Burgers vecto
In physics, the Burgers vector, named after Dutch physicist Jan Burgers,
is a vector, often denoted b, that represents the magnitude and direction
of the lattice distortion resulting from a dislocation in a crystal lattice.
Burgers vectors
The vector's magnitude and direction is best understood when the
dislocation-bearing crystal structure is first visualized without the
dislocation, that is, the perfect crystal structure. In this perfect crystal
structure, a rectangle whose lengths and widths are integer multiples of
"a" (the unit cell edge length) is drawn encompassing the site of the
original dislocation's origin. Once this encompassing rectangle is drawn,
the dislocation can be introduced. This dislocation will have the effect of
deforming, not only the perfect crystal structure, but the rectangle as well.
The said rectangle could have one of its sides disjoined from the
perpendicular side, severing the connection of the length and width line
segments of the rectangle at one of the rectangle's corners, and displacing
each line segment from each other. What was once a rectangle before the
dislocation was introduced is now an open geometric figure, whose
opening defines the direction and magnitude of the Burgers vector.
Specifically, the breadth of the opening defines the magnitude of the
Burgers vector, and, when a set of fixed coordinates is introduced, an
angle between the termini of the dislocated rectangle's length line
segment and width line segment may be specified.
The direction of the vector depends on the plane of dislocation, which is
usually on one of the closest-packed crystallographic planes. The
magnitude is usually represented by the equation:
where a is the unit cell edge length of the crystal, ||b|| is the magnitude of
Burgers vector and h, k, and l are the components of the Burgers vector, b
= <h k l>. In most metallic materials, the magnitude of the Burgers vector
for a dislocation is of a magnitude equal to the interatomic spacing of the
material, since a single dislocation will offset the crystal lattice by one
close-packed crystallographic spacing unit.
In edge dislocations, the Burgers vector and dislocation line are at right
angles to one another. In screw dislocations, they are parallel.
The Burgers vector is significant in determining the yield strength of a
material by affecting solute hardening, precipitation hardening and work
hardening.
Frank-Read Source
Animation illustrating how stress on a Frank-Read Source (center) can
generate multiple dislocation lines in a crystal.
This page has some issues
In materials science, a Frank-Read source is a mechanism explaining
the generation of multiple dislocations in specific well-spaced slip planes
in crystals when they are deformed. When a crystal is deformed, in order
for slip to occur, dislocations must be generated in the material. This
implies that, during deformation, dislocations must be primarily
generated in these planes. Cold working of metal increases the number of
dislocations by the Frank-Read mechanism. Higher dislocation density
increases yield strength and causes work hardening of metals.
It was proposed by and named after British physicist Charles Frank and
Thornton Read.
History
Charles Frank detailed the history of the discovery from his perspective
in Proceedings of the Royal Society in 1980.[1] In 1950 Charles Frank,
who was then a research fellow in the physics department at the
University of Bristol, visited the United States to participate in a
conference on crystal plasticity in Pittsburgh. Frank arrived in the United
States well in advance of the conference to spend time at a naval
laboratory and to give a lecture at Cornell University. When, during his
travels in Pennsylvania, Frank visited Pittsburgh, he received a letter from
fellow scientist Jock Eshelby suggesting that he read a recent paper by
Gunther Leibfried. Frank was supposed to board a train to Cornell to give
his lecture at Cornell, but before departing for Cornell he went to the
library at Carnegie Institute of Technology to obtain a copy of the paper.
The library did not yet have the journal with Leibfried's paper, but the
staff at the library believed that the journal could be in the recently
arrived package from Germany. Frank decided to wait for the library to
open the package, which did indeed contain the journal. Upon reading the
paper he took a train to Cornell, where he was told to pass the time until
5:00, as the faculty was in meeting. Frank decided to take a walk between
3:00 and 5:00. During those two hours, while considering the Leibfried
paper, he formulated the theory for what was later named the Frank-Read
source. A couple of days later, he traveled to the conference on crystal
plasticity in Pittsburgh where he ran into Thorton Read in the hotel lobby.
Upon encountering each other, the two scientists immediately discovered
that they had come up with the same idea for dislocation generation
almost simultaneously (Frank during his walk at Cornell, and Thornton
Read during tea the previous Wednesday) and decided to write a joint
paper on the topic. The mechanism for dislocation generation described
in that paper is now known as the Frank-Read source.
Mechanism
Figure 1: Frank-Read Source
The Frank-Read source is a mechanism based on dislocation
multiplication in a slip plane under shear stress.
Consider a straight dislocation in a crystal slip plane with its two ends, A
and B, pinned. If a shear stress is exerted on the slip plane then a force
, where b is the Burgers vector of the dislocation and x is the
distance between the pinning sites A and B, is exerted on the dislocation
line as a result of the shear stress. This force acts perpendicularly to the
line, inducing the dislocation to lengthen and curve into an arc. The
bending force caused by the shear stress is opposed by the line tension of
the dislocation, which acts on each end of the dislocation along the
direction of the dislocation line away from A and B with a magnitude of
, where G is the shear modulus. If the dislocation bends, the ends of
the dislocation make an angle with the horizontal between A and B,
which gives the line tensions acting along the ends a vertical component
acting directly against the force induced by the shear stress. If sufficient
shear stress is applied and the dislocation bends, the vertical component
from the line tensions, which acts directly against the force caused by the
shear stress, grows as the dislocation approaches a semicircular shape.
When the dislocation becomes a semicircle, all of the line tension is
acting against the bending force induced by the shear stress, because the
line tension is perpendicular to the horizontal between A and B. For the
dislocation to reach this point, it is thus evident that the equation:
must be satisfied, and from this we can solve for the shear stress:
This is the stress required to generate dislocation from a Frank-Read
source. If the shear stress increases any further and the dislocation passes
the semicircular equilibrium state, it will spontaneously continue to bend
and grow, spiraling around the A and B pinning points, until the segments
spiraling around the A and B pinning points collide and cancel. The
process results in a dislocation loop around A and B in the slip plane
which expands under continued shear stress, and also in a new dislocation
line between A and B which, under renewed or continued shear, can
continue to generate dislocation loops in the manner just described. A
Frank-Read loop can thus generate many dislocations in a plane in a
crystal under applied stress. The Frank-Read source mechanism explains
why dislocations are primarily generated on certain slip planes;
dislocations are primarily generated in just those planes with Frank-Read
sources. It is important to note that if the shear stress does not exceed:
and the dislocation does not bend past the semicircular equilibrium state,
it will not form a dislocation loop and instead revert to its original state.