
Lets pretend we are considering an actual grain
boundary. We have found a suitable transformation matrix that produces crystal
II out of crystal I with the right orientation, we have solved
the basic equation, and we have constructed a suitable Olattice. What
does that give us? 

We now must address the essential question: What
is the significance of the Olattice for grain and phase boundaries?
What is the physical meaning? There is an easy answer and a difficult implementation: 


First of all, the Olattice in
itself has no physical meaning whatsoever  in this it is
exactly like the CSL. 


However, since it always
exists (unlike the CSL) and is
defined in both crystals, if you were to design a boundary between two crystals of given orientation (and thus with
one welldefined Olattice) that intersects as many Olattice points as
possible, you will obtain the best physical fit along the boundary,
i.e. probably the lowest grain boundary
energies. 


"Best physical fit" is not a very quantitative way
of putting it. It means that the atoms to the left and right of the boundary
will not have to be moved very much to the positions they will eventually
occupy in the real boundary. This also can be expressed as "minimal strain"
situation; the expression Bollmann uses. 


If atoms happen to sit on an Olattice point, they do
not have to move at all because then then occupy equivalent positions in both
crystals; if they are close to an Olattice point, they only move very
little, because at the Opoints the fit is perfect. 


The misfit increases moving away from an Olattice
point and reaches a maximum between Olattice points. 

The crystals then can be expected
to increase the area of best fit between Olattice points and to
concentrate the misfit in the regions between
Olattice points  this will be a dislocation with Burgers vector = lattice vector. We cannot, at this
stage produce grain boundary dislocations, i.e. we are still limited to
small angle grain boundaries. 


There is a direct important consequence from this for the
basic equation: We can replace T(I) by b(I),
the set of possible Burgers vectors because they are always translation vectors of the lattice and
obtain 


(I  A^{–1})
r_{0} = b(I) 




Remember that all translations vectors of the lattice are
possible Burgers vectors; this came straight from the
Volterra
definition of dislocations. The fact that observed Burgers vectors are always the smallest
possible translation vectors does not interfere with this statement  all it
means is that a "Bollmann" dislocation with a large Burgers vector
would immediately decompose into several dislocations with smaller vectors.


Our basic equation yields the
base vectors of the Olattice
if we feed it with the base vectors, i.e. the smallest possible translation
vectors, of the crystal lattice. Since the Burgers vectors in a given lattice
are pretty much the smallest possible translation vectors, too, we may see the
Olattice as some kind of transformation of the blattice, the
lattice defined by taking the permissible Burgers vectors of a crystal as the
base vectors of a lattice. 


The crucial point now is to
realize that the lines of intersection of the the actual plane of the boundary
with the cell walls of the Olattice (which,
remember, looks like a honeycomb)),
are the dislocations in the grain boundary.
Whenever we cross over from one cell in the honeycomb structure to the next, we
moved one Burgers vector apart in the real
lattices. It is helpful at this point, to study the case of a small angle grain
boundary treated in the advanced section under
" Bollmanns view of Franks
formula"; the essential picture is reproduced below. 




The magenta lines are the
Olattice lines; the
honeycomb structure is shown in blue, and the intersection with an arbitrary
boundary plane produces the red dislocation network. 


This is why it becomes important what kind of unit cell we
pick for the Olattice as
mentioned before.
As always, there are many
possible choices. 


Bollmann gives precise directions for the choice of the
"right" unit cell of the Olattice  simply take the largest
one possible (producing as few dislocations as possible). We will not reproduce
the mathematical arguments; here we just note that it is possible to define an
optimal Olattice. 