sapphirine and wollastonite. In each case, the observed long-period structures are consistent with those predicted to be stable by the appropriate mappings onto ...
PHYSICS CIIEMISTRY MIN[RAIS
Phys Chem Minerals (1989) 16:343-351
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 examined in the case of sapphirine. Consideration of interactions 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 parameters may also be used to express stacking fault energies. This approach, combined with a simple nucleation-andgrowth model, is used to examine the possibility of metastable generation of complex polytypes in sapphirine. Statistical analysis of defect distributions and frequencies in sap~phirine suggests that interactions over several hundred Angstroms 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 modifications were first identified in silicon carbide by Baumhauer (1912), who later coined the term 'polytypism' for such behaviour. Many other synthetic materials were subsequently found to be polytypic, over 100 different structural modifications being known for SiC and CdI2, for instance, in some cases possessing regularly repeating stacking sequences with periods of several hundred structural layers. Many important groups of natural minerals, notably the micas, pyroxenes and pyroxenoids, and spineUoids, also display 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 examples. Several theoretical models have been proposed to explain 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 thermodynamic 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 carbide). In some cases, such as PbI2, these transformations are reversible, and polytype equilibria may be studied experimentally (Salje et al. 1987). The early theoretical models are comprehensively reviewed 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 microscopic observations on sapphirine are interpreted in this paper. 2. The A N N N I 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 interactions 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 neighbours, 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 orientational or positional states. In particular, the type of onedimensional polytypism found in, for instance, spinelloids, where the structural layers are mapped onto their neighbours by one of two stacking operators, may be modelled as weakly-coupled layers of relatively strongly-coupled
344
J,
{1)
{co}
Ittt
31
t122~ etc
etc
it;; Fig. L Ground-state ANNNI phase diagram
spins, which may be either 'up' or 'down'. Again, if there exist competing interactions between spins in neighbouring layers, a variety of different spin ordering patterns may be established. If exchange interactions between only adjacent and next-nearest layers are considered, this is the axial nextnearest neighbour Ising (ANNNI) model. A lot of theoretical work has been done on this model system. At low temperatures, the first-nearest neighbour interactions Jt and second-nearest neighbour interactions J2 determine a very simple phase diagram with three phases stable over extended fields in J1--J2 space (Fig. 1). These are the ferromagnetically ordered phase in which the layer spins align: T ~ ~ T T 1" the antiferromagnetic phase T $ ~ $ ~ ~ a n d a ' t w o - u p t w o - d o w n ' p h a s e 1" ~ $ $ ~ ~. These structures may be described more compactly in the band notation of Fisher and Selke (1981) as (oo), (1} and (2) respectively, where the brackets enclose a regularly repeating sequence, and each number represents the length of a band of spins in similar orientation. It should be noted that the ( 1 ) - ( 2 ) and ( o 0 ) - ( 2 ) equilibrium lines are degenerate, phases such as (12) and (3) also being stable along them. The behaviour of this model at elevated temperatures becomes complicated in that successively more complex phases are stabilised between the {2) and ( 2 ) fields and between the (oo) and ( 2 ) fields by entropic effects as the temperature increases. This behaviour is discussed in detail by Bak and yon Boehm (1980), Fisher and Selke (1980) and Duxbury and Selke (1983). An extension of the model to include third-nearest neighbour interactions has also been investigated (Selke et al. 1985). The results of these studies are collected and summarised in Yeomans and Price (1986). A detailed mathematical treatment of the A N N N I model has been published by Fisher and Selke (1981). One possible problem with applying the A N N N I model to mineralogical systems is that long-period magnetic spin structures are stabilised by spin disorder within the layers. This would correspond to the presence of line defects and stacking-fault jogs in polytypes, which are found empirically to be very rare (Salje et al. 1987). If such an analogy is valid, then polytypic materials grow in a low normalised temperature regime and no structures more complex than the four-layer ( 2 ) phase can be stable unless third-, fourth
etc. nearest-neighbour interactions are invoked. This may be justified in some cases (it will be shown below that this is true for some mineralogical systems) but explaining the very long period regular structures mentioned above remains problematical. However, enough 'spin disorder' may actually arise through lattice vibrations to render long-period structures stable (Yeomans pers. comm. 1987). Interestingly, the possible importance of vibrational entropy in stabilising SiC polytypes was anticipated by Jagodzinski (1954). Weltner (1969) deduces from his calculations on ZnS that it is probably an unimportant factor there, but it is doubtful whether it is safe to generalise his conclusions to other systems. In conclusion, the A N N N I model generates either simple phase relationships or complex, non-analytic behaviour depending only on the values of the four parameters T, Jr, J2 and J0 (the intra-layer exchange energy). The structures arising are evocatively similar to those of polytypic systems. The mapping of A N N N I spins onto polytypic structure units will be discussed below. 3. Mapping the ANNNI Model Onto Polytypic Systems The appropriate choice of structural feature onto which the A N N N I spins are to be mapped is by no means unequivocal. It will be shown below that there exist a number of possible mappings, each of which has a different set of correspondences between spin structures and stacking sequences associated with it. These mappings are appropriate to different polytypic systems, dependent on the relative magnitudes of various short-range interaction terms, and the symmetric properties of the layers. (a) The Direct Mapping Price (1983) treats the spinneloid structures as composed of layers which may be in either an 'up' or 'down' orientation. Two adjacent layers in the same orientation are mirror-related, whereas two in opposite orientations are related by a glide operator. It is equally valid to consider the layers as being translationally pseudo-equivalent, in which case 'up' and 'down' may be correlated with the z co-ordinate (0 or 0.5) of some suitable reference point in each layer. The analogy with the A N N N I spins is straightforward, and on this scheme, the spinel structure corresponds with the (1} phase, the modified spinel structure with the (2) phase, and so on. Price and Yeomans (1984) apply this mapping to a variety of different polytypic systems and show that the structures found usually correlate well with those predicted by the A N N N I model. (b) The Wollastonite Mapping Angel et al. (1985) point out, however, that problems arise for structures possessing a certain type of pseudosymmetry. Suppose, in the spinelloid example above, that the stacking vector relating two adjacent layers at the same height is termed' T', whereas that relating vectors at different heights is 'G' (terminology of Henmi et al. 1983). The spinel structure is then (G), modified spinel is (TG), and ( T ) corresponds to a fictive structure with infinite chains of tetrahedra. There exist systems, however, in which the ( T ) and ( G ) structures are twin-related, and therefore identical in energy in the absence of an applied shear stress. This is the case for wollastonite and related minerals, zoisite-clino-
345 zoisite, and sapphirine, amongst others. A look at the ANNNI phase diagram reveals that if ( T ) is to be mapped onto (oo) and ( G ) onto (1), then J1 is constrained to be equal to zero. This is proved on examination of the expression for the ground state energy per layer of a structure given in Angel et al. (1985): N layers
E=Ez-(1/N)
Z
J, slsi+,
GGGGGG
Stacking Sequence TGTGTG TTGGTTGG
tltitlt t t tlt
ttlitti tt ltt
(1)
all i, n
where Ez is the internal energy of a layer and st is a spin variable taking the values + 1 for an ' u p ' spin and - 1 for a ' d o w n ' spin. These spin variables are in turn related to translation operators ti = + 1 for a T vector, - 1 for a G vector, according to the equation:
tttlttt t ttt tttlt
