International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.2, p. 762

## Section 9.2.2.2.3. MDO polytypes

S. Ďuroviča

#### 9.2.2.2.3. MDO polytypes

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Any family of polytypes theoretically contains an infinite number of periodic (Ross, Takeda & Wones, 1966; Mogami, Nomura, Miyamoto, Takeda & Sadanaga, 1978; McLarnan, 1981a, b, c) and non-periodic structures. The periodic polytypes, in turn, can again be subdivided into two groups, the `privileged' polytypes and the remaining ones, and it depends on the approach as to how this is done. Experimentalists single out those polytypes that occur most frequently, and call them basic. Theorists try to predict basic polytypes, e.g. by means of geometrical and/or crystallochemical considerations. Such polytypes have been called simple, standard, or regular. Sometimes the agreement is very good, sometimes not. The OD theory pays special attention to those polytypes in which all layer triples, quadruples, etc., are geometrically equivalent or, at least, which contain the smallest possible number of kinds of these units. They have been called polytypes with maximum degree of order, or MDO polytypes. The general philosophy behind the MDO polytypes is simple: all interatomic bonding forces decrease rapidly with increasing distance. Therefore, the forces between atoms of adjacent layers are decisive for the build-up of a polytype. Since the pairs of adjacent layers remain geometrically equivalent in all polytypes of a given family, these polytypes are in the first approximation also energetically equivalent. However, if the longer-range interactions are also considered, then it becomes evident that layer triples such as ABA and ABC in close-packed structures are, in general, energetically non-equivalent because they are also geometrically non-equivalent. Even though these forces are much weaker than those between adjacent layers, they may not be negligible and, therefore, under given crystallization conditions either one or the other kind of triples becomes energetically more favourable. It will occur again and again in the polytype thus formed, and not intermixed with the other kind. Such structures are – as a rule – sensitive to conditions of crystallization, and small fluctuations of these may reverse the energetical preferences, creating stacking faults and twinnings. This is why many polytypic substances exhibit non-periodicity.

As regards the close packing of spheres, the well known cubic and hexagonal polytypes ABCABC and ABAB, respectively, are MDO polytypes; the first contains only the triples ABC, the second only the triples ABA. Evidently, the MDO philosophy holds for a layer-by-layer rather than for a spiral growth mechanism. Since the symmetry principle of polytypic structures may differ considerably from that of close packing of equal spheres, the OD theory contains exact algorithms for the derivation of MDO polytypes in any category (Dornberger-Schiff, 1982; Dornberger-Schiff & Grell, 1982a).

### References

Dornberger-Schiff, K. (1982). Geometrical properties of MDO polytypes and procedures for their derivation. I. General concept and applications to polytype families consisting of OD layers all of the same kind. Acta Cryst. A38, 483–491.
Dornberger-Schiff, K. & Grell, H. (1982a). Geometrical properties of MDO polytypes and procedures for their derivation. II. OD families containing OD layers of M > 1 kinds and their MDO polytypes. Acta Cryst. A38, 491–498.
McLarnan, T. J. (1981a). Mathematic tools for counting polytypes. Z. Kristallogr. 155, 227–245.
McLarnan, T. J. (1981b). The number of polytypes of sheet silicates. Z. Kristallogr. 155, 247–268.
McLarnan, T. J. (1981c). The number of polytypes in close packings and related structures. Z. Kristallogr. 155, 269–291.
Mogami, K., Nomura, K., Miyamoto, M., Takeda, H. & Sadanaga, R. (1978). On the number of distinct polytypes of mica and SiC with a prime layer-number. Can. Mineral. 16, 427–435.
Ross, M., Takeda, H. & Wones, D. R. (1966). Mica polytypes: systematic description and identification. Science, 151, 191–193.