Defect behavior in common crystal structures

3.6 Defect behavior in common crystal structures

3.6.1 Dislocation vector diagrams and the Thompson tetrahedron

The classification of defects into point, line, planar and volume is somewhat restrictive in presenting an overview of defect behavior in materials, since it is clear, even from the discussion so far, that these

130 Physical Metallurgy and Advanced Materials

Figure 3.34 Construction (a) and orientation ( b) of the Thompson tetrahedron ABCD. The slip directions in a given {1 1 1} plane may be obtained from the trace of that plane, as shown for the (1 1 1) plane in ( b).

defects are interrelated and interdependent. In the following sections these features will be brought out as well as those which relate to specific structures.

In dealing with dislocation interactions and defects in real material it is often convenient to work with a vector notation rather than use the more conventional Miller indices notation. This may be illustrated by reference to the fcc structure and the Thompson tetrahedron.

All the dislocations common to the fcc structure, discussed in the previous sections, can be rep- resented conveniently by means of the Thompson reference tetrahedron (Figure 3.34a), formed by joining the three nearest face-centering atoms to the origin D. Here ABCD is made up of four {1 1 1} planes (1 1 1), (¯1 ¯1 1), (¯1 1 ¯1) and (1 ¯1 ¯1) as shown by the stereogram given in Figure 3.34b, and the edges AB, BC, CA . . . correspond to the of the faces are labeled α, β, γ, δ, as shown in Figure 3.35a, all the dislocation Burgers vectors are represented. Thus, the edges (AB, BC . . .) correspond to the normal slip vectors, a/2 half-dislocations, or Shockley partials, into which these are dissociated have Burgers vectors of the a/6 Greek–Roman symbols γA, γB, γD, δA, δB, etc. The dissociation reaction given in the first reaction in Section 3.4.3.2 is then simply written:

BC → Bδ + δC and there are six such dissociation reactions in each of the four {1 1 1} planes (see Figure 3.35). It is

conventional to view the slip plane from outside the tetrahedron along the positive direction of the unit dislocation BC, and on dissociation to produce an intrinsic stacking fault arrangement; the Roman– Greek partial Bδ is on the right and the Greek–Roman partial δC on the left. A screw dislocation with Burgers vector BC, which is normally dissociated in the δ-plane, is capable of cross-slipping into the α -plane by first constricting Bδ + δC → BC and then redissociating in the α-plane BC → Bα + αC.

3.6.2 Dislocations and stacking faults in fcc structures

3.6.2.1 Frank loops

A powerful illustration of the use of the Thompson tetrahedron can be made if we look at simple Frank loops in fcc metals (see Figure 3.32a). The Frank partial dislocation has a Burgers vector perpendicular

Crystal defects 131

b D a [1 1 1] A 具121 B

A Thompson tetrahedron closed (a) and opened out ( b). In ( b) the notation [1 1 0

Figure 3.36 Single-faulted, double-faulted (A) and unfaulted (B) dislocation loops in quenched aluminum (after Edington and Smallman, 1965; courtesy of Taylor & Francis).

to the (1 1 1) plane on which it lies and is represented by Aα, Bβ, Cγ, Dδ, αA, etc. Such loops shown in the electron micrograph of Figure 3.36 have been produced in aluminum by quenching from about 600 ◦

C. Each loop arises from the clustering of vacancies into a disk-shaped cavity, which then forms

a dislocation loop. To reduce their energy, the loops take up regular crystallographic forms with their edges parallel to the energy by dissociating on an intersecting {1 1 1} plane, forming a stair-rod at the junction of the two

132 Physical Metallurgy and Advanced Materials

Figure 3.37 Removal of the stacking fault from a Frank sessile dislocation by stress (after Goodhew and Smallman).

{1 1 1} planes, e.g. Aα →Aδ + δα when the Frank dislocation lies along [¯1 0 1] common to both α - and δ-planes.

Some of the loops shown in Figure 3.36 are not Frank sessile dislocations as expected, but prismatic dislocations, since no contrast of the type arising from stacking faults can be seen within the defects. The fault will be removed by shear if it has a high stacking fault energy, thereby changing the sessile

Frank loop into a glissile prismatic loop according to the reaction: a/3[1 1 1] + a/6[1 1 ¯2] → a/2[1 1 0]. Stressing the foil while it is under observation in the microscope allows the unfaulting process to be

observed directly (see Figure 3.37). This reaction is more easily followed with the aid of the Thompson tetrahedron and rewritten as:

Dδ + δC → DC. Physically, this means that the disk of vacancies aggregated on a (1 1 1) plane of a metal with high

stacking fault energy, besides collapsing, also undergoes a shear movement. The dislocation loops shown in Figure 3.37b are therefore unit dislocations with their Burgers vector a/2[1 1 0] inclined at an angle to the original (1 1 1) plane. A prismatic dislocation loop lies on the surface of a cylinder, the cross-section of which is determined by the dislocation loop, and the axis of which is parallel to the [1 1 0] direction. Such a dislocation is not sessile, and under the action of a shear stress it is capable of movement by prismatic slip in the [1 1 0] direction.

Many of the large Frank loops in Figure 3.36 (for example, marked A) contain additional triangular- shaped loop contrast within the outer hexagonal loop. The stacking fault fringes within the triangle are usually displaced relative to those between the triangle and the hexagon by half the fringe spacing, which is the contrast expected from overlapping intrinsic stacking faults. The structural arrangement of those double-faulted loops is shown schematically in Figure 3.38, from which it can be seen that two intrinsic faults on next neighboring planes are equivalent to an extrinsic fault. The observation of double-faulted loops in aluminum indicates that it is energetically more favorable to nucleate a Frank sessile loop on an existing intrinsic fault than randomly in the perfect lattice, and it therefore follows that the energy of a double or extrinsic fault is less than twice that of the intrinsic fault, i.e.

γ E < 2γ I . The double loops marked B have the outer intrinsic fault removed by stress. The addition of a third overlapping intrinsic fault would change the stacking sequence from the perfectABCABCABC to ABC ↓ B ↓ A ↓ CABC, where the arrows indicate missing planes of atoms, and produce a coherent twinned structure with two coherent twin boundaries. This structure would be

energetically favorable to form, since γ twin <γ I <γ E . It is possible, however, to reduce the energy of the crystal even further by aggregating the third layer of vacancies between the two previously formed neighboring intrinsic faults to change the structure from an extrinsically faulted ABC ↓ B ↓ ABC to the perfect ABC ↓↓↓ ABC structure. Such a triple-layer dislocation loop is shown in Figure 3.39.

Crystal defects 133 C

Figure 3.38 The structure of a double dislocation loop in quenched aluminum (after Edington and Smallman, 1965; courtesy of Taylor & Francis).

0.25 ␮ m

Figure 3.39 Triple-loop and Frank sessile loop in Al–0.65% Mg (after Kritzinger, Smallman and Dobson, 1969).

3.6.2.2 Stair-rod dislocations

The stair-rod dislocation formed at the apex of a Lomer–Cottrell barrier can also be represented by the Thompson notation. As an example, let us take the interaction between dislocations on the δ- and α -planes. Two unit dislocations BA and DB, respectively, are dissociated according to:

BA → Bδ + δA (on the δ-plane) and DB → Dα + αB (on the α-plane)

and when the two Shockley partials αB and Bδ interact, a stair-rod dislocation αδ = a/6[1 0 1] is formed. This low-energy dislocation is pure edge and therefore sessile. If the other pair of partials interact then the resultant Burgers vector is (δA + Dα) = a/3[1 0 1] and of higher energy. This vector is written in Thompson’s notation as δD/Aα and is a vector equal to twice the length joining the midpoints of δA and Dα.

3.6.2.3 Stacking-fault tetrahedra

In fcc metals and alloys, the vacancies may also cluster into a three-dimensional defect, forming

a tetrahedral arrangement of stacking faults on the four {1 1 1} planes with the six of the tetrahedron, where the stacking faults bend from one {1 1 1} plane to another, consisting of stair-rod dislocations. The crystal structure is perfect inside and outside the tetrahedron, and the

134 Physical Metallurgy and Advanced Materials

Figure 3.40 Formation of defect tetrahedron: (a) dissociation of Frank dislocations; ( b) formation of new stair-rod dislocations; (c) arrangement of the six stair-rod dislocations.

three-dimensional array of faults exhibits characteristic projected shape and contrast when seen in transmission electron micrographs, as shown in Figure 3.42. This defect was observed originally in quenched gold but occurs in other materials with low stacking-fault energy. One mechanism for the formation of the defect tetrahedron by the dissociation of a Frank dislocation loop (see Figure 3.40) was first explained by Silcox and Hirsch. The Frank partial dislocation bounding a stacking fault has, because of its large Burgers vector, a high strain energy, and hence can lower its energy by dissociation according to a reaction of the type:

a/3[1 1 1] → a/6[1 2 1] + a/6[1 0 1].

The figures underneath the reaction represent the energies of the dislocations, since they are propor- tional to the squares of the Burgers vectors. This reaction is therefore energetically favorable. This reaction can be seen with the aid of the Thompson tetrahedron, which shows that the Frank partial dislocation Aα can dissociate into a Shockley partial dislocation (Aβ, Aδ or Aγ) and a low energy stair-rod dislocation (βα, δα or γα), for example Aα →Aγ + γα.

The formation of the defect tetrahedron of stacking faults may be envisaged as follows. The collapse of a vacancy disk will, in the first instance, lead to the formation of a Frank sessile loop bounding

a stacking fault, with edges parallel to the according to the above reaction into the appropriate stair-rod and partial dislocations, and, as shown

Crystal defects 135 in Figure 3.40a, the Shockley dislocations formed by dissociation will lie on intersecting {1 1 1}

planes, above and below the plane of the hexagonal loop; the decrease in energy accompanying the dissociation will give rise to forces which tend to pull any rounded part of the loop into over, because the loop will not in general be a regular hexagon, the short sides will be eliminated by the preferential addition of vacancies at the constricted site, and a triangular-shaped loop will form (Figure 3.40b). The partials Aβ, Aγ and Aδ bow out on their slip plane as they are repelled by the stair-rods. Taking into account the fact that adjacent ends of the bowing loops are of opposite sign, the partials attract each other in pairs to form stair-rod dislocations along DA, BA and CA, according to the reactions:

γ A + Aβ → γβ, δA + Aγ → δγ, βA + Aδ → βδ. In vector notation the reactions are of the type:

a/6[¯1 ¯1 ¯2] + a/6[1 2 1] → a/6[0 1 ¯1]

(the reader may deduce the appropriate indices from Figure 3.35), and from the addition of the squares of the Burgers vectors underneath it is clear that this reaction is also energetically favorable. The final defect will therefore be a tetrahedron made up from the intersection of stacking faults on the four {1 1 1} planes, so that the dislocations (Figure 3.40c).

The tetrahedron of stacking faults formed by the above sequence of events is essentially symmet- rical, and the same configuration would have been obtained if collapse had taken place originally on any other (1 1 1) plane. The energy of the system of stair-rod dislocations in the final configuration is

proportional to 6 1 1 × 1 18 = 3 , compared with 3 × 3 = 1 for the original stacking fault triangle bounded by Frank partials. Considering the dislocation energies alone, the dissociation leads to a lowering of energy to one-third of the original value. However, three additional stacking-fault areas, with energies of γ per unit area, have been newly created and if there is to be no net rise in energy these areas will impose an upper limit on the size of the tetrahedron formed of about 50 nm.

Tetrahedra may also form by the nucleation and growth of a three-dimensional vacancy cluster. The smallest cluster that is able to collapse to a tetrahedron and subsequently grow by the absorption of vacancies is a hexa-vacancy cluster. Growth would then occur by the nucleation and propagation of jog lines across the faces of the tetrahedron, as shown in Figure 3.41. The hexa-vacancy cluster may form by clustering di-vacancies and is aided by impurities which have excess positive charge relative to the matrix (e.g. Mg, Cd or Al in Au). Hydrogen in solution is also a potent nucleating agent because the di-vacancy/proton complex is mobile and attracted to ‘free’ di-vacancies. Figure 3.42 shows the increase in tetrahedra nucleation after pre-annealing gold in hydrogen.

Worked example

By considering the energy of a triangular-shaped dislocation loop and that of a tetrahedron into which it could dissociate, estimate the maximum size of the tetrahedron. (Assume that the dislocation energy

per unit length is μb 2 , where μ, the shear modulus, is 2.7 × 10 10 Nm −2 , b is the Burgers vector, a the lattice parameter is 0.4 nm and the stacking-fault energy γ is 33 mJ m −2 .)

Solution

Let ℓ = side length of triangular loop or tetrahedron. Energy E

L of loop = 3ℓ × μb 2 F + (Area) × γ, where b F = 3 < 1 1 1> (Frank dislocation)

136 Physical Metallurgy and Advanced Materials

Figure 3.41 Jog line forming a ledge on the face of a tetrahedron.

(a)

(b) Figure 3.42 Tetrahedra in gold quenched (a) and preannealed in H 2 –N 2 gas ( b) (after

Johnston, Dobson and Smallman).

Energy E T of tetrahedron = 6ℓ × μb 2 SR + 4Aγ, where b SR = a 6 < 1 1 0> (stair-rod dislocation)

= 6.68 × 10 −8 m = 668 Å. When ℓ < 668Å, E T <E L so the tetrahedron is more stable. Hence maximum size of the tetrahedron

is 668 Å, beyond which the loop is more stable.

Worked example

Explain with well-labeled sketches the dislocation reactions in the fcc structure written in the Thompson tetrahedron notation:

(i) Bδ + δC → BC → Bα + αC (ii) Aα + αD →AD (iii) Cα + δC → αδ (iv) Aα →Aδ + δα.

Crystal defects 137

Solution

The Thompson tetrahedron ABCD is made up of the four operative {111} planes in the fcc structure. These are labeled α-, β-, γ- and δ-planes; A opposite to α, etc.

(i) This is a cross-slip reaction of a dissociated dislocation Bδ + δC in the δ-plane, which recombines to BC and then dissociates to Bα + αC on moving into the α-plane. (ii) This is the removal of the stacking fault from the Frank dislocation Aα by the sweeping cross (shear) of a Shockley partial αD in the α-plane to produce a perfect dislocation AD:

Aa AD

⫹a D

(iii) The formation of a stair-rod dislocation at the intersection of two {111} planes α and δ by the coming together (interaction) of Shockley partials Cα in the α-plane and δC in the δ-plane to form αδ.

Ca

(iv) This indicates the dissociation of a Frank dislocation Aα lying in the α-plane into a Shockley partial Aδ on the intersecting δ-plane and leaving a stair-rod δα at the junction of the α-plane and the intersecting δ-plane.

Shockley Ad on d-plane

Aa

Frank dislocation

Stair-rod da, at

on a -plane

junction, of a and d

planes

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CS

Figure 3.43 Burgers vectors in the cph lattice (after Berghezan, Fourdeux and Amelinckx, 1961).

3.6.3 Dislocations and stacking faults in cph structures

In a cph structure with axial ratio c/a, the most closely packed plane of atoms is the basal plane (0 0 0 1) and the most closely packed directions direction of the vector

nitude of the vector in terms of the lattice parameters is given by a[3(u 2 2 + uv + v 2 ) + (c/a) w 2 ] 1/2 . The usual slip dislocation therefore has a Burgers vector a/3 plane. This slip vector is

be written without difficulty as a/3 c-axis, as for example referred to unit distances (a, a, a, c) along the respective axes (e.g. 1/3 Other dislocations can be represented in a notation similar to that for the fcc structure, but using a double-tetrahedron or bipyramid instead of the single tetrahedron previously adopted, as shown in Figure 3.43. An examination leads to the following simple types of dislocation:

1. Six perfect dislocations with Burgers vectors in the basal plane along the sides of the triangular base ABC. They are AB, BC, CA, BA, CB and AC, and are denoted by a or 1/3

2. Six partial dislocations with Burgers vectors in the basal plane represented by the vectors Aσ, Bσ, Cσ and their negatives. These dislocations arise from dissociation reactions of the type:

AB → Aσ + σB and may also be written as p or 1 3

3. Two perfect dislocations perpendicular to the basal plane represented by the vectors ST and TS of magnitude equal to the cell height c or

4. Partial dislocations perpendicular to the basal plane represented by the vectors σS, σT, Sσ, Tσ

of magnitude c/2 or 1 2

5. Twelve perfect dislocations of the type 1 3

which is a vector equal to twice the join of the midpoints of SA and TB. These dislocations are more simply referred to as (c + a) dislocations.

6. Twelve partial dislocations, which are a combination of the partial basal and non-basal dislo- cations, and represented by vectors AS, BS, CS, AT, BT and CT or simply (c/2) + p equal to

the resultant dislocations are imperfect because the two sites are not identical.

Crystal defects 139 Table 3.2 Dislocations in hcp structures.

Type AB, BC

AS, BS SA/TB Vector

The energies of the different dislocations are given in a relative scale in Table 3.2, assuming c/a is ideal. There are many similarities between the dislocations in the cph and fcc structure, and thus it is not necessary to discuss them in great detail. It is, however, of interest to consider the two basic processes of glide and climb.

stacking fault

stacking fault

Ss

AS SA B B B B B C

(f) Figure 3.44 Stacking faults in the cph lattice (after Partridge, 1967; by courtesy of the

(d)

(e)

American Society for Metals).

3.6.3.1 Dislocation glide

A perfect slip dislocation in the basal plane AB 1 = 3 [¯1 2 ¯1 0] may dissociate into two Shockley partial dislocations separating a ribbon of intrinsic stacking fault which violates the two next-nearest neighbors in the stacking sequence. There are actually two possible slip sequences: either a B-layer slides over an A-layer, i.e. Aσ followed by σB (see Figure 3.44a) or an A-layer slides over a B-layer by the passage of a σB partial followed by an Aσ (see Figure 3.44b). The dissociation given by

AB → Aσ + σB may be written in Miller–Bravais indices as:

140 Physical Metallurgy and Advanced Materials This reaction is similar to that in the fcc lattice and the width of the ribbon is again inversely

proportional to the stacking fault energy γ. Dislocations dissociated in the basal planes have been observed in cobalt, which undergoes a phase transformation and for which γ is considered to be low ( ≈25 mJ m −2 ). For the other common cph metals Zn, Cd, Mg, Ti, Be, etc., γ is high (250–300 mJ m −2 ). No measurements of intrinsic faults with two next-nearest neighbor violations have been made, but intrinsic faults with one next-nearest neighbor violation have been measured (see 3.7.1) and show that Mg ≈ 125 mJ m −2 , Zn ≈140 mJ m −2 , and Cd ≈150–175 mJ m −2 . It is thus reasonable to conclude that intrinsic faults associated with Shockley partials have somewhat higher energy. Dislocations in these metals are therefore not very widely dissociated. A screw dislocation lying along a [¯1 2 ¯1 0] direction is capable of gliding in three different glide planes, but the small extension in the basal plane will be sufficient to make basal glide easier than in either the pyramidal (1 0 ¯1 1) or prismatic (1 0 ¯1 0) glide (see Figure 6.18). Pyramidal and prismatic glide will be more favored at high temperatures in metals with high stacking-fault energy, when thermal activation aids the constriction of the dissociated dislocations.

3.6.3.2 Dislocation climb

Stacking faults may be produced in hexagonal lattices by the aggregation of point defects. If vacancies aggregate as a platelet, as shown in Figure 3.44c, the resultant collapse of the disk-shaped cavity (Figure 3.44d) would bring two similar layers into contact. This is a situation incompatible with the close packing and suggests that simple Frank dislocations are energetically unfavorable in cph lattices. This unfavorable situation can be removed by either one of two mechanisms, as shown in Figures 3.44e and f. In Figure 3.44e the B-layer is converted to a C-position by passing a pair of equal and opposite partial dislocations (dipole) over adjacent slip planes. The Burgers vector of the dislocation loop will be of the σS type and the energy of the fault, which is extrinsic, will be high because of the three next-nearest neighbor violations. In Figure 3.44f the loop is swept by an Aσ-type partial dislocation, which changes the stacking of all the layers above the loop according to the rule

A → B → C →A. The Burgers vector of the loop is of the type AS, and from the dislocation reaction Aσ

2 [0 0 0 1] → 6 [2 0 ¯2 3] and the associated stacking fault, which is intrinsic, will have a lower energy because there is only one next-nearest neighbor violation in

1 + σS →AS or 1

the stacking sequence. Faulted loops with b

2 c + p) have been observed in Zn, Mg and Cd (see Figure 3.45). Double-dislocation loops have also been observed when the inner dislocation

=AS or ( 1

loop encloses a central region of perfect crystal and the outer loop an annulus of stacking fault. The structure of such a double loop is shown in Figure 3.46. The vacancy loops on adjacent atomic planes are bounded by dislocations with non-parallel Burgers vectors, i.e. b

2 c + p) and b = ( 1 2 c − p), respectively; the shear component of the second loop acts in such a direction as to eliminate the

fault introduced by the first loop. There are six partial vectors in the basal plane, p 1 ,p 2 ,p 3 and the negatives, and if one side of the loop is sheared by either p 1 ,p 2 or p 3 the stacking sequence is changed according to A → B → C →A, whereas reverse shearing A → C → B →A, results from either −p 1 , −p 2 or −p 3 . It is clear that the fault introduced by a positive partial shear can be eliminated by a subsequent shear brought about by any of the three negative partials. Three, four and more layered loops have also been observed in addition to the more common double loop. The addition of each layer of vacancies alternately introduces or removes stacking faults, no matter whether the loops precipitate one above the other or on opposite sides of the original defect.

As in fcc metals, interstitials may be aggregated into platelets on close-packed planes and the resultant structure, shown in Figure 3.47a, is a dislocation loop with Burgers vector Sσ, containing a high-energy stacking fault. This high-energy fault can be changed to one with lower energy by having the loop swept by a partial, as shown in Figure 3.47b.

Crystal defects 141

Figure 3.45 Growth of single- and double-faulted loops in magnesium on annealing at 175 ◦ C for: (a) t = 0 min, (b) t = 5 min, (c) t = 15 min and (d) t = 25 min (courtesy R. Hales).

Figure 3.46 Structure of double-dislocation loop in cph lattice (Hales, Dobson and Smallman, 1968; courtesy of the Royal Society of London).

142 Physical Metallurgy and Advanced Materials

Figure 3.47 Dislocation loop formed by aggregation of interstitials in a cph lattice with high-energy (a) and low-energy ( b) stacking faults.

All these faulted dislocation loops are capable of climbing by the addition or removal of point defects to the dislocation line. The shrinkage and growth of vacancy loops has been studied in some detail in Zn, Mg and Cd, and examples, together with the climb analysis, are discussed in Section

3.6.4 Dislocations and stacking faults in bcc structures

The shortest lattice vector in the bcc lattice is a/2[1 1 1], which joins an atom at a cube corner to the one at the center of the cube; this is the observed slip direction. The slip plane most commonly observed is (1 1 0) which, as shown in Figure 3.48, has a distorted close-packed structure. The (1 1 0) planes are packed in an ABABAB sequence and three {1 1 0} type planes intersect along

a three {1 1 0} planes and for this reason the slip lines are often wavy and ill-defined. By analogy with the fcc structure it is seen that in moving the B-layer along the [¯1 ¯1 1] direction it is easier to shear

in the directions indicated by the three vectors b 1 ,b 2 and b 3 . These three vectors define a possible dissociation reaction:

The stacking-fault energy of pure bcc metals is considered to be very high, however, and hence no faults have been observed directly. Because of the stacking sequence ABABAB of the (1 1 0) planes the formation of a Frank partial dislocation in the bcc structure gives rise to a situation similar to that for the cph structure, i.e. the aggregation of vacancies or interstitials will bring either two A-layers or

Crystal defects 143

[001] b 3 b (110)

Figure 3.48 The (1 1 0) plane of the bcc lattice (after Weertman; by courtesy of Oxford University Press).

two B-layers into contact with each other. The correct stacking sequence can be restored by shearing the planes to produce perfect dislocations a/2[1 1 1] or a/2[1 1 ¯1].

Slip has also been observed on planes indexed as (1 1 2) and (1 2 3) planes, and although some workers attribute this latter observation to varying amounts of slip on different (1 1 0) planes, there is evidence to indicate that (1 1 2) and (1 2 3) are definite slip planes. The packing of atoms in a (1 1 2) plane conforms to a rectangular pattern, the rows and columns parallel to the [1 ¯1 0] and [1 1 ¯1] directions, respectively, with the closest distance of approach along the [1 1 ¯1] direction. The stacking √ sequence of the (1 1 2) planes isABCDEFAB . . . and the spacing between the planes a/

6. It has often been suggested that the unit dislocation can dissociate in the (1 1 2) plane according to the reaction:

because the homogeneous shear necessary to twin the structure is 1/

2 in a shear can be produced by a displacement a/6[1 1 ¯1] on every successive (1 1 2) plane. It is therefore believed that twinning takes place by the movement of partial dislocations. However, it is generally recognized that the stacking-fault energy is very high in bcc metals so that dissociation must be limited. Moreover, because the Burgers vectors of the partial dislocations are parallel, it is not possible to separate the partials by an applied stress unless one of them is anchored by some obstacle in the crystal.

When the dislocation line lies along the [1 1 ¯1] direction it is capable of dissociating in any of the three {1 1 2} planes, i.e. (1 1 2), (¯1 2 1) and (2 ¯1 1), which intersect along [1 1 ¯1]. Furthermore, the a/2[1 1 ¯1] screw dislocation could dissociate according to:

to form the symmetrical fault shown in Figure 3.49. The symmetrical configuration may be unstable, and the equilibrium configuration is one partial dislocation at the intersection of two {1 1 2} planes and the other two lying equidistant, one in each of the other two planes. At larger stresses this unsymmetrical configuration can be broken up and the partial dislocations induced to move on three neighboring parallel planes, to produce a three-layer twin. In recent years an asymmetry of slip has been confirmed in many bcc single crystals, i.e. the preferred slip plane may differ in tension and compression. A yield stress asymmetry has also been

144 Physical Metallurgy and Advanced Materials

Figure 3.49 Dissociated a/2 [1 1 1] dislocation in the bcc lattice (after Mitchell, Foxall and Hirsch, 1963; courtesy of Taylor & Francis).

noted and has been related to asymmetric glide resistance of screw dislocations arising from their ‘core’ structure.

An alternative dissociation of the slip dislocation proposed by Cottrell is

The dissociation results in a twinning dislocation a/6[1 1 ¯1] lying in the (1 1 2) plane and an a/3[1 1 2] partial dislocation with Burgers vector normal to the twin fault, and hence is sessile. There is no reduction in energy by this reaction and is therefore not likely to occur except under favorable stress conditions.

Another unit dislocation can exist in the bcc structure, namely a[0 0 1], but it will normally be immobile. This dislocation can form at the intersection of normal slip bands by the reaction:

with a reduction of strain energy from 3a 2 / 2 to a 2 . The new a[0 0 1] dislocation lies in the (0 0 1) plane and is pure edge in character and may be considered as a wedge, one lattice constant thick, inserted between the (0 0 1) and hence has been considered as a crack nucleus. a[0 0 1] dislocations can also form in networks of a/2

3.6.5 Dislocations and stacking faults in ordered structures

When the alloy orders, a unit dislocation in a disordered alloy becomes a partial dislocation in the superlattice with its attached anti-phase boundary interface, as shown in Figure 3.50a. Thus, when this dislocation moves through the lattice it will completely destroy the order across its slip plane. However, in an ordered alloy, any given atom prefers to have unlike atoms as its neighbors, and consequently such a process of slip would require a very high stress. To move a dislocation against the force γ exerted on it by the fault requires a shear stress τ = γ/b, where b is the Burgers vector; in β-brass, where γ is about 0.07 N m −1 , this stress is 300 MN m −2 . In practice, the critical shear stress of β-brass is an order of magnitude less than this value, and thus one must conclude that slip occurs by an easier process than

Crystal defects 145

(a)

(b)

Figure 3.50 Dislocations in ordered structures.

the movement of unit dislocations. In consequence, by analogy with the slip process in fcc crystals, where the leading partial dislocation of an extended dislocation trails a stacking fault, it is believed that the dislocations which cause slip in an ordered lattice are not single dislocations but coupled pairs of dislocations, as shown in Figure 3.50b. The first dislocation of the pair, on moving across the slip plane, destroys the order and the second half of the couple completely restores it again, the third

dislocation destroys it once more, and so on. In β-brass 10 and similar weakly ordered alloys such as AgMg and FeCo, the crystal structure is ordered bcc (or CsCl-type) and, consequently, deformation is believed to occur by the movement of coupled pairs of a/2[1 1 1]-type dislocations. The combined slip vector of the coupled pair of dislocations, sometimes called a super-dislocation, is then equivalent to a[1 1 1] and, since this vector connects like atoms in the structure, long-range order will be maintained.

The separation of the super-partial dislocations may be calculated, as for Shockley partials, by equating the repulsive force between the two like a/2 anti-phase boundary. The values obtained for β-brass and FeCo are about 70 and 50 nm, respectively, and thus superdislocations can be detected in the electron microscope using the weak beam technique (see Chapter 4). The separation is inversely proportional to the square of the ordering parameter and superdislocation pairs ≈ 12.5 nm width have been observed more readily in partly ordered FeCo (S = 0.59).

In alloys with high ordering energies the anti-phase boundaries associated with superdislocations cannot be tolerated and dislocations with a Burgers vector equal to the unit lattice vector a operate to produce slip in as CsBr, but strongly ordered intermetallic compounds such as NiAl are also observed to slip in the

Ordered A 3 B-type alloys also give rise to superdislocations. Figure 3.51a illustrates three (1 1 1) layers of the L1 2 structure, with different size atoms for each layer. The three vectors shown give rise to the formation of different planar faults; a/2[¯1 0 1] is a super-partial producing apb, a/6[¯2 1 1] produces the familiar stacking fault and a/3[¯1 ¯1 2] produces a superlattice intrinsic stacking fault (SISF). A [¯1 0 1] superdislocation can therefore be composed of either

10 Chapter 1, Figure 1.20, shows the CsCl or β 2 structure. When disordered, the slip vector is a/2[1 1 1], but this vector in the ordered structure moves an A atom to a B site. The slip vector to move an A atom to an A site is twice the length

and equal to a[1 1 1].

146 Physical Metallurgy and Advanced Materials CSF formation with

1st N–N violation

APB formation: with 1st N–N violation

SISF formation

no N–N violation

Figure 3.51 (a) Stacking of (1 1 1) planes of the L1 2 structure, illustrating the apb and fault vectors. ( b) Schematic representation of superdislocation structure.

or

3 + SISF on(1 1 1) + 3

Each of the a/2[¯1 0 1] super-partials may also dissociate, as for fcc, according to:

The resultant superdislocation is shown schematically in Figure 3.51b. In alloys such as Cu 3 Au, Ni 3 Mn, Ni 3 Al, etc., the stacking-fault ribbon is too small to be observed experimentally but superdis- locations have been observed. It is evident, however, that the cross-slip of these superdislocations

Crystal defects 147 will be an extremely difficult process. This can lead to a high work-hardening rate in these alloys, as

discussed in Chapter 6. In an alloy possessing short-range order, slip will not occur by the motion of superdisloca- tions since there are no long-range faults to couple the dislocations together in pairs. However, because the distribution of neighboring atoms is not random the passage of a dislocation will destroy the short-range order between the atoms, across the slip plane. As before, the stress to do this will

be large, but in this case there is no mechanism, such as coupling two dislocations together, to make the process easier. The fact that, for instance, a crystal of AuCu 3 in the quenched state (short-range order) has nearly double the yield strength of the annealed state (long-range order) may be explained on this basis. The maximum strength is exhibited by a partially ordered alloy with a critical domain size of about 6 nm. The transition from deformation by unit dislocations in the disordered state to deformation by superdislocations in the ordered condition gives rise to a peak in the flow stress with change in degree of order (see Chapter 5).