(in crystals), crystal defects; lines along which and in the vicinity of which the regular arrangement of the atomic planes characteristic for the crystal is disrupted. Dislocations and other crystal defects determine many physical properties of crystals; such properties are called structurally sensitive properties. In particular, the mechanical properties of crystals—strength and plasticity—are caused by the existence of dislocations and by their special features.

Types of dislocations. The simplest types of dislocations are edge and screw dislocations. In an ideal crystal, the neighboring atomic planes are parallel over their entire width, but in a real crystal, the atomic planes frequently terminate within the crystal (Figure l,a), which gives rise to an edge dislocation, whose axis is the edge of the “extra” plane. The use of electron microscopes of high resolving power makes it possible to observe the arrangement of the atomic rows in some crystals that is characteristic of edge dislocation.

The formation of edge dislocations may be imagined by cutting the crystal along part of the plane ABCD (Figure 1 ,b),

Figure 1. Edge dislocation: (a) Disruption of the atomic plane within the crystal, (b) diagram of the formation of an edge dislocation

displacing the lower part with respect to the upper part by one atomic distance b in the direction perpendicular to AB, and then again joining the atoms at the opposite sides of the cut. The remaining extra plane terminates along the edge dislocation AB. The vector b, whose magnitude is equal to the interatomic distance, is called the characteristic slip vector (the Burgers vector). The plane passing through the slip vector and the line of dislocation is called the glide plane for the edge dislocation.

If the slip direction b is parallel rather than perpendicular to the boundary AB of the cut, a screw dislocation results (Figure 2,a). In contrast to the edge dislocation, the glide plane for a screw dislocation may be any crystallographic plane passing through the line AB. A crystal with a screw dislocation no longer consists of parallel atomic planes but should be regarded as consisting of a single atomic plane twisted into the shape of a helicoid, or spiral staircase without steps (Figure 2,b). Figure 2,c shows the arrangement of atoms above (open circles) and below (solid circles) the glide plane in a simple cubic lattice with a screw dislocation. If the screw dislocation emerges at the external surface of the crystal, then the step AD, one atomic layer in height, is terminated at the point of emergence A (Figure 2,b). This step plays an active role during the crystallization process. Atoms of material precipitating from the vapor phase or solution readily add to the step at the surface of the growing crystal. The number of atoms included in the step and the rate of displacement of the step at the crystal surface are greater in the area of the emergence of the dislocation; therefore, the step is twisted about the axis of the dislocation. The step gradually rises from one crystal “level” to another, which leads to the spiral growth of the crystal.

Between the extreme cases of pure edge and screw dislocations, many intermediate cases are possible in which the dislocation line forms an arbitrary angle with the slip vector (mixed dislocation). The dislocation line must not necessarily be straight; it may be an arbitrary curve. The dislocation lines cannot terminate within the crystal. They must be closed, forming a loop; branched, to give several dislocations; or emerging at the surface of the crystal. The dislocation density in a crystal is defined as the average number of dislocation lines intersecting an area of 1 sq cm drawn within the body or as the total dislocation length in 1 cu cm. The dislocation density usually varies from 10² to 10³ per sq cm in the most

Figure 2. Screw dislocation: (a) Diagram of the formation of a screw dislocation, (b) arrangement of atoms in a crystal with a screw dislocation (atoms are located at the corners of the cubes), (c) arrangement of atoms in the glide plane of the screw dislocation

perfect single crystals and reaches 10¹¹-10¹² per sq cm in strongly distorted (work-hardened) metals.

Dislocations as sources of internal stresses. The regions of the crystal in the vicinity of dislocations are in an elastically stressed state. The stresses decrease in inverse proportion to distance from the dislocation. The stress fields of the individual dislocations are made visible (in transparent crystals with low dislocation density) by polarized light. Depending on the orientation of the glide vectors of two dislocations, they may be either repelled or attracted. The approach of two dislocations with the same glide vectors (Figure 3,a) increases the compression of the crystal on one side of the glide plane and the stretching of the crystal on the other side of the glide plane. The approach of dislocations with opposite slip vectors leads to a compensation of the stretching and compression on both sides of the glide plane (Figure 3,b,c,d).

Figure 3. (a) and (b) Attracting and repelling dislocations; (c) and (d) annihilation of attracting dislocations

The magnitude of the elastic energy caused by the dislocation stress field is proportional to b² and is usually ~ 10⁻⁴ erg per cm of dislocation length.

Motion of dislocations. Dislocations can move within the crystal, causing plastic deformation. Motion of the dislocation in the glide plane is called glide. Glide of a single dislocation through the crystal leads to a plastic shear by one interatomic distance b (Figure 4). Upon displacement of a dislocation within the glide plane, interatomic bonds are broken and re-formed at any given moment not between all of the atoms in the glide plane (Figure 4,a) but only between the atoms located near the axis of the dislocation (Figure 4,b). For this reason, dislocation glide occurs at relatively small external stresses. These stresses are several orders of magnitude lower than the stress leading to a plastic deformation of a perfect crystal without dislocations (theoretical shear strength). Filament crystals (whiskers) without dislocations may have shear strengths approaching the theoretical.

Figure 4. Motion of a dislocation in the glide plane is accompanied by rupture and reformation of the interatomic bonds. In a dislocation-free crystal, shear in the glide plane requires simultaneous rupture of all interatomic bonds

Motion of an edge or mixed dislocation in the direction perpendicular to the glide plane is called climb. It occurs as a result of the diffusion of atoms, or the movement of vacancies in opposite directions, from the crystal toward the edge of the extra plane forming the dislocation (Figure 5). Since the rate of diffusion decreases very sharply (exponentially) with decreasing temperature, climb occurs with appreciable velocity only at rather high temperatures. If the crystal containing the dislocation is under load, the fluxes of atoms and vacancies are directed in such a way that the elastic stresses are minimized. This results in plastic deformation of the crystal caused not by glide but by climb of the dislocation. Thus, the plastic deformation of a crystal with a dislocation is always motion of the dislocation. In this case the rate of plastic deformation of the crystal is found to be directly proportional to the density of moving dislocations and to their mean velocity. Plastic deformation of a crystal without dislocations occurs through diffusion of point defects.

Figure 5. Climb of an edge dislocation. Atoms of the extra plane migrate to the vacant lattice points.

Mobility of dislocations. Glide of dislocations is resisted not only by the strength of the interatomic bonds that are being broken but also by the scattering of the thermal oscillations of atoms and conduction electrons (in metals) in the elastically distorted region of the crystal surrounding the moving dislocation, by elastic interaction with other dislocations and with atoms of impurity elements in solid solutions, by grain boundaries in polycrystals, and by particles of another phase in separating alloys (twins). Part of the work of external forces is expended on overcoming these obstacles; as a result, the fewer defects the crystal contains, the greater the dependence of the mobility of dislocations on the lattice structure. The rate of dislocation glide increases sharply with stress but does not exceed the speed of propagation of sound in the crystal. The rate of climb is proportional to the stress.

Formation and disappearance of dislocations. Dislocations usually arise during the formation of crystals from melts or the gaseous phase. Methods of growing single crystals containing no dislocations at all are very complex and have been developed only for a few crystalline materials. After careful annealing, the crystals usually contain 10⁴-10⁵ dislocations per sq cm. The smallest plastic deformation of such a crystal leads to an intensive “multiplication” of dislocations (Figure 6), without which significant plastic deformation of the crystal is impossible. If new dislocations were not generated within the crystal, the deformation would terminate after the emergence onto the surface of all dislocations present in the crystal.

Figure 6. Diagram of a Frank-Read dislocation source. A section of the dislocation is fixed at points A and B. The section is bent under the influence of the external force (arrow) and successively assumes configurations (a) through (g) until the closed dislocation loop is separated, with the regeneration of the initial section AB. The attracting regions m and n of the loop undergo annihilation at stage (f).

Attracting dislocations with opposing glide vectors, located in the same glide plane, annihilate each other on approach (Figure 3,b, c, d). If such dislocations are located in different glide planes, climb is required for their annihilation. For this reason, high-temperature annealing, which favors climb, lowers the density of dislocations.

Dislocations as a source of lattice curvature. Crystal regions separated by rows of dislocations (Figure 7) or by dislocation

Figure 7. Dislocations forming a grain boundary

networks have a different orientation of the atomic planes and are called crystalline blocks (grains). If dislocations are uniformly distributed throughout the crystal volume, the block structure does not exist, but the lattice is curved (Figure 8).

Figure 8. Bent crystal

The curvature of the atomic planes and distortion of the interplane distances near dislocations increase the intensity of X-ray and electron scattering. This is the basis of the X-ray and electron-microscope methods for the study of dislocations.

Dislocation structure of deformed crystals. Fracture. The distribution of dislocations in deformed crystals is usually nonuniform. At a low degree of deformation (usually up to 10 percent), dislocations are frequently located along selected glide planes. An increase in deformation generates (usually in metals) a block structure, which may be detected with an electron microscope or by X-ray scattering. The size of the blocks decreases with increasing deformation. Multiplication of the dislocations decreases the mean distances between dislocations, their elastic stress fields overlap, and glide is made more difficult (work hardening of crystals). To continue the slippage, the externally applied stress must be increased.

Upon further multiplication of dislocations, the internal stresses may attain values approaching the theoretical strength. In this case, fracture of the crystal takes place through the nucleation and propagation of microscopic cracks. Destruction may also be facilitated by thermal oscillations.

Effect of dislocations on the physical properties of crystals. Dislocations affect not only such mechanical properties of solids as plasticity and strength, for which the existence of dislocations is essential, but also other physical properties of crystals. For example, an increase in the number of dislocations leads to a decrease in crystal density and internal friction, a change in the optical properties, and an increase in electrical resistance. Dislocations increase the mean rate of diffusion within the crystal and accelerate aging and other processes involving diffusion. Dislocations decrease the chemical stability of the crystal; as a result, treatment of the surface with various materials (etching agents) leads to the formation of visible etch pits at the sites of the emergence of dislocations at the surface. This forms the basis for detecting dislocations in opaque materials by the selective etching method.

irregularities in the shapes of the initial bedding of rock caused by tectonic movements of the earth’s crust, magmatic activity, metamorphism, and exogenous processes such as glacial movement, landslides, karst, and river erosion. Dislocations are usually subdivided into plicate and rupture (disjunctive) dislocations, and sometimes injection dislocations are also distinguished.