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Defects in solids and their Applications

defects and defect processes in nonmetallic solids defects in organic solids intrinsic and extrinsic defects in solids radiation effects and defects in solids abbreviation
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Published Date:25-10-2017
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Chapter 4 Defects in solids 4.1 Types of imperfection Real solids invariably contain structural discontinuities and localized regions of disorder. This heterogene- ity can exist on both microscopic and macroscopic scales, with defects or imperfections ranging in size from missing or misplaced atoms to features that are visible to the naked eye. The majority of materials used for engineering components and structures are made up from a large number of small interlocking grains or crystals. It is therefore immediately appro- priate to regard the grain boundary surfaces of such polycrystalline aggregates as a type of imperfection. Other relatively large defects, such as shrinkage pores, gas bubbles, inclusions of foreign matter and cracks, may be found dispersed throughout the grains of a metal or ceramic material. In general, however, these large-scale defects are very much influenced by the processing of the material and are less fundamental to the basic material. More attention will thus be given to the atomic-scale defects in materials. Within each grain, atoms are regularly arranged according to the Figure 4.1 (a) Vacancy–interstitial, (b) dislocation, basic crystal structure but a variety of imperfections, (c) stacking fault, (d) void. classified generally as crystal defects, may also occur. A schematic diagram of these basic defects is shown crystal defects. Defects in crystalline macromolecular in Figure 4.1. These take the form of: structures, as found in polymers, form a special subject and will be dealt with separately in Section 4.6.7. ž Point defects, such as vacant atomic sites (or simply vacancies) and interstitial atoms (or simply intersti- tials) where an atom sits in an interstice rather than 4.2 Point defects a normal lattice site ž Line defects, such as dislocations 4.2.1 Point defects in metals ž Planar defects, such as stacking faults and twin Of the various lattice defects the vacancy is the only boundaries species that is ever present in appreciable concen- ž Volume defects, such as voids, gas bubbles and trations in thermodynamic equilibrium and increases cavities. exponentially with rise in temperature, as shown in In the following sections this type of classification Figure 4.2. The vacancy is formed by removing an will be used to consider the defects which can occur atom from its lattice site and depositing it in a nearby in metallic and ceramic crystals. Glasses already lack atomic site where it can be easily accommodated. long-range order; we will therefore concentrate upon Favoured places are the free surface of the crystal, aDefects in solids 85 Figure 4.2 Equilibrium concentration of vacancies as a function of temperature for aluminium (after Bradshaw and Pearson, 1957). grain boundary or the extra half-plane of an edge dislo- cation. Such sites are termed vacancy sources and the vacancy is created when sufficient energy is available Figure 4.3 Variation of the energy of a crystal with addition (e.g. thermal activation) to remove the atom. If E is f of n vacancies. the energy required to form one such defect (usually expressed in electron volts per atom), the total energy increase resulting from the formation ofn such defects the Boltzmann factor exp E/kT, the effect of the f is nE . The accompanying entropy increase may be f vacancy on the vibrational properties of the lattice calculated using the relations SD k ln W,where W also leads to an entropy term which is independent is the number of ways of distributing n defects and of temperature and usually written as exp S/k. The f N atoms on NCn lattice sites, i.e. NCn/nN fractional concentration may thus be written Then the free energy, G, or strictly F of a crystal of cDn/ND exp S/kexpE/kT f f n defects, relative to the free energy of the perfect crystal, is DA exp E/kT 4.3 f FDnE  kTlnNCn/nN (4.1) f The value of the entropy term is not accurately known but it is usually taken to be within a factor 1 which by the use of Stirling’s theorem simplifies to of ten of the value 10; for simplicity we will take it to be unity. FDnE  kT NCn lnNCnn ln nN ln N f The equilibrium number of vacancies rises rapidly 4.2 with increasing temperature, owing to the exponential The equilibrium value of n is that for which form of the expression, and for most common metals dF/dnD 0, which defines the state of minimum free 4 has a value of about 10 near the melting point. For 2 energy as shown in Figure 4.3. Thus, differentiating example, kT at room temperature (300 K) is³1/40 eV equation (4.2) gives and for aluminium E D 0.7 eV, so that at 900 K we f have 0DE  kT lnNCn ln n f   7 40 300 cD exp  ð ð DE  kT ln NCn/n f 10 1 900 so that 9.3/2.3 4 D exp 9.3D 10 ³ 10 n D exp E/kT f As the temperature is lowered, c should decrease NCn in order to maintain equilibrium and to do this the Usually N is very large compared with n,sothat vacancies must migrate to positions in the structure where they can be annihilated; these locations are then the expression can be taken to give the atomic concen- known as ‘vacancy sinks’ and include such places as tration, c, of lattice vacancies, n/ND exp E/kT. f the free surface, grain boundaries and dislocations. A more rigorous calculation of the concentration of The defect migrates by moving through the energy vacancies in thermal equilibrium in a perfect lat- maxima from one atomic site to the next with a tice shows that although c is principally governed by frequency     1 Stirling’s approximation states that ln NDN ln N. S E m m D exp exp  0 2 dF/dn or dG/dn in known as the chemical potential. K KT86 Modern Physical Metallurgy and Materials Engineering where  is the frequency of vibration of the defect be retained in the structure. Moreover, if the cool- 0 in the appropriate direction,S is the entropy increase ing rate of the metal or alloy is particularly rapid, m and E is the internal energy increase associated with as, for example, in quenching, the vast majority of m the process. The self-diffusion coefficient in a pure the vacancies which exist at high temperatures can be metal is associated with the energy to form a vacancy ‘frozen-in’. E and the energy to move it E , being given by the Vacancies are of considerable importance in gov- f m expression erning the kinetics of many physical processes. The industrial processes of annealing, homogenization, pre- E DE CE SD f m cipitation, sintering, surface-hardening, as well as oxi- dation and creep, all involve, to varying degrees, the Clearly the free surface of a sample or the grain transport of atoms through the structure with the help boundary interface are a considerable distance, in of vacancies. Similarly, vacancies enable dislocations atomic terms, from the centre of a grain and so to climb, since to move the extra half-plane of a dislo- dislocations in the body of the grain or crystal are cation up or down requires the mass transport of atoms. the most efficient ‘sink’ for vacancies. Vacancies are This mechanism is extremely important in the recov- annihilated at the edge of the extra half-plane of atoms ery stage of annealing and also enables dislocations to of the dislocation, as shown in Figure 4.4a and 4.4b. climb over obstacles lying in their slip plane; in this This causes the dislocation to climb, as discussed way materials can soften and lose their resistance to in Section 4.3.4. The process whereby vacancies creep at high temperatures. are annihilated at vacancy sinks such as surfaces, In metals the energy of formation of an interstitial grain boundaries and dislocations, to satisfy the atom is much higher than that for a vacancy and thermodynamic equilibrium concentration at a given is of the order of 4 eV. At temperatures just below temperature is, of course, reversible. When a metal the melting point, the concentration of such point is heated the equilibrium concentration increases and, 15 defects is only about 10 and therefore interstitials to produce this additional concentration, the surfaces, are of little consequence in the normal behaviour of grain boundaries and dislocations in the crystal reverse metals. They are, however, more important in ceramics their role and act as vacancy sources and emit because of the more open crystal structure. They are vacancies; the extra half-plane of atoms climbs in the also of importance in the deformation behaviour of opposite sense (see Figures 4.4c and 4.4d). solids when point defects are produced by the non- Below a certain temperature, the migration of vacan- conservative motion of jogs in screw dislocation (see cies will be too slow for equilibrium to be main- Section 4.3.4) and also of particular importance in tained, and at the lower temperatures a concentration materials that have been subjected to irradiation by of vacancies in excess of the equilibrium number will high-energy particles. 4.2.2 Point defects in non-metallic crystals Point defects in non-metallic, particularly ionic, struc- tures are associated with additional features (e.g. the requirement to maintain electrical neutrality and the possibility of both anion-defects and cation-defects existing). An anion vacancy in NaCl, for example, will be a positively-charged defect and may trap an elec- tron to become a neutral F-centre. Alternatively, an anion vacancy may be associated with either an anion interstitial or a cation vacancy. The vacancy-interstitial pair is called a Frenkel defect and the vacancy pair a Schottky defect, as shown in Figure 4.5. Interstitials are much more common in ionic structures than metal- lic structures because of the large ‘holes’ or interstices that are available. In general, the formation energy of each of these two types of defect is different and this leads to different defect concentrations. With regard to vacancies, when  C E E , i.e. the formation will initially produce f f more cation than anion vacancies from dislocations and boundaries as the temperature is raised. However, the electrical field produced will eventually oppose the production of further cations and promote the formation of anions such that of equilibrium there Figure 4.4 Climb of a dislocation, (a) and (b) to annihilate, (c) and (d) to create a vacancy. will be almost equal numbers of both types and theDefects in solids 87 Figure 4.5 Representation of point defects in two-dimensional ionic structure: (a) perfect structure and monovalent ions, (b) two Schottky defects, (c) Frenkel defect, and (d) substitutional divalent cation impurity and cation vacancy. combined or total concentration c of Schottky defects can only be transferred by the diffusion of the charge 4 carrying defects through the oxide. Bothp-andn-type at high temperatures is¾10 . semiconductors are formed when oxides deviate from Foreign ions with a valency different from the host stoichiometry: the former arises from a deficiency of cation may also give rise to point defects to maintain cations and the latter from an excess of cations. charge neutrality. Monovalent sodium ions substituting Examples ofp-type semiconducting oxides are NiO, for divalent magnesium ions in MgO, for example, PbO and Cu O while the oxides of Zn, Cd and Be are 2 must be associated with an appropriate number of n-type semiconductors. either cation interstitials or anion vacancies in order to maintain charge neutrality. Deviations from the sto- ichiometric composition of the non-metallic material 4.2.3 Irradiation of solids as a result of excess (or deficiency) in one (or other) There are many different kinds of high-energy radi- atomic species also results in the formation of point ation (e.g. neutrons, electrons, ˛-particles, protons, defects. deuterons, uranium fission fragments, -rays, X-rays) An example of excess-metal due to anion vacancies and all of them are capable of producing some form is found in the oxidation of silicon which takes place at of ‘radiation damage’ in the materials they irradiate. the metal/oxide interface. Interstitials are more likely While all are of importance to some aspects of the to occur in oxides with open crystal structures and solid state, of particular interest is the behaviour of when one atom is much smaller than the other as, for materials under irradiation in a nuclear reactor. This is example, ZnO (Figure 4.6a). The oxidation of copper because the neutrons produced in a reactor by a fis- to Cu O, shown in Figure 4.6b, is an example of non- 2 sion reaction have extremely high energies of about stoichiometry involving cation vacancies. Thus copper 2 million electron volts (i.e. 2 MeV), and being elec- vacancies are created at the oxide surface and diffuse trically uncharged, and consequently unaffected by the through the oxide layer and are eliminated at the electrical fields surrounding an atomic nucleus, can oxide/metal interface. travel large distances through a structure. The resul- Oxides which contain point defects behave as semi- tant damage is therefore not localized, but is distributed conductors when the electrons associated with the throughout the solid in the form of ‘damage spikes.’ point defects either form positive holes or enter the The fast neutrons (they are given this name because 7 1 conduction band of the oxide. If the electrons remain 2 MeV corresponds to a velocity of 2ð 10 ms locally associated with the point defects, then charge are slowed down, in order to produce further fission, Figure 4.6 Schematic arrangement of ions in two typical oxides. (a) Zn O, with excess metal due to cation interstitials and 1 (b) Cu O, with excess non-metal due to cation vacancies. 288 Modern Physical Metallurgy and Materials Engineering by the moderator in the pile until they are in thermal equilibrium with their surroundings. The neutrons in a pile will, therefore, have a spectrum of energies which ranges from about 1/40 eV at room temperature (thermal neutrons) to 2 MeV (fast neutrons). However, when non-fissile material is placed in a reactor and irradiated most of the damage is caused by the fast neutrons colliding with the atomic nuclei of the material. The nucleus of an atom has a small diameter (e.g. 10 10 m), and consequently the largest area, or cross- section, which it presents to the neutron for collision Figure 4.7 Formation of vacancies and interstitials due to particle bombardment (after Cottrell, 1959; courtesy of the is also small. The unit of cross-section is a barn, i.e. 28 2 Institute of Mechanical Engineers). 10 m so that in a material with a cross-section 9 of 1 barn, an average of 10 neutrons would have to 19 2 pass through an atom (cross-sectional area 10 m ) annihilating each other by recombination. However, it for one to hit the nucleus. Conversely, the mean free is expected that some of the interstitials will escape 9 path between collisions is about 10 atom spacings or from the surface of the cascade leaving a correspond- about 0.3 m. If a metal such as copper (cross-section, ing number of vacancies in the centre. If this number 5 4 barns) were irradiated for 1 day 10 s in a neutron is assumed to be 100, the local concentration will be 17 2 1 flux of 10 m s the number of neutrons passing 3 100/60 000 or³2ð 10 . through unit area, i.e. the integrated flux, would be Another manifestation of radiation damage concerns 22 2 10 nm and the chance of a given atom being hit the dispersal of the energy of the stopped atom into 6 Dintegrated fluxð cross-section would be 4ð 10 , the vibrational energy of the lattice. The energy is i.e. about 1 atom in 250 000 would have its nucleus deposited in a small region, and for a very short struck. time the metal may be regarded as locally heated. To For most metals the collision between an atomic distinguish this damage from the ‘displacement spike’, nucleus and a neutron (or other fast particle of mass where the energy is sufficient to displace atoms, this m) is usually purely elastic, and the struck atom heat-affected zone has been called a ‘thermal spike’. To mass M will have equal probability of receiving any ° raise the temperature by 1000 C requires about 3Rð kinetic energy between zero and the maximumE D max 4.2 kJ/mol or about 0.25 eV per atom. Consequently, 2 4E Mm/ MCm ,where E is the energy of the fast n n a 25 eV thermal spike could heat about 100 atoms neutron. Thus, the most energetic neutrons can impart of copper to the melting point, which corresponds an energy of as much as 200 000 eV, to a copper to a spherical region of radius about 0.75 nm. It is atom initially at rest. Such an atom, called a primary very doubtful if melting actually takes place, because ‘knock-on’, will do much further damage on its sub- 11 the duration of the heat pulse is only about 10 to sequent passage through the structure often producing 12 10 s. However, it is not clear to what extent the secondary and tertiary knock-on atoms, so that severe heat produced gives rise to an annealing of the primary local damage results. The neutron, of course, also con- damage, or causes additional quenching damage (e.g. tinues its passage through the structure producing fur- retention of high-temperature phases). ther primary displacements until the energy transferred Slow neutrons give rise to transmutation products. in collisions is less than the energy E (³25 eV for d Of particular importance is the production of the noble copper) necessary to displace an atom from its lat- gas elements, e.g. krypton and xenon produced by fis- tice site. sion in U and Pu, and helium in the light elements The damage produced in irradiation consists largely B, Li, Be and Mg. These transmuted atoms can cause of interstitials, i.e. atoms knocked into interstitial posi- severe radiation damage in two ways. First, the inert tions in the lattice, and vacancies, i.e. the holes they gas atoms are almost insoluble and hence in association leave behind. The damaged region, estimated to con- with vacancies collect into gas bubbles which swell tain about 60 000 atoms, is expected to be originally and crack the material. Second, these atoms are often pear-shaped in form, having the vacancies at the cen- created with very high energies (e.g. as ˛-particles tre and the interstitials towards the outside. Such a or fission fragments) and act as primary sources of displacement spike or cascade of displaced atoms is knock-on damage. The fission of uranium into two shown schematically in Figure 4.7. The number of new elements is the extreme example when the fis- vacancy-interstitial pairs produced by one primary sion fragments are thrown apart with kinetic energy knock-on is given by n'E /4E , and for copper max d ³100 MeV. However, because the fragments carry is about 1000. Owing to the thermal motion of the a large charge their range is short and the damage atoms in the lattice, appreciable self-annealing of the restricted to the fissile material itself, or in materi- damage will take place at all except the lowest tem- peratures, with most of the vacancies and interstitials als which are in close proximity. Heavy ions can beDefects in solids 89 accelerated to kilovolt energies in accelerators to pro- an easier process and the activation energy for migra- duce heavy ion bombardment of materials being tested tion is somewhat lower than E for single vacancies. m for reactor application. These moving particles have a Excess point defects are removed from a mate- short range and the damage is localized. rial when the vacancies and/or interstitials migrate to regions of discontinuity in the structure (e.g. free sur- faces, grain boundaries or dislocations) and are annihi- 4.2.4 Point defect concentration and lated. These sites are termed defect sinks. The average annealing number of atomic jumps made before annihilation is Electrical resistivity  is one of the simplest and given by most sensitive properties to investigate the point defect nDAzvt exp E /kT (4.5) m a concentration. Point defects are potent scatterers of electrons and the increase in resistivity following where A is a constant ³1 involving the entropy of quenching  may be described by the equation migration, z the coordination around a vacancy, v the 13 Debye frequency ³10 /s ,t the annealing time at the DA exp E/kT (4.4) f Q ageing temperatureT andE the migration energy of a m the defect. For a metal such as aluminium, quenched where A is a constant involving the entropy of to give a high concentration of retained vacancies, the formation, E is the formation energy of a vacancy f annealing process takes place in two stages, as shown and T the quenching temperature. Measuring the Q in Figure 4.8; stage I near room temperature with an resistivity after quenching from different temperatures 4 activation energy³0.58 eV and n³ 10 ,and stage II enables E to be estimated from a plot of  f 0 ° in the range 140–200 C with an activation energy of versus 1/T . The activation energy, E ,for the Q m ¾1.3eV. movement of vacancies can be obtained by measuring Assuming a random walk process, single vacancies the rate of annealing of the vacancies at different p would migrate an average distance ( nð atomic spac- annealing temperatures. The rate of annealing is ing b)³30 nm. This distance is very much less than inversely proportional to the time to reach a certain either the distance to the grain boundary or the spac- value of ‘annealed-out’ resistivity. Thus, 1/t DA 1 ing of the dislocations in the annealed metal. In this exp E /kT and 1/t D exp E /kT and by m 1 2 m 2 case, the very high supersaturation of vacancies pro- eliminating A we obtain ln t /t DE 1/T  2 1 m 2 duces a chemical stress, somewhat analogous to an 1/T /k where E is the only unknown in the 1 m osmotic pressure, which is sufficiently large to create expression. Values ofE andE for different materials f m new dislocations in the structure which provide many are given in Table 4.1. new ‘sinks’ to reduce this stress rapidly. At elevated temperatures the very high equilibrium The magnitude of this chemical stress may be concentration of vacancies which exists in the structure estimated from the chemical potential, if we let dF gives rise to the possible formation of divacancy and represent the change of free energy when dn vacancies even tri-vacancy complexes, depending on the value are added to the system. Then, of the appropriate binding energy. For equilibrium between single and di-vacancies, the total vacancy dF/dnDE C kT ln n/N DkT ln c C kT ln c f 0 concentration is given by D kT ln c/c 0 c Dc C 2c v 1v 2v where c is the actual concentration and c the equilib- 0 rium concentration of vacancies. This may be rewrit- and the di-vacancy concentration by ten as 2 c DAzc exp B /kT 2v 1v 2 where A is a constant involving the entropy of forma- tion of di-vacancies,B the binding energy for vacancy 2 pairs estimated to be in the range 0.1–0.3 eV and z a configurational factor. The migration of di-vacancies is Table 4.1 Values of vacancy formation .E / and migration f .E / energies for some metallic materials together with m the self-diffusion energy .E / SD Energy Cu Al Ni Mg Fe W NiAl (eV) E 1.0–1.1 0.76 1.4 0.9 2.13 3.3 1.05 f Figure 4.8 Variation of quenched-in resistivity with E 1.0–1.1 0.62 1.5 0.5 0.76 1.9 2.4 m temperature of annealing for aluminium (after Panseri and E 2.0–2.2 1.38 2.9 1.4 2.89 5.2 3.45 D Federighi, 1958, 1223).90 Modern Physical Metallurgy and Materials Engineering dF/dVD Energy/volume stress 3 D kT/b ln c/c 4.6 0 where dV is the volume associated with dn vacancies 3 and b is the volume of one vacancy. Inserting typical values, KT ' 1/40 eV at room temperature, bD 2 3 0.25 nm, shows KT/b ' 150 MN/m . Thus, even a moderate 1% supersaturation of vacancies i.e. when c/c D 1.01 and ln c/c D 0.01, introduces a 0 0 2 chemical stress  equivalent to 1.5MN/m . c The equilibrium concentration of vacancies at a Figure 4.9 Variation of resistivity with temperature produced by neutron irradiation for copper (after Diehl). temperature T will be given by c D exp E/kT 2 2 f 2 and at T by c D exp E/kT . Then, since 1 1 f 1   1 1 around room temperature and is probably caused by ln c /c D E/k  2 1 f the annihilation of free interstitials with individual T T 1 2 vacancies not associated with a Frenkel pair, and also the chemical stress produced by quenching a metal the migration of di-vacancies. Stage IV corresponds to from a high temperature T to a low temperature T the stage I annealing of quenched metals arising from 2 1 is vacancy migration and annihilation to form dislocation   loops, voids and other defects. Stage V corresponds T 1 3 3  D kT/b ln c /c D E/b 1 to the removal of this secondary defect population by c 2 1 f T 2 self-diffusion. For aluminium, E is about 0.7 eV so that quench- f ing from 900 K to 300 K produces a chemical stress 4.3 Line defects 2 of about 3 GN/m . This stress is extremely high, sev- eral times the theoretical yield stress, and must be 4.3.1 Concept of a dislocation relieved in some way. Migration of vacancies to grain All crystalline materials usually contain lines of struc- boundaries and dislocations will occur, of course, but tural discontinuities running throughout each crystal it is not surprising that the point defects form addi- or grain. These line discontinuities are termed dislo- tional vacancy sinks by the spontaneous nucleation of 10 12 cations and there is usually about 10 to 10 mof dislocations and other stable lattice defects, such as 1 dislocation line in a metre cube of material. Disloca- voids and stacking fault tetrahedra (see Sections 4.5.3 tions enable materials to deform without destroying the and 4.6). basic crystal structure at stresses below that at which When the material contains both vacancies and the material would break or fracture if they were not interstitials the removal of the excess point defect present. concentration is more complex. Figure 4.9 shows A crystal changes its shape during deformation by the ‘annealing’ curve for irradiated copper. The the slipping of atomic layers over one another. The resistivity decreases sharply around 20 K when the theoretical shear strength of perfect crystals was first interstitials start to migrate, with an activation calculated by Frenkel for the simple rectangular-type energy E ¾ 0.1 eV. In Stage I, therefore, most m lattice shown in Figure 4.10 with spacing a between of the Frenkel (interstitial–vacancy) pairs anneal out. Stage II has been attributed to the release of 1 interstitials from impurity traps as thermal energy This is usually expressed as the density of dislocations 10 12 2 supplies the necessary activation energy. Stage III is D 10 to 10 m . Figure 4.10 Slip of crystal planes (a); shear stress versus displacement curve (b).Defects in solids 91 the planes. The shearing force required to move a plane of atoms over the plane below will be periodic, since for displacementsxb/2, where b is the spacing of atoms in the shear direction, the lattice resists the applied stress but forxb/2 the lattice forces assist the applied stress. The simplest function with these properties is a sinusoidal relation of the form D  sin2x/b '  2x/b m m where  is the maximum shear stress at a m displacementD b/4. For small displacements the elastic shear strain given by x/a is equal to / from Hooke’s law, where is the shear modulus, so that  D /2 b/a (4.7) m and since b' a, the theoretical strength of a perfect Figure 4.11 Schematic representation of (a) a dislocation crystal is of the order of /10. loop, (b) edge dislocation and (c) screw dislocation. This calculation shows that crystals should be rather strong and difficult to deform, but a striking exper- imental property of single crystals is their softness, the dislocation line is rarely pure edge or pure screw, which indicates that the critical shear stress to pro- but it is convenient to think of these ideal dislocations 5 2 since any dislocation can be resolved into edge and duce slip is very small (about 10 µ or³50gf mm ). screw components. The atomic structure of a simple This discrepancy between the theoretical and experi- edge and screw dislocation is shown in Figure 4.13 mental strength of crystals is accounted for if atomic and 4.14. planes do not slip over each other as rigid bodies but instead slip starts at a localized region in the structure and then spreads gradually over the remainder of the 4.3.3 The Burgers vector plane, somewhat like the disturbance when a pebble is It is evident from the previous sections that the Burgers dropped into a pond. vector b is an important dislocation parameter. In any In general, therefore, the slip plane may be divided deformation situation the Burgers vector is defined by into two regions, one where slip has occurred and the constructing a Burgers circuit in the dislocated crystal other which remains unslipped. Between the slipped as shown in Figure 4.12. A sequence of lattice vectors and unslipped regions the structure will be dislocated is taken to form a closed clockwise circuit around (Figure 4.11); this boundary is referred to as a dislo- the dislocation. The same sequence of vectors is then cation line, or dislocation. Three simple properties of taken in the perfect lattice when it is found that the a dislocation are immediately apparent, namely: (1) it circuit fails to close. The closure vector FS (finish- is a line discontinuity, (2) it forms a closed loop in the start) defines b for the dislocation. With this FS/RH interior of the crystal or emerges at the surface and (right-hand) convention it is necessary to choose one (3) the difference in the amount of slip across the dis- direction along the dislocation line as positive. If this location line is constant. The last property is probably direction is reversed the vector b is also reversed. the most important, since a dislocation is characterized The Burgers vector defines the atomic displacement by the magnitude and direction of the slip movement produced as the dislocation moves across the slip associated with it. This is called the Burgers vector, plane. Its value is governed by the crystal structure b, which for any given dislocation line is the same all because during slip it is necessary to retain an identical along its length. lattice structure both before and after the passage of the dislocation. This is assured if the dislocation has 4.3.2 Edge and screw dislocations It is evident from Figure 4.11a that some sections of the dislocation line are perpendicular to b,others are parallel to b while the remainder lie at an angle to b. This variation in the orientation of the line with respect to the Burgers vector gives rise to a difference in the structure of the dislocation. When the dislocation line is normal to the slip direction it is called an edge dislocation. In contrast, when the line of the dislocations is parallel to the slip direction the Figure 4.12 Burgers circuit round a dislocation A fails to dislocation line is known as a screw dislocation. From close when repeated in a perfect lattice unless completed by the diagram shown in Figure 4.11a it is evident that a closure vector FS equal to the Burgers vector b.92 Modern Physical Metallurgy and Materials Engineering Figure 4.13 Slip caused by the movement of an edge dislocation. Figure 4.14 Slip caused by the movement of a screw dislocation. a Burgers vector equal to one lattice vector and, since Figure 4.13 if the crystal is of side L. The force on the 2 the energy of a dislocation depends on the square of top face stressð area is ðL . Thus, when the two the Burgers vector (see Section, its Burgers halves of the crystal have slipped the relative amountb, vector is usually the shortest available lattice vector. the work done by the applied stress forceð distance 2 This vector, by definition, is parallel to the direction is L b. On the other hand, the work done in mov- of closest packing in the structure, which agrees with ing the dislocation total force on dislocation FLð experimental observation of the slip direction. 2 distance moved is FL , so that equating the work The Burgers vector is conveniently specified by done givesFforce per unit length of dislocation D its directional co-ordinates along the principal crys- b. Figure 4.13 indicates how slip is propagated by the tal axes. In the fcc lattice, the shortest lattice vector movement of a dislocation under the action of such a is associated with slip from a cube corner to a face force. The extra half-plane moves to the right until centre, and has components a/2, a/2, 0. This is usu- it produces the slip step shown at the surface of the ally written a/21 1 0, where a is the lattice parameter crystal; the same shear will be produced by a negative and 1 1 0 is the slip direction. The magnitude of the 1 dislocation moving from the right to left. vector, or the strength of the dislocation, is then given  p The slip process as a result of a screw dislocation is 2 2 2 2 by fa 1 C 1 C 0 /4gD a/ 2. The correspond- shown in Figure 4.14. It must be recognized, however, ing slip vectors for the bcc and cph structures are that the dislocation is more usually a closed loop and bD a/21 1 1 and bD a/31 1 2 0 respectively. slip occurs by the movement of all parts of the dislo- cation loop, i.e. edge, screw and mixed components, 4.3.4 Mechanisms of slip and climb as shown in Figure 4.15. A dislocation is able to glide in that slip plane The atomic structure of an edge dislocation is shown which contains both the line of the dislocation and in Figure 4.13a. Here the extra half-plane of atoms is its Burgers vector. The edge dislocation is confined above the slip plane of the crystal, and consequently the dislocation is called a positive edge dislocation and to glide in one plane only. An important difference is often denoted by the symbol?. When the half-plane between the motion of a screw dislocation and that of is below the slip plane it is termed a negative disloca- 1 tion. If the resolved shear stress on the slip plane is  An obvious analogy to the slip process is the movement of and the Burgers vector of the dislocation b, the force a caterpillar in the garden or the propagation of a ruck in a on the dislocation, i.e. force per unit length of dislo- carpet to move the carpet into place. In both examples, the cation, is FD b. This can be seen by reference to effort to move is much reduced by this propagation process.Defects in solids 93 which contains both the dislocation line and its Burg- ers vector. However, movement of the dislocation line in a direction normal to the slip plane can occur under certain circumstances; this is called dislocation climb. To move the extra half-plane either up or down, as is required for climb, requires mass transport by diffu- sion and is a non-conservative motion. For example, if vacancies diffuse to the dislocation line it climbs up and the extra half-plane will shorten. However, since the vacancies will not necessarily arrive at the disloca- tion at the same instant, or uniformly, the dislocation climbs one atom at a time and some sections will lie in one plane and other sections in parallel neighbouring planes. Where the dislocation deviates from one plane to another it is known as a jog, and from the diagrams of Figure 4.17 it is evident that a jog in a dislocation may be regarded as a short length of dislocation not lying in the same slip plane as the main dislocation but having the same Burgers vector. Jogs may also form when a moving dislocation cuts 1 through intersecting dislocations, i.e. forest disloca- tions, during its glide motion. In the lower range of temperature this will be the major source of jogs. Two examples of jogs formed from the crossings of dislo- cations are shown in Figure 4.18. Figure 4.18a shows a crystal containing a screw dislocation running from top to bottom which has the effect of ‘ramping’ all the planes in the crystal. If an edge dislocation moves Figure 4.15 Process of slip by the expansion of a through the crystal on a horizontal plane then the screw dislocation loop in the slip plane. dislocation becomes jogged as the top half of the crys- tal is sheared relative to the bottom. In addition, the screw dislocation becomes jogged since one part has to an edge dislocation arises from the fact that the screw take the upper ramp and the other part the lower ramp. dislocation is cylindrically symmetrical about its axis The result is shown schematically in Figure 4.18b. with itsb parallel to this axis. To a screw dislocation all Figure 4.18c shows the situation for a moving screw crystal planes passing through the axis look the same cutting through the vertical screw; the jog formed in and, therefore, the motion of the screw dislocation is each dislocation is edge in character since it is per- not restricted to a single slip plane, as is the case pendicular to its Burgers vector which lies along the for a gliding edge dislocation. The process thereby a screw axis. screw dislocation glides into another slip plane having A jog in an edge dislocation will not impede the a slip direction in common with the original slip plane, motion of the dislocation in its slip plane because it as shown in Figure 4.16, is called cross-slip. Usually, can, in general, move with the main dislocation line the cross-slip plane is also a close-packed plane, e.g. by glide, not in the same slip plane (see Figure 4.17b) f111g in fcc crystals. The mechanism of slip illustrated above shows that the slip or glide motion of an edge dislocation is restricted, since it can only glide in that slip plane Figure 4.17 Climb of an edge dislocation in a crystal. 1 A number of dislocation lines may project from the slip Figure 4.16 Cross-slip of a screw dislocation in a crystal. plane like a forest, hence the term ‘forest dislocation’.94 Modern Physical Metallurgy and Materials Engineering Figure 4.19 (a) Formation of a multiple jog by cross-slip, and (b) motion of jog to produce a dipole. 1 jogs will make a contribution to the work-hardening of the material. Apart from elementary jogs, or those having a height equal to one atomic plane spacing, it is possible to have multiple jogs where the jog height is several atomic plane spacings. Such jogs can be produced, for example, by part of a screw dislocation cross- slipping from the primary plane to the cross-slip plane, as shown in Figure 4.19a. In this case, as the screw dislocation glides forward it trails the multiple jog behind, since it acts as a frictional drag. As a result, two parallel dislocations of opposite sign are cre- ated in the wake of the moving screw, as shown in Figure 4.19b; this arrangement is called a dislocation dipole. Dipoles formed as debris behind moving dis- locations are frequently seen in electron micrographs Figure 4.18 Dislocation intersections. (a) and taken from deformed crystals (see Chapter 7). As the (b) screw–edge, (c) screw–screw. dipole gets longer the screw dislocation will eventu- ally jettison the debris by cross-slipping and pinching off the dipole to form a prismatic loop, as shown in Figure 4.20. The loop is capable of gliding on the sur- but in an intersecting slip plane that does contain the face of a prism, the cross-sectional area of which is line of the jog and the Burgers vector. In the case that of the loop. of a jog in a screw dislocation the situation is not so clear, since there are two ways in which the jog can move. Since the jog is merely a small piece of edge dislocation it may move sideways, i.e. conservatively, along the screw dislocation and attach itself to an edge component of the dislocation line. Conversely, the jog may be dragged along with the screw dislocation. This latter process requires the jog to climb and, because Figure 4.20 Formation of prismatic dislocation loop from it is a non-conservative process, must give rise to screw dislocation trailing a dipole. the creation of a row of point defects, i.e. either vacancies or interstitials depending on which way the 1 When material is deformed by straining or working the jog is forced to climb. Clearly, such a movement is flow stress increases with increase in strain (i.e. it is harder difficult but, nevertheless, may be necessary to give the to deform a material which has been strained already). This dislocation sufficient mobility. The ‘frictional’ drag of is called strain- or work-hardening.Defects in solids 95 are the normal stresses , along thex-andy-axes 4.3.5 Strain energy associated with xx yy respectively, and the shear stress  which acts in the xy dislocations direction of they-axis on planes perpendicular to thex- Stress fields of screw and edge axis. The third normal stress  D  C where zz xx yy dislocations  is Poisson’s ratio, and the other shear stresses  yz and  are zero. In polar coordinates r,  and z,the zx The distortion around a dislocation line is evident from stresses are  ,  ,and  . rr  r Figure 4.1 and 4.13. At the centre of the dislocation the Even in the case of the edge dislocation the dis- strains are too large to be treated by elasticity theory, placement b has to be accommodated round a ring of but beyond a distance r , equal to a few atom spacings 0 length 2r, so that the strains and the stresses must Hooke’s law can be applied. It is therefore necessary contain a term in b/2r. Moreover, because the atoms to define a core to the dislocation at a cut-off radius r 0 in the region 0 are under compression and for ³b inside which elasticity theory is no longer appli-  2 in tension, the strain field must be of the cable. A screw dislocation can then be considered as a form b/2r f ,where f is a function such as cylindrical shell of lengthl and radiusr contained in an sin  which changes sign when  changes from 0 to elastically isotropic medium (Figure 4.21). A discon- 2. It can be shown that the stresses are given by tinuity in displacement exists only in the z-direction, i.e. parallel to the dislocation, such that uD vD 0,  D  DD sin/r;  D D cos/r; rr  r wD b. The elastic strain thus has to accommodate 2 2 2 2 y3x Cy yx y a displacement wD b around a length 2r.Inan  DD ;  D D 4.9 xz yy elastically isotropic crystal the accommodation must 2 2 2 2 2 2 x Cy x Cy occur equally all round the shell and indicates the 2 2 xx y simple relation wD b/2 in polar (r, , z) coor-  D D xy 2 2 2 dinates. The corresponding shear strain  D  D z z x Cy b/2r and shear stress D  D ub/2r which acts z z where DD b/21 . These equations show that on the end faces of the cylinder with  and equal to rr r 1 the stresses around dislocations fall off as 1/r and zero. Alternatively, the stresses are given in cartesian hence the stress field is long-range in nature. coordinates (x, y, z) 2 2  D  D by/2x Cy xz zx Strain energy of a dislocation 2 2 A dislocation is a line defect extending over large  D  D bx/2x Cy 4.8 yz zy distances in the crystal and, since it has a strain energy 1 with all other stresses equal to zero. The field of a per unit length (J m ), it possesses a total strain screw dislocation is therefore purely one of shear, energy. An estimate of the elastic strain energy of having radial symmetry with no dependence on.This screw dislocation can be obtained by taking the strain 1 mathematical description is related to the structure of energy (i.e. ð stressð strain per unit volume) in an 2 a screw which has no extra half-plane of atoms and annular ring around the dislocation of radius r and cannot be identified with a particular slip plane. 1 thickness dr to be ð b/2r ðb/2r ð 2rdr. 2 An edge dislocation has a more complicated stress The total strain energy per unit length of dislocation is and strain field than a screw. The distortion associated then obtained by integrating from r the core radius, 0 with the edge dislocation is one of plane strain, since to r the outer radius of the strain field, and is there are no displacements along thez-axis, i.e. wD 0.    2 r 2 b dr b r In plane deformation the only stresses to be determined ED D ln (4.10) 4 r 4 r 0 r 0 With an edge dislocation this energy is modified by the term 1 and hence is about 50% greater than a screw. For a unit dislocation in a typical crystal r ' 0.25 nm, r' 2.5 µ mand ln r/r ' 9.2, so that 0 0 2 the energy is approximately b per unit length of 2 dislocation, which for copper taking D 40 GNm , 19 bD 0.25 nm and 1 eVD 1.6ð 10 J is about 4 eV 2 for every atom plane threaded by the dislocation. If the reader prefers to think in terms of one metre of dis- Figure 4.21 Screw-dislocation in an elastic continuum. location line, then this length is associated with about 10 2ð 10 electron volts. We shall see later that heavily- 1 3 This subscript notation z indicates that the stress is in the 16 deformed metals contain approximately 10 m/m of -direction on an element perpendicular to the z-direction. The stress with subscript rr is thus a normal stress and 2 The energy of the core must be added to this estimate. The denoted by  and the subscript r a shear stress and rr 1 2 denoted by  . core energy is about b /10 or eV per atom length. r 296 Modern Physical Metallurgy and Materials Engineering Figure 4.22 Interaction between dislocations not on the same slip plane: (a) unlike dislocation, (b) like dislocations. The arrangement in (b) constitutes a small-angle boundary. dislocation line which leads to a large amount of on the angle between the Burgers vector and the line energy stored in the lattice (i.e.³4J/g for Cu). Clearly, joining the two dislocations (Figure 4.22a). because of this high line energy a dislocation line will Edge dislocations of the same sign repel and oppo- always tend to shorten its length as much as possi- site sign attract along the line between them, but the ble, and from this point of view it may be considered component of force in the direction of slip, which gov- 2 to possess a line tension, T³ ˛b , analogous to the erns the motion of a dislocation, varies with the angle 1 surface energy of a soap film, where ˛³ . . With unlike dislocations an attractive force is expe- 2 ° ° rienced for 45 but a repulsive force for 45 , and in equilibrium the dislocations remain at an angle Interaction of dislocations ° of 45 to each other. For like dislocations the converse The strain field around a dislocation, because of its ° applies and the position D 45 is now one of unstable long-range nature, is also important in influencing the equilibrium. Thus, edge dislocations which have the behaviour of other dislocations in the crystal. Thus, it same Burgers vector but which do not lie on the same is not difficult to imagine that a positive dislocation ° slip plane will be in equilibrium when D 90 ,and will attract a negative dislocation lying on the same consequently they will arrange themselves in a plane slip plane in order that their respective strain fields normal to the slip plane, one above the other a distance should cancel. Moreover, as a general rule it can be h apart. Such a wall of dislocations constitutes a small- said that the dislocations in a crystal will interact with angle grain boundary as shown in Figure 4.22b, where each other to take up positions of minimum energy to the angle across the boundary is given by D b/h. reduce the total strain energy of the lattice. This type of dislocation array is also called a sub- Two dislocations of the same sign will repel each grain or low-angle boundary, and is important in the other, because the strain energy of two dislocations annealing of deformed metals. 2 on moving apart would be 2ðb whereas if they By this arrangement the long-range stresses from combined to form one dislocation of Burgers vec- the individual dislocations are cancelled out beyond a 2 2 tor 2b, the strain energy would then be 2b D 4b ; distance of the order of h from the boundary. It then a force of repulsion exists between them. The force follows that the energy of the crystal boundary will is, by definition, equal to the change of energy with be given approximately by the sum of the individual position dE/dr and for screw dislocations is simply 2 energies, each equal to fb /4 1 g lnh/r  per 0 2 FD b /2 r where r is the distance between the two unit length. There are 1/h or /b dislocations in a unit dislocations. Since the stress field around screw dislo- length, vertically, and hence, in terms of the misorien- cations has cylindrical symmetry the force of inter- tation across the boundary D b/h, the energy  per gb action depends only on the distance apart, and the unit area of boundary is above expression for F applies equally well to par-   allel screw dislocations on neighbouring slip planes. 2 b h For parallel edge dislocations the force–distance rela-  D ln ð gb 4 1  r b 0 tionship is less simple. When the two edge dislocations     lie in the same slip plane the relation is similar to that b b 2 D ln for two screws and has the formFD b /1 2 r, 4 1  r but for edge dislocations with the same Burgers vector but not on the same slip plane the force also depends D E A ln 4.11 0Defects in solids 97 where E D b/4 1  and AD lnb/r ;thisis 0 0 4.4 Planar defects known as the Read–Shockley formula. Values from it 4.4.1 Grain boundaries give good agreement with experimental estimates even ° up to relatively large angles. For ¾ 25 , ¾ b/25 gb The small-angle boundary described in Section 2 or¾ 0.4J/m , which surprisingly is close to the value is a particular example of a planar defect interface in for the energy per unit area of a general large-angle a crystal. Many such planar defects occur in materials ranging from the large-angle grain boundary, which is grain boundary. an incoherent interface with a relatively high energy of 2 ¾0.5J/m , to atomic planes in the crystal across which there is a mis-stacking of the atoms, i.e. twin interfaces 4.3.6 Dislocations in ionic structures and stacking faults which retain the coherency of the The slip system which operates in materials with NaCl 2 packing and have much lower energies 0.1J/m . structure is predominantly a/2h110if110g. The clos- Generally, all these planar defects are associated with est packed planef100g is not usually the preferred slip dislocations in an extended form. plane because of the strong electrostatic interaction that A small-angle tilt boundary can be described ade- would occur across the slip plane during slip; like ions quately by a vertical wall of dislocations. Rotation of are brought into neighbouring positions across the slip one crystal relative to another, i.e. a twist boundary, plane for 100 but not for the 110. Dislocations in can be produced by a crossed grid of two sets of screw these materials are therefore simpler than fcc metals, dislocations as shown in Figure 4.24. These bound- but they may carry an electric charge (the edge disloca- aries are of a particularly simple kind separating two tion onf110g, for example). Figure 4.23a has an extra crystals which have a small difference in orientation, C ‘half-plane’ made up of a sheet of Na ions and one of whereas a general grain boundary usually separates  Cl ions. The line as a whole can be charged up to a crystals which differ in orientation by large angles. In maximum ofe/2 per atom length by acting as a source this case, the boundary has five degrees of freedom, or sink for point defects. Figure 4.23b shows different three of which arise from the fact that the adjoin- jogs in the line which may either carry a charge or be ing crystals may be rotated with respect to each other about the three perpendicular axes, and the other two uncharged. The jogs at B and C would be of charge from the degree of freedom of the orientation of the Ce/2 because the section BC has a net charge equal boundary surface itself with respect to the crystals. to e. The jog at D is uncharged. ° Such a large-angle 30–40  grain boundary may sim- ply be regarded as a narrow region, about two atoms thick, across which the atoms change from the lattice orientation of the one grain to that of the other. Nev- ertheless, such a grain boundary may be described by an arrangement of dislocations, but their arrangement will be complex and the individual dislocations are not easily recognized or analysed. The simplest extension of the dislocation model for low-angle boundaries to high-angle grain boundaries is to consider that there are islands of good atomic fit surrounded by non-coherent regions. In a low-angle boundary the ‘good fit’ is perfect crystal and the ‘bad Figure 4.23 Edge dislocation in NaCl, showing: (a) two extra half-sheets of ions: anions are open circles, cations are shaded; (b) charged and uncharged jogs (after Kelly and Figure 4.24 Representation of a twist boundary produced Groves, 1970). by cross-grid of screw dislocations.98 Modern Physical Metallurgy and Materials Engineering fit’ is accommodated by lattice dislocations, whereas special orientation between them. Such intrinsic sec- for high-angle boundaries the ‘good fit’ could be an ondary dislocations must conserve the boundary struc- interfacial structure with low energy and the bad fit ture and, generally, will have Burgers vectors smaller accommodated by dislocations which are not neces- than those of the lattice dislocations. sarily lattice dislocations. These dislocations are often When a polycrystalline specimen is examined in termed intrinsic secondary grain boundary dislocations TEM other structural features apart from intrinsic grain (gbds) and are essential to maintain the boundary at boundary dislocations (gbds) may be observed in a grain boundary, such as ‘extrinsic’ dislocations which that mis-orientation. have probably run-in from a neighbouring grain, and The regions of good fit are sometimes described by the coincident site lattice (CSL) model, with its interface ledges or steps which curve the boundary. At development to include the displacement shift complex low temperatures the run-in lattice dislocation tends (DSC) lattice. A CSL is a three-dimensional super- to retain its character while trapped in the interface,  lattice on which a fraction 1/ of the lattice points whereas at high temperatures it may dissociate into in both crystal lattices lie; for the simple structures several intrinsic gbds resulting in a small change in there will be many such CSLs, each existing at a misorientation across the boundary. The analysis of particular misorientation. One CSL is illustrated in gbds in TEM is not easy, but information about them Figure 4.25 but it must be remembered that the CSL will eventually further our understanding of important boundary phenomena (e.g. migration of boundaries is three-dimensional, infinite and interpenetrates both during recrystallization and grain growth, the sliding crystals; it does not in itself define an interface. How- of grains during creep and superplastic flow and the ever, an interface is likely to have a low energy if way grain boundaries act as sources and sinks for point it lies between two crystals oriented such that they defects). share a high proportion of lattice sites, i.e. preferred misorientations will be those with CSLs having low  values. Such misorientations can be predicted from 4.4.2 Twin boundaries the expression Annealing of cold-worked fcc metals and alloys, such   p as copper, ˛-brass and austenitic stainless steels usu- b 1 D 2tan N ally causes many of the constituent crystals to form a annealing twins. The lattice orientation changes at the 2 2 2 twin boundary surface, producing a structure in which where b and a are integers and ND h Ck Cl ;the  2 2 one part of the crystal or grain is the mirror-image of value is then given by a CNb , divided by 2 until the other, the amount of atomic displacement being an odd number is obtained. proportional to the distance from the twin boundary. The CSL model can only be used to describe cer- The surfaces of a sample within and outside an tain specific boundary misorientations but it can be annealing twin have different surface energies, because extended to other misorientations by allowing the pres- of their different lattice orientations, and hence respond ence of arrays of dislocations which act to preserve a quite differently when etched with a chemical etchant (Figure 4.26). In this diagram, twins 1 and 2 both have two straight parallel sides which are coherent low-energy interfaces. The short end face of twin 2 is non-coherent and therefore has a higher energy content Figure 4.25 Two-dimensional section of a CSL with  ° 536.9 100 twist orientation (courtesy of Figure 4.26 Twinned regions within a single etched grain, P. Goodhew). produced by deformation and annealing.Defects in solids 99 Dissociation into Shockley partials The relationship between the two close-packed struc- tures cph and fcc has been discussed in Chapter 2 where it was seen that both structures may be built up from stacking close-packed planes of spheres. The shortest lattice vector in the fcc structure joins a cube corner atom to a neighbouring face centre atom and defines the observed slip direction; one such slip vec- tor a/21 0 1 is shown as b in Figure 4.28a which 1 is for glide in the 111 plane. However, an atom Figure 4.27 Grain boundary/surface triple junction. which sits in a B position on top of the A plane would move most easily initially towards a C position and, consequently, to produce a macroscopical slip move- per unit surface. Stacking faults are also coherent ment along 1 0 1 the atoms might be expected to and low in energy content; consequently, because of take a zigzag path of the type B C B following this similarity in character, we find that crystalline the vectors b D a/62 1 1 and b D a/61 1 2 alter- materials which twin readily are also likely to contain 2 3 nately. It will be evident, of course, that during the many stacking faults (e.g. copper, ˛-brass). initial part of the slip process when the atoms change Although less important and less common than slip, from B positions to C positions, a stacking fault in the another type of twinning can take place during plas- 111 layers is produced and the stacking sequence tic deformation. These so-called deformation twins changes from ABCABC... to ABCACABC....Dur- sometimes form very easily, e.g. during mechanical ing the second part of the slip process the correct polishing of metallographic samples of pure zinc; this stacking sequence is restored. process is discussed in Chapter 7. To describe the atoms movement during slip, dis- The free energies of interfaces can be determined cussed above, Heidenreich and Shockley have pointed from the equilibrium form of the triple junction where out that the unit dislocation must dissociate into two three interfaces, such as surfaces, grain boundaries 1 half dislocations, which for the case of glide in the or twins, meet. For the case of a grain boundary 111 plane would be according to the reaction: intersecting a free surface, shown in Figure 4.27, a/21 0 1 a/62 1 1Ca/61 1 2  D 2 cos /2 (4.12) gb s Such a dissociation process is (1) algebraically correct, and hence  can be obtained by measuring the dihe- gb since the sum of the Burgers vector components of the dral angle and knowing  . Similarly, measurements s two partial dislocations, i.e. a/62C 1, a/61C 1, can be made of the ratio of twin boundary energy to a/61C 2, are equal to the components of the Burg- the average grain boundary energy and, knowing either ers vector of the unit dislocation, i.e. a/2, 0, a/2,  or  gives an estimate of  . s gb T and (2) energetically favourable, since the sum of the strain energy values for the pair of half dislocations is less than the strain energy value of the single unit 4.4.3 Extended dislocations and stacking dislocation, where the initial dislocation energy is pro- faults in close-packed crystals 2 2 portional to b D a /2 and the energy of the resultant 1 2 2 2 Stacking faults partials to b Cb D a /3. These half dislocations, or 2 3 Shockley partial dislocations, repel each other by a Stacking faults associated with dislocations can be force that is approximately FDb b cos 60/2 d, 2 3 an extremely significant feature of the structure of and separate as shown in Figure 4.28b. A sheet of many materials, particularly those with fcc and cph stacking faults is then formed in the slip plane between structure. They arise because to a first approximation the partials, and it is the creation of this faulted region, there is little to choose electrostatically between the which has a higher energy than the normal lattice, that stacking sequence of the close-packed planes in the prevents the partials from separating too far. Thus, if fcc metals ABCABC... and that in the cph metals 2  J/m is the energy per unit area of the fault, the force ABABAB... Thus, in a metal like copper or gold, the per unit length exerted on the dislocations by the fault atoms in a part of one of the close-packed layers may is  N/m and the equilibrium separation d is given by fall into the ‘wrong’ position relative to the atoms of equating the repulsive force F between the two half the layers above and below, so that a mistake in the dislocations to the force exerted by the fault, .The stacking sequence occurs (e.g. ABCBCABC...). Such equilibrium separation of two partial dislocations is an arrangement will be reasonably stable, but because some work will have to be done to produce it, stacking 1 faults are more frequently found in deformed metals The correct indices for the vectors involved in such than annealed metals. dislocation reactions can be obtained from Figure 4.37.100 Modern Physical Metallurgy and Materials Engineering Figure 4.28 Schematic representation of slip in a 111 plane of a fcc crystal. then given by   µ b b cos 60 2 3 dD 2    a a 1 µ p p 2   a 2 6 6   D D 4.13   2  24  Figure 4.29 Edge dislocation structure in the fcc lattice, (a) and (b) undissociated, (c) and (d) dissociated: (a) and (c) are viewed normal to the 11 1 plane (from from which it can be seen that the width of the stacking Hume-Rothery, Smallman and Haworth, 1969; courtesy of fault ‘ribbon’ is inversely proportional to the value of the Institute of Metals). the stacking fault energy  and also depends on the value of the shear modulus . strain energy along a line through the crystal associ- Figure 4.29a shows that the undissociated edge dis- ated with an undissociated dislocation is spread over location has its extra half-plane corrugated which may a plane in the crystal for a dissociated dislocation (see be considered as two 10 1 planes displaced relative Figure 4.29d) thereby lowering its energy. to each other and labelled a and b in Figure 4.29b. On A direct estimate of  can be made from the obser- dissociation, planes a and b are separated by a region vation of extended dislocations in the electron micro- of crystal in which across the slip plane the atoms are in the wrong sites (see Figure 4.29c). Thus the high scope and from observations on other stacking faultDefects in solids 101 defects (see Chapter 5). Such measurements show that and copper the partials are separated to about 12 and the stacking fault energy for pure fcc metals ranges 6 atom spacings, respectively. For nickel the width is 2 2 about 2b since although nickel has a high  its shear from about 16 mJ/m for silver to ³200 mJ/m for modulus is also very high. In contrast, aluminium has nickel, with gold ³30, copper ³40 and aluminium 2 2 alower  ³ 135 mJ/m but also a considerably lower 135 mJ/m , respectively. Since stacking faults are value for  and hence the partial separation is limited coherent interfaces or boundaries they have energies to about 1b and may be considered to be unextended. considerably lower than non-coherent interfaces such 2 Alloying significantly reduces  and very wide dislo- as free surfaces for which  ³ b/8³ 1.5J/m and s 2 cations are produced, as found in the brasses, bronzes grain boundaries for which  ³  /3³ 0.5J/m . gb s and austenitic stainless steels. However, no matter how The energy of a stacking fault can be estimated narrow or how wide the partials are separated the two from twin boundary energies since a stacking fault half-dislocations are bound together by the stacking ABCBCABC may be regarded as two overlapping fault, and consequently, they must move together as a twin boundaries CBC and BCB across which the next unit across the slip plane. nearest neighbouring plane are wrongly stacked. In fcc The width of the stacking fault ribbon is of impor- crystals any sequence of three atomic planes not in tance in many aspects of plasticity because at some the ABC or CBA order is a stacking violation and stage of deformation it becomes necessary for disloca- is accompanied by an increased energy contribution. tions to intersect each other; the difficulty which dislo- A twin has one pair of second nearest neighbour cations have in intersecting each other gives rise to one planes in the wrong sequence, two third neighbours, source of work-hardening. With extended dislocations one fourth neighbour and so on; an intrinsic stacking the intersecting process is particularly difficult since fault two second nearest neighbours, three third and no the crossing of stacking faults would lead to a com- fourth nearest neighbour violations. Thus, if next-next plex fault in the plane of intersection. The complexity nearest neighbour interactions are considered to make may be reduced, however, if the half-dislocations coa- a relatively small contribution to the energy then an lesce at the crossing point, so that they intersect as approximate relation  ' 2 is expected. T perfect dislocations; the partials then are constricted The frequency of occurrence of annealing twins gen- erally confirms the above classification of stacking together at their jogs, as shown in Figure 4.31a. fault energy and it is interesting to note that in alu- The width of the stacking fault ribbon is also impor- minium, a metal with a relatively high value of , tant to the phenomenon of cross-slip in which a annealing twins are rarely, if ever, observed, while they dislocation changes from one slip plane to another are seen in copper which has a lower stacking fault intersecting slip plane. As discussed previously, for energy. Electron microscope measurements of  show glide to occur the slip plane must contain both the that the stacking fault energy is lowered by solid solu- Burgers vector and the line of the dislocation, and, tion alloying and is influenced by those factors which consequently, for cross-slip to take place a dislocation affect the limit of primary solubility. The reason for this is that on alloying, the free energies of the ˛-phase and its neighbouring phase become more nearly equal, i.e. the stability of the ˛-phase is decreased relative to some other phase, and hence can more readily tolerate mis-stacking. Figure 4.30 shows the reduction of  for copper with addition of solutes such as Zn, Al, Sn and Ge, and is consistent with the observation that anneal- ing twins occur more frequently in ˛-brass or Cu–Sn than pure copper. Substituting the appropriate values for,a and in equation (4.13) indicates that in silver Figure 4.31 (a) The crossing of extended dislocations, Figure 4.30 Decrease in stacking-fault energy  for copper (b) various stages in the cross-slip of a dissociated screw with alloying addition (e/a). dislocation.102 Modern Physical Metallurgy and Materials Engineering must be in an exact screw orientation. If the dislo- bounding the collapsed sheet is normal to the plane a cation is extended, however, the partials have first to with bD 1 1 1, where a is the lattice parameter, and 3 be brought together to form an unextended dislocation such a dislocation is sessile since it encloses an area as shown in Figure 4.31b before the dislocation can of stacking fault which cannot move with the dislo- spread into the cross-slip plane. The constriction pro- cation. A Frank sessile dislocation loop can also be cess will be aided by thermal activation and hence the produced by inserting an extra layer of atoms between cross-slip tendency increases with increasing temper- two normal planes of atoms, as occurs when interstitial ature. The constriction process is also more difficult, atoms aggregate following high energy particle irradi- the wider the separation of the partials. In aluminium, ation. For the loop formed from vacancies the stacking where the dislocations are relatively unextended, the sequence changes from the normal ABCABCA... to frequent occurrence of cross-slip is expected, but for ABCBCA..., whereas inserting a layer of atoms, e.g. low stacking fault energy metals (e.g. copper or gold) an A-layer between B and C, the sequence becomes the activation energy for the process will be high. Nev- ABCABACA.... The former type of fault with one ertheless, cross-slip may still occur in those regions violation in the stacking sequence is called an intrin- where a high concentration of stress exists, as, for sic fault, the latter with two violations is called an example, when dislocations pile up against some obsta- extrinsic fault. The stacking sequence violations are cle, where the width of the extended dislocation may conveniently shown by using the symbol4 to denote be reduced below the equilibrium separation. Often any normal stacking sequence AB, BC, CA but5 for screw dislocations escape from the piled-up group by the reverse sequence AC, CB, BA. The normal fcc cross-slipping but then after moving a certain distance stacking sequence is then given by4444...,the intrinsic fault by44544... and the extrinsic fault in this cross-slip plane return to a plane parallel to the by445544.... The reader may verify that the original slip plane because the resolved shear stress is higher. This is a common method of circumventing fault discussed in the previous Section is also an intrin- obstacles in the structure. sic fault, and that a series of intrinsic stacking faults on neighbouring planes gives rise to a twinned struc- Sessile dislocations ture ABCABACBA or 444455554. Elec- tron micrographs of Frank sessile dislocation loops are The Shockley partial dislocation has its Burgers vec- shown in Figures 4.38 and 4.39. tor lying in the plane of the fault and hence is glissile. Another common obstacle is that formed between Some dislocations, however, have their Burgers vector extended dislocations on intersecting f111g slip not lying in the plane of the fault with which they are planes, as shown in Figure 4.32b. Here, the associated, and are incapable of gliding, i.e. they are combination of the leading partial dislocation lying in sessile. The simplest of these, the Frank sessile dislo- the 111 plane with that which lies in the 11 1 cation loop, is shown in Figure 4.32a. This dislocation plane forms another partial dislocation, often referred is believed to form as a result of the collapse of the to as a ‘stair-rod’ dislocation, at the junction of the lattice surrounding a cavity which has been produced two stacking fault ribbons by the reaction by the aggregation of vacancies on to a 111 plane. As shown in Figure 4.32a, if the vacancies aggregate a a a 112C 2 1 1 1 0 1 6 6 6 on the central A-plane the adjoining parts of the neigh- bouring B and C planes collapse to fit in close-packed The indices for this reaction can be obtained from formation. The Burgers vector of the dislocation line Figure 4.37 and it is seen that there is a reduction in Figure 4.32 Sessile dislocations: (a) a Frank sessile dislocation; (b) stair-rod dislocation as part of a Lomer-Cottrell barrier.Defects in solids 103 2 2 2 energy from a /6 C a /6 to a /18. This trian- gular group of partial dislocations, which bounds the wedge-shaped stacking fault ribbon lying in a h101i direction, is obviously incapable of gliding and such an obstacle, first considered by Lomer and Cottrell, is known as a Lomer-Cottrell barrier. Such a barrier impedes the motion of dislocations and leads to work- hardening, as discussed in Chapter 7. Stacking faults in ceramics Some ceramic oxides may be described in terms of fcc or cph packing of the oxygen anions with the cations occupying the tetrahedral or octahedral intersti- tial sites, and these are more likely to contain stacking faults. Sapphire, ˛-Al O , deforms at high tempera- 2 3 tures on the 0001 h1120i basal systems and dis- Figure 4.33 A diamond cubic lattice projected normal to 1 1 sociated h1010iC h0110i dislocations have been 1 10 . represents atoms in the plane of the paper andC 3 3 represents atoms in the plane below. 11 1 is observed. Stacking faults also occur in spinels. Stoi- perpendicular to the plane of the paper and appears as a chiometric spinel (MgAl O or MgO. nAl O , nD 1) 4 2 3 2 horizontal trace. deforms predominantly on thef111gh110i slip system ° at high temperature ¾1800 C with dissociated dislo- 2 cations. Non-stoichiometric crystals (n 1) deform at faults ¾50 mJ/m . Dislocations could slip between lower temperatures when the f110gh110i secondary the narrowly spaced planes Ba, called the glide set, slip system is also preferred. The stacking fault energy or between the widely spaced planes bB, called the decreases with deviation from stoichiometry from a shuffle set, but weak beam microscopy shows dis- 2 value around 0.2J/m for nD 1 crystals to around sociation into Shockley partials occurs on the glide 2 ° ° set. A 60 dislocation (i.e. 60 to its Burgers vec- 0.02 J/m for nD 3.5. 1 tor ah110i) of the glide set is formed by cutting 2 Stacking faults in semiconductors out material bounded by the surface 1564 and then joining the cut together. The extra plane of atoms Elemental semiconductors Si, Ge with the diamond terminates between a and B leaving a row of dan- cubic structure or III-V compounds InSb with the spha- gling bonds along its core which leads to the electrical lerite (zinc blende) structure have perfect dislocation effect of a half-filled band in the band gap; plastic Burgers vectors similar to those in the fcc lattice. ° deformation can make n-type Ge into p-type. The 60 Stacking faults onf111g planes associated with partial dislocation BC and its dissociation into υC and Bυ dislocations also exist. Thef111g planes are stacked in is shown in Figure 4.34. The wurtzite (ZnS) structure the sequence AaBbCcAaBb as shown in Figure 4.33 has the hexagonal stacking sequence AaBbAaBb.... and stacking faults are created by the removal (intrin- sic) or insertion (extrinsic) of pairs of layers such Similarly, stacking faults in the wurtzite structure are as Aa or Bb. These faults do not change the four thin layers of sphalerite BbAaBbCcAA... analogous nearest-neighbour covalent bonds and are low-energy to stacking faults in hexagonal metals. ° Figure 4.34 (a) The 60 dislocation BC, (b) the dissociation of BC into υC and Bυ.