- Email: [email protected]

227

The Use of the Burgers Vector Octahedron in the Face-centered Cubic Lattice

V. K. LINDROOS

Department of Physical Metallurgy, Helsinki University of Technology, Otaniemi, Finland

The use of a geometrical device called the Burgers vector octahedron is illustrated for the notation of the Burgers vectors of those dislocations in the face-centered cubic lattice which cannot be visualized by the conventional Thompson tetrahedron. Furthermore, a combination of the tetrahedron and octahedron called the Burgers vector tetra-octahedron is presented as a means of visualizing all of the fcc Burgers vector notations.

Introduction The tetrahedron devised by Thompson [1] has long been the standard tool for analyzing dislocation reactions in face-centered cubic crystals. However, such a device can be used to represent only those Burgers vectors of the fcc dislocations which are listed in Table 1. Because there are other dislocations, including some such as the Hirth dislocation [2-4] which were not known at the time of the presentation of the Thompson tetrahedron, whose Burgers vectors cannot be visualized with the tetrahedron, there is need for an additional visualization de~dce. I t is the purpose of the present contribution to present the Burgers vector octahedron 1 as a device to fulfill this need.

The Burgers Vector Octahedron Consider the Burgers vector octahedron A B C D E F shown in Fig. 1. The following Burgers vectors can be directly visualized: the edges of the octahedron represent all six Burgers vectors of type a/2 (110 }; the diagonals 1The convenience of the Burgers vector octahedron for representing perfect and partial Hirth dislocations has been shown earlier [5]. © American Elsevier Publishing Company, Inc., 1975

V. K. Lindroos

228 TABLE 1 The fce Burgers vectors represented by the Thompson tetrahedron Dislocation type

Burgers vector

Perfect dislocation Shockley partial dislocation Frank partial dislocation Lomer-Cottrelllock dislocation

a/2 (110) a/6 (112) a/3 (111) a/6 (110)

Energy

a~/2 a2/6 a2/3 a2/18

AC, BD and EF represent all three Burgers vectors of Hirth dislocations of type a(100); the connecting lines of adjacent sides, e.g. ~.~, represent the Burgers vectors of the Hirth lock dislocations of type a/3 (100); and those of opposite adjacent sides, e.g., ~ , represent the Burgers vectors of a stair-rod dislocation of type a/3 (110) at an acute bend. The presentation of the Shockley partials with the octahedron is the same as with the Thompson tetrahedron. Because of the necessity to represent the Burgers vectors of the Hirth dislocations, the other Burgers vectors and the {111} slip planes are represented in duplicate. Furthermore, the connecting lines of the opposite parallel sides of the octahedron, such as ~y, reveal Burgers vectors of the type a/3 (111 ), which,

~x FIG. 1. The Burgers vector octahedron for the notation of the fcc Burgers vectors which cannot be visualizedby the Thompson tetrahedron.

Burgers Vector Octahedron

229 TABLE 2

The fcc Burgers Vectors Represented by the Burgers vector Octahedron Shown in Fig. 1 Symbol

Dislocation type

Burgers vector

Energy

AB AC Fu 3"~ ~ ~3"

Perfect dislocation* Perfect Hirth dislocation Shockley partial dislocation* Hirth lock dislocation Stair-rod at acute bend see Characterization of Dislocations

a/2 (110) a (100 } a/6 (112 } a/3 (100 } a/3 (110) a/3 (11i }

a2/2 a2 a2/6 a:/9 2a2/9 a2/3

* also revealed by the Thompson tetrahedron. even though they have the same indices, differ from the conventional Frank partials as represented by the Thompson tetrahedron, because neither the starting nor the ending point of the Burgers veetor is at a stable atomic position. Such Greek-Greek Frank partials, ~-, might result from the reaction between a Hirth partial and a stair-rod dislocation at an acute bend: i.e., a/3[001] + a/3[l-/0] --* a/311X1]. The Burgers vectors revealed by the oetahedron as described above are summarized in the Table 2.

The Burgers Vector Tetra-Octahedron Examination of Tables 1 and 2 shows that a stair-rod dislocation at an obtuse bend with a Burgers vector of the type a/6 (301 } cannot be visualized by either the tetrahedron or octahedron. However, this Burgers vector can be directly visualized by combining the tetrahedron and oetahedron, Fig. 2, where it is represented b y the vector a~,. Furthermore, such a tetra-oetahedron reveals the Burgers vector ~ , connecting two metastable atomic positions and having the same indices as that of a Shoekley partial, viz., a/6 (112); again Table 3 summarizes these representations. I t should be noted that a Shoekley partial having a Burgers vector of the type ~ , is presumably not glissile contrary to the nature of the conventional Shoekleys such as As, Fig. 2. This is because the atomic motion ~ , instead of producing a low energy stacking fault . . . ABCA [ C A B C . . . , causes creation of high energy stacking fault of the type . . . ABCA [ ABC

230

V. K. Lindroos Z

F

~+ . . . . . ~J

I

C

",a X

FIG. 2. The Burgers vector tetra-octahedron for the notation of the fcc Burgers vectors which cannot be visualized by either the Thompson tetrahedron or the Burgers vector octahedron.

. . . . Therefore, Shockley partials of the type u ~ resemble sessile dislocation locks having Burgers vectors connecting two metastable atomic positions, e.g., the Lomer-Cottrell and partial Hirth locks. Such 'sessile' Shockleys can form by the dissociation of a high energy stair-rod dislocation at an obtuse bend: i.e., a/6[130] --~ a/6[121] + a/6[011]. Characterization of Dislocations Some of the vagueness existing in the literature in distinguishing between dislocations, which have different types of Burgers vectors, could be avoided TABLE 3 The fce Burgers Vectors Revealed only by the Tetra-Octahedron Shown in Fig. 2 Symbol

Dislocation type

Burgers vector

Energy

otT ot~

Stair-rod dislocation at obtuse bend see Characterization of Dislocations

a/6 (301) a/6 (112)

a~/6

5a~/18

a (100)

Hirth dislocation

0

T, 0

Visualization

Type

T - O = Burgers vector tetra-octahedron

O = Burgers vector octahedron

Frank partial

Shockley partial

T, O T

a/3 (111)

Visualization

a/6 (112)

vector

Burgers

Partial dislocations

(Roman-Greek)

T = Thompson tetrahedron

a/2 (110)

vector

Burgers

Unit dislocation

(Roman-Roman)

Type

Perfect dislocations

O O

a/3 (111) a/3 (100 }

Frank subpartial Hirth subpartial

T 0 T-O

a/6 (110) a/3 (110 } a/6 (301)

acute stair-rod obtuse stair-rod

LomerCottrell sub-partial

T-O

Visualization

a/6 (112)

vector

Burgers

Shockley sub-partial

(Greek-Greek)

Type

Sub-partial dislocations

The Characterization of Dislocation and Their Burgers Vectors in fcc Crystals

TABLE 4

bO 5¢ b--a

O~

~v

232

V. K. Lindroos

by making use of the following system of dislocation characterization. Referring to Figs. 1 and 2, it can be seen that there are altogether three different kinds of dislocations corresponding to the three different types of Burgers vectors in face-centered cubic crystals: (1) dislocations having Burgers vectors connecting two equilibrium atomic positions corresponding to the Roman-Roman notation, such as AB; these are perfect dislocations, (2) dislocations having Burgers vectors connecting equilibrium atomic positions to metastable positions corresponding to the Roman-Greek notation, such as Aa; these are partial dislocations; and finally, (3) dislocations having Burgers vectors connecting two metastable atomic positions corresponding to the Greek-Greek notation, such as a ~, these are sub-partial dislocations. According to this system of characterization, the dislocations occurring in face-centered cubic crystals are those listed in the Table 4. Finally, it should be noted that the Burgers vector devices described here have been found particularly useful in analyzing dislocation networks, as pointed out b y Miekk-oja and Lindroos [6].

The author would like to express appreciation to Professor J. P. Hirth for a helpful discussion on the topic of this contribution. The support of this work from the Finnish Academy is gratefully acknowledged. References 1. N. Thompson, Dislocation nodes in face-centered cubic lattice, Proc. Phys. Soc. B. 66, 481 (1953). 2. J. Friedel, On the linear work hardening rate of face-centered cubic single crystals, Phil. Mag. 46, 1169 (1955). 3. J. P. Hirth, On dislocation interactions in the fcc lattice, J. Appl. Phys. 32, 700 (1961). 4. F. R. N. Nabarro, Z. S. Basinski and D. B. Holt, The plasticity of pure single crystals, Adv. Phys. 13, 193 (1964). 5. V. K. Lindroos, Graduate Thesis, Helsinki University of Technology, 1965. 6. H. M. Miekk-oja and V. K. Lindroos, 1974, Constitutive Equations in Plasticity, edited by A. S. Argon and B. H. Kear, MIT Press, Cambridge, in press.

Received August, 1974