ORDERING PROCESSES IN ALLOYS¹)

F. C. Nix and W. Shockley, New York, USA

INTRODUCTION

Ordered substitutional alloys

Among alloys of various types there are known the so-called solid solutions of the substitutional type, obtained by adding various amounts of metal B to metal A. The crystals of such an alloy are very similar to the crystals of the pure metal, with the only difference that in some sites of the lattice atoms A are replaced by atoms B. For many years it was thought that this substitution was a matter of pure chance and that there was no order whatsoever in the placement of atoms at the lattice sites. Recently,²) however, with the aid of the chief arbiter in questions of crystal structure—X-ray scattering—it has been established that in a large number of cases the atoms of metals are arranged in the alloy lattice in just as definite a manner as the different ions in the crystal lattice of a salt.

The first suggestion of the existence of ordered structures was made by Tammann, but it was based not on X-ray data, but on the results of the following chemical experiments. Tammann found that suitably treated Cu—Au alloys containing more than 50 at. % Cu are acted upon by nitric acid, which dissolves copper but not gold. The acid does not act on specimens containing 50 at. % or less Cu. From this he concluded that the 50% Cu—Au alloy represents an ordered arrangement of atoms and that the excess atoms are easily removed, since they do not enter into this regular arrangement.

At the present time the conclusive evidence in favor of the existence of ordered arrangements, or, as they are called, “superstructures,” is provided by “superstructure lines” on X-ray photographs. Fig. 1 gives a schematic representation of a superstructure and of the beam of X-rays scattered by it. Atoms A and B form—

¹) Rev. Mod. Phys., 10, 1, 1938. Translated by B. N. Finkelstein.
²) The bibliography will be given at the end of the article in the following issue.

form a regular lattice; in this case, at some nodes, which we shall call $\alpha$-nodes, there must be atoms A, and at other—

Fig. 1. Superstructure and the origin of a superstructure line

nodes—$\beta$-nodes—atoms B. We shall consider an atom A to be in its “proper” place when it is located at an $\alpha$-node, and to be in an “improper” place when it is situated at a $\beta$-node; analogous definitions also apply to atoms B. In the completely ordered state in Fig. 1 all atoms are in their “proper” places. Now let us imagine that a beam of X-rays falls on the crystal and is reflected in such a way that the path difference of the rays reflected by two planes containing $\alpha$-nodes is equal to the wavelength, while that of the rays reflected by neighboring $\alpha$- and $\beta$-planes is half the wavelength. Then all rays scattered by atoms A will be in the same phase with one another and in the opposite phase to the rays scattered by atoms B. If the scattering factors of the two atoms are different, then these rays will not extinguish one another, and a diffraction line will appear on the photographic plate. Let us consider what will be obtained in the case when the atoms are arranged at random, so that equal numbers of atoms A and B are located at $\alpha$- and $\beta$-nodes. Then the $\alpha$-nodes give rise to scattered waves whose amplitudes are equal to the mean value of the amplitudes obtained respectively from atoms A and B; the same is true for the $\beta$-nodes. The waves scattered by the $\alpha$- and $\beta$-nodes, now having identical amplitudes, completely extinguish one another, and no line is obtained on the photographic plate.

Such lines, whose appearance is caused by the presence of a superstructure, are known as “superstructure lines.” Their presence on a radiograph is proof of the existence of a superstructure, while their intensity is, to a certain extent, a measure of the degree to which the superstructure approaches perfection. Fig. 2 shows the radiograph of $\mathrm{Cu_3Au}$ in the disordered state (on the right), in the completely ordered state (on the left), and in an intermediate state (in the middle). The superstructure lines are indicated on the left.

Superstructure lines were first observed by Bain$^{25A}$ for $\mathrm{Cu_3Au}$ and by Friauf$^{25B}$ for $\mathrm{Fe_3Si}$, and the first X-ray investigation of an ordered structure was carried out by Yogan-

son and Linde\({}^{25A}\) for the alloy CuAu. In the cases indicated, the experiment was considerably facilitated by the large difference in scattering (atomic) factors. In some cases the atoms have almost identical scattering powers, and the superstructure lines are so weak that they are almost impossible to detect. The recently developed method has made it possible to study such alloys; a remarkable example of such a study is the work of Jones and Sykes\({}^{37}\) on β-brass, in which both elements Cu and Zn are adjacent in the periodic table and therefore scatter with approximately the same strength. A survey of ordered structures studied by X-ray methods up to the present time is given in Part II, § 16.

Fig. 2. X-ray photograph with superstructure lines (retouched)

All superstructures studied so far are characterized by the following general feature: atoms of one kind tend to be surrounded by atoms of the other kind.

For the time being we shall confine ourselves to this assertion, which is quite sufficient for us, and shall postpone a more detailed consideration until §§ 5 and 16.

The order–disorder transformation

Interesting investigations of superstructures can be carried out by studying the phenomena that occur when an alloy is heated. Let us trace this process, taking as our starting point a completely ordered superstructure like that shown in Fig. 1, beginning at low temperature.

As the temperature increases, at first there appears only a large amplitude of thermal vibrations of the atoms about their equilibrium positions. When this effect becomes appreciable, random pairs or small groups of atoms acquire sufficient energy to leave their places in the lattice and exchange places with one another. This exchange leads to a certain number of atoms occupying “illegal” sites: atoms A in β-sites and atoms B in α-sites.

Thus, the disorder in the arrangement of atoms manifests itself in two closely related ways: from the local point of view the neighbors of some atoms belong to the same kind, whereas—in accordance with the tendency to have unlike neighbors—they ought to be atoms of the other kind; from the point of view of large distances in the crystal, this disorder manifests itself in the fact that some \(\beta\)-sites are occupied by \(A\) atoms. To characterize the ordering of an alloy in both of the indicated senses the following definitions have been adopted: the “near,” or “local,” order, denoted by the letter \(\sigma\), whose quantitative definition is given in § 2, is a measure of how well, on average, each atom is surrounded by neighbors of the other kind; \(\sigma = 1\) represents the best ordered arrangement, and \(\sigma = 0\) complete disorder. The “long-range order,” or “order at large distances,” denoted by the letter \(S\) and quantitatively defined in § 1, is a measure of how completely the \(\alpha\)-sites are occupied by \(A\) atoms and the \(\beta\)-sites by \(B\) atoms; \(S = 1\) represents complete order, and \(S = 0\) disorder.

The tendency of atoms to be surrounded by atoms of the other kind manifests itself, at very low temperatures, in the formation of an ideal superstructure. At higher temperatures, when, owing to thermal motion, atoms fall into illegal sites, this tendency leads to the return of such atoms to legal sites. At any temperature below a certain critical temperature considered below, a definite state of equilibrium is established, characterized by a definite number of atoms located on illegal sites; in this state the number of cases per unit time in which atoms, owing to thermal motion, pass from legal sites to illegal ones is equal to the number of cases in which atoms pass from illegal sites to legal ones owing to thermal motion and to the ordering influence exerted on the atoms in illegal sites by their neighbors located on legal sites. Since, with increasing temperature, the number of atoms on “illegal” sites increases, some of the neighbors of such atoms will also turn out to be on “foreign” sites and will tend to remain there, instead of passing into the correct position.

Consequently, with increasing temperature not only does disorder increase, but also the ease of transition to a disordered arrangement. This process therefore proceeds with ever increasing speed until a certain critical temperature \(T_c\) is reached, at which the “long-range order” \(S\) disappears discontinuously. By analogy with the theory of ferromagnetism, this temperature has been called the “Curie point of ordering.” In Fig. 3 one possible form of this process is presented; there also exists another type of process, described in § 1, in which the quantity \(S\) undergoes no discontinuity but decreases continuously down to zero.

However, even above \(T_c\) there exists some local order, due to the tendency of atoms to have neighbors of the other kind. Although this tendency is now no longer capable of creating “long-range”

“order” throughout the entire crystal; nevertheless—in opposition to the disordering action of thermal motion—it prevents the establishment of complete disorder and ensures the presence of a considerable number of pairs of unlike neighbors. To destroy them, the crystal must be heated to an even higher temperature.

Consideration of the theories that explain and predict the various stages of ordering phenomena is the principal subject of the theoretical—first—part of the present article. Such theories have been developed by numerous investigators; among the pioneers are Gorsky[^28B], Borelius, Johansson and Linde[^28A], Wagner and Schottky[^31Г], and Dellinger and Graf[^34А]. Subsequently Borelius[^34А] considerably expanded the theory and, in particular, considered the question of temperature hysteresis. Bragg and Williams[^34C] again developed the theory, extending and simplifying it in many respects. Further development is associated chiefly with Bethe’s work[^35B]. Other questions considered in this part concern the transition of the system to an equilibrium state and effects connected with changes in the composition of the alloy.

Fig. 3. Dependence of the degree of order at long and short distances on temperature

Energy Considerations

To create disorder, to transfer atoms to “foreign” positions and to overcome the ordering force, energy is required. This additional energy appears in the “anomalous heat capacity,” i.e., in an additional heat capacity over and above that obtained from the Dulong–Petit law. This energy and heat capacity are connected with the arrangement of atoms in the lattice and are often denoted by the terms “configurational energy” and “configurational heat capacity.” The rate of disordering increases from zero to a maximum value at a temperature slightly below $T_c$. Figure 4 shows the temperature dependence of the heat capacity. The additional heat capacity above $T_c$

Fig. 4. Dependence of heat capacity on temperature for $\beta$-brass with 48.9% (at.) Zn. The dashed straight line is calculated from the heat capacities of Cu and Zn according to the rule of mixtures.

is due to local order, which requires an expenditure of energy for its diminution. Energy measurements leading to curves analogous to those shown in Fig. 4 constitute an important tool for the study of superstructures. Where X-rays give no answer, the existence of such heat-capacity curves gives indirect confirmation of the presence of a superstructure (the danger of confusing this with heat capacity due to ferromagnetism can easily be eliminated). We shall consider such energy measurements in § 14, Part II.

Other manifestations of ordering

Besides thermal phenomena, there are also other physical phenomena on which the ordered state of an alloy has a profound influence. They are of interest for two reasons: first, like anomalous heat capacities, they are those indirect indications which prove useful in studying ordering transformations and in establishing the existence of ordered structures; and, second, they make it possible to use ordering phenomena to obtain materials with new desirable properties. One of the most striking indirect manifestations of ordering is the behavior of the specific resistance represented in Fig. 5[^36]. We see that, superposed on the normal linear dependence on temperature, there is a rapid increase ending in a jump at \(T_c\); the reader will at once note the close connection of this fact with the increase of disorder at long distances that occurs as the temperature rises (Fig. 3).

Fig. 5. Dependence of electrical resistance on temperature for \(\mathrm{Cu_3Au}\). The alloy was in an equilibrium state at temperatures above \(350^\circ\mathrm{C}\).

Magnetic and mechanical properties also depend on ordering.

A general survey of the influence exerted by ordering on these and various other properties occupies a number of sections in the second part. The question is also considered there of how these changes lead to the establishment of ordered structures in cases that have not yet been finally investigated with the aid of X-rays.

Cooperative Phenomena

Phenomena of ordering in alloys belong to the general class of phenomena known as “cooperative phenomena.” Physical systems in which cooperative effects are observed contain a certain number of units which, by their joint action, give rise to a certain “cooperative” property, the magnitude of which is a measure of the degree of cooperation; in our case these units are the atoms forming the superstructure. This process has, in general, the following character: the ability of individual units to oppose, by combining, destructive actions (for example, thermal motion) increases considerably as the existing degree of cooperation increases, and so considerably that, when the destructive action is not too great, a “cooperative” state is established.

Among cooperative phenomena, three are best known: ferromagnetism, the rotation of molecules in solids, and phenomena of ordering in alloys. Of these phenomena, the last began to be studied later than the others, but to a greater extent than the others it is supported by a satisfactory theory. Recent successes in this respect have, it seems to us, brought forward a new point of view which will contribute to an understanding of the whole field of these phenomena; indeed, at the present time an attempt is being made to extend certain methods of the theory of ordering in alloys to other cooperative phenomena^37.

PART I. THEORY OF THE PHENOMENON OF ORDERING

A. Equilibrium theory for alloys of simple constant composition

In the following four sections we shall deal with alloys of a definite constant simple composition. Further, we shall assume that for all states of ordering—from complete order to an entirely random arrangement—there is no appreciable change in the positions of the lattice sites; thus we consider only the question of the arrangement of atoms in an unchanged lattice. Some theories prove to be sufficiently general for them to be applicable to a large number of different lattices. Others are valid only for simple, body-centered, and face-centered cubic lattices. In these cases all sites are equivalent, and only a definite arrangement of atoms in the lattice makes some site more suitable for atoms of one kind than for atoms of another kind.

In reality, there are sometimes small distortions of the lattice, usually associated with the degree of ordering; in other cases, upon ordering, a complete change of the lattice type occurs. We shall postpone consideration of these phenomena until Part II and for the present…

we shall neglect these effects. We shall also leave aside the troublesome questions of terminology that arise in the case of such alloys when the absence, in the strict sense of the word, of a repeating elementary cell makes it incorrect to use the term “lattice.”

We shall consider only binary alloys (the two chemical elements are denoted respectively by the letters A and B), and only of two definite compositions: one—with equal numbers of atoms A and B; its chemical formula is AB; the other—with three atoms B to one atom A, described by the formula AB$_3$. Further, when generalizing our considerations to other alloys, an alloy will be characterized by the atomic percentage of component A; the cases indicated above correspond to 50 and 25%. For the cases under consideration one can always choose $\alpha$-sites—one for each atom A—so that they themselves form a spatial lattice. The remaining sites will be $\beta$-sites, and there will be enough of them to accommodate the atoms B. Owing to the equivalence of all sites, the $\alpha$-sites may be chosen in different ways, and it is immaterial how this choice is made.

The energy, heat capacity, and other properties of an alloy, apart from the arrangement of atoms or the state of ordering, depend also on many other factors. For example, the energy of an alloy depends not only on this arrangement but also on thermal vibrations. In what follows, unless a special reservation is made, it will be assumed that the effects due to changes in ordering and the effects due to thermal vibrations can be separated from one another, and that this separation has been carried out. The energy, entropy, and heat capacities calculated here are connected with the state of ordering. They represent the so-called “anomalous” or “configurational” parts of the corresponding quantities. The usual quantities connected, for example, with the Dulong–Petit law will not be considered.

§ 1. Bragg and Williams Theory$^{34C,\,35B,\,35V}$

The basic concept of the Bragg and Williams theory is the notion of “long-range order.” Our first task will be to give a quantitative definition of this concept. We shall see how from this there follows a theory predicting the existence of a critical temperature and the existence, and sometimes also the absence, of latent heat. The absence in this theory of the concept of “short-range order” is its shortcoming, which is removed by Bethe’s theory, considered in the next paragraph.

Suppose that the $\alpha$-sites have been chosen in some definite way and that atoms A have been placed in them. Then the remaining $\beta$-sites are occupied by atoms B. Let the total number of atoms, equal to the total number of sites, be $N$. In the present article we shall everywhere consider a gram-atom of substance; consequently, $N = 6.06 \cdot 10^{23}$ and $Nk = R = 1.986\ \text{cal}/\text{degree}\cdot\text{g-atom}$. Let us denote by $F_A$ the concentration of ato-

ms A, and consequently also the concentration of $\alpha$-sites; then $F_B=1-F_A$ represents the corresponding quantity for B and $\beta$.

When the ideally ordered arrangement is disturbed, some of the A atoms pass into $\beta$-sites, displacing an equal number of B atoms, which pass into $\alpha$-sites. To describe such cases we indicate the concentration of $\alpha$-sites still occupied by “their own” atoms; denote this quantity by $r_\alpha$. The concentration of incorrectly occupied $\alpha$-sites will be $w_\alpha=1-r_\alpha$. In the same way, $r_\beta$ and $w_\beta$ represent the concentration of correctly and incorrectly occupied $\beta$-sites. The number of A atoms in $\beta$-sites is equal to $w_\beta F_\beta N$ and is equal to the number of B atoms in $\alpha$-sites, $w_\alpha F_\alpha N$. Hence the equations follow

\[ 1=F_A+F_B=r_\alpha+w_\alpha=r_\beta+w_\beta, \tag{1,1} \]

\[ w_\alpha F_A=w_\beta F_B . \tag{1,2} \]

In the state of complete ordering, $r_\alpha$ and $r_\beta$ are equal to unity, while $w_\alpha$ and $w_\beta$ are equal to zero. In the completely disordered state the probability that a given site is occupied by an A atom is equal to $F_A$; consequently, the concentration of $\alpha$-sites occupied by A atoms is also equal to $F_A$, and in our notation one obtains $r_\alpha=w_\beta=F_A$ and $w_\alpha=r_\beta=F_B$. The parameter $S$, introduced by Bragg and Williams to characterize the state of ordering, must be defined in such a way that it is equal to unity for complete ordering and zero in the opposite case.

	Order	Disorder
$r_\alpha$	$1$	$F_A$
$w_\alpha$	$0$	$F_B$
$r_\beta$	$1$	$F_B$
$w_\beta$	$0$	$F_A$
$S$	$1$	$0$

The table given indicates the possibility of expressing $S$ in various ways through the other parameters; as equations (1,1) and (1,2) show, these methods are equivalent, and we shall put

\[ S=\frac{r_\alpha-F_A}{1-F_A}=1-\frac{w_\alpha}{F_B} =\frac{r_\beta-F_B}{1-F_B}=1-\frac{w_\beta}{F_A}. \tag{1,3} \]

Degree of ordering as a function of ordering energy and temperature. Let us first consider thermal equilibrium at some constant temperature $T$. Above all, we assume that in the state of equilibrium there exists a definite ordering energy $V$; when an A atom occupying an $\alpha$-site and a B atom occupying a $\beta$-site exchange places, two atoms are obtained in “foreign” positions, and the energy of the crystal increases by $V$. With respect to this energy it is assumed that it is identical for all pairs of lattice sites and does not depend on fluctuations of the local order. This assumption is not consistent with exp—

by the transformation of an ordered structure into a disordered one, which was given in the introduction and, as we shall see below, is based on Bethe’s theory (§ 2).

At the end of this section we shall give a rigorous derivation of the equilibrium condition, using the principles of statistical mechanics. But before doing so we shall first consider the simpler method, first applied by Bragg and Williams, who obtained their results with its aid. We may confine ourselves to considering only the atoms A, because once their positions are specified, the distribution of the atoms B is thereby determined. Suppose that all atoms A, with the exception of one, are immobile, and let us determine the probability that this atom is in its own or in a foreign site. This atom will move in the lattice, exchanging places with neighboring atoms B. It will spend part of the time \(f_{\alpha}\) in \(\alpha\)-positions, and part of the time \(f_{\beta}\)—in \(\beta\)-positions; our problem reduces to finding the ratio

\[ \frac{f_{\alpha}}{f_{\beta}}, \]

i.e. the ratio of the probability for this atom to be in the correct position to the probability of being in an incorrect position. Fortunately, one may disregard the dynamics of the process of exchanging positions and, knowing only that such processes exist, find the answer in classical statistics. For the atom under consideration there are \(w_{\alpha}F_A N\) available (i.e. not occupied by other atoms A) correct positions and \(r_{\beta}F_B N\) available incorrect positions. These incorrect positions, however, are under unfavorable conditions because of the Boltzmann factor \(e^{-V/kT}\), corresponding to the additional energy \(V\) associated with the residence of an atom A in an incorrect position. Consequently, for the atom A under consideration the ratio of the probability of being in a correct position to the probability of being in an incorrect position is equal to

\[ \frac{w_{\alpha}F_A N}{r_{\beta}F_B e^{-V/kT}} . \tag{1,4} \]

If the alloy is in an equilibrium state, then the behavior of the atom A chosen by us must be typical for all atoms A, and the distribution of the latter between the \(\alpha\)- and \(\beta\)-sites must be expressed by the above ratio.

The number of atoms A in \(\alpha\)- and \(\beta\)-positions is respectively equal to \(r_{\alpha}F_A N\) and \(w_{\beta}F_B N\); hence for the condition of statistical equilibrium one obtains

\[ \frac{r_{\alpha}F_A N}{w_{\beta}F_B N} = \frac{w_{\alpha}F_A N}{r_{\beta}F_B N} e^{V/kT} \tag{1,5} \]

or, if we express our results in words,

\[ \frac{\text{number of } \alpha\text{-sites occupied by atoms } A} {\text{number of } \beta\text{-sites occupied by atoms } A} = \frac{\text{number of } \alpha\text{-sites not occupied by atoms } A} {\text{number of } \beta\text{-sites not occupied by atoms } A} \cdot e^{\frac{V}{kT}} \tag{1,6} \]

These expressions are easily brought to the form

\[ \frac{r_\alpha r_3}{w_\alpha w_\beta}=e^{\frac{V}{kT}} \tag{1,7} \]

or, expressing through \(S\),

\[ \left\{\left[\frac{1}{F_B(1-S)}\right]-1\right\} \left\{\left[\frac{1}{F_A(1-S)}\right]-1\right\} = e^{\frac{V}{kT}} . \tag{1,8} \]

It is easy to see that the quantities in brackets are positive, and that the value \(S=0\) corresponds to the value \(\dfrac{V}{kT}=0\), i.e. to zero ordering energy or to making the temperature infinite; \(S\) takes a value equal to unity when \(\dfrac{V}{kT}\) tends to infinity. Intermediate values of \(\dfrac{V}{kT}\) will give values of \(S\) lying in the interval between ideal order and complete disorder.

It is essential to note that in equation (1,8) \(S\) is a function of one variable, \(\dfrac{V}{kT}\). The expression obtained by solving (1,8) with respect to \(S\) will be written in the form

\[ S=S\left(\frac{V}{kT}\right)=S(X), \tag{1,9} \]

\[ X=\left(\frac{V}{kT}\right). \tag{1,10} \]

In the simple case \(AB\) \(\left(F_A=F_B=\frac{1}{2}\right)\), it follows from (1,8) that

\[ S=\operatorname{tgh}\left(\frac{X}{4}\right). \tag{1,11} \]

Dependence of the ordering energy on the degree of order. We have just shown how the ordering energy and the temperature \(T\), by means of classical statistics, determine the degree of order \(S\). However, this accomplishes only half the task, because the ordering energy \(V\) is itself directly determined by the degree of order of the alloy. Let us for the moment postpone further consideration of the relation \(S(X)\) and turn to the question of the dependence of \(V\) on the degree of order.

If the alloy is completely ordered, then the formation of a pair of atoms occupying illegal positions requires the expenditure of a certain energy, which we shall denote by \(V_0\). However, as the alloy becomes disordered, the amount of energy expended on such an exchange of positions becomes smaller. This is easiest to see for a state in which all atoms are arranged completely at random (complete disorder). In this case the division of sites into \(\alpha\)- and \(\beta\)-sites has no physical meaning; the distinction between them is purely formal and reduces to remembering the position that existed when the superstructure was present. Consequently, in this case no energy will be expended on the exchange of atoms of two sites, and \(V\) will be equal to zero. Thus \(V\) depends on the degree of order in such a way that it assumes its maximum value \(V_0\) at \(S=1\) and reaches a minimum—zero—at \(S=0\). The simplest assumption about the relation between \(V\) and \(S\), satisfying these conditions, which we shall adopt in this paragraph and which was made by Bragg and Williams, is expressed by the formula

\[ V = V_0 S. \tag{1,12} \]

It is, as Bragg and Williams pointed out, a rather crude approximation. Indeed, the ordering energy for any pair of atoms depends on the arrangement of their nearest neighbors and only indirectly depends on the distribution of atoms over \(\alpha\)- and \(\beta\)-sites throughout the whole lattice. This shortcoming is very largely removed in Bethe’s theory, considered in § 2, in which the entire energy of the superstructure is due to the interaction of nearest pairs of atoms.

The existence of the dependence \(V(S)\) once again emphasizes the idea underlying the theory being described: the force tending to create order in the lattice depends on the degree of ordering already achieved. An essential feature distinguishing the relation \(V(S)\) from \(S(X)\) is its definiteness. The function \(V(S)\) has a definite value independently of temperature, independently of whether the system is in equilibrium or not; on the other hand, if the system is not in equilibrium, then the relation \(S(X)\) between \(S\), \(V\), and \(T\) no longer holds.

The assumption made by Bragg and Williams concerning the ordering energy leads to a simple expression for the energy of the alloy. In order to pass from an alloy characterized by order \(S\) to an alloy with order \(S+dS\), we must move a certain number of atoms A from incorrect positions to correct ones. This number is found from relation (1,3), which defines the order, and is equal to

\[ F_A N\, dr_\alpha = F_A N F_B\, dS. \]

For each of these displacements the energy decreases by the amount \(V=V_0S\); consequently, the change in the energy of the alloy is equal to

\[ dE = - V N F_A F_B dS = - N V_0 F_A F_B S dS. \]

By integrating, taking the energy in the completely ordered state to be zero, we obtain

\[ E(S)=-\frac{1}{2}NV_0F_AF_B(1-S^2)=E_0(1-S^2) \tag{1,13} \]

\[ E_0=-\frac{1}{2}NV_0F_AF_B. \tag{1,14} \]

Here \(E_0\) represents the total energy of transformation of one gram-atom of the alloy from the ordered to the disordered state. In this paragraph, irrespective of which theory we are considering, by \(E_0\) we shall always mean the energy required to transfer the alloy from the state of complete order to the completely disordered state. For the cases AB and \(AB_3\) we obtain \(E_0=\dfrac{NV_0}{8}\) and, respectively, \(\dfrac{NV_0}{32}\); these values are given in the first line of Table 1 on p. 361.

Statistical equilibrium, case AB. If the temperature is assigned a constant value \(T_1\), then we have two equations for the two unknowns \(V\) and \(S\). The simultaneous solution of these equations leads to the equilibrium condition.

For clarity, let us write the equations in a somewhat modified form

\[ S=S\left(\frac{V}{kT_1}\right)=S(X), \tag{1,15} \]

\[ S=\frac{V}{V_0}=\left(\frac{kT_1}{V_0}\right)\left(\frac{V}{kT_1}\right)=\frac{kT_1}{V_0}X. \tag{1,16} \]

Now we see that the right-hand sides of these equations may be regarded as functions of \(X\left(=\dfrac{V}{kT_1}\right)\). The equilibrium condition corresponds to the value of \(X\) for which they are equal, i.e. \(X=S(X)\), with the corresponding values of \(V\) and \(S\) being the equilibrium values. In order to solve these equations, Bragg and Williams plotted the right-hand sides as functions of \(X\). The point of intersection of the two curves gives the required condition.

Fig. 6. \(V(S)\) for different temperatures and \(S(X)\); case AB

Fig. 6 is such a graph for the case AB. The curve \(S(X)\) shows that for large values of \(X\), i.e. for large ordering energies or low temperatures, \(S\) approaches unity—complete order—whereas for small \(V\) or large \(T\), \(S\) approaches zero. The function \(V(S)\), of course, is represented by a straight line passing through the origin and form-

...forming with the abscissa axis an angle whose tangent is equal to \(\frac{kT_1}{V_0}\). The point of intersection \(P_1\) gives the equilibrium values of \(S\) and \(V\) for the temperature \(T_1\). For higher temperatures, for example \(T_2\), the slope of \(V(S)\) is steeper, while the curve \(S(X)\) remains unchanged. The point of intersection moves to \(P_2\) in the direction of smaller degrees of ordering. With increasing temperature \(P\) moves ever lower along the curve \(S(X)\) until, finally, for example at \(T_3\), the straight line \(V(S)\) no longer intersects \(S(X)\)—except at the origin.

It is of interest to find the temperature at which \(P\) reaches the origin; this is the critical temperature, above which “long-range order” does not exist. When the straight line \(S = \frac{kT}{V_0}X\) is tangent to the curve \(S(X)\) at the origin, then \(\frac{kT}{V_0}\) is equal to the value of \(\frac{dS(X)}{dX}\) at \(X = 0\). From equation (1,11) we easily find, for \(X = 0\),

\[ \frac{dS}{dX} = \frac{1}{4}. \]

Hence we obtain \(\frac{kT}{V_0} = \frac{1}{4}\), or, introducing \(E_0 = \frac{NV_0}{8}\),

\[ T_c = \frac{V_0}{4k} = \frac{2E_0}{R}, \tag{1,17} \]

where \(T_c\) denotes the critical temperature. This value is placed in the second row of Table 1.

Nonequilibrium state. Under certain circumstances the alloy will not be in a state of equilibrium; its degree of ordering will change with time. In one of the following paragraphs we shall discuss this question from a quantitative point of view; for now we shall confine ourselves to showing toward what the system will tend.

Suppose that the alloy is at a temperature \(T\), corresponding to the straight line \(V(S)\) in Fig. 7. Since this straight line is a graphical representation of the relation \(V = V_0S\), which according to the assumption made is always valid, every possible state of the system is represented by a point on this line. Suppose that the system is characterized by an order \(S_1\), indicated on the lower part of the line. Then its state is represented by the point \(Q\) with abscissa \(X_1\). If the temperature and the energy of ordering had values corresponding to \(X = X_1\), then the equilibrium value of the order would be \(S_2 = S(X_1)\). Consequently, when the system is in a state characterized by the value \(X = X_1\), then it

will tend toward the state \(S_2\). In the case under consideration \(S_3\) is greater than \(S_1\); consequently, the ordering in the system will increase with time, and the point \(Q\) will approach \(P\). By analogous reasoning one can show that \(Q\) will move upward in all those cases when \(S_2\) is above \(S_1\), and downward when \(S_2\) is below \(S_1\), i.e. upward in all those cases when \(S(X)\) passes above \(V(S)\).

We see, therefore, that \(P\) represents a state of stable equilibrium. This, however, cannot be said of the origin of coordinates, because although it also represents a solution of equations (1,15) and (1,16), we see that any fluctuation deviation from it leads to a situation where \(S(X)\) is greater than \(V(S)\), as a result of which the system will tend toward larger values of \(S\) until the point \(P\) is reached.

Statistical equilibrium, case AB\(_3\). Bragg and Williams also considered alloys of other formulas, different from AB. The type AB\(_3\) corresponds to a series of alloys of interest from the physical point of view, such as, for example, Cu\(_3\)Au; for it the curve \(S\) as a function of \(X\) is shown in Fig. 8. This figure is only a schematic one, since it was necessary to depict, on an enlarged scale, the change in curvature of the curve \(S(X)\) in order to show it on a drawing of such dimensions. We see an essential difference between this case and the case AB, consisting in the fact that now three simultaneous solutions of equations (1,15 and 1,16) can be obtained:

Fig. 8. \(V(S)\) for various temperatures and \(S(X)\); case AB\(_3\)

\[ S = S(X), \]

\[ S = \frac{V}{V_0} = \frac{kT}{V_0}X. \]

The straight line \(V(S)\), corresponding to the temperature \(T_2\), intersects the curve \(S(X)\) three times: at the origin of coordinates, at \(P'\), and at \(P\). If we apply here the theory of nonequilibrium states developed in the preceding paragraph, we shall see that \(P'\) is unstable and that the initial state corresponding to the point \(Q\) will tend toward \(P\), if it is above \(P'\), and toward the origin of coordinates if it is below \(P'\).

From this reasoning we must conclude that, at any temperature to which three intersections of the curves correspond, there must be two stable states of the alloy, one of which corresponds to complete disorder. These two states, possessing different degrees of ordering and different reserves of energy, must be regarded as different phases of the substance; the assumption that they can be in equilibrium over a wide interval

temperatures, contradicts Gibbs’ phase rule. By means of arguments based on consideration of the free energy and given below in the present section, it can be shown that for temperatures below a certain definite critical temperature \(T_c\), the system at the point \(P\) possesses a smaller free energy than for \(S=0\), and therefore represents a state of thermodynamic equilibrium; above \(T_c\) the equilibrium state is the state with \(S=0\). The straight line corresponding to \(T_c\) is indicated in the figure; for it the areas of the loops \(A\) and \(A'\) are equal, and, as will be shown below, this condition contains within it the equality of the free energies. For the critical temperature one obtains

\[ S = 0.467 \]

\[ T_c=\frac{0.205 V_0}{k}=\frac{2.18 E_0}{R}. \tag{1,18} \]

Some predictions of the theory. Order. If all the calculations are carried out to determine the dependence of order on temperature, the Bragg—Williams curves shown in Figs. 9—14 are obtained. Other curves pertain to theories that will be considered later. In all cases the temperature is given in reduced units: the quotient of \(RT\) divided by the total configurational energy corresponding to the transition of the system from the state of complete order to the disordered state.

Fig. 9. Dependence of long-range order on temperature for the case AB. Simple cubic lattice for the Bethe and Kirkwood curves

We see that the order in the alloy AB falls to zero at a definite critical temperature, taking all values between unity and zero. However, in the case \(AB_3\), at \(T_c\) a jump occurs, and the order falls at once from the value 0.467 to zero. The reader can easily verify that these results are in agreement with the qualitative predictions based on consideration of Figs. 6 and 8.

Energy. The dependence of order on temperature can be sub—

substituted into equation (1.13) \(E(S)=E_0(1-S^2)\), and from this the curves in Figs. 11 and 12 were constructed, showing the dependence of the energy on temperature.

Fig. 10. Dependence of long-range order on temperature for the case \(AB_3\). Face-centered lattice for the Payerls curve

Fig. 11. Dependence of the configurational energy on temperature for the case \(AB\). Simple cubic lattice for the Bethe and Kirkwood curves

In the case \(AB_3\), at \(T_c\) there is a discontinuity in the order, and consequently also in the energy. This change in energy appears in the form of latent heat, which must be supplied to the alloy in order to transform it at the critical temperature from the state of order \(S=0.467\) to the state of order \(S=0\). The magnitude of the latent heat is easily calculated from the formula for \(E(S)\)

\[ Q=E(0)-E(0.467)=(0.467)^2E_0=0.218E_0=0.0205NV_0= =0.100RT_c. \tag{1.19} \]

As we have already said, in the Bragg—Williams theory all the transformation energy \(E_0\) is necessary in order to obtain an alloy from the state-

TABLE 1¹)

All quantities refer to 1 gram-atom	AB Bragg—Williams	AB Simple cubic, \(Z=6\) Bethe, 1st approx.	AB Simple cubic, \(Z=6\) Bethe, 2nd approx.	AB Simple cubic, \(Z=6\) Kirkwood	AB Body-centered cubic, \(Z=8\) Bethe, 1st approx.	AB Body-centered cubic, \(Z=8\) Kirkwood	AB\(_3\) Bragg—Williams	AB Face-centered cubic, \(Z=12\) Peierls
\(E_0\)	\(\dfrac{NV_0}{8}\)	\(\dfrac{3N\nu}{2}\)	\(\dfrac{3N\nu}{2}\)	\(\dfrac{3N\nu}{2}\)	\(2N\nu\)	\(2N\nu\)	\(\dfrac{3N\nu_0}{32}\)	\(\dfrac{3N\nu}{4}\)
\(\dfrac{RT_c}{E_0}\)	2	1,644	1,581	1,577	1,738	1,707	2,19	1,33
\(\dfrac{E(T_c-)}{E_0}\)	1	0,800	0,754	0,789	0,857	0,854	0,792	0,18
\(\dfrac{Q}{E_0}\)	0	0	0	0	0	0	0,218	0,26
\(\dfrac{E(T_c+)}{E_0}\)	1	0,800	0,754	0,789	0,857	0,854	1,00	0,54
\(\dfrac{E_c(\sigma)}{E_0}\)	0	0,200	0,246	0,211	0,143	0,146	0	0,46
\(\dfrac{RT_c}{E(T_c+)}\)	2	2,055	2,097	2	2,028	2	2,19	2,38
\(\dfrac{\varphi(T_c-)}{R}\)	0,693	0,633	0,628	0,626	0,652	0,650	0,462	0,19
\(\dfrac{\varphi(Q)}{R}\)	0	0	0	0	0	0	0,100	0,27
\(\dfrac{\varphi_c(\sigma)}{R}\)	0,693	0,633	0,626	0,626	0,652	0,650	0,562	0,46
\(\dfrac{\varphi(\infty)}{R}\)	0	0,0604	0,065	0,067	0,0411	0,043	0	0,10
\(\dfrac{}{}\)	0,693	0,693	0,693	0,693	0,693	0,693	0,562	0,562
\(\dfrac{C(T_c-)}{R}\)	1,50	1,90	2,14	4,233	1,78	2,207	2,36	—
\(\dfrac{C(T_c+)}{R}\)	0	0,119	0,203	0,134	0,081	0,0858	0	0,16

¹) \(E_0\) — energy of transformation from perfect order to complete disorder; \(T_c\) — critical temperature; \(E(T_c)\) — configurational energy immediately below \(T_c\); \(Q\) — latent heat at \(T_c\); \(E(T_c+)\) — configurational energy immediately above \(T_c\); energy necessary for the destruction of superstructure; \(E_c(\sigma)\) — energy of order at short distances immediately above \(T_c\); \(\varphi(T_c)\) — entropy immediately below \(T_c\); \(\varphi(Q)\) — change of entropy due to latent heat; \(\varphi(T_c+)\) — entropy immediately above \(T_c\); \(\varphi_c(\sigma)\) — entropy associated with short-range order immediately above \(T_c\); \(\varphi(\infty)\) — total change of entropy; \(C(T_c)\) — heat capacity immediately below \(T_c\); \(C(T_c+)\) — heat capacity immediately above \(T_c\).

of order \(S=0\) at a temperature immediately above \(T_c\). This ceases to be true for the theories considered below, which introduce the concept of order at short distances: there, for this purpose only a part of \(E_0\) is required, denoted by the symbol \(E(T_{c+})\). We shall denote by \(E(T_{c-})\) the energy required to reach

Fig. 12. Dependence of the configurational energy for the case \(AB_3\). Face-centered lattice for the Peierls curve

the critical temperature without completing the transition. Then
\[ E(T_{c+})-E(T_{c-})=Q \]
represents the latent heat. These quantities are also given in Table 1. The energy which will be obtained through the destruction of local order above \(T_c\) is equal to
\[ E_0-E(T_{c+})=E_c(\sigma). \]

It is of interest to compare the critical temperature with the energy \(E(T_{c+})\) required to bring the alloy into a state with long-range order equal to zero (\(S=0\)). For this purpose the ratio
\[ \frac{RT_c}{E(T_{c+})} \]
has been calculated. All theories give approximately the same values for this quantity—about 2.

Heat capacity. The configurational heat capacity is found by differentiation:
\[ C=\frac{dE}{dT}. \]
This quantity has, as is shown below, the following interesting property: for all values of \(V_0\) it is one and the same function of \(S\) or of \(\frac{RT}{E_0}\). From dimensional considerations or from the calculations given above one can conclude that the equilibrium value of \(S\) must be a function of
\[ \frac{RT}{E_0}=y. \]
Combining this result with equation (1,13), we find
\[ C=\frac{d}{dT}E_0(1-S^2)=-2RS\frac{dS(y)}{dy}. \tag{1,20} \]
Thus neither \(E_0\), nor \(V_0\), nor \(T\) enters here explicitly, and \(C\) depends only on the analytic form of \(S\) as a function of \(\frac{RT}{E_0}\).

The calculation of \(\dfrac{dS}{dy}\) is rather complicated, and for details we refer the reader to the corresponding work \(^{38c}\). The results for AB and \(\mathrm{AB}_3\) are shown in Figs. 13 and 14.

Entropy. With the increase of energy and disorder upon raising the temperature there is also associated an increase of entropy. It can be calculated in two completely equivalent ways: using the relation \(d\Phi = \dfrac{dQ}{T} = \dfrac{dE}{T}\), or the expression \(\Phi = k \ln W\), where \(W\) is the a priori probability. For our purposes it is more convenient to use the latter relation.

For a given value of \(S\), different from unity, there are many ways in which the atoms can be arranged. We define the a priori probability of a state with order \(S\) as a quantity proportional to the number of ways of creating the order \(W(S)\). Then we obtain the relation

\[ \Phi = k \ln W(S). \tag{1,21} \]

For finding \(W(S)\), consider \(F_A N\) atoms on \(\alpha\)-sites. The number of ways in which they can be arranged there (regarding as indistinguishable arrangements differing only by permutations of A atoms among themselves and B atoms among themselves) is given by the known expression

\[ W_\alpha = \binom{F_A N}{r_\alpha F_A N} = \frac{(F_A N)!}{(r_\alpha F_A N)!\,(w_\alpha F_A N)!}. \tag{1,22} \]

Fig. 13. Dependence of the configurational heat capacity on temperature for the AB case. Simple cubic lattice for the Bethe and Kirkwood curves

Labels in the graph: ordinate—configurational heat capacity \(C/R\); abscissa—temperature \(RT/E_0\). Legend: Bragg–Williams; Bethe 1; Bethe 2; Kirkwood.

Fig. 14. Dependence of the configurational heat capacity on temperature for the \(\mathrm{AB}_3\) case

Labels in the graph: ordinate—configurational heat capacity \(C/R\); abscissa—temperature \(RT/E_0\). Legend: Bragg–Williams.

An analogous expression determines the number of ways in which the atoms can be arranged at the \(\beta\)-sites. The a priori probability of the state \(S\) is determined by the expression

\[ W(S)=W_\alpha W_\beta; \tag{1,23} \]

application of Stirling’s formula gives

\[ \Phi(S)=k\ln W(S)=-kN\{F_A(r_\alpha\ln r_\alpha+w_\alpha\ln w_\alpha)+ \]
\[ +F_B(r_\beta\ln r_\beta+w_\beta\ln w_\beta)\}. \tag{1,24} \]

This relation may be expressed as

\[ \Phi(S)=k\ln W(S)=-R\{F_A[1-F_B(1-S)]\ln[1-F_B(1-S)]+ \]
\[ +F_AF_B(1-S)\ln F_B(1-S)+ \]
\[ +F_B[1-F_A(1-S)]\ln[1-F_A(1-S)]+ \]
\[ +F_BF_A(1-S)\ln F_A(1-S). \tag{1,25} \]

The reader will easily see that every expression whose logarithm is taken is positive and does not exceed unity; consequently, each term of the sum in square brackets is negative, while \(\Phi\) is positive. Further consideration shows that the limiting values for \(S=1\) and \(S=0\) are

\[ \Phi(1)=0, \]

\[ \Phi(0)=-R(F_A\ln F_A+F_B\ln F_B). \tag{1,26} \]

Hence the following values are obtained for the changes of entropy in passing from order to disorder:

\[ \mathrm{AB}\quad \Delta\Phi=R\ln 2=0.693\,R=1.37\ \text{cal/deg. g-atom}, \tag{1,27} \]

\[ \mathrm{AB}_3\quad \Delta\Phi=R\frac{1}{4}(4\ln4-3\ln3)=0.562\,R= \]

\[ =1.11\ \text{cal/deg. g-atom}. \tag{1,28} \]

These two values must follow from any theory which assumes that at low temperatures there is complete order, and at high temperatures—disorder, and are not a particular result of the Bragg—Williams approximation.

We can also calculate the entropy for a temperature immediately below the critical one; putting \(S=0.467\) in equation (1,25), we obtain \(\Phi(0.467)=0.462\,R\). In this case it is easy to calculate the entropy also by another method. The entropy above \(T_c\), when \(S=0\), has already been calculated and was found equal to \(0.562\,R\). The latent heat is equal to \(0.100\,RT_c\), and thus the change of entropy at \(T_c\) is equal to \(0.100\,R\). Consequently, the entropy immediately below \(T_c\) is

\[ 0.562-0.100\,R=0.462\,R. \tag{1,29} \]

The circumstance that the Bragg—Williams theory leads to results consistent with thermodynamics is a consequence of a theorem proved in the following paragraphs; the Bragg—Williams method is equivalent to the method of statistical mechanics based-

...based on a consideration of the free energy; as is known, such an interpretation always leads to results consistent with thermodynamics, and can be applied to the derivation of the basic thermodynamic principles.

Derivation of the Bragg—Williams equations from the principle of free energy. Let us use the expression for the entropy, obtained in the preceding section, in order to obtain anew, and more rigorously, the main results of the Bragg—Williams theory. In our exposition we shall follow Williams\(^{35}\) and Fowler\(^{36}\).

The statistical weight of some state is equal to the product of its a priori probability and the Boltzmann factor. Thus the weight of a state characterized by the degree of order \(S\) is equal to

\[ \gamma(S)=W(S)e^{-\frac{E(S)}{kT}}, \tag{1,30} \]

where \(W(S)\) was calculated above (equation 1,25), and \(E(S)\) is determined by equation (1,13)

\[ E(S)=\frac{1}{2}NV_0F_AF_B(1-S^2). \tag{1,31} \]

The equilibrium state at a given temperature corresponds to the maximum value of \(\gamma(S)\). In order to determine this state, we shall seek the \(S\) which maximizes the quantity \(\ln\gamma(S)\) instead of \(\gamma(S)\). For \(\ln\gamma(S)\) we obtain the expression

\[ \ln\gamma(S)=\ln W(S)-\frac{E(S)}{kT}. \tag{1,32} \]

Although we shall be dealing with \(\ln\gamma(S)\) in the form given here, let us rewrite this expression once more in the following form:

\[ -kT\ln\gamma(S)=E(S)-kT\ln W(S)=E(S)-T\Phi(S)= \]

\[ =F(S). \tag{1,33} \]

As the written relations show, the quantity \(\ln\gamma(S)\) is very simply connected with the free energy \(F\). Thus our reasoning represents a somewhat simplified proof that the thermodynamic condition of equilibrium as a minimum of the free energy is a consequence of the statistical requirement of maximum probability\(^{1}\).

We shall seek the value of \(S\) which makes \(\ln\gamma(S)\) a maximum. The expressions on the right-hand side of equation (1,32) are known functions given in formulas (1,25) and (1,13).

\(^{1}\) Since a rigorous consideration of thermodynamic analogies is impossible, in the present review we shall confine ourselves to this rough illustration. For an exhaustive exposition of such questions the reader should consult \(^{367}\).

Carrying out the calculations, we obtain

\[ 0=\frac{d\ln \gamma(S)}{dS}=\frac{d\ln W(S)}{dS}-\frac{1}{kT}\frac{d\ln E(S)}{dS}= \]

\[ =NF_A F_B\left[-\ln\left(\frac{1}{F_A(1-S)}-1\right)-\ln\left(\frac{1}{F_B(1-S)}-1\right)+ \frac{1}{kT}NV_0F_AF_BS\right] \]

\[ =NF_AF_B\left[-\ln\left(\frac{1}{F_A(1-S)}-1\right)- \ln\left(\frac{1}{F_B(1-S)}-1\right)+\frac{V_0S}{kT}\right]. \tag{1,34} \]

The factor \(NF_AF_B\) may be transferred to the other side and we may write

\[ L(S)=\frac{\ln \gamma(S)}{NF_AF_B}, \tag{1,35} \]

\[ X_1(S)=S\frac{V_0}{kT}, \tag{1,36} \]

\[ X_2(S)=\ln\left[\frac{1}{F_A(1-S)}-1\right] \left[\frac{1}{F_B(1-S)}-1\right]. \tag{1,37} \]

The equilibrium condition is equivalent to the quantity \(L\) attaining a maximum,

\[ \frac{dL}{dS}=X_1(S)-X_2(S)=0. \tag{1,38} \]

In order to solve these equations, let us construct graphs for \(X_1(S)\) and \(X_2(S)\) as functions of \(S\). The results for the case \(AB_3\) are shown in Fig. 15. For reasons that will soon become clear, we put \(S\) on the ordinate axis, and \(X_1\) and \(X_2\) on the abscissa axis. We see that curves are obtained of the same kind as in Fig. 8 for \(S(X)\) and \(V(S)\). We shall give proof of their identity somewhat below; for now let us study these curves in their new aspect.

Fig. 15. Comparison of the free-energy method with the method \(\upsilon(S)\) and \(S(X)\)

For any value of \(S\), for example \(S_1\), the derivative \(\frac{dL}{dS}\) is equal to the horizontal distance \(X_1-X_2\) between the two curves. When this difference is positive, an increase of \(S\) is associated with an increase in probability; the change in \(L\) in going, for example, from \(S_2\) to \(S_3\) is represented by the shaded area between the curves.

Continuing this reasoning, one can show that \(P'\) always gives a smaller value of \(L\) and, consequently, of \(\ln \gamma(S)\), than \(S=0\) or \(P\), and

that \(S=0\) and \(P\) give the same value for \(\ln \gamma(S)\) only in the case when \(T\) is chosen so that the areas of the two loops between \(X_1\) and \(X_2\) are equal. When \(T\) is below this critical temperature, the state \(P\) is more probable, while when \(T\) is above \(T_c\), the state \(S=0\) proves more probable.

We must also check that these curves differ in no way from the curves \(S(X)\) and \(V(S)\). Recalling the previous definition [equation (1,10)] of the quantity

\[ X=\frac{V}{kT}, \]

we see that

\[ V(S)=V_0S=kTX \]

coincides with

\[ X=X_1(S)=\frac{V_0}{kT}\,S . \]

The old equation (1,8), from which the expression for \(S(X)\) was derived, read

\[ \left(\frac{1}{F_A(1-S)}-1\right) \left(\frac{1}{F_B(1-S)}-1\right) =e^{\frac{V}{kT}}=e^x . \]

Taking the logarithm of this expression obviously gives the relation between \(S\) and \(X\), expressed by the formula for \(X_2(S)\).

Thus we see that the requirement of equality of the areas enclosed between the intersections of the curves \(V(S)\) and \(S(X)\) is equivalent to the condition of a maximum for \(\ln \gamma(S)\), or to the condition of a minimum for the free energy.

§ 2. Bethe’s Theory\(^{35\mathrm{B}}\)

Introduction. The weakness of the Bragg—Williams method is its macroscopic character. The ordering energy acting on any individual atom is assumed to depend on the distribution in the crystal of all the other atoms, and not, as it would seem it should be, only on its nearest neighbors. Therefore one had to expect the appearance of a theory based on taking account of individual atoms and of the interaction forces between them. Significant progress in this direction was achieved by Bethe\(^{35\mathrm{B}}\), whose work was continued by Peierls\(^{36\mathrm{B}}\) and others. Bethe assumed that atoms interact pairwise, so that every two atoms possess a mutual potential energy which, however, rapidly decreases with increasing distance between the atoms. He supposed this decrease to be so rapid that he regarded as different from zero only the potential energy of two atoms that are nearest neighbors (Bethe’s theory can be generalized by means of

considering interactions with atoms located, relative to the given one, in the next coordination sphere. This possibility was investigated by Chang.^{37}

The alloys which we consider in this paragraph have simple lattices; in them it is easy to determine the number of nearest neighbors for each atom (we shall here neglect the effect of small distortions of the lattice, as a result of which the nearest neighbors may pass into the next coordination sphere). The number of nearest neighbors of each atom in the lattice1 depends on the type of the latter. We shall denote it by \(z\). The dependence of \(z\) on the type of lattice is as follows:

Lattice type	\(z\)
Simple cubic	6
Body-centered cubic	8
Face-centered cubic	12
Hexagonal close-packed	12
Two-dimensional square net	4

Assumption concerning the energy. Thus each atom is surrounded by \(z\) nearest neighbors, to which correspond \(z\) bonds with nonzero values of the potential energy. Since each bond belongs to a pair of atoms, the total energy accounted for by this is one half of the energy corresponding to a single bond. If both atoms belong to species A, we shall denote their mutual potential energy by \(v_{AA}\); if both atoms are of species B, then by \(v_{BB}\); if one atom is of species A and the other of species B, then by \(v_{AB}\). In the present article it is assumed that these energy values are constants of the two metals and that the influence on them of order, composition, or mechanical state is excluded. This assumption (differing from Bethe’s analogous assumption only by an explicit indication of the limits of its applicability) we shall call the “nearest-neighbor interaction hypothesis.”

In this case one can obtain a simple expression for the energy of the alloy. Let the numbers of pairs (in what follows, a pair formed by neighboring atoms will simply be called a pair) of types AA, BB, and AB be respectively \(Q_{AA}\), \(Q_{BB}\), and \(Q_{AB}\). Then, assigning to each pair the corresponding potential energy and summing over all pairs in the lattice, we obtain for the total energy

\[ E = v_{AA}Q_{AA} + v_{BB}Q_{BB} + v_{AB}Q_{AB}. \tag{2,1} \]

It may seem that an advantage of Bethe’s theory is the presence of three parameters \(v_{AA}\), \(v_{BB}\), and \(v_{AB}\), which can be varied, whereas the Bragg—Williams theory had only one, \(V\). But such an assumption, as will be shown below, is based on a misconception; in reality Bethe’s parameters have

meaning not by themselves, but only in a certain linear combination

\[ v=\frac{1}{2}(v_{\mathrm{AA}}+v_{\mathrm{BB}})-v_{\mathrm{AB}}. \tag{2,2} \]

The reason for the importance of \(v\) is that the interchange of any two atoms in a crystal is associated with a change in energy by an amount that is a multiple of \(v\). Consider, for example, the interchange of atoms A and B that are not nearest neighbors.\(^1\) Let the first atom have \(a\) neighbors belonging to type A and \((z-a)\) to type B, and let the B atom have \(a'\) neighbors A and \((z-a')\) neighbors B. The total energy associated with these two atoms is equal to

\[ a v_{\mathrm{AA}}+(z-a+a')v_{\mathrm{AB}}+(z-a')v_{\mathrm{BB}}. \tag{2,3} \]

If the two atoms are interchanged, the energy obtained is

\[ a'v_{\mathrm{AA}}+(z-a'+a)v_{\mathrm{AB}}+(z-a)v_{\mathrm{BB}}, \tag{2,4} \]

and the change in energy is equal to

\[ (a'-a)v_{\mathrm{AA}}+(a'-a)v_{\mathrm{BB}}-2(a'-a)v_{\mathrm{AB}} =2(a'-a)v. \tag{2,5} \]

Here we have a special example of the general result given in Appendix 1. It is shown there that, for an alloy of a given constant composition,

\[ Q_{\mathrm{AB}}=\mathrm{const}-2Q_{\mathrm{AA}} =\mathrm{const}'-2Q_{\mathrm{BB}}. \tag{2,6} \]

Using these equations, one may represent \(E\) in one of the following three forms:

\[ E= \begin{cases} 2v Q_{\mathrm{AA}}+\mathrm{const},\\ 2v Q_{\mathrm{BB}}+\mathrm{const},\\ -v Q_{\mathrm{AB}}+\mathrm{const}. \end{cases} \tag{2,7} \]

The values of \(\mathrm{const}\) here are different, but they are of no interest to us, since they do not affect changes in the configurational energy.

Now we see that, for positive \(v\), smaller energy values are obtained when pairs of unlike atoms are formed at the expense of pairs of like atoms. This agrees with the observed fact that in ordered structures like atoms tend to be far from one another. A negative value of \(v\) would make like atoms tend toward one another and would lead, at low temperatures, to the separation of pure metals.

It should be noted that these relations have one feature that is important in practice. When carrying out calculations it is sometimes considerably easier to count the number of AA pairs than the number of AB pairs. Thus, applying the first of the three expressions for \(E\), we must count only the number of AA pairs; applying the third, only the number of AB pairs. The same result may be reached in another way:

\(^1\) The reader can easily convince himself that this restriction is immaterial.

Since only the magnitude \(v\) is important, the results obtained with the aid of \(v_{AA}\), \(v_{BB}\), and \(v_{AB}\) are also obtained in the case where one sets \(v_{AB}=v_{BB}=0\) and \(v_{AA}=2v\). In this case only the interaction between atoms forming \(AA\) pairs is taken into account. An analogous device can be used to exclude other types of pairs.

Short-range order. Bethe introduces into his theory a new parameter \(\sigma\), characterizing order. This parameter is defined in the same way as in Bragg and Williams, in the sense that it is chosen equal to unity for a state of complete order and zero for a completely disordered state. However, it differs in that it is connected not with the \(\alpha\)- and \(\beta\)-sites of the lattice, but with the behavior of nearest neighbors. Let \(Q\) denote the total number of pairs in the lattice. In terms of \(N\) and \(z\) it is expressed by the following relation:

\[ Q=\frac{z}{2}N. \tag{2,8} \]

Then the fraction of pairs formed by different atoms is

\[ q=\frac{Q_{AB}}{Q}. \tag{2,9} \]

In the state of complete order, \(q\) reaches its maximum value \(q\) (max), equal to unity in some simple cases, while in a completely disordered state it has the smaller value \(q\) (random).

Bethe defines the parameter \(\sigma\) by the relation

\[ \sigma=\frac{q-q(\text{random})}{q(\text{max})-q(\text{random})}; \tag{2,10} \]

this parameter assumes the limiting values—one and zero—respectively for completely ordered and completely disordered states.

The parameter \(\sigma\) indicates how, on average, each atom is surrounded by neighbors; i.e., it is a measure of the order immediately around each atom and therefore received the name “short-range order” or “local” order—in contrast to the long-range order \(S\), which indicates the distribution of atoms over the \(\alpha\)- and \(\beta\)-sites throughout the entire lattice.

Limits restricting the application of Bethe’s theory. In discussing Bethe’s theory we shall restrict ourselves, as he himself did in his original work, to alloys of type \(AB\). Further, we shall assume that in this case all nearest neighbors of a given \(\alpha\)-site are \(\beta\)-sites and conversely. This holds for simple and body-centered cubic lattices and for the two-dimensional square net. An example of an ordered structure in the case of a simple cubic lattice is the rock-salt lattice, in which the \(\alpha\)-sites form a face-centered lattice occupied by ions of one sign, while the \(\beta\)-sites are occupied

ions of the other sign. The cesium chloride lattice presents an analogous example for a body-centered lattice. The ordered structure of a two-dimensional net is shown in Fig. 16(a), where the sites occupied by atoms A may be regarded as α-nodes, and the sites occupied by atoms B as β-nodes.

For these simple cases it is easy to determine the limiting values of \(q\) (max) and \(q\) (random). For perfect order all pairs belong to the AB type and \(q\) (max) \(=1\). For a disordered distribution the probability that, for a given atom, any neighbor will be an atom of the other kind is equal to one half; therefore half of the pairs belong to the AB type and \(q\) (random) \(=\frac{1}{2}\).

Hence we find

\[ \sigma=2\left(q-\frac{1}{2}\right). \tag{2,11} \]

Applying the third expression for the energy, we obtain for the energies of the best ordered and the completely disordered states:

Complete order

\[ -vQ_{\mathrm{AB}}(\text{max})=-vQ= \]

\[ =-\frac{1}{2}Nzv. \tag{2,12} \]

Complete disorder

\[ -vQ_{\mathrm{AB}}(\text{random})=-vQq(\text{random})= \]

\[ =-\frac{1}{4}Nzv. \tag{2,13} \]

Consequently, the transformation energy, denoted in the same way as in the Bragg—Williams theory, is equal to

\[ E_0=\frac{1}{4}Nzv, \tag{2,14} \]

and the energy of intermediate states, expressed through \(\sigma\), is equal to

\[ E=E_0(1-\sigma). \tag{2,15} \]

Fig. 16. Representation of various degrees of long-range and short-range order

Comparing the various theories with one another, we shall ascribe the same value to the quantity \(E_0\). Equating the values of \(E_0\) following from the Bragg—Williams and Bethe theories, we obtain

\[ V_0=2zv. \tag{2,16} \]

This equation has a quite definite physical meaning. Let us consider a state differing from ideal order only

that two atoms are in “illegal” sites. Although such a state can be realized by exchanging nearest neighbors, in reality it is much more probable that two atoms that have found themselves in “illegal” sites were not nearest neighbors: the first variant can be realized in \(\frac{1}{2}Nz\) ways, and the second in \(\frac{1}{4}N^2\) ways. Consequently, the “first excited state” of the alloy corresponds to the appearance of two atoms in “illegal” sites in different parts of the lattice. In the Bragg—Williams theory the lattice energy is, by definition, equal to \(V_0\); in Bethe’s theory each atom decreases the number of pairs of unlike atoms by \(z\), and therefore the energy is equal to \(2zv\), or—when \(E_0\) is the same—is equal to \(V_0\).

Thus we see that, for equal values of the total transformation energy, both theories lead to concordant results for states close to the completely ordered state. We shall see further that both theories coincide in the region of low temperatures.

Relation between \(S\) and \(\sigma\). Figure 16(a) shows the completely ordered state, when the number of possible \(AB\) pairs is greatest. Consequently, both \(S\) and \(\sigma\) are equal to unity. It is easy to see that for such a lattice these two conditions are equivalent: complete order at long distances implies the same at short distances, and conversely. In Fig. 16(b), however, we see that, from the point of view of order at long distances, half the atoms are in the correct and half in the incorrect sites, and, consequently, \(S=0\). This case is discussed in more detail in Part 2. It may be described by saying that the crystal consists of two regions “shifted in phase” relative to one another. In each of them there is perfect order, but for the crystal as a whole \(S=0\). The situation with \(\sigma\) is somewhat different; in the case considered \(\sigma=0.7\). This deviation of \(\sigma\) from unity is evidently due to the existence of a boundary surface between the regions. The larger the regions, the smaller the relative value of the boundary surface, and the closer \(\sigma\) approaches unity, although \(S\) remains equal to zero.

In the Bragg—Williams theory, a small energy corresponds to a high degree of order at long distances, and, consequently, the state characterized by order at long distances is stable at low temperatures. In Bethe’s theory the connection between the energy and order at long distances is not so close; indeed, we have just shown the possibility of attaining almost the minimum energy, i.e. \(\sigma=1\), in the absence of order at long distances. These considerations compel one to raise the following question: will the existence of order at long distances follow from Bethe’s assumptions as the result of all the mathematical derivations? He removes this difficulty by considering probabil-

...ness of the appearance of an interface. The ratio of the probability of the existence of an interface to the probability of its absence is equal to the product of the number of ways in which this interface can be realized and the Boltzmann factor into which enters the additional energy due to its presence. After an estimate has been obtained for the magnitude of this probability, two conclusions can be drawn: the probability of the appearance of an interface is vanishingly small at low temperatures; it reaches a large value at a certain critical temperature, independent of the dimensions of the crystal, provided only that the number of atoms is large. This calculation is analogous to the proof of the existence of a critical temperature in Bethe’s theory. However, in order to calculate the critical temperature one must resort to the indirect method, also proposed by Bethe.

Fig. 17. Selection of a set of sites for Bethe’s indirect method

Fig. 18. Two Bethe approximations:
a—first; b—second

If the Bragg—Williams and Bethe theories coincided, then we could equate the expressions for the energy in these theories [equations (1,13) and (2,15)] and put

\[ E = E_0(1-\sigma)=E_0(1-S^2)\quad \text{or}\quad \sigma=S^2. \]

In reality, however, \(\sigma>S^2\) (for example, above \(T_c\), \(\sigma>0=S\)), except for \(T=0\), when \(S=\sigma=1\), and \(T=\infty\), when \(S=\sigma=0\). Thus, in order to attain a state with a given degree of long-range order, characterized by the quantity \(S\), Bethe’s theory requires less energy than the Bragg—Williams theory.

Bethe’s indirect method. The concepts of interaction of nearest neighbors, short- and long-range order must be combined in order to construct an equilibrium theory. To this end we choose an arbitrary set of sites, as indicated in Fig. 17, and subject it to detailed consideration. We shall henceforth distinguish between two parts of this set: the internal part, which may consist of one or several sites, and the boundary part. The choice is made in such a way that the nearest neighbors of the internal sites which do not themselves belong to them form the boundary. All other sites

the lattice form the external region. Bethe carried out the calculations for sets of two sizes. In the smaller set, Fig. 18(a), considered as the first approximation, the internal part is formed by one \(\alpha\)-site, and the boundary consists of \(z\) neighboring \(\beta\)-sites. In the larger set, Fig. 18(b), considered as the second approximation, the internal part is formed by the entire smaller set, and the boundary is formed by neighboring \(\alpha\)-sites.

The set of sites chosen by us for consideration is physically in no way different from any other similar set in the lattice. Consequently, the conclusions obtained from considering it may with equal success be applied to any other set. Further, as was established earlier, there is complete symmetry between atoms \(A\) and \(B\) and \(\alpha\)- and \(\beta\)-sites. Consequently, if some statement concerning the distribution of atoms \(A\) over \(\alpha\)-sites is true, then the analogous statement concerning atoms \(B\) in \(\beta\)-sites must also be true. However, in the set considered by Bethe, one \(\alpha\)-site is singled out as its center. Therefore consideration of an arbitrarily chosen group, unless an indication is added of how the possibility of arbitrary choice is ensured, may lead to conclusions incompatible with the physical symmetry between \(\alpha\)- and \(\beta\)-sites. As we shall see below, the solution of Bethe’s equation is obtained from the requirement that this incompatibility not occur.

If the atoms are in some way arranged over the boundary sites, it proves possible to calculate the probability of a definite arrangement inside. Indeed, since all the nearest neighbors of the internal atoms (or, in the first approximation, of a single atom) are known, the energy can be determined from this and the Boltzmann factor calculated. Since the total number of atoms in the whole crystal is very large, the a priori probability that an arbitrarily chosen atom turns out to be an atom \(A\) or an atom \(B\) is equal to \(\frac{1}{2}\), independently of how many atoms had been chosen before. Thus the probabilities are completely determined by the Boltzmann factor considered above. However, we still do not know what arrangement to adopt for the boundary atoms.

Let us suppose that in the external region there exists long-range order. This will influence what occurs at the boundary in the sense that it will force atoms \(A\) to occupy \(\alpha\)-positions, and atoms \(B\) — \(\beta\)-positions. To characterize this effect we introduce the ordering energy \(u\), whose value we shall determine later; owing to the influence of the external region, the energy of an atom occupying an “illegitimate” place at the boundary is greater by the amount \(u\) than the energy of an atom in its “own” place. Now we can calculate the probabilities of the various arrangements, since all the necessary energy values are known. The results of these calculations will be expressed as functions of three variables: two of them—\(v\) and \(T\)—are regarded as known, while the third—\(u\)—is unknown.

At first glance it might seem that one would have to introduce a set of values of \(u\), and that it would be impossible to stop at just one of them. However, we shall see that contradictions can be avoided only if a definite value of \(u\) is chosen. This value must precisely correspond to equilibrium long-range order in the outer region, and with its help we can compute, for the equilibrium state, all the data pertaining to it.

In order to find the true value of \(u\), let us first note that at the center of the group there is an \(\alpha\)-site and that it has as its nearest neighbors \(z\) \(\beta\)-sites. Next, we shall assume that in the whole crystal there is an equal number of atoms and sites of both kinds. Every correct theory must treat \(\alpha\)- and \(\beta\)-sites symmetrically, and, consequently, the probability \(r_{\alpha}\) of finding an atom in its proper place at the central \(\alpha\)-site is equal to the probability \(r_{\beta}\) of finding an atom in its proper place at one of the neighboring \(\beta\)-sites. Next we can compute \(r_{\alpha}(v,T,u)\) and \(r_{\beta}(v,T,u)\) as functions of the known quantities \(v\) and \(T\), and of the unknown quantity \(u\). The equation

\[ r_{\alpha}(v,T,u)=r_{\beta}(v,T,u) \tag{2,17} \]

can serve to determine \(u\) as a function of the other two variables.

Thus, taking definite values of \(v\) and \(T\), one can use equation (2,17) to determine \(u\). In the first approximation each boundary atom has one neighbor in the inner region and \((z-1)\) in the outer region. The theory shows that at low temperatures \(u\) is equal to \((z-1)v\), in accordance with the influence of the \((z-1)\) outer neighbors that are in their proper places. Analogous results are obtained also in the second approximation.

The probability of any arrangement in the inner region and on the boundary, expressed in terms of \(v\) and \(T\) and in terms of the now known quantity \(u\), is completely determined. Thanks to this, one can determine the long-range order throughout the whole lattice; let us denote by \(r\) the common value of the quantities \(r_{\alpha}(v,T,u)\) and \(r_{\beta}(v,T,u)\). Then \(r\) is the probability that the central site, or any one of its nearest-neighbor sites, is occupied by an atom in its proper place. As we know, the group under consideration is typical for the whole lattice, and, consequently, the probability that any site is occupied by an atom in its proper place is equal to \(r\). Hence, by the definition of long-range order [equation (1,3)], we find that

\[ S=2\left(r-\frac{1}{2}\right). \]

We can also compute the probability that some pair consists of different atoms; this probability \(q\) is determined by equation (2,9); we shall also cite equations (2,11) and (2,15):

\[ \sigma=2\left(q-\frac{1}{2}\right) \quad\text{and}\quad E=E_0(1-\sigma). \]

These quantities, like \(S\), pertain to the entire lattice.

The results of both Bethe “approximations” differ only slightly from one another. The second approximation is considered more accurate,

because for it the boundary lies farther from the central atoms. In this case all the errors introduced as a result of several simplifying assumptions concerning the influence of the external region will have a smaller effect on the central atoms, to which equation (2.17) is applied, because the intermediate atoms are taken into account more accurately. The insignificant difference between the results of the two approximations makes it possible to suppose that the influence of such errors is small and that both approximations are sufficiently accurate.

Results of the Bethe theory for an AB alloy. By means of the method described above, Bethe calculated the dependence of \(S\) and \(\sigma\) on temperature for a simple cubic lattice in the first and second approximations. The results are shown in Figs. 9, 11, and 13 and in Table 1. Some details of the calculations in the first approximation are given in Appendix 2.

Bethe’s theory predicts for the case AB that, just as in the theory of Bragg and Williams, the quantity \(S\) becomes zero at a definite critical temperature \(T_c\), without undergoing any jump. Therefore, at \(T_c\) there is no latent heat of transformation. In Bethe’s theory, at the critical temperature \(u\) disappears; this means that, for the external region, the distinction between atoms A and B disappears. Therefore the possibility of finding an atom of either kind at the boundary is equally probable, and \(S=0\). However, owing to the interaction between the atoms of the group, each atom tends to have as its neighbor an atom of the other kind to an even greater degree than one of the same kind; and although \(S\) becomes zero, \(\sigma\) remains different from zero. Therefore only part of the energy \(E_0\) is necessary to obtain the alloy at \(T_c\); since in this case the latent heat is absent, the energies immediately below and above \(T_c\) are equal, and the desired energy is expressed by the relation \(E(T_c-)=E(T_c+)\). The ratio of this quantity to the critical temperature proves to be almost the same as in the Bragg—Williams theory, as is shown by the row \(\dfrac{RT_c}{E(T_c)}\) in Table 1.

In order to heat the alloy above \(T_c\) and destroy the order at short distances, additional energy is necessary. It gives rise to the appearance of an anomalous heat capacity above the critical temperature. In this respect Bethe’s theory represents a considerable improvement over the Bragg and Williams theory, in which there is no anomalous heat capacity above the critical temperature; as we shall see in § 14, Part 2, experiment reveals a considerable anomalous heat capacity above \(T_c\). Slightly below the critical temperature the heat capacity has a peak, as was the case in the Bragg—Williams theory, owing to the rapid disappearance of long-range order. Immediately above the critical temperature its value is considerably smaller, in accordance with the slow decrease above \(T_c\) of the order at short distances. The limiting values of the heat capacity immediately below and above the jump are shown in Table 1.

Entropy changes. Up to now we have presented no considerations justifying the use of Bethe’s approximate method. With regard to the Bragg and Williams method, based on the use of the relations \(V(S)\) and \(S(X)\), it was shown that it gives a mathematically rigorous solution consistent with the physical premises; this followed from the consideration of the principle of free energy at the end of § 1. Bethe’s approximate method, it would seem, does not allow such a simple illustration by means of the concept of free energy, and in order to check its accuracy as a method of mathematical solution of the problem, based on the idea of interaction of nearest neighbors, we must turn to other methods. Since for Bethe’s method there is no corresponding statistical-mechanical model leading to the consideration of free energy, satisfactory numerical agreement of Bethe’s theory with the predictions of thermodynamics is therefore not trivial, as in the case of the Bragg and Williams theory, but serves to a greater or lesser degree as a measure of the absolute accuracy of his approximation method.

For a thermodynamic check of Bethe’s theory it is necessary to calculate the change in entropy on passing from complete order to a completely disordered state. As was indicated in § 1, this quantity should be determined by equation (1,27)

\[ \Delta \Phi = R \ln 2 = 0.693\,R. \tag{2,18} \]

In Bethe’s theory it can be obtained by integrating the heat capacity

\[ \Delta \Phi = \int_{0}^{\infty} c\,\frac{dT}{T}. \tag{2,19} \]

In the second approximation one obtains

\[ \Delta \Phi = 0.689\,R \quad \text{(Bethe)}. \tag{2,20} \]

This good agreement, together with the insignificance of the difference between the results of the first and second approximations, leads one to suppose that this approximation is sufficiently good.

It would also be interesting to know the change in entropy on passing from complete order to the critical temperature. Unfortunately, Bethe does not give this. However, it can be estimated from his data relating to the change in entropy between the critical temperature and the state of complete disorder. Combining these quantities with the exact value of the entropy for the completely disordered state, we calculated the entropy values given in Table 1.

Peierls’ application of Bethe’s theory to the case \(AB_3\) \(^{36}\). Peierls applied the hypothesis of interaction of nearest neighbors to alloys containing unequal numbers of atoms A and B. His results can be applied to \(\mathrm{Cu}_3\mathrm{Au}\). The following two

these features considerably increase the computational difficulties of his work in comparison with Bethe’s. First, there are three times as many atoms B as atoms A, as a result of which the symmetry between the \(\alpha\)- and \(\beta\)-nodes, which lies at the foundation of Bethe’s theory, disappears. Further, the face-centered lattice of the Cu—Au system turns out to be more complicated than a simple cubic or a body-centered cubic lattice. In the latter, the nearest neighbors of any atom are never nearest neighbors of one another; in a face-centered lattice, for each nearest neighbor of a given atom, four atoms among its own nearest neighbors are also nearest neighbors of the given atom.

These difficulties are of a purely mathematical character. It is necessary to introduce several values of the energy \(u\), and compatibility relations more complicated than the equality \(r_{\alpha}=r_{\beta}\) also prove necessary. In physical respects, however, the premises and the course of the solution remain the same as in the case considered by Bethe. For details of Peierls’ work we refer the reader to the corresponding sources \(^{36a,\,37a}\). His results are given below.

Like Bragg and Williams, Peierls finds that for the case \(AB_3\) there is an abrupt change in the degree of order at a definite critical temperature, with which a latent heat is associated. Like Bethe, he finds that the state above the critical temperature is not completely disordered, but that a relatively high degree of local (short-range) order is preserved. The decrease of the latter with increasing temperature accounts for the anomaly of the heat capacity above the critical temperature.

The temperature dependence of long-range order in Peierls’ theory is shown in Figs. 10 and 12. The two vertical dotted lines in these figures attract attention. The point is that Peierls’ calculations do not allow him to determine the critical temperature exactly; however, by means of the dotted straight lines he indicates the limits within which it is enclosed. The same computational obstacles did not allow him to give heat-capacity curves. The values in Table 1 represent a rough estimate based on his data.

It would also be of interest to compute the entropy from Peierls’ theory in order to estimate the accuracy of his method, as was done with Bethe’s theory. This was not carried out because of computational difficulties. Using Peierls’ tables, we estimated the change in entropy in the transition from \(T_c\) \(\left(\dfrac{RT_c}{E_0}=1.33\right.\) was chosen arbitrarily for this purpose\()\) to \(T=\infty\), caused by the disappearance of short-range order. It is also necessary to estimate the change in entropy at the critical temperature. By combining with the theoretical value for the entropy of the disordered state, the quantities in Table 1 were obtained.

§ 3. Representation of Entropy as a Function of Energy1

Let us analyze the problem of ordering anew, with the aim of finding its real solution. In the results presented in the preceding paragraph, changes in energy, heat capacity, and the state of order were given. However, none of these quantities is in itself especially important; rather, we are interested in something that, as far as possible, would turn out to be sufficiently self-contained.

The answer to the question that interests us is given by the relative probability considered in statistical mechanics; this probability is the product of the a priori probability by the Boltzmann factor and has already been considered by us in connection with the question of the free energy at the end of § 1. In our case the a priori probability is determined by the number of ways in which the atoms can be arranged. We shall for the time being neglect ordering at large distances \(S\) and restrict ourselves to consideration of the energy. According to Bethe, the energy changes by amounts that are multiples of \(v\). The lowest energy, corresponding to perfect order, may be taken equal to zero, and then all the remaining values of the energy will be multiples of \(v\). For each possible value of the energy \(E\) there exists a certain number of ways \(W(E)\) to arrange the parts of the system so that it possesses the energy \(E\). The relative probability of finding the system in a state with this energy \(E\) is equal to \(\gamma(E)\), where

\[ \gamma(E)=W(E)e^{-\frac{E}{kT}}=e^{\ln W(E)-\frac{E}{kT}} . \tag{3,1} \]

If \(W(E)\) were a known function of \(E\), it would be possible to find the maximum of \(\gamma(E)\) and thus to find, for each temperature, the most probable or equilibrium state. Or, conversely, we could find the statistical sum

\[ \Gamma=\sum_E \gamma(E) \tag{3,2} \]

and, with its aid, calculate all equilibrium properties of the system.2

Thus the determination of the functional dependence \(W(E)\) on \(E\) is equivalent to solving the problem. We shall therefore make various assumptions concerning the form of \(W(E)\) and see to what dependences of the system’s properties on temperature they lead. In doing so it is most convenient to introduce the entropy \(\Phi(E)\) by means of Boltzmann’s relation

\[ \Phi(E)=k\ln W(E). \tag{3,3} \]

Substituting this expression into the equation for \(\gamma\), we obtain

\[ \gamma(E)=e^{\ln\left[w(E)-\frac{E}{kT}\right]} = e^{\frac{T\Phi(E)-E}{kT}} = e^{-\frac{F}{kT}}; \tag{3,4} \]

the latter result shows that the maximum probability corresponds to the minimum of the free energy

\[ F=E-T\Phi . \tag{3,5} \]

Graphical representation. The curve \(C\) in Fig. 19 represents the behavior of the configurational energy and entropy of the alloy. The lowest energy corresponds to a high degree of order and, consequently, to a small number of ways of arrangement, i.e., to a small value of the entropy. For higher energies there exists a larger number of ways of arranging the atoms, and the entropy increases. The maximum of entropy corresponds to the energy of the completely disordered state, which can be realized in the greatest number of ways. However, we can create arrangements to which there correspond values of energy exceeding \(E_0\), by bringing together identical atoms. This process reaches its limit when all atoms A have been brought together and form a pure crystal A, while the remaining separate atoms B form a pure crystal B. Such an arrangement can be realized only in a small number of ways, and to it there corresponds a low value of entropy. This explains the turn of the curve above \(E_0\).

Fig. 19. Schematic representation of the dependence of configurational energy on configurational entropy

The slope of the tangent to the curve at each point is determined by the quantity

\[ \frac{dE}{d\Phi}\;{}^\circ\mathrm{K}. \tag{3,6} \]

As is easy to see, it has the dimension of temperature, and we shall use this fact to study the conditions of equilibrium.

Each point of the curve corresponds to two values \(E\) and \(\Phi\), representing some state of the system. To determine which point corresponds to equilibrium at some temperature \(T\), we proceed as follows. Draw a straight line having slope equal to \(T\) and intersecting the curve at some point \(P'\), for which \(E\) and \(\Phi\) have respectively the values \(E'\) and \(\Phi'\). It is easy to see that the segment cut off by this straight line on the \(E\)-axis,

is equal to \(E' - T\Phi'\), i.e., to the free energy which this system would possess if it could exist at temperature \(T\), having energy \(E'\) and entropy \(\Phi'\). But among all paired values \(E'\) and \(\Phi'\) representing points of the curve, those values for which thermal equilibrium at \(T\) takes place correspond to the minimum of the free energy. Consequently, they are represented by the point at which the straight line of slope \(T\) is tangent to the curve. At this point

\[ \frac{dE}{d\Phi}=T. \tag{3.7} \]

This condition is also obtained if the derivative of equation (3.5) is set equal to zero.

Let us trace the behavior of the system as the temperature changes. At \(T=0\) the stable state is represented by the point at the origin, with \(E=\Phi=0\). For large values intermediate points are obtained, as, for example, \(P\). Finally, as \(T\) approaches infinity, the point \(G\) is reached, at which the entropy has its maximum value. The part of the curve above \(G\), obviously, cannot correspond to stable states.

Latent heat. Curve \(D\) represents a case qualitatively different from the one just considered; it has a concave segment between \(P_1\) and \(P_2\), owing to which one and the same straight line can be tangent to it at two points. However, only for a certain slope \(T_c\) does the free energy have identical values for the points \(P_1\) and \(P_2\). For temperatures below \(T_c\), stable states correspond to points below \(P_1\); above \(T_c\), the stable states are those represented by points above \(P_2\). Thus \(T_c\) is the transition temperature, and here there occurs a jump \(\Delta E\) in the energy and \(\Delta\Phi\) in the entropy. These quantities are connected by the relation

\[ \Delta E = T_c \Delta \Phi. \tag{3.8} \]

Heat capacity. The smooth increase of energy and entropy with increasing temperature determines the magnitude of the heat capacity, which is easily expressed through the derivatives of the function analytically representing curve \(C\). If the equation of the latter is represented in the form \(E=E(\Phi)\), then

\[ \frac{dE}{d\Phi}=T,\qquad \frac{d^2E}{d\Phi^2}=\frac{dT}{d\Phi}. \tag{3.9} \]

Hence, for the heat capacity one obtains

\[ C=\frac{dE}{dT}=T\frac{d\Phi}{dT}=\frac{dE}{d\Phi}:\frac{d^2E}{d\Phi^2}. \tag{3.10} \]

Relation between the Bragg–Williams and Bethe hypotheses concerning energy. Suppose that the system is characterized by a certain degree of long-range order. Let us denote by \(W(E,S)\) the number of ways in which it is possible

to realize a state with energy \(E\). Note that \(W(E,S)\) is less than \(W(E)\). According to the Bragg—Williams theory, all arrangements of atoms with a prescribed order have one and the same energy \(E\). We know, however, that according to Bethe the energy depends only on how the atoms are arranged in small regions, and that the energy can be expressed in the following way with the aid of this local, or short-range, order [equation (2.15)]:

\[ E=E_0(1-\sigma). \]

Let us accept this Bethe assumption and try, with its aid, to interpret the Bragg—Williams formulation.

Among the various ways in which the atoms can be arranged for a given order \(S\), some will correspond to large, and some to small, values of \(\sigma\). Obviously, one can introduce some mean value of \(\sigma\); and since the number of atoms is assumed to be very large, the probability of a considerable deviation of \(\sigma\) from this mean value will be small. In the present paragraph, by averaging we mean a purely statistical process; no connection with the Boltzmann factor is assumed.

For a simple lattice of type AB, considered in connection with Bethe’s theory, it is easy to find the relation between \(S\) and the mean value of \(\sigma\). Consider any pair consisting of one \(\alpha\)-site and its nearest-neighboring \(\beta\)-site; the probability that the two atoms at these sites are in their proper places is \(r_\alpha r_\beta\), and the probability of the opposite event is \(w_\alpha w_\beta\). Consequently, the probability that the pair is formed by unlike atoms is \(r_\alpha r_\beta+w_\alpha w_\beta\). This quantity is the mean value \(q\), defined by equation (2, 9), as the fraction of pairs belonging to the type AB. From the relation (1, 3) between \(r\), \(w\), and \(S\) we find

\[ q_{\mathrm{av}}=\frac{1}{2}\left(1+S^2\right), \tag{3,11} \]

whence, with the aid of (2, 11), it follows for \(\sigma\) that

\[ \sigma_{\mathrm{av}}=S^2. \tag{3,12} \]

Hence for the mean value of the energy in the presence of order \(S\) in the system we obtain

\[ E_{\mathrm{av}}(S)=E_0(1-\sigma_{\mathrm{av}})=E_0(1-S^2). \tag{3,13} \]

This result quite clearly reveals the connection between the Bragg—Williams and Bethe theories. In the former it is assumed that the energy for a given \(S\) has a completely definite value; in the latter this energy is not exactly determined, but may fluctuate about a mean value which is just as definite as in the former theory.

The entropy for a state with order \(S\) was defined in § 1 by counting the number of ways in which the atoms can be arranged for the given degree of order, and equation (1,25) represents a formula for \(\Phi(S)\). Since \(E_{\mathrm{av}}(S)\) and \(\Phi(S)\) are known ...

function \(S\), then one can construct a graph expressing one of these quantities as a function of the other, taking \(S\) as a parameter. The results of such a construction are shown schematically in Fig. 20; here the convex part of the curve for \(AB_3\) is presented on an enlarged scale. These curves break off for the disordered state \(S=0\), which in the Bragg—Williams theory gives the maximum value of the energy. The critical temperatures are determined by the slope of the dashed straight lines; the values of \(S\) are indicated on one of the curves. For temperatures above the critical one, in the stable state \(E=E_0\) and \(S=0\); consequently, the energy above \(T_c\) does not increase, owing to which the anomalous heat capacity turns to zero. In this respect these curves differ from the curves in Fig. 19, where the state with \(E=E_0\) is reached only at infinitely high temperature. An analytic study of the curve \(AB_3\) would make it possible to determine the exact slope of the dashed straight line which is tangent to the curve near

\[ S=\frac{1}{2} \]

and passes through the point \(S=0\). The results of these calculations were given in § 1.

Curves of constant \(S\). In the Bragg—Williams theory, when constructing the curves, to each value of \(S\) there corresponded a definite energy and entropy and, in this way, a point on the curve was obtained. In reality, however, as was found above, to each value of \(S\) there corresponds a multitude of values of the energy and entropy \(\Phi(E,S)=k\ln W(E,S)\). Therefore each value of \(S\) gives not a point, but an entire curve. This curve gives the maximum entropy for the most probable or mean energy at the same value of \(S\), and therefore also \(E_{\mathrm{av}}(S)\), as does the Bragg—Williams curve. Further, as we shall see below, the magnitude of this maximum entropy is \(\Phi(S)\). Consequently, the curve for a definite value of \(S\), which henceforth we shall call the “curve of constant \(S\),” has maximum entropy at the point which on the Bragg—Williams curve corresponds to the same value of \(S\). A part of the curve of constant \(S\) for

\[ S=\frac{3}{4} \]

is shown in Fig. 20 for the case \(AB_3\).

Fig. 20. Dependence of the energy on the entropy according to the Bragg—Williams theory

At first it may seem surprising that the maximum entropy on the curve of constant \(S\) coincides with the entropy calculated by Bragg—Williams for the same value of \(S\). The point is that in the Bragg—Williams case the entropy corresponds to all arrangements for order \(S\), whereas for the curve of constant \(S\) the maximum entropy corresponds only to those of them which possess-

yield the mean energy \(E_{\mathrm{av}}(S)\) or the nearest allowed value of the energy. Therefore one should expect that the maximum entropy on the curve of constant \(S\) will turn out to be somewhat lower than the Bragg–Williams value. In reality, however, the difference proves to be vanishingly small owing to the presence of a large number of atoms. By virtue of the latter circumstance, the number of ways in which the atoms can be arranged for an order \(S\) is, by equation (1, 25), determined by an expression of the form

\[ e^{N f(S)}, \tag{3,14} \]

where \(f(S)\) is of the order of unity. The number of allowed energy values of the system is a quantity of order \(N\). Therefore the most frequently occurring value of the energy can be realized in at least

\[ \frac{1}{N} e^{N f(S)} \tag{3,15} \]

ways. The entropy corresponding to it is equal to

\[ \Phi = kN f(S) - k \ln N. \tag{3,16} \]

The second term may be neglected in comparison with the first, and therefore the entropy is practically the same for all arrangements characterized by the degree of order \(S\).

In the following paragraph we shall consider the method developed by Kirkwood, and shall see how its results can be applied to elucidating the nature of the “curves of constant \(S\).”

§ 4. Kirkwood’s Method

Kirkwood proposed a very ingenious and elegant method that can be applied to finding the form of the “curves of constant \(S\).” Since the effectiveness of the method is connected with the use of certain mathematical devices, we shall present here only the results, and give the details in Appendices 3, 4, and 5.

As was explained in the preceding paragraph, a state with degree of order \(S\) is characterized not by any particular value of the energy, but has a broad distribution over energies. By statistical averaging (considering only the a priori probability, and not the Boltzmann factor) we found the mean energy \(E_{\mathrm{av}}(S)\). Other energy values are also possible, and therefore we are interested in the distribution over energies; a solution of this problem would be equivalent to finding the “curve of constant \(S\)” expressing the energy as a function of the entropy. There are no methods that would allow one to find this distribution directly1; however, we can find various quantities, more

or less closely connected with it. One of them is the mean square deviation of the energy from the mean value1

\[ \Delta_2=[(\Delta E)^2]_{\mathrm{av}}=[(E-E_{\mathrm{av}})^2]_{\mathrm{av}} =\sum \frac{(E-E_{\mathrm{av}})^2}{W(S)} . \tag{4,1} \]

Here \(W(S)\) denotes the total number of ways in which atoms characterized by the degree of order \(S\) can be arranged, and is determined by equation (1,25); the summation extends over all these arrangements. \(\Delta_2\) is a measure of the width of the distribution. It can be calculated for any lattice and any state—by direct, but very tedious, computations analogous to those encountered in mathematical statistics. A simple case is considered in Appendix 4.

For the application of Kirkwood’s method it is in general necessary to calculate all quantities of the type

\[ [(\Delta E)^2]_{\mathrm{av}},\quad [(\Delta E)^3]_{\mathrm{av}},\quad [(\Delta E)^4]_{\mathrm{av}} \]

and so on. These quantities are usually called moments and are denoted symbolically as follows:

\[ \Delta_n=[(E-E_{\mathrm{av}})^n]_{\mathrm{av}} . \tag{4,2} \]

Like \(E_{\mathrm{av}}\), they are functions of \(S\). When they are known, one can obtain the expansion of the free energy as a function of these moments and of temperature. In Appendix 3 the following expression is obtained:

\[ F=E_{\mathrm{av}}-T\Phi(S)-\frac{\Delta_2}{2!\,kT} +\frac{\Delta_3}{3!(kT)^2} -\frac{\Delta_4-3\Delta_2^{\,2}}{4!(kT)^3}+\cdots . \tag{4,3} \]

Here \(\Phi(S)=k\ln W(S)\) is the entropy calculated on the basis of the Bragg—Williams theory. The numerators of the higher terms of the expansion are not simple “moments,” but are the so-called “semi-invariants” of Thiele \(^{22a}\).

As was indicated in the preceding paragraph, we regard the principal aim as obtaining the curve of the dependence of energy on entropy; for this purpose we transform Kirkwood’s result in the corresponding way. For the solution of any practical problem this transformation is not necessary and may even lead to more complicated results; our chief aim in carrying out this transformation is to demonstrate the method. With the aid of the general equation given in Appendix 5, it is shown that from the expression for the free energy in the form of a power series in powers of \(1/T\) one can obtain an expression for the entropy in the form of a power series in powers of \(E\). After carrying out all the calculations, equation (4,3)

transforms into

\[ \Phi=\Phi(S)+k\left[-\frac{1}{2!\Delta_2}(E-E_{\mathrm{av}})^2+ \frac{\Delta_3}{3!\Delta_2^3}(E-E_{\mathrm{av}})^3+ \frac{\Delta_4\Delta_2-3\Delta_3^2-3\Delta_2^3}{4!\Delta_2^5}(E-E_{\mathrm{av}})^4-\cdots\right]. \tag{4,4} \]

Here we are dealing with the analytic form of the result established in the preceding paragraph, namely that the entropy on the “curve of constant \(S\)” is equal to \(\Phi(S)\) when \(E=E_{\mathrm{av}}(S)\).

Thus, in principle—if only we wish to calculate a sufficient number of moments—we can obtain the “curve of constant \(S\)” with any desired accuracy. However, the calculation of moments is quite tedious, and it has been carried out only for the first two moments, although for the simple lattice Kirkwood also calculated the third moment.

Fig. 21. Dependence of energy on entropy for a simple cubic lattice, AB case, according to Kirkwood’s method. The dotted curve is the curve \(S=0\) for the centered lattice, AB case.

An important feature of Kirkwood’s method is its generality. It can be applied not only to simple lattices of the type considered by Bethe.

It is also possible not to restrict oneself to the stoichiometric ratio of the elements: the mean fluctuations are hardly more difficult to find for a ratio of the elements \(3.1:1\) than for \(3:1\).

Results of the application of Kirkwood’s method. We shall confine ourselves here to consideration of the AB case for a simple cubic lattice. In Fig. 21 the energy curves as functions of the entropy are shown for constant \(S\), as well as the Bragg–Williams curve. For the construction only the first two terms of equation (4,4) were taken; the “curves of constant \(S\)” turn out to be parabolas with vertices lying on the Bragg–Williams curve. This means that the energy distribution

\[ W(E,S)=e^{\frac{\Phi(E,S)}{k}}, \]

has the form of a Gaussian error curve. This approximation was introduced earlier into the theory of cooperative phenomena: into the theory of ferromagnetism\(^{32K}\) and the theory of rotation of polar molecules in solids\(^{37S}\).

Stable states always correspond to a maximum of the entropy at constant energy, i.e. to points on the envelope of the “curves of constant \(S\).” For small values of the energy and entropy this envelope almost completely coincides with the Bragg–Williams curve,

This is because, for small energies, only a small number of atoms are not in their own positions; as was indicated in the discussion of the relation \(V_0 = 2zv\) in § 2, the probability that they will meet together and create an energy fluctuation is so small that it may be neglected. Therefore the Bragg—Williams approximation proves sufficient. For large energies the envelope deviates more from the Bragg—Williams curve; for each value of the energy, the maximum of the energy occurs at smaller values of \(S\) than is given by the Bragg—Williams curve. If one moves upward along the envelope, a condensation of the values of \(S\) begins, down to \(S = 0\). The last part of the curve is not an envelope, but rather part of the parabola \(S = 0\).

On the curve there is no abrupt change in the slope of the tangent at the point where the curve \(S = 0\) joins the envelope, or near it. Therefore there is also no discontinuity in the energy that would determine a latent heat. The theory predicts the same continuous character of the disappearance of order as do the Bragg—Williams and Bethe theories for the case of an \(AB\) simple cubic lattice. Near the point where the curve \(S = 0\) joins the envelope, the latter has very slight curvature; therefore a small change in slope is associated with a large change in energy and gives a large heat capacity. Immediately above this point the curve \(S = 0\) has considerably greater curvature and gives a smaller heat capacity. Mathematically this is connected with \(\dfrac{d^2 E}{d \Phi^2}\) in equation (3, 10).

Since the Kirkwood method is based on considerations connected with free energy, compatibility with thermodynamics is fulfilled automatically.

The results to which the Kirkwood method leads for the simple cubic lattice are shown in Figs. 9, 11, 13 and in Table 1. Here, as in the Bragg—Williams theory,

\[ \frac{kT_c}{E(T_c+)} = 2. \]

We have also calculated the quantities in Table 1 for a space-centered lattice. As Kirkwood\(^{58D}\) informed us, the value 4.23 for the heat capacity immediately below \(T_c\), after taking into account equation (4,3), decreases to 1.7.

Other applications of the Kirkwood method. It would be of considerable interest to apply Kirkwood’s method to the case \(AB_3\) and to compare it with the results of Payerls’s work. This would require consideration of a face-centered lattice, since the hypothesis of interaction of nearest neighbors does not lead, for an alloy of such composition, to the formation of a superstructure in a simple or body-centered cubic lattice. The authors made an attempt at such an application with allowance for the second-order moment. It turned out that this approximation is insufficient; in constructing parabolas corresponding to \(S = \mathrm{const}\), it appears that the parabola for \(S = 0\) lies far from all the others, as is shown in Fig. 21 by the dashed curve. In this

approximations; consequently, no superstructure is obtained, and therefore it is necessary to pass to the calculation of moments of higher orders, which is associated with considerable difficulties.

B. Equilibrium theories for alloys of arbitrary composition

The theories considered above were limited to very simple alloys with such concentrations of metals that there are enough atoms of each kind to fill some fraction of the sites in the lattice, for example one half in the case of AB and one quarter or three quarters in the case of AB$_3$. For metallurgy these cases are very special; in general, broader concentration ranges must be considered.

There are a few theoretical investigations for the case of arbitrary concentration. Borelius$^{35\mathrm{C}}$ gave a classification of possible results, useful for a general understanding of the problem. Recently two authors have used the nearest-neighbor interaction hypothesis: Istkhop$^{37\mathrm{C}}$ used Bethe’s first approximation to find the dependence of the critical temperature on concentration, dwelling in particular detail on alloys close to AB$_3$; Shockley$^{38\mathrm{C}}$ used the Bragg—Williams approximation to consider the phase diagram in the region of the ordering transformation for alloys which, like Cu—Au, form face-centered lattices.

5. Energy of formation of alloys

In this section we shall consider alloys in which, with change in concentration, only slight changes of the lattice occur. Thus the transition of a face-centered lattice into a body-centered one is excluded from consideration. We shall consider the transition of a face-centered lattice into a slightly distorted tetragonal lattice, but we shall neglect the distortion caused by the splitting of the nearest neighbors of a given site into nearest neighbors and next-to-nearest neighbors. This simplification will make it possible to apply the nearest-neighbor interaction hypothesis with ease, because the number of the latter is, for a given site, a constant quantity, and only the change in the number of pairs of like and unlike atoms need be taken into account. In calculating the energy of the alloy it proves convenient to take the energy of the pure components as the zero of energy. Then expression (2, 1) for the energy of an alloy of arbitrary concentration and degree of order can be simplified by the methods indicated in Appendix 1 and reduced to the form

\[ E=-vQ_{AB}, \tag{5,1} \]

where \(Q_{AB}\), according to the definition given in paragraph 2, is the total number of \(AB\) pairs.

The energy of the most ordered and of the completely disordered states as a function of concentration. The energy of the disordered state can readily be calculated by the method applied in § 3; this gives the smooth curves in Fig. 22—for body-centered and face-centered cubic lattices. A substantially more difficult problem is to determine, for each concentration, the minimum possible energy, because it is necessary to find the arrangement of atoms corresponding to this minimum energy and to prove that no other arrangement will give a smaller value of the energy. The solution of this problem is outlined in Appendix 1 and leads to the curves, consisting of rectilinear segments, in Fig. 22. For purposes of further comparison with experimental data, let us note that for a 50% atomic concentration the best arrangement in the body-centered lattice corresponds to a structure of the CsCl type, in which each atom has eight nearest neighbors of the other kind, as shown in Fig. 39 (C). For the face-centered lattice the best arrangement leads to the structure indicated below for the alloy CuAu (Fig. 39, B), where each atom has eight neighbors of the other kind and four of the same kind. For the body-centered lattice and composition \(AB_3\), the condition of minimum energy does not give a single lowest energy state; here it proves possible in several ways to arrange the atoms so as to correspond to the smallest value of the energy; however, for the face-centered cubic lattice a definite superstructure is obtained, as shown in Fig. 39 (A), when each atom is surrounded by twelve \(B\) atoms. In Part II we shall see that, in a number of cases, these three ordered

Fig. 22. Energy of formation from pure \(A\) and \(B\) of a disordered alloy (solid curve) and of a completely ordered alloy (straight lines)

structures; but in other cases structures are found which, in a fundamental way, contradict the hypothesis of interaction of nearest neighbors.

A striking feature of the state diagram is its symmetry with respect to 50% concentration. This circumstance follows in an obvious way from equation (5, 1) and again emphasizes the fact that only a single parameter \(v\) is of importance, and not three parameters \(v_{AA}\), \(v_{BB}\), \(v_{AB}\). It is just as obvious that equation (5, 1) must lead to the conclusion that all properties connected with the formation of a superstructure are symmetric with respect to 50% concentration1. Experiment does not confirm this conclusion and thus shows that the hypothesis of interaction of nearest neighbors proves to be insufficient.

It is somewhat difficult to explain the existence of long-range order for concentrations that differ considerably from simple ratios of the type \(1:1\) or \(1:3\). Thus, if there are sufficiently many excess atoms to fill some plane in the alloy (i.e., to obtain a plane passing through the alloy or separating pairs AB), then one may have atoms on one side of the plane “out of phase” with atoms on the other side of the plane and obtain a state of minimum energy characterized by the absence of long-range order. However, the number of ways by which such a state can be realized is small in comparison with the number of ways by which an ordering scheme coherent throughout the whole crystal is realized, when the excess atoms are distributed at random over “foreign” sites, and such a state also corresponds to an energy minimum. For any particular lattice there exists a certain minimum concentration of atoms A, below which the energy minimum can be realized by a larger number of ways for distributions corresponding to the absence of long-range order. According to Ishton’s theory, which uses Bethe’s first approximation, even in such alloys in which the concentration of atoms A reaches only \(\frac{1}{z}\) (\(z\) is the number of nearest neighbors of each atom), a superstructure can still form. This number appears to be rather small—it means that when each atom B has, as a neighbor, on the average only one atom A, there are nevertheless enough atoms A for the formation of a superstructure. In Shokley’s work, which considers the face-centered lattice in the Bragg—Williams approximation, the energy minimum always corresponds to the best possible order, and, consequently, a superstructure must be obtained at all concentrations. We do not think, however, that this implausible conclusion can discredit the other, more substantial conclusions at which Shokley arrives.

§ 6. Dependence of the Critical Temperature on Concentration

Bethe’s theory, extended by Peierls, can be applied to the calculation of the critical temperature for an arbitrary concentration. It is necessary only, in calculating the probability of any configuration of atoms, to find the a priori probability in accordance with the relative number of atoms of each kind; the calculations, however, are very complicated.

Ising carried out calculations for two cases²⁷⁰. The first case corresponds to a lattice in which, as in a body-centered or simple cubic lattice, one can choose a set of $\alpha$- or $\beta$-sites such that the nearest neighbors of every $\alpha$-site are $\beta$-sites and conversely (for arbitrary concentrations of the number of atoms A and $\alpha$-sites, or atoms B and $\beta$-sites, in general, they are not equal to one another, and therefore even for the best ordered arrangements there will be a certain number of atoms in “foreign” positions). Further, he performed his calculations under the assumption that for an arbitrary concentration there should be no latent heat in the transformation. He believes that this probably introduces only a small error into the results. With this restriction he was able to examine the question exactly in the first approximation of Bethe’s theory (the “internal” region consists of only one atom). The dependence of the critical temperature on concentration is shown in Fig. 23. The limiting case $z=\infty$ is also shown, for which the Bethe and Bragg—Williams approximations coincide. In Bethe’s theory only part of the transformation energy is expended in reaching the critical temperature. This part is shown in Fig. 24. Here as the unit there has been chosen the amount of energy,

Fig. 23. Dependence of the critical temperature on composition

Fig. 24. Dependence on composition of the energy required for the destruction of the superstructure

necessary to bring 50% of the alloy to the critical temperature; it is \(\frac{z-2}{z-1}\) times greater than the total transformation energy of 50% of the alloy, i.e. \(0.800\,\frac14 Nzv\;(=0.800\,E_0)\) for the simple cubic lattice, \(0.855\,\frac14 Nzv\;(=0.855\,E_0)\) for the body-centered lattice, and \(\frac14 Nzv\;(=E_0)\) for the Bragg—Williams approximation. Since Ising rules out the possibility of the appearance of latent heat, we shall denote this energy not by \(E(T_c+)\), but by \(E(T_c)_P\). In the Bragg—Williams approximation all the transformation energy is expended in reaching \(T_c\). Therefore the Bragg—Williams curve represents the difference between the curves in Fig. 22 corresponding to the best ordered and completely disordered states.

Ising also made calculations for the face-centered lattice for concentrations close to \(AB_3\). In some experiments it was observed that the critical temperature reaches a maximum for the stoichiometric ratio \(3:1\). Instead of determining the dependence of \(T_c\) on concentration for this case, Ising calculates the derivative of \(T_c\) with respect to concentration at this point. He finds that the derivative does not vanish, as it should for a maximum, and that higher critical temperatures should be expected in the region of composition \(1:1\).

§ 7. Phase diagrams for the ordering phenomenon

Various authors have noted the similarity between the disappearance of long-range order and the melting of a solid. J. H. Williams\(^{35}\) even calls the state above \(T_c\) having “liquid order.” In accordance with these ideas, we may imagine a solid solution as consisting of an ordered and a disordered phase. It is therefore natural to raise the question of the phase diagram for such an alloy.

Shockley solved this problem in the Bragg—Williams approximation for the face-centered lattice. The copper—gold system has a face-centered lattice and possesses ordered structures for \(\mathrm{Cu}_3\mathrm{Au}\) and \(\mathrm{CuAu}\). The first of these is cubic, and the second tetragonal. Shockley’s work shows that here we are dealing with special cases of two ordered phases, so that in reality the problem includes three phases: the disordered phase \(\xi\), the ordered cubic phase \(\eta\), and the ordered tetragonal phase \(\zeta\). The phase diagram for these three phases is shown in Fig. 25. The relation between the critical temperature \(T_{50}\) for 50% composition and the ordering energy for the same composition is taken from the Bragg—Williams theory for the case \(AB\); in accordance with Fig. 22, the transformation energy \(E_0\) is equal to \(Nv\).

In considering this system, Shockley finds it necessary to generalize the definition of order. This is achieved by dividing the face-centered lattice into four simple cubic lattices, as shown in Fig. 26. For each of these lattices an ordering parameter is introduced, defined as a function of the fraction of lattice sites occupied by atoms A. Numbering the lattices from 1 to 4 and

![Figure 25 and Figure 26]

Fig. 25. State diagram for a face-centered cubic lattice; solid curves are critical temperatures for an alloy homogeneous in composition; dashed lines

Fig. 26. Division of a face-centered cubic lattice into four simple cubic lattices; all corresponding sites of each simple lattice are marked with the same symbol

denoting these fractions by \(f_1, f_2, f_3, f_4\), we obtain the following definition of the ordering parameters:

\[ \begin{aligned} S_1 &= 2\left(f_1-\frac{1}{2}\right),\\ &\cdots \cdots \cdots \qquad (7,1)\\ S_4 &= 2\left(f_4-\frac{1}{2}\right). \end{aligned} \]

These expressions were chosen by analogy with the relation \(S=2\left(r_\alpha-\frac{1}{2}\right)\), which was used in § 1 for the case AB. The following values of these parameters may be assigned to the ordered and disordered states AB and AB\(_3\):

	\(S_1\)	\(S_2\)	\(S_3\)	\(S_4\)
AB ordered	1	1	\(-1\)	\(-1\)
AB disordered	0	0	0	0
AB\(_3\) ordered	1	\(-1\)	\(-1\)	\(-1\)
AB\(_3\) disordered	\(-\frac{1}{2}\)	\(-\frac{1}{2}\)	\(-\frac{1}{2}\)	\(-\frac{1}{2}\)

The values \(+1\) and \(-1\) correspond to simple lattices occupied only by atoms A and B, respectively. Thus, for example,

for the ordered system AB, two simple lattices are occupied only by atoms A and two—only by atoms B. In this case it is immaterial how these two simple lattices are chosen, and the values of \(S_i\) may be interchanged.

Such a structure is observed in the case of CuAu and is shown in Fig. 39 (B). Analogously, with the ordered system \(\mathrm{AB}_3\) one may associate three simple lattices consisting only of atoms B and one simple lattice consisting of atoms A, which corresponds to the Cu—Au system known from experiment (Fig. 39, A). This scheme is easily applied to an arbitrary concentration: for example, the system \(\mathrm{A}_3\mathrm{B}_5\), lying midway between AB and \(\mathrm{AB}_3\), gives

\[ \begin{array}{rcccc} \mathrm{A}_3\mathrm{B}_5 \text{ completely ordered}\ . . & 1 & 0 & -1 & -1\\ \mathrm{A}_3\mathrm{B}_5 \text{ disordered}\ . . . . . & -\dfrac{1}{4} & -\dfrac{1}{4} & -\dfrac{1}{4} & -\dfrac{1}{4} \end{array} \]

The treatment of this system by means of order parameters is carried out analogously to that described at the end of § 1. The entropies of all the simple lattices are calculated and added to obtain the total entropy; the mean energy is likewise determined. Both quantities turn out to be explicit functions of the four variables \(S_1, S_2, S_3, S_4\), which determine the state of order. Hence the free energy \(E - T\Phi\) is obtained as a known function of the state of order, and its minimum can be found by the usual mathematical methods. The computational work required for this increases considerably, because the state of order cannot be determined with the aid of a single parameter. It turns out, moreover, that the dependence of the energy on composition is different for three phases, characterized by definite relations between the parameters,

\[ \begin{array}{rll} \text{Phase } \xi \text{ disordered cubic} & S_1 = S_2 = S_3 = S_4, & (7,2)\\ \text{Phase } \eta \text{ ordered cubic} & S_1 \ne S_2 = S_3 = S_4, & (7,3)\\ \text{Phase } \zeta \text{ ordered tetragonal} & S_1 = S_2 \ne S_3 = S_4, & (7,4)\\ & \text{and } S_1 \ne S_2 \ne S_3 = S_4 \ne S_1. & (7,5) \end{array} \]

In constructing the free energy as a function of concentration at constant temperature, for each of the three phases one obtains results analogous to those represented in Fig. 27 (this figure gives only a qualitative picture; portions of the curve near the points of intersection are given on a greatly enlarged scale). It is seen from this that, in different regions of concentration, different phases possess the least free energy and are stable. Near the points of intersection of the free-energy curves, common tangents can be constructed, as shown in the left half of Fig. 27. For concentrations between the points of tangency, the most stable state of the alloy is a mixture of two phases having compositions corresponding to the points of tangency1. After such curves

found for other temperatures as well, one can construct the state diagram in Fig. 25.

The existence of two phases of different concentration, of which one is ordered and the other disordered, in equilibrium with one another, has not yet been observed experimentally. Although such a case follows inevitably from the principles of thermodynamics, it is possible that its practical realization is difficult because equilibrium is established slowly.

If the time for segregation into two phases is insufficient, then the curve should be constructed as indicated in the right-hand part of Fig. 27, where the phases that are stable for the given concentrations are shown. The results are shown by the dashed line in Fig. 25, corresponding to the Ising curve for \(z=\infty\) in Fig. 23, with the sole exception that here we are dealing with another type of lattice. We see that for the composition \(3:1\) no maximum is obtained for the critical temperature.

Fig. 27. Dependence of the free energy on composition for three phases of a face-centered cubic lattice

As the authors believe, there are no theoretical grounds to expect a maximum or minimum of the critical temperature for any composition other than \(1:1\). Here the curve must have a horizontal tangent because of the symmetry of the curve on both sides of this point. This conviction is based on the following consideration. At the critical temperature only a very small degree of order remains. Therefore an excess or deficiency of atoms of one kind gives a continuous change in the state of the system, and therefore there is no reason for any composition, for example \(3:1\), to acquire special significance. The situation is quite different at low temperatures; there the ratio of atoms of different kinds is very important, and a deficiency or excess of atoms of one kind will cause various changes in energy; note the special point of the energy curve as a function of concentration for the ordered face-centered lattice in Fig. 22.

We see, further, that in the concentration interval \(3:1\) and \(1:1\) there are two critical temperatures: one corresponding to the transition from the ordered tetragonal structure to the ordered cubic one, and the other corresponding to the transition from the cubic ordered structure to the cubic disordered one. As shown in § 16 of Part 2, for the alloy Cu—Au both transformations were observed at one concentration \(^{86e}\). However, one

of the two observed ordered structures is more likely orthorhombic than cubic. With each of these transformations there are associated a latent heat and a kink in the heat-capacity curve. Fig. 28 shows the dependence of the latent heat on concentration, and in Fig. 29 the heat-capacity curve for \(A_3B_5\) is depicted. Thus there is confirmed the conclusion, based on the work of Bragg—Williams, Bethe, and Peierls, that the latent heat is absent only for alloys of composition \(1:1\). We shall return briefly to this point in § 13.

Fig. 28. Latent heat of transformation for a face-centered cubic lattice

Fig. 29. Dependence of heat capacity on temperature for a face-centered cubic lattice; case \(A_3B_5\)

§ 8. Investigations not using the concept of nearest-neighbor interaction

Borelius does not use the concept of nearest-neighbor interaction, but instead introduces a more general expression for the energy, which—in our notation—may be represented as follows:

\[ E=E_0(a+bS^2+cS^4+dS^6+\ldots), \tag{8,1} \]

and for the entropy,

\[ \Phi=\Phi(S). \tag{8,2} \]

This expression is obtained in the Bragg—Williams approximation; the constant coefficients in the expression for the energy must be determined from experiment. This method allows Borelius to consider the question in a more general way than is permitted by the nearest-neighbor interaction concept; however, at the same time it has the drawback that it forces one to restrict oneself only to general results and thus does not provide data for direct comparison with experiment.

The energy—entropy—composition surface. Borelius considers this question using the notation introduced in § 3.

processes of ordering in alloys

by drawing curves expressing the dependence of the energy on the entropy. These curves corresponded to a definite composition; if the concentration is laid off along a third axis, perpendicular to the axes of entropy and energy, then the family of energy—entropy curves for different concentrations forms a surface. Just as complete knowledge of the energy—entropy curve gave the solution of the problem for the case of a definite composition, so now complete knowledge of this surface gives all the necessary information about the system.

Borelius dwells in detail on the study of such a surface, including even the case in which the disordered state has a lower energy than the ordered one. This interesting case applies to metals with limited mutual solubility at low temperatures. It was recently considered by R. Becker ^{37Z}, using the representation of the interaction of nearest neighbors and introducing a negative value of \(v\). Further consideration of this question would be out of place in the present review; it is mentioned here only as an example of the use of the ideas developed here in other branches of physical metallurgy.

§ 9. Thermodynamic Potentials of Ordered Phases

Tammann based his assumptions on the existence of ordered structures on chemical data concerning the Cu—Au system. In the introduction we briefly considered these arguments. He found two concentration limits for the action of substances dissolving Cu: one at \(\mathrm{Cu}_3\mathrm{Au}\) and the other at \(\mathrm{CuAu}\). They correspond to a sharp change, near \(\mathrm{Cu}_3\mathrm{Au}\) and \(\mathrm{CuAu}\), in the slope of the tangent to the curve representing the dependence of the free energy on the concentration, which for low temperatures is very similar to the curve in Fig. 23 for the fully ordered state. Using the graphical method for determining thermodynamic potentials, it is easy to see that at these concentrations there must occur a sharp change in the thermodynamic potential of Cu, as a result of which the extraction of Cu from alloys not very rich in Cu becomes more difficult.

This method of investigating superstructures is at present little used. For further details we refer to the works of Tammann ^{19A, 21A}.

C. Behavior of Alloys Not in Thermal Equilibrium

Up to the present time, very few theoretical investigations have been made of the behavior of alloys not in equilibrium. The two main works concern different aspects of the question. Bragg and Williams investigated the rate of approach to equilibrium of allo-

of alloys lying very close to equilibrium; Borelius dealt chiefly with questions of thermal hysteresis, i.e. with effects associated with ordering, which are very similar to the supercooling of liquids or to the phase shift in the transition of white tin into gray.

The rate at which an alloy approaches equilibrium is determined by the frequency with which atoms exchange places. This exchange, in turn, is connected with the activation energy, and, like many chemical processes characterized by an activation energy, it “freezes” at a definite temperature, i.e. at this temperature the relaxation time reaches the order of several years, while a little above it it is of the order of hours. In some alloys, for example in β-brass, the “freezing point” lies considerably below the critical ordering temperature \(T_c\). These alloys usually become ordered upon cooling from a temperature above \(T_c\); during cooling they reach temperatures below \(T_c\), at which a superstructure should form. On the other hand, if \(T\) is considerably above \(T_c\), then a superstructure cannot be obtained by heat treatment: at temperatures at which the superstructure would be thermodynamically stable, the relaxation time is so large that it does not allow ordering; at temperatures that give a short relaxation time, the superstructure is unstable. The “freezing” phenomenon just considered may prevent the formation of a considerable number of superstructures which otherwise would be expected.

§ 10. Relaxation Time

Bragg and Williams, using the ideas set forth in § 1, developed a theory of relaxation for an alloy approaching a state of thermal equilibrium. In our exposition we adhere rather closely to the original work \(^{24c}\).

As was indicated in § 7, when considering the curves \(S(X)\) and \(V(S)\), when the alloy is not in a state of equilibrium, the degree of order differs from that which should exist in the presence of the ordering force. Let us again focus on the atoms A and consider how they pass from \(\alpha\)-sites to \(\beta\)-sites and back.

Let us denote by \(n_w\) the number of A atoms in \(\alpha\)-sites which, per second, displace B atoms from \(\beta\)-sites, i.e. the number of atoms that find themselves in “foreign” positions, and by \(n_r\) the corresponding number of A atoms returning to \(\beta\)-positions and displacing B atoms from \(\alpha\)-sites. In the state of equilibrium we have

\[ n_w = n_r. \tag{10.1} \]

In the absence of equilibrium this equality is violated, and the total number of A atoms located in their “own” positions will change. In accordance with the definitions of § 1, the total number of atoms located in their own positions is equal to \(F_A N r_\alpha\), and the rate of change of this

the quantity is equal to

\[ F_A N \frac{d r_\alpha}{dt}=n_r-n_\alpha. \tag{10,2} \]

Let us suppose that the temperature \(T\) remains constant; denote by \(r_{\alpha e}\) the corresponding equilibrium value of \(r_\alpha\), and let the instantaneous value of \(r_\alpha\) be

\[ r_\alpha=r_{\alpha e}+\delta, \]

where \(\delta\) denotes the deviation from equilibrium. When \(\delta\) is zero, we have equilibrium, and the difference \(n_r-n_w\) vanishes. If \(\delta\) is positive, then there are more atoms on their own sites than there should be, and therefore a change takes place through an increase of the atoms on foreign sites. At constant temperature \(T\), both quantities \(n_w\) and \(n_r\) are functions of \(r_\alpha\), and their difference, for small values of \(\delta\), may be represented in the form

\[ n_w-n_r=\left[\frac{d}{d r_\alpha}(n_w-n_r)\right]_e \delta=C\delta, \tag{10,4} \]

where the index \(e\) indicates that the derivative is evaluated at \(r_\alpha=r_{\alpha e}\), and

\[ C\equiv \left[\frac{d}{d r_\alpha}(n_w-n_r)\right]_e . \tag{10,5} \]

Substituting into equation (10,2), we find

\[ \frac{d\delta}{dt}=-\frac{C}{F_A N}\delta . \tag{10,6} \]

We have obtained the usual equation for a quantity changing with time according to an exponential law. If we introduce the relaxation time

\[ \tau=\frac{N F_A}{C}, \tag{10,7} \]

then the solution may be represented in the form

\[ \delta=\delta_0 e^{-\frac{t}{\tau}}, \tag{10,8} \]

where \(\delta_0\) is the deviation from equilibrium at the instant \(t=0\). The problem facing Bragg and Williams consisted in estimating the magnitude of \(C\).

Let us consider a pair of neighboring atoms A and B. Owing to thermal motion, there is a probability that these two atoms will acquire sufficient energy and exchange places. In this process the atom A, passing from an \(\alpha\)-site to a \(\beta\)-site, must overcome the poten-

cial barrier of height \(W\), and therefore its energy of thermal motion at the beginning of this process must be at least equal to \(W\). On returning it must again overcome this barrier, but since at the beginning it possessed potential energy \(V=V_0S\), because both atoms were in “foreign” places, now only the initial thermal energy \(W-V\) will be needed.

We shall approximate the conditions necessary for two adjacent atoms A and B to exchange places and occupy “foreign” places by making the following three assumptions:

1. When atom A crosses the plane passing through the \(\alpha\)-site and perpendicular to the straight line connecting the pair of sites, it possesses kinetic energy exceeding \(\dfrac{W}{2}\), and at the same time moves in a direction lying within a certain solid angle \(4\pi\omega\).

2. Atom B satisfies the same condition (with atom A and the \(\alpha\)-site replaced by atom B and the \(\beta\)-site).

3. These events are separated in time only by an amount equal to a fraction \(\varphi\) of the period of oscillation, which may be taken to be the same for both atoms.

“In other words, the exchange will take place during one period of oscillation if both atoms possess sufficient energy, move in the proper direction, and almost simultaneously approach the barrier.” Denoting the frequency of oscillation by \(\nu\), and by \(F_{\alpha\beta}\) the probability that two adjacent atoms A and B will in unit time find themselves in foreign places, we find

\[ f_{\alpha\beta}=16\nu\varphi\omega^2\left(1+\frac{W}{2kT}\right)e^{-\frac{W}{kT}}. \tag{10,9} \]

Similarly, the probability that two neighboring atoms located in foreign places will in unit time pass to their own places is equal to

\[ f_{\beta\alpha}=f_{\alpha\beta}e^{\frac{V}{kT}}. \tag{10,10}\,^1 \]

Let us next estimate the number of ways in which an atom A, located in its own place, can pass to a foreign place by exchanging places with neighboring atoms B. The number of atoms A in \(\alpha\)-sites is equal to \(r_\alpha F_A N\). Let each of these \(\alpha\)-sites be surrounded by \(y\) neighboring \(\beta\)-sites. Each of the latter is occupied by an atom B with probability \(r_\beta\).

Thus, on the average, each of the \(r_\alpha F_A N\) atoms A located in their own places can pass to a foreign place in \(yr_\beta\) ways; therefore the total number of ways in which atoms A can

\(^1\) Bragg and Williams establish this relation by means of an argument analogous to that used in deriving the law of mass action.

to pass to a foreign site is

\[ r_{\alpha}F_{\mathrm A}Ny r_{\beta}=NF_{\mathrm A}y r_{\alpha}r_{\beta}. \tag{10,11} \]

Similarly, the number of ways in which atoms A can return to \(\alpha\)-sites is

\[ NF_{\mathrm A}y w_{\alpha}w_{\beta}. \tag{10,12} \]

Multiplying these expressions by the probability that each of these interchanges of places will occur per unit time, we obtain

\[ n_w=NF_{\mathrm A}y r_{\alpha}r_{\beta}f_{\alpha\beta}, \tag{10,13} \]

\[ n_r=NF_{\mathrm A}y w_{\alpha}w_{\beta}f_{\beta\alpha}. \tag{10,14} \]

It is easy to see that the equilibrium condition \(n_r=n_w\) leads to the equation

\[ \frac{r_{\alpha}r_{\beta}}{w_{\alpha}w_{\beta}} = \frac{f_{\beta\alpha}}{f_{\alpha\beta}} = e^{\frac{V}{kT}}, \tag{10,15} \]

which is the equilibrium condition (1,7) § 1.

All the quantities entering into \(n_w\) and \(n_r\) are either constants or known functions of \(r_{\alpha}\). Therefore the derivative \(C\), equation (10,5), can be found, and from equation (10,7) the quantity \(\tau\) computed. We have

\[ n_w-n_r = NF_{\mathrm A}y r_{\alpha}r_{\beta}f_{\alpha\beta} \left[ 1-\frac{w_{\alpha}w_{\beta}}{r_{\alpha}r_{\beta}}e^{\frac{V}{kT}} \right]. \tag{10,16} \]

From (1,7), (1,8), and (1,37) we find that

\[ \frac{r_{\alpha}r_{\beta}}{w_{\alpha}w_{\beta}}=e^{X_2(S)}; \]

from (1,12) and (1,36) it follows that

\[ \frac{V}{kT}=X_1(S). \]

Differentiating (10,16) with respect to \(S\) and noting that the equilibrium value \(S_e\) satisfies the relation \(X_1(S_e)=X_2(S_e)\), we find

\[ \left[\frac{d}{dS}(n_w-n_r)\right]_e = F_{\mathrm B}\left[\frac{d}{dr_{\alpha}}(n_w-n_r)\right]_e = F_{\mathrm B}C = \]

\[ = NF_{\mathrm A}y r_{\alpha}r_{\beta}f_{\alpha\beta} \left[X'_2(S_e)-X'_1(S_e)\right] = \]

\[ = NF_{\mathrm A}y f_{\alpha\beta} \left(\frac{1+S_e}{1-S_e}\right) \left(1-\frac{X'_1}{X'_2}\right). \]

Substitution of the value of \(C\) into equation (10,5) leads to

\[ \frac{1}{\tau} = \left(1-\frac{X'_1}{X'_2}\right) \frac{(1+S_e)y}{(1-S_e)F_{\mathrm B}}\, 16\nu\omega^2 \left(1+\frac{W}{2kT}\right) e^{-\frac{W}{kT}}. \tag{10,17} \]

The term containing \(\left(1-\dfrac{X'_1}{X'_2}\right)\) is due to the dependence of \(V\) on \(S\) and, consequently, on \(r_a\). If \(V\) did not depend on \(r_a\), then \(\left(1-\dfrac{X'_1}{X'_2}\right)\) would be equal to unity; this dependence is such as to increase the relaxation time. This is easy to see by imagining a deviation in the direction of disordering; in this case the degree of order decreases, and consequently the ordering force also decreases. Thus the rate at which equilibrium is restored is not as great as in the case where the ordering force would remain constant. Near the critical temperature the curves \(X_1(S)\) and \(X_2(S)\) intersect at a small angle, and consequently the ratio of the slopes of the tangents is close to unity. Therefore Brett and Williams think that the magnitude of the relaxation time may account for the lag in the rearrangement of the lattice when the temperature changes near the critical temperature. They cite cases in which such a lag was observed1. The dominant term in equation (10,17) is the exponential one, and changes in the other factors may be neglected except, perhaps, in the case when \(\left(1-\dfrac{X'_1}{X'_2}\right)\) is close to zero. Thus, in general, one may write

\[ \tau = A e^{\frac{W}{kT}}, \tag{10,18} \]

Putting \(\nu = 10^{13}\ \mathrm{sec}^{-1}\) (which corresponds to the characteristic temperature \(500^\circ\mathrm{K}\)),

\[ \omega^2 \varphi = 5 \cdot 10^5, \]

\[ y = 4, \]

\[ \frac{W}{kT} = 600, \]

they obtain

\[ A = 10^{-12}\ \mathrm{sec}. \]

They also obtained an estimate of the magnitude of \(A\) for a transition along a path in which three atoms take part, and this gives quantities of the same order.

For \(W\) a reasonable order of magnitude has been chosen. It must lie between the energy required to melt the alloy and the energy required to separate it into free atoms. The latter quantity is determined by the binding energy of the alloy and is about \(80\ \mathrm{kg\ cal}\) per 1 gram-atom for the system

Cu—Au, whence for \(\frac{W}{k}\) one obtains about \(40\,000^\circ\mathrm{K}\). From the energy required in order to melt the alloy, one obtains about \(5000^\circ\mathrm{K}\). The temperatures of interest for ordering phenomena in the Cu—Au system lie around \(600^\circ\mathrm{K}\); consequently, \(\frac{W}{kT}\) should lie between 8 and 70; thus the value 24, adopted by Bragg and Williams, is quite reasonable as an order of magnitude.

The results of experimental investigations in this field are contained in \(^{36}\) and are described in § 18 of Part 2.

§ 11. Theory of temperature hysteresis

Borelius \(^{24A,\,37B}\) developed a theory of temperature hysteresis based on consideration of the “incoming” part of the curve expressing energy as a function of entropy. For a visual representation of this theory, let us choose the scale so that a straight line at an angle of \(45^\circ\) corresponds to a unit of temperature. Then a straight line forming an angle \(\theta\) with the abscissa axis (on which the entropy values are laid off) will correspond to the temperature \(T=\operatorname{tg}\theta\). But instead of drawing this straight line, let us rotate the whole drawing through the angle \(\theta\), as shown in Fig. 30. It is easy to see that, for a given temperature, the ordinate is proportional to the free energy. Therefore, if one allows a heavy ball to roll along the curve, its lowest position will correspond to the stable state. In Fig. 30 the ball is in a position corresponding to a state with large energy and, consequently, with a high degree of disorder. Such a state is not stable, because on the curve there are points lying lower. However, the system cannot attain these states without passing through intermediate sta-

Fig. 30. Illustration of the origin of temperature hysteresis

states with higher free energy; since the latter is thermodynamically impossible, and from the point of view of statistical mechanics highly improbable, the sphere will remain in a position that may be called “metastable.” Cooling the system lowers the hump on the free-energy curve; the latter finally disappears at some temperature \(T_1\), at which the sphere would simply roll to the left. If the system is now heated, the hump will appear again, and at some temperature \(T_2\) the depressions on both sides will have the same depth. It follows from thermodynamics that this should be the transition temperature; however, the system is now to the left of the hump and cannot make the transition from its metastable state until some temperature \(T_3\) has again been reached, at which the sphere could roll freely. In Fig. 31 the solid curve indicates the equilibrium behavior, while the dashed lines and arrows show the hysteresis loop predicted by this theory.

Fig. 31. Hysteresis loop for temperature hysteresis

By introducing various values of the coefficients into his expression (8.1) for the energy as a function of order, Borelius was able so to modify the energy curves as functions of entropy as to obtain agreement with those observations in which hysteresis had been found.

As Bragg and Williams point out \(^{35E,\,37}\), there is no need for the whole specimen to pass over the free-energy hump; this is so improbable that such a transition may not occur at all. Instead, it is sufficient for a nucleus of the stable state to form, after which the transition from one phase to the other will proceed in the manner of recrystallization. The additional free energy required for the formation of such a nucleus is small, and the probability of its release is very great. According to Bragg and Williams, they estimated the probability of formation per unit time of such a nucleus; its order of magnitude is such that significant hysteresis should not be expected.

The experimental data relating to these questions, and not yet sufficiently convincing in character, are presented in § 18 of Part 2.

D. Adaptation of the theory to reality

Up to now, in our theory, the atoms have been represented as somewhat indefinite individuals. It was assumed that atoms of different kinds, denoted by the letters A, B, etc., can be distinguished from one another; all their physical properties were characterized by one

or by several of the three quantities \(z\), \(v\), and \(V_0\). However, alloys in reality are systems that are too complex for such a simple description to be sufficient; and, as we shall see in Part 2, their behavior often deviates from the conclusions of a theory constructed on this basis. In order to bring the theory closer to reality it is necessary to introduce a larger number of characteristic properties of the metals forming the alloy. In the following paragraphs we shall describe some preliminary steps taken in this direction.

§ 12. Origin of the Energy of Ordering

The theory of the origin of the energy of ordering has not so far achieved notable successes. Hume-Rothery \(^{35N,\,36L}\) created a valuable qualitative picture based on ideas about deformations in the lattice caused by the presence of two kinds of atoms. He notes, as we have already done at the beginning of this section, that a characteristic feature of superstructures is the tendency of identical atoms to keep away from one another. This can be understood by assuming that atoms of one kind distort the lattice of atoms of the other kind and that the energy of distortion reaches a maximum when the deformations are distributed as uniformly as possible. This leads to a repulsive force between identical atoms that decreases rapidly with distance and gives the arrangement of atoms characteristic of superstructures. Hume-Rothery relates the deformation in the lattice to the difference in the sizes of the atoms of the two kinds. If this difference is small, then the deformation will be small and the ordering force so insignificant that a superstructure will not form. On the other hand, if this difference is too large, then the mixing energy of the metals will be positive, and they will be practically insoluble in one another (the case of “unfavorable sizes”); under these conditions the solution will be dilute, and no superstructure will form. Only in the case when the difference in atomic sizes lies within some limited range will both the energy of ordering and the solubility favor the formation of a superstructure. The available experimental data confirm Hume-Rothery’s qualitative picture. However, the possibility of making any quantitative predictions on its basis appears remote, and we believe that exact theories will be based on the quantum-mechanical electronic theory of metals.

Mott \(^{37L}\) made a quantum-mechanical calculation of the energy of the superstructure in \(\beta\)-brass. A detailed discussion of Mott’s work, which would require the application of the modern electronic theory of metals, lies beyond the scope of the present review. Mott finds that the principal terms in the expression for the energy are due to exchange repulsion between ions and electrostatic actions associated with the fact that the Cu and Zn ions possess different charges in the alloy. He obtains the following quantities, calculated

per gram-atom:

\[ \begin{array}{lr} \text{Electrostatic energy} \ . . . . . . & 620\ \text{cal} \\ \text{Energy of exchange repulsion} \ . . . & 300\ \text{cal} \\ \text{Total energy (calculated)} \ . . . . . & 920\ \text{cal} \\ \begin{array}{l} \text{From experiment it is found that the total}\\ \text{energy (observed) is less} \end{array} \ . . . . . & 990\ \text{cal} \end{array} \]

In his work he also has the opportunity to discuss the validity of the hypothesis of interaction of nearest neighbors; he comes to the conclusion that taking account of the interaction energy between nearest neighbors is a good approximation, but that the magnitude of the interaction energy cannot be regarded as independent of the state of order, and it must be assumed that it decreases with decreasing order. This would lead to a sharper approach to disordering with increasing temperature and could explain the fact that the peak of \(5.1\) cal on the anomalous heat-capacity curve for \(\beta\)-brass, shown in Figs. 4 and 35, is considerably higher than the values \(1.78\) and \(2.21\) cal obtained for a body-centered lattice from a theory that takes into account nearest-neighbor interaction (see Table 1). Further, it also follows from Mott’s work that the interaction energy must depend on concentration. Further study of this question should be of considerable interest for explaining curves of the dependence of the critical temperature on concentration, such as, for example, Fig. 39 in § 15 of Part 2.

An improvement in the agreement between theory and experiment was achieved by Chang\(^{37V}\) by introducing into Bethe’s theory an interaction with atoms lying in the next configurational shell. In this case two parameters are obtained which can be used to alter the conclusions of the theory. The results obtained by introducing further parameters in the present state of the theory should be regarded rather as an indication of possible types of behavior than as an explanation of the experimental data.

§ 13. The influence of lattice vibrations

In the introduction to Part 1 we assumed the absence of a connection between lattice vibrations and the state of order. This assumption was dictated by considerations of simplicity and by the absence of any information whatsoever about the possible character of this connection. Some experimental data, cited in § 20 of Part 2, show that the state of order affects the elastic properties and, consequently, also the frequencies of lattice vibrations. In this paragraph we shall see how different kinds of dependence of the Debye temperature on the state of order can alter the symmetry with respect to \(50\%\) (atomic) concentration or affect the latent heat.

Entropy of lattice vibrations. If lattice vibrations depend on the state of order, then their free energy must be added to the configurational free energy and, in order to find the equilibrium state, one must seek the minimum of the total free energy. However, in the expression for the free energy of lattice vibrations, only one term can depend on the state of order, namely the term arising from that part of the entropy which is associated with the Debye temperature. It is easily found from the expression for the free energy of a harmonic oscillator1. Let \(\Theta_1\) denote the value of the Debye temperature \(\Theta\) for the completely ordered state. Taking the entropy of this state to be zero, we obtain

\[ \Phi_L(\Theta)=3R\ln\frac{\Theta_1}{\Theta}. \tag{13,1} \]

This expression is of very great importance for the following remarks; therefore we shall derive it by means of simple physical arguments. We are interested in temperatures above the Debye temperature, so that we may take the energy associated with each of the frequencies to be equal to \(kT\), independently of the value of \(\Theta\). Consequently, the energy must not depend on the order, and it may be left out of consideration. Further, the number of frequencies in a given interval is proportional to \(\Theta\); we assume that all frequencies change in the same ratio as \(\Theta\) changes with order. Therefore, if two states of the system are denoted by 1 and 2, then for state 2 the number of frequencies in a given interval will be greater and, consequently, the number of ways in which the energy can be distributed among the various frequencies will also increase.

Indeed, for each frequency there will be more ways by a factor \(\frac{\Theta_1}{\Theta}\), and in all—by

\[ \left(\frac{\Theta_1}{\Theta_2}\right)^{3N} \tag{13,2} \]

more ways. Therefore the entropy difference for states 1 and 2 is

\[ \Phi_2-\Phi_1 = k\ln\left(\frac{\Theta_1}{\Theta_2}\right)^{3N} =3R\ln\frac{\Theta_1}{\Theta_2}. \tag{13,3} \]

This expression is equivalent to that given above.

Dependence of the energy on the total entropy. The relation between the configurational energy and the entropy, which from now on we shall denote by \(\Phi_c(E)\), was considered in §§ 3 and 4. Owing to the assumption made concerning the connection between the Debye temperature and the state of order (we assume here that the effect due to thermal expansion can be separated out), we may regard \(\Phi_L(\Theta)\) as a function of the configurational ener-

gies and to depict this dependence graphically. Curves 1—6 in Fig. 32 give a schematic representation of several possible cases. The curves AB and AB\(_3\) correspond to the configurational curves for these concentrations. They are constructed, as indicated in § 3, in accordance with the ideas underlying the theories of Bethe and Kirkwood, so that the maximum entropy is obtained for the energy \(E_0\), corresponding to the completely disordered state with short-range order \(\sigma = 0\). Energies higher than \(E_0\) correspond to negative values of \(\sigma\); for them the entropy values are smaller than for the disordered state, since in such states identical atoms gather together.

Fig. 32. Possible types of behavior of the entropy of lattice vibrations and its influence in the cases AB and AB\(_3\)

In order, for a given configurational energy, to obtain the entire entropy that depends on ordering, it is necessary to add to the configurational entropy \(\Phi_c(E)\) the entropy of lattice vibrations \(\Phi_L(E)\)

\[ \Phi_T(E)=\Phi_L(E)+\Phi_c(E). \tag{13,4} \]

The simplest dependence of \(\Phi_L\) on \(E\) is a straight line (curves 2 and 5). In this case the influence of lattice vibrations reduces to a change in the temperature at which the various stages of disordering are reached. In case 2 the increase of disordering is associated with an increase in entropy; this promotes the destruction of the superstructure and lowers the temperature. In accordance with equation (3,7) we find that the temperature corresponding to a given value of the configurational energy is determined from the relation

\[ \frac{1}{T}=\frac{d\Phi_T}{dE}=\frac{d\Phi_c}{dE}+\frac{d\Phi_L}{dT}=\frac{1}{T_0}+\frac{1}{T_L}, \tag{13,5} \]

where \(T_0\) represents the temperature that would be obtained if the lattice vibrations were neglected, and \(T_L\) is the slope of the tangent to the curve \(E(\Phi_L)\). We see that \(T\) is less than \(T_0\) for positive \(T_L\), so that the influence of lattice vibrations reduces to shifting disordering toward lower temperatures. \(T=\infty\) corresponds to

\[ T_0=-T_L. \tag{13,6} \]

Thus in this case the state at high temperatures is not completely disordered, but corresponds to a negative value of \(T_0\); this means that on the configurational curve it is represented by a point above the state of maximum entropy, corresponding to negative short-range order.

A negative slope of the straight line corresponds to the fact that the more ordered states are more stable, and thus the transformation is shifted toward high temperatures. In this case \(T = \infty\) corresponds to a definite positive value of \(T_0\), so that a completely disordered state will never be reached, and the alloy will retain a certain degree of order up to very high temperatures.

A very interesting effect is associated with the curvature of the \(\Phi_L\) curve. It was shown above that the latent heat is connected with the curvature of the “incoming” part of the curve, expressing the relation between energy and entropy, as shown on a greatly enlarged scale on the curve AB\(_3\). Calculations in the Bragg—Williams approximation show that this curvature is very small, and consequently joining curves such as 1 or 4 to the AB\(_3\) curve can completely eliminate the “incoming” part and lead to the absence of latent heat. An illustration of this is the curve AB \(+\) 1 in Fig. 32. On the other hand, \(\Phi_L\) curves such as 3 or 6 will give an increase of the “incoming” part and, consequently, of the latent heat as well. Thus a curve of the AB type, which by itself gives no latent heat, can give such heat in combination with a curve such as 3 or 6, which increases the “incoming” part of \(\Phi_T\). Other possible cases could also be indicated; however, at present it would be inexpedient to develop these considerations in greater detail.

Some preliminary calculations by the authors\(^{38c}\) show that a number of features of the Cu—Au system can be explained by the following considerations. The curve \(\Phi_L\) for Cu\(_3\)Au approaches type 4. This gives an increase in the critical temperature, a decrease in the latent heat, and prevents complete disordering at high temperature. On the other hand, \(\Theta\) for CuAu\(_3\) is almost independent of order. Therefore the critical temperature for CuAu\(_3\) will be lower than for Cu\(_3\)Au; it may be so low that the ordering process “freezes” at temperatures above the critical temperature, which explains the absence of a pronounced superstructure in CuAu\(_3\).

(To be continued)

1. See, for example, 36 M, p. 2. ↩↩↩↩↩↩↩↩

2. See, for example, R. H. Fowler[^26]. ↩