Any point x within a crystal may be represented as the sum of a lattice vector u (which has
integral components) and a small vector β whose components are fractional and less than
unity. The internal co-ordinates of the point x are then defined to be the components of the
vector β. The O–lattice method takes account of all coincidences, between non-lattice sites of
identical internal co-ordinates as well as the coincidence lattice sites.
All lattice points in a crystal are crystallographically equivalent and any lattice point may be
used as an origin to generate the three dimensional crystal lattice. To identify points of the
CSL we specify that any lattice vector u of crystal A must, as a result of the transformation
to crystal B, become another lattice vector x of A – i.e. x = u + v where v is a lattice vector
of A. The CSL point x can then be considered to be a perfect fit point in an interface between
A and B, because it corresponds to a lattice point in both crystals.
Non–lattice points in a crystal are crystallographically equivalent when they have the same
internal co-ordinates. To identify O–points
64,65
, we specify that any non-lattice vector x of
crystal A must, as a result of transformation to crystal B, become another non-lattice vector
y of A such that y = x + v where v is a lattice vector of A; the points x and y thus have
the same internal co–ordinates in A. The O–point y can then be considered to be a perfect fit
point in the interface between A and B. Note that when x becomes a lattice vector, y becomes
a CSL point. The totality of O–points obtained by allowing crystals A and B to notionally
interpenetrate forms the O–lattice
64,65
, which may contain the CSL as a sub-lattice if A and B
are suitably orientated at an exact CSL orientation. Any boundary between A and B cuts the
O–lattice and will contain regions of good fit which have the periodicity corresponding to the
periodicity of a planar net of the O–lattice along which the intersection occurs. Boundaries
parallel to low-index planes of the O–lattice are in general two dimensionally periodic with
relatively small repeat cells, and those with a high planar O–point density should have a
relatively low energy.
Consider two crystals A and B which are related by the deformation (A S A) which converts
the reference lattice A to that of B; an arbitrary non-lattice point x in crystal A thus becomes
a point y in crystal B, where
[A; y] = (A S A)[A; x]
If the point y is crystallographically equivalent to the point x, in the sense that it has the same
internal co-ordinates as x, then y is also a point of the O–lattice, designated O. This means
that y = x + u, where u is a lattice vector of A. Since y is only an O–point when y = x + u,
we may write that y is an O–lattice vector O if
64,65
[A; y] = [A; O] = [A; x] + [A; u] = (A S A)[A; x]
or in other words,
[A; u] = (A T A)[A; O] (30a)
where (A T A) = I − (A S A)
−1
. It follows that
[A; O] = (A T A)
−1
[A; u] (30b)
By substituting for u the three basis vectors of A in turn
5
, we see that the columns of (A T A)
−1
define the corresponding base vectors of the O–lattice.
Since O–points are points of perfect fit, mismatch must be at a maximum in between neigh-
bouring O–points. When O is a primitive O–lattice vector, equation 30b states that the
amount of misfit in between the two O–points connected by O is given by u. A dislocation
83