Advances in Our Knowledge in the Field of the Structure of Crystals

G. V. Wulff.

1. Crystals and X-rays. — 2. Crystals and the space lattice. — 3. Laue’s experiment. — 4. “Reflection” of X-rays. — 5. W. H. Bragg’s spectrometer. — 6. “Monochromatic” X-ray tubes. — 7. Structure of sodium chloride. — 8. Determination of wavelengths. — 9. Ions as structural units. — 10. Spectrometric determination of structure. — 11. The Debye–Scherrer method (powder method). — 12. Laue radiograph method. — 13. Analysis of some studied structures. — 14. General criteria for establishing structure. — 15. Forces acting between the elements of structure. — 16. Conclusion.

1. Crystals and X-rays. We had long already had a clear conception of the possible structure of crystals and knew that, although the rules according to which their structural elements may be arranged in a crystal are numerous, their strictly determined number is 230. These rules were known to us exactly, and their derivation formed a special chapter of crystallography—the theory of the structure of crystals. We lacked, however, one very essential thing: we had no means of determining precisely according to which one of all these rules each given crystal was built, and we were forced to remain in the realm of mere conjecture. Such a means we obtained only in the phenomena observed when a beam of X-rays passes through crystals, in the phenomena of diffraction of X-ray light, discovered in 1912 by M. Laue with his collaborators—Friedrich and Knipping. Here crystallography came to the aid of physics, helped to clarify the nature of X-rays, which from that moment ceased to be X-rays, and, making use of these successes, physics itself made a colossal step forward by applying the knowledge we had gained to the experimental determination of the structure of crystals. Crystallography led physics to the spectral analysis of X-rays, to an expansion of our knowledge in the study of electromagnetic waves; physics gave crystallography the possibility of penetrating into the arrangement of atoms in a solid substance and of clarifying the nature of the forces that hold atoms in a crystal in astonishing order.

2. Crystal and space lattice. A space lattice is an infinite geometrical figure consisting of an infinite number of points—the vertices of an infinite number of identical parallelepipeds, placed together without gaps in such a way that each parallelepiped forms, as it were, a continuation of the neighboring parallelepipeds adjacent to it. The points, as elements of the lattice, are called nodes. Through the nodes of the lattice one may draw, in various directions, systems of parallel straight rows, studded with nodes at equal intervals, and systems of equally spaced parallel planes—nets, likewise of various directions, in which the nodes are situated at the vertices of parallelograms. The space lattice contains within itself all the geometrical properties of the crystal, since a crystal is defined by the relative direction of its edges and faces, and not by the extent of one or another of them. This is not a hypothesis, but a law of nature, a fact that can serve as the subject of a simple lecture demonstration: the mutual direction of the edges and faces of a crystal of one and the same substance is the same as the mutual direction of the rows and nets of one and the same lattice. At the same time, in the lattice expressing the geometrical properties of the crystal, the absolute distances of its nodes are not important—these distances may be changed in any ratio whatever, common to rows of different directions.

A hypothesis, though one that is highly probable, is the assertion that the structural material elements of the crystal—atoms or molecules—are situated in the crystal at the nodes of a space lattice. The regularity of the bounding of a crystal can arise only from the regularity of its structure, and the discontinuous geometrical image of a space lattice, which determines the laws of the external bounding of a crystal, can most simply express also the laws of its internal structure. This hypothesis was one of the two hypotheses that formed the basis of Laue’s experiment. The other hypothesis consisted in the assumption that X-ray light consists of electromagnetic waves whose length is commensurable with interparticle distances.

3. Laue’s experiments. Proceeding from these two hypotheses, Laue arrived at the discovery of the phenomenon which he called the diffraction of X-rays. He took the crystal, built of regularly arranged structural units, as a diffraction grating of three dimensions, passed through it a beam of parallel X-rays, selected from the radiation of an X-ray tube by a consecutive series of narrow apertures in lead screens, and placed a photographic plate in the space behind the crystal, where the result of the interference of the scattered rays was expected. As is known, his collaborators, Friedrich and Knipping, after a prolonged exposure, found on the plate a series of spots, which proved to be the traces of a series of practically parallel beams of rays into which the primary ...

beam in passing through a crystal. The positions of the spots on the photographic plate were, in general, in agreement with the calculation.

In Laue’s experiment diffraction should be represented to oneself in the following way: it is known that, on falling upon a substance, X-rays excite in it so-called secondary rays, while the electrons of the atoms of the substance are set into vibration. The atoms become sources of X-radiation, and the waves emitted by them interfere in the surrounding space, dividing it into bright and dark intervals. Knowing the arrangement of the light sources and the wavelength of the light emitted by them, it is not difficult to calculate the relative arrangement of regions of both kinds; thus in Laue’s experiment we have a relation between three kinds of quantities: the quantities determining the arrangement of the sources, the wavelength, and the relative directions of the scattered beams of rays. Only the latter are subject to direct determination. The ultimate aim of Laue’s experiment was to determine the wavelength of X-rays, and therefore he had to take as known the arrangement of the atoms in the zinc blende crystal that served him for the experiments. However, as was said above, it was impossible to determine the structure of crystals accurately by the former method; it was determined by Laue not quite correctly, and therefore the wavelength was also not determined accurately by him. In addition, in his experiments he used a tube giving a large quantity of “white” radiation with a continuous spectrum, which also introduced uncertainty into the value of the computed wavelength. To sort all this out fell to the lot of subsequent investigations.

4. “Reflection” of X-rays. Simultaneously and independently of one another, W. L. Bragg (son), Terada, and the author came to the conclusion that the distribution of beams of rays scattered by a crystal is such as if these beams arose from reflection of the primary beam on the lattice planes of the crystal (on its atomic planes). Having interference as its basis, this reflection is distinguished by selectivity: atomic planes of a crystal, layered at distances \(d\) from one another, reflect from a beam of rays incident upon them at an angle \(\theta\) (measured from the plane of incidence) only waves whose wavelength \(\lambda\) satisfies the relation

\[ 2d \sin \theta = n\lambda, \ldots \ldots (1) \]

where \(n\) is an integer. Thus, if the incident beam is “white,” then at a given angle \(\theta\) only a definite harmonic series of waves is reflected; but if it is homogeneous, “monochromatic,” i.e. \(\lambda\) has one definite value, then reflection can occur only at a definite series of angles \(\theta\) corresponding to a series of values of \(n\). In the latter case one speaks of reflections or spectra of different orders, corresponding to the values of \(n\).

Since the quantity \(d\) varies in crystals with direction, the conditions of reflection likewise vary depending on the position of the reflecting faces and atomic planes in the crystal.

W. L. Bragg derived these laws from a consideration of Laue radiograms. The author derived them both from radiograms and from Laue’s formulas, revealing their crystallographic meaning. Terada obtained traces of the beams scattered by a crystal not on a photographic plate, but on a fluorescent screen, and was able to verify that, when the crystal is rotated, the bright spots on the screen move as if they were “reflections” of reflected rays.

It is not difficult to appreciate the enormous importance of such simplicity in the laws of scattering of X-rays by a crystal for the purposes of formal crystallography, which uses the reflection of a light ray from the faces of a crystal to measure the angles between its faces. In using X-rays, we do not need the external boundaries of the crystal, since these rays are reflected from the atomic planes inside the crystal in exactly the same way as light rays from its external faces. This circumstance was used for the determination of crystals by Conac in France. In Russia, independently and simultaneously, K. V. Vasil’ev, on the author’s idea, constructed a universal goniometer of the theodolite type, making it possible to use both visible and X-rays for measuring crystals. In addition, in Russia N. E. Uspenskii constructed a simple apparatus for direct observation of the “reflections” on a fluorescent screen.

The simple theory given by the author makes it possible to explain one phenomenon, first noticed by N. E. Uspenskii during his joint work with the author, and subsequently described by Friedrich. This phenomenon consists of bright bands passing through the spots of radiograms. The author showed that these bands coincide with the traces of the light cones along which X-rays are scattered by rows of atoms in the crystal. Corresponding to diffraction spectra of zeroth order, these bands must necessarily pass through the trace of the primary beam on the screen or on the photographic plate and through the spots of the radiogram.

5. Measurement of the angle of reflection. W. H. Bragg’s (the father’s) spectrometer.

Knowing the distance on the photograph from the trace of the reflected ray to the trace of the central ray and the distance of the photograph from the crystal, one can determine the angle \(\theta\) for each reflected ray.

W. H. Bragg proposed using an ionization chamber to measure the angle of reflection and constructed an instrument that he called a spectrometer for X-rays. The collimator of this spectrometer consists of a consecutive series of slits in lead plates, and the telescope is replaced by an ionization chamber. With

on the side of the axis of the instrument the chamber is closed by a mica plate with a slit-shaped aperture, covered by an aluminum leaf. The walls of the chamber are charged to a potential of about 200 volts, while the inner electrode, connected with the gold leaf of a sensitive electroscope, is kept at earth potential at the beginning of the experiment. The X-rays entering the chamber change the potential of the leaf, and the intensity of the rays is measured by the degree of deflection of the leaf. On the table of the instrument the crystal is placed in such a way that the axis of rotation of the instrument passes through the face being tested. On the limb of the instrument the angles of rotation both of the crystal and of the ionization chamber are read off.

6. “Monochromatic” X-ray tubes. By means of his spectrometer W. H. Bragg proved that the so-called characteristic X-rays emitted by the anticathodes of tubes consist of a series of bright spectral lines which produce particularly strong ionization in the ionization chamber of the spectrometer. The presence of such a bright wave, quite definite in its length, in the beam incident upon the crystal brings great definiteness into the use of the formula given above, making it possible to compare with one another different thicknesses of atomic layers even in the case when the absolute wavelength of the monochromatic X-ray light employed remains unknown to us; and if we were able to measure also the absolute wavelength, then we would obtain a means of determining the said thicknesses in absolute measure. Hence the great importance of the so-called monochromatic X-ray tubes for determining the structure of crystals becomes clear. In such tubes the anticathode is made of palladium, rhodium, silver, or copper. Copper has proved especially convenient with respect to the wavelength of its characteristic radiation, belonging to the \(K\) series; but a tube with a copper anticathode must be kept under the action of an air pump throughout the experiment. By fitting such a copper tube with a window of thin nickel leaf, one can completely absorb the weaker line from the bright doublet, and then the radiation emerging from the tube will be quite monochromatic.

7. Structure of sodium chloride. The first crystals chosen for the determination of structure were sodium chloride (common or rock salt) and the related potassium chloride, bromide, and iodide, which crystallize in cubes. Investigating reflection from the principal faces of the cube, octahedron, and dodecahedron, W. H. Bragg and W. L. Bragg found that for all these substances, with the exception of potassium chloride, the sines of the angles of reflection of the first order on the cube, dodecahedron, and octahedron stand in the ratio:

\[ 1 : \sqrt{2} : \sqrt{3}, \]

which indicates that the structural elements in these crystals are arranged as in a cube with centered faces, i.e. at the vertices

of the cube and at the centers of its faces. Such a cube is one of the parallelepipeds of the lattice of these crystals. Let us note that such a lattice is, in essence, an aggregate of four lattices of the cubic type, inserted one into another. Potassium chloride exhibits another ratio of sines, equal to

\[ 1:\sqrt{2}:\sqrt{3} \]

characteristic of a lattice with simple cubic cells, and is, as it were, an exception to the general rule. This contradiction is removed, and the interference phenomena in crystals of this kind are fully explained, if one assumes that the structural units are not particles, but atoms. By arranging the sodium atoms and the chlorine atoms in two separate lattices with cells in the form of cubes with centered faces, and by inserting one lattice into the other so that along the edges of the cubes the sodium and chlorine atoms alternate at equal intervals, we obtain the structure of crystals of sodium chloride proposed by the Braggs. In this case the exception represented by potassium chloride is only apparent. Being close in atomic weight, the potassium and chlorine atoms scatter X-rays with almost equal strength, and therefore in interference phenomena they take part almost as identical sources arranged in a simple cubic lattice.

Having found the relative arrangement of the atoms, one can also determine their absolute distances. The entire arrangement of atoms in sodium chloride can be divided into cubes, at the vertices of which there are alternately placed one chlorine atom and one sodium atom. The number of atoms of each of these elements falling within the internal volume of this cube will be equal to one half. Taking the absolute weight of the hydrogen atom as \(1.66\cdot 10^{-24}\) g and the specific gravity of rock salt as 2.17, we obtain the volume of the substance in our cube, equal to \(\{\frac{1}{2}(23+35.5)\cdot 1.66\cdot 10^{-24}\}:2.17\).

The side of the cube will be equal to the cube root of this number, i.e. \(2.81\cdot 10^{-8}\) cm. The unit \(10^{-8}\) cm is the ångström, \(\overset{\circ}{A}\), adopted in spectral analysis, and therefore now all atomic distances are usually expressed in ångströms. Thus the structure of rock salt and its absolute dimensions were determined.

8. Determination of the wavelength. The next step consisted in the exact determination of the wavelength. In formula (1) one must substitute for \(d\) the value found, \(2.81\overset{\circ}{A}\), and for \(\theta\) the angle of reflection of the first order from the face of the cube, which, according to the Braggs’ determination for the characteristic rays of a palladium anticathode, turned out to be \(5^\circ 54'\); and we obtain the wavelength \(0.577\overset{\circ}{A}\) for the palladium characteristic radiation. Subsequently Moseley, and then Siegbahn, obtained much more accurate values both for the lattice

sodium chloride, as well as for the wavelengths of X-rays of various origins as determined by them.

Having at our disposal the value of the wavelength, we can determine the distance \(d\) in formula (1) from measurements of the angles \(\theta\) in all cases. Thus, for the first time the problem of determining the actual structure of crystals was solved.

9. Ions as structural units. In the structure of sodium chloride, parallel to the faces of the octahedron, layers of sodium and chlorine alternate at equal distances from one another, so that midway between the brighter chlorine layers there are less bright sodium layers. Because of this, the first spectrum on the octahedral face is weakened. It disappears completely in potassium chloride. The same is observed in crystals of \(MgO\) and \(CaO\), built according to the sodium-chloride type: in \(MgO\) crystals the first spectrum on the octahedral face is entirely absent, whereas in \(CuO\) it is present, but weakened. This can depend only on the fact that in \(KCl\) and \(MgO\) the sources of radiation in the alternating layers are completely identical. Considering secondary radiation as resonance and the electrons in atoms as resonators, we arrive at the conclusion that the identity of the sources of radiation must be attributed to the identity of the number of resonators, i.e. electrons, in these sources. The same number of electrons in our cases \(KCl\) and \(MgO\) is obtained if we assume that the sources of radiation are not atoms, but ions. Indeed, the ion \(K^+\) with one positive charge contains 18 electrons; the ion \(Cl^-\) with one negative charge likewise contains 18; the ion \(Mg^{++}\) with two positive charges contains 10 electrons; \(O^{--}\) with two negative charges likewise contains 10. The first to draw attention to this was Debye, who established, by precise experiments on interference in lithium fluoride, that the structural units in crystals are electrically polar ions, and that the forces holding these units in equilibrium are electrostatic. These conclusions are in complete agreement with the work of A. F. Ioffe and M. V. Kirpicheva on the electrical conductivity of crystals.

In seeking a possible arrangement of the structural units, the fact just stated is of enormous importance, since what must be arranged in space are no longer individual atoms, but whole atomic groups. For example, in order to find the structure of \(CaCO_3\), one must place the ions \(Ca\) and \(CO_3\); for \(NH_4Cl\), the ions \(NH_4\) and \(Cl\), and so on.

10. Determination of structure by means of a spectrometer. Determination of structure by means of a spectrometer is achieved by determining the angles and brightness of reflections of various orders on various faces of a crystal that is crystallographically quite definite.

From the angles of reflection we first of all determine the number of chemical molecules in the unit volume of the structure. This is achieved by the follow-

in the following way. Having determined the crystal crystallographically, i.e., having assigned all its faces to the three faces chosen as the planes of the crystallographic axes, we determine the thickness of the layers parallel to these planes from observations of the angles of reflection of the first order. Let these distances be \(d_1\), \(d_2\), and \(d_3\), and the angles between the axes \(\alpha\), \(\beta\), and \(\gamma\). Then the volume \(V\) of the structural unit will be:

\[ V=d_1d_2d_3\sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}. \]

Knowing the molecular weight \(M\) of the substance of the crystal, its specific gravity \(D\), and the absolute weight of a hydrogen atom, \(1.66\cdot 10^{-24}\) g, we shall find the absolute volume \(v\) of one molecule:

\[ v=\frac{M\cdot 1.66\cdot 10^{-24}}{D}. \]

Dividing the value \(V\) by \(v\), we find the required number of particles.

A comparison of the angles of reflection of the first order on the faces of the cube, dodecahedron (or their analogues in less symmetric crystals) will give us a general idea of the form of the crystal lattice. We have already seen above that, for the lattice of a simple cubic cell, the ratio of the sines of these angles is \(1:\sqrt{2}:\sqrt{3}\), while for a lattice with a cell having atoms at the centers of the faces this ratio is \(1:\sqrt{2}:\frac{1}{2}\sqrt{3}\). A third type, not yet encountered by us, with nodes at the center of the cubic cell, is characterized by the ratio \(1:\frac{1}{2}\sqrt{2}:\sqrt{3}\).

Having armed ourselves with these data, we proceed to the placement of the atoms within the structural unit. For this we must know the symmetry of the crystal. As is known, according to their symmetry crystals are divided into 32 classes. To each of the classes there belongs a series of structural groups from among the 230 mentioned at the beginning of this article. This series is further subdivided into three groups according to one of the three kinds of lattice. Each of these latter groups contains a small number of structural groups, from which it is no longer very difficult to choose the one most suitable for the given case. The guiding rules were worked out in detail by P. Niggli in Germany, S. Nishikawa in Japan, and R. Wyckoff in America. Niggli even compiled a determinative key in the form of tables. The essence of the matter comes down to the following. Each structural group is essentially a definite complex of elements of symmetry—simple mirror planes and glide planes, axes of symmetry—simple and screw (helical)—and centers of symmetry. All 230 groups differ from one another in the number of these elements and in their mutual arrangement. In the final analysis each group may be reduced to one of the three types of lattice, through whose cells the elements of symmetry pass in one way or another. Each element of symmetry causes a point lying outside it to be repeated the corresponding number of times, giving a system of equivalent points. Points lying on the elements of sym-

metry, are not repeated by these elements, but may be repeated by others. This reduces the number of equivalent points, which in a particular case may be reduced to a single point. Since the geometrical equivalence of points must also correspond to the chemical equivalence of the atoms expressed by the point, it is natural that, in placing the atoms entering into the particle, one must take account of their chemical meaning. Thus, the oxygen atoms in the ion \(CO_3\) are all identical and must be placed symmetrically around an axis of the third order, while the carbon atom, as non-repeated, must be placed on this axis. In the ion \(NH_4\), all hydrogen atoms are identical and must be situated at the vertices of a regular tetrahedron around the nitrogen atom, whose place is at the center of this tetrahedron. The whole tetrahedron must occupy a symmetrical position in the structure. It becomes clear what decisive importance considerations of a chemical character have in choosing a possible structure.

Having found the probable arrangement of the atoms in the unit volume of the structure, we must determine their mutual distance. For this it is necessary to compute the share of participation of each atom entering into the structural unit in the total result of the interference, to sum these shares for all atoms of the unit, and to compare the results of the calculation with the observed intensities of the spectra.

Let us refer the structure to the edges of the unit cell as coordinate axes, and let us suppose that at the origin of coordinates there is an atom with \(A_0\) electrons. The amplitude determined by it at an external point \(M\) will be proportional to \(A_0\), and the coefficient of proportionality will depend on the direction; for the present we shall leave it out of consideration. Another atom with \(A_1\) electrons and with coordinates \(x, y, z\) (in general oblique-angled), situated very close to the first as compared with the point \(M\), will give at it an amplitude proportional to \(A_1\) with the same value of the coefficient of proportionality. Suppose that we observe a reflection of order \(n\) on an atomic plane passing through the second atom at a distance \(\delta\) from the first atom. If \(d\) is the distance of such a plane, passing at distances of the structural period from the first atom, then the share of the second atom in the interference is expressed by an amplitude proportional to

\[ A'_1 = A_1 \cos 2\pi n \frac{\delta}{d}. \]

If by \(\alpha, \beta, \gamma\) we denote the angles of the normal to the plane with the coordinate axes, then the equation of the plane passing through the second atom will be

\[ \delta = x \cos \alpha + y \cos \beta + z \cos \gamma \]

If the edges of the structural unit are denoted by \(a, b, c\), then this equality may be represented in the form

\[ \frac{\delta}{d} = \frac{x}{a}\cdot\frac{a \cos \alpha}{d} + \frac{y}{b}\cdot\frac{b \cos \beta}{d} + \frac{z}{c}\cdot\frac{c \cos \gamma}{d}. \]

\(\frac{x}{a}, \frac{y}{b}, \frac{z}{c}\) represent the relative coordinates of the second atom, and the fractions \(\frac{a \cos \alpha}{d}\), \(\frac{b \cos \beta}{d}\), and \(\frac{c \cos \gamma}{d}\) may be replaced by the numbers \(h, k, l\), which characterize the atomic plane crystallographically, by its indices, always integers. In the present case these numbers will be relatively prime. In view of what has been said, the amplitude determined by the second atom will be

\[ A'_1 = A_1 \cos 2\pi a (mh + nk + pl) \]

if by \(m, n, p\) we denote the relative coordinates of the atom.

The total amplitude of all the atoms in the unit volume, including also the atom situated at the origin of coordinates, is expressed by the sum

\[ \Sigma A \cos 2\pi(mh + nk + pl)\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\quad (2) \]

The amplitude at the point \(M\) will be proportional to this polynomial. As for the coefficient of proportionality, it represents an as yet unexplained, complex function of very many quantities. Unfortunately, knowledge of this function is also quite necessary for the determination of structure, since only such knowledge will make it possible to compare the calculated brightness of spectra of different orders with the observed intensities. The first approximate and simple value of this function was given by W. H. Bragg, assuming that the brightness of spectra decreases inversely proportional to the square of the order of the spectrum. More exact investigations in this field have been undertaken by W. L. Bragg. For the time being one has to use an approximate rule. Having calculated the polynomials (2) for different faces and different orders of spectra and compared their squares with the observed intensities, one can obtain a sufficient number of equations for determining the coordinates of all the atoms entering into the structural unit. The structure is thus determined.

II. The Debye–Scherrer method (powder method). This method differs from the preceding one by its generality and by the simplicity of the experimental arrangement. The inner lateral surface of a lead cylindrical chamber about 8 centimeters in diameter is lined with photographic film; along the axis of the chamber is placed a paper or collodion tube about 2 millimeters in diameter, filled with a fine powder of the crystal, and through a narrow opening in the side wall of the chamber a narrow beam of monochromatic rays is passed through the powder. Reflection, say of the first order, of such a beam can occur on some face of the crystal—for example, on a face of the cube—only at a definite angle. The beam of rays will find in the powder a sufficient number of crystallites whose cubic faces are favorably oriented, and the rays will be reflected from them in all planes passing through the primary beam of rays, forming

hollow cone, whose intersection with the lateral surface of the cylinder is imprinted on the film in the form of a spectral curved line. The lines will be of different brightness and will correspond to reflections of different orders on different faces of the crystal.

The powder method gives at once, for all faces of the crystal, the same data as the spectrometric method gives only successively. This constitutes its enormous advantage.

The same advantage must also be considered the fact that there is no need for us to have whole and large crystals; it is sufficient to have only a small heap of crystal powder, of a volume the size of a large pinhead. The disadvantages of the method include: 1) the weakness of the energy of the reflected rays, which leads to a long exposure; 2) the impossibility of indicating at once which line belongs to a given face of the crystal, and conversely; 3) the coincidence of lines for different faces characterized by the same distance between atomic planes; and 4) the indistinctness of the photograph for crystals of low symmetry, owing to the partial superposition of one line upon another for faces with a small difference in the distances between atomic planes. This method is also distinguished by the greater complexity of the computations of the results of measurement. Methods of computation were developed, on the one hand, by Runge, and on the other by Templer and Jonsson. We shall describe them in general terms.

The result of measuring a roentgenogram is very simple. The ratio of the mutual distance of two symmetrical lines to the radius of the camera gives four times the angle of reflection, the latter being measured from the reflecting plane. Comparing the sines of these angles, it is easy to single out the orders of spectra belonging to one and the same face. For any spectral line formula (1) is valid, which we shall write in the form

\[ \frac{1}{d}=\frac{2\sin\theta}{n\lambda}. \]

We shall determine the distance \(d\) as the quotient obtained by dividing the volume \(V\) of the parallelepiped of the lattice by the area of its base \(S\), and we shall write that

\[ S^2=\frac{4V^2\sin^2\theta}{n^2\lambda^2}. \]

The quantity \(S^2\), as crystallography teaches, has the form of a quadratic form:

\[ S^2=h^2A^2+k^2B^2+l^2C^2+2klBC\cos\alpha+2hlAC\cos\beta+2hkAB\cos\gamma, \]

where \(h, k, l\) are integers determining the face of the crystal, and \(A, B, C, \alpha, \beta, \gamma\) are quantities determining the lattice. The volume \(V\) is a quantity constant for all atomic layers and can also be simply expressed through the six lattice parameters mentioned. The quantities \(\theta_n\) and \(\lambda\) are known. The photograph gives us a series of equalities, from which also

it is necessary to determine these six parameters. The problem would be solved very simply if it were known which triple of numbers \(h, k, l\) corresponds to the given line; for this, however, we have no indications, and only because \(h, k, l\) are integers and quite small is the problem soluble.

The experimental side of the powder method was improved by H. Bohlin.

The method is indispensable in a great many cases. Very often we have to do with substances that crystallize poorly and, finally, with colloidal substances whose particles may possess a crystalline structure. This method extends even to liquids. Indeed, Debye and Scherrer obtained interference in benzene and established in it the existence of structural rings with sides of \(6.02 \mathring{A}\) and a thickness of \(1.19 \mathring{A}\). It was also applied to the so-called liquid crystals, which gave interference, like liquids, in the form of a single blurred broad line, so that they turned out not to be crystals at all, as the author of this article had maintained ever since O. Lehmann published his discovery of these remarkable substances.

O. Lehmann maintained that a space lattice is not a feature essentially distinguishing crystals from liquids, since crystals are characterized by residual elastic deformation, under which they nevertheless remain homogeneous, while their lattice is substantially disturbed. A. F. Ioffe’s experiments on the residual deformation of rock-salt crystals, in which the course of deformation was observed by means of the diffraction of X-rays, brilliantly showed that, on passing the elastic limit, a crystal breaks up into a multitude of small crystals displaced relative to one another according to definite geometrical laws.

12. The Laue radiogram method. In contrast to both preceding methods, this method is based on the use of “white” radiation. Laue radiograms at once give a multitude, sometimes up to a thousand, of “spots” from various atomic planes. Many of the spots are formed by reflection from planes of one and the same crystallographic form. This method was developed by S. Nishikawa in Japan and R. Wyckoff in America because, by giving an abundant number of reference points for determining the structure, it makes it possible to dispense with both of the basic laws underlying the spectrometric method and the powder method—namely, without the law of proportionality of the amplitude to the number of electrons (the atomic number) of the atom and without the law of inverse proportionality of the amplitude to the order of reflection—as laws not yet precisely established. R. Wyckoff therefore considers the results of this method the most accurate and capable even of serving as a real basis for testing both of these laws.

The relative position of the “spots” from different planes is determined by the symmetry of the crystal, while their absolute position on the photograph depends on the distance of the plate from the crystal. Knowing the symmetry of the crystal and its orientation with respect to the photographic plate, one can easily determine the position of the “spot” for any atomic plane specified by its characteristic indices \(h, k, l\). The inverse problem is solved just as easily. Crystallography teaches how to calculate the thickness of an atomic layer from the data \(h, k, l\). The distance from the crystal to the photographic plate and the distance from the “spot” to the central spot give the angle of reflection. These data make it possible, by formula (1), to calculate the value \(n\lambda\), after one of the absolute measurements has been made with the aid of a spectrometer. By placing the crystal so that the beam of rays does not pass along any one of its important axes, we shall give different inclinations to different planes of one and the same form, which will therefore reflect rays of different wavelengths. Plotting on a diagram the brightness of reflection on the planes of one and the same form as a function of the reflecting wavelength, we obtain a curve depicting the effect of the rays on the plate. Constructed for different reflecting planes, such curves may be compared at one and the same wavelength. As Nishikawa showed, the data obtained in this way may serve for choosing various arrangements of atoms compatible with the symmetry of the crystal.

The author of this article has shown that Laue radiographs can be used even without any aid from a spectrometer, if the distribution of brightness in the white radiation of the tube and the wavelength of the brightest wave are known to us. Several of the brightest spots owe their brightness to the reflection of precisely this wave, since the brightness usually changes rapidly with wavelength. Knowing the crystallographic significance of the reflecting planes, one can calculate also the absolute dimensions of the thickness of their atomic layers, and hence the number of molecules in the unit volume of the structure.

Sometimes, as P. Ewald has shown, it is sufficient to compare the brightnesses of the spots qualitatively in order to determine the positions of the atoms quite accurately.

13. Analysis of some studied structures. It is natural to suppose that the bodies simplest in chemical composition will also possess the simplest structure. The simplest of such bodies are the chemical elements, among them the metals. We know the structure of a whole series of metals, and it proves in fact to be very simple. Taking identical spheres and packing them closely into a triangular heap, we shall see that the centers of the outer spheres will lie on the edges of a regular tetrahedron. Such a heap will possess the symmetry of the cubic system of crystals, and the centers of its spheres will be arranged according to a lattice with cells in the form of cubes, with nodes at the vertices and at the centers of the faces. This structure is possessed by the metals—aluminium, silver, go-

gold, copper, lead, calcium, indium, palladium, iridium, platinum, and thorium. The thought suggests itself that the atoms of these metals may be likened to touching spheres. From the specific and atomic weights one can also calculate the diameters of these spheres. Such a calculation was made by the author for mercury long before the discovery of the diffraction of X-rays. Assuming that in solid mercury the atoms are spherical and arranged in a tetrahedral heap, we obtain for the diameter of the atom \(3.0\text{ Å}\), whereas from the diffusion of mercury vapor the kinetic theory of gases gives the value \(2.8\text{ Å}\). Of course, the meaning of the word diameter remains indeterminate. The author, however, still does not know whether his assumption is correct that mercury crystallizes in the same lattice as the metals listed above.

Alongside the closest packing of spheres described, there exists another of the same density, but having the symmetry of the hexagonal system of crystals. If, after placing upon one close-packed layer of spheres another such layer, the spheres of the third layer are laid so that each sphere covers the spaces between the spheres of the lower layer that are visible through the second layer, then we obtain the above-mentioned system; but if the spheres of the third and subsequent layers are arranged so that these spaces remain visible all the time, then the second closest-packed—hexagonal—system of spheres is obtained. Metals built according to this type are zinc, cadmium, magnesium, and ruthenium.

Chromium, titanium, sodium, tantalum, and tungsten crystallize in a less dense system; their lattice is composed of cubes with nodes at the vertices and at the center of the cubes. Nickel and iron, depending on the conditions of crystallization, occur in a cubic lattice of both of the indicated types.

Of the simple bodies of nonmetallic nature, let us point to carbon. The structure of both its varieties—diamond and graphite—is now known to us and appears highly instructive.

The structure of diamond was established by the Braggs and can be represented most simply in the form of two identical cubic lattices with centered faces, pushed into one another so that one is obtained from the other by a displacement along the diagonal of the cube by one quarter of the length of this diagonal. Here we are dealing with two closest-packed cubic systems. If each cube with centered faces of one system is divided into eight smaller cubes, then the atoms of the other system will be placed at the centers of these small cubes, but not in all of them—only in every other one. Each of the atoms of one system will be at the center of a tetrahedron formed by atoms of the other system and will therefore be surrounded by four atoms, corresponding to its tetravalency. If the centers of neighboring atoms are joined by rods, then these rods will be arranged like the rods of the models of carbon atoms used in organic chemistry. Following the course of the rods from atom to atom, we note that in the diamond system they connect atoms into

hexagons that do not lie in a plane. The length of the rods (the distance between atoms) is equal to \(1.54\ \text{Å}\).

The structure of graphite was established by Debye and Scherrer by the powder method. It is analogous to the structure of diamond and may be derived from the structure of diamond in the following way. Choose one of the diagonals of the cube of the diamond structure and leave the rods parallel to it in their former positions, lengthening them to \(3.41\ \text{Å}\); all the rods not parallel to it are placed in planes perpendicular to this diagonal, making their length equal to \(1.45\ \text{Å}\). The structure obtained belongs to the rhombohedral system of crystals.

By their powder method Debye and Scherrer established that soot consists of graphite crystals. A comparison of the structures of diamond and graphite suggests that their difference depends on the fact that the properties of the atoms themselves in these varieties of carbon are different. The difference in the properties of the atoms is reflected in a difference of their symmetry, and the latter leads to a different mutual arrangement of them. This, evidently, concerns not only diamond and graphite, but also all polymorphic varieties of other substances, so that the cause of polymorphism should be sought in those abrupt changes in the properties of atoms which we now discern in the processes accompanying the quantum emission and absorption of energy.

Chemical compounds of the type \(MR\), where \(M\) is a cation and \(R\) an anion, possess a very simple structure. We have already described the structure of the halides of monovalent metals \(NaCl\), \(KCl\), \(KBr\), and \(KJ\). We shall not repeat what has been said. The oxides of divalent metals—magnesium, calcium, cadmium, and nickel—and lead sulfide have the same structure. Zinc sulfide has another structure. It can easily be derived from the structure of diamond if one of its nets is made of zinc atoms and the other of sulfur atoms. In these structures the atoms of one element are equally surrounded by atoms of the other.

Zinc oxide and cadmium sulfide have the same structure as the hexagonal variety of zinc sulfide, consisting of two hexagonal lattices inserted one into the other; but in them too the atoms of one element are surrounded by four atoms of the other.

The structures of pyrite (iron disulfide) differ in that the sulfur atoms are arranged in pairs, and if only the centers of gravity of the atoms are taken into account, one obtains the rock-salt type. Here each sulfur atom is surrounded by four equidistant iron atoms, and each iron atom by six sulfur atoms.

In cuprous oxide \(Cu_2O\), the oxygen is arranged according to the lattice of a body-centered cube; the copper atoms are situated, four in number, along the diagonals of this cube, midway between the oxygen atoms, at the vertices of a tetrahedron. An oxygen atom is surrounded by four copper atoms; a copper atom lies between two oxygen atoms.

In the magnetite $Fe_3O_4$ (properly, $Fe''Fe_2'''O_4$), and in other spinels of composition $R''R_2'''O_4$, each divalent metallic atom is surrounded by four, and each trivalent one by six, equidistant atoms.

As we see, the rule is observed that the number of atoms surrounding a given atom is equal to, or twice as great as, the valence of that atom.

Iceland spar, calcium carbonate $CaCO_3$, is built, in the manner of rock salt, from ions $Ca$ and $CO_3$. The $CO_3$ ions consist of a central atom $C$ and three atoms $O$ symmetrically arranged around it. Placing, in the rock-salt structure, $Ca$ in place of $Na$ and the $CO_3$ ions in place of $Cl$, we arrange the $CO_3$ ions with their planes perpendicular to straight lines parallel to one of the diagonals of the cube, along this diagonal. We shall not discuss the orientation of the $CO_3$ ions in their planes. The sodium nitrate salt $NaNO_3$ and one of the polymorphic varieties of $NaClO_3$ have the same structure.

Another variety of the latter salt, of the cubic system, has the $ClO_3$ ion enclosed within tetrahedra of $Na$ atoms, this ion being situated with its plane parallel to one of the faces of the tetrahedron. In its plane the ion has some freedom to rotate to the right or to the left, coming into contact by its oxygen atoms with one or the other side with the three $Na$ atoms forming the base of the tetrahedron. Corresponding to such an asymmetric and, moreover, “right” or “left” position of the ion, crystals of this substance rotate the plane of polarization in one direction or the other.

Ammonium chloride is also dimorphous. Its ions $NH_4$ and $Cl$ may have a double position. One may place nitrogen at the center of a cube, arrange the four hydrogens around it along the diagonals of the cube, and the chlorines at the vertices of the cube. Chlorine will touch two hydrogen atoms, hydrogen likewise two chlorine atoms on one side and nitrogen on the other; nitrogen will be surrounded by four hydrogen atoms. In the structural unit of volume there will be contained one particle $NH_4Cl$, as is also determined by spectrometric measurements. Another variety of sal ammoniac is built according to the sodium-chloride type.

The structure of isomorphous mixtures (solid solutions) is of interest. This question was studied by the Norwegian physicist Vegard in collaboration with Schjelderup. He investigated the structure of mixed crystals of potassium chloride and potassium bromide, potassium chloride and ammonium chloride, potassium sulfate and ammonium sulfate, and found that solid solutions are not mechanical mixtures, but true homogeneous crystals. As is easily verified experimentally, a mechanical mixture even of very fine crystal powders gives an indistinct spectrogram, since the lines are superposed one upon another. Solid solutions, however, give an entirely distinct spectrogram, from which the lattice dimensions can be accurately determined. It turns out that these dimensions, in their linear measurements, are the averages of the dimensions of the lattices of the mixing ...

crystals; moreover, in order to compute these averages one has to take into account the relative number of molecules of the substances being mixed. How precisely, however, this law of additivity of the linear dimensions of lattices in solid solutions may be regarded as established is difficult to say, since Vegard and Schjelderup derived it from only a single fact.

To Vegard we owe the establishment of a whole series of structures. We, however, have no possibility of examining all, or even many, of the structures determined up to the present. The more complex the chemical particle of the crystal, the more difficult it is to establish its actual structure. The more atoms are contained in the unit volume of the structure, the more unknowns enter into the equations that determine the coordinates of the atoms by their participation in the intensity of reflections of various orders from different planes, and the more equations are required, along with spectrometric data. If one takes into account that the unknowns enter as arguments into periodic functions of the form (2), it is clear that one cannot rely solely on the results of solving these equations, and must resort to the aid of various guiding and controlling criteria.

14. General criteria for establishing structures. We have already spoken in the course of the exposition about some of such criteria. These criteria are of two kinds—some are a priori, while others follow from certain facts.

At first the notion arose that the crystal is built of separate atoms in such a way that there is no possibility of arranging these atoms into molecules. The concept of the chemical molecule, it seemed, lost its significance for the crystal. Since all the atoms are nevertheless held in the crystal by forces of chemical affinity, the whole crystal seemed to be one molecule. The first of the structures analyzed, such as, for example, sodium chloride, seemed to speak quite definitely in favor of this view. That the crystal is built of atoms, and moreover atoms arranged in a definite order, is, of course, beyond doubt; and this idea was expressed quite definitely by P. Groth even before the possibility of determining the structure of crystals had been discovered. It is, however, quite improbable that in crystallization a destruction of chemical molecules should take place, since such destruction would require an enormous expenditure of energy, whereas we see that in crystallization not only is there no absorption of energy, but, on the contrary, its release in the form of heat is observed. In addition, we observe in the crystal the presence of molecular forces of cohesion, which are by no means smaller in magnitude than the forces of chemical affinity. Indeed, one cannot destroy the forces of chemical affinity by mechanical forces—for example, by a blow to a crystal—but one can overcome the molecular forces; and we know that crystals readily cleave under a blow along the directions of cleavage. Obviously, these directions pass in the crystal along the boundaries of groups

atoms corresponding to molecules. On this consideration the author constructed a structural theory of cleavage, which we shall set forth below as one of the criteria for determining structures. Thus, if atoms are arranged in a crystal in such a way that the boundaries of the molecules are not clear to us, this should not mean that such molecules, in the sense of the distribution of interatomic forces, do not exist. Indeed, we already see that crystals must be assumed to be built of ions. This is evident from the very phenomena of interference, and this will be the first of our criteria. We already know how to use ions as structural units. Here, however, the question arose of what to do with crystals of metals. If metals too are built of ions, then where is the other negative ion? Such an ion may be the electron, whose participation in interference is as weak as that of the hydrogen ion, and which cannot be detected by our still rather crude methods, except perhaps in crystals of hydrogen, if they should prove accessible to investigation. If this supposition concerning the structure of a metal should turn out to be correct, then in metals we would have the simplest chemical compounds; the electrical conductivity of metals would appear to us as electrolysis, and we would have no need to assume, in electrically neutral metals, a special “gas” of free electrons in order to explain electrical and thermal conductivity. Besides ions, the independent existence of water of crystallization was discovered. This was done by Clement Schaefer and Martha Schubert, by means of residual ultraviolet rays, on very many crystals.

Vegard observed that in many crystals studied by him it is possible to distinguish special “molecular axes,” along which the atoms composing the molecule are arranged. In structures of the sodium chloride type as well, such axes can be distinguished if the diagonals of the cubes of the structure are drawn so that they do not intersect and are not parallel. Then along these diagonals we obtain rows of alternating atoms of two different elements, and the arrangement of these rows will satisfy the symmetry of the crystal.

The next criterion will be cleavage. Since only those elements of the structure that are most weakly bound to one another can be separated by cleavage, the planes of division must pass either between identical ions or between layers of the structure that are “chemically saturated,” i.e. that contain the full number of the molecule’s constituent atoms. Other things being equal, the distance between layers affects the magnitude of the cohesive force, as A. Bravais already indicated. Proceeding from these considerations, the author was able to explain the cleavage of a number of structures and to indicate why diamond and zinc blende, despite the complete similarity of their structure, possess different directions of cleavage, and even to indicate wherein this difference should consist.

Consideration of the structures investigated leads to yet another very important and simple criterion. If the atoms of two elements

turn out to be adjacent in the structures of different chemical compounds of these elements, then the distance between these atoms always remains the same. This gave W. L. Bragg grounds for speaking of the diameters of atoms, and from the dimensions of the structures studied he calculated these diameters. We present a table taken from him, in which the diameters of atoms are expressed in angströms. The whole numbers denote the ordinal number of the element.

The quantities marked with an asterisk belong to electronegative atoms.

W. L. Bragg determines the accuracy of the numbers as up to 10%.

Knowledge of atomic diameters greatly facilitates the choice of a possible structure, since what must be placed in the structural unit of volume are no longer points, but bodies of definite dimensions.

1. The forces acting between the elements of a structure, as we now know, are electrical. According to the modern conception, the atom consists of an electrically positive nucleus and of electrons surrounding this nucleus, the number of which corresponds to the charge of the nucleus and determines the atomic number of the element in the periodic system. The chemical inertness of a series of elements—the rare atmospheric gases—and the fact that atoms possessing, in comparison with the atoms of these elements, an excess of one, two, three, and in some cases four electrons show a tendency to lose exactly that same number of electrons and to assume the configuration of the atoms of the inert gases, indicate the special stability of the structure of the latter. For example, sodium, with one electron in excess relative to neon, and magnesium—with two—readily lose their electrons and become charged, the former with one positive charge, the latter with two, being transformed into sodium and magnesium ions. In the same way, when two electronegative atoms approach one another, there is observed a tendency to complete the number of their electrons to the stable number of them in the atoms of the nearest inert gas following them in the periodic system.

gas. If one of the atoms of a compound is distinctly electropositive, and the other distinctly electronegative, then the latter tends to take from the former its electrons that are, so to speak, over-complete in relation to the chemically stable norm. A compound of such elements proves to be an aggregate of an equal number of positive and negative “ions.” Such a compound is called a polar bond. We find it, for example, in compounds of the type of the halides of the alkali metals. If the atoms that unite are distinctly electronegative, or else of indifferent or weakly expressed electrical character, then they tend to take electrons from one another, in order to complete the number of their electrons to a chemically neutral complement; as a result, electrons common to both atoms are obtained, which gives rise to a bond of another type—a valence bond. Since a full valence unit (in the former chemical sense) is determined both by the tendency of one atom to acquire one electron from another and by the tendency of the other to do the same from the first, the number of electrons binding the two atoms will be twice the number of valence units. To such compounds of atoms belong the structures of diamond, carborundum, and others similar to them.

Compounds of the third type, the type of molecular bonds, characterized by aggregates of chemical molecules held together by comparatively weak forces, have not yet been sufficiently studied structurally.

M. Born calculated the distances of ions in crystals of the halides of monovalent metals and the mechanical properties of these crystals, proceeding from the assumption of purely electrostatic forces between ions. A. Landé showed that greater agreement with experiment is obtained if the orbits of the electrons in the atom are placed not in a plane, according to Bohr, but on spherical surfaces. K. Fajans checked Born’s calculations in the field of thermochemical properties.

16. Conclusion. From the foregoing it is clear what great importance our knowledge of the structure of crystals has for knowledge of the properties of atoms, to which the chief attention and chief interest are now directed. Bohr’s planar model of the atom is difficult to reconcile, or even wholly irreconcilable, with the conditions of the arrangement of atoms in a crystal; and we see that it is already yielding place to a three-dimensional model of the atom. Knowledge of the properties and structure of the atom is equally important for both physics and chemistry.

Apart from its theoretical significance, knowledge of the structure of crystals has enormous practical significance, since solid bodies usually prove to be crystals. Metallography and metallurgy are beginning to take a serious interest in this new method of investigating solid bodies.

If, for the discovery of the diffraction of X-rays and its application to the investigation of the structure of crystals, we are wholly indebted

foreign researchers, nevertheless one can point to a number of Russian investigators who have made a notable contribution to this field. Here it is necessary to mention the works of A. F. Ioffe and M. V. Kirpicheva, the works of G. V. Wulff and N. E. Uspensky, N. E. Uspensky and S. T. Konobeevsky, and the works of G. V. Wulff. During the recent years of Russia’s isolation from the outside world, these investigators arrived at conclusions which partly supplemented and partly coincided with the conclusions published in the foreign literature, of which we were completely deprived until very recently and which even now reaches us extremely irregularly. Many results of the work of Russian scholars could not be published owing to the conditions of the time. Because of this latter circumstance, the author is not sure that no essential facts or theories have escaped him, and he cannot assume responsibility for the completeness of his survey.

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