Dislocation Science 
1. The Basics 

Perfect Dislocations  
First we need to make dislocations in
a general and scientific way. It's easy but tends to boggle the mind a bit so I
will introduce the procedure bit by bit. For starters, we need an essential if trivial insight. Let's imagine that we cut a crystal lattice along one of its major planes with an imaginary knife into two parts as shown below: 



The point of the exercise is to make you realize
that any movements involving only the
red base vectors (including any combination) will
retain the perfect fit between the two
halves of the lattice. This measn we can join the two halves again without any
problems. I won't say more. If you don't get this, look at the pictures below.
Before we look at this in more detail, we need a rule for translations with components perpendicular to the cut plane (involving the blue base vectors). The rule is: 

If you do this you must take out or add lattice points / unit cells as required.  
Obviously, there are always integer amounts of
lattice points or units cells of the lattice that are need to be inserted or
must be taken out to generate a smooth lattice once more after joining the two
halves. Here is the illustrations for this: 



So far it is easy. Now let's make a dislocation in the most general way possible. We only need to modify our game from above a little bit:  


We have made an edge dislocation. The cut line defines the dislocation line and the base vector used for the shifting will be what we call the Burgers vector of the dislocation from now on. We just as well could have named it "shove vector" but we honor the work of Johannes Martinus Burgers this way (so it's not Burger's vector!)  
If you look at the lattice picture above sideways, it looks like this:  


This is a variant of the "picture" of an edge dislocation that I used in the main text to introduce dislocations. It also shows that just having a picture like that, which may not be just a figure somebody has drawn on paper but a real picture from an electron microscope, allows you to reconstruct the Burgers vector, by doing a socalled Burgers circuit:  
Just run a loop around the
dislocation core (that's why it's called Burgers
circuit) that would close in a perfect lattice (e.g. 5 up, 4 left, 5
down, 4 right in the example above). The same kind of loop around a dislocation
will not close, and the vector left over
(or needed to close the loop) is the Burgers
vector of the dislocation. We won't worry about the sign here, i.e.
if the Burgers vector points left or right, and that means we don't have to
worry about doing the circuit clockwise or counterclockwise, closed in the
perfect lattice or around the dislocation, and so on. For our edge dislocation the Burgers vector is obviously at right angles to the dislocation line direction. Whenever we find that, we call the dislocation an edge dislocation. 

So far this hasn't been a big deal.
We have "made" edge dislocations in a
less abstract way
before. However, we are now in a position to make far stranger kinds of
dislocations than you ever wanted to know. So let's get to work. To make things a bit easier, you might imagine the figures that follow to depict a crystal and not a lattice. All you need to do is to imagine that whatever atomic base defines our crystal, is now in place at every point of whatever Bravais lattice we have. In the simplest case, instead of a lattice point you have now one atom. Schematic drawings look the same in both cases. We don't even draw the full lattice anymore. Here are a few simple examples. 



Screw dislocations
have their Bugers vector parallel to the dislocations line. In terms of the
atomic configuration, screw dislocations are far trickier to draw than edge
dislocations. There is no ending lattice plane and no simple perspective view.
Here is one way of giving some idea of the atomic arrangement in a simple cubic lattice, or if you take the spheres to be atoms, cubic primitive crystal (that does not exist in nature, however). 



We are not talking theory here but reality. Here is a highresolution transmission electron microscope picture of screw dislocations. Shown are the lattice planes above and perpendicular to the "cutting" plane or the plane where the dislocation line resides. It's like looking just at the pink lines in the figure above  


We see a set of {111} lattice planes
above or below (take your own choice) of the "cutting" plane. One of
the dislocation lines is indicated in yellow; the red line traces one lattice
plane across a dislocation. This was the very first highresolution picture of a screw dislocation; it was taken by me around 1979 with the Siemens Elmiskop 102 of the Mat. Science and Eng. Department of Cornell University. 

We could go on exploring all the
possible shifts or Burger vectors that come with one straight cut. We could
make many different kinds of perfect
dislocations this way. But all we would get as long as we keep the
Burgers vector in the plane of the cut, are "mixtures" of edge and
screw dislocations with some angle other than 90^{o} or 0^{o}
between dislocation line and Burgers vector. So let's be bold and go for
another step in generalizing dislocations. Just realize that there is no need that your cut ends in a straight line. With a virtual knife it is easy to make cuts of any shape. You might, for example, cut in a quarter circle as shown below, and then shift as shown. 

With a little care and hard
thinking, one can still draw a "picture" of what the perfect dislocation produced looks like in terms of
the atom arrangement. On the left hand side, it is a screw dislocation, on the
righthand side it is an edge dislocation, in between it is "in
between" or mixed dislocation. We call this and all the other ones made so far a "perfect" dislocation because its Burgers vector is a translation vector of the lattice. A perfect dislocation can exist all by itself in its host lattice, and the crystal around it stays perfect if we don't count a little elastic deformation. This implies that there are also imperfect dislocations. Yes, there are. Don't worry, we will encounter them soon enough. 



The figure helps to understand the
basic description of a general perfect dislocation:


It is easy to see that this business
of exploring all kinds of cuts and all kinds of shifts gets tiresome after a
while. Generating schematic pictures gets more and more involved but there is
less and less one can learn from this. Time to invoke the First law of applied Science. Let's forget about drawing qualitative pictures of all the kinds of dislocations we can think of, and just express the relative displacement of atoms in the vicinity of all dislocations, including the ones we can't think of, by some general equations. 

In other words, we are now going for
the
strain or
stress tensor associated with a dislocation. Sorry about that. If you haven't looked at the "science of deformation" module, the time to do that has definitely come. 

Stress and Strain Tensor of a Dislocation  
While the edge dislocation is easier
to conceive than the screw dislocations, it is the other way around if you go
for equations. Let's just jump into cold water; explanation will come later.
Much of what follows is based on the
marvellous
book of D. Hull and
D.J. Bacon:
"Introduction to dislocations". Here is the strain and stress tensor that "goes" with a screw dislocation that extends in the zdirection of a Cartesian x, y, z coordinate system. It is only an approximation and not valid right at the dislocation core. 



Using the red
cylinder coordinates as shown in the picture seems to make the equations
simpler so let's switch to cylidrical coordinates completely. The relations
between the two coordinate systems are obvious
( 



How does one get the strain tensor? It's actually not all that difficult but I will defer this question to a later module. How does one get the stress tensor if one has the strain tensor? In the equations above its obviously done by multiplying the strain tensor by G/2 (G is the shear modulus) but is that always true?  
Well  no. The general relation between stress
and strain in linear elasticity theory is Mercifully only two components of c_{ijkl} are nonzero for cubic crystals and with luck only one shows up as in the stress expression for the screw dislocation above. 

Just for completeness, here is the stress tensor for the edge dislocation  


Of course, nobody in her right mind would choose to express stress and strain around a dislocation in a Cartesian coordinate system, when quite obviously a cylinder coordinate system is much better suited to the task. Changing to cylinder coordinates as shown, we get rather simple equations for the two extremes screw and edge dislocation:  
It's kind of hard to illustrate a tensor. You would need a perspective drawing for each of its components. I can't do that, so here is a standard version showing only some aspects.  


Since there is no stress perpendicular to the
image plane, a twodimensional representation is sufficient. On the left half
of the picture, the forces acting on elementary cubes around the dislocation
are shown. They give qualitatively the components of the stress tensor. Green
or red denote compressive or tensile normal stresses, respectively. On the right half, contours of equal stress are shown for the normal component and the shear components of the stress tensor. 

As far as mixed dislocations are
concerned, you simply (haha) combine the equations for the extreme case in a
kind of weighted average. What you get covers everything that exists in terms
of perfect dislocations. Just be aware of the fact that the equations above are actually approximations for cubic crystals. They are only valid if you are a bit off from the center of the dislocation (around one or two atomic distances) . At the very core, you need to go to far more complex formulations. 



Why do I subject you to all those
equations? I daresay that very few of you will now be close to an orgasm,
hoisting out the champagne and exclaiming: "Isn't that wonderful and
exciting! Now I have seen it all and my life finally starts to make
sense!" Well, I did it for the following reasons:


But now let's look at the simple and extremely useful quantities: the energy of a dislocation and the forces between them.  
Energy of and Forces between Dislocations  
When you "make" a dislocations you need to shove a lot of atoms around a bit and that takes work or energy. The work invested after all is done is stored in the dislocation, or, to be more precise, in all the bond "springs" that needed to be pushed or pulled a bit relative to their normal length when we make a dislocation.  
Work equals force times distance. Specific work, i.e. work per unit of length is proportional to stress times strain. For the elastic energy part E_{el} of the dislocation energy, which is the part outside the core region with a radius r_{o}, we obtain with the equations from above  


Doing the integral takes a bit of fuzzing around
with tensor multiplication, integration, and so on, but the result is (almost)
easy. The energy depends somewhat on the dimension R of crystal
or grain of a poly crystal but we do not need to worry about that. After doing the same thing for an edge dislocation, coming up with some expression for the energy contained in the core regions (more difficult) one eventually realizes that the energy per unit length (» length of the Burgers vector) of any dislocation is in a fair enough approximation simply given by 



A really simple formula with a lot of power. It
tells us, for example:


With the last point I introduced forces between dislocations. The simple rule is that you add the strain fields as a function of distance. If the combined strain field gets larger as the dislocations get closer, there is a repellent force. If it gets smaller, there is an attractive force.  
In general equations it tends to get messy so I will not delve into that all that much. The basic situation is shown in the figure below. All dislocations are perpendicular to the viewing plane, inverted T's or circles symbolize edges or screw dislocations, respectively; change in color=change in sign of the Burgers vector.  


Of course, we now could also calculate how a dislocation interacts with all kinds of point defects and so on. We won't. It's enough to know that somebody could do it.  
The next logical step in
generalizing our cutandshove business would be to cut not in a plane but in
whatever irregular shape comes to mind. The cut "plane" then would
not be planar and the cut line=dislocation line could be as irregular as a
stock market curve. Be happy that we don't have to deal with that. We go right away for things even worse. We will now consider one parameter we have totally neglected so far: the crystal itself. 

Dislocations in Crystals  
All of the above is valid for lattices. You might have imagined atoms or crystals all the time, but they weren't really needed. Essentially, most of what I did so far was an exercise in tricky geometry and the math going with it. In fact, what we now call "dislocations" was already part of a general mathematical treatise from Vito Volterra around 1900, i.e. about 30 years before the "invention" of dislocations in real crystals.  
So far the the crystal only sneaked
in when we looked at energies and forces. Formally that happens whenever in the
equations a material property like the
shear modulus G comes up. But just considering energies and forces does not do justice to the full impact a real crystal will have on the properties of dislocations 

I won't go into details here but only
point out three farreaching rules that
crystals add to the business of dislocations:


Did you notice what that means for
the cutandshove procedure? Let's give it a look:



There is now a limited number of socalled
glide systems, meaning combinations of a
glide plane (=densely packed plane) and a Burger vector (=shortest possible
lattice translation vector). That is of prime importance because the more glide
systems you have, the easier it is to deform the crystal. The illustration module shows this for the three prominent lattice types. 

Of course, if you belong to the
dislocation liberation movement, you will not heed these offending rules but
make your cut as a you like. You can do that, no problem. All that will happen
is:


The second statement is a bit vague.
Why didn't I just say: "...until the extend in the favorite
direction"? Because I sneaked in a puzzle. We actually now have two mutually conflicting conditions for the
direction of dislocation line:


Both rules are not absolute but just "recommendations". While in electron microscope pictures dislocations tend to be rather straight, there are plenty of examples where they are curved, witness the two examples in the backbone.  
Well, this is getting complicated. And we are still rather looking at the appetizer and not yet eating it.  
If you are ready for a full dinner
now, order the full menu known as "The Bible": J.P. Hirth and J. Lothe, Theory of Dislocations, second edition (Krieger Publishing, Malabar Florida 1982). 

If you are content with just looking at appetizers, you might continue with this module.  
Back to Dislocation Science  
On to  
1. The Basics  
2. The reality  
.3. Specialities  
Myths and Bullshit Around Quenching
Spring Model and Properties of Crystals
Transmission Electron Microscopes
Twinning, Shear Deformation and Martensite Formation
Dislocation Science  2. The Reality
Dislocation Science  3. Specialities
© H. Föll (Iron, Steel and Swords script)