Section 9.2.1.6. Crystallographic uses of Zhdanov symbols

From the Zhdanov symbols of a close-packed structure, it is possible to derive information about the symmetry and lattice type (Verma & Krishna, 1966). Let n₊ and n₋ be the number of positive and negative numerals in the Zhdanov sequence of a given structure. The lattice is rhombohedral if n₊ − n₋ = ±1mod3, otherwise it is hexagonal. The + sign corresponds to the reverse setting and − to the obverse setting of the rhombohedral lattice. Since this criterion is sufficient for the identification of a rhombohedral structure, the practice of writing three units of identical Zhdanov symbols has been abandoned in recent years (Pandey & Krishna, 1982a). Thus the 15R polytype of SiC is written as (23) rather than (23)₃.

As described in detail by Verma & Krishna (1966), if the Zhdanov symbol consists of an odd set of numbers repeated twice, e.g. (22), (33), (221221) etc., the structure can be shown to possess a 6₃ axis. For the centre of symmetry at the centre of a sphere or an octahedral void, the Zhdanov symbol will consist of a symmetrical arrangement of numbers of like signs surrounding a single even or odd Zhdanov number, respectively. Thus, the structures (2)32(4)23 and (3)32(5)23 have centres of symmetry of the two types in the numbers within parentheses. For structures with a symmetry plane perpendicular to [00.1], the Zhdanov symbols consist of a symmetrical arrangement of a set of numbers of opposite signs about the space between two succession numbers. Thus, a stacking |522|225| has mirror planes at positions indicated by the vertical lines.

The use of abridged symbols to describe crystal structures has sometimes led to confusion in deciding the crystallographic equivalence of two polytype structures. For example, the structures (13) and (31) are identical for SiC but not for CdI₂ (Jain & Trigunayat, 1977a, Jain & Trigunayat, 1977b).