
The math and physics of the Olattice is
not particularly easy because there are some tricky details to keep in mind. In
this paragraph some of the problems, tricks and helpful definitions are just
summarized; in due time they may be specified in more details. 

It is useful in many cases to decompose the
transformation matrix A into
matrices that describe the volume
deformation (elongating or shortening only the axes of the crystal),
the shear deformation (only changing the
angles between the axes), and the rotation
of the coordinate system of crystal I separately. 


This may allow a better grasp of the real situation and helps,
if necessary, to use approximations only for suitable parts of the system. 


The main reason, however, lies in the fact that the pure rotation is not unambiguously defined.
Depending on the basic symmetries of the system, the same final state of
orientation can be obtained by many different rotations  but only one (or one
set) may make sense physically. This leads to the next point: 

The choice between various possible
transformations A. There are many
possible ambiguities, not only with respect to the rotation part, but also,
e.g., in the relations between pure shear and pure rotation; an example is
shown below. 


Starting from a given lattice I, identical lattices
II can be produced either by pure shear or by pure rotation: 




Mathematically, there is no difference, but
physically the two transformations are not
the same because the atoms involved have to move in quite different ways. Which
one is the physically sound one? As it turns out, the criterion is to preserve nearest neighbor relationships. 


Mathematically, this means that from all
possible transformation matrices T, the particular one that has to be chosen is the one
with the smallest numerical value of its determinant
T. 


This ensures that the unit cell of the Olattice
generated will have the largest possible
value (it is directly given by 1/T), which will give the smallest possible
dislocation content. This requires, of course, that you know all the possibilities for A in the first place  not a satisfying
condition for a mathematician. 


It may be noted in passing, that this ambiguity limits the usefulness of the
Olattice theory. There are cases, where the choice of the
transformation matrix following the rules of Olattice theory, does
not lead to the "correct"
solution as ascertained by looking at what the crystal does (by
TEM). 

Another generalization comes from looking at the
essentials of solutions to matrix equations. Consider the solutions of the
basic equation 





From basic
matrix algebra we know that the type of
solution depends on the
rank
of the matrix A. 


We have the following cases:
 Rank (A) = 3
The solutions define points in
Ospace, i.e. an Olattice.
 Rank (A) = 2
the solutions are Olines.
 Rank (A) = 1
The solutions are Oplanes.
 Rank (A) = 0,
we have the trivial case of identity
1.


This is an issue of prime
importance!. 


Since we can produce all grain boundaries (but not all phase boundaries) by just rotating crystal
II around one properly chosen axis,
the rank of the transformation matrix does not have to be larger than
2. 


What does this mean? Well 
for grain boundaries, there is no such thing as a Opoint lattice
 it is rather a lattice of lines. We have
essentially a twodimensional problem. 


Nevertheless, for the sake of generality, we will continue to
discuss the "Opoint lattice", knowing that it often is just a
line lattice. 

Solving the basic equation produces the
Olattice and therefore also the unit cell of the Olattice.
However, the "natural" unit cell obtained by simply connecting
Olattice points in some "obvious" manner may not be the
physically most sensible one! 


As taught in
basic
crystallography, there are many ways of defining unit cells  we have
another ambiguity! 


We will tend to take the WignerSeitz cell. Why?
Who knows at this point  just go along. What this means is illustrated
below: 





Again, the right choice must come from the
physical meaning of the Olattice. This we will discuss in the next
paragraph. Here we note that Olattice defined in this way resembles
nothing so much as a honeycomb  just
remember again, that the Opoints are lines. An illustration that comes fairly close in
a slightly different context ("Bollmanns view of Franks formula") can
be accessed via the link. 