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Note: For ease
of writing /reading in this module, variables are not in italics;
instead vectors are underlined |
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Frank's
Formula Reconsidered |
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Franks formula relates the sum
B of all Burgers vectors cut by a vector r (which
is required to be in the plane of the boundary) to the (small) rotation angle
α around an arbitrary polar vector
l that generates the second crystal from the first one. It
states: |
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Franks formula at this point is a
continuum equation, it gives a value of
B for every α and
r |
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Burgers vectors, however, are discrete. This requires the vector B
to be discrete, too. |
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Since Burgers vectors are translation vectors of the lattice,
B can only be a sum of Burgers vectors. |
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If l is a lattice vector so that the
"b-plane", the plane perpendicular to l
that contains the possible Burgers vectors, is a lattice plane, too (i.e. it
can be indexed with {hkl}, with h, k, l = integers). It contains
lattice points that define the possible Burgers vectors in this plane. |
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Note that the Burgers vectors defined in this
way must not necessarily be the shortest possible Burgers vectors
bmin, i.e. the Burgers vectors of real dislocations.
It is, however, always possible to decompose the b vectors of the
b-lattice into e.g., a/2<110> type Burgers vectors
of the fcc lattice. This may require
bmin-vectors that are not contained in the b- plane - but
all we have to do then is to imagine the net of b-vectors in this
plane to be "puckered" as shown below. |
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In the plane of the boundary, an
arbitrary r would intersect the projection of the
b-lattice onto the boundary plane along the
l-direction. In a schematic view we have the following
situation: |
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Franks formula can now be understood
as a discrete imaging of points in a two-dimensional
"Burgers vector space" onto a plane in real space. |
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The Burgers vector count along r (after
translating it to smallest possible vectors bmin)
gives the number of dislocations that are found if going along r
in the boundary plane. If even spacing is assumed, we also know the spacing in
the particular direction given by r. |
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Now comes something new. Remember
that Franks formula did not make any statement about
the arrangement of the dislocations, or - in other words - their line
direction. |
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Bollmanns view is different: The line direction of the
dislocation is obtained by probing the whole two-dimensional grain boundary
space by sweeping r around. What happens then can be understood
in purely geometrical terms, as we will see below. |
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First of all it is important to
realize that the crossing of a dislocation by the "probing" vector
r in the b-plane is directly imaged by the
Moirée pattern of the superimposed
two crystals obtained by rotating the {hkl} planes perpendicular to
l on top of each other by α as
shown below for three different αs with
pictures from Bollmanns book. |
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In the whitish (bright) areas, there is a high
degree of coincidence of lattice points, whereas in the black areas the misfit
is largest. These are of course the "O-points" in the full
O-lattice theory. |
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Whenever a vector from the origin crosses a black
area to reach a whitish area again, the translation relative to an equivalent
vector in the other lattice is just a lattice vector of the underlying plane,
which is the b-plane in our definition. In other words, if you
move to the same white area in crystal I and crystal II, the two
vectors are on top of each other. But their tips would be separated by just a
lattice vector if you now rotate the crystals back to a no-boundary
situation. |
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If the crystal now introduces a boundary, it will
increase the whitish areas, the areas of best fit, and concentrate the misfit
in the black areas - which correspond to the dislocations. A periodic structure
results which we can describe as a lattice - the (2-dimensional)
b-lattice. |
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Since the Moirée pattern does not depend
on the position along l, we can extend the
b-lattice along the l direction and obtain a
3-dimensional structure with lattice lines instead of points. If we
enclose the lattice points (respectively lines) of the b-lattice
in Wigner-Seitz cells, we obtain a kind
of honeycomb structure. |
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The decisive point
now is that the boundary plane, which can
have any position relative to
l, intersects this "honeycomb" b-lattice
somehow, for an arbitrary case we obtain the following picture (again taken
from Bollmanns book) |
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The b-lattice consists of the
yellow lattice points. It is turned into the three-dimensional
"honeycomb" lattice by introducing Wigner-Seitz cells (blue lines)
and continuing it along the l direction (magenta arrows).
The lines would be the
O-lattice in the full theory. |
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An inclined boundary plane will have a
dislocation wherever the boundary plane intersects the honeycombs. The
resulting dislocation network is shown with red lines. The points of best fit
(red points) are in the center of the network as it must be. |
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The final network may still be different because
the Burgers vectors of the dislocations now defined by the "red
lines" might be too large and decompose, as pointed out
above. |
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The final interpretation now is as
follows: |
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Wherever the boundary plane
intersects a cell wall of the (three-dimensional) b-lattice, we
have a dislocation with the Burgers vector as defined by the translation
vectors in the b-lattice. The lines defined by the intersection
of the boundary plane and the cell walls then directly define the dislocation
lines - we get a direct rendering of the dislocation network in the boundary.
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Of course, the geometry of the
dislocation network obtained in this way depends on the kind of unit cell we
chose for the "honeycomb" b-lattice. Wigner-Seitz
cells, while universal, may not be best choice possible. But it is always
possible now to "develop" the network obtained to a
network with minimum energy by using the
rules of dislocation interaction as in the example with the small angle twist boundary
on a {111} plane. |
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These and other complications need
more considerations. However, remembering that Franks formula is an
approximation and covers only small angle grain boundaries, it is not worth the
effort to improve this limited theory. It is a better at this point to unleash
the full power of O-lattice theory
which contains Franks formula as a special case. |
© H. Föll