THE DEVELOPMENT OF X-RAY ANALYSIS OF CRYSTALS

W. L. Bragg, Manchester *

In 1932 twenty years had passed since the discovery of the interference of X-rays by Laue, Friedrich, and Knipping. Immediately following the discovery, this phenomenon was applied by W. H. Bragg and his son W. L. Bragg to the analysis of the structure of crystals. The outstanding successes achieved since then by X-ray analysis, which has developed into an independent discipline of enormous scientific and practical importance, are well known. The article printed below, kindly sent by Prof. W. L. Bragg for our journal, is of particular interest, since its author not only was one of the principal participants in the discovery of X-ray analysis, but has ever since continued to play a leading role in the development and application of methods for analyzing the structure of matter by means of X-rays.

The Editors

1. The investigation of the structure of crystals constitutes a far more important branch of science than might appear at first glance, for the crystalline state is characteristic of practically every form of solid matter. We may even go further and say that any strict definition of the solid state is bound up with the concept of crystalline structure. The earth’s crust consists of mixtures of crystals of various kinds, which may be very small and intermingled with one another, but nevertheless must belong to types that can be determined and that, through the labors of mineralogists, have now been classified into an orderly system. The crystalline state is here so universal that mineralogy has always been closely related to crystallography, which in the past was almost completely ignored by other branches of science. When by chemical means we prepare some pure substance, organic or inorganic, the product obtained, if it is not a liquid or a gas, will in almost all cases be

* Translated from the author’s manuscript by N. A. Shiptsov.

crystalline. All metals and alloys are built up from a conglomerate of crystallites having the most varied orientation, which can be detected by polishing and etching the surface. Metals have properties so different from those of other bodies that it would be quite appropriate to debate whether they should be assigned to the class of solids or whether the category of metals should be considered entirely separately. The resemblance of metals to true solids consists rather in the fact that they are a conglomerate, and not in the fact that properties similar to those of solids are found in single crystals of metals. Both the one and the other are characterized by a crystalline structure. Even those substances that are products of organic processes have an ordered arrangement in their atomic structure, which therefore resembles a crystal. As striking examples one may cite cellulose from the vegetable kingdom and keratin from the animal kingdom, which is the basis of such substances as horn, wool, and hair. Such substances can acquire great strength when an inorganic crystalline substance is included in them, as, for example, shells, teeth, and bone. Thus we may say that the study of the solid state is in reality the study of crystalline or ordered arrangement.

By a crystal one usually understands a regularly constructed solid body, bounded by plane faces meeting at sharp angles. But the principal property of a crystal is the regular lattice in which the atoms are arranged. The external form is only a manifestation of this internal lattice, so that even a small particle of such a substance, having irregular boundaries, is a true crystal, just as is a body possessing a perfect geometrical form.

1. The crystalline form is so universal because it corresponds to a state with lower potential energy as compared with an irregular arrangement. When molecules of a solution or of a molten substance come together to form a solid body, then, in the case of a regular arrangement, the greatest amount of potential energy is liberated, and for this reason such a form will be the most stable. A molecule approaching the face of a growing crystal is held more firmly when it joins an already existing lattice. Cases in which a liquid passes into an amorphous form correspond to the formation of strong bonds between neighboring molecules of the liquid before the solid state is reached.

For example, in molten quartz, oxygen exerts a strong attraction on two silicon atoms, and silicon on four oxygen atoms; moreover, the corresponding temporary bonds must constantly form and break as long as the quartz remains molten. As the liquid cools, these bonds begin to have a more prolonged existence, and the liquid passes through a very viscous state and finally becomes amorphous quartz glass. The structural irregularities that existed in it pass into the solidified state before they have time to assume the correct form. The same, generally speaking, also occurs in the case of glass; only after a prolonged interval of time do the atoms acquire the possibility of assuming the correct arrangement, which corresponds to the “weathering” of glass. Boerner’s pearls, quickly frozen water, and amorphous sugar represent other cases of such premature, incorrect formation of bonds. On the other hand, in many cases molecules or atoms regroup until, finally, they fall into the position most convenient for them, i.e., until the formation of a crystal occurs.

1. The existence of a regular arrangement in a crystal has been recognized since ancient times, since the very external form of the crystal betrays its secret. Every regular three-dimensional lattice has its units arranged in such a way that they are found in plane layers, and the faces of the crystal always correspond to the simplest of these layers. To illustrate this, one may take the sides of a pyramid, in which pairs are arranged when their layers are superimposed upon one another. The number of ways in which symmetrical lattices of this kind may be constructed is a purely geometrical question, and for that reason it was solved many years before actual crystalline structures became known. The famous Russian crystallographer Fedorov, together with Schoenflies in Germany and Barlow in England, were pioneers in this field of research. Since all crystals possess symmetry of one type or another that had been determined by these investigators, this may be considered proof of the correctness of the theory they constructed.

However, the actual arrangement of atoms was found only when Laue initiated the development of a new branch of science with his discovery of the diffraction of X-rays in 1911. In this discovery a very important role was played by Ewald’s doctoral dissertation; Ewald has always been an outstanding worker in the field of X-ray crystallography. As the object of his investigations

Ewald chose the action of a crystalline medium on light; moreover, in order to solve this problem he considered the scattering of light by regularly arranged points—atoms—instead of interpreting this medium as a continuum. The story goes on to say that Laue, considering these theses, asked himself what the effect would be if the waves were shorter than the interatomic distances, and not longer, as is the case with light waves. At that time it was believed that the wavelengths of X-rays must lie in this very short region, i.e., be of the order of \(10^{-8}\) cm, and therefore Laue proposed trying to obtain diffraction of X-rays by a crystal. The experiment, performed by Friedrich and Knipping, brilliantly confirmed these expectations. When an X-ray beam was passed through a crystal, a whole system of regularly arranged spots was obtained on a photographic plate, which was evidently the result of the diffraction of X-rays by the internal lattice of the crystal. In the history of science there are not many such cases in which so simple an experiment opened up so vast a field for investigation.

1. At the time when Laue was publishing these results, my father, William Bragg, was a resolute supporter of the idea that \(\gamma\)-rays, and consequently X-rays as well, have a corpuscular nature. He arrived at this view on the basis of experiments with the ionization of gases by X-rays and \(\gamma\)-rays, from which he concluded that these rays do not ionize directly, but produce explosions at separate places, as a result of which an electron is ejected from the atom or a “cathode ray” is emitted. He attributed these explosions to collisions with atoms of those corpuscles which constitute \(\gamma\)-rays and X-rays. Direct experiments, such as Wilson’s experiments with the cloud chamber, show that ionization is indeed caused only by cathode rays. But it must be said that at that time this was the only representation one could adhere to, all the more since it was being vigorously developed in many places. My father’s views gave a quantitative explanation of the phenomenon, so that logical reasoning based on known facts compelled him to adhere to this extreme position.

The results obtained by Laue apparently contradict this view, speaking decisively in favor of X-rays being electromagnetic waves. At that time I had just completed my university education and, of course, was a resolute supporter of my father’s views. I undertook the study of Laue patterns and some experiments in order to determine whether they could not

whether these diagrams could be explained by a corpuscular theory of X-rays, according to which corpuscles or cathode rays fly through the empty spaces between atoms in the crystal structure. Of course, it soon became clear that the explanation of this effect by diffraction, which had been given by Laue, should be considered correct, but I was able to establish that the effect is obtained in a much simpler way than Laue had imagined it, and that with the aid of such an effect one can determine the arrangement of atoms in a crystal.

The question at that time stood approximately as follows. Laue obtained diffraction photographs with a cubic crystal of zinc blende, ZnS, and he succeeded in showing that the positions of the spots on the photographic plate are in agreement with the arrangement of scattering centers at the nodes of a cubic lattice. If one imagines cubes stacked together so that the atoms or molecules are placed at the corners of the cubes, this will correspond to the arrangement considered by Laue. However, such an arrangement should have given many more spots than were actually present in the photograph. Laue assumed that this limitation of the pattern is due to the existence of only a few wavelengths or lines in the X-ray spectrum, since it can be shown that the conditions for interference determine not only the direction of the diffracted ray, but also the wavelength that can be diffracted. From another point of view, I imagined that the beam consists of irregular pulses of radiation, which is equivalent to the idea that the beam consists of all wavelengths contained in a broad interval, like “white light” in optics. Such pulses are reflected by layers of atoms in the crystal structure, and the reflected rays form Laue’s pattern. In the different ways of considering the phenomenon of diffraction there is no essential difference, so that one may adhere to the more convenient notion of reflection. The essential difference in our interpretations consisted in the fact that Laue attributed the features of the diffraction pattern to the structure of the radiation (groups of monochromatic waves), whereas I was able to show that these features are caused by the structure of the crystal itself. If the diffracting centers in the structure of zinc blende are placed both at the corners of the cube and at the centers of its faces, and not only at the corners of the cube, then the principal features of the diffraction pattern are fully explained. William Pope in Cambridge, who together with Barlow had proposed theories of the structure of crystals, advised me to study Laue patterns obtained in the case of NaCl and KCl. These patterns proved susceptible to a complete explanation,

and, moreover, led to the first analysis of the atomic arrangement shown in Fig. 1.

The interpretation of this problem on the basis of reflection leads to the following law. When an X-ray falls upon a layer of atoms situated in some plane, the diffracted waves form the front of the reflected wave in accordance with Huygens’ well-known principle. The atoms of a crystal are arranged in groups of parallel layers, which can be drawn in an infinite number of ways in the most varied directions of the lattice. Let us consider the reflection of one of these groups of layers, which are situated at a distance \(d\) from one another. If the angle of incidence of the ray on such a layer is \(\theta\), then the path difference for waves reflected from two successive layers will be \(2d\sin\theta\), as follows from the well-known law of optics. Therefore, for wavelength \(\lambda\), reflection will occur under the condition

\[ n\lambda = 2d\sin\theta . \]

Conversely, if we regard the X-rays as impulses, then the layers will reflect a stream of such impulses with spacing \(2d\sin\theta\). This stream can be resolved by Fourier’s theorem into components whose wavelengths are equal to:

\[ 2d\sin\theta;\quad \frac{1}{2}(2d\sin\theta);\quad \frac{1}{3}(2d\sin\theta)\ \text{and so on.} \]

Fig. 1.

We may consider that the crystal either selects these wavelengths from some interval of incident “white” X-rays, or produces monochromatic streams from a series of irregular impulses, exactly as in the corresponding optical problem of the grating. Every face of a crystal is parallel to some group of crystalline planes, and therefore it can be used for the “reflection” of X-rays, as I have succeeded in showing experimentally.

1. Experiments were carried out by my father in order to determine whether the rays which give Laue’s pattern and which, evidently, are electromagnetic waves are indeed X-rays, and not some accompanying radiation. He reflected an X-ray from a crystal face and measured the intensity of the reflected ray by means of an ionization chamber.

(ionization spectrometer). At the same time, measurements were made of the absorption by various screens placed in the path of the beam. It turned out that the reflected radiation was in all respects similar to the original X-ray radiation.

During these experiments, when radiation reflected at various angles was being measured, a tube with a platinum anticathode was used. It turned out that the reflected waves consist both of a continuous band of “general” X-ray radiation, analogous to white light, and of three distinct monochromatic lines. The latter were immediately identified by means of absorption measurements, and it turned out that they represented the characteristic \(L\)-radiation of platinum. Later, tubes with nickel, silver, radium, and palladium anticathodes were used. In each of these cases there appeared a pair of X-ray spectral lines corresponding to the characteristic \(K\)-radiation. These were the first analyses of X-ray spectra. Moseley and Darwin also tried to detect this effect, but because of the design of their apparatus, in which they used excessively narrow slits, they were unable to find these peaks of characteristic radiation. Subsequently Moseley developed this new discovery, and, thanks to his brilliant technique in constructing an X-ray tube with a large number of replaceable anticathodes, he succeeded in finding the relationship between the X-ray spectra of the elements and in discovering the connection between the number of electrons and the ordinal number.

These original experiments served as the basis for two tremendous branches of science, whose growth began with Laue’s discovery. One of these branches concerns X-ray spectra. The study of these spectra is very important from the standpoint of understanding atomic structure, since it makes it possible to obtain knowledge of the energy levels of the tightly bound electrons in the inner part of the atom, whereas optical spectra arise only through reactions in the outer, freely bound electrons. On the other hand, the study of the reflection of monochromatic radiation by the various faces of a crystal proved to be a far more powerful method of analyzing crystals than the study of Laue photographs. One of the first analyses was the analysis of diamond, whose excellent cubic structure is characterized by the fact that in it each carbon atom is symmetrically surrounded by four other atoms. Thus, for the first time, the existence of tetrahedral bonds around the carbon atom was proved, something that had previously had to be assumed on the basis of the data of organic chemistry. Soon after this, as a result of studies carried out—

...given by my father together with me; there followed also the analysis of a whole series of other simple crystals.

1. A crystal structure is a lattice whose constituent parts are atoms. These atoms may be of different kinds; they may be situated at different distances from one another, but we can always distinguish in the structure a “unit of the lattice.” The latter consists of a group of atoms which can be repeated constantly, like some part of a wallpaper pattern or of a fabric pattern, of course in this case not in two but in three dimensions. Heavy atoms scatter X-rays more effectively than light atoms. When an X-ray beam falls on such a lattice, diffracted rays are observed. The task of X-ray analysis consists precisely in determining the arrangement of the atoms by observing diffraction effects. Here one may make three kinds of brief remarks concerning the results that X-rays provide.

First of all, they make it possible to carry out measurements of the scale of repetition of the lattice, or to find what are called the “dimensions of the cell.” This process is entirely analogous to calibrating an optical grating by means of a known wavelength. The distances between the lines in this grating can be measured on the basis of the observed angle at which diffraction of a given wave of wavelength \(\lambda\) occurs. In exactly the same way, the known wavelength of X-rays can be used to measure those intervals through which the repetition of the crystalline lattice proceeds in different directions in space. In this case the geometrical problem is more complicated, but the principle is the same.

Further, X-rays make it possible to determine the symmetry of the crystal. To give a brief illustration of this question would be somewhat more difficult, but Fig. 2 can to some extent explain the matter. Figs. 2,a and 2,b show two types of lattice in which the right and left parts are symmetrical. In Fig. 2,a this occurs because the units (which may have a completely arbitrary form) are reflected in vertical lines. In Fig. 2,b there are no such planes of reflection, but nevertheless it is obvious that the right and left parts have an entirely identical appearance. This symmetry is due to a “glide plane,” which reflects the units and at the same time displaces them. If the crystal as a whole has a plane of symmetry, then we may conclude that its lattice is built according to the type of one of these drawings, but which one in particular cannot be said in advance. It is precisely X-rays that at once make it possible to solve this problem.

If a ray is reflected from those planes of the crystal which correspond to the dotted lines in the figures, then the cell dimensions found will constitute only half of the true period of the lattice in case 2,b and, conversely, they will correspond to the full period in case 2,a. In the case of a three-dimensional lattice the possible types of symmetry become far more complicated. Fedorov was the first to study this question completely, and he succeeded in showing that there exist 230 types of lattices in three dimensions. On the basis of the same principle that has been illustrated here by a simple example, X-rays can indicate to which of these 230 types any given crystal belongs.

Finally, X-rays make it possible to reveal the structure of the elementary cell itself, which consists of a group of atoms. This is achieved on the basis of observing the intensities of the diffracted rays. Waves from the atoms in this cell interfere, reinforcing one another in the case of some rays and weakening one another in other cases. Each atomic configuration gives its own definite effect. By means of guesses of various kinds it is possible to construct such a lattice as will fully explain the observed phenomena.

Fig. 2.

1. X-ray analysis was developed through the efforts of many investigators, but certain individual achievements stand out especially in the history of its development. In the earlier stages of development, large single crystals were used for study, with the X-rays being reflected by various faces, both those existing in their natural form and those prepared artificially. However, it proves no easy matter to obtain the majority of simple substances in the form of large crystals. The simplest compounds, generally speaking, exist in the form of conglomerates of very small crystals, whereas a whole series of complex bodies can be grown in the form of large specimens, such as potassium ferrocyanide, alum, etc. The discovery of the “powder method,” made simultaneously by Debye and Scherrer in Germany and by Hull in America, marked very great successes; this method makes it possible to dispense

and without single crystals. A monochromatic beam of X-rays here falls upon a mass of crystal ground into powder. Each type of diffraction requires a certain definite orientation of the reflecting crystal, but in the general mass of the smallest crystals some of them will always be oriented in such a way that this necessary condition is satisfied. The diffracted rays emerge from the powder in the form of a series of conical surfaces, which may intersect a photographic film bent around the powder and be recorded on it. Each crystalline material gives its own characteristic powder photograph, by means of which the material can be identified.

One of the results of the application of this new method was a general revision of the crystalline structure of simple substances, and the enormous credit in this respect belongs to Goldschmidt and his school. This revision led to an extraordinarily great increase in our knowledge of the forces acting between atoms in a crystal. Another result was the emergence of an entirely new method for studying metals and alloys. When an entire series of alloys is obtained by gradually increasing the quantity of metal $B$ in metal $A$, then $B$ at first enters into the crystal lattice of metal $A$. This process, generally speaking, ceases at some definite percentage content, after which there follows the appearance of a new “phase,” which has a higher content of $B$ than the preceding solid solution. Polishing and etching the surface of the metal show these two phases in the form of two kinds of crystals. As the composition changes from pure metal $A$ to pure metal $B$, a whole series of phases is obtained. Formerly these latter had to be inferred on the basis of thermal measurements and observations of the appearance after etching. X-ray measurements, however, make it possible to determine each phase as a characteristic crystalline structure, since each of them gives its own type of powder photograph; thus X-ray analysis proves to be the most powerful instrument of research in metallurgy. However, we can not only determine each phase with certainty, but can also find similar phases in alloys of different metals, and hence above all derive generalizations about the structure and properties of alloys. This work is still only beginning, since the number of different alloys is extraordinarily large, but even now a new chapter has been written into metallurgy. The technical significance of this new metallurgy can hardly be overestimated.

1. All the work described thus far relates to pro-

…crystals, since even the structures of alloys, generally speaking, belong to a very simple type. A further important development of the analysis was its application to ever more complex crystals, so that at the present time no structure seems so complex that it cannot be studied.

One of the most important conditions for success was the auxiliary role of logical methods, by means of which the symmetry of a crystal can be determined. As was indicated above, there exist 230 different types of symmetry. In earlier analyses, where one had to deal with simple specimens, the determination of symmetry was carried out directly. The compilation of general rules and the development of a logical scheme for determining any one of the 230 groups was accomplished thanks to the work of Niggli in Germany, Wyckoff in America, and Astbury and Yardley in England. If first the elementary cell of a crystal is measured and then it is determined which series of spectra characteristic of this cell is absent owing to the influence of symmetry, then every crystal can be assigned to one of the 230 groups. In this way we obtain the skeleton of planes, axes, and centers of symmetry by which the structure of the crystal is determined. Such a determination of the “space group” of symmetry, as it is now customary to say, plays a very substantial role in the analysis.

Beginning in 1924, I undertook attempts to extend the analysis to very complex structures characteristic of many crystals. Silicates were chosen as the material; these are complex inorganic compounds and occur in nature in the form of well-developed crystals. The technique of X-ray measurements with an ionization spectrometer was improved, and we were able to show that direct measurements can be carried out with great success on extraordinarily complex atomic arrangements. The silicates, which constitute the principal component part of the earth’s crust, were successfully classified in the form of a regular scheme. The complexity of a crystal is expressed by the large number of coordinates, or “parameters,” required in order to determine the positions of its atoms. Rock salt (Fig. 1), for example, has no parameters, since all atoms are located exactly at the corners of a cube. Earlier analyses were carried out only on crystals having one, or perhaps two, parameters. But silicates have a number of parameters from 10 to 60, whence arises the need to develop a new method for their analysis.

As our knowledge of these inorganic structures expanded, those …

principles by which the existence of these structures is determined. These principles were summarized in a very important article by Pauling in 1928. The first analyses involved very great difficulty, since we knew too little about the probable arrangement of atoms and had to move forward with great caution, stopping at the proof of each step. Now the answers to problems of structure have begun to seem incredibly simple and obvious, as often happens in other fields of knowledge as well. Using Pauling’s principles, in many cases we can predict in advance the nature of the structure, giving certain hints, for example, concerning the space group that forms the basis of the structure, after which X-rays are needed only to verify the accuracy of our conjecture about the structure or to decide which of several conjectures is correct.

1. In all methods of X-ray analysis one has to deal with the determination of the positions and intensities of diffracted rays. The most widely used of these is the rotating-crystal method, proposed by Polanyi and Schiebold. A small crystal is placed in the path of a monochromatic X-ray beam and is then subjected to slow rotation by means of a clockwork mechanism about an axis perpendicular to the beam. During such rotation, one series of planes after another comes into the position in which reflection of the incident beam occurs. These reflected rays are recorded either on a photographic plate or on a film bent in the form of a cylinder around the axis of rotation. In this way a multitude of reflections is obtained, and their intensities can be roughly estimated from the intensities of the spots on the plate or film. Such rough measurements are sufficient for many purposes of analysis, which under these conditions does not require a great expenditure of time.

This method has been applied to the analysis of crystals more often than other methods. In doing so, excellent results can be obtained with very small pieces of crystals, even having a mass of about \(10^{-6}\) g.

1. A further extension of X-ray analysis, promising to yield results of extraordinarily great importance, consists in applying it to the study of the structure of very complex organic molecules. Chemical methods in the past have told us much more about the atomic arrangement of organic molecules than in the case of inorganic substances. Organic molecules, having strong internal bonds, constitute stable structures. If we add some new group to

at some point of the molecule, the structure of the original part does not change very much. Thus there arose stereochemistry, based on the conception of four bonds around the carbon atom, directed toward the vertices of a tetrahedron, and of the six-membered benzene ring. X-ray analysis confirmed the correctness of both of these conceptions. But there also exist many such complex molecules as strychnine, whose structure still has to be guessed, insofar as this can be done on the basis of the data of chemical analysis. In doing so, the molecule has to be destroyed and the structure of its fragments determined. However, there are various ways of assembling these fragments into a single whole. In such cases X-ray analysis often provides the additional information needed for the complete determination of the structure. Although the molecule may be too complex for direct analysis, the size and shape of the space occupied by it in the crystal structure are nevertheless determined directly. By testing in this way the various possible structures, one can find that model of the molecule which occupies just the right space.

Another success fell to the study of such structures as cellulose. On the basis of chemical considerations Haworth concludes that cellulose is a chain of glucose rings linked with one another, and this was confirmed by the work of Polanyi, Mark, and others. These chains are arranged parallel to the cellulose fibers. It must be said that cellulose has an entirely crystalline character in comparison with wool, which gives only very blurred diffraction effects. Nevertheless, the study of the structure of wool led Astbury to very interesting results. It was possible, for example, to show that mechanical stretching changes the form of the wool molecule, whose fibers consist of chains of amino acids. If these fibers are subjected to such stretching and then allowed to rest, one can observe changes in the semicrystalline structure between two completely different forms. It is obvious that these observations are of very great importance, since wool is closely related to the proteins that form part of living matter.

In conclusion, a few words should be said about the applications of X-ray analysis, in particular to the solution of technical problems. X-ray methods constitute new means of obtaining information about bodies whose properties we are interested in knowing. In the past only methods of chemical analysis and optical observations were used. Chemical analysis proceeds by destroying complex substances into elements or simpler ones.

groups and the determination of these constituent parts. But if, in order to obtain the necessary knowledge, the original substance has to be destroyed, then along with this destruction there also goes the destruction of many possibilities for obtaining such knowledge. Optical observations tell us about what may be called the geography of the structure of a substance, but they are too crude to determine its atomic composition. X-ray methods are a supplement to these methods. The latter do not require the destruction of the substance, so that its atomic arrangement is determined precisely at the time when it fully reveals its properties.

Technologists who set themselves the goal of improving certain processes by using materials of better quality now rely on mechanical, chemical, and optical tests. The latter cannot lead to a direct solution of the given problem, but provide only knowledge on which the technologist can build his intuitive guesses about the causes of poor quality and about ways to improve it. In order that X-ray investigation may find application alongside the older methods, it is necessary that technologists understand what possibilities are opened up by this application and what hopes may be placed in the results of the corresponding experiments. Such investigation is a slow process, but nevertheless the new analysis is finding more and more application in laboratories of applied knowledge, so that there can now be no doubt that it will gain universal recognition in the very near future.