ARRANGEMENT OF ATOMS IN METALS AND ALLOYS*

U. L. Bragg

I. Introduction

Thanks to X-ray analysis, which has made it possible to determine the arrangement of atoms in solid bodies, a number of problems in many areas of science have received new illumination. One of the fields in which X-ray analysis has proved especially fruitful is metallurgy. In response to your kind invitation to read before you your May lecture, I should like to subject to discussion those conclusions about the internal structure of metals and alloys that can be obtained by studying their atomic lattices. In doing so, I intend not to dwell on the technique of this area of physical research itself, with which my own activity is especially connected; I have set myself the more ambitious task of discussing before an audience of specialists the very foundations of metallurgical science. I hope that you will treat my exposition with due indulgence, taking into account the fact that I am a physicist, not a metallurgist.

II. Definition of a Metal and an Alloy

Metals are bodies distinguished by the characteristic properties of malleability, ductility, high optical reflectivity, and electrical conductivity. The melting together of two metals likewise leads to the formation of substances that differ in characteristic fashion from ordinary chemical compounds. On a phase diagram, in passing from pure metal $A$ to pure metal $B$, regions of pure phases alternate with mixed two-phase regions. In single-phase regions the composition of the alloy is homogeneous; it changes as one moves forward through the region, and all the properties of the alloy show a corresponding continuous change. In a two-phase region the alloy consists of two parts, which are phases situated to the left

* The 25th May Lecture at the Institute of Metals; Journ. of the Inst. of Metals 55, 1935. Translated by E. G. Ananiashvili.

and to the right of the region; both phases are constant in composition, but their relative amounts in the alloy differ at different points of the region. Single-phase regions may be extensive, embracing wide limits of composition; they may, however, also be so narrow that they can be taken for a definite chemical compound. By analogy with chemical compounds, it is usually customary to assign to each phase a definite formula of the type \(A_mB_n\), expressing, as it were, the ideal composition of the given phase, and to explain the continuous change in its composition by excesses of the pure metals \(A\) or \(B\), dissolved in the pure phase \(A_mB_n\). I do not intend to dispute this generally accepted and convenient method of describing the phenomena, but I should like to point out the necessity of reconsidering its significance in the light of new information about the nature of alloys. In any case it is clear that alloys with their variable composition differ from chemical compounds precisely defined in composition, although intermediate cases do exist, making the transition gradual.

III. PHASE LATTICE

X-ray analysis has made it possible to establish the important fact that each phase corresponds to its own characteristic atomic lattice. Westgren was the pioneer in this field. This lattice remains unchanged in each single-phase region, apart from a small variation in size. In the transition from one phase to another it changes completely. In a copper-zinc alloy, for example, the \(\alpha\)-phase has the face-centered cubic lattice of copper; the \(\beta\)-phase has a body-centered cubic lattice; the \(\gamma\)-phase is characterized by a complex cubic lattice with 52 atoms in the unit cell; the \(\varepsilon\)-phase corresponds approximately to hexagonal closest packing; and, finally, the \(\eta\)-phase has a lattice related to the hexagonal lattice of zinc.

The number of lattice types discovered in various systems is limited. Ever newer alloys reveal, one after another, the same lattice types or slight modifications of them. The existence has also been established of several further lattice types—such as a cubic structure with 20 atoms in the unit cell, found, for example, in the silver-aluminum system—and in addition there is still a certain number of complex structures not yet solved. The structures encountered most often are those found in almost all pure metals, namely: face-centered cubic (cubic closest packing), body-centered cubic, and hexagonal closest packing. The remaining types may be obtained from these by a slight deformation of them, or by removal of some of the atoms and distribution of the remainder in a somewhat altered order.

IV. Differences Between an Alloy Phase and a Chemical Compound

If each phase corresponded to a definite chemical compound distinct from the others, it would be natural to expect for it also a special structure. In reality the composition of a phase is not constant, but its change can be explained by the partial replacement of the atoms of one metal by atoms of another. In typical alloys, the change in composition occurs by the replacement of some atoms by others, and not by the penetration of extra atoms into the interstices of the lattice. In other words, an X-ray check of the density shows that the number of atoms of any kind in the elementary cell is constant and is characteristic for the given phase. The penetration of atoms into the interstices of the lattice occurs in alloys of certain metals with hydrogen, boron, carbon, and nitrogen, which must be assigned to a special class.

I shall give here some X-ray conclusions which, in my opinion, vividly illustrate the difference existing between an alloy and a chemical compound of the ordinary type. The same conclusion can, of course, be reached by many other paths, but the path I set forth shows how important the investigation of the arrangement of atoms can be. Certain phases of the systems copper—zinc, silver—zinc, gold—zinc, and copper—cadmium show a composition approximately expressed by the formulas \((\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au})_5\mathrm{Zn}_8\) and \(\mathrm{Cu}_5\mathrm{Cd}_8\) (for example, \(\gamma\)-brass). These phases are very similar to one another both in their hardness and brittleness and in their external appearance. In accordance with this similarity, one and the same

Fig. 1.

the same γ-cubic structure with 52 atoms in the unit cell, first investigated by Bradley and Thewlis¹. A more thorough investigation by Bradley and Gregory² showed, however, that there is a very considerable difference between them, the essence of which is clear from Fig. 1. Despite the fact that the phase lattices are almost identical, differing only slightly in the coordinates of the atoms, the distribution of the atoms among the different lattice positions is completely different in Ag₅Zn₈ and Au₅Zn₈, on the one hand, and Cu₄Cd₈ on the other. As for Cu₅Zn₈, Bradley and Gregory give arguments in favor of the view that its structure is similar to that of Ag₅Zn₈ and Au₅Zn₈, although a direct X-ray verification of this fact proves impossible because the zinc and copper atoms, having almost identical scattering power, cannot be distinguished by means of X-rays. The figure shows both the similarity of the two lattices and the non-identity of the positions occupied by the silver and gold atoms in the structures on the left and right sides of the figure. These atoms are marked by hatching.

In isomorphous chemical compounds, the atoms of the corresponding elements invariably occupy corresponding positions. This is a necessary consequence of the circumstance that the bonds forming the structures allow the replacement of one element by another element similar to it, since the nature of these bonds remains essentially unchanged. In the case under consideration, the replacement of zinc by cadmium does not change the phase lattice, although the zinc atoms in the first case and the cadmium atoms in the second are surrounded by entirely different neighbors.

We can only conclude that the phase lattice common to both structures depends essentially on no direct bonds whatsoever between neighboring atoms.

The idea of the existence of a special phase lattice, independent of the manner in which atoms are distributed among its various positions, was first advanced by Tammann in one of his hypotheses, published in 1919. In order to explain the phenomenon of a change in electrical conductivity during prolonged annealing, he supposed that in an alloy there occurs a transition from a disordered distribution of atoms in the lattice to an ordered one. Similar transitions were discovered by Johansson and Linde⁴ in the gold–copper system by means of X-ray analysis and were studied extensively thereafter. In all such cases the phase lattice proves to be completely independent of the manner in which atoms of different kinds are distributed within it.

V. Hume-Rothery Rule

After the above-described definition of a phase had been established, the next step forward was the brilliant generalization known as the “Hume-Rothery rule.” As the number of investigated phase lattices increased,

it was found, as mentioned above, that the same types of lattices recur in different alloys. Hume-Rothery¹⁸ noted that if some metal, for example copper, is alloyed with other metals of different valency, then often on the diagram of each alloy a series of identical phases is observed. The higher the valency of the added metal, the smaller is its atomic percentage at the corresponding point of the phase diagram. Expressed in quantitative form, this rule states that identical phase lattices are characterized by a constant ratio of the number of free electrons to the number of atoms. After Hume-Rothery formulated this rule, a considerably larger number of examples confirming it was found. This rule is not always fulfilled, but there is no doubt that it represents the expression of a certain principle of fundamental significance.

The Hume-Rothery rule is so often mentioned that I shall not venture to dwell on it here at length. I shall confine myself only to a few examples. In the following scheme, to each metallic atom one should assign the following number of free electrons:

\[ \begin{array}{rlr} \mathrm{Cu},\ \mathrm{Ag},\ \mathrm{Au},\ \ldots\ldots\ldots\ldots & & 1\\ \mathrm{Mg},\ \mathrm{Zn},\ \mathrm{Cd},\ \mathrm{Hg},\ \ldots\ldots\ldots & & 2\\ \mathrm{Al},\ \ldots\ldots\ldots\ldots\ldots & & 3\\ \mathrm{Sn},\ \mathrm{Si},\ \ldots\ldots\ldots\ldots & & 4\\[0.8em] \left. \begin{array}{l} \mathrm{Fe},\ \mathrm{Co},\ \mathrm{Ni},\\ \mathrm{Ru},\ \mathrm{Rh},\ \mathrm{Pd},\\ \mathrm{Os},\ \mathrm{Ir},\ \mathrm{Pt}. \end{array} \right\} \ \ldots\ldots\ldots\ldots & & 0 \end{array} \]

Body-centered cubic structure (β); ratio of the number of electrons to the number of atoms \(3:2 = 1.5\): \((\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au})\ \mathrm{Zn}\), \(\mathrm{Cu}_3\mathrm{Al}\), \(\mathrm{Cu}_5\mathrm{Sn}\), \(\mathrm{CuBe}\), \(\mathrm{AgMg}\), \((\mathrm{Co}, \mathrm{Ni}, \mathrm{Fe})\ \mathrm{Al}\).

Complex cubic structure with 20 atoms in the unit cell (β); ratio of the number of electrons to the number of atoms, \(1.5\): \(\mathrm{Ag}_3\mathrm{Al}\), \(\mathrm{Au}_3\mathrm{Al}\), \(\mathrm{Cu}_5\mathrm{Si}\), \(\mathrm{CoZn}_3\).

Complex cubic structure with 52 atoms in the unit cell (γ); ratio of the number of electrons to the number of atoms \(21:13 = 1.61\): \((\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au})_5(\mathrm{Zn}, \mathrm{Cd})_8\), \(\mathrm{Cu}_9\mathrm{Al}_4\), \(\mathrm{Cu}_{31}\mathrm{Sn}_8\), \((\mathrm{Fe}, \mathrm{Co}, \mathrm{Ni}, \mathrm{Pd}, \mathrm{Pt})_5\mathrm{Zn}_{21}\).

Hexagonal structure with maximum close packing (ε); ratio of the number of electrons to the number of atoms \(7:4 = 1.75\): \((\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au})(\mathrm{Zn}, \mathrm{Cd})_3\), \(\mathrm{Cu}_3\mathrm{Sn}\), \(\mathrm{Cu}_3\mathrm{Sb}\), \(\mathrm{CuBe}_3\), \(\mathrm{Cu}_5\mathrm{Ge}\), \(\mathrm{Ag}_5\mathrm{Al}_3\), \(\mathrm{Au}_5\mathrm{Al}_3\), \(\mathrm{Au}_3\mathrm{Hg}\), \(\mathrm{Ag}_3\mathrm{In}\).

Exact observance of the ratios \(3:2\), \(21/13\), \(7/4\) is not obligatory. The number of atoms in the unit cell, of course, is a definite whole number, but the ratio of the number of elec-

the number of electrons to the number of atoms varies in different alloys, and we can say only that the characteristic phase appears when it approaches one of the cited values: 1.5, 1.6, or 1.75.

The conclusion from the Hume-Rothery rule is that the phase lattice is in some way determined by the number of free electrons per atom, which must be packed into the structure; combining this conclusion with our preceding generalization, we shall note that this packing apparently depends not so much on the distribution of atoms among the positions of the phase lattice as mainly on the lattice itself.

VI. G. Jones’s Theory of Phase Lattices

The Hume-Rothery rule is a most important generalization concerning the lattices of alloys; the theoretical justification of this rule is given by Jones’s theory; this circumstance already emphasizes its significance. I consider it necessary to dwell briefly on certain points of this theory, since it is precisely here that the theory of metals is connected with the results of X-ray investigations, which is very essential for our problem.

In developing any problem of atomic structure with the aid of the new mechanics, we follow known laws. The difficulty lies in the complexity of the problem. It is possible to calculate the energy states of one electron in a field of forces, for example the electron of a hydrogen atom in the field of its nucleus; however, already in the case of two electrons the problem is solved only approximately. The problem of the atomic structure of a solid is much more complex. We regard every solid as a complex of atomic nuclei and electrons with a known interaction between them; what is sought is such a configuration of the nuclei as will lead to a structure with minimal potential energy—this will be the true structure. Both the very nature of the problem and the methods of solving it are the same for the case of a copper crystal as for the case, say, of a crystal of anthracene or rock salt. However, owing to the complexity of a complete solution, we proceed by way of approximations, already different for crystals of different types. Practice has shown that such a path leads to very useful results. The conception of ionic, homopolar, and metallic bonds in various bodies, explaining the sharp difference in the physical properties of solids with interatomic bonds of different types, is based on the fact that, in the general solution of the problem for the structure of each type, there exists a number of factors that may be neglected, and a number of factors that acquire dominant significance. Let us compare, for example, the case of an ionic crystal, say, potassium chloride, with the case of a metal crystal: in both cases the electronic states are a function of the structure. However, the individual ions K+ or Cl− hold their electrons so strongly that, when the ions approach and form a crystal lattice,

the electronic configuration around each ion changes quite insignificantly. It is possible, with sufficient approximation, to calculate the energy released in the formation of a crystal lattice as the decrease in electrostatic energy when two charges of opposite sign approach each other. In the case of the formation of a crystal by metallic atoms, however, the problem looks different, since each individual atom possesses one or several weakly bound valence electrons. Since, upon formation of a metallic crystal, the atoms are at distances smaller than the distance of the valence electron from the nucleus of its atom in the free state, a radical change in the energy states of the valence electron is natural when it enters the metallic system. Thus, in discussing the problem of the structure of a metallic system, the center of interest is transferred to the scheme of common valence electrons. The remaining electrons are firmly bound to the atomic nucleus and may, together with the latter, be regarded as a field of forces acting on the outer electrons.

Experiments in the field of electron diffraction give us the possibility of treating moving electrons as a system of waves. The classical experiments of G. P. Thomson and of Davisson and Germer showed that the scattering of electrons by crystals can be regarded as the diffraction of a wave flux by the three-dimensional lattice of a crystal. The effects observed are reminiscent of the basic experiments that laid the foundation for X-ray structural analysis of crystals. The effective wavelength that must be substituted into the formula for calculating the angle of diffraction is equal to \(\lambda = \frac{h}{mv}\), where \(h\) is Planck’s constant, and \(mv\) is the momentum of the electron. In other words:

\[ E_{\text{kin}} = \frac{mv^{2}}{2} = \frac{h^{2}}{2m\lambda^{2}} . \tag{1} \]

Let us now consider the free electrons contained in \(1\ \text{cm}^{3}\) of metal, representing them as a system of waves possessing different lengths and propagating in different directions*. These electrons belong to the whole piece, just as in a free atom a certain number of electrons belong to one and the same nucleus. In a free atom the electronic states may be represented by oscillations of various types, with two electrons in each state.

In the same way, it is necessary to find definite types of oscillations in a piece of metal—namely, those that are permitted by the presence of definite bonds in this piece—and then to assign two electrons to each state as well. The analysis of oscillations of this kind

* Lack of space does not allow us to cite many other authors, whose work is headed by Sommerfeld, who developed the modern theory of the electronic structure of metals.

is similar to finding the fundamental tones and overtones of an organ pipe, or, rather, to the study of vibrations of a solid in Debye’s theory of specific heat. Fig. 2 shows the solution of the problem for two dimensions. The plus and minus signs refer to parts that are in opposite phase; the dotted lines represent equivalent waves forming stationary vibrations. The more nodes fit into the length of each edge, the smaller the corresponding wavelength.

Let us suppose that, per unit volume, \(N\) electrons are to be distributed, two to each state. First of all the lowest energy states are filled, possessing a small number of nodes, i.e. a large wavelength and, consequently, a small kinetic energy and strong bonds. Next, in turn, states corresponding to ever shorter waves are filled, until all \(N\) electrons have been accommodated, the last electrons possessing the shortest wavelength \(\lambda_{\min}\). It can be shown that

\[ N=\frac{8\pi}{3}\cdot\frac{1}{\lambda_{\min}^{3}}. \tag{2} \]

Fig. 2. Standing waves in a unit area.

Since the number of free electrons \(N\) is known, \(\lambda_{\min}\) can be calculated for each typical metal; the corresponding kinetic energy of the electron can likewise be calculated. The order of magnitude of the kinetic energy, as calculations show, is \(5\ eV\). This energy is enormously greater than the energy of thermal vibrations at ordinary temperature; namely, it corresponds to a temperature above \(40\,000^\circ\text{C}\). Thus the electron theory of metals resolves the old problem of why free electrons do not alter the heat capacity. At ordinary temperatures the energy of almost all electrons is subject to the condition of being in the lowest energy states compatible with the structure; only a few of them possess additional thermal energy.

The minimum wavelength is on the average equal to \(5\ \text{Å}\), or \(5\cdot10^{-8}\ \text{cm}\). This is approximately twice as large as the mean interatomic distance. The nodes of these shortest waves are situated at approximately the same distance from one another as the atomic rows of the crystal lattice of the metal. This, of course, is true only approximately. I have dwelt on these considerations because the relation of the shortest wavelengths to the interatomic distances is the basis of Jones’s theory of alloy lattices. It is precisely because the shortest wavelengths possess the

in order of magnitude, they are reflected (electron diffraction) by systems of planes with the largest interplanar spacings. The relation between wavelength and energy, expressed by equality (1), as well as by the parabolic curve of Fig. 3, is valid only for free electrons. The interaction of atoms in the crystal destroys this curve. In addition, one must note a special effect for electron waves moving in such a direction and having such a length that they satisfy the law of reflection from a family of crystallographic planes, namely \(\lambda = 2d \sin \vartheta\), where \(d\) is the interplanar spacing and \(\vartheta\) is the glancing angle. Longer waves correspond to electron energies significantly smaller than the energy values calculated from the simple equality; for shorter waves the opposite is true. The solid line in Fig. 3 shows the modified relation.

Fig. 3.

Let us again consider the distribution of electrons among the various states in a crystal. First of all, they can be placed in states with low energy and large wavelength (on the left in Fig. 3). If, in placing the electrons, we do not leave the branch \(OA\) of the curve, then all the electrons will be located in states with comparatively low energies. This advantage is lost in passing to the branch \(BC\), for \(B\) lies much higher than the normal curve. The interval \(AB\) corresponds to approximately 1 eV; this is a very considerable quantity. The critical value of the wavelength depends, naturally, on the direction of propagation of the waves; this factor must be taken into account.

The interplanar spacings in a crystal, which determine its periodicity, are a consequence of the structure of the phase lattice. The essence of Jones’s theory may be expressed by the assertion that all alloys possess phase lattices, as far as possible, of such a type that the electrons can be distributed along the branch \(OA\) of the curve and thus are in states of low energy. Such a structure is therefore stable.

It should be noted that this distribution of electrons depends only on the phase lattice, and not on the dimensions of the structure. It is true that, with an increase in the size of the elementary cell, a smaller number of

electrons fall per unit volume, but, on the other hand, the interplanar spacing of each system of planes increases, and thus the critical limit corresponds to a wave of shorter length. The two effects just balance each other.

Thus the theory predicts the dependence of the phase lattice on the ratio of the number of electrons to the number of atoms in the structure. This is the Hume-Rothery rule.

The first example proposed by Jones is the $\gamma$-structure with 52 atoms in the unit cell. This is a very unusual structure, and therefore the question has always arisen as to why the atoms occupy such peculiar positions. Previous X-ray studies have shown that the most intense rays (both X-ray and electron) are reflected from planes with indices $(411)$ and $(330)$; both systems of planes have an interplanar spacing equal to $2.08\ \text{\AA}$. Other planes with larger interplanar spacings give reflected rays of insignificant intensity in comparison with the two mentioned, which is a direct result of the peculiarity of the atomic configuration. The totality of the planes $411$, $141$, $114$, $\bar{4}11$, $\bar{1}41$, $\bar{1}14$, etc., and $330$, $033$, $303$, etc., forms a large number of directions in the crystal, and in whatever direction the electron wave may move, there will always be found a plane to which it will be approximately normal. For any direction of waves, therefore, the critical point of Fig. 3 occurs at a wavelength $2d$, or $4\ \text{\AA}$. On the other hand, the $\gamma$-structure is found when the ratio of the number of electrons to the number of atoms is approximately equal to $21/13$, or when there are 84 electrons for 52 atoms in the unit cell. Applying relation (2), we find that, if the electrons are placed on different levels, they begin to be reflected from the planes $411$ and $330$ when their number in the cell increases to 80, and waves moving normally to these planes are situated above point $A$ of the curve in Fig. 3. On the branch $OA$, for any directions of waves, only 90 electrons can be accommodated. A structure with low energy should be expected when the actual number of electrons is contained between these limits, which agrees well with the approximate number 84 observed in practice.

I hope that I have succeeded in explaining to you the essence of the matter and in doing justice to this beautiful theory. The positions of the atoms prove to be chosen in such a way that the planes $411$ and $330$ give intense reflected rays; at the same time the break $AB$ of the curve in Fig. 3 occurs at such a value of the wavelength that on the branch $OA$ of the curve the maximum number of electrons is accommodated, and the structure possesses low energy and, consequently, high stability. If, when the composition is changed, the ratio of the number of electrons to the number of atoms varies beyond a certain limit, the appearance of a new family of planes is necessary in order that the structure may remain stable, and therefore—

a new phase arises, possessing the corresponding planes.

Further, these planes retain their reflecting power even if the atoms are distributed at random among the positions of the phase lattice. Hence it is clear why the phase lattice does not depend on the distribution of the atoms: as has become evident from experiment as well, the decisive significance belongs not to interatomic bonds, but to the interaction between the lattice of metallic atoms, on the one hand, and the general system of valence electrons, on the other. The phase is the lattice of atomic positions.

I should like once again to emphasize that what has been set forth above is not a completed theory, but is only a starting point in the investigation of energy states which promise to be exceptionally fruitful for explaining the properties of such complex structures as γ-alloys. Such an assessment, however, does not in the least diminish its significance.

VII. Distribution of Atoms among the Positions of the Phase Lattice

I should now like to touch upon the factors that determine the distribution of the atoms of the alloy components in the phase lattice. This problem is extremely interesting from both the theoretical and the practical points of view.

It has long been discovered that the distribution of atoms in the lattice of a solid homogeneous phase can change. What takes place here is not the formation of a new phase growing at the expense of the old one from centers of crystallization, but a homogeneous change in the internal structure of the entire phase, a redistribution (re-sorting) of atoms within each element of the structure. The possibility of such a process, as already mentioned, was considered by Tammann; it was discovered by Johansson and Linde in copper–gold alloys 3,4. This process in the alloy AuCu was studied in detail by Dehlinger and Graf 6. A more complex example of the same process in the iron–aluminum system was studied by Bradley and Jay 7. The nature of the lattice change occurring in this process is clear from Fig. 4.

Fig. 4a depicts the lattice of an alloy of iron with aluminum whose composition corresponds to the formula Fe₃Al. If this alloy is obtained by rapid cooling of the melt, its structure will be as shown in the left part of the figure. The iron atoms are situated at the vertices of the cubes. In the centers of the cubes various numbers of iron and aluminum atoms are distributed at random. The number of centers is equal to the number of vertices, and therefore, if half of the hatched centers are occupied by aluminum, the composition of the whole alloy corresponds to the formula Fe₃Al.

If, instead of quenching, the alloy is subjected to slow cooling or is annealed, the aluminum atoms arrange themselves in a regular system, and the structure shown in the right part of the figure is obtained. The centers of the cubes in the direction parallel to an edge are occupied

alternately by iron and aluminum atoms, and the whole structure possesses a more regular symmetry. The lower part of the figure depicts the alloy \(\mathrm{Cu}_3\mathrm{Au}\). In the disordered structure the atoms of both metals are distributed at random in the nodes of a face-centered cubic lattice. Upon annealing, the gold atoms are located at the vertices of the cubes, while the centers of the faces, the number of which is three times the number of vertices, are occupied by copper atoms. This alloy was among the first studied by Johansson and Linde. A more complex transformation of the lattice occurs in the alloy \(\mathrm{CuAu}\). The disordered state is here also represented by the left-hand part of

Fig. 4.

Fig. 4b, although in this case there is an equal number of gold and copper atoms, distributed arbitrarily among the lattice sites. Upon ordering of the structure, the gold and copper atoms are arranged alternately, in layers parallel to one of the three mutually perpendicular faces of the cube. The crystal of the alloy possesses tetragonal symmetry, since the structure now has a single fourfold axis, perpendicular to the alternating layers of Cu and Au atoms.

Such a regular distribution of atoms among the positions of a simpler basic lattice is called a superstructure (Überstruktur). The presence of a superstructure is revealed by the presence of additional lines in the Debye photograph, as shown in Fig. 5. In all three Debye diagrams of the figure there is one and the same system of intense lines. These lines belong to the cubic body-centered lattice characteristic of all iron–aluminum alloys with compositions from Fe to FeAl. If the iron atoms and the aluminum atoms were distributed in the lattice in an arbitrary manner, then at any ratio of Fe and Al

the same simple X-ray pattern would be obtained, identical with the upper X-ray pattern in the figure, obtained from the alloy Fe₃Al subjected to quenching. The middle X-ray pattern corresponds to an annealed alloy with the same composition Fe₃Al. The additional lines present in the X-ray pattern indicate that, as a result of the ordered distribution of unlike atoms in the lattice, new reflecting planes have appeared. The appearance of the new planes is perhaps best illustrated by Fig. 4b. In the lattice shown on the left side of the figure, in the direction perpendicular to the face of the cube, identical planes are repeated at intervals of

\[ \frac{a_{\alpha}}{2}, \]

where \(a_{\alpha}\) is the edge of the cube, whereas in the structure shown on the right, where the separated gold atoms occupy the vertices of the cubes, the distance between identical planes is already equal to \(a_{\alpha}\). Translating this into the language of X-ray diffraction, in the first case we obtain reflections 200, 400, etc., while in the second case reflections 100, 200, 300, … of all orders are obtained. The additional lines in the middle X-ray pattern of Fig. 5 owe their origin to the superstructure shown in Fig. 4a (right). The lower X-ray pattern was obtained from the alloy FeAl.

Fig. 5. Debyegrams of Fe—Al alloys, taken with iron \(K\)-radiation. 1. Fe₃Al 25% Al, quenching at 700°, body-centered cubic lattice. Random distribution of Fe and Al. 2. FeAl 25% Al. Annealed alloy. Composition ideally Fe₃Al. 3. FeAl 50% Al. Structure, CsCl both for the quenched and for the annealed alloy.

In this alloy the iron atoms occupy the vertices of the cubes, and the aluminum atoms—the centers. As a result of the separation of atoms according to this rule, a new system of additional lines appears on the X-ray photograph.

If such a separation of atoms occurs only partially, then the superstructure lines become less intense; with a completely arbitrary distribution they disappear altogether. By photometering the superstructure lines and comparing their intensity with the lines of the basic (phase) lattice, one can form a judgment about the degree of ordering of the structure. Bradley and Jay, in the above-mentioned investigation, traced the process of separation of atoms for all iron–aluminum alloys whose composition varies in the interval between Fe and FeAl.

Such processes of transition from an ordered structure to a disordered one and back again (order-disorder transformations) are distinguished by certain very interesting features. First of all, they do not resemble ordinary phase transformations. In them we do not have two sharply delimited phases—completely ordered and completely disordered—with an abrupt transition from one to the other, as in the case of the transition from the solid state to the liquid. On the contrary, all intermediate stages are observed. On the other hand, this problem is of great interest theoretically. Atoms are distributed among certain positions in different ways, which recalls the usual school problems in which it is required to find all possible ways of placing colored balls in a number of boxes. To each such arrangement there corresponds a definite value of the entropy, an expression for which can be written down. With the aid of some extremely simple assumptions one can also calculate the internal energy of the structure. Thus it becomes possible to find a way toward a thermodynamic treatment of the problem—a way that is extremely easy to interpret and that, I admit, has given me personally a considerably clearer idea of the meaning of such terms as “free energy” and “thermodynamic equilibrium.”

VIII. Thermodynamics of the Processes of Ordering of Structure

The course of the process and its relation to phase transformations are considered in the works of Borelius, Johansson and Linde[^8], Gorsky[^9], and also Dellinger and Graf[^10]. Recently, E. J. Williams and I[^11] published a theoretical study based on certain very simple assumptions concerning the energies of various atomic configurations; the works of Borelius[^12] and Dellinger[^13], which appeared at approximately the same time, summarize the conclusions reached by the investigators mentioned. If in the present section I adhere to our method of exposition, this is only while fully giving due credit to the works of the authors named.

In the general case, as has also become clear from experiments with quenched and annealed alloys, it is evident that the ordered

the superstructure is a stable state at low temperatures, since the internal energy for an ordered distribution of atoms is less than for a disordered one. At high temperatures the thermal motion of the atoms becomes too strong, and the order is disrupted.

Let us trace the process of transformation in an alloy containing two kinds of atoms, \(A\) and \(B\), assuming throughout that the alloy is in thermodynamic equilibrium at any temperature. The degree of ordering of the structure may be defined by a parameter \(S\), varying from 1 for the case of a completely ordered structure to 0 for a completely disordered distribution. The quantity \(S\) is a linear function of the probability of finding a given atom in the “proper” plane, i.e. in the position which it should occupy in a completely ordered structure.

At low temperatures all atoms occupy the proper positions. Suppose that, in order to move one atom \(A\) from its proper position to some other, “improper” one (including also the displacement of an atom \(B\) from some other position into its place, “improper” because an atom \(A\) should be there), it is necessary to expend work \(W_0\). The question is: how many atoms, as the temperature rises, will at any moment be in improper positions?

Fig. 6.

If the magnitude of the work were constant, the matter would reduce to a very simple thermodynamic problem. The answer is represented by the continuous curve in Fig. 6. As the temperature increases, an ever greater number of atoms are found outside the proper positions. At very high temperatures the curve approaches \(S=0\), and the distribution becomes entirely arbitrary.

However, we have no right to assume that \(W_0\) is constant. Initially any atom of the lattice occupies some position in a completely ordered structure, and in order to displace it, it is necessary to overcome the forces of its interaction with quite definite atoms surrounding it. The energy required for this is \(W_0\). As the disorder increases, some of the surrounding atoms will be in improper positions, and the work of displacing \(A\) will decrease. Ultimately, when the distribution of atoms becomes arbitrary, the distinction between “proper” and “improper” positions disappears, and atoms on the average pass from one position to another without expenditure of work. We may assume that, in a state with degree of ordering \(S\), the work of displacement is equal to \(W = W_0 S\).

If now one represents on the diagram the state of equilibrium at

at any temperature, we obtain the dashed curve of Fig. 6. At low temperatures both curves coincide, but as the temperature rises the structure offers ever less resistance to the thermal motion that destroys it, and toward the end the curve drops steeply down to the value \(S=0\) (complete disorder). Figuratively speaking, one may say that “complete disorder reigns in the demoralized structure.” It is very important to note that the final destruction of order occurs at a certain definite critical temperature \(T_k\). In the approximate considerations for the case of an alloy with equal numbers of atoms \(A\) and \(B\), given in our work, the critical temperature is determined by the relation

\[ \frac{kT_k}{W_0}=0.25; \tag{3} \]

where \(k\) is Boltzmann’s constant. Bethe’s more exact calculation gives a somewhat smaller value of \(T_k\).

The fact that there is a critical point is not isolated among a number of similar phenomena. An analogous argument was applied to explain the Curie point in ferromagnetic substances; recently Fowler\(^{14}\), following a similar path, explained the existence of a critical point at which the molecules of a solid cease to rotate. This property is a characteristic thermodynamic feature of an ordered structure, in which the force holding some element in its proper position depends on the degree to which neighboring elements are in the positions belonging to them—on “public opinion,” if one may use such an analogy. Every such structure is subject to a sudden “demoralization,” reaching its culmination at the critical point.

Let us now trace the transition described in the ordinary processes of phase transformations. We shall again begin with low temperatures, still assuming the alloy to be in a state of equilibrium; the heat supplied to the alloy is expended not only on increasing the thermal motion, but also on displacing atoms from regular positions into irregular ones. The heat capacity of the alloy must therefore be higher than normal, taking as the normal value the heat capacity in accordance with the Dulong and Petit law. This excess heat capacity, increasing with rising temperature, reaches its greatest value in the immediate vicinity of the critical point, where the change in order proceeds most intensively; then the heat capacity suddenly decreases to the normal value and remains so above the critical temperature, since any further change in the degree of order of the structure is already impossible.

Fig. 7 shows the change in heat capacity as a function of temperature for \(\beta\)-brass of approximate composition CuZn. Around \(470^\circ\) this alloy undergoes a well-known transformation, with the modification \(\beta\) existing above this temperature and the modification \(\beta'\) below it. There can be no doubt that here we are dealing with a transition from an ordered structure to a disordered one.

At high temperatures the Cu and Zn atoms are arbitrarily distributed among the sites of a body-centered cubic lattice. Below \(470^\circ\) the atoms begin to separate, becoming distributed between the vertices and the centers of the cubes. The shape of the curve coincides exactly with that which we derived theoretically. This curve was obtained by Sykes\(^{15}\), who constructed an ingenious apparatus for testing the predictions of our theory—an apparatus which will in all probability find wide application.

In this process, neither absorption nor evolution of latent heat occurs at the critical point. The degree of order, and consequently also the internal energy, are continuous functions of temperature, although the rate of its change with temperature (the heat capacity) undergoes a discontinuity at the critical point. On the transition curve of \(\beta\)-brass to \(\beta'\)-brass there is no horizontal segment characteristic of ordinary phase transformations; only a slowing of the rate of cooling is observed.

Fig. 7. Heat capacity of \(\beta\)-brass containing
\(49.06\%\) Zn

A quantitative verification of the theory may be carried out as follows: we have seen that the critical temperature is determined by the relation

\[ \frac{kT_b}{W_0} \sim 0.25. \]

It is also possible to calculate the total change in energy in the transition from disorder to order (without taking thermal vibrations into account). It is given by the equation

\[ E_0 - E_1 = \frac{NW_0}{8}, \tag{4} \]

in which \(N\) is the number of atoms in the piece of alloy. For any intermediate degree of order \(S\),

\[ E_0 - E_1 = \frac{NW_0S^2}{8}. \]

Substituting \(W_0\) from (3),

\[ E_0 - E_1 = \frac{NkT_k S^2}{2} = \frac{RT_k S^2}{2}, \tag{5} \]

if \(N\) is taken to be Loschmidt’s number, so that \(Nk = R\).

Knowing \(T_k\), it is thus possible to calculate the change in energy upon ordering of the structure. The magnitude of this change can also be obtained experimentally by measuring, in the diagram of Fig. 7, the areas bounded by the curve of the observed heat capacity and the curve representing the normal heat capacity, due in its origin only to the thermal motion of the atoms. The latter quantity can be determined with good accuracy. A comparison of theory with experiment is given in Fig. 8. The results of the comparison are in general favorable for the theory. Experiment shows a somewhat more rapid increase of the energy than the theory predicts, but the resulting change is close to the theoretical one. Thus, at the critical temperature \(470^\circ\), the theory gives the figure \(740\ \text{cal}\), whereas Sykes’s measurements gave \(640\ \text{cal}\) for a piece of brass of weight

\[ \frac{A_1 + A_2}{2}, \]

where \(A_1\) and \(A_2\) are the atomic weights of zinc and copper.

Fig. 8.

The process of transition from disorder to order can also be traced by measuring the electrical conductivity. The electrical resistance for a disordered distribution of atoms considerably exceeds the magnitude of the resistance that is due in origin to the thermal vibrations of the atoms alone; moreover, this excess does not depend on temperature. The cause of the electrical resistance is the same in both cases: namely, constant disturbances of the lattice due to the irregularities of a random distribution in one case, and fluctuating disturbances of the lattice due to thermal motion in the other. Fig. 9 shows the difference between theory and experiment.

Fig. 9. AuCu alloy. 1. \(\rho_0 = \rho_a + \alpha_0 T\), complete disorder. 2. Experimental curve of Kurnakov and Ageev. 3. Theoretical curve \((\rho_s)\). Degree of ordering corresponding to equilibrium. 4. \(\rho_1 = \alpha_1\), complete order.

In the case of the alloy Cu\(_3\)Au the transformation process proceeds somewhat differently. A theoretical investigation shows that at the critical temperature the structure suddenly passes from disorder—above the temperature \(T_k\)—to partial order—below \(T_k\) (\(S \approx 0.4\)); and upon further lowering of the temperature it becomes more and more ordered. Sykes, who measured the heat capacity of the alloy Cu\(_3\)Au, found that the latter reaches a very high maximum near the critical temperature, and that at this point there is a large change in energy. Apparently, the change in heat capacity here occurs discontinuously, but the transition proves to be somewhat smoothed by temperature gradients in the specimen. The total change in energy in this case as well is in good agreement with the theoretically calculated value.

IX. Comparison of the processes of transition from a disordered structure to an ordered one with phase transformations

The equilibrium state of an alloy can be determined either by expressing the condition that, during any interval of time, equal numbers of atoms pass from ordered positions to disordered ones and conversely, or else by forming an expression for its “free energy” \(F\) and using the circumstance that at equilibrium the free energy is a minimum, i.e. \(\dfrac{\partial F}{\partial S} = 0\). Borelius and Dellinger adhere to such a formal approach. It is appropriate here to touch upon the relation between the processes of ordering of the structure and ordinary phase transitions. The free energy is defined as

\[ F = U - T\varphi, \tag{6} \]

where \(U\) denotes the internal energy, \(T\) the temperature, and \(\varphi\) the entropy of the structure. As in all similar questions, here we are dealing only with differences of the quantities \(F\), \(U\), and \(\varphi\) for two different states. At low temperatures the condition of a minimum for \(F\) coincides with the condition of a minimum for \(U\) and requires \(S = 1\), i.e. complete ordering of the structure. At high temperatures the term containing \(\varphi\) becomes significant, since the probability of a disordered distribution increases, and \(S\) decreases.

The principal difficulty of the problem lies in obtaining an expression for \(U\). In doing this we have based ourselves on the interactions in the structure of the alloy, whereas Borelius and Dellinger adopt general expressions for the internal energy in terms of the degree of order and show that, by choosing the proper expression, one can explain the behavior of the alloy. We shall take

\[ U_s = U_0 aS^2, \tag{7} \]

as was indicated above. The difference of entropies for states with different degrees of order is independent (in the-

in the approximation under consideration) on the temperature and is given by the expression

\[ \varphi_s-\varphi_0 = k \lg \frac{P_s}{P_0}, \tag{8} \]

in which \(P_s\) and \(P_0\) denote the probabilities of the two states and can be obtained directly from the geometry of the structure as functions of \(S\). For the disordered state \(P_0\) is greater than \(P_s\) for the state with degree of order \(S\).

In the case of a phase transformation, for example the transition from the liquid state to the solid, the \(F,T\) curve for \(F_{\text{тв}}-F_{\text{ж}}\) corresponds to that shown in Fig. 10a. At a definite temperature \(T_{\text{пл}}\) the two curves intersect. Since a substance in equilibrium possesses the lowest possible free energy, the course of its change is represented by the solid curve in Fig. 10a; moreover, at the melting point \(T_{\text{пл}}\) a transition from the solid state to the liquid occurs. A sudden

Fig. 10.

change in the direction of the curve corresponds to the liberation of latent heat, numerically equal to the change in \(T \dfrac{dF}{dT}\). Let us now compare this diagram with the diagram of the transition from the ordered state to the disordered one. In this case there is still a variable parameter \(S\), which determines the degree of order of the structure. Instead of two curves, one corresponding to the liquid state and the other to the solid state, we have a family of curves corresponding to different values of the parameter. They are shown in Fig. 10b.

Each of the \(F,T\)-curves is a straight line, determined by the equation

\[ F_s - F_0 = U_s - U_0 - T(\varphi_s-\varphi_0), \]

beginning below the abscissa axis, since the difference \(U_s-U_0\) is negative, and directed upward, since \(\varphi_s-\varphi_0\) is also negative (for the entropy value for the disordered state is greater than for the ordered one). The equilibrium state at all temperatures is the state of the lowest free energy. This state is represented by the envelope of the family of curves, as is clear from the figure. At any temperature the degree of order-

ness of the alloy structure corresponds to a straight line tangent to the envelope at the point corresponding to the given temperature. As the temperature increases, the curve ultimately merges with the line of complete disorder \(S = 0\), and therefore, since no change of \(T \dfrac{dF}{dT}\) occurs, the transition of the alloy to an arbitrary distribution is not accompanied by the release of latent heat.

In the case of the alloy \(\mathrm{Cu}_3\mathrm{Au}\), the envelope intersects the straight line \(S = 0\), rather than merging with it, and therefore latent heat is released.

Ehrenfest called phase transformations that occur without the release of latent heat “phase transformations of the second kind.” The transformations in alloys described above, as was already indicated by Dehlinger, are an example of the Ehrenfest concept. Their relation to the processes of ordinary phase transformations is very clearly illustrated with the aid of \(F, T\)-diagrams.

X. Relaxation Time of an Alloy

An alloy subjected to annealing is not initially in a state of equilibrium. At high temperature the thermal motion becomes sufficiently intense to cause a redistribution of the atoms, and the alloy gradually approaches the state of equilibrium. In most cases the processes occurring during annealing are extremely complex from the theoretical point of view, including, for example, crystal growth or separation of a new solid phase. In the special case of a transition to an ordered structure the process appears comparatively simple, so that it proves possible to give a theoretical interpretation of the various effects of heat treatment.

We assumed above that the alloy is in a state of equilibrium and that the atoms readily change position. For this it is necessary that each atom be able to acquire the amount of thermal energy \(W\) needed for the transition. The atoms of a metal vibrate with a frequency approximately equal to \(10^{13}\) vibrations per second, so that in the course of one second they make an enormous number of attempts to change position. If in one of these attempts an atom happens to possess thermal energy of order \(W\) and, moreover, to move in the proper direction, then a change of position will occur.

In the work of Williams and Betts the question is subjected to theoretical consideration, the results of which may ultimately be represented as follows. The relaxation time of an alloy \(\tau\) is defined as the time in which its deviation from the state of equilibrium reaches \(\dfrac{1}{e}\) of its initial value; \(\tau\) is determined by the formula

\[ \tau = Ae^{\frac{W}{kT}}, \]

where \(W\) is the energy barrier that must be overcome, while \(A\) is a constant, the order of magnitude of which may be

computed. \(A\) has a value of about \(10^{-12\pm 2}\). If the relaxation time at some one temperature is known to us, then this approximate value of \(A\) makes it possible to compute relaxation times for other temperatures. If, for example, \(\tau\) at temperature \(T_1\) is equal to one second, then we find that for

\[ \frac{T}{T_1}=1.2 \quad \tau = 0.01 \text{ sec} \]

\[ \frac{T}{T_1}=1 \quad \tau = 1 \text{ sec. (by definition)} \]

\[ \frac{T}{T_1}=0.8 \quad \tau = 17 \text{ min.} \]

\[ \frac{T}{T_1}=0.6 \quad \tau = 3 \text{ years.} \]

\[ \frac{T}{T_1}=0.5 \quad \tau = 30\,000 \text{ years.} \]

This in turn makes it possible to compare the effects of the most rapid quenching permissible in practice and of the slowest possible annealing. If annealing is carried out with cooling \(10^5\) times slower than in quenching, then the alloy remains in a state characteristic of a temperature 30% lower than that which was observed during quenching. Very rough determinations of rates serve to determine their limits, since \(\tau\) changes with temperature with extraordinary rapidity.

It is useful to note that, when \(\tau = 1\) sec, then \(kT\) is equal to \(W/30\). In other words, the barrier energy \(W\) is approximately 30 times greater than the mean energy of one degree of freedom.

The small number of diverse states attainable by annealing and quenching explains the rarity of cases, such as \(\mathrm{Fe_3Al}\), \(\mathrm{Cu_3Au}\), \(\mathrm{CuAu}\), in which an alloy can be preserved in an ordered or disordered state by means of heat treatment.

The critical temperature of order of magnitude \(T\) must fall within this narrow range. In the majority of cases either the alloy passes into the ordered state at any quenching rate (high \(T_k\)), or atomic rearrangement ceases before the critical temperature is reached, so that it constantly remains in the disordered state (low \(T_k\)).

In conclusion I wish to consider the picture of the structure of an alloy to which this whole series of considerations leads. X-ray study of the solid state has made it necessary to reconsider, from new points of view, many of the generalizations of chemistry and in particular the idea of the chemical molecule and of valence bonds; the new mechanics has explained the various types of atomic associations. One of the earliest results of X-ray analysis—the structure of sodium chloride—refuted the existence of “NaCl molecules”

—an idea already doubtful at that time. It was established that equality in the number of sodium and chlorine atoms in the solid structure is a consequence of the “correct” alternation of \(Na^+\) and \(Cl^-\) ions with equal and opposite electrostatic charges. The idea of the molecule as a certain whole is still retained in its application to organic molecules, where direct bonds are formed between atoms; but where inorganic compounds are concerned, it must be revised. The term “valence” acquires a different meaning.

The concept of molecules is even less applicable in the case of compounds formed by metals. In inorganic compounds, since there are no molecules there, the balancing of definite electrostatic charges of positive and negative ions requires that the proportion of these ions be expressed by whole numbers. The composition of a compound is expressed by a definite formula. I must qualify this statement somewhat, since many cases are known where, for example, one divalent ion may be replaced to one degree or another by two monovalent ions, so that the rule of ratios of whole numbers is violated. In any case, the equilibrium of electrostatic charges subjects the composition to definite laws.

The situation is otherwise with alloys. All components of an alloy are electropositive. Although the Hume-Rothery rule and Jones’s theory indicate the presence of a definite number of atoms in the elementary cell, they still do not require that the ratio of dissimilar atoms be expressed by simple whole numbers. These whole numbers enter into the picture of the structure only in the event that ordered superstructures can be formed. An ideal superstructure requires simple ratios. It is precisely for this reason that characteristic features appear in phases with a composition corresponding to these ideal ratios, such as \(1:1\) or \(1:3\). A small excess of one component or another spoils the ideal superstructure and disproportionately changes the properties of the alloy. For example, one may expect a sharp minimum of electrical resistance at a composition that permits an ideal superstructure.

If one concludes from these features that there exists a “compound” \(CuZn\) or \(AuCu_3\), and that excess atoms of one kind or another, in the case of a nonideal proportion, are “dissolved” in this compound, then, if understood too literally, such a conclusion will be erroneous. Such formulas serve as convenient labels for series of phases, but it does not follow from them that groups \(CuZn\) or \(AuCu_3\) and separate dissolved atoms exist. All atoms of every metal play the same role in the structure.

Naturally, the theory, as has also been emphasized here, exaggerates some points and neglects others. Transitional cases may be encountered which indicate the approach of an alloy to a chemical compound of the usual type, when other factors acquire predominant importance.

Our understanding of the chemical nature of the structure of alloys

is at an exciting stage, when broad horizons of new knowledge are opening ahead, and I shall consider myself satisfied if I have succeeded in outlining them before you with due breadth.

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