Burgers vectors are accordingly noted B=(n1,…,n6), and the same for diffraction vectors G.

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From: Dislocations in Solids, 2008

## Related terms:

D. Hull, D.J. Bacon, in Introduction to Dislocations (Fifth Edition), 2011

### Burgers Vectors and Stacking Faults

Some of the important metals of this structure are given in Table 6.1. As explained in section 1.2, the (0001) basal plane is close-packed and the close-packed directions are 〈112¯0〉. The shortest lattice vectors are 13〈112¯0〉, the unit cell generation vectors **a** in the basal plane. It may be anticipated, therefore, that dislocation glide will occur in the basal plane with Burgers vector b=13〈112¯0〉: this slip system is frequently observed. None of the metals has a structure represented by ideally close-packed atomic spheres, which would require the lattice parameter ratio *c/a* to be (8/3)1/2=1.633, although magnesium and cobalt have a *c/a* ratio close to ideal. This indicates that directionality occurs in interatom bonding. As a consequence of this, it is found that some metals slip most easily with b=13〈112¯0〉 on the first-order prism planes 101¯0 (see Table 6.1). Basal and prism planes, and a first-order pyramidal plane with a common <1¯21¯0> axis are shown in Fig. 6.1.

*Metal*

*Be*

*Ti*

*Zr*

*Mg*

*Co*

*Zn*

*Cd*

c/a ratio | 1.568 | 1.587 | 1.593 | 1.623 | 1.628 | 1.856 | 1.886 |

Preferred slip | basal | prism | prism | basal | basal | basal | basal |

plane for b=a | (0001) | 101¯0 | 101¯0 | (0001) | (0001) | (0001) | (0001) |

Burgers vectors for the structure may be described in a similar fashion to the Thompson tetrahedron for face-centered cubic metals by using the bi-pyramid shown in Fig. 6.2. The important dislocations and their Burgers vectors are as follows.

Figure 6.2. Burgers vectors in the hexagonal close-packed lattice.

(From Berghezan, Fourdeux and Amelinckx,*Acta Metall.*9, 464, 1961.)

(a)

Perfect dislocations with one of six Burgers vectors in the basal plane along the sides of the triangular base *ABC* of the pyramid, represented by **AB, BC, CA, BA, CB** and **AC**.

Perfect dislocations with one of two Burgers vectors perpendicular to the basal plane, represented by the vectors **ST** and **TS**.

Perfect dislocations with one of twelve Burgers vectors represented by symbols such as **SA**/**TB**, which means either the sum of the vectors **ST** and **AB** or, geometrically, a vector equal to twice the join of the midpoints of *SA* and *TB*.

Imperfect basal dislocations of the Shockley partial type with Burgers vectors **A*** σ*,

**B**

*,*

**σ****C**

*,*

**σ**

**σ****A**,

**σ****B**and

**σ****C**.

Imperfect dislocations with Burgers vectors perpendicular to the basal plane, namely, **σ****S**, **σ****T**, **S*** σ* and

**T**

*.*

**σ**Imperfect dislocations which are a combination of the latter two types given by **AS, BS**, etc. Although these vectors represent a displacement from one atomic site to another the associated dislocations are not perfect. This is because the sites do not have identical surroundings and the vectors are not translations of the lattice.

The Miller–Bravais indices and length *b* of these Burgers vectors **b** are given in Table 6.2. The value of *b*2 for ideal close-packing (c2=83a2) is also given: appropriate adjustments are required when dealing with non-ideal packing.

Table 6.2. Dislocations in Hexagonal Close-Packed Structures. The Values of *b*2 are for the Case c2=83a2

*Type*

*AB*

*TS*

*SA/TB*

*Aσ*

*σS*

*AS*

b | 13〈112¯0〉 | <0001> | 13〈112¯3〉 | 13〈1¯100〉 | 12<0001> | 16〈2¯203〉 |

b | a | c | (c2+a2)12 | a/√3 | c/2 | (a23+c24)12 |

b2 | a2 | 83a2 | 113a2 | 13a2 | 23a2 | a2 |

A number of different stacking faults are associated with the partial dislocations listed in Table 6.2. According to the hard-sphere model of atoms, three basal-plane faults exist which do not affect nearest-neighbor arrangements of the perfect stacking sequence *ABABAB…* Two are intrinsic and conventionally called *I*1 and *I*2. Fault *I*1 is formed by removal of a basal layer, which produces a very high energy fault, followed by slip of 13〈101¯0〉 of the crystal above this fault to reduce the energy:

These faults introduce into the crystal a thin layer of face-centered cubic stacking (*ABC*) and so have a characteristic stacking-fault energy *γ*. The main contribution to *γ* arises from changes in the second-neighbor sequences of the planes. There is one change for *I*1, i.e. the *A* sequence changes to *C*, two for *I*2 and three for *E*, and so to a first approximation γE≈32γI2≈3γI1. Experimental estimates of *γ* for the basal-slip metals are generally higher than those quoted earlier for the face-centered cubic metals. However, theoretical values obtained by *ab initio* methods (section 2.4) are now considered to be more reliable. Values for *γ* for the *I*2 basal fault in magnesium are in the range 30 to 40 mJm−2 and calculations for zirconium give 200 mJm−2.

The symmetry of the face-centered cubic structure guarantees that the 111 *γ*-surface has an extremum (maximum or minimum) for the fault vector 16〈112¯〉 because three mirror planes perpendicular to the fault plane exist at that position (see Figs 5.3 and 5.4Fig 5.3Fig 5.4). The same condition applies to the *I*2 fault on the basal plane of the hexagonal close-packed metals. It is not satisfied for the first-order prism planes 101¯0 in these metals, however.

See more: How Many Electrons In The Third Shell S, Electron Shells

Geometrical models used to consider possible faults on the 101¯0 planes have led to proposals that stable faults may exist with vectors 118〈2463¯〉 or 19〈112¯0〉. The first produces a fault which is stable in a lattice consisting of hard spheres. The second results in prism planes adjacent to the fault ribbon adopting the stacking of 112 planes in body-centered cubic metals, the relevance being that this is the stable crystal structure for titanium and zirconium at high temperature. There is no evidence that these faults exist in real metals, however. Treatment based on *ab initio* calculations for zirconium have shown that a stable fault exists with vector 16〈112¯α〉, where *α* is between 0 and 0.6. The corresponding value of *γ* is approximately 140 mJm−2. Prism-plane faults in metals such as magnesium that slip predominantly on the basal plane are believed to have very high energy and are probably unstable.