
We now generalize the present view of
dislocations as follows: 


1. Dislocation lines may be
arbitrarily curved  never mind that we
cannot, at the present, easily imagine the atomic picture to that. 


2. All lattice
vectors can be Burgers vectors, and as we will see later, even
vectors that are not lattice vectors are
possible. A general definition that encloses all cases is needed. 

As ever so often, the basic ingredients needed for
"making" dislocations existed before dislocations in crystals were
conceived. Volterra, coming from the mechanics of the
continuum (even crystals haven't been discovered yet), had defined all possible
basic deformation cases of a continuum (including crystals) and in those
elementary deformation cases the basic definition for dislocations was already
contained! 


The link shows
Volterra' basic deformation
modes  three can be seen to produce edge dislocations in crystals, one generates a
screw dislocation. 


Three more cases produce defects called "disclinations". While of theoretical
interest, disclinations do not really occur in "normal" crystals, but
in more unusual circumstances (e.g. in the twodimensional lattice of flux
lines in superconductors) and we will not treat them here. 

Volterra's insight gives us the tool to define
dislocations in a very general way. For this we invent a little contraption
that helps to imagine things: the "Volterra
knife", which has the property that you can make any conceivable
cut into a crystal with ease (in your mind). So lets produce dislocations with
the Volterra knife: 

1. Make a
cut, any cut, into the crystal using the Volterra knife. 


The cut is always defined by some plane inside the crystal (here the plane indicated
by he red lines). 


The cut does not have to be on a flat plane, but
we also do not gain much by making it "warped". The picture shows a
flat cut, mainly just because it is easier to draw. 


The cut is by necessity completely contained
within a closed line, the
red cut
line (most of it on the outside of the crystal). 


That part of the cut line that is inside the crystal will define the line vector
t of the dislocation to be formed. 




2. Move the two
parts of the crystal separated by the cut relative to each other by
a translation vector of the lattice;
allowing elastic deformation of the lattice in the region around the
dislocation line. 


The translation vector chosen will be the Burgers vector b of the
dislocation to be formed. The sign will depend on the convention used. Shown
are movements leading to an edge dislocations
(left) and a screw dislocation (right). 




3. Fill in material or take some out, if
necessary. 


This will always be necessary for obvious reasons whenever
your chosen translation vector has a component perpendicular to the plane of
the cut. 


Shown is the case where you have to fill in
material  always preserving the structure of the crystal that was cut, of
course. 


Left: After
cut and movement. Right: After filling up
the gap with crystal material. 




4. Restore the
crystal by "welding" together the surfaces of the cut.



Since the displacement vector was a translation vector of the lattice, the surfaces will
fit together perfectly everywhere  except
in the region around the dislocation line defined as by the cut line. 

A onedimensional defect was produced, defined by
the cut line (= line vector
t of the dislocation) and the displacement vector which we call
Burgers vector b.



It is rather obvious (but not yet proven) that the
Burgers vector defined in this way is identical to the one
defined before. This will become
totally clear in the following paragraphs. 

From the Volterra construction of a
dislocation, we can not only obtain the simple edge and screw dislocation that
we already know, but any dislocation.
Moreover, from the Volterra construction we can immediately deduce a new list
with more properties of dislocations: 


1. The Burgers
vector for a given dislocation is always the same, i.e. it does not
change with coordinates, because there is only one displacement for every cut. On the other hand,
the line vector may be different at every
point because we can make the cut as complicated as we like. 


2. Edge and screw
dislocations (with an angle of 90° or 0°, resp.,
between the Burgers and the line vector) are just special cases of the general case of a
mixed dislocation, which has an
arbitrary angle between b and t that may even
change along the dislocation line. The illustration shows the case of a curved
dislocation that changes from a pure edge dislcation to a pure screw
dislocation. 



We are looking at the plane of the cut (sort of a semicircle
centered
in the lower left corner). Blue circles denote atoms just below,
red circles atoms just above the cut. Up on the right the dislocation is
a pure edge dislocation, on the lower left it is pure screw. In between
it is mixed. In the link this dislocation is shown moving in an
animated illustration. 



3. The Burgers vector must be independent
from the precise way the Burgers circuit is done since the Volterra
construction does not contain any specific rules for a circuit. This is easy to
see, of course: 




Old circuit 
Two arbitrary alternative Burgers circuits.
The colors serve to make it easier
to keep track of the steps. 



4. A dislocation cannot end in the interior of an otherwise perfect
crystal (try to make a cut that ends internally with your Volterra knife), but
only at



5. If you do not have to add matter or to
take matter away (i.e. involve interstitials or vacancies), the Burgers vector
b must be in the plane of the cut
which has two consequences:
 The cut plane must be planar; it is defined by the line vector and the
Burgers vector.
 The cut plane is the glide plane of the
dislocation; only in this plane can it move without the help of interstitials
or vacancies.
. 


The glide plane is thus the plane
spread out by the Burgers vector b and the line vector t. 


6. Plastic deformation is promoted by the
movement of dislocations in glide planes. This is easy to see: Extending your
cut produces more deformation and this is identical to moving the dislocation!



7. The magnitude of
b (= b) is a measure for the
"strength" of the dislocation, or the amount of elastic
deformation in the core of the dislocation. 

A not so obvious, but very important
consequence of the Voltaterra definition is 


8. At a dislocation knot the sum of all Burgers vectors is zero, Sb = 0, provided all line vectors point
into the knot or out of it. A dislocation knot is simply a point where three or
more dislocations meet. A knot can be constructed with the Volterra knife as
shown below. 

Statement 8. can be proved in
two ways: Doing Burgers circuits or using the Voltaterra construction twice. At
the same time we prove the equivalence of obtaining b from
a Burgers circuit or from a Voltaterra construction. 


Lets look at a dislocation knot formed by three
arbitrary dislocations and do the Burgers circuit  always taking the direction
of the Burgers circuit from a "right hand" rule 





Since the sum of the two individual circuits must
give the same result as the single "big" circuit, it follows: 





Or, more generally, after reorienting all
t vectors so that they point into the knot: 




Now lets look at the same situation
in the Voltaterra construction: 


We make a first cut with a Burgers vector
b_{1} (the green one in the illustration below).



Now we make a second cut in the same plane that
extends partially beyond the first one with Burgers vector
b_{2} (the red line). We have three different
kinds of boundary lines: red and green where the cut lines are distinguishable,
and black where they are on top of each other. And we have also produced a
dislocation knot! 





Obviously the displacement vector for the black
line, which is the Burgers vector of that dislocation, must be the sum of the
two Burgers vectors defined by the two cuts: b =
b_{1} + b_{2}. So we
get the same result, because our line vectors all had the same "flow"
direction (which, in this case, is actually tied to which part of the crystal
we move and which one we keep "at rest"). 

If we produce a dislocation knot by two cuts that
are not coplanar but keep the Burgers
vector on the cut plane, we produce a knot between dislocations that do not
have the same glide plane. As an immediate consequence we realize that this
knot might be immobile  it cannot move.



A simple example is shown below (consider that the
Burgers vector of the red dislocation may have a glide plane different from the
two cut planes because it is given by the (vector) sum of the two original
Burgers vectors!). 




We can now draw some conclusion about
how dislocations must behave in circumstances not so easy to see directly: 


Lets look at the glide plane of a dislocation loop. We can easily produce a loop with
the Volterra knife by keeping the cut totally inside the crystal (with a
real knife that could not be done). In the
example the dislocation is an edge dislocation. 


The glide plane, always defined by Burgers and
line vector, becomes a glide cylinder! The
dislocation loop can move up or down on it, but no lateral movement is
possible. 





What would the glide plane of a screw dislocation
loop look like? Well there is no such thing as a screw
dislocation loop  you figure that one out for yourself! 


A pure (straight) screw dislocation has no particular glide plane
since b and t are parallel and thus
do not define a plane. A screw dislocation could therefore (in principle) move
on any plane. We will see later why there are still some restrictions. 

This leaves the touchy issue of the
sign convention for the line vector t. This is important! The sign of the line vector
determines the sign of the Burgers vector, and the Burgers vector, including
sign, is what you will use for many calculations. This is so because for a
Burgers circuit you must define if you go clockwise or counterclockwise around
the line vector, using the righthand convention. We
will go clockwise! 


The easiest way of dealing with this is to
remember that the sum of the Burgers vectors must be zero if all line vectors
either point into the knot or away from it. 


As long as only three dislocations meet at one
point, there is no big problem in being consistent in the choice of line vector
and Burgers vectors, once you started assigning signs for the line vectors, you
can throw in the Burgers vector. There is however no principal restriction to
only three dislocations meeting at one point; in this case the situation is not
always unambiguous; we will deal with that later. This is not as easy as it
seems. We will do a little exercise for that. 


Last we define: The circuit is to close around the
dislocation; the circuit in the reference crystal then defines the Burgers
vector. 


