
The determination of the precise
dislocation structure needed to transform a nearcoincidence boundary into a true coincidence boundary with some superimposed
grain boundary dislocation network can be exceedingly difficult (to you, not to
the crystal), especially when the steps possibly associated with the grain
boundary dislocations must be accounted for, too. 


Nevertheless, the structure thus
obtained is what you will see in a TEM picture  the crystal has no
problem whatsoever to "solve" this problem! 

In order to get familiar with the
concept, it is easiest, to consider the environment of the S= 1 grain boundary, i.e. the boundary between two
crystals with almost identical orientation.



This kind of boundary is known as "smallangle grain boundary"
(SAGB) , or, as already
used above as "S1 boundary". 


We can easily imagine the two extreme cases: A
pure tilt and a pure
twist boundary; they are
shown below. 








Obviously, we are somewhat off the S1 position. Introducing grain boundary dislocations
now will establish the exact S1 relation
between the dislocations (and something heavily disturbed at the dislocation
cores). The DSClattice as well the CSL are identical with the
crystal lattice in this case, so the grainboundary dislocations are simple
lattice dislocations. 


Introducing a sequence of edge dislocations in the
tilt case and a network (not necessarily square) of screw dislocations in the
twist case will do the necessary transformation; this is schematically shown
below 







This may not be directly obvious, but
we will be looking at those structures in great detail in the next paragraph.
Here we note the important points again: 


Between the dislocation lines we now have a
perfect S1
relation (apart from some elastic bending). 


All of the misfit relative to a perfect S orientation is concentrated in the grain boundary
dislocations. 


We thus lowered the grain boundary energy in the area
between the dislocations and raised it
along the dislocations  there is the possibility of optimizing the grain
boundary energy. The outcome quite generally is: 

Grain boundaries
containing grain boundary dislocations which account for small misfits relative
to a preferred (low) S orientation, are in
general preferable to dislocationfree boundaries. 


The Burgers vectors of the grain boundary
dislocations could be translation vectors of one of the crystals, but that is
energetically not favorable because the Burgers vectors are large and the
energy of a
dislocation scales with Gb^{2} and there is a much
better alternative: 


The dislocations accounting for small deviations
from a low S orientation are dislocations in the DSC lattice belonging to
the CSL lattice that the grain boundary
S endeavors to assume. Why should that be so?
There are several reasons: 


 Dislocations in the DSC lattice belong to both crystals since the DSC lattice is
defined in both crystals.
 Burgers vectors of the DSC lattice are smaller than Burgers vectors
of the crystal lattice, the energy of several DSC lattice dislocations
with a Burgers vector sum equal to that of a crystal lattice dislocations thus
is always much smaller. With
S_{i}b_{i}(DSC) =
b(Lattice), we always have
S_{i}b_{i}^{2}(DSC)
<< b^{2}(Lattice).
This is exactly the same consideration as in the
case of lattice dislocations split into partial dislocations.
 A dislocation arrangement with the same "Burgers vector count" along some arbitrary
vector r produces exactly the same displacement (remember
the basic Volterra definition and the
double cut
procedure).


In other words: We can always imagine
a low angle boundary of crystal lattice dislocations that produces exactly the
small misorientation needed to turn an arbitrary boundary to the nearest low
S position and superimpose it on this
boundary. 


Next, we decompose the crystal lattice
dislocations into dislocations of the DSC lattice belonging to the low
S orientation. 


This will be the dislocation
network that we are going to find in the real boundary! 

Lets illustrate this. First we
construct another kind of DSC lattice dislocation, very similar, but
different to the one we had before.
The coincidence points are marked in blue, atoms of the two crystal lattices in
green and red. 


The plane of the cut now is perpendicular to the
boundary and extends, by necessity, all the way to the boundary. We produced a
DSC edge dislocation with a Burgers vector perpendicular to the boundary
plane (and a step of the boundary plane). 





If we were to repeat this procedure
at regular intervals along the boundary, we obtain the structure schematically
outlined below. 





In essence, we superimposed a tilt
component with a tilt angle a that for small
angles is given by is given by 







with d = spacing of the DSC
lattice dislocations and b = Burgers vector of the DSC
lattice dislocations. 

In short, we can do everything with
DSC lattice dislocations in a grain boundary that we can do with crystal
lattice dislocations. This leads to the crucial question
alluded to before: 


How do we calculate the DSClattice? As an example for
the most general case of grain boundaries in triclinic lattices? Or even worse:
For phase boundaries between two different
lattices (with different lattice constants)? 

The answer is: Use the "Bollmann
theory or "Olattice theory"  it covers (almost)
everything. 


However, unless you are willing to devote a few
months of your time in learning the concept and the math of the
Olattice theory, you will encounter problems  it is not an easy
concept to grasp. 


We will deal with the
Olattice theory in a
backbone II section, here we note that the most important cases have been
tabulated. Some solutions for fcc crystals
are given in the table: 





S 
Generation 
b from DSClattice 
1 
"Smallangle GB" 
a/2 <110>, possibly split into partials 
3 
Twin 
a/6 <112>, a/3 <111> 
5 
37° around <100> 
a/10 <130> 
9 
39,9° around <110> 
1/18<114>, 1/9<122>, 1/18<127>, 
19 
26,5° around <110> 
a/38 <116>, a/19<133>, a/19 <10,9,3> 
41 
12,7° around <100> 
a/82<41,5,4>, a/82<910>, ... 






Interestingly (and very satisfyingly), the
DSC lattice vectors belonging to the S =
3 boundary are our
old
acquaintances, the partial Burgers vectors associated with stacking faults
in the crystal lattice. 


This is but natural  a S = 3 twin boundary is after all a very close relative
to stacking faults. 

Now a question might come up:
S = 41 is not exactly a "low
S" value; and Burgers vectors of
a/82<41,5,4> appear to be a bit odd, too. So does this still make
sense? Are boundaries close to a S41
orientation still special and bound to have grain boundary dislocations? 


Only the experiment can
tell. The following TEM picture shows a S41 grain boundary (from Dingley and Pond, Acta Met.
27, 667, 1979) 