Dissociation into Shockley partials The relationship between the two close-packed struc-

4.4.3.2 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/2[1 0 1] is shown as b 1 in Figure 4.28a which Figure 4.27 Grain boundary/surface triple junction .

is for glide in the ⊲1 1 1⊳ plane. However, an atom which sits in a B position on top of the A plane would move most easily initially towards a C position and,

per unit surface. Stacking faults are also coherent consequently, to produce a macroscopical slip move- and low in energy content; consequently, because of

ment along [1 0 1] the atoms might be expected to this similarity in character, we find that crystalline

take a zigzag path of the type B ! C ! B following materials which twin readily are also likely to contain

the vectors b 2 D a/6[2 1 1] and b 3 D a/6[1 1 2] alter- many stacking faults (e.g. copper, ˛-brass).

nately. It will be evident, of course, that during the Although less important and less common than slip,

initial part of the slip process when the atoms change another type of twinning can take place during plas-

from B positions to C positions, a stacking fault in the tic deformation. These so-called deformation twins

⊲ 1 1 1⊳ layers is produced and the stacking sequence sometimes form very easily, e.g. during mechanical

changes from ABCABC . . . to ABCACABC . . .. Dur- polishing of metallographic samples of pure zinc; this

ing the second part of the slip process the correct process is discussed in Chapter 7.

stacking sequence is restored. The free energies of interfaces can be determined

To describe the atoms movement during slip, dis- from the equilibrium form of the triple junction where

cussed above, Heidenreich and Shockley have pointed three interfaces, such as surfaces, grain boundaries

out that the unit dislocation must dissociate into two 1 or twins, meet. For the case of a grain boundary

half dislocations, which for the case of glide in the intersecting a free surface, shown in Figure 4.27,

⊲ 1 1 1⊳ plane would be according to the reaction:

gb D 2 s

a/ 2[1 0 1] ! a/6[2 1 1] C a/6[1 1 2] and hence gb can be obtained by measuring the dihe-

Such a dissociation process is (1) algebraically correct,

since the sum of the Burgers vector components of the can be made of the ratio of twin boundary energy to

s . Similarly, measurements

two partial dislocations, i.e. a/6[2 C 1], a/6[1 C 1], the average grain boundary energy and, knowing either

a/ 6[1 C 2], are equal to the components of the Burg- s or gb gives an estimate of T .

ers vector of the unit dislocation, i.e. a/2, 0, a/2, and (2) energetically favourable, since the sum of the strain energy values for the pair of half dislocations

4.4.3 Extended dislocations and stacking

is less than the strain energy value of the single unit dislocation, where the initial dislocation energy is pro-

faults in close-packed crystals

portional to b 2 1 ⊲D a 2 / 2⊳ and the energy of the resultant

3. These half dislocations, or Stacking faults associated with dislocations can be

4.4.3.1 Stacking faults

partials to b 2 2 Cb 3 2 Da 2 /

Shockley partial dislocations, repel each other by a an extremely significant feature of the structure of

2 b 3 many materials, particularly those with fcc and cph

and separate as shown in Figure 4.28b. A sheet of structure. They arise because to a first approximation

stacking faults is then formed in the slip plane between there is little to choose electrostatically between the

the partials, and it is the creation of this faulted region, stacking sequence of the close-packed planes in the

which has a higher energy than the normal lattice, that fcc metals ABCABC . . . and that in the cph metals

prevents the partials from separating too far. Thus, if ABABAB . . . 2 Thus, in a metal like copper or gold, the J/m is the energy per unit area of the fault, the force

atoms in a part of one of the close-packed layers may per unit length exerted on the dislocations by the fault fall into the ‘wrong’ position relative to the atoms of

is N/m and the equilibrium separation d is given by the layers above and below, so that a mistake in the

equating the repulsive force F between the two half stacking sequence occurs (e.g. ABCBCABC . . .). Such

dislocations to the force exerted by the fault, . The an arrangement will be reasonably stable, but because

equilibrium separation of two partial dislocations is some work will have to be done to produce it, stacking

faults are more frequently found in deformed metals 1 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 ⊲1 1 1 ⊳ plane of a fcc crystal .

then given by µ b 2 b 3 cos 60

dD  a a 1 µ   p p

 D ⊲ 4.13⊳

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 ⊲1 1 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 Figure 4.29a shows that the undissociated edge dis-

strain energy along a line through the crystal associ- location has its extra half-plane corrugated which may

ated with an undissociated dislocation is spread over

a plane in the crystal for a dissociated dislocation (see to each other and labelled a and b in Figure 4.29b. On

be considered as two ⊲1 0 1⊳ planes displaced relative

Figure 4.29d) thereby lowering its energy. dissociation, planes a and b are separated by a region

A direct estimate of can be made from the obser- of crystal in which across the slip plane the atoms are

vation of extended dislocations in the electron micro- in the wrong sites (see Figure 4.29c). Thus the high

scope and from observations on other stacking fault

Defects 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

about 2b since although nickel has a high its shear nickel, with gold ³30, copper ³40 and aluminium

from about 16 mJ/m 2 for silver to ³200 mJ/m 2 for

modulus is also very high. In contrast, aluminium has 135 mJ/m 2 2 , respectively. Since stacking faults are a lower ³ 135 mJ/m but also a considerably lower

coherent interfaces or boundaries they have energies considerably lower than non-coherent interfaces such

to about 1b and may be considered to be unextended. as free surfaces for which s ³

8 ³ 1.5 J/m 2 and

Alloying significantly reduces and very wide dislo-

cations are produced, as found in the brasses, bronzes The energy of a stacking fault can be estimated

grain boundaries for which gb ³ s / 3 ³ 0.5 J/m 2 .

and austenitic stainless steels. However, no matter how from twin boundary energies since a stacking fault

narrow or how wide the partials are separated the two ABCBCABC may be regarded as two overlapping

half-dislocations are bound together by the stacking twin boundaries CBC and BCB across which the next

fault, and consequently, they must move together as a nearest neighbouring plane are wrongly stacked. In fcc

unit across the slip plane.

crystals any sequence of three atomic planes not in The width of the stacking fault ribbon is of impor- the ABC or CBA order is a stacking violation and

tance in many aspects of plasticity because at some is accompanied by an increased energy contribution.

stage of deformation it becomes necessary for disloca-

A twin has one pair of second nearest neighbour tions to intersect each other; the difficulty which dislo- planes in the wrong sequence, two third neighbours,

cations have in intersecting each other gives rise to one one fourth neighbour and so on; an intrinsic stacking

source of work-hardening. With extended dislocations fault two second nearest neighbours, three third and no

the intersecting process is particularly difficult since fourth nearest neighbour violations. Thus, if next-next

the crossing of stacking faults would lead to a com- nearest neighbour interactions are considered to make

plex fault in the plane of intersection. The complexity

a relatively small contribution to the energy then an may be reduced, however, if the half-dislocations coa- approximate relation ' 2 T is expected.

lesce at the crossing point, so that they intersect as The frequency of occurrence of annealing twins gen-

perfect dislocations; the partials then are constricted 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

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 cation is extended, however, the partials have first to

with b D a 3 [1 1 1], where a is the lattice parameter, and

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 symbol 4 to denote

be reduced below the equilibrium separation. Often any normal stacking sequence AB, BC, CA but 5 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 by 4 4 4 4 . . ., the in this cross-slip plane return to a plane parallel to the

intrinsic fault by 4 4 5 4 4 . . . and the extrinsic fault original slip plane because the resolved shear stress

by 4 4 5 5 4 4 . . .. The reader may verify that the 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-

4.4.3.3 Sessile dislocations ture ABCABACBA or 4 4 4 4 5 5 5 5 4. Elec- The Shockley partial dislocation has its Burgers vec-

tron micrographs of Frank sessile dislocation loops are tor lying in the plane of the fault and hence is glissile.

shown in Figures 4.38 and 4.39. Some dislocations, however, have their Burgers vector

Another common obstacle is that formed between not lying in the plane of the fault with which they are

extended dislocations on intersecting f1 1 1g slip associated, and are incapable of gliding, i.e. they are

planes, as shown in Figure 4.32b. Here, the sessile. The simplest of these, the Frank sessile dislo-

combination of the leading partial dislocation lying in cation loop, is shown in Figure 4.32a. This dislocation

the ⊲1 1 1⊳ plane with that which lies in the ⊲1 1 1⊳ is believed to form as a result of the collapse of the

plane forms another partial dislocation, often referred lattice surrounding a cavity which has been produced

to as a ‘stair-rod’ dislocation, at the junction of the by the aggregation of vacancies on to a ⊲1 1 1⊳ plane.

two stacking fault ribbons by the reaction As shown in Figure 4.32a, if the vacancies aggregate

6 [1 1 2] C 6 [2 1 1] ! a 6 on the central A-plane the adjoining parts of the neigh- [1 0 1] 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

energy from [⊲a 2 / 6⊳ C ⊲a 2 / 6⊳] to a 2 /

18. This trian-

gular group of partial dislocations, which bounds the wedge-shaped stacking fault ribbon lying in a h1 0 1i 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.