THEORY OF ATOMIC DIFFUSION IN ALLOYS

M. A. Krivoglaz, A. A. Smirnov

§ 1. INTRODUCTION

The diffusion of atoms in alloys of metals, as is well known, plays an essential role in many practically important processes, for example in heat treatment.

The study of diffusion is also of great importance for the theory of the solid state, since it makes it possible to elucidate the mechanism of atomic mobility, to obtain information about defects of the crystal lattice, about the energies of interaction between atoms, etc.

To explain diffusion occurring in solids, three basic mechanisms of atomic displacement in the crystal lattice have been proposed. The first of these reduces to the exchange of atoms with one another. However, this mechanism could not explain the phenomenon of electrical conductivity in ionic crystals. In connection with this, a mechanism of atomic displacement through interstices was proposed, as well as a mechanism carried out by the replacement of vacant lattice sites (holes). These mechanisms were made by Ya. I. Frenkel[^1] the basis of the theory of diffusion developed by him. On the basis of the three simplest types of atomic displacement indicated above, more complex models of diffusion in solids were constructed. These include, for example, the simultaneous exchange of positions by a group of three or more atoms moving along a certain closed contour, the displacement of a group of atoms shifted around an extra atom relative to their equilibrium positions along the direction in which the nearest lattice sites are located, etc. In what follows we shall confine ourselves to considering only the simplest mechanisms of diffusion.

Of these, the exchange of positions by neighboring atoms is improbable, since it requires a large activation energy. Therefore the indicated mechanism plays a small role in diffusion in most crystals. This conclusion has been confirmed for some metal alloys by sufficiently convincing experiments. In the work of [[unclear: word at page break]] brass[^2]

a rectangular rod was coated with a layer of copper. Thin molybdenum wires were placed between the brass and the copper. After a 56-day anneal at a temperature of \(785^\circ\mathrm{C}\), during which diffusion took place across the brass—copper boundary, the distance between the wires lying on opposite faces of the rod decreased. An effect of this kind was studied in a number of works\(^{2a}\) for various alloys (Cu—Ni, Cu—Au, Ag—Au, Ag—Pd, Ni—Co, Ni—Au, Fe—Ni, Sn—Cu, Al—Cu). The result obtained rules out the mechanism of exchange of places (of two or several atoms), since under this mechanism the number of atoms inside the boundary marked by the wires should remain unchanged, while the decrease in the lattice constant due to the change in the alloy composition gives an effect roughly 10 times smaller than that observed.

On the other hand, the indicated experiments can be readily explained both with the aid of the vacancy mechanism of diffusion and with the mechanism of diffusion by interstitials. In both cases the diffusion of zinc from brass can proceed faster than that of copper into brass, while the holes remaining in the brass can partially disappear, for example as a result of their coalescence and the closing of the voids formed during plastic deformation. In this way the experiments mentioned were explained. The unsuitability of the exchange-of-places model for explaining the phenomenon of diffusion in alloys also follows from the experiments of B. Ya. Pines\(^{3}\).

However, for substitutional alloys there is no direct experimental evidence in favor of either of these two mechanisms. One can only say with certainty that diffusion of interstitial atoms is carried out by their motion through interstices. In substitutional alloys (and in pure metals), the main role is apparently played by vacancy diffusion.

In studying diffusion in alloys one should distinguish the case in which there is no concentration gradient of any substance in the alloy (self-diffusion) from the case in which such a gradient is present (chemical diffusion).

The study of self-diffusion by the tracer-atom method, begun in the works of G. Hevesy\(^{4}\), A. M. Zagrubskii\(^{5}\), and others, has by now yielded extensive experimental material on diffusion in pure metals and alloys. It has become clear that the coefficient of self-diffusion in alloys differs from the coefficient of chemical diffusion. This difference, from the phenomenological point of view, was explained in works\(^{6}\).

As follows from the thermodynamic theory of diffusion (see, for example, the review\(^{7}\)), the flux of diffusing atoms is proportional to the gradient of the chemical potential, which in some cases may be directed in the direction opposite to

gradient. In this case we are dealing with so-called uphill diffusion, in which atoms of the given kind move predominantly into regions with a higher concentration of them. An interesting case of uphill diffusion was considered by S. T. Konobeevskii$^8$, who took into account the influence of nonuniform elastic deformation on diffusion in alloys. Later B. Ya. Lyubov and N. S. Fastov$^9$ investigated this problem within the framework of a phenomenological theory.

A special case of diffusion is heterophase (reaction) diffusion, accompanied by the formation of a compound through which the diffusing atoms subsequently pass. The case of mutual diffusion of metals through a layer of an intermetallic compound was considered theoretically by Ya. I. Frenkel and M. I. Sergeev$^{10}$, who used for this purpose a place-exchange model.

Among the numerous experimental works on the study of heterophase diffusion, we shall mention the work of V. Z. Bugakov$^{11}$, who investigated this phenomenon in binary metal alloys, and that of V. I. Arkharov$^{12}$, who studied heterophase diffusion through an oxide layer. In these works one can also find a detailed bibliography on the questions indicated.

The experimental study of diffusion in pure metals and the theoretical consideration of this question (see, for example,$^1$) led to the well-known formula for the dependence of the diffusion coefficient $D$ on the absolute temperature $T$:

\[ D = D_0 e^{-\frac{Q}{kT}}, \tag{1,1} \]

where the constants $D_0$ and $Q$ are usually called, respectively, the pre-exponential factor and the activation energy.

Let us note that in alloys there is no reason to expect, for the dependence of $D$ on $T$, a formula of the type (1,1). Indeed, in this case different diffusing atoms of the given kind are in different conditions and must overcome potential barriers of different heights depending on the character of their environment by neighboring atoms. Different energies are also required for the formation of holes at different sites. Therefore the diffusion process cannot be characterized by a single activation energy. Nevertheless, experimentally, a rectilinear course of the dependence of $\lg D$ on $\dfrac{1}{T}$ is usually obtained with a sufficient degree of accuracy. The reasons for this will be clarified below.

The experimental study of the dependence of the diffusion coefficient on the composition of the alloy (see, for example,$^{13}$) has shown that for different alloys this dependence may have different forms. Cases are often encountered in which, at one edge of the concentration

diagrams the change of the diffusion coefficient with composition is relatively small, while in the other it is considerably stronger.

It is interesting to note that in a number of cases a small amount of an impurity can substantially change the diffusion coefficient. Such a phenomenon was observed, for example, in the experiments of S. D. Gertsriken and I. Ya. Dekhtyar \(^{14}\). In the experiments of P. L. Gruzin, Yu. V. Kornev, and G. V. Kurdyumov \(^{15}\) it was found that the addition of a small amount (up to 4.5 at. %) of carbon to \(\gamma\)-iron greatly reduces the activation energy of self-diffusion of iron and reduces the pre-exponential factor by several orders of magnitude.

The question of the temperature and concentration dependence of the diffusion coefficient in alloys from the microscopic point of view has been considered only recently, and its exposition is the aim of the present article*). In particular, effects caused by the influence of ordering on diffusion are considered in detail.

These effects should usually appear at not very high temperatures, at which the diffusion coefficient can be measured by modern methods, for example those based on the study of the damping of vibrations of an alloy specimen \(^{15б}\).

It should be noted that the phenomenon of diffusion in alloys is very complex and cannot be treated rigorously within the framework of the contemporary theory of the solid state. Therefore, to investigate this phenomenon one has to use a highly simplified model, which nevertheless makes it possible to clarify the main qualitative features of the temperature and concentration dependence of the diffusion coefficient. In what follows, a number of assumptions are adopted which are usually used in the statistical theory of alloys.

It is assumed that the energy of a crystal can be represented as the sum of the interaction energies of pairs of nearest-neighbor atoms, and the dependence on composition and degree of order of other kinds of energy (the energy of conduction electrons, the energy of vibrations of the crystal lattice, etc.) is not taken into account. It is also assumed that the interaction energies of the diffusing atom with neighboring atoms do not depend on the composition of the alloy. These energies (or certain combinations of them) are parameters of the theory. Geometrical distortions of the crystal lattice are not explicitly taken into account.

*) In the work of G. P. Il’kevich \(^{15а}\) an attempt was made at a theoretical study of the concentration dependence of the diffusion coefficient in disordered alloys. However, the author based his calculations on a mechanism of exchange of positions of neighboring atoms, and also used the manifestly invalid assumption that the probability of exchange of positions is independent of the composition of the alloy.

I. THEORY OF DIFFUSION OF ATOMS INTRODUCED INTO THE INTERSTICES OF A CRYSTAL LATTICE

§ 2. DIFFUSION OF INTRODUCED ATOMS IN ALLOYS WITH A BODY-CENTERED CUBIC LATTICE16, 17, 18

a) Determination of the diffusion coefficient in β-brass type alloys without taking correlations into account

Of the two most probable diffusion mechanisms indicated above, which may occur in alloys, we shall first investigate the simpler mechanism of diffusion of atoms of a third element through the interstices of the crystal lattice of a binary alloy.

Consider a substitutional alloy of metals \(A\) and \(B\), having a body-centered cubic lattice of the β-brass type. This alloy may be in both an ordered and a disordered state. Let us call the lattice sites in the ordered state that are proper for atoms \(A\) sites of the 1st kind, and the sites proper for atoms \(B\) sites of the 2nd kind. Let the sites of the 1st kind be located at the vertices of the cubic cells, and the sites of the 2nd kind at their centers. Into the interstices of the \(A—B\) alloy lattice atoms of some third element \(C\) may be introduced. We shall assume that atoms \(C\) have positions of stable equilibrium at the centers of faces and at the midpoints of edges of the cubic elementary cells.* We shall call the former interstices of type \(O_1\) and the latter \(O_2\) (Fig. 1). Interstices \(O_1\) have four neighboring sites of the 1st kind at a distance

\[ \frac{a_0}{\sqrt{2}} \]

and two neighboring sites of the 2nd kind at a distance

\[ \frac{a_0}{2} \]

(\(a_0\) is the length of the edge of the cubic cell). Interstices \(O_2\) have four neighboring sites of the 2nd kind

Fig. 1.

* For example, for a solution of carbon in \(\alpha\)-iron this assumption has been confirmed by X-ray investigations.19

at a distance \(\dfrac{a_0}{\sqrt{2}}\) and two neighboring sites of the 1st kind at a distance \(\dfrac{a_0}{2}\). Here one takes into account not only the two nearest neighboring atoms, which would give results that are too crude, but also the next four atoms in order of distance.

Let us denote the positions of the interstitial atoms \(C\), corresponding to the vertices of the potential barriers for transition from one interstitial site to a neighboring one, by the letter \(P\). Taking into consideration that substitutional alloys consist of atoms \(A\) and \(B\), which do not differ greatly in atomic radii, it may be assumed with sufficient accuracy that the position \(P\) lies midway along the straight line joining the neighboring interstitial sites \(O_1\) and \(O_2\). Then the positions \(P\) have four neighboring sites at a distance \(\dfrac{\sqrt{5}}{4}a_0\). Two of them will be sites of the first kind and two of the second kind.

Let us introduce the following notation for the energies \(v_{CA}\) and \(v_{CB}\) of interaction of the pairs of atoms \(CA\) and \(CB\) at different distances:

\[ \begin{aligned} v_{CA}\left(\frac{a_0}{\sqrt{2}}\right)&=-A; & v_{CA}\left(\frac{a_0}{2}\right)&=-a; & v_{CA}\left(\frac{\sqrt{5}}{4}a_0\right)&=-\alpha,\\ v_{CB}\left(\frac{a_0}{\sqrt{2}}\right)&=-B; & v_{CB}\left(\frac{a_0}{2}\right)&=-b; & v_{CB}\left(\frac{\sqrt{5}}{4}a_0\right)&=-\beta. \end{aligned} \tag{2,1} \]

Let us choose the direction of the \(x\)-axis along an edge of the cubic cell and suppose that in this direction a concentration gradient of the interstitial atoms \(C\) is established. Such a choice of the direction of the gradient does not limit the generality of the reasoning, since diffusion is isotropic in crystals of cubic structure. Consider two neighboring atomic planes \(I\) and \(II\), perpendicular to the \(x\)-axis. Let plane \(I\) pass through the centers of the cubes, and plane \(II\) through their vertices. Denote by \(n\) the number of atoms \(C\), referred to \(1\ \mathrm{cm}^3\) of alloy. Then \(n=n_1+n_2\), where \(n_1\) is the concentration of atoms \(C\) in the interstitial sites \(O_1\), and \(n_2\) in \(O_2\). It is not difficult to verify that on \(1\ \mathrm{cm}^2\) of plane \(I\) there are \(\dfrac{2}{3}a_0 n_1(x)\) atoms \(C\) in interstitial sites of type \(O_1\), and on \(1\ \mathrm{cm}^2\) of plane \(II\) there are \(\dfrac{2}{3}a_0 n_2\left(x+\dfrac{a_0}{2}\right)\) atoms \(C\) in interstitial sites of type \(O_2\).

As was indicated in the introduction, the diffusing atoms \(C\), which occupy different interstitial sites \(O_1\) (and also \(O_2\)), are under different energy conditions, since they are surrounded by different numbers of atoms \(A\) and \(B\) at the six sites neighboring the interstitial site. In what follows, all possible configurations of atoms \(A\) and \(B\) of this type will be explicitly taken into account. Let us divide all \(n_1\) atoms \(C\) into groups, to each of which we assign atoms \(C\) with the same configu-

of atoms \(A\) and \(B\) at the six neighboring sites around the interstice \(O_1\). The number of atoms \(C\) in such groups will be denoted by \(n_{1m}\) (\(m\) is the number of the configuration). We shall confine ourselves to the usual case in which the mobility of atoms \(C\) in the alloy is considerably greater than the mobilities of atoms \(A\) and \(B\). Then, when the state of order of the alloy \(A — B\) changes, the atoms \(C\) have time to reach an equilibrium state. If we restrict ourselves to the case where the number of interstitial atoms is small compared with the number of interstices of the crystal, then the numbers \(n_{1m}\) can be determined by means of the Boltzmann distribution as follows:

\[ n_{1m}=\lambda W_{O_1}^{m}\exp\frac{U_{O}^{m}}{kT}. \tag{2,2} \]

Here \(\lambda\) is a normalization factor, \(W_{O_1}^{m}\) is the probability of realization of the \(m\)-th configuration around the interstice \(O_1\), and \(U_{O}^{m}\) is the potential energy of the atom \(C\) in this interstice, taken with the opposite sign (Fig. 2). Similarly, the numbers \(n_{2m}\) of atoms \(C\) surrounded by neighboring atoms with configurations \(m\) around interstices \(O_2\), are equal to

Fig. 2.

\[ n_{2m}=\lambda W_{O_2}^{m}\exp\frac{U_{O}^{m}}{kT}, \tag{2,3} \]

where \(W_{O_2}^{m}\) is the probability of the \(m\)-th configuration of neighbors around \(O_2\).

For transition to a neighboring interstice the atom \(C\) must overcome a potential barrier of height \(U_{O}^{m}-U_{P}^{m}\), where \(U_{P}^{m}\) is the potential energy of the atom \(C\) at position \(P\), taken with the opposite sign. In this case, for the given structure, the neighbors of the point \(P\) are simultaneously neighbors of \(O_1\), so that both energies \(U_{O}^{m}\) and \(U_{P}^{m}\) are determined uniquely by specifying the configuration \(m\). The probability \(w^{m}\) of transition of an atom \(C\) from the given interstice with the \(m\)-th configuration of the surrounding-

...of its atoms into a neighboring one is determined by the known formula\(^1\)

\[ w^m=\frac{1}{\tau_0}\exp\left[-\frac{U_0^m-U_P^m}{kT}\right], \tag{2.4} \]

where \(\tau_0\) is a constant having the dimension of time and equal, in order of magnitude, according to\(^1\), to \(10^{-13}\) sec. We shall approximately regard the value \(\tau_0\) as the same for all configurations\(^*\).

The number of atoms \(C\) having the \(m\)-th configuration of neighbors and passing per unit time from \(1\ \mathrm{cm}^2\) of plane \(I\) into plane \(II\) can then be written in the following form (transitions from sites \(O_2\), located in plane \(I\), are obviously impossible):

\[ S^m_{I\to II}=\frac{2}{3}a_0 n_{1m}w^m =\frac{2}{3}\frac{a_0}{\tau_0}\lambda(x)W^m_{O_1}\exp\frac{U_P^m}{kT}. \tag{2.5} \]

The total number of atoms \(C\) passing per unit time from \(1\ \mathrm{cm}^2\) of plane \(I\) into plane \(II\) will be equal to

\[ \begin{aligned} S_{I\to II}&=\frac{2}{3}\frac{a_0}{\tau_0}\lambda(x)R,\\ \text{where}\qquad R&=\sum_m W^m_{O_1}\exp\frac{U_P^m}{kT}, \end{aligned} \tag{2.6} \]

and the summation is carried out over all 64 configurations \(m\). Analogously, an expression may be found for the reverse flux of atoms \(C\) from plane \(II\) to plane \(I\):

\[ S_{II\to I}=\frac{2}{3}\frac{a_0}{\tau_0}\lambda\left(x+\frac{a_0}{2}\right)R. \tag{2.7} \]

The normalization factor \(\lambda\) is determined from the condition

\[ \sum_m n_{1m}+\sum_m n_{2m}=n. \]

With the aid of formulas (2.2) and (2.3) we obtain:

\[ \begin{aligned} \lambda(x)&=\frac{n(x)}{\sigma},\\ \text{where}\qquad \sigma&=\sum_m (W^m_{O_1}+W^m_{O_2})\exp\frac{U_0^m}{kT}. \end{aligned} \tag{2.8} \]

\(^*\) Strictly speaking, this assumption means that we restrict ourselves to cases where \(\tau_{0A}\) for the diffusion of atoms \(C\) in the pure metal \(A\) differs little from \(\tau_{0B}\) for diffusion in the pure metal \(B\). However, there is reason to suppose that for substitutional alloys of the \(A-B\) type, which are formed by atoms \(A\) and \(B\) with close atomic radii and force fields of the same type, \(\tau_0\) will not depend strongly on the type of configuration \(m\).

From formulas (2.6), (2.7), and (2.8) it follows that the resultant diffusion flux of atoms \(C\) is equal to

\[ S=S_{I\to II}-S_{II\to I}=-\frac{1}{3}\frac{a_0^2}{\tau_0}\frac{R}{\sigma}\frac{dn}{dx}. \tag{2.9} \]

Comparing expression (2.9) with the equality

\[ S=-D\,\operatorname{grad} n, \tag{2.10} \]

which in the present case is the definition of the diffusion coefficient, and introducing the notation

\[ D_0=\frac{1}{3}\frac{a_0^2}{\tau_0}, \tag{2.11} \]

we obtain the following expression for the diffusion coefficient of atoms \(C\):

\[ D=D_0\frac{R}{\sigma} = \frac{ \sum\limits_m W_{O_1}^{m}\exp\!\left(-\frac{U_p^{m}}{kT}\right) }{ \sum\limits_m \left(W_{O_1}^{m}+W_{O_2}^{m}\right)\exp\!\left(-\frac{U_O^{m}}{kT}\right) }. \tag{2.12} \]

To determine the dependence of the diffusion coefficient on the composition of the alloy, temperature, and degree of order, it is necessary to obtain explicit expressions for the probabilities \(W_{O_1}^{m}\) and \(W_{O_2}^{m}\). If correlations between the occupation of sites in the alloy are not taken into account, then the quantities \(W_{O_1}^{m}\) and \(W_{O_2}^{m}\) are expressed as products of six a priori probabilities of occupation, by the given atoms, of lattice sites neighboring the interstice. These probabilities can be expressed in terms of the concentrations of the alloy atoms and the degree of long-range order

\[ \eta=\frac{p_A^{(1)}-c_1}{\gamma} \]

by the formulas:

\[ \begin{aligned} p_A^{(1)}&=c_1+\gamma\eta; & p_B^{(1)}&=c_2-\gamma\eta;\\ p_A^{(2)}&=c_1-\gamma\eta; & p_B^{(2)}&=c_2+\gamma\eta, \end{aligned} \tag{2.13} \]

where \(p_A^{(1)}, p_B^{(1)}, p_A^{(2)}, p_B^{(2)}\) are the probabilities of occupation of sites of the 1st and 2nd kind by atoms \(A\) and \(B\), \(c_1\) and \(c_2\) are the relative atomic concentrations of atoms \(A\) and \(B\) in the alloy \(A—B\) \((c_1+c_2=1)\), and

\[ \gamma= \begin{cases} c_1, & \text{for } c_1\leq \dfrac{1}{2},\\[4pt] c_2, & \text{for } c_1>\dfrac{1}{2}. \end{cases} \tag{2.14} \]

Taking into account that the quantities \(U_O^m\) and \(U_p^m\) are sums, taken with the opposite sign, of the interaction energies of atom \(C\) with the neighboring atoms of the alloy \(A—B\), it is not difficult to compute the sums,

entering into formula (2.12). Using formulas (2.1) and (2.13), we obtain:

\[ D=D_0 \frac{K_\alpha^{(+)2}K_\alpha^{(-)2}} {K_A^{(+)2}K_\alpha^{(-)2}+K_A^{(-)2}K_\alpha^{(+)2}}, \tag{2.15} \]

where

\[ \begin{aligned} K_\alpha^{(+)}&=(c_1+\gamma\eta)e^{\frac{\alpha}{kT}}+(c_2-\gamma\eta)e^{\frac{\beta}{kT}},\\ K_\alpha^{(-)}&=(c_1-\gamma\eta)e^{\frac{\alpha}{kT}}+(c_2+\gamma\eta)e^{\frac{\beta}{kT}},\\ K_A^{(+)}&=(c_1+\gamma\eta)e^{\frac{A}{kT}}+(c_2-\gamma\eta)e^{\frac{B}{kT}},\\ K_A^{(-)}&=(c_1-\gamma\eta)e^{\frac{A}{kT}}+(c_2+\gamma\eta)e^{\frac{B}{kT}},\\ K_a^{(+)}&=(c_1+\gamma\eta)e^{\frac{a}{kT}}+(c_2-\gamma\eta)e^{\frac{b}{kT}},\\ K_a^{(-)}&=(c_1-\gamma\eta)e^{\frac{a}{kT}}+(c_2+\gamma\eta)e^{\frac{b}{kT}}. \end{aligned} \tag{2.16} \]

In particular, for a disordered alloy, when \(\eta=0\), from (2.15) follows the formula

\[ D=D_{\mathrm{d.u}}=\frac{D_0}{2} \frac{\left(c_1 e^{\frac{\alpha}{kT}}+c_2 e^{\frac{\beta}{kT}}\right)^4} {\left(c_1 e^{\frac{A}{kT}}+c_2 e^{\frac{B}{kT}}\right)^4 \left(c_1 e^{\frac{a}{kT}}+c_2 e^{\frac{b}{kT}}\right)^2}. \tag{2.17} \]

To obtain the temperature and concentration dependence of the diffusion coefficient, it must be taken into account that in (2.15) the long-range order parameter \(\eta\) in the equilibrium state of the \(A—B\) alloy is also a function of temperature and composition. For an approximate estimate of this dependence in alloys of the given structure, where the order—disorder transition is a second-order phase transition, one may use the formula known in the statistical theory of ordering, obtained by neglecting correlations:

\[ 16\frac{\varepsilon}{kT}\gamma\eta= \ln\frac{(c_1+\gamma\eta)(c_2+\gamma\eta)} {(c_1-\gamma\eta)(c_2-\gamma\eta)}. \tag{2.18} \]

The constant \(\varepsilon\) entering into (2.18) is related to the interaction energies of neighboring atoms of the \(A—B\) alloy by the relation

\[ \varepsilon=2|v_{AB}|-|v_{AA}|-|v_{BB}|. \tag{2.19} \]

The temperature \(T_0\) of the order—disorder transition is expressed in the aforementioned statistical theory by means of the formula

\[ T_0=\frac{8\varepsilon}{k}\,c_1(1-c_1). \tag{2.20} \]

b) Features of the temperature and concentration dependence of the diffusion coefficient

Let us first consider the case of a disordered alloy. As was indicated in the introduction, for alloys there is no reason to expect a linear dependence of $\ln D$ on $\dfrac{1}{T}$, although such a dependence is usually observed experimentally with a sufficient degree of accuracy.

It follows indeed from formula (2.17) that the indicated dependence does not occur. However, when the constants (2.1) are chosen to be of the same order of magnitude as is usually encountered in real alloys, the deviations from the straight line given by (2.17) turn out to be small and noticeable only over a wide temperature interval (usually not covered in experimental investigations). The dependence of $D$ on $c_1$ according to (2.17) may be of various types. Apparently, a fairly widespread case should be that in which at one end of the concentration diagram there is a weak dependence of $D$ on composition, while at the other it is considerably stronger.

Let us proceed to the consideration of the case of ordered alloys. In ordering alloys that are in an equilibrium state, according to (2.18), for $T \leqslant T_0$ $\eta$ increases continuously from 0 to 1 as the temperature is lowered, first rapidly and then more and more slowly. This leads to the fact that the diffusion coefficient $D$, defined by (2.15), for $T \leqslant T_0$ in the vicinity of $T_0$ undergoes a characteristic sharp change. The curve of the dependence of $\ln D$ on $\dfrac{1}{T}$ at $T = T_0$ has a break; then, as $T$ decreases, it deviates strongly from the almost rectilinear course characteristic of disordered alloys, and at sufficiently low temperatures again has a form only slightly different from a straight line.

Fig. 3.

Figure 3 shows a typical curve representing such a dependence (for the case $c_1 = 1/2$, $a = b = 0.75$ eV, $\alpha = 0.4$ eV, $\beta = 0.25$ eV, $A = 0.30$ eV, $B = 0.20$ eV, $\varepsilon = 0.0352$ eV).

For a more detailed study of the temperature range lying near \(T_0\) (for \(T \leq T_0\)), where \(\eta \ll 1\), let us expand expression (2.15) for \(D\) in powers of \(\eta\) and restrict ourselves to quadratic terms. This gives

\[ D = D_{\text{n.u.}}\left(1-L\gamma^{2}\eta^{2}\right), \tag{2.21} \]

where \(D_{\text{n.u.}}\) is the diffusion coefficient for the disordered alloy, determined by formula (2.17), and

\[ \begin{aligned} L={}&6\left( \frac{ e^{\frac{A}{kT}}-e^{\frac{B}{kT}} }{ c_1 e^{\frac{A}{kT}}+c_2 e^{\frac{B}{kT}} } \right)^2 +2\left( \frac{ e^{\frac{\alpha}{kT}}-e^{\frac{\beta}{kT}} }{ c_1 e^{\frac{\alpha}{kT}}+c_2 e^{\frac{\beta}{kT}} } \right)^2 \\ &+\left( \frac{ e^{\frac{a}{kT}}-e^{\frac{b}{kT}} }{ c_1 e^{\frac{a}{kT}}+c_2 e^{\frac{b}{kT}} } \right)^2 -8\, \frac{ \left(e^{\frac{A}{kT}}-e^{\frac{B}{kT}}\right) \left(e^{\frac{a}{kT}}-e^{\frac{b}{kT}}\right) }{ \left(c_1 e^{\frac{A}{kT}}+c_2 e^{\frac{B}{kT}}\right) \left(c_1 e^{\frac{a}{kT}}+c_2 e^{\frac{b}{kT}}\right) }. \end{aligned} \tag{2.22} \]

From (2.22) it is seen that the sign of \(L\) may be either positive or negative, although the most probable, i.e. typical, case will be that of positive \(L\), corresponding to the kind of curve for the dependence of \(\ln D\) on \(\dfrac{1}{T}\) shown in Fig. 3. For \(L<0\) this curve would lie above the dashed curve giving the dependence of \(\ln D_{\text{n.u.}}\) on \(\dfrac{1}{T}\) (under the assumption that the given alloy remains disordered also for \(T<T_0\)).

In order to characterize the slope of the curve of the dependence of \(\ln D\) on \(\dfrac{1}{T}\), one may introduce the effective activation energy by the formula

\[ Q=-\frac{\partial \ln D}{\partial \dfrac{1}{kT}} . \tag{2.23} \]

In the case when this dependence is linear, the \(Q\) thus defined obviously coincides with the activation energy.

The derivative of \(\eta^2\) with respect to \(\dfrac{1}{kT}\) at \(T=T_0\), as the temperature is lowered, changes discontinuously from zero to a certain finite value, equal, according to (2.18), to

\[ \left. \frac{\partial \eta^2}{\partial \dfrac{1}{kT}} \right|_{T=T_0-0} = \frac{24\varepsilon}{\gamma^2}\, \frac{c_1^3}{ 1+\left(\dfrac{c_1}{1-c_1}\right)^3 }. \tag{2.24} \]

As a consequence of this, \(Q\) undergoes a jump, the magnitude of which according to

according to formulas (2.23), (2.21), and (2.24) is equal to:

\[ \Delta = Q\big|_{T_0-0}-Q\big|_{T_0+0} =24L(T_0)\frac{c_1^3}{1+\left(\frac{c_1}{1-c_1}\right)^3}\,\varepsilon . \tag{2.25} \]

To estimate the possible values of the jump \(\Delta\), let us consider the particular case when, at all distances, the interaction energy of atoms \(C\) with atoms \(A\) is so much greater than with atoms \(B\) that

\[ e^{\frac{A}{kT_0}}\gg e^{\frac{B}{kT_0}},\qquad e^{\frac{\alpha}{kT_0}}\gg e^{\frac{\beta}{kT_0}},\qquad e^{\frac{a}{kT_0}}\gg e^{\frac{b}{kT_0}} . \tag{2.26} \]

Then we may approximately put \(L(T_0)=-\dfrac{1}{c_1^2}\), and from (2.25)

\[ \Delta=\frac{24\varepsilon c_1}{1+\left(\frac{c_1}{1-c_1}\right)^3}. \tag{2.27} \]

For example, at \(c_1=\dfrac{1}{2}\) we obtain from this \(\Delta=6\varepsilon\). Choosing for \(T_0\) a value close to those observed experimentally, for instance \(T_0=810^\circ\mathrm{K}\), from (2.20) at \(c_1=\dfrac{1}{2}\) we find \(\varepsilon=0.035\ \mathrm{ev}\), which gives \(\Delta=0.21\ \mathrm{ev}\), i.e. a value sufficiently large for it to be detectable experimentally with the present accuracy of experiment. It is not difficult to convince oneself that formula (2.25) may in particular cases lead to still larger values of \(\Delta\) (for example, for \(a=b\), \(e^{\frac{A}{kT_0}}\gg e^{\frac{B}{kT_0}}\), \(e^{\frac{\alpha}{kT_0}}\gg e^{\frac{\beta}{kT_0}}\), we obtain \(\Delta\) eight times larger than in the case just considered). Let us now turn to the study of the concentration dependence (at \(T=\mathrm{const}\)) of the diffusion coefficient. First of all, it is not difficult to see that at the point \(c_1=\dfrac{1}{2}\) the curve \(D(c_1)\) has no break. This circumstance follows from the fact that at the point \(c_1=\dfrac{1}{2}\) the derivative \(\dfrac{\partial(\gamma\eta)}{\partial c_1}=0\) (although \(\dfrac{\partial\gamma}{\partial c_1}\) and \(\dfrac{\partial\eta}{\partial c_1}\) have a discontinuity at this point), while \(\dfrac{\partial D}{\partial c_1}\) is continuous. Therefore

\[ \frac{dD}{dc_1} = \frac{\partial D}{\partial c_1} + \frac{\partial D}{\partial(\gamma\eta)} \frac{\partial(\gamma\eta)}{\partial c_1} \]

at \(c_1=\dfrac{1}{2}\) is also continuous. However, the curves \(D(c_1)\) have breaks at such concentration values \(c_1=c_0\) and \(c_1=1-c_0\) for which the given temperature \(T\) is the critical temperature of the order–disorder transition. At these concentration values, as follows from formula (2.21), \(\dfrac{dD}{dc_1}\) changes by the amount \(\delta\):

\[ \delta= \left[\frac{dD}{dc_1}\right]_{\mathrm{ordered}} - \left[\frac{dD}{dc_1}\right]_{\mathrm{disordered}} = -\left[ D_{\mathrm{n.u.}}\,L\, \frac{d(\gamma\eta)^2}{dc_1} \right]_{\substack{c_1=c_0\\ c_1=1-c_0}} . \tag{2.28} \]

As already mentioned above, the typical case is that in which \(L\) is positive, i.e. the curve \(D(c_1)\) at the points \(c_1=c_0\) and \(c_1=1-c_0\), on passing into the region of ordered alloys, has a break toward the axis \(c_1\). A curve of the dependence of \(D\) on \(c_1\) of this type is shown in Fig. 4 (where \(T=650^\circ\mathrm{K}\), \(a=b=0.825\ \text{eV}\), \(\alpha=0.4\ \text{eV}\), \(\beta=0.3\ \text{eV}\), \(A=0.3\ \text{eV}\), \(B=0.19\ \text{eV}\), \(\varepsilon=0.0352\ \text{eV}\)). The magnitudes

Fig. 4.

of the jumps \(\Delta\) (at \(c_1=c_0\)) and \(\delta\) (at \(T_0(c_1)\)) turn out to be related to one another. To clarify this relation it is more convenient to consider the jump not of \(\dfrac{dD}{dc_1}\), but of the quantity \(\dfrac{d\ln D}{dc_1}\):

\[ \delta^*= \left[\frac{d\ln D}{dc_1}\right]_{\mathrm{ord}} - \left[\frac{d\ln D}{dc_1}\right]_{\mathrm{disord}} = \frac{1}{D_{\mathrm{n.\,u}}}\,\delta . \tag{2.29} \]

The jump \(\delta^*\), as follows from (2.29), (2.28), (2.25), and (2.24), is related to \(\Delta\) by a relation from which all energy constants except \(\varepsilon\) have dropped out:

\[ \varepsilon\,\frac{\delta^*}{\Delta} = -\varepsilon\, \frac{\dfrac{d(\gamma\eta)^2}{dc_1}} {\dfrac{d(\gamma\eta)^2}{d\frac{1}{kT}}} = \frac{d\,\dfrac{\varepsilon}{kT_0}}{dc_1}. \tag{2.30} \]

Using formula (2.20), we obtain:

\[ \frac{\delta^*}{\Delta} = \left[ \frac{2c_1-1}{8\varepsilon c_1^2(1-c_1)^2} \right]_{c_1=c_0} = \left[ \frac{1}{kT_0} \frac{2c_1-1}{c_1(1-c_1)} \right]_{c_1=c_0}. \tag{2.31} \]

This relation can be checked experimentally.

Let us now consider binary alloys or compounds in which the order of alternation of atoms of the components is practically independent of

THEORY OF ATOMIC DIFFUSION IN ALLOYS

temperature. To this same category may be assigned ordering alloys that are in a quenched state with a definite degree of order $\eta$ independent of temperature, but investigated at sufficiently high temperatures so that the diffusion of interstitial atoms C is still noticeable. Such a case may apparently occur in the diffusion of hydrogen. In substances of the indicated type, on the curve of the dependence of $\ln D$ on $\dfrac{1}{T}$, there will obviously be no break corresponding to the transition to the disordered state, and no sharp deviations from a rectilinear course connected with a rapid change of $\eta$ with temperature will be encountered.

The question of the diffusion of interstitial atoms in an ordering alloy was first considered by one of us $^{15\text{v}}$ with the aid of an approximate method, often used in alloy theory, which may be called the method of mean energies. In this method the various configurations are not taken into account, and it is assumed that all atoms situated in positions $O_1$ or $O_2$, or $P$, have the same potential energy, equal to the mean potential energy over all positions of the type under consideration. In this way, for the diffusion coefficient in an alloy of the structure considered, the following expression was obtained:

\[ D = \frac{D_0}{2}\exp\left\{\frac{2}{kT}\left[2(c_1A+c_2B)+c_1a+c_2b-2(c_1\alpha+c_2\beta)\right]\right\}\times \]

\[ \times \frac{1}{\operatorname{ch}\dfrac{2(2A-2B-a+b)\eta}{kT}} . \tag{2,32} \]

This formula makes it possible to obtain qualitatively most of the numerical results given above, with the exception that, for disordered alloys, the dependence of $\ln D$ on $\dfrac{1}{T}$ obtained from (2,32) is always linear, and the breaks, upon transition to the ordered state, in the curves of the temperature and concentration dependences of the diffusion coefficient according to formula (2,32) are always directed toward the abscissa axis.

Let us now clarify the question of the limits of applicability of the method of mean energies. First of all, note that for $\eta = 1$ and $c_1 = \dfrac{1}{2}$ formula (2,32) coincides with formula (2,15), as should be expected, since in this case there is only one quite definite configuration of atoms around all points of type $O_1$, as well as $O_2$ and $P$.

In the general case of arbitrary $c_1$ and $\eta$, an approximate agreement of these formulas can be expected only when the inequalities

\[ \frac{|A-B|}{kT}\ll 1,\qquad \frac{|a-b|}{kT}\ll 1,\qquad \frac{|a-\beta|}{kT}\ll 1. \tag{2,33} \]

Expansion of the expressions for \(D\) in series in powers of these quantities shows that the formulas mentioned prove to coincide in the terms of first order. However, these terms do not contain \(\eta\), and therefore the given result can be applied only to the case of disordered alloys (\(\eta = 0\)).

The largest terms containing \(\eta\) turn out to be terms of second order with respect to the small quantities (2.33), and their coincidence in the formulas mentioned for \(D\) is obtained when the additional condition is satisfied

\[ 2(\alpha-\beta)^2 = 2(A-B)^2 + (a-b)^2 . \tag{2.34} \]

Thus, the method of mean energies can be approximately applied to the diffusion problem under consideration in the case of disordered alloys when (2.33) is fulfilled, and in the case of ordered alloys when the additional condition (2.34) is also fulfilled, i.e., for example, for the particular case when \(|A-B| \simeq |\alpha-\beta|\) and \(a \simeq b\).

Nevertheless, as was shown above, the principal qualitative features of the temperature and concentration dependences of the diffusion coefficient can already be revealed by using the method of mean energies.

The expression for the diffusion coefficient also assumes a simple form in another limiting case, when the \(C\) atoms have strongly differing interaction energies with the \(A\) and \(B\) atoms. Thus, for example, if the conditions (2.26) are fulfilled, then formula (2.15) goes over into the following expression:

\[ D = \frac{D_0}{2}\,\frac{1}{c_1^2 + \gamma^2\eta^2}\exp\frac{4a-4A-2\alpha}{kT}. \tag{2.35} \]

In this case, the breaks in the curves representing the dependences of \(\ln D\) on \(1/T\) and of \(D\) on \(c_1\), when the alloy passes into the ordered state, are always obtained in the direction of the abscissa axis.

c) Allowance for correlation in the alloy

In an ordering alloy, where \(\varepsilon > 0\), each atom has a tendency to surround itself with atoms of the other kind. As a consequence, if it is reliably known that an atom of a definite kind is located at a lattice site neighboring the given one, then the probabilities of occupation of the given site by an \(A\) or \(B\) atom (a posteriori probabilities) are no longer equal to the a priori probabilities of its occupation (2.13). This means that there exists a correlation between the occupation of different sites of the alloy. In the case of complete order (and stoichiometric composition) the correlation does not manifest itself, since each atom is surrounded by neighboring atoms having the same configuration.

Consider the problem of determining the diffusion coefficient under the same assumptions as were adopted above, but taking correlation into account*). In this case the probabilities \(W^m_{O_1}\) and \(W^m_{O_2}\) entering formula (2,12) can no longer be calculated as the product of six a priori probabilities. To determine \(W^m_{O_1}\), we shall successively fill the sites \(1—6\) (Fig. 5) with atoms \(A\) and \(B\) corresponding to the given configuration \(m\). Strictly speaking, \(W^m_{O_1}\) is equal to the product of the a posteriori probabilities of occupation of these sites, calculated under the certain occupation of the previously filled sites (see, for example, \(^{20}\)).

Fig. 5.

However, approximately one may take into account only the correlation between nearest neighbors. Then the correlation may be neglected when filling sites \(1\) and \(2\), and also when filling sites \(3, 4, 5, 6\). Therefore the quantity \(W^m_{O_1}\) may be determined as the product of two a priori probabilities of occupation of sites \(1\) and \(2\) and four a posteriori probabilities of occupation of sites \(3, 4, 5, 6\), under the certain occupation of sites \(1\) and \(2\). The probability \(W^m_{O_2}\) is calculated analogously. The a posteriori probabilities needed for calculating \(W^m_{O_1}\) and \(W^m_{O_2}\) may be determined as functions of the alloy composition, the degree of long-range order, and the temperature. In doing so, considering their explicit dependence on \(T\), we retain only terms linear with respect to \(\dfrac{\varepsilon}{kT}\) \(\left(\dfrac{\varepsilon}{kT}\right.\) plays the role of a small parameter\()\).

After this one can calculate the sums entering formula (2,12).

Likewise, retaining only terms linear with respect to \(\dfrac{\varepsilon}{kT}\), we obtain the following expression for the diffusion coefficient:

\[ D = D_0 \frac{K_a^{(+)2}K_a^{(-)2}} {K_A^{(+)4}K_a^{(-)2}+K_A^{(-)4}K_a^{(+)2}} \times \]

\[ \times \left[ 1+4\frac{\varepsilon}{kT} \left(c_1^2-\gamma^2\eta^2\right) \left(c_2^2-\gamma^2\eta^2\right) \times \right. \]

\[ \left. \times \left( -\frac{R_a^2}{K_a^{(+)}K_a^{(-)}} + 2\, \frac{K_A^{(+)3}K_a^{(-)}+K_A^{(-)3}K_a^{(+)}} {K_A^{(+)4}K_a^{(-)2}+K_A^{(-)4}K_a^{(+)2}} R_A R_a \right) \right], \tag{2,36} \]

* ) Correlation is taken into account by means of the method developed by I. M. Lifshitz \(^{20}\).

where

\[ \left. \begin{aligned} R_{\alpha}&=e^{\frac{\alpha}{kT}}-e^{\frac{\beta}{kT}},\\ R_A&=e^{\frac{A}{kT}}-e^{\frac{B}{kT}},\\ R_a&=e^{\frac{a}{kT}}-e^{\frac{b}{kT}}. \end{aligned} \right\} \tag{2.37} \]

In formula (2.36) the dependence of \(\eta\) on the composition of the alloy and its temperature, according to \(^{20}\), is determined by the relation

\[ \frac{(c_1+\gamma\eta)(c_2+\gamma\eta)} {(c_1-\gamma\eta)(c_2-\gamma\eta)} = \exp\left\{ 2z\gamma\eta\,\frac{\varepsilon}{kT} \left[ 1-\frac{\varepsilon}{2kT} \left(c_1^2+c_2^2-2\gamma^2\eta^2\right) \right] \right\}, \tag{2.38} \]

where \(z\) is the coordination number.

In particular, in the case of disordered alloys the expression for the diffusion coefficient has the form

\[ D_{\mathrm{d.\,a.}} = \frac{1}{2}D_0\, \frac{K_\alpha^4}{K_A^2K_a^2} \left[ 1+ 4c_1^2c_2^2\frac{\varepsilon}{kT} \left( 2\,\frac{R_AR_a}{K_AK_a} - \frac{R_\alpha^2}{K_\alpha^2} \right) \right], \tag{2.39} \]

where

\[ K_\alpha=K_\alpha^{(+)}\big|_{\eta=0},\qquad K_A=K_A^{(+)}\big|_{\eta=0},\qquad K_a=K_a^{(+)}\big|_{\eta=0}. \tag{2.40} \]

Formulas (2.36) and (2.39) pass into formulas (2.15) and (2.17) if, in the square brackets, the second term is neglected. It follows from (2.39) that the correction introduced by taking correlation into account tends to zero when the temperature is raised or when the concentration of one of the alloy components tends to zero. The correlation correction to the diffusion coefficient is also small if the diffusing atoms \(C\) interact almost identically with atoms \(A\) and \(B\), i.e., the differences \(R_A, R_a\), and \(R_\alpha\) are small. Finally, as is seen from formula (2.36), the correlation correction to the diffusion coefficient becomes insignificant when the degree of long-range order in the alloy tends to unity.

To estimate the magnitude of the change in the diffusion coefficient when correlation is established in a disordered alloy, one may suppose, for example, that atoms \(C\) interact appreciably more strongly with atoms \(A\) than with atoms \(B\), so that inequalities (2.26) are satisfied.

For an alloy of stoichiometric composition \(\left(c_1=c_2=\dfrac{1}{2}\right)\) at the temperature \(T_0\) of the order–disorder transition \(\left(\dfrac{\varepsilon}{kT_0}=0.59\right)\)

the expression in square brackets in formula (2.39) is equal to 1.59 (without taking correlation into account it is equal to unity). Thus, with the establishment of short-range order in a disordered alloy (as a result of annealing at a temperature somewhat higher than \(T_0\)), the diffusion coefficient of interstitial atoms may change appreciably.

All the qualitative features of the temperature and concentration dependence of the diffusion coefficient indicated in the preceding section are preserved when correlation is taken into account. However, the estimates of the quantities characterizing the discontinuities on the curves \(D(T)\) at \(c_1=\mathrm{const}\) and \(D(c_1)\) at \(T=\mathrm{const}\) at the points corresponding to the order–disorder transition change. For small \(\eta\), the diffusion coefficient has the form (2.21) also when correlation is taken into account. Here, however, \(D_{\mathrm{n.u.}}\) is determined by formula (2.39), and \(L\) is equal to

\[ L = 2\,\frac{R_\alpha^2}{K_\alpha^2} + 6\,\frac{R_A^2}{K_A^2} - 8\,\frac{R_\alpha R_A}{K_\alpha K_A} + \]

\[ + \frac{R_a^2}{K_a^2} + 4\,\frac{\varepsilon}{kT}(c_1^2+c_2^2) \left( 2\,\frac{R_A R_a}{K_A K_a} - \frac{R_a^2}{K_a^2} \right) + \]

\[ + 4\,\frac{\varepsilon}{kT}\,c_1^2c_2^2 \left[ \frac{R_a^4}{K_a^4} + 2\,\frac{R_A R_a}{K_AK_a} \left( 3\,\frac{R_A^2}{K_A^2} - 5\,\frac{R_aR_A}{K_aK_A} + \frac{R_a^2}{K_a^2} \right) \right]. \tag{2.41} \]

When formula (2.38) is used instead of (2.18) to determine the degree of long-range order, the magnitude of the jump of the derivative \(\eta^2\) with respect to \(\frac{1}{kT}\) will also change (increase). From formula (2.38), the following value is obtained for this jump:

\[ \left. \frac{d\eta^2}{d\frac{1}{kT}} \right|_{T_0-0} = 12\,\frac{c_1^2c_2^2}{\gamma^2}\, \varepsilon\, \frac{\sqrt{6c_1^2c_2^2-c_1c_2}} { 1-3c_1c_2-24c_1^3c_2^3 \left(\frac{\varepsilon}{kT_0}\right)^2 }, \tag{2.42} \]

where

\[ \frac{\varepsilon}{kT_0} = \frac{1}{1-2c_1c_2} \left( 1-\frac{\sqrt{6c_1c_2-1}}{2\sqrt{c_1c_2}} \right). \tag{2.43} \]

It follows from (2.43) that the curve of the dependence of \(T_0\) on \(c_1\) no longer passes through the points \(c_1=0\) and \(c_1=1\) (as was the case when correlation was not taken into account), i.e., the ordered state of the alloy can exist only in a limited concentration interval in the middle part of the concentration diagram.

The magnitude of the jump in the effective activation energy will be equal to

\[ \Delta = 12c_1^2c_2^2 \frac{ \sqrt{6c_1^2c_2^2-c_1c_2} }{ 1-3c_1c_2-24c_1^3c_2^3\left(\frac{\varepsilon}{kT_0}\right)^2 } \,\varepsilon L(T_0). \tag{2,44} \]

To estimate this quantity one may again consider the case when, at \(T=T_0\), the inequalities (2,26) are satisfied. Then \(L(T_0)\simeq \dfrac{1}{c_1^2}+4\dfrac{\varepsilon}{kT_0}\), and for \(c_1=c_2=\dfrac12\) the quantity \(\Delta\simeq 14\varepsilon\simeq 8kT_0\).

If \(T_0\simeq 800^\circ\mathrm{K}\), then \(\Delta\simeq 0.6\) eV. Thus, allowance for correlation leads in the present case to a considerable (almost threefold) increase in \(\Delta\).

Fig. 6.

Taking correlation into account, the relation between the jump \(\delta^*\) of the derivative \(\dfrac{d\ln D}{dc_1}\) upon transition to the ordered state and \(\Delta\) has the form

\[ \varepsilon \frac{\delta^*}{\Delta} = \frac{ d\,\dfrac{\varepsilon}{kT_0} }{ dc_1 } = \]

\[ = \left[ \frac{ (2c_1-1) \left( 24c_1^2c_2^2 - 6c_1c_2 + 1 - 8c_1c_2\sqrt{6c_1^2c_2^2-c_1c_2} \right) }{ 4c_1c_2(1-2c_1c_2)^2\sqrt{6c_1^2c_2^2-c_1c_2} } \right]_{c_1=c_0}. \tag{2,45} \]

The graph of the dependence of \(\varepsilon\dfrac{\delta^*}{\Delta}\) on \(c_1\) is shown in Fig. 6 (\(z=8\)).

d) Diffusion of interstitial atoms in alloys of the Fe\(_3\)Al type

Alloys with a body-centered cubic lattice may also have a structure of the Fe\(_3\)Al type. Let us consider an \(A—B\) alloy having the indicated structure (Fig. 7). Let there be more atoms \(A\) in the alloy than atoms \(B\). Then the lattice sites lying, for example, at the vertices of the cubic cells (with edge \(a_0\)) are constantly occupied by atoms \(A\). At the centers of the cells are located the remaining atoms \(A\), as well as atoms \(B\). In the case of the stoichiometric composition of the alloy \(A_3B\), under complete ordering the atoms located at the centers of the cells form a lattice of the NaCl type. In the disordered state these atoms are distributed over the lattice sites at random, while atoms \(A\) still remain at the vertices of the cubes. As in the case of alloys with a \(\beta\)-brass-type lattice, the interstitial atoms \(C\) have positions of stable equilibrium at the centers of the faces of the cubic cells—in points \(O_1\)—and at the midpoints of the cell edges—in points \(O_2\). Midway between the interstices \(O_1\) and \(O_2\) lie the vertices of the potential barriers for atoms \(C\)—the points \(P\).

Fig. 7.

Carrying out the same calculation with allowance for correlation as in the case of alloys with a \(\beta\)-brass-type lattice, one can obtain the following final expression for the diffusion coefficient of atoms \(C\):

\[ D = D_0 \frac{ e^{\frac{2a}{kT}} K_a^{(+)}K_a^{(-)} }{ e^{\frac{4A}{kT}}K_a^{(+)}K_a^{(-)} + e^{\frac{2a}{kT}}K_A^{(+)2}K_A^{(-)2} } \times \]
\[ \times \left[ 1+ \frac{\varepsilon}{kT} \left(c_1^2-\gamma^2\eta_1^2\right) \left(c_2^2-\gamma^2\eta_2^2\right) \times \right. \]
\[ \left. \times \left( - \frac{R_a^2}{K_a^{(+)}K_a^{(-)}} + \frac{ e^{\frac{4A}{kT}}R_a^2 + 4e^{\frac{2a}{kT}}K_A^{(+)}K_A^{(-)}R_A^2 }{ e^{\frac{4A}{kT}}K_a^{(+)}K_a^{(-)} + e^{\frac{2a}{kT}}K_A^{(+)2}K_A^{(-)2} } \right) \right]. \tag{2,46} \]

Here \(c_1\), \(c_2\), and \(\eta\) denote the concentrations of atoms \(A\) and \(B\) and the degree of long-range order at the sites located at the centers of the cubic cells. The remaining notation is the same as in the case of alloys of the \(\beta\)-brass type. The diffusion coefficient without allowance for correlation is obtained from formula (2.46) when the second term in the square bracket of this formula is neglected. Investigation of formula (2.46) shows that the temperature and concentration dependences of the diffusion coefficient in alloys of the \(\mathrm{Fe}_3\mathrm{Al}\) type have the same qualitative features as in alloys of the \(\beta\)-brass type. However, in alloys of the \(\mathrm{Fe}_3\mathrm{Al}\) type the graph of the dependence of \(\ln D\) on \(\dfrac{1}{T}\) apparently more often has a bend in the direction away from the abscissa axis. The magnitude of the jump \(\Delta\) in most cases turns out to be considerably smaller than in the case of alloys with a \(\beta\)-brass-type lattice. Correlation in the alloy also has a smaller effect on \(D\). This is connected with the fact that the ordering process in a lattice of the \(\mathrm{Fe}_3\mathrm{Al}\) type affects only half the sites. In addition, in this lattice each of the interstices \(O_1\) and \(O_2\) is surrounded by the same number of sites of the 1st and 2nd kinds, which also weakens the influence of ordering on the diffusion of interstitial atoms.

The dependence of the quantity \(\varepsilon \dfrac{\delta^*}{\Delta}\) on the alloy composition for alloys of the structure under consideration has the following form:

\[ \varepsilon \frac{\delta^*}{\Delta} = \left[ \frac{ (2c_1-1) \left( 20c_1^2c_2^2-6c_1c_2+1-4c_1c_2\sqrt{15c_1^2c_2^2-3c_1c_2} \right) }{ 2(1-2c_1c_2)\,2c_1c_2\sqrt{15c_1^2c_2^2-3c_1c_2} } \right]_{\substack{c_1=c_0\\ c_1=1-c_0}} \tag{2.47} \]

The graph of this function is shown in Fig. 6 (\(z=6\)).

§ 3. DIFFUSION OF INTERSTITIAL ATOMS IN ALLOYS WITH A FACE-CENTERED CUBIC LATTICE \(^{21}\)

Let us investigate the problem of the diffusion of interstitial atoms in an alloy having a face-centered cubic lattice of the \(\mathrm{AuCu}_3\) type, within the framework of the simplified alloy model adopted earlier. Consider a binary ordering substitutional alloy of metals \(A\) and \(B\) in a state of thermodynamic equilibrium; for definiteness, suppose that the number of atoms \(A\) is smaller than the number of atoms \(B\). Then the sites of the 1st kind (regular for atoms \(A\)) will be located at the vertices of the cubic cells, and the sites of the 2nd kind (regular for atoms \(B\)) at the centers of their faces. We shall assume that the atoms \(C\) introduced into the alloy have positions of stable equilibrium in interstices of type \(O_1\), located at the centers of the cubic…

cells, and of type \(O_2\)—at the midpoints of their edges (see Fig. 8)\(^*\). The interstice \(O_1\) is surrounded by six nearest neighbors—sites of the 2nd kind, situated at a distance \(a_0/2\). The interstice \(O_2\) is surrounded by two sites of the 1st kind and four sites of the 2nd kind, situated at a distance \(a_0/2\). Choosing the direction of the \(x\)-axis and singling out two adjacent atomic planes \(I\) and \(II\) perpendicular to it, as shown in Fig. 8, we find that atom \(C\) can pass from the interstice \(O_1\) in plane \(I\) to the neighboring interstices \(O_2\) in plane \(II\) by four different, but geometrically equivalent, paths. We shall take the top of the potential barrier for such a transition to be at the point \(P_1\), situated midway between \(O_1\) and \(O_2\). An atom \(C\) occupying an interstice \(O_2\) in plane \(I\) can pass into an interstice \(O_2\) in plane \(II\) likewise by four equivalent paths. In doing so it must pass through the top of the potential barrier—the point \(P_2\), situated midway between two neighboring interstices \(O_2\). An atom \(C\) can pass from an interstice \(O_2\) in plane \(II\) to an interstice \(O_1\) in plane \(I\) by two paths, passing through positions of type \(P_1\), and to interstices \(O_2\) in plane \(I\) also by two paths, passing through positions of type \(P_2\). The position \(P_1\) has two neighboring sites of the 2nd kind at a distance \(\frac{\sqrt{2}}{4}a_0\), and also two neighboring sites of the 1st kind and two sites of the 2nd kind at a distance \(\frac{\sqrt{6}}{4}a_0\). The position \(P_2\) has one neighboring site of the 1st kind and one of the 2nd kind at a distance \(\frac{\sqrt{2}}{4}a_0\), and also four neighboring sites of the 2nd kind at a distance \(\frac{\sqrt{6}}{4}a_0\).

Fig. 8.

Let us denote the interaction energies of atom \(C\) with atoms \(A\) and \(B\), \(v_{AC}\) and \(v_{BC}\), at distances \(\frac{\sqrt{2}}{4}a_0\), \(a_0/2\), and \(\frac{\sqrt{6}}{4}a_0\), respectively, by the following

\(^*\) Such an arrangement of interstices for solutions of carbon and nitrogen in \(\gamma\)-iron was proved by X-ray analysis\({}^{22}\).

as follows:

\[ \left. \begin{aligned} v_{AC}\left(\frac{\sqrt{2}}{4}a_0\right)&=-a';& v_{AC}\left(\frac{a_0}{2}\right)&=-A';\\ v_{AC}\left(\frac{\sqrt{6}}{4}a_0\right)&=-a',\\[4pt] v_{BC}\left(\frac{\sqrt{2}}{4}a_0\right)&=-\beta';& v_{BC}\left(\frac{a_0}{2}\right)&=-B';\\ v_{BC}\left(\frac{\sqrt{6}}{4}a_0\right)&=-b'. \end{aligned} \right\} \tag{3.1} \]

Next, let us introduce the a priori probabilities of replacement of lattice sites by atoms \(A\) and \(B\), which for the given structure are related to the concentrations \(c_1\) and \(c_2\) of these atoms and to the degree of long-range order \(\eta\) by the formulas:

\[ \left. \begin{aligned} p_A^{(1)}&=c_1+\gamma\eta;& p_B^{(1)}&=c_2-\gamma\eta,\\ p_A^{(2)}&=c_1-\frac{1}{3}\gamma\eta;& p_B^{(2)}&=c_2+\frac{1}{3}\gamma\eta, \end{aligned} \right. \]

where

\[ \left. \gamma= \begin{cases} 3c_1, & \text{for } c_1\leqslant \dfrac{1}{4},\\[4pt] c_2, & \text{for } c_1\geqslant \dfrac{1}{4} \end{cases} \quad (c_1=1-c_2). \right\} \tag{3.2} \]

Computing, by the method indicated in § 2, the counter-flows of atoms \(C\) between planes \(I\) and \(II\) and finding their resultant flow, we obtain (without taking correlations into account, but allowing for the presence of different configurations of atoms \(A\) and \(B\)) the following formula for the diffusion coefficient of atoms \(C\):

\[ D=D'_0 \frac{ \left(K'_a-\frac{1}{3}\gamma\eta R'_a\right) \left(K'_a-\frac{1}{3}\gamma\eta R'_a\right)^2 }{ \left(K'_A-\frac{1}{3}\gamma\eta R'_A\right)^4 } \times \]

\[ \times \left[ \frac{ \left(K'_a-\frac{1}{3}\gamma\eta R'_a\right) \left(K'_a+\gamma\eta R'_a\right)^2 }{ \left(K'_A-\frac{1}{3}\gamma\eta R'_A\right)^2 +3\left(K'_A+\gamma\eta R'_A\right)^2 } + \frac{ \left(K'_a-\frac{1}{3}\gamma\eta R'_a\right)^2 \left(K'_a+\gamma\eta R'_a\right) }{ \left(K'_A-\frac{1}{3}\gamma\eta R'_A\right)^2 +3\left(K'_A+\gamma\eta R'_A\right)^2 } \right], \tag{3.3} \]

where

\[ D'_0=2\frac{a_0^2}{\tau_0}, \tag{3,4} \]

\[ \left. \begin{aligned} K'_\alpha&=c_1 e^{\alpha'/kT}+c_2 e^{\beta'/kT};& R'_\alpha&=e^{\alpha'/kT}-e^{\beta'/kT},\\ K'_A&=c_1 e^{A'/kT}+c_2 e^{B'/kT};& R'_A&=e^{A'/kT}-e^{B'/kT},\\ K'_a&=c_1 e^{a'/kT}+c_2 e^{b'/kT};& R'_a&=e^{a'/kT}-e^{b'/kT}. \end{aligned} \right\} \tag{3,5} \]

In the case of a disordered alloy the diffusion coefficient is equal to

\[ D_{\text{d.u.}}=\frac{D'_0}{2}\, \frac{ \left[c_1 e^{\alpha'/kT}+c_2 e^{\beta'/kT}\right]^2 \left[c_1 e^{a'/kT}+c_2 e^{b'/kT}\right]^4 }{ \left[c_1 e^{A'/kT}+c_2 e^{B'/kT}\right]^6 }. \tag{3,6} \]

Examination of formula (3,6) shows that the temperature and concentration dependences of the diffusion coefficient in disordered alloys of the structure under consideration have the same qualitative features as in alloys with a body-centered lattice. However, substantially different results are obtained for these two structures when studying the passage of the alloy through the critical ordering temperature \(T_0\). In an alloy with a face-centered cubic lattice of the type AuCu\(_3\), a first-order phase transition takes place, and upon ordering the degree of long-range order changes discontinuously from zero to the value \(\eta_0\). Therefore the diffusion coefficient at the temperature \(T_0\) also changes discontinuously, in contrast to alloys of the \(\beta\)-brass type, where it is continuous and only the effective activation energy undergoes a jump. The latter effect, evidently, also occurs in alloys of the AuCu\(_3\) type.

For a qualitative clarification of the temperature dependence of the degree of long-range order at temperatures below \(T_0\), one may use an approximate formula obtained in the statistical theory of ordering, which does not take into account correlations in the alloy:

\[ \ln \frac{ (c_1+\gamma\eta)\left(c_2+\frac{1}{3}\gamma\eta\right) }{ \left(c_1-\frac{1}{3}\gamma\eta\right)(c_2-\gamma\eta) } = \frac{16}{3}\frac{\varepsilon}{kT}\gamma\eta. \tag{3,7} \]

Here \(\varepsilon\) is determined by formula (2.19) and is related to the temperature \(T_0\) at \(c_1=\frac14\) by the relation \(\varepsilon \simeq 1.22\,kT_0\).

With a further decrease in temperature, the degree of long-range order increases smoothly from \(\eta_0\) to unity. In this case the graph of the dependence of \(\ln D\) on \(\frac{1}{T}\) at first deviates appreciably from a straight line, and then is usually well approximated by a straight line. As an example, Fig. 9 shows the graph of the dependence of \(\ln D\) on \(\frac{1}{T}\) for \(c_1=\frac{1}{4}\) and the following values of the parameters: \(\alpha'=0.95\ \text{eV}\), \(\beta'=0.75\ \text{eV}\), \(A'=B'=0.70\ \text{eV}\), \(a'=0.29\ \text{eV}\), \(b'=0.09\ \text{eV}\), \(T_0=800^\circ\text{K}\).

The formula for the diffusion coefficient is considerably simplified in the limiting cases when the atoms \(C\) interact much more strongly with the atoms of one component of the alloy than with the atoms of its other component.

Fig. 9.

In these limiting cases, upon ordering (at the point \(T=T_0\)) \(D\) decreases discontinuously by 20–50%. A still more significant decrease (by several times) in the diffusion coefficient upon ordering is also possible, as, for example, in the case shown in Fig. 9. Formula (3.3) also allows for an increase in the diffusion coefficient upon ordering. We note that taking correlation into account would lead to a larger value of the jump in \(\eta\), and consequently also in \(D\). A large jump in the degree of long-range order \(\eta\) in the alloy \(\mathrm{AuCu}_3\) was in fact observed in the experiments of A. P. Komar and N. N. Buinov \(^{23}\), the magnitude of the jump exceeding the value that follows from the theory of ordering without taking correlation into account.

On the curve of the dependence \(D(c_1)\) (for \(T=\mathrm{const}\)) there are also jumps of the diffusion coefficient. These jumps must occur at such values of the concentration \(c_1=c'_0\) and \(c_1=c''_0\), at which the temperature of the alloy is equal to the ordering temperature. Figure 10 shows the curve

\[ \frac{D(c_1)}{D'_0} \quad \left(\text{in the region } c_1 \leq \frac{1}{4}\right), \]

constructed for the parameter values \(\alpha'=0.95\ \text{eV}\), \(\beta'=0.866\ \text{eV}\), \(A'=B'=0.7\ \text{eV}\), \(a'=0.29\ \text{eV}\), \(b'=0.248\ \text{eV}\), \(\varepsilon=0.084\ \text{eV}\), and at temperature \(T=700^\circ\mathrm{K}\).

Fig. 10.

Thus, for the given structure, ordering exerts a stronger influence on the diffusion of interstitial atoms than in the case of alloys with a body-centered cubic lattice.

II. THEORY OF DIFFUSION EFFECTED BY THE VACANCY MECHANISM

§ 4. DETERMINATION OF THE EQUILIBRIUM CONCENTRATION OF VACANCIES AT THE SITES OF A BODY-CENTERED CUBIC LATTICE

a) Calculation by the method of mean energies \({}^{24}\)

For the investigation of the problem of diffusion in alloys effected by the vacancy mechanism, it is first of all necessary to find the equilibrium number of vacancies at the sites of the crystal lattice. If the alloy at a given temperature is in an ordered state, then it is necessary to know the concentrations of vacancies at sites of each kind. In this section the indicated problem will be considered by the method of mean energies and without taking into account the effect of enrichment of the coordination shells neighboring the vacancies by atoms of some kind (the latter effect will be taken into account in the next section).

Using the simplified alloy model adopted earlier, let us consider a binary, in the general case ordering, solid solution of metals \(A\) and \(B\) of substitutional type with a lattice of the \(\beta\)-brass type. For the atoms of the alloy we shall regard transitions into interstices as impossible. Let \(\mathfrak{N}\) denote the total number of lattice sites of the alloy and \(N\) the total number of atoms (of these, \(N_A\) atoms \(A\) and \(N_B\) atoms \(B\)). The numbers of atoms \(A\) and \(B\) at sites of the 1st and 2nd kind will be denoted respectively by \(N_A^{(1)}\), \(N_B^{(1)}\), \(N_A^{(2)}\), and \(N_B^{(2)}\), and the number of vacancies

at sites of the 1st and 2nd kind by \(n_1\) and \(n_2\). Then

\[ \left. \begin{aligned} N_A^{(1)}+N_A^{(2)}&=N_A,\qquad N_B^{(1)}+N_B^{(2)}=N_B,\\ N_A^{(1)}+N_B^{(1)}+n_1&=N_A^{(2)}+N_B^{(2)}+n_2=\frac{\mathfrak N}{2}. \end{aligned} \right\} \tag{4.1} \]

Introduce the concentration of atoms \(A\) in the alloy \(c_1=\dfrac{N_A}{N}\) and the probabilities of occupation of sites of the 1st and 2nd kind by atoms \(A\), \(B\), or by a hole:

\[ \left. \begin{aligned} p_A^{(1)}&=2\frac{N_A^{(1)}}{\mathfrak N},\qquad p_B^{(1)}=2\frac{N_B^{(1)}}{\mathfrak N},\qquad p_{\mathrm h}^{(1)}=2\frac{n_1}{N},\\ p_A^{(2)}&=2\frac{N_A^{(2)}}{\mathfrak N},\qquad p_B^{(2)}=2\frac{N_B^{(2)}}{\mathfrak N},\qquad p_{\mathrm h}^{(2)}=2\frac{n_2}{N}. \end{aligned} \right\} \tag{4.2} \]

Using (4.1), the probabilities (4.2) can be represented in the form

\[ \left. \begin{aligned} p_A^{(1)}&=(c_1+\gamma\eta_*)\left(1+\frac{n_1+n_2}{N}\right)^{-1},\\ p_A^{(2)}&=(c_1-\gamma\eta_*)\left(1+\frac{n_1+n_2}{N}\right)^{-1},\\ p_B^{(1)}&=(c_2-\gamma\eta_*-\delta)\left(1+\frac{n_1+n_2}{N}\right)^{-1},\\ p_B^{(2)}&=(c_2+\gamma\eta_*+\delta)\left(1+\frac{n_1+n_2}{N}\right)^{-1},\\ p_{\mathrm h}^{(1)}&=2\frac{n_1}{N}\left(1+\frac{n_1+n_2}{N}\right)^{-1},\\ p_{\mathrm h}^{(2)}&=2\frac{n_2}{N}\left(1+\frac{n_1+n_2}{N}\right)^{-1}, \end{aligned} \right\} \tag{4.3} \]

where

\[ \eta_*=\frac{2\dfrac{N_A^{(1)}}{N}-c_1}{\gamma}, \]

\(\gamma\) is determined by formula (2.18), and \(\delta=\dfrac{n_1-n_2}{N}\). The quantity \(\eta_*\), obviously, becomes the long-range order parameter \(\eta\) in the case when there are no holes in the alloy.

Let the interaction energies of neighboring pairs \(AA\), \(BB\), and \(AB\), taken with the opposite sign, be denoted by \(V_{AA}\), \(V_{BB}\), and \(V_{AB}\). Then the configurational part of the energy \(E\) of the alloy in the adopted model can be written in the form

\[ E=-8\left[N_A^{(1)}\left(p_A^{(2)}V_{AA}+p_B^{(2)}V_{AB}\right)+N_B^{(1)}\left(p_A^{(2)}V_{AB}+p_B^{(2)}V_{BB}\right)\right] \tag{4.4} \]

or, taking (4.3) into account,

\[ E=-4N\left(1+\frac{n_1+n_2}{N}\right)^{-1} \left\{(c_1^2-\gamma^2\eta_*^2)V_{AA}+\right. \]

\[ \left.+\left[c_2^2-(\gamma\eta_*+\delta)^2\right]V_{BB} +2(c_1c_2+\gamma^2\eta_*^2+\gamma\eta_*\delta)V_{AB}\right\}. \tag{4.5} \]

The number \(W\) of different configurations of the crystal in the given approximation is evidently equal to

\[ W= \frac{\left(\dfrac{\mathcal N}{2}\right)!}{N_A^{(1)}!N_B^{(1)}!n_1!}\cdot \frac{\left(\dfrac{\mathcal N}{2}\right)!}{N_A^{(2)}!N_B^{(2)}!n_2!}. \tag{4.6} \]

Expressing the configurational part of the free energy of the crystal
\(\psi=E-kT\ln W\) as a function of \(\eta_*, n_1\), and \(n_2\), from the equilibrium conditions
\(\dfrac{\partial\psi}{\partial\eta_*}=0\),
\(\dfrac{\partial\psi}{\partial n_1}=0\), and
\(\dfrac{\partial\psi}{\partial n_2}=0\), we find expressions for the equilibrium, at a given temperature, quantities \(\eta_*, n_1\), and \(n_2\).

The condition \(\dfrac{\partial\psi}{\partial\eta_*}=0\), after neglecting values \(n_1\) and \(n_2\) small in comparison with \(N\), leads to formula (2.18). The conditions
\(\dfrac{\partial\psi}{\partial n_1}=0\) and
\(\dfrac{\partial\psi}{\partial n_2}=0\), upon discarding terms of order
\(\left(\dfrac{n_1}{N}\right)^2\), \(\left(\dfrac{n_2}{N}\right)^2\), and smaller, give the equilibrium values of \(n_1\) and \(n_2\) in the form

\[ n_1=\varphi_1 n;\qquad n_2=\varphi_2 n, \tag{4.7} \]

where

\[ n=\sqrt{n_1n_2}=\frac{N}{2}e^{-\frac{\bar u}{kT}}, \tag{4.8} \]

\[ u=4\left[(c_1^2-\gamma^2\eta^2)V_{AA} +(c_2^2-\gamma^2\eta^2)V_{BB} +2(c_1c_2+\gamma^2\eta^2)V_{AB}\right], \tag{4.9} \]

\[ \varphi_1= \sqrt{\frac{c_1+\gamma\eta}{c_1-\gamma\eta}}\, e^{-\frac{8(V_{AB}-V_{AA})}{kT}\gamma\eta} = \sqrt{\frac{c_2-\gamma\eta}{c_2+\gamma\eta}}\, e^{\frac{8(V_{AB}-V_{BB})}{kT}\gamma\eta}, \tag{4.10} \]

\[ \varphi_2=\frac{1}{\varphi_1}, \]

and in this approximation the quantity \(\eta_*\) has been replaced by the degree of long-range order \(\eta\).

From formulas (4.7)—(4.10) it follows that for disordered alloys \((\eta=0)\), \(\varphi_1=\varphi_2=1\), and consequently,

\[ n_1=n_2=n_0=\frac{N}{2}e^{-\frac{u_0}{kT}}, \]

where

\[ u_0=4\left[V_{BB}+2(V_{AB}-V_{BB})c_1-\varepsilon c_1^2\right] \tag{4.11} \]

and

\[ \varepsilon=2V_{AB}-V_{AA}-V_{BB}. \]

In an ordered alloy, however, the numbers of holes \(n_1\) and \(n_2\) at sites of the first and second kinds are different.

In a disordered alloy, owing to the fact that the method of mean energies was used, \(\ln n_1\) and \(\ln n_2\) depend linearly on \(\frac{1}{T}\). The total number of holes in the alloy \(n_1+n_2\) is an even function of \(\eta\). Since, as was indicated in § 2, the derivative of \(\eta^2\) with respect to \(\frac{1}{T}\) at \(T=T_0\) undergoes a jump (but remains finite), upon transition to the ordered state the graph of the dependence of \(\ln(n_1+n_2)\) on \(\frac{1}{T}\) has a kink. On the other hand, \(n_1\) and \(n_2\) contain linear terms of the expansion in powers of \(\eta\). Since

\[ \frac{d\eta}{d\frac{1}{T}} \]

tends to infinity as \(T\to T_0\) \((T<T_0)\), the graphs of the dependences of \(\ln n_1\) and \(\ln n_2\) on \(\frac{1}{T}\) will have a vertical tangent at the point \(T=T_0\). In the immediate vicinity of the point \(T=T_0\), one of these curves goes upward and the other downward upon transition into the region of ordered alloys.

Turning to consideration of the dependence of the hole concentration on the composition of the disordered alloy, we note that the form of the dependence of \(u_0\) on \(c_1\) according to (4.11) will be different for alloys ordering at lower temperatures \((\varepsilon>0)\) and for decomposing alloys \((\varepsilon<0)\).

In the first case the curve \(u_0(c_1)\) is convex away from the \(c_1\) axis, and in the second—toward this axis. If, moreover, \(V_{AB}>V_{AA}\) and \(V_{AB}>V_{BB}\), then the function \(u_0(c_1)\) has a maximum, and \(n_0(c_1)\) a minimum. If instead \(V_{AB}<V_{AA}\) and \(V_{AB}<V_{BB}\), then \(u_0(c_1)\) has a minimum, and \(n_0(c_1)\) a maximum.

Upon transition to the ordered state, kinks occur on the curves of the dependence of \(n_1\) and \(n_2\) on \(c_1\) at those concentration values for which the given temperature is equal to the critical temperature \(T_0\).

b) Investigation of the question of hole formation in alloys by the quasi-chemical method

In the preceding section the concentration of holes in an alloy was determined without taking correlation into account. Now the correlation in the alloy will be taken into account by means of the quasi-chemical approximation \({}^{25}\) for the same case that was considered in the preceding section.

We denote by \(N_{AA},\ N_{BB},\ N_{\text{дд}},\ N_{AB},\ N_{BA},\ N_{A\text{д}},\ N_{\text{д}A},\ N_{B\text{д}},\ N_{\text{д}B}\) the numbers of pairs of neighboring atoms \(A,\ B\), and holes, where the first subscript indicates that the given atom or hole is located at a site of the first kind, and the second subscript at a site of the second kind. Then the configurational part of the alloy energy is equal to

\[ E=-N_{AA}V_{AA}-N_{BB}V_{BB}-(N_{AB}+N_{BA})V_{AB}. \tag{4.12} \]

The quasichemical approximation consists in treating individual pairs of atoms as independent units, so that the number of distinct configurations \(W\) is assumed proportional to the number of ways in which the \(4N\) pairs can be divided into nine groups containing the pairs \(AA,\ BB,\ \text{дд},\ AB,\ BA,\ A\text{д},\ \text{д}A,\ B\text{д},\ \text{д}B\):

\[ W=h\frac{(4N)!}{N_{AA}!N_{BB}!N_{\text{дд}}!N_{AB}!N_{BA}!N_{A\text{д}}!N_{\text{д}A}!N_{B\text{д}}!N_{\text{д}B}!}. \tag{4.13} \]

Here the factor \(h\) does not depend on the numbers of pairs, but does depend on the distribution of atoms \(A,\ B\) and holes over sites of the first and second kind. Determining this factor in the way usually done in the quasichemical approximation, we obtain:

\[ h= \frac{\left(N_A^{(1)}+N_B^{(1)}+n_1\right)!}{N_A^{(1)}!N_B^{(1)}!n_1!} \cdot \frac{\left(N_A^{(2)}+N_B^{(2)}+n_2\right)!}{N_A^{(2)}!N_B^{(2)}!n_2!} \times \]

\[ \times \frac{ N_{AA}^{0}!N_{BB}^{0}!N_{\text{дд}}^{0}!N_{AB}^{0}!N_{BA}^{0}!N_{A\text{д}}^{0}!N_{\text{д}A}^{0}!N_{B\text{д}}^{0}!N_{\text{д}B}^{0}! }{(4N)!}, \tag{4.14} \]

where

\[ N_{ik}^{0}=16\,\frac{N_i^{(1)}N_k^{(2)}}{N} \]

\[ (i,k=A,B,\text{д}). \tag{4.15} \]

Using formulas (4.12), (4.13), (4.14), and (4.15) to determine the free energy of the alloy \(\psi=E-kT\ln W\), and taking into account that at thermodynamic equilibrium the free energy is minimal, one may obtain, to within terms of higher order of smallness relative to \(\frac{n_1}{N}\) and \(\frac{n_2}{N}\), the following expressions for the numbers of holes at sites of the first and second kind:

\[ n_1=\frac{N}{2} \left(\frac{c_1-\eta}{c_1+\eta}\right)^{1/2} P_{AA}^{4}e^{-\frac{4V_{AA}}{kT}} \left[ 1+\frac{P_{AB}}{P_{AA}} e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^8, \tag{4.16} \]

\[ n_2=\frac{N}{2} \left(\frac{c_1+\eta}{c_1-\eta}\right)^{1/2} P_{AA}^{4}e^{-\frac{4V_{AA}}{kT}} \left[ 1+\frac{P_{BA}}{P_{AA}} e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^8. \tag{4.17} \]

Here \(P_{AA}=\dfrac{N_{AA}}{4N}\) is determined by the formula

\[ P_{AA}= \frac{-\left[(c_2-c_1)e^{\frac{\varepsilon}{kT}}+2c_1\right]} {2\left(e^{\frac{\varepsilon}{kT}}-1\right)} + \frac{ \sqrt{ \left[(c_2-c_1)e^{\frac{\varepsilon}{kT}}+2c_1\right]^2 +4\left(c_1^2-\gamma^2\eta^2\right)\left(e^{\frac{\varepsilon}{kT}}-1\right) } } {2\left(e^{\frac{\varepsilon}{kT}}-1\right)}, \tag{4.18} \]

\[ \left. \begin{aligned} P_{AB}&=\frac{N_{AB}}{4N}=c_1+\gamma\eta-P_{AA},\\ P_{BA}&=\frac{N_{BA}}{4N}=c_1-\gamma\eta-P_{AA}, \end{aligned} \right\} \tag{4.19} \]

and the degree of long-range order \(\eta\) in the quasichemical approximation is determined by the simultaneous solution of equation (4.18) and the equation

\[ 7\ln\frac{(c_1+\eta)(c_2+\eta)}{(c_1-\eta)(c_2-\eta)} = 8\ln\frac{c_1+\eta-P_{AA}}{c_1-\eta-P_{AA}}. \tag{4.20} \]

In particular, for a disordered alloy in which correlations may be neglected (but \(\dfrac{|V_{AA}-V_{AB}|}{kT}>1\)),

\[ n_1=n_2=\frac{N}{2}e^{-\frac{4V_{AA}}{kT}} \left[ c_1+c_2 e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^8 . \tag{4.21} \]

If formulas (4.18) and (4.19) for \(P_{AA}\), \(P_{AB}\), and \(P_{BA}\) are taken into account and \(n_1\) in (4.16) and \(n_2\) in (4.17) are expanded in series in powers of \(\dfrac{V_{AB}-V_{AA}}{kT}\) and \(\dfrac{V_{AB}-V_{BB}}{kT}\), then, on retaining the linear terms, these expansions coincide with the corresponding expansions of expressions (4.7) for \(\dfrac{n_1}{N}\) and \(\dfrac{n_2}{N}\).

If, however, the energy differences \(V_{AB}-V_{AA}\) and \(V_{AB}-V_{BB}\) are not small in comparison with \(kT\), then the dependence of \(\ln n\) on \(\dfrac{1}{T}\) for a disordered alloy according to (4.21) is not exponential. In the general case there is no unambiguous relation between the sign of \(\varepsilon\) and the direction of convexity of the concentration curve. Thus, for example, if the conditions for the applicability of formula (4.21) are satisfied, then the concentration curve is always convex toward the abscissa axis. In this case the dependence of the hole concentration on the alloy composition according to formula (4.21) may be very strong.

§ 5. DETERMINATION OF THE EQUILIBRIUM CONCENTRATION OF HOLES
AT THE SITES OF A FACE-CENTERED CUBIC LATTICE \(^{26}\)

Let us consider the question of the formation of holes at the sites of a face-centered cubic lattice of the type \(\mathrm{AuCu}_3\), under the same assumptions as were adopted in subsection a of § 4. In contrast to the case examined there, now, out of the total number \(\mathfrak{N}\) of lattice sites, \(\frac{1}{4}\mathfrak{N}\) are sites of the 1st kind and \(\frac{3}{4}\mathfrak{N}\) are sites of the 2nd kind. Carrying out, for this case, the determination of the equilibrium number of holes \(n_1\) and \(n_2\) at sites of the 1st and 2nd kind by a method analogous to that used in subsection a of § 4, we find:

\[ \begin{aligned} n_1 &= \chi_1 n',\\ n_2 &= 3\chi_2 n', \end{aligned} \tag{5,1} \]

where

\[ \left. \begin{aligned} \chi_1 &= \left( \frac{c_2-\gamma\eta}{c_2+\frac{1}{3}\gamma\eta} \right)^{3/4} e^{\frac{4\left(V_{AB}-V_{BB}\right)}{kT}\eta},\\ \chi_2 &= \chi_1^{-1/3}, \end{aligned} \right\} \tag{5,2} \]

\[ n'=\frac{N}{4}e^{-6\frac{u'}{kT}}, \tag{5,3} \]

\[ u'= \left(c_1^2-\frac{1}{9}\gamma^2\eta^2\right)V_{AA} + \left(c_2^2-\frac{1}{9}\gamma^2\eta^2\right)V_{BB} + 2\left(c_1c_2+\frac{1}{9}\gamma^2\eta^2\right)V_{AB}. \tag{5,4} \]

Here the degree of long-range order \(\eta\) is determined by formula (3,7), \(\gamma\) by formula (3,2), and \(N\), as before, denotes the total number of atoms in the alloy.

The temperature and concentration dependence of the concentration of holes in a disordered alloy, as follows from formulas (5,1)—(5,4), has the same qualitative features as for alloys with a body-centered cubic lattice. Upon transition to the ordered state, in contrast to alloys of the \(\beta\)-brass type, not only the derivatives of the hole concentration with respect to \(T\) and \(c_1\) change discontinuously, but also the concentrations themselves.

In this case, a situation is possible in which the concentration of holes on sites of one type increases, while on sites of another type it decreases (Fig. 11) (but a situation is impossible in which the concentrations of holes on sites of both types increase).

\[ \ln \frac{n_1}{N},\ \ln \frac{n_2}{3N} \]

\[ \frac{1}{T}10^3 \]

\[ \left(c_1=0.2,\ V_{AA}=0.26138;\ V_{BB}=0.26738;\ V_{AB}=0.30638\right) \]

Fig. 11.

§ 6. Self-diffusion in alloys with similar interaction energies of atoms27

Below we shall consider self-diffusion (diffusion of labeled atoms of one of the components) in ternary and binary disordered alloys with a body-centered cubic lattice, occurring by the vacancy mechanism. In order to clarify only the qualitative aspect of the phenomenon, one may use the approximate method of mean energies. Analogously to how this was done in subsection a of § 4 (where the absolute values of the differences of the interaction energies of neighboring atoms were assumed small in comparison with \(kT\)), one can obtain an expression for the concentration of holes in a ternary disordered alloy \(A-B-C\):

\[ n = Ne^{-\frac{U}{kT}}, \tag{6,1} \]

where

\[ U = 4\left[c_1^2V_{AA}+c_2^2V_{BB}+c_3^2V_{CC}+2c_1c_2V_{AB} +2c_1c_3V_{AC}+2c_2c_3V_{BC}\right], \tag{6,2} \]

\(V_{ik}\) are the interaction energies of neighboring atoms \(i\) and \(k\) \((i,\ k=A,\ B,\ C)\), taken with the opposite sign, and \(c_1,\ c_2\), and \(c_3\) are the relative atomic concentrations of atoms \(A,\ B\), and \(C\).

Consider, for example, an atom \(A\) located at a site of the crystal lattice, and a hole at the site adjacent to it. We shall take into account only the apparently most probable type of transitions of atom \(A\), when it replaces a hole at the nearest site. In the body-centered cubic lattice two neighboring sites are at a distance

\[ d=\frac{\sqrt{3}}{2}a_0 \simeq 0.86a_0. \]

The curve characterizing the change in the potential energy of atom \(A\) as it moves into a hole along the space diagonal of the elementary cell has minima at the sites and a maximum which, as above, we shall assume to be located midway between the sites. Denote by \(U_0\) the potential energy of atom \(A\) at a lattice site, taken with the opposite sign, and by \(U_p\) that at the point \(P\) corresponding to the top of the potential barrier. Then the barrier height will be equal to \(\Delta U=U_0-U_p\). Atom \(A\), located at point \(P\), has six nearest neighbors at a distance

\[ \frac{\sqrt{11}}{4}a_0 \simeq 0.83a_0, \]

which almost coincides with \(d\). Therefore, in what follows we shall approximately regard these distances as equal. Then, in our approximation,

\[ U_0=7(c_1V_{AA}+c_2V_{AB}+c_3V_{AC}),\quad U_p=6(c_1V_{AA}+c_2V_{AB}+c_3V_{AC}) \]

and, consequently,

\[ \Delta U=c_1V_{AA}+c_2V_{AB}+c_3V_{AC}. \tag{6.3} \]

Let a concentration gradient of labeled atoms \(A\) be created in the direction of the \(x\)-axis, parallel to an edge of the cubic cell (under the condition that the total concentration of all atoms \(A\) remains constant). As before, consider two neighboring atomic planes \(I\) and \(II\), perpendicular to the \(x\)-axis. Each site on one of these planes has four neighboring sites on the other. Denote by \(\nu(x)\) the number of labeled atoms \(A\) per \(1\ \mathrm{cm}^2\) of the plane with coordinate \(x\). The probability that a hole will be found next to atom \(A\) on the neighboring plane is, in the adopted approximation, \(4n/N\). Taking into account that the probability \(w\) of transition of atom \(A\) into a neighboring hole per unit time, as above, is given by the formula \(w=\dfrac{1}{\tau_0}e^{-\frac{\Delta U}{kT}}\), it is not difficult to find an expression for the number of labeled atoms \(A\) passing per unit time from \(1\ \mathrm{cm}^2\) of plane \(I\) to plane \(II\):

\[ S_{I\to II}=\frac{4n}{N}\,\nu(x)\,\frac{1}{\tau_0}\,e^{-\frac{\Delta U}{kT}}. \tag{6.4} \]

Similarly, the expression for the counterflow has the form

\[ S_{II\to I}=\frac{4n}{N}\,\nu\left(x+\frac{a_0}{2}\right)\frac{1}{\tau_0}\,e^{-\frac{\Delta U}{kT}}, \tag{6,5} \]

Noting that the number of labeled atoms \(A\) in \(1\ \mathrm{cm}^3\) of the crystal \(\nu_0\) is equal to \(\dfrac{2}{a_0}\nu\), we obtain the following formula for the resulting flux:

\[ S=S_{I\to II}-S_{II\to I} = -\frac{a_0^2}{\tau_0}\frac{n}{N}e^{-\frac{\Delta U}{kT}}\frac{d\nu_0}{dx}. \tag{6,6} \]

Hence, taking (6,1) into account, one can determine the diffusion coefficient \(D_A\) of the labeled atoms \(A\):

\[ D_A=\frac{a_0^2}{\tau_0}e^{-\frac{Q}{kT}}, \tag{6,7} \]

where the activation energy \(Q\) has the form

\[ Q=U+\Delta U = 4(c_1^2V_{AA}+c_2^2V_{BB}+c_3^2V_{CC}+2c_1c_2V_{AB} +2c_1c_3V_{AC}+2c_2c_3V_{BC}) +c_1V_{AA}+c_2V_{AB}+c_3V_{AC}. \tag{6,8} \]

Let us consider some special cases.

1) Let, in a series of alloys, \(c_1\) remain constant, while \(c_3\) (and therefore also \(c_2\)) takes various values from \(0\) to \(1-c_1\). Eliminating \(c_2\) from (6,8), we find:

\[ Q=q_0+q_1c_3+q_2c_3^2, \]

where

\[ \begin{aligned} q_0&=4(V_{AA}+V_{BB}-2V_{AB})c_1^2+(7V_{AB}+V_{AA} \\ &\qquad -8V_{BB})c_1+4V_{BB}+V_{AB},\\[4pt] q_1&=8(V_{BB}-V_{AB}+V_{AC}-V_{BC})c_1+8V_{BC} \\ &\qquad -8V_{BB}+V_{AC}-V_{AB},\\[4pt] q_2&=4(V_{BB}+V_{CC}-2V_{BC})\equiv -4\varepsilon_{BC}. \end{aligned} \tag{6,9} \]

It follows from this that the curve \(Q(c_3)\) is convex in the direction opposite to the \(c_3\) axis if \(q_2<0\), i.e. \(\varepsilon_{BC}>0\). This occurs if the binary alloy of metals \(B\) and \(C\) (whose diffusion is not being investigated) can be in an ordered state at lower temperatures*). If, however, \(\varepsilon_{BC}<0\), i.e. the \(B-C\) alloy is decomposing, then the curve \(Q(c_3)\) is convex toward the \(c_3\) axis.

* The conclusion made here and in analogous places below will be valid if the binary and ternary alloys have the same structure and close lattice constants.

At small concentrations of atoms \(C\), when \(c_3 \ll 1\), \(Q \simeq q_0 + q_1 c_3\). In this case \(Q\) will increase, and \(D_A\) decrease, with increasing \(c_3\) (i.e., for example, upon adding a small impurity of metal \(C\) to the alloy \(A—B\)), if \(q_1 > 0\), and will decrease if \(q_1 < 0\). When, moreover, \(c_1\) is sufficiently small (which often occurs in the study of diffusion), the first of these cases (\(q_1 > 0\)) is realized if the impurity atoms \(C\) have a larger (in absolute value) interaction energy with the atoms of the base alloy \(A—B\) than the interaction energy of the atoms of this alloy with one another: \(V_{BC} > V_{BB}\), \(V_{AC} > V_{AB}\). The second case (\(q_1 < 0\)) occurs (at small \(c_1\)) if the reverse inequalities hold.

Analogously, one may investigate the problem of the diffusion of tagged atoms \(A\) in alloys \(A—B—C\) with a constant concentration of atoms \(C\) (see \(^{27}\)).

2) Let us now consider the special case of a binary alloy. Let \(c_3 = 0\). Then \(c_2 = 1 - c_1\), and (6,7) gives the diffusion coefficient of tagged atoms \(A\) in the binary alloy \(A—B\). In this case the activation energy takes the form:

\[ Q = q'_0 + q'_1 c_1 + q'_2 c_1^2, \]

where

\[ \left. \begin{aligned} q'_0 &= 4V_{BB} + V_{AB},\\ q'_1 &= 7V_{AB} + V_{AA} - 8V_{BB},\\ q'_2 &= 4(V_{AA} + V_{BB} - 2V_{AB}) \equiv -4\varepsilon . \end{aligned} \right\} \tag{6,10} \]

As in the case discussed above, the curve \(Q(c_1)\) is convex toward the side opposite to the \(c_1\)-axis for ordering alloys \(A—B\) \((\varepsilon > 0)\), and toward the \(c_1\)-axis for decomposing alloys \((\varepsilon < 0)\).

Let us also clarify the question of the influence of small impurities of metal \(B\) in metal \(A\) on its self-diffusion. For this purpose, consider the segment of the curve \(Q(c_1)\) where \(c_1 \simeq 1\), and \(c_2 \ll 1\). Then

\[ Q = 5V_{AA} + 9(V_{AB} - V_{AA})c_2 . \tag{6,11} \]

Thus, if \(V_{AB} > V_{AA}\), then \(Q\) increases, and \(D_A\) decreases with increasing \(c_2\), i.e., for example, upon adding metal \(B\) to metal \(A\).

Let us note that the dependences of \(Q\) on composition obtained, which are characteristic for different types of alloys, could possibly be used for selecting those additions to an alloy or to a pure metal which would cause a change in the diffusion coefficient in the required direction. For this, however, it is necessary to know the relation between the interaction energies of neighboring atoms (which in this paragraph were assumed not to differ greatly).

§ 7. SELF-DIFFUSION IN ALLOYS WITH STRONGLY DIFFERING
ENERGIES OF ATOMIC INTERACTION

In this section, in contrast to the preceding one, in determining the self-diffusion coefficient we shall take into account the effect of enrichment of the first coordination sphere around a hole by atoms \(A\) or \(B\), as well as the presence of different configurations around the diffusing atom. At the same time, for simplicity, we shall not take into account correlations between the atoms of the alloy, i.e., we shall neglect the terms containing the factor \(\varepsilon/kT\) in the formulas obtained by the quasichemical method. On the other hand, we shall assume the quantities \(\left|V_{AB}-V_{AA}\right|/kT\) and \(\left|V_{AB}-V_{BB}\right|/kT\) to exceed unity (therefore, \(V_{AB}-V_{AA}\) and \(V_{AB}-V_{BB}\) have different signs). Such a state of affairs occurs for a number of alloys, including, apparently, β-brass. The calculation given below applies to alloys with a body-centered cubic lattice of the β-brass type, which may be in both the disordered and the ordered state.

In § 4, by means of the quasichemical method, the concentrations of holes on sites of the first and second kind were obtained (see formulas (4,16) and (4,17)). In the same approximation one can determine the a posteriori probabilities \(p_{AA}^{(\alpha)}\) and \(p_{AB}^{(\alpha)}\) \((\alpha=1,2)\) that next to a hole on a site of kind \(\alpha\) there is an atom \(A\) or \(B\). Then the numbers of holes on sites of the first and second kind, surrounded by \(i\) atoms \(A\) and \(8-i\) atoms \(B\), can be determined from the formulas:

\[ n_{1i}=\frac{8!}{i!(8-i)!}\,n_1\left[p_{AA}^{(1)}\right]^i\left[p_{AB}^{(1)}\right]^{8-i}= \]

\[ =\frac{N}{2}\frac{8!}{i!(8-i)!} \left(\frac{c_1-\gamma\eta}{c_1+\gamma\eta}\right)^{1/2} P_{AA}^4 e^{-\frac{4V_{AA}}{kT}} \left[ \frac{P_{AB}}{P_{AA}}e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^{8-i}, \tag{7,1} \]

\[ n_{2i}=\frac{N}{2}\frac{8!}{i!(8-i)!} \left(\frac{c_1+\gamma\eta}{c_1-\gamma\eta}\right)^{1/2} P_{AA}^4 \times \]

\[ \times e^{-\frac{4V_{AA}}{kT}} \left[ \frac{P_{BA}}{P_{AA}}e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^{8-i}, \tag{7,2} \]

where \(P_{AA}\), \(P_{AB}\), and \(P_{BA}\) are determined by formulas (4,18) and (4,19).

If correlations between the occupation of alloy sites are not taken into account, then in formulas (7,1) and (7,2) one may put

\[ P_{AA}=c_1^2-\gamma^2\eta^2,\qquad P_{AB}=(c_1+\gamma\eta)(c_2+\gamma\eta); \]

\[ P_{BA}=(c_1-\gamma\eta)(c_2-\gamma\eta). \]

Then

\[ n_{1i}=\frac{N}{2}\,\frac{8!}{i!(8-i)!}\left(\frac{c_1-\gamma\eta}{c_1+\gamma\eta}\right)^{7/2} (c_1^2-\gamma^2\eta^2)^4 \times \]

\[ {}\times e^{-\frac{4V_{AA}}{kT}} \left[ \frac{c_2+\gamma\eta}{c_1-\gamma\eta} e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^{8-i}, \tag{7.3} \]

\[ n_{2i}=\frac{N}{2}\,\frac{8!}{i!(8-i)!}\left(\frac{c_1+\gamma\eta}{c_1-\gamma\eta}\right)^{7/2} (c_1^2-\gamma^2\eta^2)^4 \times \]

\[ {}\times e^{-\frac{4V_{AA}}{kT}} \left[ \frac{c_2-\gamma\eta}{c_1+\gamma\eta} e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^{8-i}. \tag{7.4} \]

As in the preceding section, let us consider two atomic planes \(I\) and \(II\), passing respectively through the faces of the cubic cells and through their centers. On each square centimeter of plane \(I\) there are \(a_0 n_{1i}\) holes surrounded by \(i\) atoms \(A\) and \(8-i\) atoms \(B\), and on \(1\ \mathrm{cm}^2\) of plane \(II\) there are \(a_0 n_{2i}\) holes with the indicated surroundings. The flux \(S_{II\to I}\) of atoms \(A\) from \(1\ \mathrm{cm}^2\) of plane \(II\) into holes located in plane \(I\) is determined by the formula

\[ S_{II\to I}=4a_0\sum_{i,m} n_{1i}\,\varpi^m W_{mi}. \tag{7.5} \]

Here the index \(m\) indicates the configuration of atoms \(A\) and \(B\) surrounding the site of the second kind \(O_2\), in which the atom \(A\) was located before the jump, and the point \(P\), at which the potential energy of atom \(A\) on its path from \(O_2\) into the neighboring hole has a maximum; \(W_{mi}\) denotes the probability of the indicated configuration of atoms (\(W_{mi}\) is taken to be zero if an atom \(B\) is located at the point \(O_2\)); \(\varpi^m\) denotes the probability, per unit time, of transition of an atom \(A\) with the \(m\)-th configuration of neighbors from the position \(O_2\) into the neighboring hole. The factor 4 in (7.5) arises because a transition is possible from any of the four sites adjacent to the given hole and lying in plane \(II\). The summation in (7.5) is carried out first over all configurations \(m\) corresponding to a given \(i\), and then over \(i\) from \(i=0\) to \(i=8\).

Let us divide the ten sites determining the configuration \(m\) into three groups: in the first we include the four sites neighboring \(O_2\) but not neighboring \(P\); in the second group we include the three sites neighboring both \(O_2\) and \(P\); and in the third group—the three sites of the nearest environment of the hole, neighboring only \(P\). Suppose that in the first group there are \(k\) atoms \(A\) and \(4-k\) atoms \(B\), and in the second group \(l\) atoms \(A\) and \(3-l\) atoms \(B\). The probability \(W_{kl}\) of occurrence of configurations in which the first group contains \(k\) atoms \(A\), and the second group \(l\) atoms \(A\), is determined by the formula

\[ W_{kl}=\frac{4!}{k!(4-k)!}\,[p_A^{(1)}]^k[p_B^{(1)}]^{4-k} \frac{3!}{l!(3-l)!}\,[p_A^{(1)}]^l[p_B^{(1)}]^{3-l}. \tag{7.6} \]

It remains to know the probability of realizing some prescribed configuration of atoms \(A\) and \(B\) in the third group. The probability \(W_{ri}\) that, at four of the eight sites surrounding the hole under consideration, there are \(r\) atoms \(A\) and \(4-r\) atoms \(B\), and that at one prescribed site belonging to this quartet (at the site \(O_2\)) there is certainly an atom \(A\), is determined by the following formula:

\[ W_{ri}= \frac{4!}{r!(4-r)!}\, \frac{4!}{(i-r)!\,[(8-i)-(4-r)]!}\, \frac{i!(8-i)!}{8!}\, \frac{r}{4}. \tag{7,7} \]

The probability of realizing a distribution of atoms at the ten sites surrounding the points \(O_2\) and \(P\), which is characterized by the numbers \(k\), \(l\), \(r\), and \(i\), is equal to the product \(W_{kl}\) and \(W_{ri}\).

Determining the energy of atom \(A\) in the positions \(O_2\) and \(P\), we obtain for the probability, per unit time, of transition of atom \(A\) from \(O_2\) through the point \(P\) into a neighboring hole the following expression:

\[ w^m=\frac{1}{\tau_0}\exp\left[ -(k+l)\frac{V_{AA}}{kT} -(7-k-l)\frac{V_{AB}}{kT} + \right. \]

\[ \left. +(l+r-1)\frac{\alpha}{kT} +(6-l-r+1)\frac{\delta}{kT} \right], \tag{7,8} \]

where \(\alpha\) and \(\delta\) are the interaction energies, taken with the opposite sign, of the atom \(A\) located at the point \(P\) with the neighboring atoms \(A\) and \(B\).

Calculating the flux (7,5), taking into account expressions (7,3), (7,8), (7,7), and (7,6), and summing over \(k\), \(l\), \(r\), and \(i\), we obtain the following expression for \(S_{II\to I}\):

\[ S_{II\to I} = 2\frac{a_0}{\tau_0} N \left( \frac{c_1-\gamma\eta}{c_1+\gamma\eta} \right)^{7/8} (c_1^2-\gamma^2\eta^2)^4 e^{\frac{4V_{AA}}{kT}} \times \]

\[ \times \left[ (c_1+\gamma\eta)e^{-\frac{V_{AA}}{kT}} + (c_2-\gamma\eta)e^{-\frac{V_{AB}}{kT}} \right]^4 \times \]

\[ \times \left[ (c_1+\gamma\eta)e^{\frac{\alpha-V_{AA}}{kT}} + (c_2-\gamma\eta)e^{\frac{\delta-V_{AB}}{kT}} \right]^3 \times \]

\[ \times \left[ e^{\frac{\alpha}{kT}} + \frac{c_2+\gamma\eta}{c_1-\gamma\eta} e^{\frac{\delta+V_{AA}-V_{AB}}{kT}} \right]^3 \left[ 1+ \frac{c_2+\gamma\eta}{c_1-\gamma\eta} e^{\frac{V_{AA}-V_{AB}}{kT}} \right]^4, \tag{7,9} \]

where, in accordance with § 6, one may put \(\alpha=V_{AA}\), \(\delta=V_{AB}\). The flux \(S^{*}_{II\to I}\) of labeled atoms from plane \(II\) to plane \(I\) is

\[ \frac{ N_A^{*}\left(x+\frac{a_0}{2}\right) }{ N_A } \]

times smaller than the flux \(S_{II\to I}\), while the flux \(S^{*}_{II\to I}\) from plane \(I\) to plane \(II\) is

\[ \frac{N_A^{*}(x)}{N_A} \]

times smaller than the flux \(S_{II\to I}\). Determining the resultant flux of labeled atoms, and then, in the same way as above, the self-diffusion coefficient \(D_A\), after

after simple transformations we obtain:

\[ D_A = D^0 e^{-\frac{5V_{AA}}{kT}} \frac{\left(c_1^2-\eta^2\gamma^2\right)^2}{c_1} \left[1+\frac{c_2+\gamma\eta}{c_1-\gamma\eta} e^{\frac{V_{AA}-V_{AB}}{kT}}\right]^4 \times \]

\[ \times \left[1+\frac{c_2-\gamma\eta}{c_1+\gamma\eta} e^{\frac{V_{AA}-V_{AB}}{kT}}\right]^4, \tag{7,10} \]

where

\[ D^0=\frac{a_0^2}{\tau_0}. \]

For a disordered alloy \(\eta=0\), and the self-diffusion coefficient is equal to

\[ D_A=D'=D^0 e^{-\frac{5V_{AA}}{kT}} \left[c_1+c_2 e^{\frac{V_{AA}-V_{AB}}{kT}}\right]^8 . \tag{7,11} \]

From formula (7,11) it is seen that, as in the case of diffusion of interstitial atoms, the dependence of \(\ln D_A\) on \(\frac{1}{T}\), generally speaking, is not linear even in a disordered alloy. However, over small intervals of reciprocal temperatures, in which measurements are usually made, this dependence may be approximated by a linear function. This approximation is especially accurate when

\[ \frac{\left|V_{AA}-V_{AB}\right|}{kT}\gg 1 \]

and one of the terms in the square brackets of formula (7,11) is considerably smaller than the other term (the latter case can also occur if the concentration of one of the alloy components tends to zero).

The dependence of the self-diffusion coefficient on the alloy composition, according to formula (7,11), can be very strong, provided only that the ratio \(\frac{\left|V_{AA}-V_{AB}\right|}{kT}\) is sufficiently large. Let us note that in the case under consideration the curve of the dependence of \(D_A\) on \(c_1\) is always convex toward the abscissa axis, independently of the sign of the ordering energy \(\varepsilon\). If, however, the energies \(V_{AA}\), \(V_{BB}\), and \(V_{AB}\) are close, then, as was indicated in the preceding paragraph, the direction of convexity of the curve representing the dependence of \(Q\) on \(c_1\) is determined by the sign of \(\varepsilon\).

In order to determine how the addition of a small amount of atoms \(B\) to pure metal \(A\) affects the self-diffusion coefficient of atoms \(A\), we expand expression (7,11) in a series in powers of \(c_2\) and retain the linear terms of the expansion:

\[ D_A=D^0 e^{-\frac{5V_{AA}}{kT}} \left[ 1+8c_2\left( e^{\frac{V_{AA}-V_{AB}}{kT}}-1 \right) \right]. \tag{7,12} \]

It follows from formula (7,12) that the increase of \(D_A\) with increasing concen-

tration \(c_2\) can be very large if the quantity \(e^{\frac{V_{AA}-V_{AB}}{kT}}-1\) is large. On the other hand, the indicated quantity cannot be less than \(-1\), so that the decrease in the self-diffusion coefficient of atoms \(A\) upon the addition of atoms \(B\) cannot be very significant (for homogeneous solid solutions). Let us note that the aforementioned sharp influence of an impurity on diffusion is observed experimentally\({}^{14}\).

Upon the transition of an alloy into the ordered state, as in the case of diffusion of interstitial atoms, characteristic features of the temperature dependence of the diffusion coefficient must be observed. To investigate the diffusion coefficient near the order–disorder transition point, one should expand the function \(D_A(T,c_1,\eta)\) in a series in powers of \(\eta\). Restricting ourselves to the quadratic terms of the expansion, we obtain:

\[ D_A=D'(1-\gamma^2\eta^2 L'). \tag{7,13} \]

Here \(D'\) is determined by formula (7,11), and

\[ L'=\frac{9}{2c_1^2}-\frac{8\zeta}{c_1^3\left(1+\frac{c_2}{c_1}\zeta\right)} +4\frac{\zeta^2}{c_1^4\left(1+\frac{c_2}{c_1}\zeta\right)^2}, \tag{7,14} \]

where

\[ \zeta=e^{\frac{V_{AA}-V_{AB}}{kT}} . \tag{7,15} \]

Since the derivative

\[ \frac{d\eta^2}{d\frac{1}{kT}} \]

at the point \(T=T_0\) changes discontinuously (see formula (2,24)), the effective activation energy at \(T=T_0\) also changes discontinuously by the amount

\[ \Delta=Q|_{T_0-0}-Q|_{T_0+0} =L'\gamma^2\frac{d\eta^2}{d\frac{1}{kT}}, \tag{7,16} \]

where \(L'\) is determined by formula (7,14), and

\[ \frac{d\eta^2}{d\frac{1}{kT}} \]

by formula (2,24).

In particular, for alloys of stoichiometric composition

\[ \Delta=27\varepsilon f(\zeta), \tag{7,17} \]

where

\[ f(\zeta)=27-\frac{96\zeta}{(1+\zeta)^2}. \tag{7,18} \]

Investigating the function \(f(\zeta)\) in the interval \(0<\zeta<\infty\), we find that it takes only positive values lying in the interval

\[ 3\leq f(\zeta)<27. \]

Thus, upon ordering, the effective activation energy always increases, and the graph of the dependence of \(\ln D_A\) on \(\frac{1}{T}\) always has a break in the direction toward the abscissa axis. The quantity \(\Delta\) assumes its largest value at \(\zeta=0\) and at \(\zeta=\infty\), i.e., when

\[ \frac{V_{AA}-V_{AB}}{kT}=\pm\infty . \]

Thus, for the experimental detection of a break in the curve of the dependence of \(\ln D\) on \(\frac{1}{T}\), favorable alloys are those in which the difference between \(V_{AA}\) and \(V_{AB}\) is large in comparison with \(kT\) and \(\varepsilon\) (consequently, \(V_{AA}\) and \(V_{BB}\) also differ greatly from one another). In such alloys the jump in the effective activation energy is, in order of magnitude, equal to \(13kT_0\), i.e., \(\sim 1\) eV, if \(T_0\sim 700—800^\circ\) K. A sufficiently sharply expressed break should apparently be expected for self-diffusion in \(\beta\)-brass, since the activation energies of self-diffusion in pure Zn and Cu differ by almost a factor of three (they are, respectively, \(Q_{\mathrm{Zn}}\simeq 20 \frac{\mathrm{kcal}}{\mathrm{g\text{-}atom}}\), \(Q_{\mathrm{Cu}}\simeq 57 \frac{\mathrm{kcal}}{\mathrm{g\text{-}atom}}\)). Indeed, the activation energy is determined to a considerable extent by the vacancy-formation energy, and the latter is connected with the energies of interaction between atoms.

The theoretically predicted characteristic features of the temperature dependence of the diffusion coefficient in ordering alloys are apparently confirmed by preliminary experiments of S. D. Gertsriken and I. Ya. Dekhtyar, \(^{28}\) the results of which can be interpreted in accordance with the point of view indicated above. However, the absence of a sufficient number of points in the disordered region does not make it possible to draw this conclusion quite definitely.

On the concentration curves of the diffusion coefficient one should also expect breaks (in the direction toward the abscissa axis), analogous to those considered in § 2.

On the temperature and concentration curves of the coefficients of self-diffusion in alloys where the order—disorder transition is a first-order phase transition (for example, in alloys with a crystal lattice of the AuCu\(_3\) type), jumps are obtained analogous to those which were investigated in § 3.

§ 8. ON THE INFLUENCE OF INTERSTITIAL ATOMS ON THE SELF-DIFFUSION OF A METAL \(^{29}\)

Let us proceed to consider the question of the influence of a small amount of an impurity element \(C\), dissolved in a metal \(A\), on the self-diffusion of this metal. For definiteness we shall assume that the metal \(A\) has a face-centered cubic lattice,

and the atoms \(C\) are introduced into its interstices (the centers of the elementary cubic cells and the midpoints of their edges).

Let us denote by \(N_C\) the number of atoms \(C\), by \(N_A\) the number of atoms \(A\), and by \(c\) their ratio \(c=\dfrac{N_C}{N_A}\). Further, let \(E_0\) denote the energy of the aforementioned solid solution, having no holes at the sites of the crystal lattice.

Let us take into account that at a nonzero temperature there are holes in the crystal, which may be surrounded by different numbers \((i)\) of neighboring atoms \(C\) \((0 \leqslant i \leqslant 6)\). Denote by \(n_i\) the number of holes in the crystal whose neighbors are \(i\) atoms \(C\). Then the energy of a solution of atoms \(C\) in the metal \(A\), in the presence of holes, in the nearest-neighbor approximation, will have the form:

\[ E=E_0(c)+u_A\sum_{i=0}^{6} n_i-u_{AC}\sum_{i=0}^{6} i n_i, \tag{8.1} \]

where \(u_A\) is the energy required to form a hole in the pure metal \(A\) \((u_A>0)\), and \(u_{AC}\) is the interaction energy of atoms \(A\) and \(C\) at the distance \(\dfrac{a_0}{2}\).

To determine the free energy of the crystal, let us find the number of different configurations of atoms \(A\) and holes at sites and atoms \(C\) in interstices, for given values of \(N_A\), \(N_C\), and all \(n_i\):

\[ W= \frac{ \left( N_A+\sum_{i=0}^{6} n_i \right)! }{ N_A!\prod_{i=0}^{6} n_i! } \times \]

\[ \times \frac{ \left( N_A-5\sum_{i=0}^{6} n_i \right)! }{ \left( N_C-\sum_{i=0}^{6} i n_i \right)! \left[ N_A-5\sum_{i=0}^{6} n_i-\left( N_C-\sum_{i=0}^{6} i n_i \right) \right]! } \times \prod_{i=0}^{6} \left[ \frac{6!}{i!(6-i)!} \right]^{n_i}. \tag{8.2} \]

From formulas (8.1) and (8.2) we obtain, in the usual way, the expression for the free energy of the alloy:

\[ \psi = E - kT \ln W . \]

The equilibrium concentrations \(\frac{n_i}{N_A}\) of holes at the sites of the crystal lattice can be determined from the equilibrium conditions

\[ \frac{\partial \psi}{\partial n_i}=0 \quad (i=0,1,\ldots,6), \]

and, taking into account that the concentrations of holes are sufficiently small, we obtain:

\[ \frac{n_i}{N_A} = \frac{1}{1-c} e^{-\frac{u_A}{kT}} \frac{6!}{i!(6-i)!} \left( c e^{\frac{u_{AC}}{kT}} \right)^i (1-c)^{6-i}. \tag{8.3} \]

Summing this expression over all \(i\), it is not difficult to find the total concentration of holes \(\frac{n}{N_A}\) at the sites of the crystal lattice of the alloy:

\[ \frac{n}{N_A} = \sum_{i=0}^{6} \frac{n_i}{N_A} = \frac{1}{1-c} e^{-\frac{u_A}{kT}} \left[ 1 + c\left(e^{\frac{u_{AC}}{kT}}-1\right) \right]^6 . \tag{8.4} \]

It follows from formula (8.4), among other things, that even if the energy \(u_{AC}\) is equal to zero, the addition of atoms \(C\) leads to a certain increase in the number of holes, since each new hole increases the number of interstitial positions in the alloy, and therefore also the entropy of the atoms \(C\). This, in turn, will lead to a decrease in the free energy of the crystal.

Proceeding to the determination of the diffusion coefficient of atoms \(A\), let us draw atomic planes \(I\) and \(II\) as indicated in Fig. 8. The number of holes surrounded by \(i\) atoms \(C\), on \(1 \text{ cm}^2\) of plane \(II\), is equal to \(\frac{a_0}{2} n_i\). Around these holes in plane \(I\) there are \(2a_0 n_i\) neighboring atoms \(A\). The probability \(w_n\) of transition of an atom \(A\) into a neighboring hole per unit time is equal to

\[ w_n = \frac{1}{\tau_0} e^{-\frac{u_{Pn}-u_{On}}{kT}} . \tag{8.5} \]

Here \(u_{On}\) and \(u_{Pn}\) are the energies of an atom \(A\), respectively, in the initial position \(O\) (at a site) and at the top of the potential barrier (at point \(P\)). The index \(n\) denotes the number of atoms \(C\) adjacent to point \(P\) \((n=0,1,2)\). The height of the potential barrier \(u_{Pn}-u_{On}\), obviously, is equal to \(\Delta U_A + nb\), where \(\Delta U_A\) is the barrier height in a pure

in metal \(A\), and \(b\) is the difference of the interaction energies of atoms \(A\) and \(C\) at distances \(\dfrac{\sqrt{2}}{4}a_0\) and \(\dfrac{a_0}{2}\). The probability \(W_{ni}\) that, of the \(i\) atoms \(C\) surrounding a given hole, \(n\) atoms \(C\) are located at the two interstitial sites adjacent to point \(P\), is determined by the formula

\[ W_{ni}=\frac{2!}{n!(2-n)!}\,\frac{4!}{(i-n)!(4-i+n)!}\,\frac{i!(6-i)!}{6!}. \tag{8,6} \]

The flux \(S_{I\to II}\) of atoms \(A\) passing per unit time from \(1\ \mathrm{cm}^2\) of plane \(I\) into plane \(II\), equal to the opposite flux \(S_{II\to I}\), is evidently expressed by the formula

\[ S_{I\to II}=\sum_{i=0}^{6}\sum_{n=0}^{2}2a_0 n_i W_{ni}\,\mathfrak{w}_n . \tag{8,7} \]

Let a concentration gradient of labeled atoms \(A\) be created in the direction of the \(x\)-axis, perpendicular to planes \(I\) and \(II\). Calculating the sum in (8,7) and determining their diffusion coefficient, we find:

\[ D=D_A^0\frac{1}{1-c} \left[ 1+c\left(e^{\frac{u_{AC}}{kT}}-1\right) \right]^4 \left[ 1+c\left(e^{\frac{u_{AC}-b}{kT}}-1\right) \right]^2 , \tag{8,8} \]

where

\[ D_A^0=D_0 e^{-\frac{Q_A}{kT}} \]

\((Q_A=u_A+\Delta U_A)\) is the self-diffusion coefficient of atoms \(A\) in pure metal \(A\), and

\[ D^0=\frac{a_0^2}{\tau_0}. \]

It follows from formula (8,8) that a small amount of impurity atoms \(C\) can significantly increase the diffusion coefficient \(D\), provided only that the quantities \(e^{\frac{u_{AC}}{kT}}\) or \(e^{\frac{u_{AC}-b}{kT}}\) considerably exceed unity. From this formula it is also clear that the self-diffusion coefficient does not depend exponentially on \(\dfrac{1}{T}\), as occurs in diffusion in pure metals. For alloys, as was shown above, there is no reason to expect such a dependence of \(D\) on \(\dfrac{1}{T}\). However, if measurements of the diffusion coefficient in alloys are carried out in a small temperature interval \((200—400^\circ)\), lying near some temperature value \(T_1\), then, as before, the graph of the dependence of \(\ln D\) on \(\dfrac{1}{T}\) may, with sufficient accuracy—

THEORY OF ATOM DIFFUSION IN ALLOYS

…can be approximated with sufficient accuracy by a straight line. As an example, Fig. 12 shows such a graph, constructed for the values

Fig. 12.

of the parameters \(u_{AC}=0.45\ \text{eV}\), \(u_{AC}-b=0\), \(c=0.05\). Let us introduce the effective activation energy \(Q\) at temperature \(T_1\):

\[ Q=-\left.\frac{\partial\ln D}{\partial\frac{1}{kT}}\right|_{T=T_1}, \]

and also the effective pre-exponential factor \(D'\), which is determined by the formula

\[ D'=De^{\frac{Q}{kT}}. \tag{8,9} \]

Obviously, in the case under consideration the quantity \(Q\) is equal to

\[ Q(T_1)=Q_A-2\left[ \frac{2e^{\frac{u_{AC}}{kT_1}}u_{AC}} {1+c\left(e^{\frac{u_{AC}}{kT_1}}-1\right)} + \frac{e^{\frac{u_{AC}-b}{kT_1}}(u_{AC}-b)} {1+c\left(e^{\frac{u_{AC}-b}{kT_1}}-1\right)} \right]c. \tag{8,10} \]

At small concentrations of atoms \(C\), when

\[ c\left(e^{\frac{u_{AC}}{kT_1}}-1\right)\ll 1,\qquad c\left(e^{\frac{u_{AC}-b}{kT_1}}-1\right)\ll 1, \tag{8,11} \]

the effective activation energy depends linearly on \(c\):

\[ Q=Q_A-Q'c = Q_A-2\left[ 2e^{\frac{u_{AC}}{kT_1}}u_{AC} + e^{\frac{u_{AC}-b}{kT_1}}(u_{AC}-b) \right]c. \tag{8,12} \]

If one of the ratios $\dfrac{u_{AC}}{kT_1}$ or $\dfrac{u_{AC}-b}{kT_1}$ exceeds unity, then a small amount of impurity atoms $C$ can significantly reduce $Q$. On the other hand, in the model adopted, a sharp increase of $Q$ upon the addition of atoms $C$ is impossible, since an increase of $Q$ occurs for negative energies $u_{AC}$ and $u_{AC}-b$ (one of these energies may be positive, but small), when $e^{\frac{u_{AC}}{kT_1}}$ and $e^{\frac{u_{AC}-b}{kT_1}}$ are small.

As the concentration of atoms $C$ is increased, when at least one of the conditions (8.11) is violated, the dependence of $Q$ on $c$ begins to deviate

Fig. 13.

from the linear dependence, showing a tendency toward saturation, if $e^{\frac{u_{AC}}{kT_1}}$ or $e^{\frac{u_{AC}-b}{kT_1}}$ is much greater than unity. In Fig. 13 a typical curve of the dependence of $Q$ on $c$ is given, constructed for the parameter values $Q_A=3$ ev, $u_{AC}=0.45$ ev, $u_{AC}-b=0$, $T_1=1400^\circ\mathrm{K}$.

From formulas (8.9), (8.8), and (8.10) it follows that the effective pre-exponential factor $D'$ is equal to

\[ D'= \frac{D^0}{1-c} \left[ 1+c\left(e^{\frac{u_{AC}}{kT_1}}-1\right) \right]^4 \left[ 1+c\left(e^{\frac{u_{AC}-b}{kT_1}}-1\right) \right]^2 \times \]

\[ \times \exp\left\{ -2\left[ \frac{2e^{\frac{u_{AC}}{kT_1}}\dfrac{u_{AC}}{kT_1}} {1+c\left(e^{\frac{u_{AC}}{kT_1}}-1\right)} + \frac{e^{\frac{u_{AC}-b}{kT_1}}\dfrac{u_{AC}-b}{kT_1}} {1+c\left(e^{\frac{u_{AC}-b}{kT_1}}-1\right)} \right]c \right\}. \tag{8.13} \]

If the concentration of atoms \(C\) is sufficiently small (\(c \ll 1\)), then in expression (8.13) one may discard the terms quadratic in \(c\), as well as the terms linear in \(c\) which do not contain the factors

\[ e^{\frac{u_{AC}}{kT_1}} \quad \text{or} \quad e^{\frac{u_{AC}-b}{kT_1}}, \]

which may be considerably greater than unity. As a result, \(D'\) may be approximately represented in the form of an exponential function

\[ D' \simeq D^0 e^{-\xi(T_1)c}, \]

where

\[ \xi(T_1)=2\left[ 2e^{\frac{u_{AC}}{kT_1}} \left(\frac{u_{AC}}{kT_1}-1\right) + e^{\frac{u_{AC}-b}{kT_1}} \left(\frac{u_{AC}-b}{kT_1}-1\right) \right]. \tag{8.14} \]

In this case the quantity \(\xi(T_1)\cdot c\) may be sufficiently large if the ratios

\[ \frac{u_{AC}}{kT_1} \quad \text{or} \quad \frac{u_{AC}-b}{kT_1} \]

considerably exceed unity.

Thus, the formulas obtained make it possible to give a qualitative explanation of the recently discovered, by P. L. Gruzin, Yu. V. Korneev, and G. V. Kurdyumov\({}^{15}\), linear decrease of the effective activation energy and exponential decrease of the effective pre-exponential factor in the coefficient of self-diffusion of iron upon addition to \(\gamma\)-iron of a small amount of carbon. The correct orders of magnitude of \(Q'(T_1)\) and \(\xi(T_1)\) are obtained if, for example, the following parameter values are chosen: \(u_{AC}=0.45\) eV, \(u_{AC}-b=0\). It should be noted that with these parameter values, at \(c\sim 2\text{–}5\) atomic %, criterion (8.11) ceases to be satisfied and deviations from the linear dependence \(Q(c)\) should be observed. However, the curve \(Q(c)\) can be approximated in this interval by a straight line in such a way that the deviations from the straight line remain within the limits of experimental accuracy. Further, in accordance with experiment, the effective pre-exponential factor decreases sharply upon addition of carbon to iron. For example, at \(c=0.02\) the quantity \(D'\simeq 10^{-2}D^0\). Thus, for the initial portion of the curve of the dependence of \(D'\) on \(c\) (in the interval from \(c=0\) to \(c=0.025\)), with the chosen values of the constants, one obtains approximately a dependence of the effective pre-exponential factor \(D'\) on \(c\) of the form \(D'=10^{-c}D^0\), in agreement with the experimental data\({}^{15}\). However, at larger values of \(c\) there is a slower decrease of \(D'\) with increasing \(c\). Nevertheless, already in the aforementioned small interval of concentrations, when \(c\) increases from 0 to 0.025, \(D'\) decreases by approximately a factor of 100. The theory set forth in § 8 thus makes it possible to explain the main experimental results obtained in \({}^{15}\).

§ 9. CONCLUSION

Let us recall once more the principal conclusions that follow from the theory of diffusion in alloys considered above.

1. In contrast to the widely held view according to which the diffusion coefficient \(D\) in alloys depends exponentially on the reciprocal temperature, we see that in fact, even for a disordered alloy, a more complicated dependence of \(D\) on \(T\) must occur. Nevertheless, an investigation of the formulas given above shows that, for disordered alloys, the graph of the dependence of \(\ln D\) on \(\frac{1}{T}\) can be approximated with sufficient accuracy by a straight line in those small temperature intervals in which measurements are usually carried out. However, when diffusion is studied over a sufficiently wide temperature interval, deviations from the indicated rectilinear dependence may be observed. In this case diffusion in the alloy cannot be characterized by a single constant activation energy.

2. Upon transition to the ordered state, characteristic features must appear in the curves of the temperature dependence of the diffusion coefficient. If ordering is a second-order phase transition, then the diffusion coefficient changes continuously, but the curve of the dependence of \(\ln D\) on \(\frac{1}{T}\) has a kink at the transition temperature \(T_0\). If, however, the transition to the ordered state is a first-order phase transition, then at \(T = T_0\) not only the effective activation energy but also the diffusion coefficient itself changes discontinuously. In the ordered state, near the transition temperature \(T_0\), deviations of the graph of the indicated dependence from a straight line must be especially noticeable.

3. On the concentration curve of the diffusion coefficient there must also occur jumps and kinks at those compositions at which the alloy passes into the ordered state (in the case \(T = \mathrm{const}\)).

4. Investigation of the concentration dependence of \(D\) in the case of a small concentration of one of the components of the alloy makes it possible to explain the sometimes observed strong influence of a small amount of impurity on the diffusion coefficient and on the diffusion parameters (the effective pre-exponential factor and the effective activation energy).

5. If the energies of interaction of atoms in the alloy are known, the theory makes it possible in some cases to predict how a given impurity should affect the diffusion coefficient.

In conclusion, let us note that with the present-day experimental technique it is possible to detect the indicated effects and, in particular, the effects associated with the transition to an ordered state. Carrying out such experimental studies would not be without interest.

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