
8.4 Image contrast for selected defects 499
More complex defects can have characteristic features of more than one of the
defect types introduced in this section. In ordered alloys, for instance, stacking
faults may be formed by the dissociation of perfect dislocations into more than two
partial dislocations. The stacking fault will then consist of regions that have pure
SF nature, and regions where the ordered lattice on one side of the fault plane is
shifted by a lattice centering vector with respect to the lattice on the other side. This
means that the resulting fault is a stacking fault and at the same time an anti-phase
boundary. Such a defect is commonly known as a complex fault.
In summary, we have defined the following planar defects.
r
Stacking faults: the displacement vector is not a Bravais lattice translation vector.
r
Anti-phase boundaries: the displacement vector is a lattice vector of the
disordered
phase, but not of the ordered phase.
r
Orientation variant interfaces: the variants are related by a symmetry operation of the
point group of the parent phase, but not of the point group of the product phase. This
includes domain boundaries in ferroelectric
and magnetic materials, and (transformation)
twin boundaries. The displacement vector is typically a linear function of the distance to
the interface.
8.4.3.2 Planar defect contrast formation (two-beam case)
In Section 6.5.2, we introduced the two-beam scattering matrix formalism, in which
the general solution of the two-beam case is written by means of the scattering
matrix S(s, z
0
) (equation 6.49 on page 380):
S(s, z
0
) ≡ e
−(π/ξ
0
)z
0
TSe
−iθ
g
Se
iθ
g
T
(−)
. (8.40)
This matrix is also known as the response matrix of the crystal. For the centro-
symmetric case, the phase factors on the off-diagonal elements are equal to unity.
For simplicity, we will ignore the normal absorption exponential, although it should
be taken into account for all numerical work.
In the presence of a defect with displacement field R, the two beam equa-
tions (6.14) and (6.15) are modified by a phase factor:
dT
dz
+ π is
g
T =
iπ
q
−g
e
iα
g
S; (8.41)
dS
dz
− π is
g
S =
iπ
q
g
e
−iα
g
T. (8.42)
Consider a crystal with a planar defect as shown schematically in Fig. 8.16(a). The
top section (I) of the crystal is defect-free and is, hence, described by the scattering
matrix S(s
1
, z
1
). Section II has a slightly different orientation (different excitation