Stacking-fault tetrahedra fault, i.e. E < 2 I . The double loops marked B have
4.6.2.3 Stacking-fault tetrahedra fault, i.e. E < 2 I . The double loops marked B have
the outer intrinsic fault removed by stress. In fcc metals and alloys, the vacancies may also clus- The addition of a third overlapping intrinsic fault
ter into a three-dimensional defect, forming a tetra- would change the stacking sequence from the per-
hedral arrangement of stacking faults on the four fect ABCABCABC to ABC # B # A # CABC, where
f 1 1 1g planes with the six h1 1 0i edges of the tetrahe- the arrows indicate missing planes of atoms, and pro-
dron, where the stacking faults bend from one f1 1 1g duce a coherent twinned structure with two coherent
plane to another, consisting of stair-rod dislocations. twin boundaries. This structure would be energetically
The crystal structure is perfect inside and outside the
tetrahedron, and the three-dimensional array of faults sible, however, to reduce the energy of the crystal
favourable to form, since twin < I < E . It is pos-
exhibits characteristic projected shape and contrast even further by aggregating the third layer of vacan-
when seen in transmission electron micrographs as cies between the two previously-formed neighbouring
shown in Figure 4.44. This defect was observed orig- intrinsic faults to change the structure from an extrin-
inally in quenched gold but occurs in other materials sically faulted ABC # B # ABC to perfect ABC ###
with low stacking-fault energy. One mechanism for the ABC structure. Such a triple-layer dislocation loop is
formation of the defect tetrahedron by the dissociation shown in Figure 4.41.
of a Frank dislocation loop (see Figure 4.42) was first explained by Silcox and Hirsch. The Frank partial dis-
4.6.2.2 Stair-rod dislocations location bounding a stacking fault has, because of its large Burgers vector, a high strain energy, and hence
The stair-rod dislocation formed at the apex of a can lower its energy by dissociation according to a Lomer-Cottrell barrier can also be represented by the
reaction of the type
Thompson notation. As an example, let us take the interaction between dislocations on the υ- and ˛-
C a/6[1 0 1] planes. Two unit dislocations BA and DB, respectively,
a/ 3[1 1 1] ! a/6[1 2 1]
are dissociated according to
The figures underneath the reaction represent the ener- BA ! Bυ C υA ⊲on the υ-plane⊳
gies of the dislocations, since they are proportional to and DB ! D˛ C ˛B ⊲on the ˛-plane⊳
the squares of the Burgers vectors. This reaction is, therefore, energetically favourable. This reaction can
and when the two Shockley partials ˛B and Bυ inter-
be seen with the aid of the Thompson tetrahedron, act, a stair-rod dislocation ˛υ D a/6[1 0 1] is formed.
which shows that the Frank partial dislocation A˛ can This low-energy dislocation is pure edge and there-
dissociate into a Shockley partial dislocation (Aˇ, Aυ fore sessile. If the other pair of partials interact then
or A ) and a low energy stair-rod dislocation (ˇ˛, υ˛ the resultant Burgers vector is ⊲υA C D˛⊳ D a/3[1 0 1]
or ˛) for example A˛ ! A C ˛.
108 Modern Physical Metallurgy and Materials Engineering
Figure 4.42 Formation of defect tetrahedron: (a) dissociation of Frank dislocations. (b) formation of new stair-rod dislocations, and (c) arrangement of the six stair-rod dislocations .
The formation of the defect tetrahedron of stacking to 6 ð 1 D 1 , compared with 3 ð 18 1 3 3 D 1 for the orig- faults may be envisaged as follows. The collapse of
inal stacking fault triangle bounded by Frank partials.
a vacancy disc will, in the first instance, lead to the Considering the dislocation energies alone, the disso- formation of a Frank sessile loop bounding a stacking
ciation leads to a lowering of energy to one-third of fault, with edges parallel to the h1 1 0i directions.
the original value. However, three additional stacking Each side of the loop then dissociates according to
fault areas, with energies of per unit area, have been the above reaction into the appropriate stair-rod and
newly created and if there is to be no net rise in energy partial dislocations, and, as shown in Figure 4.42a, the
these areas will impose an upper limit on the size of Shockley dislocations formed by dissociation will lie
the tetrahedron formed. The student may wish to ver- on intersecting f1 1 1g planes, above and below the
ify that a calculation of this maximum size shows the plane of the hexagonal loop; the decrease in energy
side of the tetrahedron should be around 50 nm. accompanying the dissociation will give rise to forces
De Jong and Koehler have proposed that the tetra- which tend to pull any rounded part of the loop
hedra may also form by the nucleation and growth into h1 1 0i. Moreover, because the loop will not in
of a three-dimensional vacancy cluster. The smallest general be a regular hexagon, the short sides will be
cluster that is able to collapse to a tetrahedron and eliminated by the preferential addition of vacancies at
subsequently grow by the absorption of vacancies is a the constricted site, and a triangular-shaped loop will
hexa-vacancy cluster. Growth would then occur by the form (Figure 4.42b). The partials Aˇ, A and Aυ bow
nucleation and propagation of jog lines across the faces out on their slip plane as they are repelled by the stair-
of the tetrahedron, as shown in Figure 4.43. The hexa- rods. Taking into account the fact that adjacent ends
vacancy cluster may form by clustering di-vacancies of the bowing loops are of opposite sign, the partials
and is aided by impurities which have excess positive attract each other in pairs to form stair-rod dislocations
change relative to the matrix (e.g. Mg, Cd or Al in along DA, BA and CA, according to the reactions
Au). Hydrogen in solution is also a potent nucleating
B, υA C A ! υ , ˇA C Aυ ! ˇυ agent because the di-vacancy/proton complex is mobile and attracted to ‘free’ di-vacancies. Figure 4.44 shows
A C Aˇ !
In vector notation the reactions are of the type the increase in tetrahedra nucleation after preannealing gold in hydrogen.
a/ 6[1 1 2]
C a/6[1 2 1] ! a/6[0 1 1]