The Burgers vector b is the vector which
defines the magnitude and direction of slip. Therefore, it is the most
characteristic feature of a dislocation. For a pure edge dislocation the
Burgers vector is perpendicular to the dislocation line, while for a pure screw
dislocation the Burgers vector is parallel to the dislocation line. Actually,
dislocations in real crystals are rarely straight lines and
rarely lie in a
single plane. In general, a dislocation will be partly edge and partly screw in
character. Dislocations will ordinarily take the form of curves or loops, which
in three dimensions form an interlocking dislocation network. In considering a
dislocation loop in a slip plane any small segment of the dislocation line can
be resolved into edge and screw components. Considering the diagram, the
dislocation loop is pure screw at point A and pure edge at point B, while along
most of its length it has mixed edge and screw components. Note, however, that
the Burgers vector is the same along the entire dislocation loop. If this were
not so, part of the crystal above the slipped region would have to slip by a
different amount relative to another part of the crystal and this would mean that
another dislocation line would run across the slipped region.
A convenient way of defining the Burgers vector
of a dislocation is by means of the Burgers circuit. Consider an atomic
arrangement with an edge dislocation. Starting at a lattice point, imagine a
path traced from atom to atom, an equal distance in each direction, always in
the direction of one of the vectors of the unit cell. If the region enclosed by
the path does not contain a dislocation, the Burgers circuit will close.
However, if the path encloses a dislocation, the Burgers circuit will not close.
The closure failure of the Burgers circuit is the Burgers vector b. The closure
failure of a Burgers circuit around several dislocations is equal to the sum of
their separate Burgers vectors.
Because a dislocation represents the boundary
between the slipped and unslipped region of a crystal, topographic
considerations demand that it either must be a closed loop or else must end at
the free surface of the crystal. In general, a dislocation line cannot end
inside of a crystal. The exception is at a node, where three or four
dislocation lines meet. A node can be considered as two dislocations with
Burgers vectors b1 and b2 combining to produce a resultant dislocation b3. The
vector b3 is given by the vector sum of b1 and b2.
Since the periodic force field of the crystal
lattice requires that atoms must move from one equilibrium position to another,
it follows that the Burgers vector must always connect one equilibrium lattice
position with another. Therefore, the crystal structure will determine the possible
Burgers vectors. A dislocation with a Burgers vector equal to one lattice spacing
is said to be a dislocation of unit strength. Because of energy considerations
dislocations with strengths larger than unity are generally unstable and
dissociate into two or more dislocations of lower strength. The criterion for
deciding whether or not dissociation will occur is based on the fact that the
strain energy of a dislocation is proportional to the square of its Burgers
vector. Therefore, the dissociation reaction b1—>b2 + b3 will occur when b12
> b22 + b32, but not if b12
< b22 + b32.
Dislocations with strengths less than unity are
possible in close-packed lattices where the equilibrium positions are not the
edges of the structure cell. A Burgers vector is specified by giving its
components along the axes of the crystallographic structure cell. Thus, the
Burgers vector for slip in a cubic lattice from a cube corner to the centre of
one face has the components ao/2, ao/2, 0. The Burgers
vector is [ao/2, ao/2, 0], or, as generally written, b =
(ao/2) [110].
