
Periodic Olattices are
clearly special; and it is selfevident that every CSL orientation must
correspond to a periodic Olattice. But there
is more. 


At any Olattice
point in a periodic Olattice, we have a certain arrangement of the
crystal atoms around that point, a specific pattern. Since in a periodic Olattice
there are only a finite number of different equivalence points, there is only a
finite number of distinct patterns, too. 


An individual
pattern is called a pattern element. There
are as many pattern elements as there are equivalent points in the reduced
Olattice. 


This is a crucial concept in
Olattice theory, unfortunately it is not explained very well in
Bollmanns book. Let's see what is meant by pattern: 





Shown are two lattices (blue and
magenta which are superimposed) and one Opoint (red). A representation of the geometry of the atoms that you
may put into the lattices is given by the yellow triangles. They are simply
constructed by connecting the lattice points of the two lattices
"around" the Opoint with the Opoint. 


The picture also demonstrates (but
does not prove) an universal theorem: Any
Opoint can be chosen as the origin
for the transformation that produces lattice 2 from lattice 1
(here it is a simple rotation). 

In a nonperiodic Olattice,
the representation of patterns in the way shown is different at any
Opoint  this is also rather difficult to draw. 


This is where Olattice theory
gets hard to illustrate. Nobody surpasses Bollmann who provides complicated
drawings of patterns (done by hand!) in his book,
one example is shown in the
link. 

The question now is: Which
orientations provide periodic
Olattices? It appears that there is no simple formula coming up with
transformation matrices or angles for rotations that produce periodic Olattices. We have to go the other
way and ask two questions for any possible
orientation: 


Is the corresponding Olattice
periodic? 


If yes, how many pattern elements
(= N) are contained in the reduced Olattice? 

What we want is N as a
function of some misorientation angle for some simple geometries. This needs
some numerical calculations; lets look at the results for rotations on the
{110} planes of cubic crystals 


The following picture shows N
as a function of the misorientation angle: 





This again is one of Bollmanns trickier
pictures (with some color added), because it is only understandable if you read and understood much
of what has been said before in his book (it is neither explained what the
difference is between the two curves  they have after all an identical
coordinate system, nor what the bold lines (here dark blue) implicate). 


Well, the Nvalues are given
for two independent kinds of
transformations which both include the same rotation T (upper and lower curve), but one
inlcudes a socalled "unimodular transformation" in addition. The one
with the smallest determinant which, as
we have seen, is the one you should use, changes from the upper curve to
the lower curve at T = 30^{o}
and this explains the bold (or dark blue) lines. Since the two curves are
different, you now see that it matters, indeed, which transformation matrix you
pick. 


Don't worry; it is not necessary to
understand that in detail. Just acknowledge, that N can be computed and
that unambiguities with respect to different transformation matrices can be
dealt with somehow. 


Also note that the "real"
curve would be a fractal with N = ¥ for
most values; it is smoothed here by only counting the equivalence points in
100 "pixels" of the Olattice (so N
³ 100 applies) and stopping the numerical
procedure after some time if it does not turn periodic anyway. 

It is clear that there are several
"special" orientations for this geometry with small values of
N. This looks good, however, we are not yet done. We are really looking
for Olattices that are periodic on a short scale, i.e. the patterns
should repeat after a short distance. This requires three ingredients: 


Periodicity as a starter, i.e. N is finite
(or in reality e.g. N < 100 for numerical calculation). 


Small
values of N, because the pattern repeats after N steps
 the larger N, the longer it takes for a repetition. To give an
example: For N = 10 you have to go out 10 lattice constants of
the Olattice before the same pattern is encountered again. 


This immediately calls for small lattice constants of the Olattice,
too. Or, to be more general, for small volumes
V_{O} of the Olattice cells. 

The real measure for
the periodicity of the Olattice patterns is therefore not N, but
the density N' of periodic
equivalence points given by 





With T meaning the determinant of the transformation
matrix, since V_{O} = 1/ T follows from basic
matrix algebra together
with the definition of the
Olattice. 