[IUCr Home Page] [Commission Home Page]

next up previous
Next: (d) The h-c notation Up: Notations Used for Representing Close-Packed Structures Previous: (b) The classical ABC notation

(c) Zhdanov notation

If the layers in a close-packed structure are projected on to one of the close- packed planes, the atoms fall into one of the three possible positions A, B and C with xy coordinates 00, $\frac{1}{3}$ $\frac{2}{3}$ and $\frac{2}{3}$ $\frac{1}{3}$ respectively. The passage from A$\rightarrow$ B$\rightarrow$ C$\rightarrow$ A involves a vector translation of $\frac{1}{3}$ $\frac{2}{3}$ in the basal plane, whereas the passage from A$\rightarrow$ C$\rightarrow$ B$\rightarrow$ A involves a vector translation of $\frac{2}{3}$, $\frac{1}{3}$ = $-\frac{1}{3}$, $-\frac{2}{3}$ (fig. 1). H$\ddot{\rm a}$gg9 therefore denoted the former by a plus sign (+) and the latter by a minus sign (-). A structure such as ABCB is thus represented as ++-. The relationship between the three orientations, A, B and C of the close-packed layers may also be visualized in terms of clockwise or anticlockwise rotation about [00.1] through 60$^{\circ}$. Frank10 used the symbols $\bigtriangleup$ and $\bigtriangledown$ for the two rotations. Thus the $\bigtriangleup$ symbol implies a cyclic change A$\rightarrow$ B$\rightarrow$ C$\rightarrow$ A and the symbol $\bigtriangledown$ implies an anticyclic change. No compactness results from the use of these + and - or $\bigtriangleup$ and $\bigtriangledown$ symbols for representing a close-packed structure because their number remains the same as the number of layers in the ABC sequence of the structure. Zhdanov11 therefore suggested summing up the consecutive + (or $\bigtriangleup$) and - (or $\bigtriangledown$) signs and putting them down in numeral figures. Thus the 6H SiC structure having the ABC sequence ABCACB and a H$\ddot{\rm a}$gg9 sequence +++-- is denoted by the symbol (33) in the Zhdanov notation.

Ramsdell8 interpreted the Zhdanov symbols in terms of the zig-zag sequence of Si and C atoms in the (11$\overline{2}$0) planes of SiC structure. These planes contain all the atoms of the structure since the three symmetry axes parallel to [001] all lie in this plane. Figure 10 illustrates the meaning of the zig-zag sequence, taking the 6H (33) structure of SiC as example. If a Si or C atom lies on A in one layer, the next must be either to the right on B, or to the left on C. If to the right, the third layer may have its atom continue to the right or it may change direction and go to the left. Because of these repeated changes, a zig-zag pattern results. Such an arrangement can be described in terms of the number of layers added in each direction in succession and has been called the `zigzag sequence` by Ramsdell. The unit cell is completed after arriving at an identical atom having the same environment as the atom from which one started. Thus in Fig. 10 the unit-cell of 6H is completed at 2 and not at 1. The Zhdanov notation is by far the most convenient and concise notation to describe close-packed structures.


next up previous
Next: (d) The h-c notation Up: Notations Used for Representing Close-Packed Structures Previous: (b) The classical ABC notation

Copyright © 1981, 1997 International Union of Crystallography

IUCr Webmaster