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In the CSL model of grain boundaries the
DSC lattice was introduced to
account for small deviations from a perfect lattice coincidence orientation. It
was the lattice of all translations of one of the
crystals that conserved the given CSL. Translations other
then those of the DSC lattice would destroy the coincidence of lattice
points. |
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The lattice vectors of the DSC lattice therefore could
also be interpreted as the set of possible Burgers vectors for dislocations
allowed in a grain boundary without destroying the coincidence. |
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While the simple recipe for constructing the DSC
lattice in the simple cases usually shown (two-dimensional, cubic lattices) is
rather straight forward, it was
neither mathematically justified, nor is it immediately clear how it should be
constructed in complicated cases. |
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The DSC lattice, in fact, comes from the
O-lattice theory and was simply adopted to the "easy"
CSL model. |
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Obviously we now
must ask ourselves: What happens to an O-lattice, particularly a periodic one, if
we translate one of the crystals? |
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This is actually one of the more complicated questions to ask,
especially for the rank of the transformation matrix A < 3 (as we
expect for grain boundaries). |
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We will not go into details here because in
this rendering of O-lattice theory we omitted some more mathematical
points considering what happens to the O-lattice in a given situation if
you shift (= translate) crystal I or crystal II. Or, in a
reversed situation, how you must shift crystal I or II if you
translate the given O-lattice. |
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The first answer to the question
above is: |
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In general (i.e. rank A = 3), the O-lattice is preserved, but shifted by
some amount that depends on the (arbitrary) magnitude of the translation of the
crystal chosen.This is in contrast to the
CSL, where arbitrary shifts not contained in the DSC lattice will
completely destroy the CSL. |
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This does not help, we obviously must find a more specific
criterion than just conservation of the O-lattice in general in order to
find specific translations that correspond to Burgers vectors of grain boundary
dislocations. We therefore ask more specifically: |
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What happens to the
pattern
elements associated with every equivalence point in a reduced
periodic O-lattice upon shifting one
of the crystals? |
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Since we have seen (without proving it) that any
O-point can be taken as the origin for the rotation transforming crystal
I into crystal II; we should be able to shift lattice I by
any vector pointing to an equivalence point in the reduced O-lattice
without changing pattern elements. In other words we simply change the origin
of the rotation (we only look at rotations in this examples). |
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The O-lattice then will also be shifted by some other
vector which can be calculated by employing our
basic equation
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The ri are the base vectors of the
O-lattice if we take Ti to be the set of base
vectors of the crystal I lattice. |
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Now shift the crystal I
lattice by some vector e connecting equivalence points, replace
ri by ri =
r0i + Dri, with Dri = shift of the O-lattice
for a shift e of the crystal lattice, and solve the equation for
the Dri.
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Well, lets not do it, but
accept that there is a shift that can be calculated. |
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On second thoughts, this
must also be true for lattice II. We thus may also employ vectors that
translate lattice II by one of the vectors pointing to equivalence
points in the reduced O-lattice. |
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And on third thoughts (not
entirely obvious), we also must be able to translate the O-lattice
itself by any vector that connects equivalence points. This requires that the
O-lattice shifts by some vector - it is the reverse problem from the one
outlined above. |
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The trick is that all those shifts
may be different, and while they all produce the same general O-lattice,
there might be different pattern
elements. But - there is a finite
number of pattern elements and a finite
number of possible shifts. |
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Obviously, the set of all different configurations
(distinguished by pattern elements) obtainable defines the complete geometry of
the particular boundary with the periodic O-lattice considered because
no configuration is special. |
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The set of all possible displacement vectors can be expressed
as the translation vectors in a new kind of lattice, the "Complete Pattern Shift
Lattice", abbreviated by Bollmann as " DSC lattice", that we
encountered earlier (in a much
simpler form). |
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Unfortunately, it is not immediately
obvious how to calculate the DSC lattice from O-lattice theory.
In fact, the respective chapter in Bollmanns book is particularly
hermetic or obtuse. |
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Somewhat later (1979), Bollmann together with
Pond gave the old
abbreviation a new meaning: "DSC" now stands for
"Displacements which are
Symmetry Conserving".
But few people know what exactly DSC stands for - the main thing is to
understand the significance of the DSC lattice. |
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Lets see what the various
displacements discussed above really produce if applied to a simple situation.
We take the (redrawn) example from Bollmanns book. |
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First, lets construct the possible set of pattern
conserving translations by putting several reduced O-lattice cells
together (for the case of rotation around <100> of
39o 52,2´, corresponding to the S = 5 CSL). |
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The left part shows the rotation, yielding the
O-lattice. Coinciding lattice points
that are also O-points are shown in green,
the other O-points in red. On the
right-hand side the repeated reduced O-lattice is shown in the blue
crystal. |
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Now lets displace the brown lattice by a vector
pointing form the green to the red equivalence point in the above picture. Here
is what you get. |
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The O-lattice shifted down, and some new kinds of
pattern elements appear. There are no more or less special than the ones in the
picture above; both belong to the complete structure of the boundary
illustrated. |
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Note that we also obtain new equivalence points for the
boundary (in the middle of the lines defining the square lattices). |
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Now we shift the brown lattice by one of the
vectors pointing to the new equivalence point. We obtain yet another pattern
element. |
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But that's it. The pattern elements shown here are all there
are (Try to prove that yourself if you don't believe it). |
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We could now start to produce the DSC lattice, but this
will just give the same kind of
lattice we had in the simple CSL case. |
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Instead we only note that there is a sufficiently
clear procedure of how to create a DSC lattice for a given periodic
O-lattice, that is always applicable
- even to phase boundaries (in principle; of course only in principle). |
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In the next (and last) chapter, we will show how
O-lattice theory now can be applied to large angle grain boundaries and
discuss briefly its merits and limits. |
© H. Föll (Defects - Script)