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They most conspicuous issue in the
CSL theory of grain boundaries is that there are no even values for Σ!
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Try as you might - you will never find a Σ = 2 boundary or any other even number in the literature. Now why is this?
Mostly no explanation is given. |
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A rigorous proof essentially needs the full power
of the O-lattice theory, so it can
not be easily given. But the general reason for this peculiar geometric fact
can be envisioned as follows. |
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First, remember that any grain boundary can be obtained by generating
grain II out of grain I by one rotation around a suitable axis with the
rotation angle γ. |
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This means that we can produce all CSL
orientations by looking at one rotation. We
will do this for a square lattice, rotating around a <100>
axis. |
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It is, however, not obvious that we can indeed
produce all possible boundaries by this rotation, nor is it clear that the
result will be valid for grain boundaries in non-cubic crystals. But it shows
the direction of the argument. |
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From all possible rotation, some will produce
CSL structures. Which ones will do that is easily conceived: |
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The picture below shows a blue crystal I.
Taking its origin at the apex of the blue triangle on the right, we see that we
always will get a CSL orientation if we look at lattice points with the
coordinates (x, y0) which we may express
as (n, 1) if we set x0,
y0 = 1, and than rotate the crystal so the the
y-coordinate changes from 1 to +1. The shift
is indicated by the bold brown vector; we need to rotate an angle γ
given by |
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γ |
= ½ cotg |
y
x |
= 2 · cotg |
1
n |
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The red lattice has been rotated by just the
right amount to move the point (3, 1) to the position (3,
+1); the rotation center is in the middle of the crystals |
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With this procedure we created the yellow
CSL lattice. |
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Its Σ' value is
given by its area divided by the are of a unit cell of the lattice; we
have |
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Σ' |
= |
(x2 + y2)2
x0 · x0 |
= |
(3x0)2 +
(1x0)2)
x02 |
= (32 + 12) = 10 |
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Its easy to generalize for CSL sites
generated by moving the point (nx, y) on the
(nx,+y) position, we obtain for the Σ' values |
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The result will be |
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Σ' is an
odd number, if n is an even
number (The square of an even number is even plus 1 = odd)
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Σ' is an
even number, if n is an odd
number (The square of an odd number is odd plus 1 = even.) |
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So we can get even and odd numbers for Σ????. |
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Yes - but upon inspection you will find that for
n = odd, there is always an
additional coincidence point in the center of the lattice defined by the
CSL points produced by the rotation, while for even numbers of
n this is not the case. |
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In the picture above this are the green points,
and the lattice constant of the CSL lattice is now smaller. The Σ value in this case is simply |
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Σ |
= |
n2 + 12
21/2 · 21/2 |
= Σ'/2 = an odd number |
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Instead of a Σ =
10 boundary, we generated a Σ = 5
boundary and there are no even Σ values.
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q.e.d.
(sort of) |
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This, of course, is a far cry from a real
mathematical proof, but it imparts the flavor of the thinking behind it. |
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To complete this issue, the following picture shows the result
for a rotation that tranfers (2, 1) to (2, +1) |
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There is no additional coincidenc point and we end up with a
Σ = (n2 + 12) =
5 boundary, the same one as above |
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© H. Föll