
The relation between the spacing of the
dislocations and the tilt or twist angle in the special cases given was simple
enough  but what about arbitrary small angle grain boundaries with twist and
tilt components? What kind of dislocation structure and what geometry should be
expected? 


As we have seen, the detailed structure of the network can be
quite complicated and depends on materials parameters like stacking fault
energies. We can not expect to have a simple formula giving us the
answers. 


The relation giving the distance between dislocations in a boundary and the
orientation relationship for arbitrary lowangle orientations (meaning that the
two rotation angles needed for a general description are both small, lets say
£ 10^{o}  15^{o}) was
first given by Frank.
It is Franks formula referred to
before. 

Franks formula is derived in the advanced
section, here we only give the result. The lowangle grain boundary shall be
described by: 


Its dislocation network consisting of dislocations with
Burgers vectors b. 


An arbitrary vector r contained in the
plane of the boundary. 


A (small) angle a around an
arbitrary axis described by the (unit) vector l (then one
angle is enough) that describes the orientation relationship between the
grains. We may then represent the rotation by a polar vector R =
a · l 

Franks formula then is: 





with B = sum of all the specific Burgers
vectors b_{i} cut by r ; i.e.
B = S_{i}
b_{i}. 


Since the formula is formally applicable to any boundary, but
does not make much sense for large angles a
(can you see why?) we only consider lowangle boundaries. Then we can replace
sin a/2 approximately by a/2 and obtain the simplified version 




Let's illustrate this: 





Shown are two vectors r_{1} and
r_{2} contained in a boundary plane with an arbitrarily
chosen dislocation network consisting of two types of dislocations having
Burgers vectors b_{1} and
b_{2}. 


Franks formula ascertains that (r ×
l) · a equals the sum of the Burgers
vectors encountered by r, i.e. B =
2b_{2} + 3b_{1} for
r_{1}, and B =
3b_{1} for r_{1} in the
picture above. 


This is a major achievement, but not overly helpful when you
try to find out the geometry of the network for some arbitrary boundary,
because their is no simple and unique way of decomposing a sum of Burgers
vectors into its individual parts. 

This "simple" formula, however, contains
the special cases that we have considered before, and leaves enough room for
complications. It does not, however, say anything about preferred planes or network geometries. For this one
needs the full power of Bollmanns
Olattice theory. 