TRANSFORMATIONS IN ALLOYS1

F. G. Nix and W. Shockley, New York

PART II

Experimental Study of Superstructures

The theories considered in the first part dealt chiefly with two basic attributes of ordering transformations: the state of order itself and the energy quantities. We shall now consider measurements of the transformation energy, as well as determinations of heat capacities at various temperatures, and compare them with theoretical conclusions. We shall then describe the results of an X-ray study of ordered structures and compare these experimental data with the requirements of the theory, recalling that, according to the nearest-neighbor interaction picture, in an ordered structure each atom must be surrounded by the greatest possible number of atoms unlike itself.

The theory also shows how, in certain simple cases, the “Curie point for ordering,” the temperature at which long-range order appears on cooling, should vary with concentration. We shall compare these conclusions of the theory as well with experimental results. Finally, we shall consider the influence of ordering on the mechanical, magnetic, and electrical properties of alloys.

§ 14. Measurements of Heat Content

Accurate measurements of heat capacities during ordering–disordering transformations give quantities that can be compared with the theoretical conclusions presented above. The mixture method, usually used for determining the heat capacities of solids at high temperatures, does not here give results sufficiently accurate for comparison with theory, especially in the interesting region near the Curie point for ordering. Sykes[^35R][^36N] and his collaborators proposed a modification of the Nernst vacuum calorimeter for obtaining separate values of heat capacity at differ—

temperatures; to some extent an analogous modification was proposed independently of him by Moser[^36f]. The modification proposed by Sykes consists essentially in using a specimen in the form of a closed hollow cylinder, placed inside a second hollow copper cylinder and thermally insulated from it. The outer cylinder is heated by an ordinary resistance furnace wound on it, while the inner cylinder—the specimen—is heated independently by a small auxiliary heating coil. By an appropriate choice of the currents, both cylinders can be maintained at almost the same temperature, and consequently radiation losses can be reduced to a minimum. Knowing the amount of energy expended in heating the specimen and the rate of rise of its temperature, and introducing certain corrections (for radiation losses, etc.), one can obtain the true heat capacities at various temperatures.

Fig. 33. Dependence of heat capacity on temperature for the alloy Cu₃Au is shown by curve 1. Curve 2 was calculated from the heat capacities of Cu and Au according to the rule of mixtures.

Fig. 33 gives the curve of the dependence of the “anomalous” heat capacity on temperature[^36k] (the curve with open circles) for the alloy Cu₃Au, which had been previously ordered by slow cooling at a rate of 30 degrees/hour from 450°. The curve marked with black circles was obtained from the experimental heat-capacity values for Cu and Au, assuming the validity of the Kopp–Neumann rule for the alloy. The observed heat capacities differ only slightly from those calculated in this way at temperatures below 225°; at this temperature we note that a departure from complete order begins. The heat-capacity curve has a maximum at about 391°—the Curie point for ordering. Above it, the curve does not at once coincide with the Kopp–Neumann straight line, but approaches the latter at higher temperatures. Thus, anomalous heat capacity is observed both above and below the Curie point.

Fig. 34 presents an analogous curve for β-brass, taken

from the work of Sykes and Wilkinson^77P, showing in considerable detail the course of the heat capacity near the Curie point for ordering. Fig. 4 (see Part 1) was obtained by Moser^36F also for β-brass; it covers a larger temperature range than Fig. 34. On the latter curve we note that the drop in heat capacity after reaching the maximum extends over almost 10°. We shall return to this question below when comparing the experimental curve with the theory.

In Fig. 35 we give heat-capacity curves^37P for several other concentrations of the Cu—Zn alloy. The latter contain less Zn than β-brass (Fig. 4), and lie in the α + β region, so that the alloy consists not of a single β-phase, as in Fig. 4, but represents a mixture of the β phase with the α phase.

Fig. 34. Dependence of the heat capacity on temperature for β-brass containing 50.4 at. % Zn

Fig. 35. Dependence of the heat capacity on temperature for a number of Cu—Zn alloys. A—for pure α-brass containing 36.80 at. % Zn; B, C, D refer to alloys in the α + β region containing, respectively, 41.09, 43.44, and 45.64 at. % Zn

The lowest curve refers to the pure α-phase; the break in the heat-capacity curve around 200° still cannot be explained. The second curve corresponds to a mixture of α- and β-phases, consisting predominantly of the α-phase. The remaining curves correspond to a higher Zn content and, consequently, are richer in the β-phase. The peak on the curves is due to the presence of the β-phase, and its height increases with increasing content of the latter; however, since the composition of the β-phase changes only insignificantly, there is no change in the Curie point for ordering.

The energy of transformation of the alloy can be obtained by integrating the heat-capacity curve or determined directly by a simple measurement of the energy required to bring a partially ordered specimen from temperature \(T\) to a temperature lying

immediately above the Curie point for ordering. In Fig. 36 such an experimentally obtained curve\[^36К\] is presented for the alloy Cu\(_3\)Au, together with the theoretical curves considered above, obtained from the Bethe—Peierls theory. In Fig. 37 we give the dependence of the transformation energy on alloy composition for a number of Cu—Zn alloys, determined by Sykes and Wilkinson\[^37Р\]. The increase of the transformation energy with increasing Zn content in the two-phase region \(\alpha+\beta\), as before, is due to the increase in the content of the \(\beta\)-phase. The curve shows for the first time how the transformation energy varies as a function of concentration for alloys consisting of

Fig. 36. Dependence of the transformation energy on temperature for Cu\(_3\)Au; \(A\)—experimental curve, \(B\)—Bragg and Williams, \(C\)—Peierls, \(1\)—complete order according to Peierls, \(2\)—complete order according to Bragg and Williams

Fig. 37. Energy absorbed in the transformation for Cu—Zn alloys from 240 to \(500^\circ\)C

pure \(\beta\)-phase, not having the composition CuZn required for the formation of an ideal superstructure. We note that the energy necessary to transfer the alloy from the highly ordered state at \(240^\circ\) to the disordered state at \(500^\circ\) increases as the zinc content approaches 50%, i.e. CuZn. The results of theoretical work by Ishtor, presented in Fig. 25, show that the transformation energy on the side of the copper-rich \(\beta\)-phase, corresponding to approximately 45 atomic % Zn, should be 0.78 times greater than the transformation energy of the CuZn alloy. The experiments of Sykes and Wilkinson give 0.75%, which represents very good agreement.

Comparing the predictions of the theory with the experimental curves of heat capacity, we note in general qualitative agreement with the results of Bethe and Kirkwood, shown in Fig. 13 for alloys of 50—50 atomic composition, i.e. \(\beta\)-brasses, in particular with respect to the large heat capacity above the Curie point for order at long ...

distances. The Bragg—Williams theory requires, as was indicated above, that the configurational heat capacity vanish at temperatures above this point. The same general behavior is seen in Fig. 33 for the alloy \(Cu_3Au\).

The agreement between theory and experiment for heat capacities immediately below \(T_c\) is not very satisfactory. According to Sykes and Jones\(^{36H}\), the configurational heat capacity at this point for well-ordered \(Cu_3Au\), in the absence of regions between which there would be a “phase shift,” is \(62.0\,R\), whereas the Bragg—Williams theory gives (Table 1) the value \(2.4\,R\). The experimentally determined value for \(\beta\)-brass is \(5.1\,R\), while the Bragg—Williams theory gives \(1.5\,R\); Bethe’s first approximation gives \(1.78\,R\), Kirkwood’s \(2.21\,R\). We note that immediately below \(T_c\) the disordering of alloys proceeds more rapidly than follows from the theory.

One should not, however, attach serious significance to the quantitative discrepancy between theory and experiment, because the heat capacity, which is a derivative, is very strongly affected by small errors in the theory. Let us note that for the simple cubic lattice the results of Bethe and Kirkwood differ from one another by a factor of two. In § 12 an explanation due to Mott was given; on the basis of the electron theory of metals, he points out that the effective ordering force must decrease more rapidly than is required by any of these three theories. The result is a sharper disappearance of order and, consequently, a higher peak in the heat-capacity curve.

The rapid fall of the heat-capacity curve with temperature for \(\beta\)-brass in the temperature range immediately above \(T_c\) is due, one may think, to the transition of long-range order coherent throughout the whole alloy into small partially ordered regions. This explanation is analogous to that by which Mott and Potter\(^{37}\) explained similar behavior of the heat capacity of pure Ni above the magnetic Curie point. Sykes\(^{38A}\) asserts that such behavior is generally observed in order—disorder transformations.

The curves of the transformation energy as a function of temperature are more convenient for comparing theoretical values with experimental ones in the vicinity of the Curie point. In Fig. 36 are given the theoretically and experimentally determined values for such a transformation in \(Cu_3Au\). Here, too, the alloy disorders more rapidly at temperatures close to the Curie point of ordering than follows from the Bragg—Williams or Bethe—Peierls theory, while for lower temperatures better agreement is obtained. The relation between the critical temperature and the energy required to bring the alloy from some lower temperature to temperatures immediately above the critical one approximately coincides with that obtained from the theory. The experimental value

\[ \frac{RT_c}{E(T_c+-)} \]

is equal to 2.60, as compared with 2.15 according to Bragg—Williams and 2.38 according to Peierls, as indicated in Table 1 (part I).

As was indicated in §§ 1, 2, 4 and 7 of part I of the works of Bragg—Williams, Bethe, Peierls, and Shockley, there should exist a latent heat for all concentrations, with the exception of the 50—50 atomic % alloy. According to Sykes and his collaborators, latent heat was observed in the alloys Cu\(_3\)Au\({}^{36\text{н}}\) and MgCd\({}^{37\text{V}}\), whereas in the transformations in \(\beta\)-brass\({}^{37\text{P}}\) and Pd\({}^{37\text{V}}\) there is no indication of latent heat. Of these four cases, for the alloys Cu\(_3\)Au and CuZn agreement with theory is obtained; for the alloys Cu\(_3\)Pd and MgCd such agreement is not obtained.

One of the possible ways of explaining these discrepancies was indicated in § 13; it was shown there that the nature of the influence of the state of order on the Debye temperature may determine the presence or absence of latent heat. Although the correct explanation may turn out to be quite different, measurement of the Debye temperatures of these alloys would be of considerable interest.

The entropy changes associated with the destruction of the superstructure have been calculated by various theorists; the results are given in Table 1. Only the configurational part of the entropy is presented here, with the effects of lattice vibrations, electronic heat capacity, etc., neglected. Sykes and Jones\({}^{36\text{н}}\) believe that for Cu\(_3\)Au the entropy change in going from perfect order to the state at a temperature immediately above \(T_c\) is \(0.40 R\) per 1 g-atom. Table 1 gives for the corresponding quantity \(\varphi(T_c-)\) the value 0.562 according to Bragg—Williams theory and about \(0.46 R\) according to Peierls theory. The latter value is smaller than that obtained from Bragg—Williams theory because above \(T_c\) local order still exists; according to Bragg—Williams theory, however, the alloy is already completely disordered at a temperature immediately above \(T_c\). The fact that the experimental value is smaller than that obtained by Peierls may serve as an indication that the order at short distances is even greater than that which follows from Peierls theory. In Fig. 3b there is still another discrepancy with Peierls theory: the experimental latent heat is smaller than the theoretical one. As was indicated above, the latent heat may depend very strongly on the behavior of the Debye temperature, and in § 13 we have shown how this may lead to a decrease in the entropy and the latent heat. The corresponding entropy change for \(\beta\)-brass was calculated by Sykes and Wilkinson by extrapolation to an alloy of 50—50 composition and proved to be equal to \(0.51 R\). As in the preceding case, it is smaller than the value \(0.69 R\), calculated according to Bragg—Williams theory, and the value \(0.65 R\) obtained—with allowance for order at short distances—from the theory of Bethe and Kirkwood. The agreement between theory and experiment for the transformation in \(\beta\)-brass, although not complete, is nevertheless sufficient to compel most investigators to recognize the existence

superstructures; quite recently this conclusion was definitively confirmed by the X-ray data of Jones and Sykes[^37].

§ 15. Dependence of the Curie point for long-range order on alloy composition

The first investigators of the phenomena of ordering noticed that what we now call the Curie point for long-range order depends on the composition of the alloy. Even small deviations of the composition from the simple atomic ratio required for the formation of an ideal superstructure lead to a change in the temperature of the Curie point. For the composition region Cu$_3$Au the available data[^27A][^31A] indicate that an excess of either metal tends to lower this critical ordering temperature \(T_c\). Theories based on the concept of nearest-neighbor interaction do not give a maximum of the critical temperature for this composition; for the reasons considered in § 7 of Part I, we believe that this discrepancy is connected not with a mathematical approximation, but rather with certain physical properties not taken into account by the theory and characteristic of the Cu—Au system. This point of view is to some extent supported by data relating to other systems, for example to Cu—Pd, in which ordered structures are formed at concentrations close to CuPd, but, apparently, are not formed at all at the concentration[^32] exactly corresponding to CuPd.

Fig. 38. Dependence of the critical ordering temperature on composition. The dotted line represents this dependence according to Isihara’s theory.

The solid curve in Fig. 38 depicts the dependence of the critical ordering temperature on alloy composition for the \(\beta\)-phase of brass according to the data of Sykes and Wilkinson[^37P]. We note the fall of \(T_c\) with decreasing zinc content as one moves away from the composition CuZn. The dotted curve gives the dependence of \(T_c\) on composition for an alloy with a body-centered lattice, following from Isihara’s theory considered in § 6, and is in good agreement with experiment.

§ 16. Ordered structures

Before considering ordered structures which, as was shown, exist in alloys, we shall first give a brief

review of the principal conclusions of the theory on this question. Although this introduction should serve as a basis for comparing the theory with experiment, its main purpose is a detailed description of certain important superstructures.

In § 5 it was shown how, in three cases, the hypothesis of interaction of nearest neighbors leads to the existence of definite ordered structures. Briefly, the situation is as follows: according to this hypothesis, atoms tend to form pairs of unlike atoms, and the energy of the crystal proves to be \((-v)\) times the number of such pairs; consequently, the lowest-energy state corresponds to the maximum number of such pairs. For an alloy of composition AB in a body-centered lattice, this condition of a maximum number of pairs of unlike atoms leads to the CsCl structure. In the latter each atom has, as its nearest neighbors, only atoms of the other kind, and the structure shown in Fig. 39 (C) is uniquely determined. For an alloy of composition AB in a face-centered lattice, the theory leads uniquely to the arrangement of Fig. 39 (B).

Fig. 39. Ordered structures (A) Cu₃Au, (B) CuAu: open circles—Au atoms; black circles—Cu atoms; (C) CuZn: black circles—Cu atoms; open circles—Zn atoms

Here each atom is surrounded by eight atoms of the other kind and four of the same kind; here, without changing the lattice type, it is impossible to obtain surroundings in which all neighbors are of the other kind. Any arrangement in which the number of unlike neighbors is less than eight will, according to the theory, possess a higher energy and, consequently, will not represent a stable state at low temperatures. A third kind of superstructure is obtained for alloys of composition AB₃ or A₃B in a face-centered lattice; it is shown in Fig. 39 (A); each atom of the component of lower concentration is completely surrounded by atoms of the other kind—in accordance with the condition of minimum energy. For a body-centered lattice, in the case of an alloy of composition AB₃ and A₃B, the theory does not lead to a definite superstructure; here various arrangements of atoms are possible, to which there corresponds one and the same minimum value of the energy. In the subsequent discussion we shall see that in some cases these conclusions are justified, while in others

Fig. These last cases attest to the inadequacy of the simple theory.

Alloys of the Cu—Au system form an isomorphous series of solid solutions and crystallize in a face-centered lattice. According to our theory, superstructures should appear in Cu\(_3\)Au, CuAu, and CuAu\(_3\). For the first two alloys this conclusion is justified, as was shown by the X-ray investigations of Johansson and Linde\(^{27A}\); the existence of a superstructure in the third case cannot be regarded as proved\(^{36B}\). The structure of Cu\(_3\)Au is apparently such as follows from the theory. It possesses cubic symmetry, and its face-centered lattice may be represented, as was done in § 7, by means of four simple cubic lattices, of which three consist of pure copper and the fourth of pure gold. Similarly, the arrangement of atoms in the CuAu alloy may be represented by means of four such lattices: two of copper and two of gold; the arrangement of the atoms, as shown in Fig. 39 (B), has tetragonal symmetry, the fourth axis being perpendicular to the planes alternately containing Au or Cu atoms. The actual lattice possesses this tetragonal symmetry; it reveals a slight distortion in comparison with the cubic arrangement of atoms in the disordered alloy: in the ordered alloy the axial ratio \(\frac{a}{c}\) is 1.080\(^{36E}\). Similar ordered structures were found by Johansson and Linde in alloys containing from 47 to 53 at.% Au and annealed in the temperature interval from 400 to 200°. In all these alloys, with the exception, of course, of that corresponding to the 50—50 composition, there are excess Cu or Au atoms over and above those necessary for the formation of the superstructure. Such atoms apparently occupy random positions in the lattice. This question, together with closely related ones, will be discussed in connection with the dependence of the electrical resistance of an alloy on its composition.

Another superstructure, differing from those predicted by the theory, was found by Johansson and Linde\(^{36E}\), also in the Cu—Au system. It occurs in the just-mentioned alloys with a tetragonal lattice, containing from 47 to 53 at.% Au, if they are rapidly cooled after annealing between 410 and 420°, and also in alloys containing from 36 to 47 or from 53 to 65 at.% Au when they are annealed in the temperature interval from 400 to 200°. This complex structure has orthorhombic symmetry; in Fig. 40 the unit cell is shown by thick solid lines. Note that this structure may be imagined as formed from the tetragonal lattice of Fig. 39 (B) by stepwise shifts of every fifth atom along the direction \([010]\), the indicated structures extending to infinity in the directions \([001]\) and \([100]\). An alloy containing 50 at.% Au and possessing the orthorhombic structure slowly transforms into a tetragonal lat-

cell if it is annealed at temperatures between 200–380°. Since this structure is similar to the tetragonal form, it agrees with the idea of interaction between nearest neighbors; however, the boundaries along which the “phase shift” occurs introduce extra pairs formed by neighboring atoms of the same kind.

Two other systems which, like Cu—Au in the disordered state at high temperatures, form a continuous series of solid solutions are Cu—Pd and Cu—Pt.

Fig. 40. Orthorhombic lattice of CuAu. The elementary cell is outlined by heavy straight lines. Light circles — Au atoms; black circles — Cu atoms

The ordered alloy Cu$_3$Pt, like Cu$_3$Au, in accordance with the theory has the cubic structure of Fig. 39 (A). It was formerly supposed that the ordered alloy Cu$_3$Pd has exactly the same structure; however, recent observations$^{37E}$ indicate that a slight distortion of the axial ratio gives it tetragonal symmetry, but the arrangement of the atoms apparently remains the same as in Cu$_3$Au. For other compositions an even greater difference from Cu—Au is observed. The ordered alloy CuPt,

Fig. 41. A. Trigonal lattice of CuPt. Alternating planes 111) are occupied by Cu and Pt atoms. B. Illustration of the distribution of excess Pt atoms in the (111) planes of the CuPt lattice occupied by Cu atoms, for the alloy Cu$_3$Pt$_5$

although it consists of alternating planes occupied in turn by

atoms of Cu and Pt, but these planes are the planes (111), not (100). This structure was first established by Johansson and Linde^37a and possesses trigonal symmetry, as shown in Fig. 41 (A). Here each atom has only 6 neighbors of the other kind instead of 8, as in CuAu, and, obviously, the hypothesis of nearest-neighbor interaction is not justified here. Linde^37k recently extended his investigations to the range of alloys close to CuPt. He found that in alloys containing more than 50 at.% Pt, the excess Pt atoms tend to occupy privileged positions, i.e. the additional Pt atoms are located in the (111) planes which earlier were entirely occupied by Cu atoms, and, moreover, are arranged so as to be surrounded by Cu atoms in the manner indicated in Fig. 41 (B). This structure should be expected for the alloy Cu₃Pt₅. As for CuPt alloys containing an excess of copper atoms (i.e. 51–60 at.% copper), according to the available data the Cu atoms are arranged at random in those lattice planes of CuPt which contain Pt atoms. This result agrees with our conclusions from the data given below on the dependence of the electrical resistance on the composition of the alloy (Fig. 48); it turns out that the resistance of ordered alloys with an excess of Pt atoms is considerably less than the resistance of the very same alloys, but slowly cooled, whereas the resistance of annealed alloys with an excess of Cu atoms (from 51 to 60 at.% copper) approaches the resistance of rapidly cooled alloys. With Cu—Pd alloys containing from 37 to 48 at.% Pd, an even more remarkable transformation occurs^25a,27a: disordered alloys possessing a face-centered cubic lattice, on cooling, change the type of lattice and become ordered, possessing a body-centered lattice, i.e. acquire the structure of β-brass, to whose description we shall immediately turn.

The most interesting of the ordered structures obtained from a disordered body-centered cubic lattice is the structure of β-brass. When an ordered structure is produced in this system by the appropriate heat treatment, then, in accordance with the conclusions of the theory for the body-centered lattice AB, it proves to be a structure of the CsCl type (Fig. 39, C). The nature of this transformation was formerly the subject of many long discussions. Yom-Rozery emphasized the similarity between the β-phases of various alloys. On the basis of such analogies one would expect ordering transformations in β-AgMg, β-AgZn, β-AgCd, and β-AuZn. Superstructure lines were found^35j in the β-phase AgZn, which, consequently, is similar to ordered β-brass.

Other interesting ordered structures obtained from a disordered phase with a body-centered lattice are Fe₃Al and FeAl. The ordered regions in the Fe—Al system were subjected by Bradley and Jay^32a to a very ...

to thorough X-ray study. They investigated alloys containing up to 50% (atomic) Al, both in the ordered and in the disordered state. In Fig. 42 (A) the lattice of Fe\(_3\)Al is shown with designations that make it possible briefly to describe the distribution of atoms over the indicated lattice sites in passing to alloys containing a larger percentage of Al. In Fig. 42 (B) a plot according to Bradley and Jay is shown, depicting the dependence of the number of marked sites occupied by Al atoms on the composition for both quenched and annealed alloys. The solid straight line up to 18 atomic % Al, \(abcd\), indicates the identical distribution of Al atoms among

Fig. 42. (A). Enlarged cell indicating the distribution of Al atoms in ordered Fe—Al alloys. (B). Distribution of Al atoms among the four positions \(a, b, c, d\). The solid curve up to 18 atomic % Al represents the distribution both for rapidly and for slowly cooled alloys. From 18 to 38 atomic % Al, circles refer to rapidly cooled alloys, crosses—to annealed alloys. The curve from 33 to 50 atomic % Al refers both to rapidly and to slowly cooled alloys.

the positions \(abcd\), both for rapidly cooled and for annealed alloys. At 18 atomic % Al, Bradley and Jay observed broadening of the lines in the X-ray pattern, which they attributed to ordering in annealed alloys. They did not observe superstructure lines in annealed alloys up to 24 atomic % Al; they consider that at 24% Al about 62% of the \(b\) sites are occupied by Al atoms, while the remaining Al atoms occupy the position \(acd\). At 25 atomic %, on slow cooling, an ordered Fe\(_3\)Al alloy is obtained, in which about 92% of the \(b\) sites are occupied by Al atoms. When mixed in the Fe\(_3\)Al lattice along the cube diagonal, Al atoms occur through one center of the small cubes. Brett \(^{83a}\) indicated that

Al atoms tend to be located as far apart from one another as possible. In the Fe\(_3\)Al lattice we note that each Al atom tends to occupy such positions that not only its nearest neighbors, but also the next atoms would belong to the other kind. This tendency was also discussed by Yom-Rozery and Pouzlam\(^{35N}\). Turning to alloys with a high Al content, let us note that in quenched specimens the additional Al atoms are distributed over the \(b\) and \(d\) sites, and this distribution becomes uniform at approximately 40 atomic % Al. For the FeAl composition, both rapidly cooled and annealed alloys acquire an ordered structure of the β-brass type, with Al atoms at the centers of the cubes and Fe atoms at the corners. Here again the Al atoms are surrounded by the maximum number of neighbors of the other kind.

A structure similar to Fe\(_3\)Al was observed\(^{25B}\) in the Fe—Si system for the alloy Fe\(_3\)Si.

Among other ordered structures of interest is the close-packed hexagonal structure Mg\(_3\)Cd\(^{30B}\) with the following distribution of atoms:

\[ \begin{aligned} \mathrm{Mg}\left\{ \begin{array}{lll} 0\,0\,0, & \dfrac{1}{6}\ \dfrac{1}{3}\ \dfrac{1}{2} \\[6pt] \dfrac{1}{2}\,0\,0, & \dfrac{2}{3}\ \dfrac{1}{3}\ \dfrac{1}{2}; \\[6pt] 0\ \dfrac{1}{2}\ 0, & \dfrac{2}{3}\ \dfrac{5}{6}\ \dfrac{1}{2} \end{array} \right. \end{aligned} \]

\[ \mathrm{Cd}\left\{ \dfrac{1}{2}\ \dfrac{1}{2}\ 0,\quad \dfrac{1}{6}\ \dfrac{5}{6}\ \dfrac{1}{2} \right. \]

and the similar structures MgCd\(_3\)\(^{37W}\) and Ni\(_3\)Sn\(^{37O}\). Lewis and Meller\(^{37J}\) also observed a hexagonal close-packed ordered structure in the quasi-binary Mg—AgCd\(_3\) system in the range of Mg content from 37 to 60 atomic %. A somewhat more complex structure is reported by Ralph\(^{37V}\) for Ni\(_2\)MgSn. The unit cell is similar to the unit cell of β-brass, but twice as large. Bradley and Lu\(^{37C}\) recently reported a superstructure in tetragonal Cr\(_2\)Al. It forms a body-centered cubic lattice; considering the 100 planes formed alternately by the vertices and the centers of cubes, we find that every third plane contains Al atoms, and the remaining planes contain Cr atoms.

The Heusler alloy Cu\(_2\)MnAl in the ferromagnetic state is highly ordered\(^{34P}\), and according to Bradley and Rodgers\(^{34B}\) the atoms in the unit cell of the body-centered lattice are arranged as follows:

\[ \begin{aligned} \mathrm{Cu}\ \left\{ \begin{array}{cccc} 000, & 0\,\frac12\,\frac12, & \frac12\,0\,\frac12, & 110\\[2mm] \frac12\,\frac12\,\frac12, & \frac12\,00, & 0\,\frac12\,0, & 00\,\frac12; \end{array} \right. \\[2mm] \mathrm{Al}\ \left\{ \frac14\,\frac14\,\frac14,\quad \frac14\,\frac34\,\frac34,\quad \frac34\,\frac14\,\frac34,\quad \frac34\,\frac34\,\frac14 \right. \\[2mm] \mathrm{Mn}\ \left\{ \frac34\,\frac34\,\frac34,\quad \frac34\,\frac14\,\frac14,\quad \frac14\,\frac34\,\frac14,\quad \frac14\,\frac14\,\frac34 \right. \end{aligned} \]

§ 17. Phenomena That May Indicate the Degree of Order

a) The degree of long-range order in a highly ordered alloy in which there are no regions with a “phase shift” can be determined with a sufficient degree of accuracy by comparing the intensities of the X-ray superstructure lines with the lines of the normal lattice. An example of such a determination was given by Bradley and Jay\(^ {32A}\), who, in their careful study of ordered Fe—Al alloys, were able to establish the percentage ratio of each type of lattice site occupied by Al atoms. A graphical representation of their results is given in Fig. 42 (B). Ageev and Schoihet\(^ {35A}\) gave the corresponding curves for the CuAu alloy.

When the alloy is only partially ordered and there are small regions of “phase shift,” a number of disturbing factors appear which make determining the degree of order difficult, if not altogether impossible. Borelius, Johansson, and Linde\(^ {28A}\) already in 1928 examined the difficulties that arise in this case. In Fig. 43 we reproduce a drawing from their article, depicting such regions of “phase shift.”

○ — A atom  ● — B atom  ⊗ — B atom or A

Fig. 43. Regions shifted in phase relative to one another

Such a situation may readily occur in an alloy similar to β-brass. In ordering, the α-sites may be occupied by copper atoms in one part of the crystal and by zinc atoms in another. A face-centered cubic lattice, an example of which is the alloy Cu\(_3\)Au, may

can be divided into four simple lattices, any one of which may be adopted for the placement of Au atoms. Here the comparatively great complexity makes it considerably more difficult to draw a conclusion, on the basis of the X-ray photograph, about the degree of order than in the simpler case of β-brass. These so-called “phase-shift regions” are formed accidentally throughout the entire volume of the alloy; at the same time there is an increase both in the dimensions and in the degree of order within the region. When two nuclei, shifted in phase relative to one another, meet under certain conditions, they merge into one coherent ordered group. One may think that this process occurs analogously to grain growth in metals.

b) The presence of small ordered nuclei causes broadening of the superstructure lines, as a result of which it becomes difficult to determine the ratio of the intensity of the superstructure lines to the intensity of the lines of the normal lattice. Therefore, such measurements for alloys that do not possess “coherent” long-range order throughout the entire volume are of no significance for establishing the degree of order. Sykes and Jones used the broadening of the superstructure lines to determine the sizes of these nuclei, applying the Scherrer—Selyakov formula, which relates the particle size to the width of the lines on the X-ray photograph.

c) Another method proposed for estimating the degree of order is based on the fact, discovered by Johansson and Linde\(^{27A}\), that when ordering occurs the lattice of the alloy CuAu changes from cubic to tetragonal. Immediately below the Curie temperature of ordering, the ratio of the axes—the measure of “tetragonality”—is equal to \(a : c = 1.067\) and continues to increase as the temperature is lowered and, finally, when the alloy passes into the highly ordered state, reaches approximately 1.080. Borelius and his collaborators used the values of the axial ratio as a quantitative measure of the degree of order of the alloy and believe that it represents the most reliable measure of the degree of order that can be obtained from X-ray data. They proceed from the assumption that it is precisely the ordering that causes the deformation of the cubic structure leading to a tetragonal or orthorhombic lattice. The fact that tetragonality increases with increasing degree of order is an argument in favor of the Borelius hypothesis, but there are very few other data confirming it. X-ray photographs of alloys quenched from temperatures immediately below the Curie point of ordering often reveal the presence of both a disordered cubic and a partially ordered tetragonal structure, CuAu\(^{30E,\ 32H}\).

d) Another method proposed for determining the degree of order consists in measuring the electrical resistance, which, as we showed in the introduction, depends very strongly on ordering. When using such measurements one must take into account the influence on the electrical resistance of normal thermal motion. Against this method there may be

the same objections have been raised as against the use of the intensity and breadth of superstructure lines, because the electrical resistance is also affected by the size and distribution of partially ordered embryos. However, Sykes and Jones showed that, in order for this effect to become appreciable, the ordered regions must attain dimensions comparable with the mean free path of the conduction electrons. The existence of large regions “opposite in phase” will somewhat disorder the alloy and, consequently, give a greater resistance than in the case of a crystal in which there is “coherent” order throughout the whole volume over large distances. However, like measurements of the intensity of X-ray superstructure lines, measurements of electrical resistance are a satisfactory measure of long-range order for a homogeneous ordered alloy in the absence of small embryos “opposite in phase.”

e) Among the physical properties that depend on ordering is the magnetic susceptibility; this has been established for Cu—Au alloys, which we shall consider in more detail below. However, at the present time very little data has been obtained that would make it possible to establish a correspondence between the magnetic properties of an alloy and the state of order.

f) The phenomenon of alloy ordering, associated with the establishment of both long-range and short-range order, is almost always accompanied by a decrease in volume. Since changes in volume can be measured with a relatively high degree of accuracy, measurements of this kind may probably prove useful as a measure of the degree of ordering.

g) The value of the energy required to transform a partially ordered alloy, which is in equilibrium at temperature \(T\), into a disordered one at a temperature immediately above \(T_c\), and also knowledge of the energy required to transform a highly ordered alloy into a disordered one, may also be of importance for determining the degree of ordering.

§ 18. Behavior of Nonequilibrium Alloys

Relaxation time. The equilibrium ordered state of an alloy is attained by the exchange of atoms of the type that gives rise to diffusion. The theory based on this concept was set forth in § 10. It was shown there that the deviation of the system from the equilibrium state decreases according to an exponential law, which includes a certain characteristic “relaxation time” \(\tau\). This quantity depends on the temperature and is determined by the relation

\[ \tau = A e^{\frac{w}{kT}}, \tag{18.1} \]

Here \(W\) represents the potential barrier that the atom must overcome in order to exchange places.

Each state of ordering is an equilibrium one for a definite temperature \(\theta\), which in our discussion we shall use to designate the state. If the actual temperature \(T\) is greater (less) than \(\theta\), then the degree of ordering becomes greater (less) than the degree of ordering corresponding to the equilibrium temperature \(T\), and tends toward the latter. The “relaxation equation” has the following form:

\[ \frac{d\theta}{dt}=\frac{1}{\tau}(T-\theta). \tag{18.2} \]

Sykes and Evans\(^{36}\) measured resistances in order to determine the rate at which a wire of \(\mathrm{Cu_3Au}\) approaches equilibrium at various temperatures. Table 2 gives some of their values of the relaxation time for different temperatures.

TABLE 2

Temperature °C	in hours
361	8.7
350	14.4
321	64.0
300	212.0

TABLE 3

Temperature °C	in hours
340	5.0
330	6.9
320	8.3
300	12.9
280	19.0

A graph of the dependence of \(\lg \tau\) on \(\frac{1}{T}\) gives a straight line, whence one obtains \(A=10^{-8.5}\) sec. and \(\frac{W}{k}=19{,}100^\circ\mathrm{K}\) for the constants of equation (18.1). These quantities are of the order indicated by the theory of Bragg—Williams (§ 10).

Sykes and Evans also made measurements at constant cooling rates. To find the relaxation times from these it is necessary to investigate the solutions of the equation

\[ \frac{d\theta}{dt}=\frac{T-\theta}{Ae^{\frac{W}{kT}}}, \]

taking into account the dependence of the temperature \(T\) on the time \(t\), and then to apply these solutions to the determination, from the experimental data, of the relaxation time \(\tau\). The results are given in Table 3.

A graphical representation of these data also gives a rectilinear dependence between \(\lg \tau\) and \(\frac{1}{T}\), but with entirely different values for \(\frac{W}{k}\) and \(A\): \(8250^\circ\mathrm{K}\) and \(10^{0.45}\) sec. Comparison of the results, po-

obtained by two methods leads to a discrepancy that is too large for it to be explained by experimental errors. Thus, for example, at \(300^\circ\mathrm{C}\), that determined at constant temperature proved to be \(21\frac{1}{2}\) hours, whereas the determination during cooling at a constant rate gave 12.9 hours. This unpromising fact can be explained by the assumption that the alloy has two different relaxation times, which are obtained separately by different experimental methods. The theory set out above gives nothing for explaining this circumstance, perhaps because it assumed the existence of a single “coherent” long-range order throughout the entire volume.

Irreversible processes. The theories of Bethe and Peierls, based on the hypothesis of interaction of nearest neighbors, compel one to expect the formation of small ordered regions at temperatures considerably above the Curie point for long-range order. Quenching alloys from temperatures above \(T_c\) should then preserve these regions of local order in the quenched specimens even at room temperature. The higher the quenching temperature, the smaller the number of such regions of local order in the specimens should be.

Sykes and Jones\(^{36}\) measured the heat capacity of the alloy \(\mathrm{Cu_3Au}\), quenched in water from \(450^\circ\), which, according to measurements of electrical resistance, appeared to be in a completely disordered state. The resistance of the quenched alloy, measured at room temperature, fell on the continuation of the linear part of the curve of resistance versus temperature for temperatures above the Curie point of ordering, extrapolated to room temperature (Fig. 5). On the X-ray photographs no superstructure lines were found. Measurements of heat capacity showed, however, that considerable local ordering had taken place. In Fig. 44 we reproduce the curve of heat capacity versus temperature, on which the beginning of renewed release of configurational energy is visible at so low a temperature as \(60^\circ\), i.e., considerably below the point (\(230^\circ\)) where, as was previously assumed, exchange of atoms among sites ceases. The amount of energy released between 60 and \(230^\circ\mathrm{C}\) is all the greater,

Fig. 44. Dependence of heat capacity on temperature for an alloy preliminarily quenched in water from \(550^\circ\mathrm{C}\)

the higher the quenching temperature. X-ray photographs obtained by the back-reflection method (the Sachs method) from specimens which had been heated for 2 hours to 130°, as a result of which a considerable part of the configurational energy was released, nevertheless did not reveal superstructure lines. Wires of the same alloy, subjected to analogous heat treatment, showed a resistance characteristic of a disordered alloy. Sykes and Jones conclude from the breadth of the first superstructure lines appearing on the X-ray photographs that the ordered regions must have linear dimensions of the order of \(5.5\cdot 10^{-7}\) cm, or 14–20 interatomic distances.

These interesting experiments show that neither the absence of X-ray superstructure lines, nor the electrical resistance, which was considered characteristic of disordered alloys, can be regarded as proof that the alloy is completely disordered. The study of heat capacity shows that such an alloy may possess considerable local ordering, and that in some cases more than 40% of the configurational energy is released before the ordering can be detected radiographically or by means of resistance measurements.

Fig. 45. Dependence of the electrical resistance on temperature for CuAu. Curve 1 is the equilibrium curve (Fig. 5). Curve 2 depicts the change in resistance with temperature upon heating at a rate of \(2^\circ\)C per minute of a previously quenched alloy. Curve 3 represents the change in resistance with temperature upon cooling at a rate of \(3^\circ\)C per hour. Curve 4 shows the change in resistance with temperature upon reheating the same alloy at a rate of \(30^\circ\)C per hour.

Sykes and Jones believe that the growth of nuclei of the ordered phase corresponds to the initial hollow on the curve in Fig. 45; their growth proceeds up to the moment when the surfaces of the separate ordered regions come into contact. Assuming that the regions of ordering have the form of cubes with \(d\) atoms along the length of one edge, one may consider that, in the first approximation, the energy released amounts to

\[ \frac{1}{1-\frac{6}{d}} \]

of the total energy, if the energy of any atom is determined only by the character of its nearest neighbors. Proceeding from these considerations, Sykes and Jones find that

the dimensions of the region toward the end of the second trough on the heat-capacity curve in Fig. 44 amount to 12 atoms, i.e., represent a quantity comparable with the 14–20 atoms obtained from the width of the superstructure lines. In an analogous manner they find that the dimensions of the ordering region toward the end of the first trough amount to 6–3 atoms.

Figure 45 presents curves of the dependence of resistance on temperature\({}^{36}\) for the alloy \(\mathrm{Cu}_3\mathrm{Au}\) under various conditions of heating and cooling. The solid circles refer to the equilibrium state, attained at extremely slow cooling rates. By extrapolating a rectilinear segment of the equilibrium curve to the ordinate axis, one obtains the point \(\times\), corresponding to the resistance obtained by quenching the alloy in water from some temperature above \(T_c\). If this quenched alloy is heated again, then for the resistance one obtains the curve marked by open circles. The decrease in resistance due to ordering began not at \(60^\circ\), i.e., not at the temperature at which the evolution of heat became noticeable on the heat-capacity curve in Fig. 45, but only at about \(310^\circ\); this shows that the formation of small nuclei, which accounts for the evolution of heat in the interval from 60 to \(230^\circ\), does not affect the electrical resistance. On the curve marked by squares is shown the dependence of the resistance on the temperature of a wire of the same alloy, cooled at a rate of \(30^\circ\) per hour; the alloy is far from equilibrium, because the resistance is considerably higher (for example at \(250^\circ\)) than the equilibrium value. If the alloy is ordered homogeneously throughout the whole volume, then on reheating it should tend toward the corresponding equilibrium value. In reality, however, when an alloy that had previously been cooled at a rate of \(30^\circ\) per hour is heated again at a rate of \(40^\circ\) per hour, the curve (with triangles) proves to lie somewhat below the cooling curve and has the same course in the interval from 250 to \(330^\circ\). Sykes and Evans\({}^{36}\) explain this behavior by assuming that the alloy cooled at a rate of \(30^\circ\) per hour was not homogeneous in the sense of ordering, but consisted of small highly ordered regions, each of which was in equilibrium at all temperatures during the cooling period, and consequently also on heating up to temperatures of \(330^\circ\).

The appearance of different Curie points for ordering on heating and on cooling, obtained from measurements of electrical resistance, led many investigators to insist on the existence of true hysteresis. The latest work of Sykes and Evans\({}^{61}\) on the study of the alloy \(\mathrm{Cu}_3\mathrm{Au}\) showed that, if there is sufficient time for the establishment of equilibrium, the difference between the Curie points on heating and cooling is reduced to a few degrees. According to the data for \(\mathrm{Cu}_3\mathrm{Au}\), two Curie temperatures are obtained, separated by an interval of \(20^\circ\), obviously in that

case, if the rates of heating and cooling were too high for equilibrium to be established. The available data do not destroy our confidence that, if heating and cooling are carried out sufficiently slowly, the two Curie temperatures will coincide. Borelius,^37 however, thinks that in the alloys CuAu and CuPd there is a true hysteresis; thus, for the latter alloy he gives a value of 100° for the difference between the temperatures of the Curie points.

§ 19. The influence of ordering and alloy composition on electrical resistance

In Fig. 5 of the introduction (part I) the influence of homogeneous long-range order on the resistance of the alloy Cu₃Au was shown. At room temperature the resistance \(R_r\) of the rapidly cooled disordered alloy, in the absence of large regions with long-range order, proved to be \(11.4 \cdot 10^{-6}\ \mathrm{ohm\,cm}\), whereas the alloy cooled sufficiently slowly to possess a high degree of long-range order throughout its volume has a resistance

\[ R_0 = 4.3 \cdot 10^{-6}\ \mathrm{ohm\,cm}. \]

In Fig. 46 the dependence of \(R_0\) and \(R_r\) on the composition of the alloy is shown for the Cu—Au system.^36 The smooth parabolic curve \(A\) represents the dependence of the electrical resistance, measured at room temperature, on the composition for alloys quenched from 650°. Such alloys, as follows from the considerations given above, should not contain ordered regions whose linear dimensions exceed \(5.5 \cdot 10^{-7}\ \mathrm{cm}\). Such alloys should contain many small regions possessing a high degree of order at short distances, but, as we showed above, the latter should not influence such properties as electrical resistance.

Fig. 46. Dependence of the electrical resistance on composition for the Cu—Au system. \(A\) — for rapidly cooled alloys, \(B\) — for alloys annealed at 200° C; \(1\) — rapid cooling from 650° C, \(2\) — annealing at 200° C.

Curve \(B\) in Fig. 47 gives the dependence of the electrical resistance at room temperature on composition for the same alloys subjected to prolonged annealing at 200°. A similar treat-

annealing at a temperature which is below \(T_c\), but sufficiently high to ensure exchange of atoms between sites, should create “coherent” long-range order throughout the entire volume, causing the appearance of ordered structures. The figure shows that in some regions—near pure Cu and pure Au—\(R_0\) and \(R_r\) coincide, i.e., at these alloy compositions the degree of long-range order is insufficient to affect the electrical resistance. In the region from 18 to 65 at. % annealed alloys have lower, and for the compositions \(\mathrm{Cu_3Au}\) and \(\mathrm{CuAu}\) considerably lower, resistance than quenched disordered alloys. The values for 25 at. % Au coincide with the values in Fig. 5, respectively for quenched and ordered \(\mathrm{Cu_3Au}\) alloys.

The minima on the curve for annealed alloys correspond to 25 and 50 at. % Au. Annealing of alloys in the region from 65 to 85 at. % Au causes a small increase in electrical resistance in comparison with the resistance of the same alloys quenched from \(650^\circ\). This increase may be caused by segregation. Johansson and Linde\(^{36E}\) report, however, the presence of very weak superstructure lines in the region of \(\mathrm{CuAu_3}\).

Fig. 47. Dependence of electrical resistance on composition for the Cu—Pd system, 1 — rapidly cooled alloys, 2 — annealed alloys

Fig. 47 gives a graphical representation of similar measurements for a number of \(\mathrm{Cu—Pd}^{27A,32L}\) alloys. Their behavior is analogous to the behavior of the \(\mathrm{Cu—Au}\) alloys considered above, with the exception of the striking difference in symmetry near the compositions \(\mathrm{Cu_3Pd}\) and \(\mathrm{CuPd}\); prolonged annealing did not succeed in producing an ordered structure in the alloy containing 50 at. % Pd. Ordered alloys of compositions close to \(\mathrm{Cu_3Pd}\) have a structure similar to \(\mathrm{Cu_3Au}\), but according to the latest data\(^{37E}\) the lattice undergoes a slight distortion, producing tetragonal symmetry. Ordered alloys containing from 37 to 48 at. % Pd have a body-centered cubic lattice of the \(\mathrm{CsCl}^{27A,32L}\) type.

In some alloys the excess atoms (relative to those necessary for the formation of an ideal ordered

the structures themselves are distributed in an ordered manner in the lattice. One might expect that such an arrangement would affect the electrical resistance of annealed alloys, and, indeed, Fig. 48 shows such an effect in Cu—Pt alloys. In discussing in § 16 the structure of the ordered CuPt alloy, it was mentioned that in ordered alloys containing more than 50 at. % Pt, the excess Pt atoms occupy definite positions in the (111) planes of the ordered CuPt lattice occupied by Cu atoms, in such a way that at the composition corresponding to Cu$_3$Pt, each excess Pt atom is surrounded by six Cu atoms [Fig. 41 (B)]. On the other hand, the excess of Cu atoms in alloys containing more than 50 at. % Cu apparently is distributed at random in the (111) planes of the ordered CuPt structure occupied by Pt atoms. This conclusion follows from X-ray data and is in agreement with the character of the curve of the dependence of electrical resistance on composition for annealed alloys. In the region from 50 at. % Pt to almost pure Pt, annealed alloys have a noticeably lower electrical resistance than quenched or disordered alloys. However, annealed alloys containing an excess of Cu atoms show a considerable increase in resistance with an increase in the number of excess Cu atoms, and, finally, near 37 at. % Pt the resistance coincides with the resistance of the quenched alloy.

Fig. 48. Dependence of electrical resistance on composition for the Cu—Pt system:
1 — rapidly cooled and cold-worked; 2 — annealed at 300°; 3 — rapidly cooled from 900°.

Returning to the Ising theory considered in § 5, let us recall that it implies the existence of a superstructure in alloys containing even $\frac{1}{z}$ ($z$ being the number of nearest neighbors of each atom) atoms of some one kind. There are hardly any experimental cases to which this theory is strictly applicable. For a body-centered cubic lattice the limiting atomic percentage should be 12.5; the experimental value—18 at. % Al in the Fe—Al system, shown in Fig. 42—is higher than the theoretical one. Thus, in this case the interaction with atoms in the next coordination shell proves to be less effective than the theory requires. The Ising theory must

be applied to face-centered cubic and simple cubic lattices. Applying it, however, to the face-centered cubic lattice, where \(z = 12\), we note that the formation of superstructures begins at about 9, 11, and 19 at. % Pt, Pd, and Au, respectively, in alloys of these metals with copper. These values must be compared with 8.5 at. %, obtained from the formula \(\frac{1}{z}\). In the present state of the theory, especially with regard to the dependence of the ordering energy on the degree of order and composition, no substantial conclusions can be drawn from a comparison of the experimental values with the theoretical ones.

In the last paragraph we shall touch upon the question of how measurements of electrical resistance may be used as a criterion for establishing the presence of an ordered phase.

§ 20. Influence of ordering on mechanical properties

Nowak\(^{30G}\) made the observation that tempering a rapidly cooled CuPt alloy containing 50 at. % Pt caused an increase in Brinell hardness, analogous in many respects to that observed in the aging of quenched alloys.

Figure 49 shows the curve obtained in tempering a previously quenched CuPt alloy, the hardness of which increases with increasing tempering time. Quenching retains at room temperature the disordered structure characteristic of high temperatures, while tempering causes ordering of the alloy. A highly ordered CuPt alloy possesses greater hardness and tensile strength than an alloy of the same composition in the disordered state. Similar behavior was observed in CuAu, CuPd alloys and, to some extent, in the alloy \(\mathrm{Cu}_3\mathrm{Au}^{35U}\). Sachs and Weerts\(^{31E}\) report that the shear resistance at the elastic limit of the \(\mathrm{Cu}_3\mathrm{Au}\) alloy decreases on ordering in approximately the same ratio as does the electrical resistance.

Fig. 49. Aging curve for a CuPt alloy. The alloy before tempering was rapidly cooled. Annealing temperature \(500^\circ\).

Röhl\(^{33G}\), working in Grüneisen’s laboratory, carried out measurements of the influence of ordering on Young’s modulus, which is a much more fundamental property than hardness and tensile resistance.

rupture, for certain alloys with respect to which it is known that they become ordered during tempering. He found that Young’s modulus increases upon ordering in the alloys Cu$_3$Pd and Cu$_3$Au and decreases upon ordering of the alloys CuAu and CuPd.

§ 21. Influence of plastic deformation on ordering

Dellinger and Graf$^{30A}$ proved that plastic deformation of the ordered alloy CuAu destroys order, having found that superstructure lines are absent from the roentgenogram of the plastically deformed alloy. Shefer$^{33H}$ observed analogous behavior in the case of a plastically deformed ordered FeAl alloy. This was shown still more convincingly by Dahl$^{36A}$ for the alloy Cu$_3$Au and was used by him, as well as later by Linde$^{37K}$, to detect an ordered phase in a case where the existence of such a structure had escaped detection by other physical methods.

Fig. 50. Influence of plastic deformation on the electrical resistance of the Cu$_3$Au alloy; 1 — rapid cooling, 2 — annealing

Severe plastic deformation of most pure metals and disordered solid solutions causes only small changes in electrical resistance. This change is about 2% for most pure metals, such as Ag, Cu, Al, Ni, and for disordered solid solutions, such as, for example, an iron–nickel alloy containing 35 at. % Ni. There are also exceptions: a change of 18% in the case of the brittle metal tungsten and a change of 50% in the case of molybdenum. However, from such a brittle metal as tungsten one cannot expect the same behavior under mechanical deformation at room temperature as from the majority of more plastic metals and alloys. Fig. 50 shows the influence of cold working or plastic deformation on the electrical resistance of the alloy Cu$_3$Au$^{36A}$. As a measure of cold working, the percentage decrease in the cross-sectional area of a cold-drawn wire of the alloy Cu$_3$Au was taken. Curve A in Fig. 50 gives the change of resistance as a func-

... on the degree of cold working for specimens which, before plastic deformation, had been quenched in water from high temperatures in order to preserve the disordered state. We note that even after considerable deformation the change in resistance does not reach even 2%, as should have been expected in the case of cold working of a disordered solid solution. Curve B gives the change in electrical resistance as a function of the amount of plastic deformation for specimens of the same alloy which, before cold working, had been tempered at low temperature in order to create a highly ordered state. Both curves illustrate remarkably well the influence of plastic deformation on long-range order in an alloy, at least insofar as it is manifested in the electrical resistance. Judging from the latter, the “coherent” ordered structure throughout the whole volume was destroyed after a reduction of the transverse cross section by 60%. X-ray patterns taken at various stages of cold working show a decrease in the intensity of the superstructure lines with increasing degree of cold working and their final disappearance after approximately 60% rolling. A similar appreciable increase in resistance during cold working was observed in all the cases studied where a superstructure was initially present.

Fig. 51. Influence of plastic deformation on the electrical resistance of ordered, previously tempered Cu—Pt alloys: A = 80 at. % Pt, B = 25 at. % Pt

In Fig. 51 we give results recently published by Linde^37K, which show how such investigations can be applied to establishing the existence of ordered structures in the Cu—Pt system. Curve A of this figure refers to the change in resistance caused by cold working of a tempered Cu—Pt alloy containing 80 at. % Pt. Fig. 48 shows that here there is a distinctly expressed difference between the electrical resistances of an alloy of this composition in the quenched and tempered states. Curve B in Fig. 52 gives analogous measurements made for a tempered alloy containing 25 at. % Pt, which, as is known from X-ray studies, possesses a superstructure analogous to the superstructure of the alloy Cu₃Au.

Here we would like to explain the meaning of the points marked by circles and crosses on the curves for the disordered alloys in Fig. 48. Linde^37K found that the application of cold working to certain...

from quenched alloys containing more than 50 atomic % Pt produces an increase in resistance. The crosses and the open circles on one abscissa in Fig. 48 give the resistance of the same alloy, respectively before and after cold working. Thus, in alloys of these compositions even such rapid cooling as quenching in water from 900° cannot prevent some degree of ordering. The slight ordering that forms despite such cooling is evidently then destroyed by cold working, so that, apparently, only specimens subjected to cold working are truly disordered alloys.

In the Fe—Ni system, the influence of cold working on the resistance was investigated by Dalem[^36a] in order to obtain information on the existence of a superstructure. He found that Ni and Fe—Ni alloys containing less than 35 atomic % Ni undergo only small changes in resistance upon cold working, regardless of whether the specimens had been annealed or rapidly cooled; but all alloys containing from 40 to 90 atomic % Ni undergo a change in resistance greater than would be expected if the alloys had already been in a disordered state before cold working; the maximum is observed at approximately 75 atomic % Ni. For any alloy composition the change \(\Delta R\) upon cold working is greater for those alloys which had previously been annealed for a long time, i.e. subjected to treatment that promotes the formation of a superstructure, than for alloys that were slowly cooled in the furnace from 900°. Alloys quenched from 900° also reveal an analogous change, and \(\Delta R\) is greater than would be expected as a result of plastic deformation of disordered alloys. These data, together with the general behavior of the electrical resistance of Fe—Ni alloys in this concentration region for rapidly cooled and annealed alloys without cold working, indicate the existence of a superstructure in the region of \(\mathrm{Ni}_3\mathrm{Fe}\). The closeness of the scattering powers of Fe and Ni atoms for X-rays makes it almost, if not

Fig. 52. Influence of plastic deformation on the electrical resistance of \(\mathrm{Ni}_3\mathrm{Mn}\);
1 — rapid cooling from 900°; 2 — annealing at 420°.

entirely impossible to detect the superstructure radiographically.

Dahl \(^{36a}\) extended similar investigations to another interesting system, Ni—Mn, where the existence of a superstructure escapes detection owing to the almost identical scattering power for X-rays of the atoms forming this alloy. Here, for alloys initially subjected to various heat treatments, he studies the effect of cold working not only on the electrical resistance but also on the magnetic properties. Fig. 52 shows the change \(\Delta R\) of the resistance under cold working of annealed and rapidly cooled wires of a Ni—Mn alloy containing 25 atomic % Mn. It is small for the rapidly cooled wire, whereas for the annealed wire (as also in the case of the ordered alloy \(\mathrm{Cu}_3\mathrm{Au}\)) a large value is obtained.

Fig. 53. Effect of plastic deformation on the magnetic saturation \((4\pi I_\infty)\) of \(\mathrm{Ni}_3\mathrm{Mn}\);
1—annealing at \(420^\circ\), 2—rapid cooling

In Fig. 53 we give the results of Dahl’s magnetic measurements, made on the same alloy, likewise treated. The alloy \(\mathrm{Ni}_3\mathrm{Mn}\) is not ferromagnetic if it has been rapidly cooled, and remains so even after considerable cold working; but as a result of annealing it exhibits strong ferromagnetism, which is gradually destroyed with increasing degree of cold working.

Fig. 54. Dependence of the diamagnetic susceptibility on composition for the Cu—Au system;
1—rapidly cooled alloys, 2—annealed alloys

We shall return to the discussion of the influence of cold working on the superstructure later, when we consider various criteria that make it possible to judge the existence of an ordered phase.

§ 22. Effect of ordering on magnetic properties

Paramagnetic and diamagnetic alloys. The effect of ordering on the magnetic properties of alloys was first reported by Fort

and Zeeman\(^{328}\) for the alloys \(Cu_3Au\) and \(CuAu\). The solid curve in Fig. 54 shows the diamagnetic susceptibility of a series of disordered Cu—Au alloys. Ordering of the \(Cu_3Au\) alloy increased its diamagnetic susceptibility by 18%, whereas ordering of \(CuAu\) caused a decrease of almost the same magnitude. Since the measurements were made only for two states, one of which was assumed to correspond to “ideal” order and the other to complete disorder, they do not show how the magnetic susceptibility depends on the degree of long-range order; likewise it is not known whether the presence of short-range order affects the magnetic susceptibility. These magnetic measurements were carried out before the existence of local ordering had been established. It is assumed that the alloys to which the solid curve refers are in the disordered state, because no superstructure lines were observed for them, or the value of the resistance corresponded to the values for a disordered alloy.

Fig. 55. Dependence of the diamagnetic susceptibility on composition for the Cu—Pd system; 1 — rapidly cooled alloys, 2 — annealed alloys

Figure 55 shows how the magnetic susceptibility depends on composition for a series of disordered Cu—Pd alloys\(^{321}\). With the addition of strongly paramagnetic Pd to weakly diamagnetic Cu, the alloys at first become more diamagnetic, until the Pd concentration reaches approximately 25 at.%; at this point the curve turns upward and rapidly reaches the high paramagnetic value of pure Pd. Upon ordering of alloys of compositions close to \(Cu_3Pd\) and \(CuPd\), an increase in diamagnetic susceptibility is found; the greatest change in the region of the \(CuPd\) composition occurs at about 37 at.% Pd, while smaller changes occur in alloys with increasing Pd content. It is of some interest to note that the minimum on the electrical-resistance curve for ordered alloys in the region of the \(CuPd\) composition occurs at about 47 at.% Pd, whereas for this composition ordering produces the smallest change in magnetic susceptibility. On the other hand, in the region of the \(Cu_3Pd\) composition the maximum degree of ordering, determined radiographically, occurs at approximately 17 at.% Pd, i.e. at the same con-

centration, where ordering has the greatest influence both on the electrical resistance and on the magnetic susceptibility.

Influence of ordering on ferromagnetic alloys. Figs. 52 and 53 present data indicating the presence of a superstructure in the alloy Ni₃Mn. The annealed alloy exhibits the phenomenon of saturation with a magnetic moment even exceeding the magnetic moment of pure Ni, whereas the same alloy in the rapidly cooled state is not even ferromagnetic. Saturation of annealed alloys is shown in Fig. 56[^31G]. Here the magnetization at saturation is shown for a series of Ni—Mn alloys, both rapidly cooled and annealed at 450°. Annealing substantially increases the magnetization at saturation in alloys containing more than 17 atomic % Mn. In the concentration range from 24 to 35 at. % Mn, annealing transforms a nonferromagnetic alloy into a strongly ferromagnetic one. The presence of a superstructure has not yet been established roentgenographically. However, indirect data quite evidently testify to the presence of a superstructure in the annealed alloy. This structure is obviously connected with the strongly ferromagnetic state.

Fig. 56. Dependence of the magnetic saturation \((4\pi I_\infty)\) on composition for a series of Ni—Mn alloys.
1 — annealing at 450°; 2 — rapid cooling

Elinghaus[^36D] reported a number of interesting data on the relation between ordering and ferromagnetic properties in an Fe—Pd alloy containing 50 atomic % Pd. After rapid cooling the alloy had a coercive force of about 2 oersteds; after annealing for 15 hours at 500° it increased to 260 oersteds and, upon further annealing, finally reached a steady value of 150 oersteds. Annealing caused the transformation of the face-centered cubic lattice of the rapidly cooled, disordered alloy into an ordered tetragonal structure analogous to CuAu. Other observers[^36K], however, report that they find a superstructure only in the FePd₃ region. Elinghaus[^36D] also reports the presence of an ordered tetragonal structure of the CuAu type in an alloy containing 50 atomic % Fe, 45 atomic % Pt, and 5 atomic % Rh. With this ordered structure are associated a coercive force of 1400 oersteds and a residual magnetization of 3700 gauss.

Ferromagnetism was established in alloys of chromium with platinum containing from 20 to 50 atomic % Cr; the maximum on the curve of the dependence of the magnetization at saturation on composition falls at

a Cr concentration of about 30 at. %. Saturation magnetization is affected by heat treatment; in annealed alloys it is greater than in rapidly cooled ones. X-ray study did not reveal superstructure lines in alloys containing less than 40 at. % Cr; above 40 at. % such lines were observed, and their intensity increased with increasing Cr content. In this system ferromagnetism is evidently not connected with a high degree of long-range order, because an alloy containing 30 at. % Cr, in which no superstructure lines were detected, has the highest ferromagnetic saturation.

The famous Heusler alloys, as a recent study has shown, possess an ordered structure connected with their ferromagnetism. Bradley and Rodgers\(^{34B}\), Heusler\(^{34P}\), and others showed that an alloy of composition \(\mathrm{Cu}_2\mathrm{AlMn}\) is ferromagnetic if, after annealing, it is rapidly cooled from a temperature of about \(600^\circ\). We have already given the structure of this ordered phase, determined by Bradley and Rodgers.

Facko\(^{36B}\) used magnetic measurements for the detection and study of ordered structures in alloys of the systems Fe—Si, Fe—Al, Fe—Cr, Fe—V, Fe—Au, and Fe—Sn. In Fig. 57 we reproduce Facko’s measured ratio \(\frac{\sigma_0}{\sigma_{290}}\) of the magnetic saturation at \(0^\circ\mathrm{K}\) to the saturation at \(290^\circ\mathrm{K}\) for alloys of the Fe—Si system containing up to 30 at. % Si.

Fig. 57. Dependence on the composition of Fe—Si alloys of the magnetic Curie temperature \(\Theta^\circ\mathrm{C}\) and of the ratio of the magnetic saturation at \(0^\circ\mathrm{K}\) to the saturation at \(290^\circ\mathrm{K}\).

For annealed alloys, discontinuities are visible on the curve at 12.5 and 25 at. % Si. The latter composition corresponds well to the known ordered phase \(\mathrm{Fe}_3\mathrm{Si}\). On the curve of the dependence of the ferromagnetic Curie point on composition, shown in the same figure, there is likewise a kink at 25 at. % Si. Facko also attributes significance to the kink of this curve at 12.5 at. % Si. In general he considers the appearance of such special points as are observed at 12.5 and 25 at. % Si, both on the curve for \(\frac{\sigma_0}{\sigma_{290}}\) and on the curve of the magnetic Curie point, as an indication of the presence of ordered

structures. Independent X-ray data^25B^ established the presence of an ordered structure at 25 atomic % Si, but we know of no convincing X-ray data indicating the existence of a superstructure at 12.5 atomic %. Facko finds an analogous break at 25 atomic % Al on two similar curves for Fe—Al alloys, in which, as is well known, ordered structures are formed. We believe that such data can only supplement other results, but that the data presented by Facko should be regarded—at least at the present time—as insufficient for establishing, by themselves alone, the existence of a superstructure.

It has been established that some of the alloys used for making permanent magnets possess a superstructure in the state with high coercive force. An example is the so-called Mishima alloy,^63K^ containing Fe, Ni, and Al. Burgers and Snek^35F^ showed, however, that the coercive force is not proportional to the intensity of the superstructure lines. We do not regard it as established that the presence of a superstructure determines the magnetic properties, although the increase in coercive force caused by tempering is accompanied by the appearance of a superstructure.

§ 23. Criteria for Establishing the Presence of Ordered Structures

As we have already indicated, in alloys in which the atoms composing them have nearly identical scattering powers for X-rays, the presence of an ordered structure, under the usual method of X-ray investigations, escapes observation because of the weakness of the superstructure lines. These lines, as was stated in the introduction, appear because, although the X-rays scattered by the $\alpha$ sites are shifted in phase by $180^\circ$ with respect to the rays scattered by the $\beta$ sites, the difference in the scattering powers of the atoms of both kinds prevents complete weakening of the rays. If the atoms of both components have close scattering powers for X-rays, then the lines may prove too weak and cannot be distinguished; but if the difference in scattering powers is sufficiently

Fig. 58. Dependence of the atomic factor on wavelength for Zn and Cu.

if it is sufficiently large, intense lines are obtained, as in the case of the ordered alloy Cu₃Au in Fig. 2.

There is, however, the following important circumstance: although elements that stand side by side in the periodic table have almost identical scattering powers for almost all wavelengths, there nevertheless exist small intervals of wavelength for which there is an appreciable difference; by a skilful choice of the wavelength of the X-rays used it becomes possible to detect superstructure lines in certain cases in which the Cu or FeKα radiations usually employed in such investigations have proved unable to reveal the presence of a superstructure. In Fig. 58 are given the atomic-scattering curves, from the experiments of Bradley and Hope,³² for Fe, taken from Jones and Sykes,³³ together with additional data on the boundaries of the absorption bands for Cu and Zn. Jones and Sykes, on the basis of the data in Fig. 58, chose ZnKα radiation as having a wavelength that is scattered by the Cu and Zn atoms with the greatest difference in intensity. Using radiation from a zinc anticathode with a copper filter, these investigators were the first to obtain superstructure lines for ordered β-brass. Bradley and Rodgers³⁴ᵇ had earlier used the same method of selecting a suitable X-ray wavelength for studying Heusler alloys, in order to determine the positions of the Cu and Mn atoms. Exact knowledge of the atomic-factor curves should be of enormous help in finding a suitable anticathode for the study of systems in which the atoms of the components have similar scattering powers.

For establishing the presence of an ordered structure, many indirect methods also prove very useful. In alloys forming isomorphous series of solid solutions, annealing after quenching decreases the electrical resistance of those compositions in which ordered structures are formed, as shown in Figs. 46, 47, 48, and as has also been observed in Ni—Mn and Fe—Ni alloys. This phenomenon should be regarded as evidence for the formation of a superstructure in that concentration region in which a similar effect is observed. Two such regions were found in the Ni—Mn system, one near Ni₃Mn and the other near NiMn. In the Fe—Ni system this effect is observed in alloys containing from 35 to 90 at.% Ni.

Changes in the electrical resistance of a homogeneous solid solution on cooling, analogous to the change shown for Cu₃Au in Fig. 5, can also be used as an indication of the onset of the ordering process. Since analogous changes in electrical resistance are also caused by transformations in the ferromagnetic state, such data must be used with great caution.

Striking changes in electrical resistance caused by plastic deformation can also, in many cases, be used to indicate the presence of an ordered

structures. The circumstance that significant changes in resistance are also caused by cold working in tungsten and in molybdenum compels one to interpret data of this kind with caution, but we believe that the order of magnitude of the above-mentioned changes for Cu\(_3\)Au and CuPt may be taken as firm evidence that plastic deformation destroys the superstructure. It was precisely data of this kind that led Dahl\(^{36A}\), as was pointed out by one of us\(^{37T}\), to consider the existence of ordering in alloys close to Ni\(_3\)Fe possible. The behavior of the electrical resistance and of the magnetic properties upon cold working of the previously annealed alloy Ni\(_3\)Mn admits a similar interpretation.

Hypotheses concerning the nature of the \(\beta — \beta'\) transformation in the Cu—Zn system were first put forward by Tammann and Heusler\(^{26A}\) on the basis of very inaccurate heat-capacity data. The precise heat-capacity measurements of Sykes and his collaborators\(^{36R, 36H, 37P}\), together with data on the dependence of electrical resistance on temperature, agree so well with the Bragg—Williams theory that, as has come to be generally thought in recent years, this transformation belongs to the type of ordering processes. Only recently was this point of view confirmed by the above-mentioned X-ray investigations of Jones and Sykes\(^{37I}\). We believe that such changes in heat capacity are indeed of great importance for establishing the nature of many transformations.

Such magnetic measurements as those belonging to Fallot\(^{36B}\) may likewise be mentioned as significant in the event that other data are accumulated. The break in the curve \(\dfrac{\sigma_0}{\sigma_{290}}\), Fig. 57, at 12.5 at. % Si is undoubtedly analogous in its nature to the break at 25 at. % Si, where the existence of a superstructure has been established by X-ray methods. We think, however, that many other data must be obtained before real significance can be assigned to such breaks.

We are indebted to a number of persons who helped us in preparing this article. We are also grateful for discussions and communications of a special nature to J. G. Kirkwood, F. Seitz, and K. Sykes. We are especially grateful to K. K. Darrow for the efforts expended on the manuscript, which was in a disorderly state. It has given the three of us pleasure to show that a transition from disorder to order is possible not only in lattices, but also in the literature on lattices, although between these two cases there is the important and regrettable difference that in transformations from disorder to order in the metallurgical case energy is liberated, whereas in the literary case it is absorbed.

Appendix 1. Simplification of the Expression for the Energy of Interaction of Nearest Neighbors and the Energy of Formation of Alloys

Relation between the quantities \(Q\)

Consider a lattice containing \(N\) sites, each of which is surrounded by \(z\) nearest neighbors; then the total number of pairs is

\[ Q=\frac{1}{2}zN. \tag{A1.1} \]

Let \(Q\) consist of \(Q_{AA}\), \(Q_{BB}\), and \(Q_{AB}\) pairs of different types. First let us imagine that an alloy model, consisting of wooden balls and wooden sticks representing atoms and the bonds between pairs, is divided into pairs in such a way that each pair has \(\frac{1}{z}\) of a ball on each side of the sticks. Then for each \(AA\) pair there are \(\frac{2}{z}\) of an \(A\) ball, etc. If the total number of \(A\) and \(B\) balls in the model is respectively \(F_A N\) and \(F_B N\), then we must have

\[ F_A N=\frac{2}{z}Q_{AA}+\frac{1}{z}Q_{AB}, \tag{A1.2} \]

\[ F_B N=\frac{2}{z}Q_{BB}+\frac{1}{z}Q_{AB}. \tag{A1.3} \]

From these equations it follows that

\[ Q_{AA}=F_AQ-\frac{1}{2}Q_{AB}. \tag{A1.4} \]

\[ Q_{BB}=F_BQ-\frac{1}{2}Q_{AB}. \tag{A1.5} \]

Next consider the energy of the alloy in comparison with the energy of two pure crystals of \(A\) and \(B\). The energy of a crystal of pure \(A\) with \(F_A N\) atoms is \(v_{AA}\) times greater than the number of pairs, which is equal to \(\frac{z}{2}F_A N=F_AQ\). Therefore the energy of interest to us is

\[ E=v_{AA}Q_{AA}+v_{BB}Q_{BB}+v_{AB}Q_{AB}-F_Av_{AA}Q-F_Bv_{AB}Q. \tag{A1.6} \]

This expression reduces to

\[ E=\left[v_{AB}-\frac{1}{2}(v_{AA}+v_{BB})\right]Q_{AB}. \tag{A1.7} \]

Define a new quantity

\[ v=\frac{1}{2}(v_{AA}+v_{BB})-v_{AB}. \tag{A1.8} \]

Then

\[ E=-vQ_{AB}. \tag{A1.9} \]

The meaning of \(v\) is indicated in the following diagram:

\[ \begin{array}{ccc} \begin{array}{cc} A & A\\ A{-}B & A{-}B\\ A & A \end{array} & \begin{array}{c} \\ \\ \text{Difference } 2v \end{array} & \begin{array}{cc} A & A\\ A{-}A & B{-}B\\ A & A \end{array} \\ \text{Lower energy} & & \text{Higher energy} \end{array} \]

We see that the quantity \(v\) must be positive, because otherwise at low temperatures the alloy would tend to break up into the pure components A and B, in contradiction to the assumption that we are dealing with an alloy in which a superstructure forms at low temperatures.

Dependence of the Energy on Composition

Disordered state. For the disordered state, the probability that any given site is occupied by an A atom is equal to \(F_A\), and that it is occupied by a B atom is equal to \(F_B\). Consequently, the probability that two neighboring sites form an AB pair is equal to \(2F_A F_B\), where the factor 2 corresponds to the two possibilities: AB and BA. Therefore the fraction of AB pairs is \(2F_A F_B\), and the energy is

\[ E(\text{disord.})=-2vQF_A F_B=-NvzF_A F_B . \tag{A1.10} \]

Completely ordered state. Owing to the symmetry about 50 atomic %, we need consider only the case \(F_A < F_B\). The smallest possible energy corresponds to the maximum value of \(Q_{AB}\) and, on the basis of equation (A1.4), to the minimum of \(Q_{AA}\). The smallest possible value of \(Q_{AA}\) is zero, and this can be realized for a body-centered lattice: consider a completely ordered AB alloy; here there are no AA pairs and \(Q_{AA}=0\). If B atoms are substituted for A atoms without redistribution, i.e., without creating AA pairs, then all values \(F_A<\dfrac{1}{2}\) can be obtained with \(Q_{AA}=0\). Hence, from (A1.4) and (A1.9), we have

\[ E=-vQ_{AB}=-2vQF_A=-NvzF_A . \tag{A1.11} \]

This equation is applicable for \(F_A<\dfrac{1}{2}\) for the simple cubic and body-centered cubic lattices. For the face-centered lattice the situation turns out to be more complicated, because for \(F_A>\dfrac{1}{4}\) it is impossible to avoid AA pairs. This leads to a kink at 25 atomic % A in Fig. 22.

A detailed construction of the curve is given in \(^{38\mathrm{C}}\).

Addendum 2. The First Bethe Approximation

Compatibility Condition

We assume that the interior region consists of one \(\alpha\)-site and the boundary of \(z\) \(\beta\)-sites, and that there are no two \(\beta\)-sites which would be nearest neighbors. Let the ordering energy for the external region be zero, so that the relative probability of finding an A atom proves, in comparison with the probability for a B atom, to be determined by the Boltzmann factor

\[ e^{-\frac{U}{kT}}=\varepsilon . \tag{A2.1} \]

We shall choose arbitrarily \(v_{AA}=v_{BB}=v\) and \(v_{AB}=0\); this is lawful for the reasons indicated in § 2. The ratio of the probability of forming a “wrong” pair (i.e., AA or BB) to the probability of a “right” one (i.e., AB)

is equal to

\[ e^{-\frac{v}{kT}}=x=e^{-\frac{4E_0}{RTx}}. \tag{A2.2} \]

Suppose that the central atom belongs to A, i.e., is in its “own” site. Then the relative probability \(r_n\) of finding \(n\) “illegal” A atoms on the boundary is

\[ r_n=\binom{z}{n}\varepsilon^n x^n . \tag{A2.3} \]

Suppose that the central atom belongs to species B, i.e., is in an “alien” site. Then an analogous argument gives for the relative probability of finding \(n\) “illegal” A atoms on the boundary the expression

\[ w_n=\binom{z}{n}\varepsilon^n x^{z-n}, \tag{A2.4} \]

because in this case each of the \(z-n\) “illegal” B atoms on the boundary gives an interaction energy with the “illegal” internal B atom.

Accordingly, the total relative probability that the internal atom will be “legal” is

\[ r_i=\sum_{n=0}^{z} r_n=(1+\varepsilon x)^z, \tag{A2.5} \]

and the probability that it will be “illegal” is

\[ w_i=\sum_{n=0}^{z} w_n=(\varepsilon+x)^z. \tag{A2.6} \]

Hence the normalized probability that the central atom (occupying an \(\alpha\)-site) will be in a legal site is

\[ r_\alpha=\frac{r_i}{r_i+w_i}. \tag{A2.7} \]

Let us next compute the relative probability that an atom on the boundary will be “illegal.” It is equal to the mean (i.e., the statistical-mechanical average) number of “illegal” atoms on the boundary divided by \(z\). The relative probability of finding \(n\) illegal atoms on the boundary is \(r_n+w_n\). Multiplication by \(n\), averaging, and normalization give

\[ x_\varphi=zw_\beta = \frac{\displaystyle \sum_{n=0}^{z} n(r_n+w_n)} {\displaystyle \sum_{n=0}^{z} (r_n+w_n)} = z\, \frac{\displaystyle \frac{\varepsilon x r_i}{1+\varepsilon x}+\frac{\varepsilon w_i}{\varepsilon+x}} {r_i+w_i}. \tag{A2.8} \]

Hence one can determine \(r_\beta=1-w_\beta\), the probability of finding a B atom in a \(\beta\)-site. Owing to the symmetry indicated in the text, we must have \(r_\alpha=r_\beta\); this compatibility equation can now be used to determine either the equivalent quantity \(\varepsilon\).

Instead of the equality \(r_\alpha=r_\beta\), we use the relation \(w_\alpha=w_\beta\), and multiply both sides by \(r_i+w_i\). Then

\[ w_i=\frac{\varepsilon x r_i}{1+\varepsilon x}+\frac{\varepsilon w_i}{\varepsilon+x}. \tag{A2.9} \]

Dividing by \(r_i\) and collecting the terms containing \(\dfrac{w_i}{r_i}\), we obtain

\[ \frac{w_i}{r_i}\left(1-\frac{\varepsilon}{\varepsilon+x}\right) = \frac{w_i}{r_i}\,\frac{x}{\varepsilon+x} = \frac{\varepsilon x}{1+\varepsilon x}. \tag{A2.10} \]

Substituting the values of \(w_i\) and \(r_i\), we find

\[ \left(\frac{\varepsilon+x}{1+\varepsilon x}\right)^{z-1} = \varepsilon = e^{-2\delta(z-1)}. \tag{A2.11} \]

The new variable

\[ \delta=\frac{u}{2kT(z-1)} \]

has been introduced for convenience in solving equation (A2.11) with respect to \(x\). We find

\[ x=\frac{\sinh (z-2)\delta}{\sinh (z\delta)} . \tag{A2.12} \]

Fig. 59 shows the graph of the right-hand side of this equation for \(z=6\), the simple cubic lattice. For \(\delta\) exceeding 0.5, the right-hand side of equation (A2.12) differs only slightly from the function \(e^{-2\delta}\), shown by the dashed curve. Hence, for low temperatures, i.e. small \(x\) and large \(\delta\), we find

\[ \ln x=-\frac{v}{kT}\simeq -2\delta = -\frac{u}{kT(z-1)}, \]

or \(u=(z-1)v\). This is precisely the ordering energy occurring at low temperatures per atom at the boundary and due to the interaction with its \((z-1)\) “lawful” external neighbors. It is easy to see that values of \(x\) exceeding \(x_c\)

Fig. 59. Relation between \(\delta\) and \(x\) for the first Bethe approximation

\[ x_c=1-\frac{2}{z}, \tag{A2.13} \]

do not occur on the curve. To each value \(x<x_c\) there correspond two values \(\delta\) of opposite signs. This corresponds to two ordered states and is in agreement with the equivalence of the \(\alpha\)- and \(\beta\)-sites. Expressing \(x\) and \(\varepsilon\) through \(\delta\), one can calculate the quantities \(r_\alpha=r_\beta=r\) and \(w_\alpha=w_\beta=w\), and

\[ S=2\left(r_\alpha-\frac{1}{2}\right) \]

goes over into

\[ S=\tanh z\delta . \tag{A2.14} \]

Negative values of \(S\), as well as negative values of \(\delta\), correspond to interchange of \(\alpha\)- and \(\beta\)-sites.

For \(x>x_c\) there are no solutions of this type, and one must put \(\varepsilon=1\), which leads to \(r_\alpha=r_\beta=w_\alpha=w_\beta=\dfrac12\). The critical temperature is determined from the relation

\[ \frac{\vartheta}{kT_c}=-\ln x_c=-\ln\left(1-\frac{2}{z}\right) \tag{A2.15} \]

and

\[ \frac{RT_c}{E_0}=-\frac{4}{z\ln\left(1-\dfrac{2}{z}\right)}. \tag{A2.16} \]

Calculation of \(\sigma\)

Let us first consider the ordering of neighbors. Both inside and on the boundary there are \(z\) pairs. For the “regular” central atom, the number of \(AB\) pairs is \(z-n\), while for the “irregular” central atom it is \(n\). Consequently, on the average, the fraction of pairs that are \(AB\) pairs [i.e. the quantity \(q\) by definition (2.9)] is equal to

\[ q=\frac{1}{z}\, \frac{\sum (z-n)r_n+\sum n w_n}{r_i+w_i}. \tag{A2.17} \]

Therefore, according to (2.11), the quantity \(\sigma\), or, better, \(\dfrac{E}{E_0}=1-\sigma\), turns out, after some algebraic calculations, to be

\[ \begin{aligned} 1-\sigma &=\frac{4}{z}\,\frac{\sum n r_n}{r_i+w_i} =\frac{4\varepsilon x}{1+\varepsilon x}\, \frac{1}{\dfrac{z}{1+e^{z\delta-1}}} \\ &=\frac{2\sinh (z-2)\delta}{\sinh [(2z-2)\delta]\cosh z\delta}. \end{aligned} \tag{A2.18} \]

Calculation of various quantities at the critical temperature \(T_c\)

By expansion in a series near the critical temperature, Bethe finds for the heat capacity, referred to one gram-atom, at a temperature immediately below \(T_c\),

\[ \frac{C}{R}=\frac{1}{R}\frac{dE}{dT} =-\frac{z}{4}(\ln x)^2\frac{d(1-\sigma)}{dx} = \]

\[ =\frac{z-2}{4} \left(\frac{z}{z-1}\right)^2 \left(\frac{3}{2}z-1\right) \left(\lg \frac{z-2}{z}\right)^2 \tag{A2.19} \]

Above the critical temperature \(\varepsilon=1\), and

\[ 1-\sigma=\frac{2x}{1+x}. \tag{A2.20} \]

The heat capacity immediately above \(T_c\) is

\[ (3z-2) \tag{A2.21} \]

times smaller than the heat capacity immediately below \(T_c\).

To reach \(T_c\), a \((1-\sigma)\)-th part of the energy \(E_0\) is required; as a function of \(x_c\) and \(z\) we obtain

\[ \frac{E(T_c+)}{E_0}=1-c_c=\frac{2x_c}{1+x_c}=\frac{z-2}{z-1}. \tag{A2.22} \]

The entropy at \(T_c\) can be found from the change of entropy between \(T_c\) and \(T=\infty\). The authors calculated this quantity using a simple expression for \(1-\sigma\) as a function of \(x\), and obtained the following result:

\[ \begin{aligned} S(\infty)-S(T_c) &=\int_{T_c}^{\infty}\frac{dE}{T} =\left.\frac{E}{T}\right|_{T_c}^{\infty} -\int_{T_c}^{\infty} E\,d\!\left(\frac{1}{T}\right) \\ &=-\frac{E_c}{T_c} -\int_{x_c}^{1} E_0(1-\sigma)\left(-\frac{Rz}{4E_0}\right)\frac{dx}{x} \\ &=\frac{z}{2}\left[-\ln\left(1-\frac{1}{z}\right) +\frac{z-2}{2z-2}\lg\left(1-\frac{2}{z}\right)\right]R . \end{aligned} \tag{A2.23} \]

With the aid of this expression the entropy values in Table 1 were obtained.

\[ z\to\infty \]

In the Bragg—Williams theory the ordering energy depends on the order at all sites of the lattice. Physically this is equivalent to the assumption that each \(\beta\)-site is the nearest neighbor of every \(\alpha\)-site and conversely. Thus, if Bethe’s method is correct, its results should approach the results of Bragg—Williams as \(z\) tends to \(\infty\). Bethe showed in his paper that this is indeed the case.

Appendix 3. Expression for the free energy as a function of Thiele’s semi-invariants

We shall derive here the relation, established by Kirkwood\({}^{38\mathrm{B}}\), between the free energy and the semi-invariants. Let the energy of each of \(W\) possible states be \(E_i\), \(i=1,2,\ldots,W\). Then, as is known, the free energy is determined from the relation

\[ e^{-\frac{F}{kT}}=\sum_i e^{-\frac{E_i}{kT}} . \tag{A3.1} \]

Consider the equation

\[ \sum_i e^{xE_i}=e^{\lambda_0+x\lambda_1+\frac{x^2\lambda_2}{2!}+\cdots}, \tag{A3.2} \]

where, for brevity, we have put

\[ x=-\frac{1}{kT}. \tag{A3.3} \]

If this equation is regarded as an identity with respect to \(x\), then it may be used to determine the “semi-invariants” \(\lambda_i\)

as a function of the quantities \(E_i\). Expanding the exponential function on the left-hand side in a series, we obtain

\[ \sum_{i=1}^{w} 1 + x \sum_{i=1}^{w} E_i + \frac{x^2}{2!}\sum E_i^2 + \cdots = e^{\lambda_0 + x\lambda_1 + \frac{x^2}{2!}\lambda_2 + \cdots}. \tag{A3.4} \]

We shall denote sums of different powers of \(E_i\) by

\[ \left. \begin{aligned} \sum_{0} &= \sum_{i=1}^{w} 1 = w,\\ &\cdots\cdots\cdots\\ \sum_{n} &= \sum_{i=1}^{w} E_i^n,\\ &\cdots\cdots\cdots \end{aligned} \right\} \tag{A3.5} \]

Introducing the notation

\[ C(x)=\sum_{0}+x\sum_{1}+\frac{x^2}{2!}\sum_{2}+\cdots \tag{A3.6} \]

and

\[ D(x)=\lambda_0+x\lambda_1+\frac{x^2}{2!}\lambda_2+\cdots, \tag{A3.7} \]

we may write

\[ C(x)=e^D \]

and

\[ \frac{dC(x)}{dx} = e^{D(x)}\frac{dD(x)}{dx} = C(x)\frac{dD(x)}{dx}. \tag{A3.8} \]

Let us represent the right- and left-hand sides of the last equation in the form of series in powers of \(x\). Equating coefficients, we find

\[ \left. \begin{aligned} \sum_{1} &= \sum_{0}\lambda_1,\\ \sum_{2} &= \sum_{0}\lambda_2+\sum_{1}\lambda_1,\\ &\cdots\cdots\cdots\\ \sum_{n} &= \sum_{m=1}^{n} {n-1 \choose m-1}\sum_{n-m}\lambda_m \end{aligned} \right\} \tag{A3.9} \]

The value of \(\lambda_0\) is easily obtained by putting \(x=0\): \(\lambda_0=\ln\sum_0=\ln W\). Denoting the mean values over the allowed arrangements by

\[ [E^n]_{\mathrm{av}}=\frac{\sum_{n}}{\sum_{0}} \tag{A3.10} \]

and the central moments about the mean by

\[ \Delta_n=\left[(E-E_{\mathrm{cp}})^n\right]_{\mathrm{cp}} =\frac{1}{\sum\limits_0}\sum_{m=0}^n {n\choose m}\sum_m(-E_{\mathrm{cp}})^{\,n-m} \tag{A3.11} \]

we find several of the first \(\lambda\)’s:

\[ \left. \begin{aligned} \lambda_0&=\ln\sum_0=\ln W,\\[4pt] \lambda_1&=\frac{\sum_1}{\sum_0}=E_{\mathrm{cp}},\\[4pt] \lambda_2&=[E^2]_{\mathrm{cp}}-E_{\mathrm{cp}}^2=\Delta_2,\\ \lambda_3&=[E^3]_{\mathrm{cp}}-[3E^2]_{\mathrm{cp}}\cdot E_{\mathrm{cp}}+2E_{\mathrm{cp}}^3=\Delta_3,\\ \lambda_4&=\Delta_4-3\Delta_2^2. \end{aligned} \right\} \tag{A3.12} \]

Replacing \(x\) by \(-1/kT\), we find

\[ -\frac{F}{kT}=D\left(-\frac{1}{kT}\right) =\lambda_0-\frac{\lambda_1}{kT} +\frac{\lambda_2}{2!(kT)^2}+\cdots \tag{A3.13} \]

and multiplying both sides of the equation by \(-1/kT\) and denoting \(k\ln W\) by \(\Phi\), we obtain

\[ F=-T\Phi+E_{\mathrm{cp}}-\frac{\Delta_2}{2!kT} +\frac{\Delta_3}{3!(kT)^2} -\frac{\Delta_4-3\Delta_2^2}{4!(kT)^3}+\cdots, \tag{A3.14} \]

i.e., equation (4·3) of the text.

Appendix 4. Application of the Kirkwood Method

Calculation of the Second Moment

In calculating moments by the Kirkwood method it is convenient to introduce certain new variables associated with pairs of nearest neighboring sites. There are \(zN/2\) such pairs, and for each of them we introduce a variable \(p_i\) \((i=1,2,\ldots,Q)\), defined by the conditions: \(p_i=1\) if the \(i\)-th pair of sites is occupied by atoms \(A\), and \(p=0\) in all other cases. For any arrangement of atoms the total number of \(AA\) pairs is equal to

\[ Q_{AA}=\sum_{i=1}^{Q}p_i. \tag{A4.1} \]

Let us calculate the quantity

\[ \Delta_2=\left[(E-E_{\mathrm{cp}})^2\right]_{\mathrm{cp}}. \]

Averaging over all arrangements of atoms corresponding to ordering \(S\). In the notation of Appendix 1 we find

\[ \Delta_2=4\omega^2\left[\left(Q_{AA}-[Q_{AA}]_{\mathrm{cp}}\right)^2\right]_{\mathrm{cp}} \equiv 4\omega^2\Delta. \tag{A4.2} \]

Since here we are dealing with lattices such as the simple cubic or spatially centered one, we can divide all nodes into an \(\alpha\)-lattice and a \(\beta\)-lattice in such a way that each \(\beta\) corresponds to a pair of nearest neighboring nodes—one in each \(p\)-cell—and the two nodes never turn out to be in the same lattice. Restricting ourselves further to an alloy of composition \(AB\), we can write

\[ r_\alpha=r_\beta=r=\frac{1+S}{2}, \qquad w_\alpha=w_\beta=w=\frac{1-S}{2}. \tag{A4.3} \]

Let us now consider the pair associated with \(p_i\). The atom of this pair that is in the \(\alpha\)-node proves to be an \(A\) atom in the \(r'_\alpha\)-th part of all arrangements; the atom in the \(\beta\)-node proves to be an \(A\) atom in the \(w'_\beta\)-th part of the arrangements. Since the arrangements in the \(\alpha\)- and \(\beta\)-lattices are independent (we emphasize that here we are dealing with a priori probabilities), the pair \(AA\) will be obtained a number of times constituting the \(r_\alpha w_\beta = rw\)-th part of all arrangements. Consequently, \(p_i\) is equal to unity for the \(rw\)-th fraction of the arrangements and to zero in the remaining \((1-rw)\) cases. Therefore the mean value of \(p_i\) is

\[ [p_i]_{\mathrm{cp}}=rw\cdot 1+(1-rw)\cdot 0=rw=\frac{1-S^2}{4}. \tag{A4.4} \]

From equation (A4.2) we find

\[ \Delta=\sum_{i=1}^{Q}\sum_{j=1}^{Q}\left((p_i p_j)_{\mathrm{cp}}-[p_i]_{\mathrm{cp}}\cdot[p_j]_{\mathrm{cp}}\right)= \]

\[ =\sum_{i=1}^{Q}\sum_{j=1}^{Q}\left([p_i p_j]_{\mathrm{cp}}-r^2w^2\right). \tag{A4.5} \]

In the sum of the terms \([p_i p_j]_{\mathrm{cp}}\) four different cases occur:

\[ \begin{aligned} &(1)\quad i=j,\ \text{both nodes } p_i \text{ are identical with the two nodes } p_i,\\ &(2)\quad p_i \text{ and } p_j \text{ have a common } \alpha\text{-node},\\ &(3)\quad p_i \text{ and } p_j \text{ have a common } \beta\text{-node},\\ &(4)\quad p_i \text{ and } p_j \text{ have no common nodes}. \end{aligned} \]

There enter into \(\Delta\) \(Q\) terms of type (1); for each of them \(p_i^2=1\) for the \(rw\)-th part of all cases; therefore, \([p_i^2]_{\mathrm{cp}}=rw\) and

\[ \Delta(1)=Q(rw-r^2w^2). \tag{A4.6} \]

For case (2) there are \(\dfrac{N}{2}\) \(\alpha\)-nodes; for each of them one can choose a neighboring \(\beta\)-node in \(z(z-1)\) ways; therefore for this case the number of terms is \((z-1)Q\). The fraction of all arrangements giving \(p_i p_j=1\) is \(r_\alpha w_\beta w_\beta=rw^2\) (if terms of order \(\dfrac{1}{N}\) are neglected); consequently,

\[ \Delta(2)=(z-1)Q(rw^2-r^2w^2). \tag{A4.7} \]

An analogous argument gives for case (3)

\[ \Delta(3)=(z-1)Q(r^2w-r^2w^2). \tag{A4.8} \]

All the remaining \(Q^2\) terms in \(\Delta\) belong to the 4th type; however, although they are more numerous, each of them individually is sufficiently small, so that \(\Delta(4)\) is likewise of the same order of magnitude as \(\Delta(i)\) \((i=1,2,3)\). The quantity \(p_i p_j\) is equal to unity only for arrangements in which both atoms A are in \(\alpha\)- and \(\beta\)-sites. In the \(\dfrac{N}{2}\) \(\alpha\)-sites there are \(\dfrac{rN}{2}\) atoms A. Two of them will be in the indicated \(\alpha\)-sites in a fraction of the cases equal to

\[ \frac{\dfrac{1}{2}rN\left(\dfrac{1}{2}rN-1\right)} {\dfrac{1}{2}N\left(\dfrac{1}{2}N-1\right)} \tag{A4.9} \]

of all arrangements. Combining this expression with the analogous expression for the \(\beta\)-sites, we obtain

\[ [p_i p_j]_{\mathrm{av}}-r^2w^2 = \frac{r^2w^2\left(1-\dfrac{2}{rN}\right)\left(1-\dfrac{2}{wN}\right)} {\left(1-\dfrac{2}{N}\right)^2} -r^2w^2 = \]

\[ = r^2w^2\left(\frac{4rw-2}{Nrw}\right) + O\left(\frac{1}{N^2}\right). \tag{A4.10} \]

Multiplying by \(Q^2\) and neglecting in the final result quantities of order smaller than \(N\), we obtain

\[ \Delta(4)=zQ(2r^2w^2-rw). \tag{A4.11} \]

Consequently, the quantity \(\Delta\) is equal to

\[ \Delta=\Delta(1)+\Delta(2)+\Delta(3)+\Delta(4) = Qr^2w^2 = Q\,\frac{(1-S^2)^2}{16}. \tag{A4.12} \]

Calculation of the free energy and other quantities

Substituting this result into the definition of \(\Delta_2\),

\[ \Delta_2=\frac{1}{4}v^2Q(1-S^2)^3 = \frac{1}{8}Nzv^2(1-S^2)^3 \tag{A4.13} \]

and combining it with the quantity \(E_{\mathrm{av}}(S)\), calculated in § 3,

\[ E_{\mathrm{av}}(S)=E_0(1-S^2) = \frac{1}{4}Nzv(1-S^2), \tag{A4.14} \]

we obtain

\[ F=-T\Phi(S)+E_{\mathrm{av}}-\frac{\Delta_2}{2kT}+\cdots = \]

\[ = E_0(1-S^2) - TR\left[ \ln 2 -\frac{1}{2}(1+S)\ln(1+S) -\frac{1}{2}(1-S)\ln(1-S) \right] - \frac{E_0^2(1-S^2)^3}{2kT} +\cdots \tag{A4.15} \]

Kirkwood introduces the variable

\[ \alpha=\frac{zv}{2kT}=\frac{2E_0}{RT}, \tag{A4.16} \]

Then

\[ -\frac{F}{NkT}=\left[\ln 2-\frac{1}{2}(1+S)\ln(1+S)-\frac{1}{2}(1-S)\ln(1-S)\right]- \]

\[ -\frac{\alpha}{2}(1-S^2)+\frac{\alpha^2}{4z}(1-S^2)^2+\ldots \tag{A4.17} \]

The condition for a minimum of \(F\) at the given temperature,

\[ \left(\frac{\partial F}{\partial S}\right)_{\alpha}=0, \]

gives

\[ S=\operatorname{tgh}\gamma S, \tag{A4.18} \]

\[ \gamma=\alpha-\frac{\alpha^2}{z}(1-S^2). \tag{A4.19} \]

The solution of these equations leads to a value of \(S\) different from zero only for \(\alpha\) exceeding a certain critical value \(\alpha_c\), corresponding to \(S=0\) and \(\gamma=1\) in equation (A4.19),

\[ \alpha_c=\frac{1}{2}z\left\{1-(1-4z)^{\frac{1}{2}}\right\}. \tag{A4.20} \]

The values of the critical temperature calculated from this,

\[ T_c=\frac{zv}{2k\alpha_c}, \tag{A4.21} \]

are given in Table 1. Kirkwood also gives expressions for the energy, entropy, and heat capacity corresponding to the equilibrium state, which is obtained from the condition of minimum free energy:

\[ \frac{E}{NkT}=\alpha(1-S^2)-\frac{\alpha^2}{z}(1-S^2)^2, \]

\[ \frac{\Phi}{Nk}=\ln 2-\frac{1}{2}[(1+S)\ln(1+S)+ \]

\[ +(1-S)\ln(1-S)]-\frac{\alpha^2}{4z}(1-S^2)^2, \tag{A4.22} \]

\[ \frac{C}{Nk}=\alpha^2\left( \frac{S^2\left[1-\frac{2\alpha}{z}(1-S^2)\right]^2} {\cosh^2\gamma S-\gamma-\frac{2\alpha^2S^2}{z}} +\frac{(1-S^2)^2}{2z} \right). \tag{A4.23} \]

The heat capacity undergoes a discontinuity at \(\alpha=\alpha_c\). Its values on the two sides of the discontinuity are respectively equal to

\[ \frac{C}{Nk}=\alpha^2\left( \frac{\left(1+\frac{2\alpha}{z}\right)^2} {\frac{2}{3}-\frac{2\alpha}{z}} +\frac{1}{2z} \right) \tag{A4.24} \]

and

\[ \frac{C}{Nk}=\frac{\alpha^2}{2z}. \tag{A4.25} \]

Limiting expression for \(T \to \infty\)

For \(T \to \infty\) the Kirkwood series converge rapidly, and its approximate expressions become exact. The limiting expression for the energy is

\[ \frac{E}{E_0}=1-\frac{2E_0}{kT}. \tag{A4.26} \]

The reader can readily verify for himself that Bethe’s theory also leads to this expression.

Appendix 5. Calculation from the free energy of the dependence of energy on entropy

To obtain the relation between energy and entropy from the expansion for the free energy given in Appendix 3, it proves convenient to take the energy \(E_{\mathrm{cp}}\) as the origin of the energy scale. Introduce the quantity

\[ U=E-E_{\mathrm{cp}} \tag{A5.1} \]

and

\[ F'=F-E_{\mathrm{cp}}. \tag{A5.2} \]

Represent the entropy in the form of a series in powers of \(U\)

\[ \Phi(U)=\Phi_0+\Phi_1 U+\frac{\Phi_2}{2!}U^2+\cdots \tag{A5.3} \]

In order to obtain the expansion of \(F'\) in powers of \(\frac{1}{T}\), put

\[ \frac{1}{T}=G, \tag{A5.4} \]

\[ \psi(G)=\frac{F-E_{\mathrm{cp}}}{T}=\psi_0+\psi_1G+\frac{\psi_2}{2!}G^2+\cdots \tag{A5.5} \]

The coefficients of this series are very simply related to the quantities \(\lambda_i\):

\[ \left. \begin{aligned} \psi_0&=-k\lambda_0=-k\ln W,\\ \psi_1&=\lambda_1-E_{\mathrm{cp}}=0,\\ \psi_2&=-\frac{\lambda_2}{k}=-\frac{\Delta_2}{k},\\ \psi_3&=+\frac{\lambda_3}{k^2}=+\frac{\Delta_3}{k^2},\\ \psi_4&=-\frac{\lambda_4}{k^3}=-\frac{\Delta_3+3\Delta_2^2}{k^3}. \end{aligned} \right\} \tag{A5.6} \]

The quantities considered are connected by the relation

\[ F'=U-T\Phi, \tag{A5.7} \]

whence we find

\[ \psi(G)=\Phi(U)+UG. \tag{A5.8} \]

There is also another relation

\[ T=\frac{dU}{d\Phi}\quad \text{or}\quad G=\frac{d\Phi(U)}{dU}. \tag{A5.9} \]

The third relation follows from the equation

\[ - T^2 \frac{d}{dt}\left(\frac{F'}{T}\right)=\dot U \quad \text{or} \quad U=\frac{d\psi(G)}{dG}. \tag{A5.10} \]

We see that these three equations are symmetrical with respect to \(\psi(G)\) and \(G\), on the one hand, and \(\Phi(U)\) and \(U\), on the other.

In order to solve these equations, substitute (A5.10) into (A5.8)

\[ \Phi \frac{d\psi(G)}{dG} = G\frac{d\psi(G)}{dG}-\psi(G). \tag{A5.11} \]

In the expansion of both sides in power series in \(G\), one may equate the coefficients of equal powers:

\[ \begin{aligned} \Phi_0&=-\psi_0,\\ \Phi_1\psi_2&=0 \quad \text{or} \quad \Phi_1=0,\\ \frac{\Phi_2}{2!}\psi_2^2&=\frac{1}{2}\psi_2. \end{aligned} \tag{A5.12} \]

These equations may be solved with respect to \(\Phi_i\):

\[ \begin{aligned} \Phi_0&=-\psi_0=k\ln W;\\ \Phi_1&=\psi_1=0;\\ \Phi_2&=\frac{1}{\psi_2}; \qquad \psi_2=\frac{1}{\Phi_2}\\ \Phi_3&=-\frac{\psi_3}{\psi_2^3}; \qquad \psi_3=-\frac{\Phi_3}{\Phi_2^3}\\ \Phi_4&=\frac{3\psi_3^2-\psi_2\psi_4}{\psi_2^5}; \qquad \psi_4=\frac{3\Phi_3^2-\Phi_2\Phi_4}{\Phi_2^5}. \end{aligned} \tag{A5.13} \]

Substituting these values of \(\psi_i\) into \(\Delta_n\) and replacing \(U\) by \(E-E_{\mathrm{av}}\) and \(k\ln W\) by \(\Phi(S)\), we obtain the expansion (4.4) indicated in the text.

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1. Continuation; see Uspekhi fizicheskikh nauk, XX, 344, 1938. ↩