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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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Franks formula
relates B, the sum of all the specific Burgers vectors
bi cut by a vector r lying in the plane
of the boundary, to the angle α with which
one of the crystals is rotated with respect to the other one around the polar
unit vector l. It is valid for small angles (say α < 10o) and given by
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Note that we do not need three angles of rotation as required for
a general grain boundary because we do not rotate around the axis of a
coordinate system, but around the polar
vector l. |
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Note also that the grain boundary plane (and thus
r) is not required to be
perpendicular to l. r thus can have any direction and length relative to
l. |
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For the derivation of Franks formula
we consider a small angle grain boundary formed by rotating crystal 1
around an arbitrary axis l by α
and thus forming crystal 2. After that we join crystal 1 and
crystal 2 on any plane. |
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A vector r1 in the plane of the grain
boundary (to be) in crystal 1 thus gets transformed to a vector
r2 in crystal 2. Note that
r1 does not have to be perpendicular to
l. |
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Next, we make a Burgers circuit in the system
with the small angle grain boundary and a reference circuit in the perfect
crystal 1 (or crystal 2). We will move along a vector
r1 that is much longer than a lattice constant
or the spacing of the dislocations that will make up the boundary. |
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In the perfect lattice we will start from the endpoint of
r1 and move to the start of
r1 in an e.g. counter-clockwise direction. In the
crystal with the grain boundary, we do the same circuit, except that as soon as
we switch over to grain 2, we follow r2. |
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The whole procedure can be illustrated as
follows: |
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There will be a closing failure B which must be
identical to the sum of the Burgers vectors of all the dislocations contained
in the circuit. Only the components of the
b´s lying in the plane perpendicular to l are
counted, of course. |
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For clarity, the vectors r are at right angles
to l in the drawing, but this is not generally necessary. |
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From
vector calculus we know
that a rotation can be described by an axial vector given by R =
l · α. |
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The difference vector B between the two vectors
r1 and r2 (with
r2 produced from r1 by the
rotation) than can be written as |
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- and this is Franks formula from
above. |
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Note that there are two
approximations in this. First, we assume small
angles so that sin(α)
» α; and
secondly, in the same vein, we assume r1» r2 = r.
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Of course, we also assume that there is a smooth cross-over at
the boundary (or that r is so large that to give or take parts of
a lattice constant doesn't matter). |
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This is a simple formula, but like
most vector formulas, it has some hidden power. Before we look into the power
of Franks formula a little more closely, we will consider what it cannot do: |
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The formula gives the net content of Burgers vectors in a small angle
grain boundary, but not necessarily the arrangements
of the dislocations. It does not, of course, say anything about
possible splitting into partial dislocations either. This means that there
might be several arrangements of dislocations with the same B.
The one that will be observed will be (most likely) the one with lowest total
energy. |
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No elastic distortion is considered. Between the dislocation
the lattice is perfect; elastic distortion is present only in the core regions
of the dislocations. |
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Bearing this in mind, lets look at
some special cases. Since Burgers vectors are translation vectors of the
lattice, in general three sets of
non-coplanar dislocation will be required to produce the vector
B. Special cases therefore
are boundaries where only one or two sets of dislocations are needed. |
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If we have a boundary where one set of dislocations with Burgers vector
b1 is sufficient, B can be written
as |
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With N = number of dislocations cut by
r |
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This obviously, looking at Franks
formula, requires b to be perpendicular to r and
l. |
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The direction of r in the plane of the boundary
is arbitrary; this means that b must be at right angles to the
plane of the boundary or parallel to the normal n of the boundary
plane and l must be at right angles to n; it
follows that l must be contained in the boundary. |
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If we now chose the particularly simple case of r =
rp being parallel to l, we obtain
(rp × l) = 0, which means that no
dislocations are intersected by rp, implying that the
dislocation lines must be parallel to the rotation axis l. |
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This leaves room only for the
conclusion that a boundary with only one set of
dislocations must be a pure tilt boundary. |
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The spacing of the dislocations is obtained if we take
r = rra at right angles to l thus
intersecting the dislocations lines at right angles, too. In this case we can
write rra as rra = r ·
(l × n) and obtain |
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N · b |
= |
α · (r ×
l) |
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= |
α ·r · [(l ×
n) × l] |
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= |
α · r · n |
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With b = b · n and the spacing
d between the dislocations given by d = r/N, we obtain for the
spacing dtilt of dislocations in a pure tilt boundary with
the boundary plane at right angles to the Burgers vector the relation
used before: |
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Similar considerations, which are
straight forward but quite involved, can be made for the case of small angle
grain boundaries with two sets of
dislocations and the possible subsets (e.g. Burgers vectors in the
plane of the boundary for pure twist boundaries). |
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For this and more, Hull and
Bacons book can be consulted, which treats these cases in detail. |
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More important in the development of
boundary structure theories is Bollmanns
interpretation of Franks formula; which is the starting point of the
O-lattice theory as will be
discussed in the link. |
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© H. Föll