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When a material crystallizes into a number of different close-packed structures
all of which have identical layer spacings and different only in the manner of
stacking the layers, crystals of the different modifications look alike and
cannot be identified by their external morphology. In order to identify such
polytype modifications, it is necessary to determine the number of layers in the
hexagonal unit cell and the lattice type of the crystal. This can be
conveniently achieved by recording reciprocal-lattice rows parallel to *c* *
on single-crystal X-ray diffraction photographs. Since the different polytypes
of the same material have identical *a* and *b* parameters of the
direct lattice, the *a* * *b* * reciprocal lattice net is also the same.
The reciprocal lattice of these modifications differ only along the *c* *
axis which is perpendicular to the layers. For each reciprocal-lattice row
parallel to *c* * there are others with the same value of the cylindrical
coordinate . For example, the rows 10.l, 01.l, 1.l,
0.l, 0.l and 1.l all have =
*a* *. Due to symmetry, it is sufficient to record
any one of them on X-ray diffraction photographs. The number of layers,
*n* , in the hexagonal unit cell can be found by determining the *c*
parameter from *c* -axis rotation or oscillation photographs and dividing
this by the known layer-spacing *h* for that compound (*n* =
*c/h* ). The density of reciprocal points along rows parallel to *c* *
depends on the periodicity along the *c* axis. The larger the identity
period along *c* , the more closely spaced are the reciprocal-lattice points
along *c* *. In case of long-period polytypes the number of layers in the
hexagonal unit cell can be determined by using a simple alternative method
suggested by Krishna and Verma^{20}. This requires the counting of the number
of spacings after which the sequence of relative intensities begins to repeat
along the 10.l row of spots on an oscillation or Weissenberg photograph. If the
structure contains one-dimensional disorder due to a random-distribution of
stacking faults, this effectively causes the *c* lattice parameter to
become infinite (*c* *0) and results in the production of
characteristic streaks along reciprocal lattice rows parallel to *c* *. It
is therefore difficult to distinguish by X-ray diffraction between structures of
very large unresolvable periodicities and those with random disorder. Lattice
resolution in the electron-microscope has been used in recent years to identify
such structures^{21}. Figure 13 depicts the 10.l rows of some close-packed
structures of SiC as recorded on *c* -axis oscillation photographs. When
the structure has a hexagonal lattice, the positions of spots are symmetrical
about the zero layer line on the *c* -axis oscillation photograph as seen in
Fig. 13 (a) and (b) for the 6*H* and 36*H* SiC structures. However,
the intensities of the reflections on the two sides of the zero layer line are
the same for the 6*H* structure but not for the 36*H* . This is
because the 6*H* structure belongs to the hexagonal space group
*P* 6_{3}*mc* whereas the 36*H* structure belongs to the
trigonal space group *P* 3*m* 1^{2}. The apparent mirror symmetry
perpendicular to the *c* -axis in Fig. 13(a) results from the combination of
the 6_{3} screw axis with the centre of symmetry introduced by X-ray
diffraction^{22}. For a structure with a rhombohedral lattice, the positions
of X-ray diffraction spots are not symmetrical about the zero layer line because
the hexagonal unit cell is non-primitive causing the reflections *hkl* to be
absent when (*n* = 0, 1, 2,...). For the 10.l row this
means that the permitted reflections above the zero layer line are 10.1, 10.4,
10.7 etc. and below the zero-layer line 10., 10.,
10. etc. The zero layer line will therefore divide the distance
between the nearest spots on either side (namely 10.1 and 10.)
approximately in the ratio 1:2. This enables a quick identification of a
rhombohedral lattice. Thus the lattice type corresponding to Fig. 13(c) is
rhombohedral and the polytype is designated as 90*R* and belongs to the
space group *R*3*m*. Figure 13(d) depicts the 10.l row of a disordered 2*H*
SiC structure. The diffuse streak connecting the strong 2*H* reflections
is due to the presence of a random distribution of stacking faults in the
2*H* structure.

**Copyright © 1981, 1997 International Union of
Crystallography**