€3. STEADY-STATE PROCESSES NOT INVOLVING LATTICE RE-ARRANGEMENT INTRODUCTORY PAPER BY JAMES S. KOEHLER AND FREDERICK SEITZ Dept. of Physics, University of Illinois, Urbana, Illinois Received 1 1 th December, 1956 Nature provides many ways of achieving non-equilibrium states in solids. For example, one may introduce a gradient in temperature, or in isotopic or chemical composition. Rate processes involving local atomic rearrangement may then occur in such a direction as to re-establish equilibrium. Initial non-equilibrium condition can also be produced by the application of electro-magnetic fields or external stresses. Some atomic motion may occur in the equilibrium state; for example, the motion which is associated with atomic diffusion occurs normally at high temperatures. In other cases, the rearrangement is strongly influenced by the deviations from equilibrium.For instance, a strong electric field may induce local ionic motion on a much greater scale than would occur in the absence of the field. Similarly, radiation may induce lattice imperfections and may also stimulate the migration which removes them. Processes which require thermal fluctuations have been studied most widely to date and are most thoroughly understood in a quantitative manner. As a result, they will receive most of our attention. It is not necessary that the thermal fluctuations be furnished by the ambient. Under appropriate circumstances, such as when the system is being irradiated by photons, electrons, or more massive particles, significant fluctuations, commonly termed " thermal spikes ", may be induced by the radiation.In general, it appears that thermal fluctuations may act in two broad ways to produce local atomic changes. In the first, the system starts from a nearly perfect state and, in a brief time, of the order of the atomic oscillation time,undergoes a transition through an activated state to a new, nearly perfect state. For example, a normal and a radioactive atom of the same species which lie at neighbouring normal lattice sites in the lattice may interchange places as the result of motion made possible by a thermal fluctuation. In such a case the imperfection in the lattice represented by the activated, or intermediate, complex is highly transient, enduring for a time of the order of 10-13 sec.In the second case, the thermal fluctuations operate through lattice imperfec- tions which may or may not be produced by thermal fluctuations. For example, an equivalent atomic rearrangement may be produced by the thermally activated motion of vacant lattice sites or interstitial atoms generated either by independent thermal fluctuations or by external agents such as bombarding particles. The second of the two types of mechanism occurs much more commonly than the first, at least in the solids which have received most attention to the present time. For this reason, an understanding of imperfections is an essential pre- requisite for the treatment of rate processes in solid systems. Conversely, the systematic study of rate processes has provided us with a more profound insight into the nature of some of the imperfections which occur in solids.In any event, any consideration of processes which induce either equilibrium or steady-state 8586 INTRODUCTION conditions in a solid system which is not at equilibrium automatically involves a discussion of crystal imperfections and their properties. TYPES OF IMPERFECTION The lattice defects which play an important role in determining the re-establish- ment of equilibrium may be divided into three categories, namely, point, line and planar imperfections. All three may be the effective agent under appropriate circumstances; however, the second two may be active principally as catalytic agents for production of the first type. POINT IMPERFECTIONS The principal point imperfections responsible for migration are vacant lattice sites and interstitial atoms.An atom neighbouring a vacant lattice site may move into the vacancy. It, therefore, has greater freedom to move than an ordinary lattice atom. The vacancy is conserved in this process so that it may be used by a large number of atoms in succession. Similarly, an atom in a normal site which is a neighbour of an interstitial atom of the same or similar species may move into another interstitial position and allow the initial interstitial atom to take its place. Thus an interstitial atom, like a vacancy, may impart relative ease of motion to the atoms in its neighbourhood, the interstitial pattern or interstitialcy, like the vacancy, being preserved in migration. Evidently vacancy or interstitialcy motion is favoured over other mechanisms of motion only if the activation energy is sufficiently low.It should be noted that an interstitial atom may migrate through interstitial sites without changing places with a normal atom. This mode of motion is probably preferred whenever the interstitial atom is much smaller than the atoms of the host lattice (e.g. H in Pd, or C in Fe) or is of a radically different species. In such cases only the interstitial atom is transported, and the typical atom of the lattice does not gain mobility from the presence of the interstitial. In contrast, a vacancy can move only by imparting motion to at least one atom in a normal position. For energetic reasons, a vacancy or interstitial may become bound to a par- ticular distorted site or foreign atom in the lattice to produce a complex which possesses properties which are distinguished from those of either imperfection when present at a typical position in an otherwise perfect lattice.LINE IMPE~CTIONS The most important line imperfections of interest for understanding processes which effect equilibrium or steady-state conditions are dislocations.1 The typical dislocation may be envisaged, geometrically, by imagining the specimen which contains it to be cut over a surface whose boundary is a well-defined closed curve which lies partly or entirely inside the crystal. The parts of the specimen on either side of the cut are displaced by a vector distance b relative to one another (the Burgers vector) and rejoined in the new position. Wherever b has a component normal to the surface so that the material would overlap or a gap would develop as a result of displacement, material is cut away or added so that the two sides are contiguous before making the join.If the vector b is an allowed translation of the lattice, the lattice will be coherent over the surface; however, there will be an accumulated strain which grows in magnitude as one approaches the bounding line of the surface. Thus the resulting stress and strain pattern may be regarded as associated with the bounding curve which is termed the dislocation line. It can be shown that the stress-strain pattern depends only upon this line and the vector b, if the latter is an allowed translation. On the other hand, the pattern evidently depends upon the surface on which the cut is made, as wellJ .S . KOEHLER AND F. SEITZ 87 as upon the bounding line and b, if the Burgers vector is not an allowed translation, for the surface will be one of misfit in the second case. A wide range of possible dislocations can result from the variation of the line and the vector b. Two important simple cases are those in which the line is straight and b is normal to and parallel to the line. The first case is that of a Taylor-Orowan or edge dis- location. The second is that of a Burgers or screw dislocation. The first may be regarded as the result of inserting an extra portion of a crystallographic plane, lying normal to b and of thickness I b 1, which has an edge terminating at the line. The second may be regarded as the result of converting the crystal, composed of a set of lattice planes perpendicular to b, into a helical screw having pitch b.The screw is formed of the lattice planes mentioned and winds about the line of the dislocation. The intrinsic energy of a crystal evidently is increased by the presence of dislocations. This energy may be divided into a part which is associated with the length and orientation of each of the segments of the dislocation lines and a part which results from the interaction of the segments. The energy per unit length is sufficiently large that dislocations are not generated by normal ambient thermal fluctuations in typical crystals. They are usually formed as a result of an accident of growth or by large applied stresses. Dislocations can move and be generated as the result of applied stresses sufficiently great to induce plastic flow.In fact, such flow can be described in terms of the motion of appropriate dislocations. Thus the migration and gener- ation of dislocations is one of the processes which acts to re-establish equilibrium in a specimen which has been pushed away from the equilibrium state by the application of mechanical stresses. In the phenomena of creep, thermal fluctu- ations play a role in aiding the migration and generation of dislocations when stresses are applied. An edge dislocation can act as a source or sink for vacant lattice sites or interstitial atoms. For example, the edge of the added plane can be extended by depositing there either interstitial atoms or atoms taken away from normal sites.The average energy required to generate a vacancy or interstitial atom at a long edge dislocation is the same as the average energy required to create either from the surface of the specimen. It is believed that dislocations are the sources and sinks of most of the vacancies or interstitial atoms produced by ambient thermal fluctuations in typical crystals. Many of these possibilities have been found in crystals. PLANAR IMPERFECTIONS The most common planar imperfections are the boundaries between differently oriented crystals. If the angular disorientation is small, of the order of 0.1 radian or less, the boundary can be represented by an appropriate array of dislocations. It then possesses properties which can be expressed in terms of those of dislocations.On the other hand, when the disorientation is large, of the order of a radian or so, the grain boundary cannot be described uniquely in terms of a simple array of dislocations, but has a more complex individuality. In general, it may act as a source or sink for point imperfections. The stacking fault is a planar imperfection produced commonly as a result of plastic flow and other processes. The simplest stacking fault can be regarded as the result of passing a dislocation having a Burgers vector that is not an allowed translation across a surface, so that the crystal is not in registry over the area of passage, although it is in registry elsewhere. A twinned region may be produced, under proper geometrical circumstances, by passing identical dislocations which would produce stacking faults across a sequence of neighbouring lattice planes.The boundaries between the twinned and normal area evidently are planar im- perfections, closely related to stacking faults.88 INTRODUCTION THE OCCURRENCE AND INFLUENCE OF POINT IMPERFECTIONS SALTS The clearest information concerning the nature of the point imperfections which influence mass transfer is available in the salts, particularly the alkali halides.;! It is known that vacancies in the positive and negative ion lattices are generated thermally in equal numbers at elevated temperatures in the alkali halides and promote diffusion and electrolysis. Each type of vacancy is confined to migrate through its own sublattice. Ions possessing valence higher than unity which are present in substitutional solution are accompanied by vacant lattice sites.The multivalent impurity ion and its vacancy may be dissociated at elevated temperatures and contribute to the transport within the crystals. The vacancies associated with the foreign ions may impart a very high mobility to these ions, as well as to the ions of the crystal. Since the concentration product of the thermally produced defects should be constant at any given temperature, the addition of an agent which enhances one type (e.g. positive ion vacancies) will suppress the other (e.g. negative ion vacancies). It is interesting to note that the halogen ion vacancies in the alkali halides possess a characteristic absorption band in the ultra-violet portion of the spectrum, termed the alpha band. The existing evidence suggests that the vacancy can be made to migrate as a result of successive absorption and emission acts, as if the highly transient temperature fluctuations which accompany absorption and emission induce the vacancy to jump.It has been amply demonstrated that the point defects which are generated most easily by thermal fluctuations in the silver halides are interstitial silver ions and the corresponding vacant sites. The two types are present in equal numbers in the ideally pure crystal; however, each may be enhanced by the introduction of divalent negative ions or divalent positive ions, respectively. It has also been demonstrated that the point imperfections play an important role in the photo- graphic process, both in transporting the silver and halogen and in trapping electrons and holes produced by the incident light. All details of the process are not yet understood, however.There is much interest in the thermally induced point defects in many other salts, such as oxides, sulphides, and divalent halides. Extension of the techniques employed in the univalent halides should eventually provide more detailed knowledge. METALS The nature of the point imperfections generated by thermal fluctuations in metals is still somewhat obscure, although the topic is receiving much clarification at the present time. To date, most attention has been focused upon the metals having partly filled or newly filled d shells, that is, upon metals in the central parts of the long rows of the periodic system.Relatively rudimentary calculations,3 particularly in copper, have indicated that atomic migration takes place preferen- tially by the generation and migration of vacancies in cases in which ambient thermal fluctuations alone furnish the defects. The calculations suggest that the direct interchange of neighbouring atoms, without the aid of a lattice defect, and the production of interstitial atoms requires substantially more energy than the production and migration of vacancies. It is possible, however, that the situation is quite different in the alkali metals and in other metals involving newly filled rare gas shells. Experiments have established that the diffusion interface between dissimilar metals which are soluble in one another becomes displaced, usually toward the metals possessing the lowest heat of sublimation, when diffusion is permitted to occur.This Kirkendall shift demonstrates4 that the diffusion does not occur by direct interchange of atoms in the corresponding systems, but involves theJ. S . KOEHLER AND F. SEITZ 89 migration of either vacancies or interstitial atoms which move preferentially from one side of the boundary to the other. It has not yet proved possible to decide between the two modes of imperfection diffusion on the basis of experiment alone. Recent experiments 5 on the residual resistivity produced in thin wires of gold by rapid quenching from elevated temperatures seem to show that it is possible to preserve the point defects formed thermally at elevated temperatures.The atom fraction of such defects proves to be in the neighbourhood of 10-4 at 950" C . The activation energy EF for formation is found to be 1.00 5 0.05 eV (23 f 1 kcal/niole). The corresponding defect can be annealed from the specimen near room temperature and requires an activation energy EM of about 0.7 eV. The sum of the two activation energies, namely, 1.7eV, is close to the activation energy ED for self-diffusion in gold, namely, 1.71 eV. The annealing is accompanied by a contraction of the specimen. In conformity with the theoretical calculations described above, the defects are interpreted as vacancies at the present time, although this designation is not absolutely certain. Additional valuable information concerning the properties of defects in metals can be obtained by studying 6 specimens bombarded by massive energetic particles such as neutrons, protons, deuterons, and electrons. It is possible, in this way, to produce interstitial atoms and vacancies as well as more complex imperfections.The simplest disorder is produced by electrons which have only slightly more than enough momentum to dislodge atoms permanently from their normal positions. The study of the annealing of such irradiated materials shows the presence of two prominent processes. One, which has an activation energy near 0.7 eV in copper, silver and gold, appears to be almost the same as that found in the specimens of gold quenched from elevated temperatures and is probably to be associated with the migration of vacancies. The other process is found to occur at remarkably low temperatures, near 30" K, and, in accordance with Huntington's theoretical analysis, is interpreted in terms of interstitialcy migration.The activation energy is in the vicinity of 0.1 eV. It is interesting to note that many rate processes in solids, such as the ordering of disordered alloys, can be accelerated by irradiation. Presumably the lattice defects produced by irradiation serve as catalysts for the transport of atoms required for the transformation. Similarly, it appears that the high local tem- peratures achieved during bombardment, particularly with heavy ions, can produce extensive disordering in localized regions of many alloys. Less information is available for the alkali metals. The table summarizes the present data.If it is assumed that the defect observed near the melting point is also responsible for self-diffusion, the energy of motion EM shown in the table can be calculated. A very low value is obtained. MacDonald9 has attempted to quench sodium without success. This result is in agreement with a very low value of EM. Detailed theoretical calculations have not yet been made for the TABLE 1 .-ACTIVATION ENERGIES FOR DIFFUSION, DEFECT FORMATION AND DEFECT MIGRATION IN THE ALKALI METALS Li Na K self-diffusion (QsD) 0.57 eV 7 0.453 & 0.01 eV 8 energy of defect formation (EF) 0.40 & 0.02 eV 9 0.395 5 0-004 eV 9 0.395 i 0.004 eV9 energy of defect motion (EM) 0-17 & 0.04 eV 0.06 & 0.015 eV EM = QSD - EF90 INTRODUCTION alkali metals. The low values of EM may result from the fact that the body-centred cubic alkali metals are not close-packed.Thus defect motion may require less energy than in the close-packed noble metals. SILICON AND GERMANIUM At the present time we possess only a few items of information on the nature of the point defects which promote atomic migration in valence crystals. Almost all the significant investigations have been made with germanium. It is known that the activation energies for self-diffusion 10 are much larger than in metals possessing the same melting temperatures (e.g. the activation energy for self-diffusion in Ge is about 70 kcal), presumably because the energy for defect formation is large. Mayberg 11 has found that a defect may be quenched from elevated temperatures with a formation energy of about 46 kcal/mole and has suggested that this defect is that responsible for diffusion.Frank and Turnbull 12 have analysed the experiments of Tweet and Gallagher 13 on the penetration of copper into germanium and have concluded that copper may be present both interstitially and in combination with the defect responsible for self-diffusion. The latter unit is relatively tightly associated and accounts for almost all the equilibrium solubility of copper. Since the noble metals act as electron acceptors and may trap as many as three electrons, it seems most reasonable to postulate that they are normally bound substitutionally in germanium and that the defects which promote diflusion are vacant lattice sites. Studies of the annealing of the defects produced by irradiation in germanium reveals the presence of an activation energy of about 41 kcallmole, which may be that for the migration of vacancies.MOLECULAR CRYSTALS Relatively little is known concerning the nature of the thermally induced point defects which promote migration in the rare gas solids and in organic molecular crystals. By analogy with the salts and metals one might anticipate that vacancies are principally responsible for diffusion. Kanzaki 14 has estimated that the activation energy for the formation of vacancies in solid argon is almost the same as the heat of sublimation. Comparable calculations for the formation of interstitial atoms are not available. THE INFLUENCE OF LINE IMPERFECTIONS We have already seen that dislocations play an important direct influence in the re-establishment of equilibrium in specimens subject to stress as a result of their ability to move and thereby cause plastic flow.In addition, they play an exceedingly important indirect role by acting as sources and sinks for vacancies and interstitial atoms. In so acting, the associated change in dislocation pattern, designated as climb, induces strain somewhat analogous to that produced by the action of dislocations in plastic flow. These changes are responsible for the displacement of the diffusion boundary in binary systems, that is, the Kirkendall shift. The alkali halides2 expand when irradiated with X-rays and other ionizing radiations. It seems well established that the expansion is the result of the production of point defects, the most prominent of which are lattice vacancies.These defects probably are formed at dislocations, possibly as a consequence of temperature spikes produced by conversion of electronic excitation energy into vibrational energy. Under appropriate conditions the associated climb and glide of dislocations may produce relatively dramatic changes at the surface of a specimen. The whiskers formed 15 on the surfaces of solids under circumstances in which internal dislocations are induced to climb are probably the most striking example.J . S . KOEHLER AND F. SETTZ 91 I t may also be noted here that the imperfection region at a place where a dis- location terminates at a crystal surface can act as a catalyst for a variety of rate processes, among which is that of growth 16 from solution or the vapour phase.There is evidence to show 17 that moving dislocations generate point defects, either as a result of geometrical factors or high local temperatures. Thus diffusion and electrolytic conductivity can be enhanced by plastic flow. Still further, there is evidence 18 that atoms and other point imperfections may diffuse more rapidly along dislocations than through the bulk material. Such difYusion may play an important role in materials containing appropriate arrays of dislocations, particularly at low temperatures where normal volume diffusion becomes slow. THE INFLUENCE OF PLANAR IMPERFECTIONS Planar imperfections may play a wide variety of roles in promoting the re- establishment of equilibrium. Naturally, small angle boundaries will behave like arrays of dislocations, and many of their properties can be discussed in terms of the component dislocations.On the other hand, typical large-angle boundaries, such as normal grain boundaries, possess more general characteristics. For example, they may act as sources and sinks for vacancies and dislocations, as paths for easy atomic or ionic migration, and as areas for easy plastic flow, par- ticularly at elevated temperatures. The systematic study of the properties of planar imperfections, like the corresponding study of the properties of crystal surfaces, is still in its infancy and is one of the regions now being opened for extensive exploitation. 1 General discussion of the properties of dislocations may be found in the book by Cottrell, Plastic Flow and Creep (Oxford Univ. Press, 1953), and that by Read, Dislocatioizs iii Crystals (McGraw-Hill Book Company, New York, 1953). 2 A summarizing discussion of the alkali halides may be found in the article by Seitz, Rev. Mod. Physics, 1954, 26, 7. 3 Huntington, Physic. Rev., 1953, 91, 1092. 4 see, for example, Seitz, J. Physic. SOC. Japan, 1955, 10, 679. 5 Bauerle, Klabunde and Koehler, Physic. Rev., 1956, 102, 1182. 6 see, for example, Seitz and Koehler, Solid State Physics (Academic Press, New 7 Holcomb, and Norberg, Physic. Rev., 1954, 93, 919 ; see also Slichter, Bristol Con- 8 Nachtricb, Weil, Catalan0 and Lawson, J. Chem. Physics, 1952, 20, 1189. 9 MacDonald, Bristol Conference on Defects in Solids (Physic. SOC., London, 1955). 10 Letaw, Slifkin and Portnoy, Physic. Rev., 1954, 93, 892. 11 Mayberg, Physic. Rev., 1954, 95, 38. 12 Frank and Turnbull, Physic. Rev., 1956. 13 Tweet and Gallagher, Physic. Rev., 1956. 14 Kanzaki (to be published in the Phil. Mag.). 15 Hardy, Prog. Metal PJzysics (Pergamon Press, 1956), 6, p. 45. 16 Burton, Cabrera and Frank, Faraday SOC. Discussions, 1949, 5. 17 Seitz, PJzysic. Rev., 1950, 80, 239 ; Advances in Physics, 1952, 1, 43. 18 Turnbull, Bristol Coiference on Defects iiz Solids (Physic. Soc., London, 1955), p. 203. York, 1956), vol. 2, p. 305. ference on Defects in Solids (Physic. SOC., London, 1955), p. 52. p. 383.