Section 9.2.1.7.4. Determination of the stacking sequence of layers

For an nH or 3nR polytype, the n close-packed layers in the unit cell can be stacked in possible ways, all of which cannot be considered for ultimate intensity calculations. A variety of considerations has therefore been used for restricting the number of trial structures. To begin with, symmetry and space-group considerations discussed in Subsection 9.2.1.4 and 9.2.1.5 can considerably reduce the number of trial structures.

When the short-period structures act as `basic structures' for the generation of long-period polytypes, the number of trial structures is considerably reduced since the crystallographic unit cells of the latter will contain several units of the small-period structures with faults between or at the end of such units. The basic structure of an unknown polytype can be guessed by noting the intensities of 10.l reflections that are maximum near the positions corresponding to the basic structure. If the unknown polytype belongs to a well known structure series, such as (33)_n32 and (33)_n34 based on SiC-6H, empirical rules framed by Mitchell (1953) and Krishna & Verma (1962) can allow the direct identification of the layer-stacking sequence without elaborate intensity calculations.

It is possible to restrict the number of probable structures for an unknown polytype on the basis of the faulted matrix model of polytypism for the origin of polytype structures (for details see Pandey & Krishna, 1983). The most probable series of structures as predicted on the basis of this model for SiC contains the numbers 2, 3, 4, 5 and 6 in their Zhdanov sequence (Pandey & Krishna, 1975, Pandey & Krishna, 1976a). For CdI₂ and PbI₂ polytypes, the possible Zhdanov numbers are 1, 2 and 3 (Pandey & Krishna, 1983; Pandey, 1985). On the basis of the faulted matrix model, it is not only possible to restrict the numbers occurring in the Zhdanov sequence but also to restrict drastically the number of trial structures for a new polytype.

Structure determination of ZnS polytypes is more difficult since they are not based on any simple polytype and any number can appear in the Zhadanov sequence. It has been observed that the birefringence of polytype structures in ZnS varies linearly with the percentage hexagonality (Brafman & Steinberger, 1966), which in turn is related to the number of reversals in the stacking sequence, i.e. the number of numbers in the Zhdanov sequence. This drastically reduces the number of trial structures for ZnS (Brafman, Alexander & Steinberger, 1967).

Singh and his co-workers have successfully used lattice imaging in conjunction with X-ray diffraction for determining the structures of long-period polytypes of SiC that are not based on a simple basic structure. After recording X-ray diffraction patterns, single crystals of these polytypes were crushed to yield electron-beam-transparent flakes. The one- and two-dimensional lattice images were used to propose the possible structures for the polytypes. Usually this approach leads to a very few possibilities and the correct structure is easily determined by comparing the observed and calculated X-ray intensities for the proposed structures (Dubey & Singh, 1978; Rai, Singh, Dubey & Singh, 1986).

Direct methods for the structure determination of polytypes from X-ray data have also been suggested by several workers (Tokonami & Hosoya, 1965; Dornberger-Schiff & Farkas-Jahnke, 1970; Farkas-Jahnke & Dornberger-Schiff, 1969) and have been reviewed by Farkas-Jahnke (1983). These have been used to derive the structures of ZnS, SiC, and TiS_1.7 polytypes. These methods are extremely sensitive to experimental errors in the intensities.