Phys Chem Minerals (1989) 16:343-351 PHYSICS CIIEMISTRY MIN[RAIS 9 Springer-Verlag 1989 A Short-Range Interaction Model for Polytypism and Planar Defect Placement in Sapphirine Andrew G. Christy Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom Abstract. The concept of short-range interlayer interactions, fundamental to spin-analogue models for polytypism, is ex- amined in the case of sapphirine. Consideration of interac- tions out to fourth-nearest neighbours provides a rationale for the difference between the polytype suites observed for sapphirine and wollastonite. In each case, the observed long-period structures are consistent with those predicted to be stable by the appropriate mappings onto the axial next-nearest neighbour model. Short-range interaction pa- rameters may also be used to express stacking fault energies. This approach, combined with a simple nucleation-and- growth model, is used to examine the possibility of metasta- ble generation of complex polytypes in sapphirine. Statisti- cal analysis of defect distributions and frequencies in sap~phi- rine suggests that interactions over several hundred Ang- stroms must be considered if the stacking energetics are to be accurately modelled. 1. Introduction A large number of systems are known in which materials of similar or identical chemistry exhibit a range of crystal structures which may be viewed as different stackings of topologically similar structural modules. Such modifica- tions were first identified in silicon carbide by Baumhauer (1912), who later coined the term 'polytypism' for such behaviour. Many other synthetic materials were subse- quently found to be polytypic, over 100 different structural modifications being known for SiC and CdI2, for instance, in some cases possessing regularly repeating stacking se- quences with periods of several hundred structural layers. Many important groups of natural minerals, notably the micas, pyroxenes and pyroxenoids, and spineUoids, also dis- play polytypic behaviour, although the number of different structures in any one family of polytypes is usually smaller here, and the repeat lengths shorter. Wollastonite (six known structures, with repeat unit 1 7 layers; Henmi et al. 1983, Angel 1985) and sapphirine (five structures, repeat unit 1-5 layers; Christy and Putnis 1988) are typical exam- ples. Several theoretical models have been proposed to ex- plain or describe polytypism. In some, such as the screw dislocation model of Frank (1951), the observed structures are regarded as arising from specific mechanisms of crystal growth. Other workers regard them as distinct thermody- namic phases - a view whose validity is borne out by the observation of phase transformations between polytypes (cf. Jepps and Page 1983, Ness and Page 1986 on silicon car- bide). In some cases, such as PbI2, these transformations are reversible, and polytype equilibria may be studied exper- imentally (Salje et al. 1987). The early theoretical models are comprehensively re- viewed in Verma and Krishna (1966) and Trigunayat and Chadha (1971), and will not be further discussed here. A more recent model, which has proved stimulating in recent years, is outlined below since it provides the theoretical framework within which high-resolution electron micro- scopic observations on sapphirine are interpreted in this paper. 2. The ANNNI Model The approach adopted considers the crystal structure to be composed of a three-dimensional array of structural units which are topologically similar. The energetic differences between different polytypes arise through the inter-unit in- teractions varying as a function of the way in which the structural units are stacked. For the case of one-dimensional polytypism, in which stacking variation only occurs along one direction, we may, to a first approximation, consider interlayer interaction energies between neighbouring layers along this direction to be the factors directly determining polytype stability. These interaction terms may, in turn, vary with pressure, temperature, composition and so on. This approach was suggested by Hazen and Finger (1981), and was successfully applied to the NiA1204-Ni2SiO 4 spinelloid system by Price (1983). He considered interlayer interactions between first, second, and third nearest neigh- bours, and predicted a suite of short-period phases to be stable, the equilibrium stacking sequence being controlled by the signs and relative magnitudes of the interaction terms. In general, the predicted phases agreed with those actually observed in this system, and the inferred interaction terms varied in a systematic way with chemical composition, which could be ascribed to charge balance effects. An analogy may readily be made between polytypes and magnetic spin systems. In both cases, we are considering arrays of units which may be in one of a number of orienta- tional or positional states. In particular, the type of one- dimensional polytypism found in, for instance, spinelloids, where the structural layers are mapped onto their neigh- bours by one of two stacking operators, may be modelled as weakly-coupled layers of relatively strongly-coupled