and that in nature there are no forces acting at a distance without the mediation of intermediate parts of space occupied by some medium.

We regard M. Faraday as one of the most brilliant physicists precisely because, rejecting the possibility of the action of magnetic and electric forces at a distance, he was the first to form a clear conception of the manner in which these forces are transmitted from point to point in the medium surrounding electrified bodies and magnets.

Having rejected action at a distance, we also reject the existence of empty space and believe that, having pumped out from under the bell jar of the air pump the air, which is evidently necessary for the propagation of sound, we nevertheless have not pumped out something else necessary for the propagation of light and of electric and magnetic forces. This something is not like ordinary matter, such as air or water vapor; it is not material in the ordinary sense of the word, or, as they say, it is not ponderable. We call this something ether.

We shall call a medium that which, filling space, makes physical phenomena possible within it. If we could also pump out the ether, we would obtain one geometrical space in which, in our opinion, no phenomena could occur.

We see that the concept of a medium does not coincide with the concept of substance: the ether is a medium, but not a substance in the ordinary sense of the word. By virtue of this definition, we transfer to the medium certain known properties of space. Thus, we may conceive of the medium as boundless, as, for example, world space.

If a medium is bounded on all sides by surfaces separating it from the surrounding space, then we call such a portion of the medium a physical body, or simply a body.

2. Crystals and crystallography. Bodies may be gaseous, liquid, and solid. In our essay we shall deal only with solid bodies, and moreover with crystallized ones.

Considering a number of crystallized bodies—for example, crystals of quartz, rock salt, Iceland spar, feldspar—we shall immediately see that all these are bodies bounded by plane surfaces, faces, which have arisen on them of themselves, during the very formation of these bodies. This polyhedral form is essentially characteristic of crystals and must have its own special laws. We have a natural need to understand these laws, to clarify for ourselves the nature of this astonishing geometrically regular form. A science of crystals therefore had necessarily to arise, and indeed such a science exists and is called crystallography.

3. Phenomena observed in crystals. Crystals are of interest not only because of their regular form. They are also of interest for their physical properties. In a crystalline medium phenomena may occur that are in no way dependent on the form of the crystal.

Let us grind or cleave from a crystal of Iceland spar a plate and cover with it a bright little hole in a piece of cardboard: the hole will appear to us doubled. Every ray of light entering the interior of the Iceland-spar crystal undergoes such a doubling, or, as it is said, double refraction. Only a ray passing in one direction in the crystal, along the optical axis of the crystal, shows no doubling.

This phenomenon was discovered in 1669 by Erasmus Bartholin on crystals of the same Iceland spar, and served as material for H. Huygens in his classic investigation in 1672.

Double refraction is the source of many, sometimes very beautiful, phenomena observed in crystals.

Let us pass a beam of light through a thick plate of Iceland spar prepared in such a way that it is possible to make use of

only one of the two rays passing through it. Such a plate is called a Nicol prism and already has the appearance of a column rather than a plate. Let us make a beam of rays convergent by means of a system of strong converging lenses and direct its vertex into a plate of Iceland spar, cut from the crystal perpendicular to the optical axis. Then let us pass the rays through another system of strong lenses and another Nicol prism and, finally, obtain on a screen, by means of a lens, an image of that place in space where the second system of lenses can give an image of very distant objects. We shall obtain a beautiful system of rainbow circles, crossed through the center by a dark or light cross, depending on the relative position of the two Nicol prisms. The center of the figure corresponds to the optical axis of the crystal, i.e. to the direction in which there is no double refraction.

There are crystals with two optical axes, such as, for example, aragonite. If one takes a plate from such a biaxial crystal, then on the screen we obtain two eyes, surrounded by a system of rainbow curves, first closing around each of the eyes and then embracing both eyes together.

In these two examples—the interference figure in Iceland spar and in aragonite—the distinction between the concept of a medium and the concept of a substance stands out with particular clarity. Iceland spar and aragonite are one and the same substance—calcium carbonate. Nevertheless, the phenomena in Iceland spar and in aragonite are different, evidently because in them the substance calcium carbonate forms two media with different properties. We are inclined to ascribe this difference to the different arrangement of the particles of the substance in the two crystals.

The phenomena described, and the countless number of others observed in crystals, make crystals very interesting for the physicist. This interest is increased further by the fact that in crystals one can observe phenomena that cannot occur in other solid bodies. Among such phenomena is the ability of crystals to become electrified from a change in temperature, the so-called pyroelectricity.

For the physicist, a crystal appears as a special medium in which special phenomena are possible. In order to study these phenomena, the physicist makes preparations from crystals, for example, plates, Nicol prisms, and so on. For this, the crystal must first of all be deprived of its external form, without sparing its beauty.

Apparently, the form of a crystal is not of interest to the physicist, or may be of interest only insofar as it can help in studying the phenomena occurring in the crystal. In this connection, it would seem that for the physicist the science of the form of crystals, crystallography as it appears at first glance, cannot be of interest either.

Known for his work in the field of the physics of crystals, W. Voigt expresses himself thus in his textbook of crystal physics: “In order to enter the promised land of crystal physics, it is by no means necessary to traverse the whole desert of crystallography; it is by no means necessary to become acquainted with all the wealth of crystal forms, however attractive these forms may be in themselves.”

Let us see, then, whether crystallography is indeed unnecessary to the physicist, as it seems at first glance and as such eminent authorities as W. Voigt assert. We shall see from the further exposition that such an assertion is based on one fundamental misunderstanding, originating at the very foundation of crystallography, i.e. with Steno’s discovery in 1669 of his law of constancy of angles. We shall see that, when correctly interpreted, this law defines the crystal not as a physical body having a certain form, but as a medium. Before showing this, however, I shall say a few words about another misunderstanding, unfortunately also very widespread, namely that the crystal is very often called and considered indivisible in the sense of individuality.

4. In what sense may a crystal be called indivisible? Very often one has occasion to read and hear that the crystal is an indivisible (individual) of the inorganic world and that, as such, it is characterized by a form distinguishing it from the indivisibles of other species, as, for example, a dog by its form differs from a horse or from a cat, a rose from a lily, etc. Carried away by this analogy, many investigators are inclined to see a similarity between the crystal and the indivisible of the organic world in many other respects as well. For example, the ability of a crystal to restore the external form lost upon fracture, when the crystal is again placed in its mother liquor—the phenomenon of the so-called regeneration of crystals—is compared with the ability of many organisms to restore their accidentally lost organs.

At one of the annual meetings of the Moscow Physical Society named after P. N. Lebedev, A. B. Mlodzeevsky acquainted the assembled audience with a series of highly interesting and instructive cases in which inorganic nature remarkably produces such complex forms of organisms as fungi, algae, etc., imitating these organisms not only in appearance but also in the ability to grow, feed, and reproduce. He showed, for example, on the screen how a drop of chloroform “eats” a glass filament coated with shellac and, having “digested” the shellac, throws out the “undigested” filament, and many other interesting phenomena of a similar kind. However, despite all the similarity of these phenomena to the phenomena of living nature, all this is still only an analogy, not an actual identity; as for crystals, a very simple and generally known experience shows that even such an analogy cannot be extended to a crystal. A crystal possesses the ability to split along cleav—

cleavage planes, often parallel to the external faces. For example, rock salt, which crystallizes, as they say (though not quite accurately), in cubes, splits along cleavage planes parallel to the faces of the cube. In their physical properties the fragments obtained do not differ in any way from the original crystal. Here there is not even an analogy with the division of lower organisms, for in such organisms there is always an organ—the nucleus—which divides before the whole organism breaks up into parts, and each part must receive its own nucleus in order to become an independent organism.

The experiment of splitting a crystal along its cleavage shows precisely the opposite, namely, that a crystal is something opposite to the indivisible. A crystal also cannot be called an indivisible in the same way as a drop of liquid or a bubble of gas, and all analogies between a crystal and organisms built upon this designation must be regarded as a pure misunderstanding. The hope of throwing a bridge between the inorganic world and living nature with the aid of crystals must be acknowledged as an obvious utopia.

It is possible to obtain drops of liquid suspended in another liquid of the same specific gravity. As is known, drops then assume a quite definite spherical form. When broken up, they “regenerate,” i.e., their parts, at first irregular, again become spherical. In this case the drops “multiply”—there are more “indivisibles.” Two drops, upon contact, merge into one, which likewise assumes the form of a sphere; this may be compared with the so-called copulation of lower organisms. In spite of all this, we shall not seriously assert that such a drop of liquid of a quite definite form is an indivisible, an individual. If we do decide to apply this designation to a drop, it is only in a very narrow sense, in which it may be applied to any physical body of definite form and definite dimensions. We shall return to this question in the further exposition.

Just as the fragmentation of a drop shows that a drop of liquid is only a part of a liquid medium, so the experiment with cleavage reveals that a crystal too must be regarded as a solid medium, whose dimensions are accidental and whose properties do not depend on these dimensions.

1. The law of constancy of the angles of crystals. The first observers who examined the form of crystals of one and the same substance encountered considerable difficulties. It seemed that there were no two crystals resembling one another. Difference catches the eye more than similarity, and yet there must be some similarity among crystals of one and the same substance. This similarity was first established by N. Steno in 1669 in his dissertation, the title of which was given at the very beginning of our article. At the beginning of his book, after the title page, there is placed a table of drawings with explanatory text to it, of larger format than

a book, and therefore also compiled several times over. This table, as is said in the explanatory text, was made after the hurried writing of his book. In the table the first thirty figures depict longitudinal and transverse sections, of various forms, of crystals of rock crystal—“crystal,” as Steno and his contemporaries called it. These sections present great diversity. Concerning two of the most irregular longitudinal sections Steno says in his explanation that he could cite many similar examples, in which both the length and the number of sides of the section vary considerably, “but without any change of the angles”: “5 et 6 figura ex illarum genere sunt, quas innumeras afferre potui ad evincendum in plano axis laterum et numerum et longitudinem varie mutari non mutatis angulis.” He says the same concerning the appearance of the transverse section. Steno explains this diversity in the size of the sides of the sections by the different amount of matter deposited in parallel layers on the various faces of the crystal, which shows that he came close to the concept of the rates of growth of crystal faces. Thus, according to Steno, crystals of one and the same substance are characterized by identical magnitudes of the corresponding angles.

The law discovered by Steno became the foundation of all crystallography as its first law—the law of constancy of angles.

It would be a profound error to assert that Steno’s law concerns the form of a crystal or determines that form. Yet one very often has to contend with this error, and it is the chief cause of an incorrect view of crystallography and even of its incorrect presentation in many textbooks.

The point is that, when correctly understood, Steno’s law asserts that a crystal is determined not by the form of its faces, but only by the directions in which faces are found on the crystal. If we define the direction of a face by a perpendicular, a normal, drawn to the face from some point inside the crystal, then, according to Steno’s law, the crystal is determined by the aggregate of such normals, enclosing between them angles characteristic of crystals of the given substance. The length of these normals is entirely indeterminate: it is wholly immaterial at what distance from the point chosen inside the crystal we encounter the face when proceeding along the given normal.

By determining only the direction in which on the crystal we may encounter some face, Steno’s law defines the crystal as a medium of indefinite extent and, moreover, an anisotropic medium, i.e. one whose properties depend on direction. Proceeding in one direction from within the crystal, we shall meet a face; proceeding in another, we shall meet another face, at another distance, or we shall meet an edge or a vertex. The capacity to be bounded by plane faces is placed by Steno’s law in parallel with any other vector property of a crystal, for example, with the velocity of propagation of light, chang—

its magnitude with its direction in the crystal. We see how, from geometry, physics emerges; and this occurs already at the very dawn of the whole doctrine of crystals. Unfortunately, this physics, which is plainly evident in Steno’s work, remained and still remains largely unappreciated and even unnoticed, and crystallography did not receive in due time those physical foundations which correspond to its true content and which were laid down at its very inception.

Recently Steno’s law has led to a very characteristic way of representing crystals, bringing the methods of crystallography closer to those of astronomy. The astronomer, too, deals with directions along which we see the heavenly bodies. These directions appear as straight lines issuing from the observer’s eye and going toward the luminaries. The astronomer intersects these straight lines with the surface of a sphere whose center is in the observer’s eye, and represents the luminaries by the points of intersection of the straight lines with the sphere. He determines the directions of the straight lines by means of a theodolite—an instrument with two mutually perpendicular graduated circles. In accordance with the data obtained by measurement with the theodolite, the positions of the points on the sphere are determined by two numbers, the spherical coordinates. In the study of crystals they now proceed in exactly the same way, determining the relative direction of the normals to the faces by means of a theodolite goniometer and plotting the results of the determination as points on the surface of a sphere.

6. The law of whole numbers. If the fundamental law of crystallography focuses all attention on the crystal as a medium, then the other laws of this science must naturally likewise characterize the crystal as a medium. In this respect the law of whole numbers is highly instructive: it determines the entire aggregate of faces of a crystal, however many of these faces may be observed on the crystal, and makes it possible to predict the directions in which one may hope to encounter new faces.

This is not the place to go into details, but the matter concerns a question so important and so essential for the purposes of this article that I shall devote a few lines to clarifying it.

In the search for a law connecting the faces of a crystal into a single whole, we shall simplify our problem if we reduce it to a plane. If, let us say, there exists a law connecting the faces of a crystal into a single whole, then there must also exist a law connecting into a single whole the system of edges of a crystal, as a system of straight lines along which the faces of the crystal intersect. If this is so, then it is sufficient to consider one of the faces of the crystal, bounded by a sufficient number of edges, and to find the law governing the arrangement of these edges. In this way the problem will be reduced to the plane.

Let us take a face of a crystal and determine the relative direction of the edges bounding it, drawing, from some point in the plane, straight lines parallel to these edges. In the resulting pencil of straight lines—

we shall take two straight lines as axes and a third as the unit edge. On the unit edge we shall take a point at an arbitrary distance from the origin, i.e. from the point of intersection of all the straight lines, and through it draw two straight lines parallel to the axes. Between the lines drawn and the axes a parallelogram is formed. Let us repeat this parallelogram as many times as desired, drawing two series of straight lines parallel to the axes. We obtain a so-called plane parallelogrammatic net, or simply a net. Having constructed such a net, we shall easily notice that all the edges entering into our bundle are represented by the rows of the net, i.e. by straight lines passing through the vertices of the parallelograms, which in this case are called the nodes of the net. Thus the whole system of edges bounding any face of a crystal is determined by a single plane net. Generalizing this construction to three-dimensional space, we obtain, instead of a plane net, a space lattice, which unites into one system all the edges of the crystal that are rows of such a lattice. The faces of the crystal are plane nets in such a lattice, and the whole crystal can be cut out of the lattice by making plane sections along the nets of the lattice. It turns out that the denser the net expressing a face of the crystal, the more highly developed that face is on the crystal and the more often it occurs on it than other faces corresponding to less dense nets.

It turns out that this law too says nothing about the shape of the crystal, but only about the directions bound into a single whole by the space lattice. Consequently the lattice characterizes the crystal as a medium.

Moreover, the lattice characterizes the crystal as a homogeneous anisotropic medium. The homogeneity of the medium is expressed in the uniform and regular repetition of the nodes of the lattice in space; anisotropy—in the unequal spacing of the nodes along lattice rows of different directions. Like a medium, the lattice represents an image of infinite extent, having no definite boundaries.

7. Symmetry of the crystalline medium. The lattice characterizes the crystal from yet another, highly important and essential aspect. The point is that the lattice is a symmetrical image. By a symmetrical image we understand an image consisting of identical, periodically repeated parts. In a lattice such parts are the nodes, situated at equal intervals along the rows of the lattice. Since every crystal is expressed by a space lattice as though filling it, it follows that every crystal is by its nature a symmetrical medium.

In a crystal, symmetrically arranged directions must be physically equivalent, i.e. some physical property of the substance of the crystal, expressed by a vector, must have one and the same magnitude along all these directions. The faces themselves of a crystal must be regarded as identical and mutually symmetrical if

they are physically equivalent. To determine the equivalence of faces, one resorts to testing their relation to external influences, for example, their relation to solvents, obtaining so-called etch figures. If a solvent acts in the same way on two faces of a crystal, forming identical etch figures on them, then such two faces must be identical and symmetrically situated on the crystal. They are said to constitute one crystallographic form, and here again what is characteristic of this form is only the relative position of the faces entering into it.

Having precisely defined the concept of symmetry—which was done not at once, but gradually, through the combined efforts of many investigators, among whom we encounter the names of Hessel, Bravais, Minnigerode, Gadolin, Curie, Fedorov, and others—we can derive all possible cases of symmetry of a crystalline medium. There turn out to be thirty-two such cases.

If crystallography were the study of the form of crystals, then a visual representation of this result would consist of a table showing various polyhedra corresponding to each case of symmetry. Such visual tables do in fact exist, but what they depict is not polyhedra at all, but spheres divided into symmetrical parts. The number of regions into which the sphere is divided corresponds to the number of equal or equivalent directions characterizing the crystalline medium of the given symmetry. Here, directions are considered equal or equivalent if along them the magnitude of some vector property of the crystal must have the same value. A crystal is essentially characterized by the number and relative arrangement of such equivalent directions, and not by anything else. If points are marked on a sphere to indicate directions of the crystal that are equivalent to one another, and if planes tangent to the sphere are drawn at these points, then by their mutual intersections these planes form a polyhedron, the so-called simple form of the crystal; but this polyhedron, in its shape, is never manifested on the crystal. If, for example, the simple form obtained is a cube, this means that in crystals of the given symmetry there exist six equal, pairwise mutually opposite directions, situated along three mutually perpendicular straight lines. Such a system of equal directions exists, for example, in rock salt, and accordingly the crystals of this substance are clothed with six faces parallel to the planes of a cube. It is therefore said that rock salt crystallizes in cubes; however, crystals of this substance never have the form of cubes in the geometrical sense of the term.

Such a table with spheres instead of polyhedra, representing the crown of the entire doctrine of crystals, demonstrates in the most convincing way that the subject of this doctrine consists not in the form of crystals—

... but in the geometrical properties and, in particular, in the symmetry of the crystalline medium. This is what constitutes the true content of crystallography. Everything else may be assigned to that desert of which W. Voigt speaks; the sands of this desert have for two and a half centuries buried the true meaning of Steno’s law, which lies at the foundation of this entire, exceedingly clear and simple doctrine.

8. Dependence of phenomena on the symmetry of the medium. How, then, after all that has been said, can one maintain that a physicist does not need crystallography? After all, every phenomenon occurs in dependence on the symmetry of the medium.

Let us perform the following simple experiment, which clearly demonstrates this dependence. Take a plate of gypsum, cover it with a layer of wax, and touch it with a heated needle. The wax will melt around the needle, and the boundary of the melted part will prove to be elliptical in shape. Why? We shall answer this by the following experiment. Coat a copper wire mesh with square cells with paraffin so that the paraffin uniformly fills all the cells of the mesh; then, after allowing the paraffin to cool, heat the mesh at one point over a small flame and shake off the melted paraffin. A round field free of paraffin will be obtained. Let us skew another piece of the same mesh by pulling it at opposite corners. We shall obtain a mesh with rhombic cells, and if we carry out the same experiment with such a mesh, the field free of paraffin will be elliptical in appearance. Everything in the mesh has remained as before, except the symmetry, which has become different; accordingly, a different rate of heat propagation in different directions has also resulted. Evidently, in gypsum as well the matter depends on the symmetry of the medium.

Since phenomena depend on the symmetry of the medium, the physicist must know this symmetry in order to take account of the course of the phenomenon. In an isotropic medium, in which all directions are identical, there is no need to pay attention to the properties of the medium. We speak of the propagation of light, of sound in space, or of their velocity in air, water, and other substances. We leave aside the intermediate concept of the medium. But when analyzing phenomena in crystals, we must take account of the concept of the medium and must know the geometrical properties of the crystalline medium, i.e., crystallography. Thus crystallography is an introductory chapter to the physics of crystals.

9. The form of the crystal. We have seen that crystallography, as built upon Steno’s law, says nothing about the crystal as a physical body having a definite form, even one distinct from the form of other bodies of the same substance. Meanwhile this external form is something quite real; and if different crystals of one and the same substance differ in their external form, then this, obviously, depends on differences in the conditions under which these crystals were formed.

The elucidation of the conditions under which a crystal assumes one form or another was achieved comparatively recently, almost in our own time, through the work of the American physicist W. Gibbs and the French physicist P. Curie. Independently of one another, they drew attention to the fact that, in addition to the internal forces that produce the polyhedral form of a crystal, the crystal grows also under the influence of forces acting on its surface, which separates it from the external medium that nourishes it. These forces perform the work of crystal growth, under the influence of which the magnitude and the character of the crystal surface change. With respect to this energy, Gibbs and Curie formulated the principle bearing their names, which consists in the statement that the magnitude of this surface energy of a crystal must be the least possible for a given volume of the crystal. This principle is only a special case of the more general principle of economy constantly observed in natural phenomena.

Returning to our experiment with a drop of liquid suspended in another liquid, we shall now immediately see that the form of the drop obeys the same principle, only the case of the drop is much simpler. On the surface of the drop there also act forces that change its form and impart to it a shape in which its surface, and together with it the energy of this surface, will be the least possible for a given volume of the drop. For a drop of liquid whose particles move around one another with equal ease in all directions, this condition will be fulfilled when the drop takes the form of a sphere, since of all figures bounded by curved surfaces the sphere, for a given volume, has the least surface. With crystals the matter is more complicated. Owing to the anisotropy of crystals, each face of a crystal possesses its own special value of surface energy referred to a unit area of the face. This quantity is called the capillary constant of the face. Therefore, in order to calculate the total magnitude of the surface energy of a crystal, one must multiply the capillary constant of each face by the area of the corresponding face and add all the products thus obtained. This quantity, too, must have the least value for a given volume of the crystal. Differential calculus gives a method by which, from this condition, one can calculate the dimensions and form of the individual faces of the crystal, and hence determine the form of the entire crystal.

However, this method, the application of which Curie showed in two simple examples, is complicated and not visual; I, however, succeeded in finding a purely geometrical construction, very visual and leading very simply to the goal. This construction consists in the following: on the normals to the faces, drawn from a single point, segments proportional to the capillary constants of the corresponding faces are laid off from this point, and through the ends of the segments thus obtained perpendicular planes are drawn. By their mutual intersection these planes determine around the chosen point a convex polyhedron, similar to the form of the crys-

…crystal satisfying the Gibbs–Curie principle. This construction is a direct supplement to Stenon’s construction, but already in the domain of the doctrine of the form of the crystal. Stenon’s construction defines the crystal as a homogeneous unbounded medium and leaves the lengths of the normals to the faces undetermined; the construction given by me, characterizing the crystal on the basis of the Curie–Gibbs principle as a physical body, determines the distances at which Stenon’s normals must be cut off, and thereby the form of the crystal itself.

Thus, in seeking the external form of the crystal, we encounter a problem from the domain of physics.

10. Crystals and X-rays. We have seen that the directions of all possible faces and edges of a crystal, whatever may have formed on it under different conditions, are geometrically determined by a space lattice consisting of points—nodes—regularly arranged in space. This regularity in the arrangement of the nodes of the lattice long ago suggested the thought that crystals must consist of elementary particles located at these nodes. This was a very probable supposition, but it had hitherto lacked convincing proof, and therefore remained merely a hypothesis. The conception of a crystal as a space lattice lacked physical content.

Only ten years ago a discovery of great importance was made, filling the crystalline medium with such physically definite content. The desert of which Voigt speaks, and under the guise of which the doctrine of the geometrical properties of crystals presented itself to the physicist, proved to be very densely populated, and moreover not at random, but in a strictly definite, truly astonishing order.

I am speaking of the discovery of the diffraction of X-rays in crystals by M. Laue in 1912. Much was written and said about this discovery at the time, and here I shall only recall the essence of the matter.

A narrow beam of X-rays, passed through a crystal, is decomposed by the crystal into a whole series of narrow beams of low intensity. If we receive this series of beams on a photographic plate, then after development we obtain on the plate traces of the beams in the form of dark spots. The arrangement of the spots has a special regularity, indicating that the whole phenomenon takes place as though the original beam were partially reflected in the crystal from a whole series of planes arranged according to the same law by which the meshes of its lattice are arranged in the crystal. On examining more closely the action on the incident beam of X-rays of atoms arranged at the nodes of the space lattice, we indeed find that each atom becomes a source of secondary X-rays, which reinforce one another according to the laws of the interference of light in perfectly definite directions, and that these directions correspond precisely to the direc—

planes from which the initial beam would have been reflected if these planes were continuous and could actually reflect X-rays. Thus it turned out that this remarkable phenomenon, indicating the identity of X-ray light with visible light and producing interference phenomena for X-ray light, can be produced in the form described above only in crystals, owing to the fact that in crystals the atoms are arranged at the nodes of a space lattice. On the other hand, it turned out that the very form of the phenomenon, i.e., the position and brightness of the individual secondary beams, depends on the mode of arrangement of the atoms in the crystal, so that from this form of the phenomenon one may draw conclusions about the structure of the given crystal. Thus questions concerning the nature of X-rays became most closely intertwined with questions concerning the structure of crystals and their space lattice, which thereby acquired a physical significance. It turned out that to clarify the nature of X-ray light is possible only on the condition of a clear conception of the geometrical properties of the crystal. On the other hand, in X-rays we obtained a powerful method for determining the internal structure of crystals. In the introduction to the remarkable book by W. H. Bragg and W. L. Bragg, X-rays and Crystal Structure, W. H. Bragg says:

“In order to master the content and the success of the new doctrine (of the nature of X-rays), it is necessary to have knowledge both of X-rays and of crystallography. Since these branches of knowledge have hitherto not been connected with one another, it is to be expected that many who are interested in their new development will encounter an obstacle in a depressing ignorance of one or the other of them.”

Let us express confidence that henceforth this obstacle will be removed. The science of crystals must at last take the place, corresponding to its true content, among the branches of our knowledge, becoming a chapter of physics—a chapter that must be mastered by all who are interested not only in the physics of the solid state, but also in physics in general; proof of this we see in the doctrine of X-rays.