**Application of O-Lattice Theory to
Large Angle Grain Boundaries**

The basic assumption is that an arbitrary grain
boundary would prefer to be in a orientation corresponding to a periodic O-lattice with a
high density of equivalence points.
This is a fancy (i.e. more general) way for saying that boundaries prefer to be
in a low S orientation as we did in the
simple CSL model. |
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Such a particular orientation can be achieved at the expense
of generating some grain boundary
dislocations. |
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For a small angle grain boundary
we have seen that a cut of the
boundary plane through the O-lattice gives directly the geometry of the
dislocation network (give or take some adjustments to account for the
particular dislocation properties). |
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The O-lattice was obtained (by
calculations) relative to the two real crystals. Looking back, what we did was
to use one of the crystals as a reference for the
preferred state, the other one then described the deviation of the boundary from the preferred
state. |
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For large angle grain boundaries we
now do exactly the same thing - except that
the reference state is now the periodic
.O-lattice that the boundary aspires to obtain |
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The deviation of the
boundary from this preferred state is then described by the O-lattice
that comes with the orientation that describes the actual boundary . |
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The logical
consequence then is that the geometry of the dislocation network necessary to
obtain the preferred state is the described above - a so-called
O-lattice of
the two O-latticesO2-lattice or second order O-lattice. |
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That may sound a bit heavy, but it is really straight forward if you think about it. | ||

It is also clear - in principle - how we would calculate the
O2-lattice, but we are not going to look at this. |
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If we now imagine a boundary plane
cutting through our O2-lattice as
before, we now must ask how large the translation will be that the
O-lattice "crystals" experience when a O2-lattice wall
is crossed. This translation will be the Burgers vector of the second-order
dislocation forming the grain boundary dislocation network. |
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Well, as you would have guessed, it must be a
translation that conserves the underlying pattern of the O-lattices, so
the translation vector (= Burgers vector) is a vector from the
DSC-lattice of one of the primary O-lattices. |
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While our periodic reference O-lattice has
a defined DSC lattice, the other one may (and in full generality
probably will) have a non-periodic O-lattice and thus does not have a
DSC lattice. This looks like a
problem. |
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However, since O-lattices are continuous
(and smooth) functions of the misorientation angle (which the CSL is
not), we know that the two O-lattices are rather similar, and we always
can take the DSC lattice of the periodic O-lattice in a good
approximation. So there is no real problem. |
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OK. We are done. That's (almost) all there is to it. | ||

The general recipe for constructing a grain boundary with a secondary is
network is "clear". It goes exactly along the lines we derived for
small angle grain boundaires - only we work in "second order
O-lattice theory". |
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The reverse is also possible: We have a general
recipe for analyzing the structure found in
a real boundary. |
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It will just take a few months of studying the intricacies of the underlying math and some getting used to the more trickier thoughts, and you can construct and analyze all kinds of boundaries on your own. | ||

But most likely, you won't. This is due to some (sad) facts of life that will be the last thing to discuss in this context. | ||

Merits and Limitations of O-Lattice Theory |
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The merits of Bollmanns theory are clear: | ||

It nucleated a lot of work on grain boundary structures and
introduced the crucial concept of the DSC lattice and its
dislocations. |
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It was (and is) the mathematical frame work for tackling the
kind of higher-level geometry that is contained in interfaces between crystals.
(It will be interesting to see if someone sometime tackles the grain boundary
structure between single quasi-crystals, which could be done by extending
O-lattice theory into a
6-dimensional
space and then project the results back into three dimensional space). |
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It allows to conceive and analyze more complex problems, where
a CSL model is not sufficient. |
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However, there are serious problems and limitations, too. | ||

The recipe for the proper choice of the one transformation matrix you should use out of many
possible ones is not always correct. It
generally fails for (some) small angle tilt boundaries, where the
O-lattice theory would predict a twin-like structure with no
dislocations - contrary to the observations. It also fails for some other
boundaries, casting some lingering doubt on the whole thing. |
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It is still too simple to
account for real boundary structures even if the limitation referred to above
does not apply. Two examples might be mentioned. |
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1. The rigid body
translations observed in many (twin-like) boundaries, especially
in bcc lattices. |
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2. Tremendously complicated structures observed in
crystal with more than one atom in the base of the crystal - e.g. in Si.
What happens (and was first observed and then analyzed by Bollmann) is that
dislocations in the DSC lattice may split into partial dislocations
bounding a stacking fault in
the DSC lattice. While this effect may be incorporated into the
O-lattice theory, it does not make it easier. |
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Still, whereas newer theories
concerned with the structures of grain boundaries do exist, none is quite as
complete and mathematical as the O-lattice theory. A "final"
theory has not yet been proposed |
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What remains for practical work is | ||

The .
This is certainly the most important outgrowth of the DSC latticeO-lattice theory.
Grain boundaries simply cannot be discussed without reference to the DSC
lattice. For practical importance it has all but eclipsed the O-lattice.
As we have seen, it is (mostly)
easily constructed without going into heavy matrix algebra. |
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The systematic approach, always good for looking deeper into less clear situations. | ||

The good feeling, that something can be done about taking a deep look into grain boundary structures from a theoretical point of view, even if there are some limitations and unclear points. |

© H. Föll (Defects - Script)