Defects in Crystal Structure and Some Properties of Metals and Alloys

I. Ya. Dekhtyar

Introduction

Various lattice defects, such as vacancies, dislocated atoms, dislocations, and, in pure metals, also impurities of foreign atoms, play an important role in various physical processes occurring in metals and alloys. Therefore, the study of various properties of metals as a function of their structural state can provide not only valuable information about the nature of the influence of particular lattice defects on the properties of metals, but also help to clarify the mechanism of various processes.

Individual properties of metals have been considered in the literature from the point of view of the influence of defects in crystal structure. Recent years have been marked by the development of our ideas about the nature of various defects of the crystal lattice, and therefore the need has arisen to consider a number of properties of metals in a unified review.

It is known that the properties of metals are a single-valued function of the arrangement of atoms in the crystal lattice. In real solids the arrangement of atoms in the lattice is imperfect. The presence of various kinds of imperfections affects the character of physical processes occurring in solids and may completely change their properties.

In the present article only distortions of the atomic type are considered, such as, for example, vacancies, dislocated atoms, and dislocations. These lattice defects interact with one another, giving rise to the observed effects in metals. In this connection[^1] a real crystal may be regarded as a certain medium in which distortions of different types form a peculiar gas of “particles” with properties independent of one another. The indicated distortions of the atomic type may arise during the crystallization of a liquid metal, during cold and thermal treatment, and also when a metal is acted upon by corpuscular radiation of high energy.

1. Types of Atomic Distortions and Their Properties

The study of the properties of metals is complicated by the fact that it is practically difficult to realize defects of one definite kind in a metal. However, it is sometimes possible to carry out an experiment in such a way as to separate the influence of different defects. This is usually achieved by determining the activation energies for the annealing of certain processes (for example, electrical conductivity), which are then compared with the theoretical estimate of these quantities for the motion of defects of simple and complex type.

a) An estimate of the activation energy of diffusion in metals makes it possible[^2] to determine the activation energy for the motion of vacancies. For copper it turns out—

...was found to be equal to about 1 eV. It was suggested[^1] that a lower activation energy is required for the motion of a pair of vacancies. This is due to the smaller interaction of the closed electron shell of copper atoms with one of the components of the pair when they exchange places. It turned out[^3] that the activation energy for the motion of a vacancy pair is approximately half the activation energy for the motion of a single vacancy, while the dissociation energy of a pair of vacancies in copper lies within the range 0.25–0.50 eV. This indicates the possibility of coalescence of single vacancies if, at an excess concentration, they possess sufficient mobility.

A simplified method for calculating the energy of vacancy formation in monovalent metals[^4] is as follows. The metal is regarded as a certain spherical box with a uniformly distributed positive charge of the ions and freely moving electrons. If a metal ion is transferred from the center of the box to its surface, then the energy of the free electrons changes by an amount \(\Delta E_{el}\), owing to screening by electron waves of the vacancy that has formed. An estimate showed that \(\Delta E_{el}\simeq \frac{1}{6}E_F\), where \(E_F\) is the Fermi energy for the metal. For the metals Cu, Ag, and Au with a closed electron shell, the additional energy required for the formation of a vacancy is connected with the repulsion between ions and is equal to \(\Delta E_R=-0.3\) eV. Thus, for atoms with a closed shell, the energy \(U_1\) of vacancy formation is determined by the change in the energy of ionic repulsion

\[ U_1=\Delta E_{el}+\Delta E_R\simeq \frac{1}{6}E_F-0.3. \tag{1} \]

The results of theoretical calculations presented in Table I are compared with certain experimental data on the determination of the vacancy-formation energy \(U_1\). The table also gives the values of the activation energy for vacancy motion \(U_2\) and the activation energy of self-diffusion \(U\). The data presented indicate satisfactory agreement between theoretical and experimental values.

Table I

Metal	Li	Na	K	Rb	Cs	Cu	Ag	Au
\(U,\ \mathrm{eV}\)	0,57	0,45	—	0,41	—	2,03	1,96	1,96
\(U_1\)	0,40	0,39	0,39	—	—	0,9 (1,15)	0,8 (1,27)	0,67 (1,28)
\(U_2\)	0,17	0,06	—	—	—	1,13 (0,88)	1,2 (0,69)	1,29 (0,68)
\(E_F,\ \mathrm{eV}\)	3,30	3,20	2,14	1,83	1,58	7,04	5,51	5,54
\(\frac{1}{6}E_F\simeq \Delta E_{el}^{\mathrm{eV}}\)	0,55	0,53	0,36	0,31	0,26	1,17	0,92	0,92
\(\Delta E_R\)	—	—	—	—	—	−0,3	−0,3	−0,3
\(U_1=\Delta E_{el}+\Delta E_R\)	0,55	0,53	0,36	0,31	0,26	0,9	0,6	0,6

Attention is drawn to the fact that the parameters \(U_1\) and \(U_2\), according to data from different authors, differ, although the sum \(U_1+U_2\), within the limits of experimental error, always corresponds to the activation energy of self-diffusion \(U\). It may be assumed that the lower activation energy of motion (\(U_2\)) of vacancies is associated with the presence of vacancy pairs[^3].

In connection with what has been said, it is appropriate to raise the question of the manner in which vacancies are formed. It was usually assumed that, for a vacancy to be formed at some lattice site, it is necessary to transfer an atom from the site to the surface of the crystal or to the boundary between blocks. The independent formation of vacancies by diffusion into the interior of the crystal of “atoms” of emptiness was also assumed^5. However, apparently, it will be simpler to imagine that the formation of vacancies is connected with density fluctuations as a result of nonuniform thermal motion. In this case the appearance of a vacancy is connected with displacements of atoms around it, propagating in all directions with the speed of sound. At a sufficiently high cooling rate \((>10000\ \text{deg/sec}^1)\), one can, for example, fix practically all vacancies at \(1000^\circ\mathrm{C}\) in copper. However, if the cooling rate is insufficient, then they may be partially annealed out.

b) Diffusion of dissociated atoms can proceed in two ways: 1) by diffusion of a dissociated atom through interstices and 2) by displacement of a dissociated atom from an interstice into a normal lattice site, the ion located at the lattice site passing into an interstitial position. Calculations^2 have shown that the activation energy of diffusion in copper by the second method has a value of the order of \(10\) ev. The greater part of this value goes to the formation of the dissociated ion, while a small part of it corresponds to the activation energy of the motion of the dissociated atom. This means that the motion of a dissociated atom is no less probable than the motion of a vacancy. However, owing to the large energy of formation of dissociated atoms, their concentration, even at high temperature, still remains many orders of magnitude smaller than the concentration of vacancies. Taking for the energies of formation of a vacancy, a dissociated atom, and a pair of vacancies \(\sim 1\), \(\sim 10\), and \(\sim 1.6\) ev^2,^3, respectively, one can estimate the concentration of defects (the ratio of their number to the number of atoms) at various temperatures (see the table at right).

Types of defects	Concentration of defects	Concentration of defects	Concentration of defects
Types of defects	\(0^\circ\mathrm{C}\)	\(500^\circ\)	\(1000^\circ\)
Vacancies . . .	\(10^{-18}\)	\(10^{-6.5}\)	\(10^{-3.9}\)
Dissociated atoms . .	\(10^{-180}\)	\(10^{-65}\)	\(10^{-39}\)
Pairs of vacancies . . .	\(10^{-29}\)	\(10^{-10}\)	\(10^{-6.3}\)

c) Dislocations are defects of a more complex type than vacancies or dissociated atoms. As is known, they were introduced to explain the ease with which crystals can be plastically deformed. Recently dislocations have been invoked to explain the processes of catalysis, polygonization, crystal growth, the formation of color centers in alkali-halide crystals, etc. The characteristic difference between dislocations and the above-mentioned “point” defects is that they are not equilibrium distortions, do not arise as equilibrium formations below the melting point, although near it they may be formed in an almost equilibrium manner.

It may be supposed^6 that at least part of the dislocations arise during crystal growth and play a most important role in this process. In the most general form, a dislocation in a crystal is represented schematically in Fig. 1. Fig. 1 depicts a crystal cut by a surface \(S\), bounded by a closed curve \(C\); the latter may intersect the surface. The vector distance \(\mathbf{d}\) (the Burgers vector) represents the displacement of one half of the section with respect to the other, the same over the entire surface \(S\). The curve \(C\), usually called the principal dislocation line or dislocation ring, may be formed by vacancies or interstitial ...

atoms, and the vector d in these two cases will have opposite signs. Seitz\(^7\) believes that the indicated method of practical charging of dislocations is important, for example, in cooling a melt in which there is a high concentration of vacancies or interstitial atoms. Vacancies may, for example, condense on existing dislocations or, upon condensing, form new dislocations in the form of plates or spirals equivalent to dislocations. It should be noted that, by means of the condensation of vacancies or interstitial atoms, only such dislocation rings can be formed as have a definite projection onto a plane perpendicular to the Burgers vector. If, however, the ring \(C\) lies in a plane parallel to the Burgers vector, then it cannot be formed by the above method.

Fig. 1. Schematic representation of a dislocation in a crystal.

Particular cases of the dislocation considered (Fig. 1) are: the Taylor–Orowan linear dislocation and the Burgers screw dislocation.

Calculations\(^8\) have shown that if a dislocation ring lies in the slip plane, then both types of dislocations can move parallel to themselves with the aid of very small stresses, \(10^5\) dyn/cm\(^2\). This makes the crystal very plastic. As a result of the expansion of the ring during plastic flow, the length of the dislocation line in the crystal increases. It is expedient to express the dislocation density by the number of intersections of dislocation lines with a unit area in the crystal. For good natural crystals this quantity is of the order of \(10^8\) per cm\(^2\), for artificial crystals \(10^9\) per cm\(^2\), and in cold-worked material \(10^{12}\) per cm\(^2\).

Dislocation rings interact with one another. If, for example, two dislocations of different signs move in different directions, then, on meeting in one plane, they may annihilate with the release of a relatively large energy\(^9\). During their motion, dislocations can excite defects of the vacancy or displaced-atom type, which is a consequence of the high local temperature\(^ {10}\) developing during the motion of individual dislocation lines. Point defects also arise as a result of the interaction and partial annihilation of dislocations. It is believed\(^7\) that screw dislocations can act as sources or sinks for vacancies and interstitial atoms.

The idea that slip in crystals is carried out by the motion of dislocations along the slip plane assumes that sources of dislocations must exist in the material\(^ {11}\). In a metal subjected to stresses, the indicated sources can form a successive series of dislocations.

Frank and Read\(^ {12}\) proposed a hypothesis concerning the nature of these sources. They showed that a certain arrangement of dislocations makes it possible to form new dislocations in a stressed material.

In annealed material there exists a spatial network structure of dislocations of density \(D_0\) per cm\(^2\). Each element of this structure can, in principle, act as a Frank–Read source. Only those elements are important which are favorably oriented—

are oriented with respect to the applied stress and, at length \(l\), are capable of being activated. The critical shear stress required for activation of a source is expressed by the formula

\[ \tau_k = \alpha \frac{Gd}{l}, \tag{2} \]

where \(G\) is the shear modulus, \(d\) is the Burgers vector, and \(\alpha\) is a quantity (\(\approx 1\)) depending on the elastic constants of the material.

Although there is as yet no experimental evidence for the existence of the indicated sources, recently \(^{13}\) it has been possible, with the aid of an electron microscope, to photograph a linear dislocation in a crystal of platinum phthalocyanine about \(\sim 200\ \text{Å}\) long (Fig. 2).

Fig. 2. Edge dislocation in a platinum phthalocyanine crystal.

The energy of formation of dislocations in metals is practically elastic energy, which can be estimated at approximately \(2\)--\(3\ \text{eV}\) per atomic plane \(^{14}\). As regards the activation energy for the motion of dislocations, there is complete uncertainty here because very little is known about the diffusion (“creep”) of dislocations or of their parts. The difficulty here is that during dislocation motion the activation energy changes continuously as a result of the arising stressed state, which depends on the interaction of dislocations. From studies \(^{15}\) of recrystallization and polygonization of pure aluminum obtained by zone melting, it can only be assumed that dislocation motion requires a high activation energy, of the order of the activation energy of self-diffusion.

Table II gives data on the energy of formation and the activation energy for the motion of various types of defects, as well as the increase in electrical resistivity caused by them in copper.

Table II

Type of defect	Formation energy in eV	Activation energy of motion in eV	Increase in electrical resistivity
Vacancy	0.9	1.2	\(1.53 \cdot 10^{-21}\ \mu\Omega\,\text{cm}/\text{cm}^3\)
Vacancy pair	\(\sim 1.6\)	0.25—0.6	\(3 \cdot 10^{-21}\ \mu\Omega\,\text{cm}/\text{cm}^3\)
Interstitial atom	from 5 to 10	0.1—0.25	\(5.7 \cdot 10^{-21}\ \mu\Omega\,\text{cm}/\text{cm}^3\)
Dislocation	\(\sim 3\)	\(>2\)	\((0.4—1.5)\ 10^{-14}\ \mu\Omega\,\text{cm}/\text{cm}^2\)

2. METHODS FOR DETERMINING THE DENSITY OF DEFECTS OF CRYSTALLINE STRUCTURE

All methods for determining defect density are based on a regular relationship existing between the density of defects and the change in one or another property of metallic crystals.

Thus, for example, to determine the energy of formation of vacancies, and consequently also their density, the method of quenching \(^{16}\) metal from high temperatures can be used. In this case a certain excess number of vacancies is fixed. The latter, playing the role of an impurity in the metal,

cause an additional electrical resistance proportional to the concentration of vacancies. By measuring the residual electrical resistance after quenching from various temperatures, it was thus possible to determine the energy of vacancy formation. Thus, for example, for gold it turned out that \(\Delta H_f = 18.2\) kcal/g·atom, and for platinum \(\Delta H_f = 27.2\) kcal/g·atom. In observing the recovery of electrical resistance at various temperatures, the activation energies of defect motion were also determined\({}^{16}\); they proved to be 12 kcal/g·atom for gold and 25 kcal/g·atom for platinum.

The activation energy of self-diffusion for gold is \(\sim 46\) kcal/g·atom\({}^{17}\). On the other hand, this quantity may be obtained from the sum of the values of the energy of vacancy formation and the activation energy of their motion. From the values obtained\({}^{16}\) we get, for gold, \(E_a = 30.2\) kcal/g·atom, which is clearly less than the activation energy found from the study of self-diffusion and equal to 46 kcal/g·atom\({}^{17}\).

Such a discrepancy could be explained by assuming that the activation energy of defect motion found by the authors corresponds not to single vacancies, but to paired vacancies, for which the activation energy of motion may be more than a factor of two smaller than for single ones.

In work\({}^{18}\), thin gold wires (99.999%) were quenched in the temperature range 690–900° C. The cooling rate was \(\sim 80\,000\) deg/sec. The residual electrical resistance was measured at liquid-helium temperature. The authors obtained the following values for the energy of vacancy formation and the activation energy of their motion: \(\Delta H_f = 29.5\) kcal/g·atom and \(\Delta H_m = 15.7\) kcal/g·atom. Hence one can obtain a value for the activation energy of self-diffusion which agrees well with other data\({}^{17}\).

Another method for determining the value of the vacancy-formation energy, given in work\({}^{19}\), is based on measuring the deviation of the coefficient of thermal expansion from the linearity of its temperature dependence. This deviation is especially noticeable at high temperatures. It is assumed that the additional increase in the unit length of a specimen at high temperatures is caused by the appearance of vacancies at lattice sites. In this way the vacancy-formation energies for several metals were determined: \(\Delta H_f(\mathrm{Ag}) = 15.7\) kcal/g·atom, \(\Delta H_f(\mathrm{Cu}) = 17.9\) kcal/g·atom, and \(\Delta H_f(\mathrm{Al}) = 11.8\) kcal/g·atom; moreover, it turned out that these values constitute approximately 33% of the activation energy of self-diffusion.

Knowing the energy of vacancy formation, one can determine their concentration at any temperature by the formula

\[ \frac{n}{N}=\exp\left(-\frac{\Delta H_f}{kT}\right), \tag{3} \]

where \(N\) is the total number of atoms. It turned out that, at the melting temperature, the vacancy concentrations for silver, copper, and aluminum are \(1.74\cdot 10^{-3}\), \(1.35\cdot 10^{-3}\), and \(2.0\cdot 10^{-3}\), respectively.

Experiments on the study of anelastic effects\({}^{20}\) also make it possible to determine the density of defects in a metal. In work\({}^{20}\), the mobility of atoms in alloys \(\mathrm{Ag} + 30\%\ \mathrm{Zn}\) was measured after quenching excess vacancies from 400° C, the measurements being carried out at 30–70° C. From the change of the initial relaxation time with temperature, it was possible to separate the activation energy into the corresponding component parts. It turned out that the vacancy-formation energy is \(\Delta H_f = 11.8\) kcal/m, and the activation energy of their motion is \(\Delta H_m = 19.7\) kcal/m, so that the total activation energy is \(\Delta H_r = \Delta H_f + \Delta H_m = 31.5\) kcal/m, and the ratio \(\Delta H_f/\Delta H_r = 0.37\).

Mechanical properties of crystals are associated with dislocations, and their density is a parameter determining the degree of perfection of the crystal.

Determination of the density of dislocations in crystals obtained by slow growth from a solution or vapor phase indicates^21 that this quantity varies between \(10^4\) and \(10^6\) per \(\text{cm}^2\). The density of etched microscopic pits on the surface of a well-annealed aluminum crystal, slowly grown from the melt and well annealed, is likewise of the order of \(10^6\) per \(\text{cm}^2\).

To reveal dislocations in silver, thermal etching at \(600^\circ\) and low oxygen pressure was used^22. It turned out that the number of microscopic pits on the surface of a single crystal does not depend on the etching time, while their shape changes depending on the complexity of the deformation. When the surface of bent crystals is etched, the density of pits proves to be proportional to the reciprocal of the radius of bending and agrees with the calculated value of the density of linear dislocations corresponding to the given deformation of the crystal. Prolonged annealing of a crystal at high temperatures makes it possible to observe in the specimen the formation and growth of subgrains. The latter are slightly disoriented with respect to the base and relatively free of dislocations. The dislocation density determined by the authors for silver crystals, both grown from the melt and obtained by recrystallization methods, is \(2 \cdot 10^6\) per \(\text{cm}^2\). It was shown that the above method can also be applied to detecting the distribution of dislocations along slip bands in a crystal. This by no means implies that every microscopic pit represents a dislocation, or that every dislocation forms a corresponding pit. The number obtained can indicate only the order of magnitude of the density of dislocations arising under the indicated conditions.

All these values of dislocation density are considerably smaller than those values^23 which are obtained from the energy released during annealing of deformed metals Cu and Cu + 0.35 As and Ni. Specimens of these metals were subjected to various types of deformation (torsion, tension, and compression) at room temperature, followed by tempering at a constant heating rate. By the method of a differential calorimeter^24, the energy released during recrystallization was measured, and it was shown that this energy is a linear function of deformation. It is assumed^23 that the release of energy during recrystallization is connected with disappearance, through annihilation, of dislocations arising during deformation of the crystals. If this is true, then the store of energy released during recrystallization represents the energy of these dislocations. From the magnitude of the energy per unit length of individual dislocations (\(5 \cdot 10^{-4}\) erg/cm for copper and \(7 \cdot 10^{-4}\) erg/cm for nickel^14), the dislocation densities were calculated (Table III)^23. It is assumed that the dislocation density is inversely proportional to the indicated energies.

It is interesting to note that, for one and the same deformation, the calculated values of the dislocation density for different metals are close to one another.

The value of the dislocation density obtained from X-ray data^25 is likewise \(\sim 10^{11}\ \text{cm}^{-2}\).

The estimate of the dislocation density given above does not take into account the energy of interaction between them. The latter may vary depending on the density and distribution of the dislocations, on the degree of deformation, and on the method of tempering. In addition, the above values of the energies of individual dislocations are rather arbitrary. Thus,

Table III

Metal	Type of deformation	Energy released during recrystallization (cal/g)	Dislocation density (number of lines per cm²)
Electrolytic copper	Compression	0.065	\(4.8 \cdot 10^{10}\)
Electrolytic copper	Tension	0.075	\(5.6 \cdot 10^{10}\)
Electrolytic copper	Compression	0.106	\(7.9 \cdot 10^{10}\)
Electrolytic copper	Torsion	0.145	\(1.1 \cdot 10^{11}\)
Electrolytic copper	Compression	0.167	\(1.2 \cdot 10^{12}\)
Arsenical copper	Tension	0.050	\(3.7 \cdot 10^{10}\)
Arsenical copper	Torsion	0.056	\(4.2 \cdot 10^{10}\)
Arsenical copper	Torsion	0.13	\(9.7 \cdot 10^{10}\)
Arsenical copper	Torsion	0.19	\(1.4 \cdot 10^{11}\)
Arsenical copper	Torsion	0.26	\(1.9 \cdot 10^{11}\)
Nickel	Torsion	0.12	\(6.4 \cdot 10^{10}\)
Nickel	Torsion	0.18	\(9.6 \cdot 10^{10}\)
Nickel	Torsion	0.24	\(1.3 \cdot 10^{11}\)
Nickel	Torsion	0.30	\(1.6 \cdot 10^{11}\)

indicating the dependence of the dislocation density on the degree of deformation, it is necessary to note that the quantities given are only of an approximate character.

Another method for estimating the dislocation density is based on measuring the internal friction during aging of cold-worked ferrite containing a small amount of carbon \(^{26}\).

It is assumed that during aging of a deformed alloy the dissolved atoms gather at free dislocations, and the rate of segregation decreases in proportion to the amount of precipitated substance. For the fraction of segregated atoms one obtains

\[ f = 1 - \exp \left[-\alpha \rho \left(\frac{ADt}{kT}\right)^{2/3}\right], \tag{4} \]

where \(\rho\) is the dislocation density, \(D\) is the diffusion coefficient of carbon atoms, \(t\) is the aging time, \(T\) is the temperature, and \(A\) is a quantity depending on the elastic properties of the material, as well as on the degree of deviation of the position of the dissolved atom from the lattice site and on the dislocation effect. For a carbon atom in \(\alpha\)-Fe, \(A \simeq 3 \cdot 10^{-20}\ \text{dyne}\cdot\text{cm}^2\), \(\alpha \simeq 3\) \(^{27}\). Fig. 3 shows the influence of temperature on the rate of aging of ferrite \(^{26}\). The results obtained are in agreement with (4). From these data the activation energy is \(20\,000\ \text{cal/mol}\),

Fig. 3. Influence of temperature on the aging rate of deformed ferrite.

which is in agreement with the value of the activation energy for the diffusion of carbon in $\alpha$—Fe. For a material deformed by 10%, the dislocation density was found to be $2.5 \cdot 10^{11}\ \text{cm}^{-2}$.

This agrees with the estimate made on the basis of studying the aging process by the electrical-resistivity method $^{27}$. In approximate agreement with theory $^{28}$ was also the empirical dependence of the dislocation density on the magnitude of plastic deformation $(\varepsilon)$, found $^{29}$ for deformed ferrite from measurements of internal friction:

\[ \rho = 0.54 \cdot 10^{11}\varepsilon^{1/2}\ \text{cm}^{-2}. \tag{5} \]

In addition to the indicated methods, it should be noted, as was mentioned above, that the dislocation density can also be estimated from measurements of the electrical resistivity of deformed metals $^{28,30}$.

All estimates of dislocation density made on the basis of measurements of electrical resistivity give a value considerably larger than that obtained by other methods. For metals strengthened by cold working, the dislocation density measured by various methods $^{14}$ lies within the range from $10^{8}$ to $10^{12}$ lines/cm$^{2}$.

This circumstance is probably explained by the fact that, although the observed change in electrical resistivity is almost isotropic, it is possible that the dislocations are not arranged completely chaotically. In addition, the calculation carried out $^{30}$ of the electrical resistivity due to dislocations does not take into account the influence of special groups, the so-called “cluster” dislocations, which may exist in a strain-hardened metal $^{31}$. The calculations also do not take into account the fact that in close-packed metals dislocations may split into parts $^{32}$.

3. THE INFLUENCE OF DEFECTS OF CRYSTALLINE STRUCTURE ON THE ELECTRICAL RESISTIVITY OF METALS AND ALLOYS

As is known, electrical resistivity is a structure-sensitive property. This makes it possible to use this property as a sensitive indicator of subtle structural changes occurring in metals and alloys during cold and thermal treatment.

The problem arising in this connection consists in studying the change in electrical resistivity caused by different types of defects. Here there is an ambiguity connected with the fact that it is difficult to separate the influence of defects of different types. Indeed, on the one hand, vacancies, interstitial ions, and dislocations may influence electrical resistivity in one and the same direction, while, on the other hand, their simultaneous presence in a metal inevitably leads to a change in electrical resistivity caused by the interaction of different types of defects.

The influence of defects of different types on the electrical resistivity of metals was studied in works $^{33-36}$ and discussed in a review article $^{23;45}$.

The presence of vacancies in the crystal lattice should affect electrical resistivity in the same way as impurities of foreign metals do. The increase in electrical resistivity caused by one atomic percent of vacancies, according to $^{33}$, is approximately $0.4\ \mu\Omega\ \text{cm}$ for copper, silver, and gold.

This estimate, however, proves to be too low. In work $^{34}$ it is indicated that the influence of vacancies is considerably greater, and for copper the increase in electrical resistivity per one atomic percent of vacancies is $\sim 1.3\ \mu\Omega\ \text{cm}$. In $^{35}$ it is indicated that groups of vacancies of two, three, or more should lead to a decrease in electrical resistivity. Up to now

however, there are no calculations of the effect of the influence of a group of vacancies on the electrical resistivity of metals.

The only estimate of the influence of interstitial ions on the electrical resistivity of a metal was made in work \(^{33}\). For copper, silver, and gold the increase in electrical resistivity is \(\sim 0.6\,\mu\Omega\text{ cm}\) per one atomic percent of interstitial atoms.

The influence of dislocations on the scattering of conduction electrons has been the subject of theoretical investigation by many authors. This question was considered most fully in work \(^{30}\).

An estimate of the influence of dislocations on electrical resistivity must take into account their anisotropic action, as is evidenced by the electrical resistivity of strongly deformed metals.

Thus, for example \(^{37}\), it was found that the electrical resistivity of rolled metal strips in the longitudinal and transverse directions is not the same (Table IV).

Table IV

Metal (alloy composition in wt. %)	Electrical resistivity at \(20^\circ\text{C}\) (annealed state)	Increase in electrical resistivity at \(20^\circ\) in %, longitudinal	Increase in electrical resistivity at \(20^\circ\) in %, transverse
Copper (strip)	1.68	\(3.20 \pm 0.14\)	\(3.27 \pm 0.14\)
Nickel »	9.94	\(3.44 \pm 0.11\)	\(3.39 \pm 0.11\)
75/25 copper—nickel (strip)	30.56	\(3.34 \pm 0.16\)	\(4.03 \pm 0.16\)
70/30 brass (strip)	6.17	\(22.97 \pm 0.19\)	\(21.16 \pm 0.19\)
Copper (wire)	1.69	\(2.3 \pm 0.2\)	\(2.0 \pm 0.2\)
80/20 brass (wire)	5.47	\(18.4 \pm 0.1\)	\(14.9 \pm 0.3\)
94/6 aluminum bronze (wire)	11.13	\(27.4 \pm 0.1\)	\(24.2 \pm 0.3\)

The scattering of electrons in a deformed metal was investigated by means of perturbation theory, in which it is assumed that the perturbing potential is proportional to the elastic deformation. It turned out that in the direction of the axis of an edge dislocation the electrical resistivity is absent, while in the direction of slip it amounts to one third of the electrical resistivity in the direction perpendicular to the slip plane. For a screw dislocation the electrical resistivity is absent in the direction parallel to the axis, while in the direction perpendicular to the axis the scattering of electrons occurs isotropically.

However, for comparison of theory with experiment in the first approximation the anisotropy effect may be disregarded. On the other hand, since this effect is small, there is reason to consider the influence of the statistical arrangement of dislocations on the electrical resistivity.

In connection with this, work \(^{30}\) gives an estimate of the isotropic increase in electrical resistivity \((\Delta\rho)\) at a dislocation density \(N\) per \(\text{cm}^2\), assuming that they are oriented arbitrarily. The authors obtained the following values:

\[ \begin{aligned} \text{Copper:}\quad &\text{edge dislocations} && \Delta\rho_E = 0.59\cdot 10^{-14}N\,\mu\Omega\text{ cm},\\ &\text{screw dislocations} && \Delta\rho_S = 0.18\cdot 10^{-14}N\,\mu\Omega\text{ cm},\\ \text{Sodium:}\quad &\text{edge dislocations} && \Delta\rho_E = 2.10\cdot 10^{-14}N\,\mu\Omega\text{ cm},\\ &\text{screw dislocations} && \Delta\rho_S = 0.18\cdot 10^{-14}N\,\mu\Omega\text{ cm}. \end{aligned} \]

If one assumes an equal density of edge and screw dislocations, then for copper we obtain \(\Delta\rho = 0.4\cdot 10^{-14}N\,\mu\Omega\text{ cm}\). Even if one assumes that almost half of the observed change in electrical resistivity in co-

...of deformed copper is due to vacancies and interstitial atoms, then the dislocation density can be estimated. For \(\Delta \rho = 0.02\,\mu\Omega\cdot\text{cm}\) we find \(N = 5\cdot 10^{12}\) per \(\text{cm}^2\).

It should be noted, however, that to take account of distortions around a dislocation the calculations\(^{30}\) were carried out using a classical formula, which is invalid in the immediate vicinity of the dislocation line. As indicated in \(^{38}\), atoms in a linear dislocation may be arranged in two different ways; one should distinguish between so-called “open” and “close-packed” dislocations. Their formation depends on the character of the forces acting between atoms. An open dislocation apparently affects electrical resistivity more strongly than a close-packed one. An estimate of the effect of an open dislocation (1 cm of length per \(\text{cm}^3\) of metal) on the electrical resistivity of copper\(^{38}\) gave, on average, a value \(\sim 1.5\cdot 10^{-14}\,\mu\Omega\cdot\text{cm}\). Thus, in general, the change in electrical resistivity due to dislocations can be expressed as

\[ \Delta \rho = (0.4—1.5)\cdot 10^{-14}N\mu\Omega\cdot\text{cm}. \]

Table II (p. 103) gives a summary of the data obtained on the influence of various types of defects on the electrical resistivity of copper.

a) Effect of heat treatment on electrical resistivity

According to statistical thermodynamics, the equilibrium concentration of defects of a given kind is determined by the heat of formation of the defect \((\Delta H_f)\) and the change in entropy \((\Delta S_f)\). The dependence of the defect concentration on temperature is given by the expression

\[ C = \exp\left[-\frac{(\Delta H_f - T\Delta S_f)}{kT}\right]. \tag{6} \]

When the temperature is lowered, the concentration of defects decreases. However, during cooling it is practically impossible to attain equilibrium, since the defects cannot diffuse rapidly out of the lattice. Therefore they become “frozen in.”

The best “freezing-in” effect for defects can be obtained by quenching from a high temperature. It turned out that in this way it is actually possible to “freeze in” only one type of defect—vacancies (single or paired).

It should be noted that such experiments began to be carried out only recently; this is explained by the erroneous notion that, upon cooling, the vacancy concentration should supposedly always become almost equilibrated, and that it is difficult to “freeze in” vacancies in appreciable quantity.

Assuming that for vacancies, interstitial ions, and vacancy pairs in copper the energies of their formation, in accordance with the estimate\(^{2,3}\), are respectively \(1;\ 10;\ 1.6\) eV, one can compare the relative concentrations of the various types of defects at one temperature. Thus, for example, at \(1000^\circ\text{C}\) it turns out that the concentration of vacancies is \(\sim 10^{-3}\),\(^{9}\) of interstitial ions \(\sim 10^{-39}\), and of vacancy pairs \(\sim 10^{-6.3}\). Thus, the greatest effect upon quenching can be obtained mainly from “frozen-in” single vacancies.

As already noted above, with very rapid quenching, when it may be assumed that all excess vacancies are “frozen in,” the residual...

the electrical resistivity at the temperature of its measurement is assumed to be proportional to the vacancy concentration

\[ \Delta \rho_q = Ac, \tag{7} \]

then

\[ \ln \Delta \rho_q = -\frac{\Delta H_f}{kT} + \frac{\Delta S_f}{k} + \ln A. \tag{8} \]

Using the values of \(\Delta S_f^{39}\) for various metals from the theory of self-diffusion and the values of \(\Delta H_f^{16}\), measured from quenching experiments, one can estimate the coefficient \(A\) in (7).

As was shown by us \(^{39}\), the activation entropy of self-diffusion \((\Delta S_a)\) consists of the sum of the entropy change associated with the change of volume \((\Delta S_v)\), and the entropy change associated with the disordering of the structure. The latter occurs upon the formation of a vacancy and is the quantity \(\Delta S_f\) denoted above. Thus, according to \(^{39}\) we have

\[ \Delta S_a = \Delta S_v + \Delta S_f = \frac{3}{2}\alpha \chi^{-1} v_a + k \ln \left\{\frac{1}{2}\left(\frac{T}{\theta}\right)^2 [1-\Phi(y)]\right\}, \tag{9} \]

where \(\alpha\) is the expansion coefficient, \(\chi\) the compressibility, \(v_a\) the atomic volume, \(\Phi(y)\) the Gaussian function, \(y=\dfrac{0.1r}{1.41\sqrt{\overline{u_x^2}}}\), \(r\) half the interatomic distance, and \(\overline{u_x^2}\) the mean-square displacement of atoms from the equilibrium position, associated with the characteristic temperature \(\Theta\) and the Debye function \(D(x)\):

\[ 2\pi^2 \overline{u_x^2} = \frac{3h}{2m_a k\theta} \left[ \frac{D(x)}{x}+\frac{1}{4} \right]. \tag{10} \]

Let us try to estimate the increase in electrical resistivity for gold at 1 at.% vacancies. From \(^{16}\), for Au

\[ \Delta H_f = 18.2\ \text{kcal/g-at.}, \]

and, calculated from (9),

\[ \Delta S_f = 4.9\ \text{kcal/deg}\cdot\text{g-at.} \]

In work \(^{18}\) it was found that, upon quenching from \(690^\circ\mathrm{C}\), the increase in electrical resistivity is \(0.065\%\). Taking into account the specific electrical resistivity of gold \(\rho = 2.19\ \mu\Omega\,\text{cm}\), we obtain from (8)

\[ A = 1.16\ \frac{\mu\Omega\,\text{cm}}{\text{at.}\%\,\text{vac}}, \]

which agrees well with the estimate made in work \(^{34}\). Quenching experiments are complicated by a number of circumstances depending on the interaction of defects of various types. These include the interaction of vacancies with dissolved atoms or dislocations, as well as the possible formation of vacancy pairs \(^{40}\). This must undoubtedly affect the results of determining both the energy of vacancy formation and the activation energy of their motion. The results of such measurements for Au, Pt, and Ag have already been given in § 2. For alloys, moreover, a complicating circumstance is the possibility of the emergence of short-range or long-range order during heat treatment.

In work \(^{41}\) the influence of the quenching temperature \(T_q\) on the relative increase in electrical resistivity \(\Delta \rho_q/\rho\) (Fig. 4) of \(\alpha\)-brass of compositions 70/30 and 71/29 was studied. Measurements of electrical resistivity were carried out at \(-183^\circ\mathrm{C}\), and it was found that \(\rho_{-183} \simeq 4\ \mu\Omega\,\text{cm}\). It was shown that the relative

the change in electrical resistivity does not depend on the wire diameter. Thus, the observed effect cannot be attributed to quenching stresses, whose magnitude, as is known, should depend on the size and shape of the specimen. From the slope of the curve in Fig. 4 one can estimate the energy of formation of defects in the indicated alloys. It turned out that \(\Delta H_f = 8\) kcal/g-at. It is possible that this relatively small value cannot be attributed to the formation of single vacancies, but it is also quite probable that vacancies are formed predominantly near clusters of zinc atoms. However, the effect of the formation of paired vacancies should apparently manifest itself at higher temperatures. This could explain the deviation from linearity in the curve of Fig. 4 at temperatures above \(300^\circ\)C.

Fig. 4. Change in the electrical resistivity of \(\alpha\)-brass as a function of quenching temperature.

Quenched specimens of the alloy were annealed in the temperature range \(50^\circ\)—\(100^\circ\)C. The observed activation energy of the return proved to be \(1.0 \pm 0.2\) eV.

It is quite clear that during annealing the electrical resistivity decreases because the “frozen-in” defects (vacancies) relax.

As was shown in \(^{42}\), the relaxation time \(\tau \sim \dfrac{1}{D_v}\), where \(D_v\) is the vacancy diffusion coefficient. The temperature dependence of the latter is connected with the activation energy for vacancy motion \(\Delta H_m\)

\[ D_v = D_{0v}\exp\left(-\frac{\Delta H_m}{kT}\right). \tag{11} \]

Thus, from the curve of the dependence of \(\ln \tau\) on \(\dfrac{1}{T}\) one can determine the value \(\Delta H_m\). From the obtained values \(\Delta H_f = 0.34\) eV and \(\Delta H_m = 1.0\) eV it could be expected that, if vacancies exchange predominantly with zinc atoms \(^{43}\), then the activation energy of diffusion in alpha-brass should be \(E_a \simeq 1.34\) eV. For the alloy of the indicated composition, \(E_a \simeq 1.35\) eV was obtained \(^{44}\).

It should be noted that the question of the possibility of explaining the indicated experiments by the formation of short-range order in alpha-brass is controversial \(^{45}\).

Very important for understanding the mechanism of ordering in alloys are experiments on the study of electrical resistivity during annealing of quenched ordering alloys.

The dependence of the electrical resistivity of the disordered alloy \( \mathrm{Cu}_3\mathrm{Au} \) at a constant annealing temperature (\(150^\circ\mathrm{C}\)) on the quenching temperature (\(542^\circ\text{--}704^\circ\mathrm{C}\)) is shown in Fig. 5 \({}^{46}\). The initial slope of these curves is apparently proportional to the ordering rate and indicates that the ordering rate at \(150^\circ\mathrm{C}\) increases with increasing quenching temperature.

An interpretation of these data on the basis of the assumption that the ordering process is due to diffusion of vacancies is not without foundation,

Fig. 5. Dependence of the ordering rate in \(\mathrm{Cu}_3\mathrm{Au}\) on the quenching temperature.

i.e., that vacancy migration increases the degree of order in the alloy. This is facilitated by the considerable lifetime of a vacancy. During this time the vacancy is able to make a large number of jumps, in the course of which rearrangement in the positions of the atoms takes place. It then turns out that the decrease in electrical resistivity caused by an increase in the degree of order as a result of the migration of individual vacancies is much greater than the decrease in electrical resistivity associated with the disappearance of vacancies.

It has been shown \({}^{46}\) that the ordering rate decreases with time. This is what should be expected if nonequilibrium vacancies are retained during quenching. Considering the ordering and disordering of the alloy \(\mathrm{Cu}_3\mathrm{Au}\) after quenching from \(542^\circ\mathrm{C}\), it was shown \({}^{46}\) that the ordering rate depends mainly on the concentration of vacancies.

b) Influence of defects arising during deformation on the electrical resistivity of metals

There are many works on the question of the influence of elastic and plastic deformation on the electrical resistivity of metals. It has been established that elastic deformation generally affects electrical resistivity much less than plastic deformation. It has been established that electrical resistivity is a linear function of elastic deformation. However, the data of different authors are not always unambiguous. Experiments \({}^{47}\), carried out with single crystals of \(\alpha\)-brass (\(72\%\ \mathrm{Cu} + 28\%\ \mathrm{Zn}\)), showed that their electrical resistivity first increases, and with further

the slip of the crystal remains constant until slip begins along another system of planes.

In Fig. 6 it is seen that the second increase in electrical resistance begins at the moment when a kink appears on the deformation curve; in this case slip proceeds along a new system of planes. To explain this fact it is assumed\(^{45}\) that the change in electrical resistance is caused by vacancies or displaced atoms arising when screw dislocations intersect.

Experiments with polycrystalline wires of various metals have shown that deformation

\[ \left(\varepsilon=\frac{\Delta l}{l}\right) \]

at low temperatures changes the electrical resistance according to the law

\[ \frac{\Delta \rho}{\rho}\sim \varepsilon^{3/2}. \tag{12} \]

To explain this, it is proposed\(^{48}\) that, during their motion, dislocations intersect other, randomly oriented dislocations. At the obstacles thereby arising, vacancies or displaced atoms are formed. Their density is proportional to \(\varepsilon^{3/2}\). The presence only of dislocations changes the electrical resistance under deformation in proportion to \(\varepsilon^{1/2}\), so that the total increase in electrical resistance should depend on the deformation according to the law

\[ \Delta \rho=A\varepsilon^{1/2}+B\varepsilon^{3/2}, \tag{13} \]

where \(A\) and \(B\) are constants of the material. That the increase in electrical resistance during plastic deformation is to a considerable extent due to vacancies arising when dislocations intersect is also indicated by data\(^{49}\) for copper, aluminum, and iron. It was found that deformation has a much smaller effect on the electrical resistance of fully recrystallized wires. It may be assumed that during recrystallization the metal is freed of some defects. However, for alpha-brass wire annealed at various temperatures, such an effect was not observed\(^{49}\).

Fig. 6. Effect of deformation on the electrical resistance of an \(\alpha\)-brass single crystal.

A considerably greater change in electrical resistance is observed in wires after deformation by drawing or rolling. This is explained by the formation of a significantly greater concentration of defects in the lattice.

Table V gives data\(^{45}\) on the maximum change in electrical resistance of metals and alloys after considerable degrees of deformation. Molybdenum and tungsten, which have a high recrystallization temperature, give a substantial increase in electrical resistance, which is explained by the significant concentration of frozen-in defects under deformation conditions at room temperature.

Seitz\(^{35}\) suggested that during deformation vacancies and displaced atoms are formed in almost equal quantities, but under some

Table V

Effect of large degrees of deformation at room temperature on electrical resistivity

Metals and alloys	Electrical resistivity of annealed material at 20° C, in μΩ·cm	Reduction of cross section, in %	Increase in electrical resistivity, in %
1	2	3	4
b) Metals with a cubic lattice
Aluminum	2.67	88	0.5
Copper	1.67	87	2.4
Gold	2.35	98	1.6
Iron	10.19	85	1.3
Molybdenum	6.1	>99	18
Nickel	8.21	90	3.9
Palladium	10.8	96	4
Platinum	10.6	96	1
Silver	1.59	98	5
Tungsten	5.49	>99	50
b) Alloys (composition in wt. %) Solubility limit
Copper 7.5% Al ∼8.5%	12.3	75	33
Copper 10% Au 100%	∼7.8	97	1.7
Copper 20% Mn ∼15%	60	97	10
Copper 20% Ni 100%	∼26.5	84	3
Copper 3% Si ∼4%	22.4	84	10
Copper 2.4% Sn —	5.4	84	2
Copper 30% Zn ∼36%	6.3	84	26
Gold 15% Ag 100%	8.2	97	1.4
Gold 65% Ag 100%	10.9	90	1.1
Gold 75% Ag 100%	∼6.4	97	0.8
Platinum 25% Ir 100%	33	50	0.5
Silver 30% Cd∼36%	—	97	28
Silver 23% Zn∼24%	—	97	19
c) Ordering alloys
Ag₃Mn ordered	4.8	90	100
Ag₃Mn disordered	9.3	90	8
Cu₃Au ordered	6.5	70	82
Cu₃Au disordered	11.6	70	2
Ni₃Fe ordered	—	95	35
Ni₃Fe disordered	—	95	15
Ni₃Mn ordered	—	95	62
Ni₃Mn disordered	—	95	—3
CuZn ordered	5.3	∼95	∼40

under which conditions many more vacancies may form than displaced atoms.

During annealing of deformed copper, two states are observed^50 in the change of electrical resistivity: one in the region of 30° C is characterized by an activation energy of 0.7 eV, while the other in the region of 150° C is characterized by an activation energy of 1.19 eV.

This can be explained by assuming^46 that the low-temperature activation is associated with the motion of displaced atoms. In that case, upon annihilation of vacancies with displaced atoms, another annealing state is also possible at a higher temperature, associated with the migration of unannihilated vacancies.

Meanwhile, bearing in mind that the activation energy of self-diffusion in copper is \(2.07\) eV \(^{17}\), one can determine the energy of vacancy formation, equal to \(2.07-1.19=0.88\) eV. This agrees with the data obtained in work \(^{19}\). Table V gives data on the effect of large degrees of plastic deformation on the electrical resistance of single-phase alloys. In presenting these data, the authors in \(^{46}\) do not note any general tendency in their behavior. Indeed, what attracts attention is the considerable increase in electrical resistance for those \(\alpha\)-solid solutions whose composition is close to the solubility limit. If this is so, then at large degrees of deformation such alloys may be transferred into the category of supersaturated solid solutions. The decomposition of the solid solution that begins as a result of this should give an additional increase in electrical resistance. This circumstance must apparently change the general course of the increase in the electrical resistance of the \(\alpha\)-solid solution as a function of composition at large degrees of deformation. This, apparently, can explain the results given for brass in Fig. 7 \(^{46}\). Whereas at degrees of deformation up to \(\sim 60\%\) the curve of the dependence \(\Delta \rho/\rho=f(\% \mathrm{Zn})\) tends toward saturation, at deformations greater than 80% we observe a tendency toward an additional increase in electrical resistance, the greater the closer the alloy composition is to limiting saturation.

Fig. 7. Effect of the degree of deformation on the dependence of the change in electrical resistance of \(\alpha\)-brass on zinc concentration.

A large increase in the electrical resistance of deformed alloys is observed upon transition from one close-packed structure to another. This occurs, for example, in Co—Ni alloys, in which at room temperature the hexagonal structure in the range 0–30% (wt.) Ni is replaced by the structure of a face-centered cube in the range 30–100% Ni.

A considerable increase in electrical resistance is observed precisely in the region around 30% Ni.

As X-ray studies have shown, an accumulation of defects arises in this region. This, however, does not prove that they have a direct influence on the scattering of conduction electrons.

It is possible that the presence of the mentioned clusters causes the formation of defects of the vacancy type and of dislocated atoms, which are mainly responsible for the increase in electrical resistance. Although the mechanism of formation of these defects is not clear, it may nevertheless be assumed that during the transition from one quasi-equilibrium structure to another in deformed alloys, conditions should be realized that contribute to a lowering of the free energy; this is possible, for example, through the formation of vacancy-type defects.

Of great interest is the effect of deformation on the electrical resistance of ordered alloys.

It is known that plastic deformation can completely disorder an ordered alloy; in this case the electrical resistance increases considerably. It is significant that deformation increases the electrical resistance of ordered alloys much more than that of disordered alloys (Table V).

The mechanism of disordering during deformation is not yet clear; however, it may be assumed^51 that the main cause is an increase in domain boundaries as dislocations pass through them.

Fig. 8. Effect of cold working on the electrical resistance of ordered and disordered Ag$_3$Mg alloy.

The results presented in Fig. 8 show that complete disordering of the ordered Ag$_3$Mg alloy occurs at a deformation of about 90%.

c) The role of defects of the crystalline structure in recovery phenomena in metals

Recovery phenomena have been studied most fully for copper^28. In this work, besides other changes, measurements of electrical resistance were also used after various methods of producing defects in the metal: plastic deformation, irradiation*) with high-energy particles, and quenching from high temperatures close to the melting temperature. As the studies showed, recovery phenomena in copper are associated with the presence of at least five different stages. They differ in the temperature range of their existence, in the activation energy of the process, and in the magnitude of the decrease in excess electrical resistance caused by the given process.

In Fig. 9 all five stages of the return of electrical resistance in deformed, irradiated, and quenched copper are schematically presented; the axes show intervals of the measured values of activation energy and of the magnitude of the change in electrical resistance. The explanation of recovery phenomena at the various stages is based on the assumption that

*) On the effect of irradiation on the electrical resistance and other properties of metals, see^52.

they are due to various diffusion phenomena. This can be done if one proceeds from the activation-energy values given in Table II for various processes. Together with the nature of the defects in the lattice, characteristic of various methods of treatment (Table VI), the data presented in Fig. 9 make it possible to draw definite conclusions concerning the nature of the annealing stages themselves.

Fig. 9. Five stages of recovery of electrical resistivity in deformed, irradiated, and quenched copper.

Table VI

Nature of treatment	Lattice defects	Lattice defects	Lattice defects
Nature of treatment	vacancies	interstitial atoms	dislocations
Plastic deformation	Many	Very many, but fewer than vacancies	Many
Irradiation	Many	Many	Few
Quenching	Many	Absent	Few or absent

Stage I is probably characterized by diffusion of interstitial atoms and occurs in irradiated specimens, in which the formation of a large number of such defects is possible. In deformed specimens this process is unlikely owing to the high energy of formation of defects of the interstitial-atom type.

Stage II is probably characterized by paired vacancies and occurs with all methods of treatment.

If stage III is characterized by vacancy diffusion, then stages IV and V must be determined by the disappearance of dislocations excited during deformation of metals. But it is more probable to suppose that stage IV is associated with vacancy diffusion. However, the fact requires explanation that in irradiated specimens stage IV is not observed. Possibly this is explained by the fact that in the preceding stage III in irradiated specimens recombination of vacancies and interstitial atoms had already occurred.

Some authors^53 associate stage III with the diffusion of interstitial atoms; this is unlikely because of the very low activation energy for the motion of defects of this type.

It should be noted that the interpretation given above of the five recovery stages in copper is in a number of cases not unambiguous; this is perhaps also explained by the fact that the data of different authors, even when treated in the same way, are not always unambiguous. Nevertheless, it is evident that recovery phenomena must be interpreted as processes connected with the diffusion of the corresponding defects in the crystal lattice.

4. DEFECTS OF CRYSTAL STRUCTURE AND DIFFUSION IN METALS AND ALLOYS

Diffusion processes in metals are governed by the most important property of defects of the vacancy and interstitial-atom types, namely their ability to migrate relatively easily in the crystal with the aid of thermal fluctuations. In fact, defects of this type were invoked by the theory to explain the phenomena of diffusion and electrolytic conductivity in salts.

As calculation shows,^2 the energy of formation of an interstitial atom is many times greater than the energy of formation of a vacancy, and therefore, even at high temperatures, their concentration is considerably smaller than the concentration of vacancies.

Although diffusion processes in metals have been studied for more than 150 years, only in the last decade have data been obtained indicating a vacancy mechanism of diffusion.

That vacancies do indeed participate in the diffusion process could be verified from an experiment with an alloy in which it is definitely known that, owing to a special structural state, excess vacancies exist in comparison with thermal vacancies.

Alloys of the δ-phase Ni—Al proved convenient for such an experiment.^77 The diffusion of Co—60 was studied in alloys in the range of nickel compositions 47—55 at. %. It turned out (Fig. 10) that the diffusion coefficient has a minimum value at the exact stoichiometric composition, while the activation energy at this composition has a maximum value. Alloys rich in aluminum possess a large number of structural vacancies; therefore one may expect that diffusion in them is substantially determined by the mobility of vacancies. Thus, the temperature dependence of the diffusion coefficient makes it possible to determine, for alloys of this composition, the activation energy for vacancy motion. For alloys rich in nickel, however, in the region of 50 at. %, the activation energy is considerably larger, since it is determined by the sum of the vacancy-formation energy and the energy of their motion. Thus, for the alloy NiAl one can estimate—

Fig. 10. Concentration dependence of the diffusion coefficients of cobalt in alloys of the δ-phase Ni—Al.

the ratio of the vacancy-formation energy to the diffusion activation energy. It turned out to be equal to \(\sim 0.35\).

A similar result was obtained\(^{54}\) in the study of Co—60 diffusion in Co—Al alloys (Fig. 11). The transition through 50 wt. % Al was accompanied by a sharp decrease in the activation energy.

a) The mechanism of diffusion and the Kirkendall effect

More direct evidence of the participation of vacancies in the diffusion process is the effect of displacement of inert markers (usually molybdenum wires) at the boundary of contacts in a copper—brass—copper sandwich\(^{55}\). It was shown that the markers move relative to one another with increasing diffusion rate, the displacement being proportional to the square root of time. Later this effect was also observed in the case of interdiffusion in the systems: Cu—Zn, Cu—Sn, Cu—Ni, Cu—Au, and others.

Fig. 11. Diffusion parameters of cobalt in the \(\beta\)-phase Co—Al.

The phenomenological explanation of the observed displacement assumes two different diffusion coefficients for the components of the alloy. If the diffusion coefficients differ from one another, then there must be a directed flux of matter relative to the markers.

The atomistic description of the Kirkendall effect assumes that the markers are fixed relative to the crystal lattice and that there is a flux of matter relative to the lattice. Such a flux of matter would not exist if diffusion proceeded by the exchange of places of atoms or with the aid of a “ring” mechanism\(^{56}\).

If the vacancy mechanism of diffusion is accepted, then it is obvious that the flux of atoms in one direction can be compensated by an equal flux of vacancies in the opposite direction. It is assumed that vacancies are in local thermal equilibrium, and grain boundaries and dislocations can act as sources or sinks for vacancies, thereby maintaining equilibrium\(^{3,57}\). A phenomenological theory of diffusion in a binary system was developed in \(^{58}\), where the diffusion equations in the alloy are analogous to the diffusion equations in electrolytes. In the theory the general position is adopted that the driving force of diffusion is the gradient of the chemical potential, and not the concentration gradient.

Let \(I_i\) be the flux of atoms of species \(i\) (through \(1\ \mathrm{cm}^2\) in 1 sec.) relative to the lattice, \(\psi_i\) the chemical potential of atoms of species \(i\); for vacancies we introduce analogous notations \(I_v, \psi_v\), then we have

\[ I_i=-\sum_{j=1}^{n} M_{ij}\frac{\partial \psi_i}{\partial x} -M_{iv}\frac{\partial \psi_v}{\partial x}, \tag{14} \]

\[ I_v=-\sum_{j=1}^{n} M_{vj}\frac{\partial \psi_j}{\partial x} -M_{vv}\frac{\partial \psi_v}{\partial x}, \tag{14a} \]

where \(M_{ij}\) are coefficients denoting mobilities, and the summation is carried out over all types of atoms \(n\), with

\[ M_{ij}=M_{ji},\quad M_{iv}=M_{vi}. \tag{15} \]

If, in the diffusion process, the total number of lattice sites does not change, then

\[ I_v+\sum_{i=1}^{n} I_i=0, \tag{16} \]

whence

\[ \sum_{i=1}^{n} I_i=-I_v. \tag{16a} \]

The flux of vacancies will in this case lead to the Kirkendall effect, if it is assumed that the inert markers move together with the lattice. Of special interest here is the plastic deformation necessary to preserve the equilibrium concentration of vacancies.

Under simplifying assumptions one can obtain the equation

\[ I_i=-M_i\frac{\partial}{\partial x}(\mu_i-\mu_v). \tag{17} \]

When the local equilibrium concentration of vacancies is preserved, i.e. when \(\mu_v=0\), it is simplified to

\[ I_i=M_i\frac{\partial \mu_i}{\partial x}. \tag{17a} \]

If the alloy contains some fraction of “labeled” atoms, then, taking them as one of the components of the alloy, we obtain:

\[ I_T=-M_T\frac{\partial \mu_T}{\partial x}, \tag{17b} \]

where \(M_T\) is the mobility of the atoms of the radioactive isotope.

Since diffusion of the radioactive isotope is caused only by the concentration gradient, we may take:

\[ \mu_i=RT\ln\left(\frac{N_i}{N}\right), \tag{18} \]

\[ \mu_T=RT\ln\left(\frac{N_T}{N}\right), \tag{18a} \]

where \(N\) is the total number of lattice sites, \(N_i\) is the number of atoms of type \(i\); it is assumed that \(N_T\ll N_i\), and from the fact that there must be no excess flux of atoms of type \(i\), it follows that

\[ I_i=-I_T \tag{19} \]

and, consequently,

\[ M_i=\frac{N_i}{N_T}M_T. \tag{20} \]

The diffusion coefficient of the radioactive atoms \(I_T\), defined by the equation

\[ I_T=-D_T\frac{\partial N_T}{\partial x}, \tag{21} \]

is equal to

\[ D_T = kT \frac{M_T}{N_T}, \tag{22} \]

whence we obtain that

\[ M_i = N_i \frac{D_T}{kT}, \]

i.e., the mobility of the atoms is proportional to the diffusion coefficient of the radioactive isotope.

A further calculation shows that for a cubic lattice one may take

\[ D_T = 0.895\,D_{AA}. \tag{23} \]

For chemical diffusion in the lattice of a binary alloy consisting of components \(A\) and \(B\), equations (14) are written as:

\[ I_A = - M_{AA}\frac{\partial \psi_A}{\partial x} - M_{AB}\frac{\partial \psi_B}{\partial x}, \tag{24} \]

\[ I_B = - M_{BB}\frac{\partial \psi_B}{\partial x} - M_{AB}\frac{\partial \psi_A}{\partial x}. \tag{24a} \]

On the other hand,

\[ I_A = - D_A \frac{\partial N_A}{\partial x}, \tag{25} \]

\[ I_B = - D_B \frac{\partial N_B}{\partial x}, \tag{25a} \]

but, since \(d\psi_A/d\ln N_A = d\psi_B/d\ln N_B\), for the diffusion coefficients of the alloy constituents we obtain:

\[ D_A = \frac{d\psi_A}{d\ln N_A} \left( \frac{M_{AA}}{N_A} - \frac{M_{AB}}{N_B} \right), \tag{26} \]

\[ D_B = \frac{d\psi_A}{d\ln N_A} \left( \frac{M_{BB}}{N_B} - \frac{M_{AB}}{N_A} \right). \tag{26a} \]

From the mechanism of diffusion by means of vacancies and from the Kirkendall effect there follows the necessity of distinguishing between the chemical diffusion coefficients of the components, and also between the latter and the coefficient of self-diffusion of an isotope in the alloy.

The closely related phenomenon of pore formation during the sintering of metallic wires was considered in work \(^{59}\). The formation of porosity in diffusion zones was analyzed in work \(^{60}\). It is suggested that the pores are formed as a result of the separation of supersaturated vacancies arising owing to the difference in the diffusion fluxes of the atoms of the components \(^{61}\) on one side of the diffusion zone. This is confirmed by consideration of the formation of porosity during the diffusion of copper into nickel, brass, and a copper–aluminum alloy \(^{62}\). It turned out that the excess concentration of vacancies necessary for the formation of a nucleus is \(\sim 0.01\). From this point of view one can also explain the formation of porosity as a result of evaporation in vacuum of a volatile component from brass and from a number of silver alloys \(^{63}\). During the evaporation of zinc from brass \(^{64}\), the number of pores increases by more than a factor of 10, while the ratio of the total pore volume to the volume of zinc evaporated remains equal to \(0.001\). At \(700^\circ\mathrm{C}\) and a zinc loss of \(197\) mg, the mean pore radius is \(\sim 42.5\) Å, and the distance between them is \(\sim 2000\) Å. An estimate of the magnitude of the relative supersaturation of vacancies \(C/C_0\), where \(C_0\) is the equilibrium concentration of vacancies,

was found to be 1.08. This value is much higher than the value estimated in work^62.

Possible sources of vacancies providing the effects considered above are: the surface of the specimen^60, grain boundaries, and dislocations present in the specimen. If the surface of the specimen were the predominant source of vacancies, the Kirkendall effect would depend on the volume of the diffusion specimen. However, such a dependence has not been found experimentally^65.

The significance of grain boundaries as sources of vacancies could have been determined by studying the Kirkendall effect in single crystals. However, in the absence of such experiments one may assume that the simplest sources of vacancies are dislocations^10. Conversely, in order to remove vacancies from the lattice it must be assumed that dislocations act as sinks for vacancies^10.

Approximate calculations have shown^66 that the normal dislocation content in a metal is adequate to the content of the equilibrium vacancy concentration throughout the entire diffusion zone, despite the existence of an excess flux of vacancies. If this is correct, then the average distance traveled by each vacancy during its lifetime must be much less than that which occurs in diffusion, and it would seem that the assumed dislocation density should provide for this, even if not every encounter of a vacancy with a dislocation leads to their annihilation. However, the existence of porosity in many diffusion systems invalidates these conclusions.

The formation and annihilation of vacancies occur most readily at dislocation jogs. The effective number of points (the fraction of atoms along a linear dislocation) where such a process can occur may in fact be smaller than that obtained from calculations.

b) Influence of distortions of the crystal lattice on the diffusion process in metals

In § 4, a) the diffusion process was considered in connection with a vacancy mechanism, which presupposes the presence of a certain equilibrium concentration of vacancies.

However, the question of the influence of nonequilibrium distortions on the diffusion process is of great practical interest. This question was examined in detail by S. T. Konobeevskii in work^67, as well as in works^68. In the theory developed by S. T. Konobeevskii, the foundations are laid for the development of a diffusion theory of creep^69.

A substantial influence on the diffusion process in metals and alloys is exerted by stresses caused, for example, by the action of an external load or by phase transformations, as well as by the nonuniform distribution of the alloy components.

Fig. 12. Results of “uphill” diffusion in an elastically bent block.
○ Al
● Cu

The diffusion process^67 depends not only on the concentration gradient, but also on the gradient of elastic deformation. An example of such a case may be an elastically bent block (Fig. 12), in which the deformation and elastic stresses vary gradually from layer to layer. For this case Fick’s second equation may be written as:

\[ \frac{\partial c}{\partial t}=D'\frac{\partial^{2}c}{\partial x^{2}}-D''\frac{\partial^{2}\varepsilon}{\partial x^{2}}, \tag{27} \]

where \(c\) is the concentration, \(\varepsilon\) is the strain, \(t\) is the time, \(D'\) is the diffusion coefficient due to the presence of a concentration gradient, and \(D''\) is the coefficient due to the presence of a stress gradient. It was shown\(^{67}\) that, in a solid solution, \(D''\) is proportional to the relative difference in the atomic radii of the components \((r_a\) and \(r_b)\). Whereas the first term of (27) leads to equalization of the concentration, the second, which takes into account the inhomogeneous stressed state, promotes separation of the components.

The latter type of diffusion was called “uphill diffusion.” The action of uphill diffusion was used to explain the fact of strengthening of deformed specimens of brass and aluminum bronze during tempering in the region of \(270^\circ\text{C}\).\(^{70}\) S. T. Konobeevskii also considered the influence of stresses arising during phase transformations. It was shown that stresses arising during the precipitation of a new phase from a solid solution cause diffusion fluxes that accelerate the process of precipitation of the new phase, thus acting autocatalytically.

Konobeevskii’s equation was used to explain the dependence of the diffusion coefficient on concentration.\(^{71}\) This proved possible when the “concentration” stresses caused by a change in the lattice period during formation of the solid solution were explicitly taken into account. For this case equation (27) assumes the following form:

\[ \frac{\partial c}{\partial t} = D'\frac{\partial^2 c}{\partial x^2} - D'\frac{\partial^2 \varepsilon}{\partial x^2} = D'\left[ 1+ \frac{12\mu K\omega}{\left(K+\frac{4}{3}\mu\right)RT} \right]\frac{\partial^2 c}{\partial x^2} \tag{27a} \]

or, introducing an effective diffusion coefficient, we have:

\[ \frac{\partial c}{\partial t} = D_{\mathrm{eff}}\frac{\partial^2 c}{\partial x^2}, \tag{28} \]

whence

\[ D_{\mathrm{eff}}\simeq D'\left[ 1+ \frac{12\mu K\omega}{\left(K+\frac{4}{3}\mu\right)RT} \right], \tag{29} \]

where \(D'\) is the diffusion coefficient at an infinitely small concentration of the diffusing element, \(\omega=\dfrac{a-a_0}{c}\), \(a_0\) is the lattice period of the pure solvent, \(\mu\) is the shear modulus, and \(K\) is the bulk modulus.

From the theory\(^{71}\) there follows the general conclusion that, in a number of cases, “concentration distortions” create “uphill diffusion,” which at sufficiently low temperatures causes growth of concentration fluctuations up to the concentration corresponding to the new phase.

It should be noted that the equations taking into account the influence of stresses on the diffusion coefficient\(^{67,71}\) were derived under the assumption that only elastic stresses are present. It is assumed that plastic deformation does not occur. Meanwhile, it is important both practically and theoretically to elucidate the influence of plastic deformation on the diffusion process. This seems especially important to us because in this way it would be possible to shed light also on the mechanism of plastic deformation.

S. T. Konobeevskii\(^{72}\) showed that, in a deformed thin layer of copper, the diffusion coefficient of nickel into copper increases by more than 1000 times.

Ya. S. Umanskii found\(^{73}\) that the thickness of the carbide layer on the surface of a deformed tantalum plate increases 50-fold in comparison with the thickness of the layer on an undeformed plate at tem-

at a cementation temperature of 900° C; in this case the time of diffusion saturation was considerably reduced.

A very interesting example of the effect of distortions of the crystal lattice on the rate of diffusion is the fact of very intense diffusion at room temperature of the first layers of zinc deposited by condensation on the surface of a polished copper plate. As is known, the polished layer is characterized by very strong distortions of the crystalline structure.

Gerricken and Golubenko \(^{74}\) studied the effect of deformation on the rate of diffusion of zinc from \(\alpha\)-brass. The deformation was produced owing to the difference in the coefficients of expansion of brass and nickel, the two metals being in close contact. In comparison with undeformed brass, the coefficient of diffusion of zinc from \(\alpha\)-brass at temperatures of 560°, 600°, and 640° C increased, respectively, by factors of 1.7, 4.0, and 5.

In the works cited above, however, the mechanism of the influence of deformation on the diffusion process was not considered. From the standpoint of the phenomena considered in the preceding chapters, deformation associated with the formation and motion of various kinds of defects in the crystal lattice must exert a substantial influence on the mobility of atoms. Therefore, study of the influence of deformation on the mobility of atoms in metals may help to reveal the very mechanism of the phenomenon.

In work \(^{20}\) atomic mobility was studied in specimens of an Ag—Zn alloy \((70:30)\), previously quenched from 400° C. In quenched specimens with a frozen-in nonequilibrium number of vacancies the mobility of atoms should increase. This in fact occurred. Measurements of the relaxation time in the temperature range 30—70° C made it possible to separate the value of the activation energy in the alloy, which is composed of the vacancy-formation energy \(\varepsilon_0\) and the vacancy-motion energy \(\varepsilon_s\). It turned out that

\[ \varepsilon_0 = 11.8 \,\text{kcal}/\text{g-at}, \quad \varepsilon_s = 19.7 \,\text{kcal}/\text{g-at}, \quad \text{whence the ratio } \frac{\varepsilon_0}{\varepsilon_0+\varepsilon_s}=0.37 \]

has the same value as in diffusion or self-diffusion, given above.

It has repeatedly been emphasized in the literature that the mechanism of deformation of a metal at high temperatures has much in common with the mechanism of diffusion.

From what has been said so far it is clear that deformation, like quenching, leads to supersaturation in the number of vacancies. If this is so, then the diffusion coefficient must increase.

In work \(^{75}\) on the study of the fracture of metals at high temperature it is precisely assumed that, under the influence of applied stresses and thermal motion, there occurs a regular increase in the number of “nuclei” of melting or defects of the crystalline structure (vacancies). It is assumed that the rate of this reaction is proportional to the initial concentration of vacancies \((n_0)\) and depends on the stress:

\[ \left. \begin{aligned} \frac{dn}{dt} &= 2\omega n_0 \operatorname{sh}\frac{\Delta W}{RT},\\ \omega &\simeq C \exp\left[-\frac{mL}{R}\left(\frac{1}{T}-\frac{1}{T_s}\right)\right], \quad \Delta W = \frac{1}{2}qv_a\sigma. \end{aligned} \right\} \tag{30} \]

Here \(\omega\) denotes the probability of the occurrence, per unit time, of a new vacancy as a result of disordering of a certain number of atoms \((m)\); \(L\) is the heat of fusion per atom, \(T_s\) the melting temperature, \(C\) a coefficient of proportionality, \(\sigma\) the acting stress, \(q\) a certain coefficient characterizing the concentration of stresses in the region of a defect, \(v_a\) the volume per atom, and \(T\) the temperature.

The indicated assumption requires experimental confirmation. This can be accomplished by the following method.

Since stresses cause an increase in the number of vacancies, the diffusion rate should increase if the specimen, during diffusion, is subjected to the action of stresses and to a definite rate of deformation.

In the unstressed state the diffusion coefficient is equal to

\[ D = D_0 e^{-\varepsilon_\omega/RT} e^{-\varepsilon_s/RT} = D_0 n_0 e^{-\varepsilon_s/RT}. \tag{31} \]

At a constant rate of deformation the diffusion rate should increase in connection with the increase in the number of vacancies:

\[ D_\sigma = D'_0 n e^{-\varepsilon_s/RT}. \tag{32} \]

Thus,

\[ \frac{D_\sigma}{D} = \frac{D'_0}{D_0}\cdot \frac{n}{n_0}. \tag{33} \]

Integrating (30), we obtain:

\[ n = n_0[1+\omega t e^{\alpha\sigma}], \tag{34} \]

where

\[ \alpha = \frac{1}{2}\,q\,\frac{v_a}{RT}. \]

Then the ratio of the diffusion coefficients is

\[ \frac{D_\sigma}{D} = \frac{D'_0}{D_0}(1+\omega t e^{\alpha\sigma}). \tag{35} \]

In work \(^{76}\) the influence of deformation on the rate of self-diffusion in iron was studied, and it was found that

\[ D_\sigma = D(1+0.14 e^{0.0032\sigma}). \tag{36} \]

This expression is of the same type as that obtained by us \(^{35}\).

In \(^{76}\) self-diffusion was studied by the absorption method at a temperature of \(890^\circ\mathrm{C}\), at a constant time \(t=18\) hours and at various rates of deformation. Using these data, as well as the expression for the diffusion coefficient in undeformed iron

\[ D = 6.2\ \exp\left(\frac{-59800}{RT}\right)\ \text{cm}^2/\text{sec}, \]

one can, from the formulas given, estimate some of the constants entering into them. Thus, for example, it turned out that \(C=1.7\cdot 10^{-3}\ \text{sec}^{-1}\), \(q=1233\). These quantities are of the same order as those obtained \(^{77}\) for other metals (Al, Ag, Cu, and manganin) from direct measurements of the rupture time at various temperatures. Thus, the initial premise adopted in work \(^{75}\) receives experimental confirmation.

The considerations presented above, however, should be the subject of further careful study. Especially important is the study of the influence of the stressed state on the diffusion process over a wide temperature range. This would make it possible to determine the parameters of the process and thereby obtain information about the mechanism of plastic deformation, which is important for clarifying the physical nature of fracture \(^{69}\) at high temperatures.

Proceeding from the vacancy mechanism of diffusion, one may suppose that the diffusion rate can be noticeably increased under intensive

bombardment with neutrons or protons, capable of creating additional vacancies and local highly heated regions[^52]. The study of the influence of irradiation on the diffusion process is important not only for understanding the kinetics of processes occurring in solids, but is especially important for understanding the behavior of various materials in nuclear reactors.

Not only plastic but also elastic deformation must have a substantial influence on the course of diffusion processes in metals.

Usually self-diffusion in metals is studied at temperatures relatively close to the melting temperature. Under these conditions the metal softens so much that it is difficult to impose a large elastic deformation without the metal undergoing plastic deformation. This, apparently, is the reason why the influence of elastic deformations on diffusion has not been studied up to the present time. This difficulty can be avoided if diffusion is studied at low temperatures. If, in this case, the metal is subjected to the action of elastic stresses, then the activation energy for vacancy migration changes. But since the probability of vacancy displacement is very sensitive to a change in the activation energy, then, in the presence in the metal of excess vacancies (as compared with the amount in thermal equilibrium), their mobility under elastic deformation should increase noticeably. If this is true, then the diffusion coefficient will thereby increase by \(e^{\Delta E/RT}\) times (\(\Delta E\) is the decrease in activation energy).

An excess of vacancies can be created in a metal by irradiation or by cold working. The advantage of the vacancies introduced into the metal is that the diffusion process can be investigated at low temperatures, at which the material is stronger.

In work[^78] two copper wires were subjected to 20% stretching at the temperature of liquid nitrogen. After this, the recovery of electrical resistance at \(-30^\circ\text{C}\) was measured as a function of time. During the first four minutes the curves of decrease of electrical resistance coincided. Then one of the wires was loaded (\(17\ \text{kg}/\text{mm}^2\)), after which the rate of recovery in this wire increased noticeably. Assuming that the recovery is due to diffusion of vacancies, the authors, from the experimental results, calculated the ratio of the diffusion coefficients in the loaded wire as compared with the unloaded one:

\[ \frac{D_\sigma}{D} = 1.7. \]

The effect, as can be seen, is not small. These calculations make it possible to judge the influence of internal stresses on the recovery of electrical resistance after cold working.

CONCLUSION

The study of a number of properties of metals and alloys shows that the same defects of atomic type influence such processes as diffusion, electrical conductivity, plastic deformation, etc. Moreover, the study of these processes has made it possible in a number of cases to clarify the nature of defects in the crystal structure. At the same time, a more complete understanding of the mechanism of the indicated processes is possible only after clarifying the character of the interaction of defects of different types. This is all the more important because the properties of defects substantially determine the real strength of metals.

We have not at all considered such questions as the influence of defects in the crystal structure on the mechanical properties of metals, as well as the influence of defects caused by impurities on the properties of metals. These very extensive questions must be considered separately.

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