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APPENDIX 2

Examples of descriptive symbols for some polytypic structures, with dependence on choice of BLs, origins and stacking direction

A. Explanatory introduction and explicit table of symbols for two cases

(1) ZnS Family. The structures (see Fig. 1) may be considered to consist of BLs chosen as follows (Verma & Krishna, 1966).

 

Figure 1: Polytype 15R of ZnS. Structural arrangement in the ($11\={2}0$)section illustrating the orientational and displacement characters for the two choices of BLs. The S atoms of the adjacent sections which belong to the indicated tetrahedra for BL choice (b) are not drawn (see Appendix 2). Origins of BLs are chosen at the starting points of arrows indicating their displacements.

(a) Sheets of Zn-S dumb-bells projecting in close-packing positions denoted by A, B, C, with any two successive positions corresponding to different letters. Sequences AB, BC, CA are equivalent and may be considered as displacements q₊; the reverse sequences AC, CB, BA give displacements of opposite sense q_-.

(b) Sheets of ZnS₄ tetrahedra sharing common corners. The displacements of these BLs are characterized by the same sequences as above and also serve as orientational vectors Q₊ and Q_-. Since the displacement and orientation senses coincide, one kind of character is sufficient for the description.

An example is given in Table 1.

 

Substance BLs chosen Full symbol (redundant) Simplified symbol
(nonredundant) Number of BLs
per repeat Space group

ZnS
(15R) (a) |q₊q₊q₊q_-q_-|₃ |+++-|₃ 15 R3m

(b) |Q₊q₊Q₊q₊Q₊q₊Q_-q_-Q_-q_-|₃

(2) Strontium germanate : SrGeO₃. Sheets of SrO₆ octahedra (sharing edges, see Fig. 2) are linked by sheets of isolated three-membered rings of GeO₄ tetrahedra: all rings share their non-bridging O atoms with octahedra and project onto every third triangle between the bases of adjacent SrO₆ octahedra (Dornberger-Schiff, 1961). There are three choices of BLs.

 

Figure 2: Polytypism in SrGeO3. (a) General structural arrangement viewed down [001]. Sheet of SrO6 octahedra and upper sheet of Ge3O9 rings (heavily dotted - almost black - triangles). Three octahedra in the central part are left open to show how the rings of the lower tetrahedral sheet (lightly dotted triangles) are attached to the octahedral sheet. (b) Projection along [100] with the three choices of `building layers', BL(a), BL(b), and BL(c) indicated by braces. The origins for BL(a) and BL(b) are chosen in the centres of the germanate rings at points with site symmetries $\={6}2m$; for BL(c) the origin is in the octahedral sheet at a point with site symmetry 2/m, between Sr atoms. General remarks on Figs. 2(c) to 2(e): These diagrams give schematic representations of three periodic polytypes in a normal projection onto the (001) plane. The double triangle symbol represents a BL of choice (b) with layer symmetry $P(\={6})2m$; it also indicates the orientation and relative displacement of triangles drawn through the bridging oxygen atoms of the germanate rings. BLs are numbered according to sequence, 2/0 (6/0) means a coincidence of the second (sixth) and zeroth BL in projection. Displacement vectors ${\langle}j{\rangle}$ (j-displacement character) and some relevant symmetry elements of the respective space groups are also given. (c) Orthorhombic polytype |30| projected along [001]; space group Ccmm. (d) Monoclinic polytype |12|, projected along $\textbf{c}$*, which in this case is parallel to [013]; space group C2/c11 (a-axis setting). The c axis slopes upwards to the left from the cell origin O in the bottom (001) face of the cell to the origin $O^{\prime}$ in the top (001) face, i.e. $\textbf{c} = \textbf{d}_{001} -\frac{1}{3}\textbf{b}$.Hence, the c components of the glide vectors of the c and n glides are inclined to the plane of the figure. The triangles 0, 1, 2,... are related by the successive operations of the c glide planes, triangle 6 would project onto triangle 0. (e) Hexagonal polytype |345012|, projected along [001]; space group P6122.

(a) Two kinds of BLs, with BL(1) an octahedral sheet and BL(2) a sheet of isolated tetrahedral rings. Non-bridging oxygen ring atoms belong half to one, half to the adjacent BL. Since successive BLs of one kind regularly alternate with respect to their orientations, displacements (between ring centres) only are sufficient to describe the polytypes.

(b) One kind of equivalent BL: germanate sheets with planes of Sr half-atoms attached on either side. Displacements as in (a) are used.

(c) One kind of equivalent BL: octahedral sheets with planes of Ge and bridging O half-atoms attached to them on either side. Displacement vectors as in (a) and (b) are to be considered orientational vectors. These vectors have length a/3 in the normal projection on the orthohexagonal basis plane ab and are related by counterclockwise rotations of $2{\pi}n/6$ about the c axis. They are accordingly assigned the characters n = 0, 1, $\ldots$, 5. The character 0 (or 6) denotes the vector parallel to $-{\textbf a}$ of the orthogonal C-centered base $a, b = a{\surd}3$.

An example is given in Table 2.

 

Table 2: Descriptive symbols for ZnS

Substance SrGeO3	BLs chosen	Full symbol (redundant)	Simplified symbol (nonredundant)	Number of equivalent BLs per repeat	Space group
(2O)	(a)(b)	|p3p0|	|30|	2	Ccmm
	(c)	|P3P0|
(2M)	(a)(b)	|p1p2|	|12|	2	C2/c11
	(c)	|P1P2|
(6H)	(a)(b)	|p3p4p5p0p1p2|	|345012|	6	P6122
	(c)	|P3P4P5P0P1P2|

B. Illustrative papers on polytypic substances with description of symbolism used

Astrophyllite

Zvyagin & Vrublevskaya (1976).

Kaolinite-type structures

Zvyagin (1964, 1967) (tri- and di-octahedral polytypes).

Dornberger-Schiff & Durovic (1975) (tri-, di- and mono-octahedral polytypes).

Mica

Zvyagin (1964, 1967) (tri-octahedral polytypes).

Takeda (1967) (tri-octahedral polytypes).

Dornberger-Schiff, Backhaus & Durovic (1982) (tri-, di and mono-octahedral polytypes).

Vermiculites

Weiss & Durovic (1980).

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