494 Defects in crystals
equal to an integer, and therefore α = 2π n and there is no defect contrast. For
a stacking fault R is not a lattice translation vector, which means that α will be
a non-integer multiple of π. We will see in Section 8.4.3.3 that the bright field
and dark field contrast of stacking faults is entirely determined by the value of α.
The typical fringe contrast associated with stacking faults is commonly known as
α-fringe contrast.
Next, assume that, upon lowering the temperature, the disordered structure of
Fig. 8.10(a) can undergo two phase transformations: the first, at temperature T
1
,
orders the structure into the L1
2
or Cu
3
Au structure (shown in Fig. 8.10b). The
second, at temperature T
2
, distorts the unit cell into a tetragonal cell with c/a > 1
(Fig. 8.10c).
The L1
2
structure has the cP Bravais lattice;
†
the corresponding translation group
is the infinite group
T
o
={a, b, c,...},
which is a subgroup of T , i.e. T
o
⊂ T . When the ordering process begins, a partic-
ular B atom has four possible sites that it can move onto. If the B atoms occupy the
sites (0, 0, 0) + T
o
in region 1 of the crystal, and sites A + T
o
in region 2, then at
the contact plane the two ordered regions will be out-of-phase. This type of fault is
known as an anti-phase boundary or APB. The displacement vector across an APB
is a translation vector of the disordered structure, but not of the ordered structure, i.e.
R ∈ T but R ∈ T
o
[VTA74]. In the [100] projection of Fig. 8.12 the displacement
vector is the centering vector A (upper left) or B (lower right), which clearly belong
to T and not to T
o
. The two anti-phase domains are known as translation variants.
The total number of independent translation variants of an ordered structure is
equal to the ratio of the volume of the primitive unit cell of T
o
to the volume of
the primitive unit cell of T . In the example above, the volume of the primitive
unit cell of T
o
is equal to a
3
, and for T we have a volume of a
3
/4. The total
number of translation variants is therefore equal to 4; there are, then, three different
APBs possible between those translation variants (one less than the number of
translation variants) [VTA74]. The APBs can be planar or curved. If the APB
energy is isotropic, then the faults will in general be curved, since there is no
preferential orientation of the fault plane. If, on the other hand, the APB energy
is strongly anisotropic, then the APBs will be formed on particular lattice planes.
The isotropic or anisotropic nature of the APB energy can depend strongly on the
details of the crystal structure.
While stacking faults do not change the stoichiometry of the material, anti-phase
boundaries may change the local composition, depending on the direction of the
†
L1
2
has space group Pm
¯
3m, with atom A at the face-centers and B at the origin.