Structural and chemical heterogeneity of layer silicates and clay minerals

Abstract Different forms of structural and chemical heterogeneity are considered including mixed-layer minerals, disordered layer structures containing rotational and translational stacking faults, interstratification of trans-vacant (tv) and cis-vacant (cv) layers in true micas, illites and illitesmectite (I-S), short-range order in isomorphous cation distribution etc. Because determination of various structural and chemical imperfections requires elaboration of new diffraction and spectroscopic methodologies, special attention is paid to recent achievements in the development of new methodological approaches such as a multispecimen simulation of experimental X-ray diffraction (XRD) patterns from mixed-layer minerals, with account taken of layer-thickness fluctuations of the second type and possible difference between structures of outer and core layers; experimental determination of thickness distribution of illite crystals by HRTEMand the modified Bertaut-Warren-Averbach technique; XRD and thermal methods for determination of cv and tv layers in true micas, illites and I-S; generalization of Méring's rules to account for the behaviour of non-basal reflections for any defective structure in which two translations are irregularly interstratified; various ab initio calculations devoted to modelling infrared OH vibrations, octahedral cation distribution in dioctahedral 2:1 layer silicates, etc. It is shown that these recentlydeveloped methodologies have revealed new diversity in the structural and chemical heterogeneity of phyllosilicates and clay minerals, provided new insight into the structural mechanisms of their transformation in different geological environments, and discovered new natural processes.

The study of structural and chemical heterogeneity of minerals is one of the main domains in modern structural mineralogy. Determination of various imperfections that violate the uniform and periodic structure of crystals has great scientific and practical significance. As a matter of fact, the types of defects, their content and the distribution strongly modify physicochemical properties of crystals, reflect the specific physicochemical conditions of mineral formations, allow reconstruction of structural mechanisms of phase transformations, etc.
The origin of structural and chemical heterogeneity of layer minerals is predetermined by the very nature of their structure. On the one hand, different layers have similar or identical two-dimensional periodicity and similar structure of their outer basal surfaces. For these reasons, structurally and chemically different layers may coexist within one and the same crystal, leading to the formation of mixed-layer structures. On the other hand, layers in layer pairs may stack in several, energetically similar ways. Therefore, layer minerals often contain stacking faults due to irregular layer rotations, displacements, microtwinning, etc. Finally, layer minerals are usually characterized by complex chemical composition and a wide spectrum of isomorphous cation distributions. In addition, crystallization of minerals in a fine-dispersed state is also a specific type of imperfection and the study of such minerals should include analysis of structural and chemical heterogeneity of individual crystallites as well as heterogeneity in their size distribution.
It is natural that determination of various structural and chemical imperfections requires development of new diffraction and spectroscopic methodologies. In particular, a strong impetus for new theoretical and methodological developments was given by the need to interpret diffraction effects observed for defective layer structures (Méring, 1949;Brindley & Méring, 1951;James, 1965;Guinier, 1964;Kakinoki & Komura, 1952Drits & Tchoubar, 1990;Drits & Sakharov, 1976;Plançon & Tchoubar, 1977;Drits et al., 1984Drits & McCarty, 1996;Plançon, 1981Plançon, , 2001Plançon, , 2002Plançon, , 2003Reynolds, 1980Reynolds, , 1985. One remarkable feature of these effects is their ability to extract average structural characteristics of crystals deprived of 3D periodicity. On the other hand, diffraction effects from disordered structures do not obey Bragg's law and cannot be interpreted by conventional structural analysis techniques. Therefore reliable determination of structural and chemical heterogeneity of layer minerals depends on the reliable interpretation of experimental data obtained by diffraction methods. A similar situation occurs with determination of local cation environments and short-range order in the distribution of isomorphous cations by spectroscopic methods because interpretation of their spectra is often based on the 'finger print' or other semi-empirical approaches . That is why, in this paper, special attention will be paid to recent methodological approaches (combined with their practical applications) to the study of defective layer structures. The structural and chemical heterogeneity of layer silicates and clay minerals associated with interstratification of different layer types; interstratification of identical layers having different azimuthial orientations and translations; interstratification of layers having identical thickness but different inner structure and interlayer translations; and short-range order in isomorphous cation distribution, will be dealt with here.

Marcovian statistics, structural and chemical heterogeneity, Mé ring's rules
To describe the distribution of layer types in irregular mixed-layer structures, Marcovian statistics are usually used. An important parameter in this model is the short-range order factor R defined as the number of preceding layers that influence the occurrence probability for a subsequent layer of a given type. Each R value is related to a set of conditional probability parameters, and one of the main problems is to determine these parameters in order to describe the layer stacking sequence in a mineral under study (Drits & Tchoubar, 1990).
It follows from the Marcovian statistics that a mixed-layer dispersed mineral is a physical mixture, or assemblage, of statistically weighted crystallites or coherent scattering domains (CSDs). The thickness of a CSD is determined by the number of interstatified layers parallel to each other in the ab plane. The remarkable feature of such an assemblage is that CSDs in the mixed-layer sample have quite different structure and composition (Drits, 1987a(Drits, ,b, 1997. In general, the degree of heterogeneity of the composition of CSDs in mixed-layer samples is a function of the total number of layers in CSDs, the proportion of interstratified A and B layers, and the pattern in the sequence of layer types. For example, let us consider the heterogeneity in proportions of interstratified A and B layers in a mixed-layer sample in which each crystallite contains 10 layers, the average content of A and B layers are 60 and 40% and R = 0. It can be shown that the portion of crystallites consisting of 6A and 4B layers, corresponding to the average content of A and B layers in the sample, is only 25%. The remaining 75% of crystals have other compositions. If R = 1, the homogeneity of the sample increases but the portion of crystallites containing 6A and 4B layers still remains <50% (Drits, 1987a). The other source of heterogeneity is that crystallites having the same composition differ from each other in the distribution of layer types. Thus, in terms of the Marcovian model, the heterogeneity of a powder mixed-layer sample results both from the heterogeneity of the CSD compositions (A:B ratio) and from different distribution of layers in crystallites with a fixed A:B ratio. In addition, the heterogeneity of mixed-layer samples depends on the range within which thicknesses of CSDs vary and on the distribution of thicknesses within that range.
It is natural that an XRD pattern of a mixed-layer sample is a statistically weighted sum of XRD patterns, each of which corresponds to individual CSDs having the same composition and distribution of layer types. Positions and profiles of basal reflections recorded in the XRD patterns from twocomponent structures with R = 0 are determined by Méring's rules (Méring, 1949).
According to these rules, basal reflections for a random two-component structure are located between the neighbouring 00l reflections corresponding to periodic structures whose layers are interstratified. The width of the reflections increases with the distances between these neighbouring 00l reflections, and their location depends on the proportion of the interstratified layer types. Méring's rules, in combination with two rules added by Drits and Sakharov (1976) and , provide the best insights into the nature of diffraction from mixed-layer structures. In particular, they explain the origin of irrational series of basal reflections in XRD patterns for irregular mixed-layer minerals with R 5 1 and predict general regularities in the formation of diffraction effects from such structures. A simple technique based on these rules allows semiquantitative determination of proportions of layer types for two-component mixed-layer structures with any R and any thicknesses of interstratified layers . Mérings rules are an essential component of computer expert systems developed for a semi-qualitative phase analysis of clays containing periodic and mixed-layer minerals . Nadeau et al. (1984) suggested another interpretation of diffraction effects from mixed-layer structures, considering their XRD patterns to be a result of interparticle diffraction, i.e. diffraction between so-called fundamental particles. For example, in terms of this model, I-S structure can be considered as an aggregate of smectite 2:1 monolayers and illite fundamental particles or illite crystals that consist of two or more 2:1 layers bonded by interlayer K. Although simulation of XRD patterns from natural mixed-layer minerals in terms of this model is a complex task, its general conception is useful in many aspects. In particular, diagenetic evolution of I-S is accompanied not only by increasing illite layer content and structural order in layer sequences but also by increasing mean thickness of illite crystals and by modification of their thickness distribution.

Experimental measurements of thickness distribut ions of illite crystals in I-S
Recent investigations of crystal size and size distributions have shown that these characteristics can be used for reconstruction of geological history and for better understanding of crystal growth processes Merriman et al., 1990;Środoń, 2002). In particular, Eberl et al. (1998a) devel oped theoreti cal approaches, according to which each crystal growth mechanism is accompanied by a specific shape of crystal-size distribution. This means that it is possible to solve the reverse problem, i.e. to use crystal-size distribution to determine the law of crystal growth and so deduce the relative reaction history of mineral formation. One of the critical points in the solution of the problem is to provide careful and accurate crystal-size measurements. These requirements have stimulated the development of new methodologies for the determination of size parameters of clay minerals (Arkai et al., 1996;Guthrie and Veblen, 1989;Środoń et al., 1990Środoń, 2002;Elsass et al., 1998;Uhlik et al., 2000;Drits et al., 1997d).
Among XRD methods, the Bertaut-Warren-Averbach (BWA) technique is most universal because it analyses reflection profiles and determines crystal-size distributions. Drits et al. (1998a) modified this technique to apply it to clay minerals and developed the MudMaster computer program (Eberl et al., 1996). It should be emphasized that the BWA method can be applied only to clay minerals having a periodic structure at least along the c* direction. Eberl et al. (1998b) showed that saturation of I-S with polymer polyvinylpyrrolidone (PVP) is accompanied by separation of individual smectite monolayer and illite crystals or fundamental particles in such a way that they scatter X-rays independently from each other. Therefore, the PVP-treated I-S sample is a set of individual particles for which interparticle diffraction observed for the non-treated sample is eliminated. Eberl et al. (1998b) used the BWA method to determine mean thickness and thickness distribution of illite crystals in I-S from diagenetic and hydrothermal bentonites. They showed that the mean thicknesses of illite crystals are similar to those determined by the Ptshadowing TEM technique. In addition, they found that, as a rule, variations of illite crystals in the I-S samples studied follow log-normal distributions characterized by two independent parameters: the mean of the logarithms of coherent scattering domain (CSD) thickness, a, and the variance of the logarithms of CSDs, b 2 .
Recently, Dudek et al. (2002) compared thickness distribution of illite crystals in I-S using MudMaster and HRTEM. I-S samples from shales and bentonites were treated with PVP. It was found that the area-weighted distribution determined by XRD contains a larger fraction of thick (> 40 A Ê ) and a smaller fraction of thin crystals compared to the number-weighted distribution obtained by HRTEM. Nevertheless, for the fine fraction of illite crystals (< 100 A Ê ), both techniques produce similar thickness distributions. In contrast to bentonitic I-S, for shale I-S, illite crystal-thickness distributions are not lognormal and can be considered as a superposition of several lognormal distributions having different a and b 2 values. Środoń et al. (2000) studied the evolution of illite and smectite fundamental particles during smectite illitization in order to determine the structural mechanism of this reaction. Thickness distributions of these particles from bentonitic and hydrothermally altered pyroclastics were determined by a Ptshadowing TEM technique. The log-normal distribution of the area-weighted illite crystal thickness indicates the operation of a unique mechanism of the illitization process, which was simulated using the mathematical form of the Law of Proportionate Effect. It was found that illite particles grow from 20 A Ê thick illite nuclei by surface-controlled growth. Initially formed, these 20 A Ê crystals may grow from material produced by dissolution of smectite monolayers. After nucleation ceases, illite crystals continue to grow and the rate of growth is restricted by how rapidly crystallization proceeds, given a nearly infinite supply of reactants. This period of illitization is characterized by 3D growth. It was shown that Ostwald ripening, supplycontrolled growth, the coalescence of smectite layers and other crystal-growth mechanisms do not produce a log-normal distribution and the observed evolution of its parameters.

Multispecimen XRD technique
Quantitative structural analysis of mixed-layer minerals based on a trial-and-error approach requires the maximum possible agreement between the experimental XRD pattern and that calculated for a model corresponding to the actual structure of a mixed-layer sample. However, this is not a trivial task, and in practice, possibilities provided by computer simulation of experimental XRD patterns have seldom been used properly. Conventionally, in most of the publications, interpretation of the experimental XRD patterns is confined to the analysis of positions of basal reflections often ignoring their intensities and profiles. This approach employs peak migration curves which, in the form of plots or tables, represent relationships between basal reflection positions and proportions and mode of interstratification of layer types. Because this approach is restricted to two-component systems in which the thicknesses of interstratified layers are fixed, essential details of the actual structure of mixed-layer minerals may be poorly understood or even overlooked (Drits, 1997).
Therefore more reliable results are achieved when not only d values but also intensities and profiles of basal reflections coincide in the experimental and calculated XRD patterns. Even in this case, however, the actual structure of mixedlayer samples is not always unambiguously determined. The main reason is that several structural models may fit the experimental XRD pattern equally well (Sakharov et al., 1999a;Drits et al., 2002a). To increase the reliability of the results, the multispecimen XRD method was proposed by Drits et al. (1997a) and Sakharov et al. (1999a). It is based on two main requirements. First, a unique statistical model should describe the XRD patterns obtained for several specimens of the same sample subjected to different treatments: saturation by different interlayer cations, glycolation, heating etc. Second, the best possible agreement between the experimental and calculated reflections should be achieved as regards not only their positions but also intensities and profiles. The principle of the multispecimen method is that each different treatment of the same sample is an independent test of validity for the statistical structural model because each treatment changes the thickness and scattering efficiency of the swelling layers, but not the 2:1 or 1:1 layers and their distribution.
Application of the multispecimen method provides a more reliable interpretation of experimental XRD patterns and greater accuracy in the determination of structural and probability parameters (Drits et al., , 2002aSakharov et al., 1999a,b;Lindgreen et al., , 2002. A significant advantage of the method is that it can be applied to mixed-layer structures containing 3, 4 or more interstratified components with different distributions of individual layer types. Another important advantage of this approach is that it provides quantitative phase analysis of samples consisting of periodic and interstratified clay minerals (Drits et al., 2002a,b;Lindgreen et al., , 2002. The main disadvantage is that the approach is time-consuming, and one of the future tasks is to provide automation for the fitting procedure.
Two additional types of structural heterogeneity should be taken into account to describe the actual structure of mixed-layer minerals using the multispecimen method: microstrains and the potential opportunity for outer surface layers to be different from layer types of the core structure.

Layer-thickness fluctuati on or microstrains
These defects result from the presence in layer structures of structural imperfections that disturb parallel layer packings (Warr & Neito, 1998) such as edge dislocations, lateral layer terminations, bending layers, cracks, deformation, layer thickness fluctuations, etc. As a result, the actual layerstructure translations between adjacent layers may vary around the mean value. These variations may occur along the c* axis, as well as along crystallographic directions in the ab plane, modifying intensities of basal and non-basal reflections, respectively. The translations' variations may be described by a statistical distribution function of the distances between nearest layer neighbours. Depending on the nature of the interactions between the layers and the physical reasons for the translations' variations, disorder of the first and of the second type are distinguished (Guinier, 1964). Disorder of the first type along c* includes layer-thickness fluctuations which follow the same law for any layer pair, i.e. a single law describes both short-range and long-range order in a layer stack. Disorder of the second type follows a distance distribution law, in which there is no correlation between the distances of successive adjacent layers. It means that the distance distribution function between first nearest neighbours is independent of the distance fluctuations between other pairs of layer neighbours. Defects of the first type decrease the intensity of basal reflections with increasing diffusion vector (s = 2sinu/l) but not their profiles. In contrast, microstrains of the second type decrease intensity and increase broadening of basal reflections with s. The mathematical formalism describing intensity distribution for basal and non-basal reflections in periodic and mixed-layer structures containing disorders of the second type has been deduced by Drits & Tchoubar (1990). In particular, regarding the law of distribution of the translations it is assumed that it is a normal Gaussian having a specific standard deviation for each given layer type. Our experience shows that the quality of the fit significantly improved when the layer-thickness fluctuation effect in I-S samples was taken into account (Drits et al., , 2002aSakharov et al., 1999a;. Plançon (2002) described a similar approach, in which the law of distribution of translations between particular layer pairs is given as the number of particular translations occurring with particular probability. His aim was to explain an apparent contradiction between the experimental data obtained for the same smectite and I-S samples by different techniques. In particular, according to the small-angle X-ray scattering technique, I-S particles consist of large numbers of parallel layers, whereas simulation of the I-S diffraction patterns shows very thin CSDs. To illustrate the opportunity of the approach, XRD patterns were calculated for two random I-S models with W I = W S = 0.5. In one model a uniform distribution of CSD thickness between 2 and 13 layers was used and its XRD pattern was calculated using a conventional diffraction technique. In the second model, in each crystallite consisting of 50 layers, an arbitrary distribution of additional distances between each type of layers of 2, 4, 6 and 8 A Ê with respective probability of 0.05 for each of them was used. Comparison of the XRD patterns showed that the second model significantly improves the degree of resolution of the first low-angle basal reflection but led to displacement of position of the second basal reflections with respect to that observed in the XRD pattern of the first model. The problem in the practical application of this approach is that modes of distributions of translations between different layer pairs in a particular layer mineral is not known a priori. It would be interesting to compare experimental XRD patterns of mixed-layer minerals with those calculated using this approach.
Outer surface layers covering core structure of periodic and mixed-layer minerals Tsipursky et al. (1992) were the first to demonstrate that layers covering outer surfaces of crystallites and layers of the core structure may be different. In particular, it was shown by highresolution transmission electron microscopy (HRTEM) that in illite samples, the outer surface layers of the particles are represented by kaolinite layers.
Recently Ma & Eggleton (1999), using HRTEM, showed that the surface layers in kaolinite crystallites may also have different structure. In particular, a 10 A Ê pyrophyllite-like (or low-charge beidellitelike) layer is located on one side of these crystallites as a surface layer. Kaolinite particles of the other type contain one or several 10 A Ê layers on their both sides.  also observed a crondstedtite crystal terminated by a chlorite unit layer. Ma & Eggleton (1999) noted that the 10 A Ê surface layers were not detected by XRD. However, Sakharov et al. (1999c) showed that for some clay minerals, the difference between structures of outer and core layers may have significant influence on the diffraction effects. These authors developed a special mathematical formalism to simulate diffraction effects from periodic and mixed-layer structures having different outer surface layers. In particular, simulation of XRD patterns shows that chlorite crystal models consisting of the same amount of 2:1 layers but terminated by either 2:1 or brucite-like layers have significantly different distributions of basal reflection intensities, which, in turn, differ from the distribution of basal reflection intensities for periodic chlorite crystals containing the same amount of 2:1 layers and terminating with a brucite-like layer on the one side of the crystals and with a 2:1 layer on the other side. In fact, relative intensities of 001 reflections have maximum and minimum values when chlorite crystals of a given composition are terminated on both sides either by brucite-like or by 2:1 layers. Similar diffraction effects were observed for mixedlayer chlorite-smectites having different outer surface layers. It was shown that for investigated chlorite samples the best fit between experimental and calculated XRD patterns was obtained for chlorite crystallites terminated by brucite layers on both sides.

Mixed-layer minerals consisting of incommensurate layers
Depending on the dimensions and shape of unit cells of different layer types, mixed-layer structures can be grouped in three categories: (1) structures with identical two-dimensional unit cells in different layer types. Combination with a similar anion structure of the basal surface in different layers ensures stable interlayer bonds in these minerals; (2) commensurate structures in which the unit cell of one layer type is a subcell of the other type layer unit cell (e.g. chlorite); and (3) incommensurate structures, or structures whose layers have their own two-dimensional periodicities without rational relation between the cell parameters in alternating layers (Drits, 1987b;Organova, 1989;Makovicky & Hyde, 1992). Incommensurability may have different nature: the alternating layers may either differ in structure, or having a similar structure, differ in the cation and/ or anion composition. Incommensurate structures may have either ordered or random interstratification of layer types.
Selected area electron diffraction (SAED) is especially powerful for revealing mixed-layer minerals consisting of incommensurate layers (Drits, 1987b(Drits, , 1997. The combination of SAED and energy dispersion analysis (EDA) widely extends the possibilities of the structural investigation of fine-dispersed minerals, especially in the case of polymineral samples. These opportunities are illustrated by the results of the structural study of asbolanes, a very peculiar group of phyllomanganates, containing 'alien' cations (Chukhrov et al., 1982(Chukhrov et al., , 1989Manceau et al., 1982;Drits, 1987b).
It was found that asbolanes consist of two regularly interstratified incommensurate layer types having different cation compositions. Due to different lateral dimensions of the layer types, the asbolane structure can be described on the basis of two or even three sub-lattices having different a, b parameters but identical periodicity along the c axis. A very peculiar feature of asbolane structures is that the layers of one type are two-dimensionally continuous within CSDs whereas the layers of the other type have island-like structure.
EDA has shown the existence of several asbolane varieties differing in chemical composition. Application of X-ray absorbtion spectroscopy (XANES and EXAFS) has shown that cations of different nature and valency have a strong tendency to segregation (Manceau et al., 1992). For example, in the so-called Co-Ni asbolane, each layer type contains only cations of a given type, and octahedral layers MnO 2 , Ni(OH) 2 and CoO(OH) coexist within the same crystals. Among the other asbolane varieties, there is Ni-asbolane, in which Mn 4+ octahedral layers alternate with island-like Ni-containing brucite-like layers. A very exotic example is pure manganese asbolane where Mn is present in three incommensurate layers as tetra-, triand divalent cations.
Using SAED and EDA techniques, a mixed-layer incommensurate asbolane-buserite was found among oceanic Fe-Mn nodules (Chukhrov et al., 1989). The buserite structure consists of octahedral Mn 4+ layers separated by exchangeable cations and water molecules and has a period along the c axis equal to 10 A Ê . Irregular interstratification of 10 A Ê buserite and 10 A Ê asbolane layers creates 10 A Ê pseudo-periodicity along the c axis. Therefore the XRD patterns of asbolane-buserite in the air-dried state are similar to those of poorly crystallized asbolanes, buserites and todorokites. However, under the vacuum conditions of an electron microscope, the 10 A Ê buserite layers are transformed into 7 A Ê birnessite layers and the mineral is transformed into a mixed-layer asbolane-birnessite with the random interstratification of incommensurate 10 A Ê and 7 A Ê packets. This leads to a nonrational series of basal reflections which is an indication of the interstratification (Chukhrov et al., 1989).

Mixed-layer structures in which the layer thickness of one of the layers is a multiple value of the other layer thickness
A particular case of such a structure is interstratification of 14 A Ê chlorite and 7 A Ê serpentine layers (Reynolds, 1988). Figure 1 shows that even 5% of 7 A Ê layers in a mixed-layer chloriteserpentine structure significantly decreases the intensities of odd reflections and increases their width in comparison with those of even reflections (Ivanovskaya et al., 1999;. This conclusion is consistent with results obtained by Ryan and Reynolds (1997) for interstratified chlorite-serpentines. On the contrary, when the amount of 7 A Ê layers increases up to 70 -75% the diffraction effects turn out to be practically insensitive to the presence of 25 -30% of 14 A Ê layers (Fig. 2), so that the chlorite component cannot practically be detected by XRD in this case. Its presence can be determined more effectively by HRTEM (Bailey et al., 1995).

Interstratification of the layers having similar but not identical layer thicknesses
A family of mixed-layer structures in which layers and/or interlayers differ from each other in occupancy and distribution of isomorphous cations is widely spread among natural and synthetic layer compounds. In particular, mica interlayers may be occupied by K and Na or K and NH 4 . The problem is to determine orderdisorder in the distribution of these cations because they may have different distributions when interlayers have the same average composition. For example, K and NH 4 can be distributed among different interlayers according to one of the two limiting cases. In the first one, K and NH 4 are distributed homogeneously in each interlayer. In the second model each interlayer contains either K or NH 4 . In other words, this model represents a mixed-layer structure in which K-bearing 9.98 A Ê illite and NH 4 -bearing 10.33 A Ê tobelite layers are interstratified. Drits et al. (1997b) worked out a special technique to determine the average NH 4 content, as well as the distribution of NH 4 and K in illite and I-S containing fixed NH 4 . The main problem in determination of fixed NH 4 in I-S by diffraction methods is that different proportions and distribution of the layer types modify positions, intensities and profiles of basal reflections much stronger than the presence of NH 4 . It was shown that the sensitivity of diffraction effects to the presence of NH 4 cations can be significantly increased if I-S are saturated by K, subjected to several cycles of wetting and drying, and dehydrated. After such treatment, K-bearing expandable layers collapse to 9.98 A Ê and the treated I-S will have similar periodicity. It was found that the positions of the basal reflections of the treated I-S can be used to determine the average content of fixed NH 4 , whereas comparison of widths of the basal reflections with different l provides quantitative information about the distribution of K and NH 4 over different interlayers. Note that certain precautions should be taken in the application of this approach to I-S containing >50% of expandable interlayers. The reason is that even after saturation of these interlayers by K, the effect of the layer fluctuations may be significant in disturbing the relationship between the basal reflections' widths due to the interstratification of K-and NH 4 -bearing interlayers.
Application of the 2 7 Al and 2 9 Si magic-angle spinning nuclear magnetic resonance (MAS NMR) for the crystal chemical study of heterogeneous clay minerals 27 Al MAS NMR spectroscopy determines very accurate [4] Al/ [6] Al ratios and 29 Si MAS NMR may be used for fitting to obtain [4] Al/Si ratios. These data are usually used for the study of the shortrange order in distribution of [4] Al and Si in tetrahedral sheets of various layer silicates (see below). However, the data obtained by the MAS NMR method can be used to determine crystalchemical features of mixed-layer minerals having complex structures. In particular, Lindgreen et al. (2002) developed a methodology by which the [4] Al/ [6] Al ratios, in combination with the results of the chemical analysis and the multispecimen method, allowed determination of reliable averaged cation composition in octahedral and tetrahedral sheets of 2:1 layers and in brucite-like interlayers sheets in the three-component illite-smectite-ditrioctahedral chlorite (tosudite) structures found in Cretaceous-Tertiary chalk. Coexistence of I-S with kaolinite or/and chlorite is typical for many natural clays.  and Drits et al. (2002a,b) showed how the 27 Al and 29 Si spectra can be used for a quantitative analysis of such mixtures and accurate determination of structural formulae of the I-S.

Application of the multispecimen technique
To apply this technique, a mixed-layer sample is saturated by different interlayer cations (Ca, Mg or Na) and XRD patterns for each specimen are recorded in identical experimental conditions in air-dry and glycolated states. A structural model for the studied mineral can be considered as corresponding to the actual structure if the experimental XRD patterns obtained from different specimens of the sample are successfully simulated using the same set of probability parameters describing content and distribution of the interstratified layer types. If a sample is a mixture of periodic, along c axis, and mixed-layer phases then simulation of the XRD patterns includes two steps. The first uses a trial-and-error approach to determine structural and probabilities parameters that provide a satisfactory agreement between experimental and calculated positions, intensities and profiles for each phase of a sample Sakharov et al., 1999a,b). In the second step, structural and probability parameters describing each phase are fixed, and the modified program of Drits and Sakharov (1976) automatically seeks the best agreement (minimum R-factor) between experimental and calculated XRD patterns by varying the contents of the phases in a sample. As a result, both the contents of each phase and the structural and probability parameters for the mixed-layer varieties are obtained (Drits et al., 2002a;Lindgreen et al., 2002).
The efficiency of the method is illustrated by results obtained in the structural study of I-S from oil-source Upper Jurassic shales from the North Sea (Drits et al., , 2002b. The problem was to answer two questions: first, what are the crystalchemical features of I-S formed during diagenesis in oil-source shales and, second, what is the structural mechanism for the diagenetic evolution of these minerals? Generally, in oil-source rocks of sedimentary basins, oil generation takes place simultaneously with diagenetic transformation of I-S. A link between these two reactions was demonstrated: NH 3 released during maximum oil generation is fixed as NH 4 cations in the NH 4bearing mica or tobelite layers formed from smectite or vermiculite layers of the I-S. This reaction leads to formation of four-component illitetobelite-smectite-vermiculite (I-T-S-V) minerals in a diagenetic interval called the 'tobelitizationwindow' (Drits et al., 2002b). A remarkable crystal-chemical feature of the representative collection of samples studied was the fact that the proportion of K-bearing illite layers, as well as the amount of fixed K cations per O 10 (OH) 2 in the studied mixed-layer minerals were constant and equal to (65±5)% and (0.38±0.02) cations, respectively, independent of sample location, depth and degree of diagenetic transformation. On the contrary, the amount of tobelite layers and fixed NH 4 cations increases with diagenesis.
Several important conclusions concerning the structural mechanism of I-S diagenetic evolution and the initial origin of I-S follow from the crystal chemical features of I-T-S-V.
First, in oil-source Upper Jurassic shales, diagenesis is accompanied not by smectite illitization, as was generally accepted, but by smectite tobelitization. Indeed, the evolution of I-S is accompanied by selective sorbtion and fixation of NH 4 cations in former smectite and vermiculite interlayers.
Second, the constant values for the illite layers and the number of K cations can be considered as evidence of solid-phase transformation of the I-T-S-V structure. Indeed any other mechanisms would have destroyed this constancy. The conclusion concerning the structural mechanism of I-T-S-V formation is confirmed by the same shape and size of clay particles for all studied samples. The fact that the amount of illite layers does not depend on location and degree of diagenesis of the mixedlayer minerals may be considered as evidence of a common parent detrital material.
Along with the North Sea samples, I-S minerals from Cambrian Baltic shales have been studied, and formation of tobelite layers has been also found . Therefore, it is quite plausible that tobelitization of I-S and I-S-V in diagenetically altered oil-producing sedimentary basins is a common phenomenon and the presence of tobelite layers in I-T-S or I-T-S-V in oil-source rocks may be considered as independent evidence for oil generation even after migration or thermal decomposition of the oil.
The multispecimen method in combination with chemical analysis and 27 Al MAS-NMR spectroscopy was applied to the study of the structure and diagenetic transformation of illite-containing mixedlayer minerals from North Sea Cretaceous-Tertiary clays (Lindgreen et al., 2002). It was shown for the first time that, in contrast to burial diagenetic I-S transformation, early diagenesis of North Sea oil field and Danish outcrop chalk is accompanied by a solid-state process during which Mg cations were fixed in former smectite interlayers of the I-S forming brucite-like sheets and, as a result, an illitesmectite-ditrioctahedral chlorite (tosudite) structure was formed. During later diagenesis, a neoformation of a tosudite and trioctahedral chloriteberthierine took place.
Thus the new methodologies have revealed new diversity in structural and chemical heterogeneity of MLM, provided new insight into the structural mechanism of I-S evolution during oil generation and chalk rock formation, and discovered new natural processes.

Determination of the actual mutual layer arrangements in layer silicate structures
As in the case of irregular mixed-layer minerals, X-ray structural study of stacking faults in layer minerals consisting of identical layers is based on a trial-and-error approach (Drits & Tchoubar, 1990;. The nature of stacking faults depends on the structure of a mineral under study. For example, in the case of micas, stacking faults, as a rule, are confined to layer rotations, whereas in the case of chlorite, serpentine and kaolin minerals stacking faults may result from layer displacements, rotations and reflection operations. As a consequence, determination of the nature of stacking faults in these minerals by XRD is a complex problem. Investigation of stacking faults in kaolin structures is demonstrative in this respect. Different models of stacking faults based on the idealized kaolinite structure have been proposed: ±b/3 layer displacements, ±120º layer rotations, octahedral vacancy displacements. Bookin et al. (1989) showed that these models do not account for the intensity distribution in the experimental XRD patterns. These authors achieved significant progress in the solution of the problem when the actual structural features of kaolinite and dickite layers and their interlayer displacements were taken into account. According to Bookin et al. (1989), the interstratification of right-hand and left-hand enantiomorphic kaolinite layers having equivalent unit cells defines the most probable stacking faults. In terms of their models, the possibility of a continuous series between kaolinite and dickite through 'dickite'-type faults seems hardly probable. Plançon et al. (1989), using simulation of diffraction effects, confirmed the validity of the models described by Bookin et al. (1989). In addition, it was shown that many samples are mixtures of two kaolinite phases having either high or low concentrations of stacking faults. This conclusion was confirmed experimentally by Bish & Chipera (1998). Zvyagin & Drits (1996) considered, in the most general form, the theoretically possible layer stackings which may lead to stacking faults in the real kaolinite structure. To do this, the authors took into account the actual distortions of inter-and intralayer displacements of adjacent octahedral and tetrahedral sheets, deviations of unit-cell parameters and normal projections of the c axis on the ab plane observed in periodic structures of dickite, kaolinite and nacrite. It was shown that along with displacements, rotations and reflection operations, analysis of deviations of the actual structure from the idealized one has key importance not only for the understanding, but also for the explanation of the formation of different polytypes, stacking faults, polytype transitions in the kaolin structures, as well as other minerals.
Significant progress in determination of the layerstacking sequences in layer silicates has been obtained by HRTEM (Banfield & Murakami, 1998;Kogure & Banfield, 1998;Kogure & Nespolo, 1999a,b, 2001Kogure et al., 2002 Using this approach, Kogure & Nespolo (1999a,b, 2001 and Kogure et al. ( , 2002 discovered the nature of stacking faults in the actual crystal structure of Mg-rich annite, oxybiotite and two cronstedtite varieties. For example, Kogure and Nespolo (1999b) obtained HRTEM images from a Ti-containing Mg-rich annite crystal along [110] and [010]. Analysis of these images allowed determination of the layer stacking sequence containing more than 100 layers in which layers in layer pairs were rotated with respect to each other by ±60º, ±120º and 180º, and sometimes were parallel to each other. Distribution of these layer pairs shows that fragments of 1M, 2M 1 , 2M 2 and 2O polytypes coexist in the mica crystal. Analysis of HRT EM images of crondst edtite from Lostwithiel, England recorded along two directions rotated with respect to each other by 30º shows that thin fragments of the polytype groups A and C are interstratified in the crystal of this mineral. This observation shows that previous suggestions that polytype modifications of different groups do not occur together are not correct.
Very peculiar planar defects were observed in oxybiotite from the Ruiz Peak ash flow, New Mexico (Kogure & Nespolo, 2001). One of these kinds of defects is represented by tetrahedral double layers as in the hexacelsian (BaAl 2 Si 2 O 8 ) type structure, whereas the other kind corresponds to a set of brucite-like octahedral layers parallel to a (001) plane. In contrast to 2:1 layers, in the tetrahedral double layers, octahedral sheets are absent and the upper and lower tetrahedral sheets share their apical oxygen atoms.
One of the main advantages of the approach in which images are recorded in two crystallographic directions is that it provides a new insight into the structural mechanism of phase transformations (Banfield & Murakami, 1998;Murakami et al., 1999;Schmidt & Livi, 1999). For example, investigation of the chloritization mechanism of biotite in a granitic rock showed that, typically, two biotite layers transform to one chlorite layer. Less commonly, mica interlayers are replaced by brucitelike sheets without distortion of adjacent 2:1 layers. It was found that the chlorite polytype in biotitechlorite intergrowths is represented by IIbb or, more seldom, interstratification of Ibb, Iab, IIab and IIbb varieties (Kogure & Banfield, 2000). Murakami et al. (1999) showed that the saponiteto-chlorite reaction proceeds via corrensite and, thus, includes two stages: first, from saponite to corrensite, and then from corrensite to chlorite. Both stages occur though dissolution and precipitation. These data show that corrensite, as well as chlorite and smectite, should be considered as independent structural units participating in the formation of mixed-layer chlorite-corrensite-smectites. Previously these suppositions were made by Beaufort et al. (1997).
Even these several examples demonstrate the great contribution of HRTEM to the study of the structural and chemical heterogeneity of minerals. However, in some cases, opportunities for this method are limited. For example, this is probably the case of dioctahedral micas and I-S, in which structural heterogeneity is associated with interstratification of trans-vacant (tv) and cis-vacant (cv) 2:1 layers. It is more effective to study this heterogeneity by diffraction methods and thermal analysis.  showed that in the 2:1 layers of smectites, as a rule, one of two symmetrically independent cis-octahedra is vacant, whereas illite normally has tv layers. Accordingly, formation of illite layers from smectite layers during smectite illitization should lead to an increase i n proport i on of tv 2:1 l ayers. Coexistence of cv and tv layers in I-S formed in volcanic and sedimentary rocks was described recently in many papers (McCarty & Reynolds, 1995, 2001Drits et al., , 2002aAltaner & Ylagan, 1997;Cuadros & Altaner, 1998a;Ylagan et al., 2000;Lindgreen & Surlyk, 2000). However, physicochemical conditions, structural mechanisms of formation and other factors responsible for interstratification of cv and tv layers in I-S remain poorly understood. , Cuadros and Altaner (1998b) and Ylagan et al. (2000), in bentonites and hydrothermally alerted pyroclastic material, found that an increase in illite layers is a accompanied by increasing tv layers. On the contrary, Reynolds (1995, 2001) showed that in K-bentonite from the Appalachian Basin the amounts of cv layers in I-S do not correlate with expandability and rotational disorder. Moreover, interstratification of tv and cv layers may occur even in non-swelling illites. For example, an illite sample from the Amethyst Vein System, Colorado, initially described as 3T illite (Horton, 1983) in fact, contains almost equal amounts of tv and cv layers (unpublished data).

Coexistence of tv and cv layers in I-S; cvcontaining I-Sa possible indicator of their initial volcanic origin
There are contradictory data concerning the relationship between cation composition of 2:1 layers and distribution of octahedral cations over transand cis-sites. According to , in dioctahedral smectites this distribution depends on the degree of Al for Si substitution and the homogeneity of octahedral cation composition. According to their classification, beidellites and nontronites are trans-vacant, whereas montmorillonites and some Al-rich smectites, in which the layer negative charge is located in both octahedral and tetrahedral sheets of the 2:1 layers, are cis-vacant. Sainz-Diaz et al. (2001b) carried out theoretical modelling of cv and tv sheets of illites and smectites of various chemical compositions using empirical interatomic potentials. They concluded that, although the energy difference between the tv and cv layers is small, the cv layers are more stable when the cation composition of the layer is more smectitic. These results are consistent with recent experimental data showing that most smectites are cis-vacant (Drits et al., , 1998bCuadros & Altaner, 1998a,b;Ylagan et al., 2000;Cuadros, 2002). On the contrary, according to McCarty and Reynolds (2001) the amount of cv layers in the I-S increases with tetrahedral Al and decreases with octahedral Mg and Fe content, i.e. when the cv layer composition is more illitic. These observations are consistent with the fact that most structurally well ordered cv 1M illites contain 2:1 layers consisting mostly of Al and Si (Zvyagin et al., 1985;Drits et al., 1993;Lanson et al., 1996;Reynolds & Thomson, 1993). Thus cv layers should be stable in smectites, for which the layer negative charge is mostly located in the octahedral sheets of the 2:1 layer, and in illites, for which this charge is located largely in the tetrahedral sheets.
One of the plausible explanations for this tendency may be related to different interlayer structures of 1M illites consisting either of tv or cv layers. In the case of tv 1M illite the upper and lower tetrahedral sheets of the 2:1 layer are related by a mirror plane parallel to the ac plane. In the case of cv 1M illite, these sheets are rotated with respect to each other by 120º . It is well known that in dioctahedral micas the interlayer cavity has a slightly irregular form due to tetrahedral tilt and basal oxygen surface corrugation (Bailey, 1984). In the interlayer cavity of tv 1M illite, each K cation is located in a distorted octahedron in which two basal oxygen atoms of adjacent layers related by a space diagonal passing through the mirror plane are shifted in opposite directions inside the layers. Therefore even in tv 1M illite having phengitic cation composition, the distances between the K and two 'depressed' basal oxygen atoms are greater than those with the other four 'inner' oxygen atoms (Drits, 1987b). In contrast, the interlayer K in cv 1M illite has an environment which is quite similar to that of interlayer K in 2M 1 muscovite because the upper and lower tetrahedral sheets in each cv 2:1 layer are rotated with respect to each other by 120º. Therefore a very small displacement of interlayer K along the two-fold axis toward the nearest vacant octahedral site significantly equalizes the lengths of the K -O bonds. The hypothesis that almost (Mg,Fe)-free illites consisting of cv layers are more preferable to tv 1M illites having the same cation composition can be supported at least by two experimental observations. Drits et al. (1993) described an association of periodic tv and cv 1M illites from hydrothermal alterations around uranium deposits located in the Athabasca basement (Canada). It was shown that cv 1M illites in coarser size fractiona are significantly more abundant than in the fine fraction. Lanson et al. (1996) studied kaolin illitization processes during burial diagenesis of Rotliegend sandstones and observed the coexistence of tv and cv 1M illites, as well as the increase in the abundance of cv 1M illite with temperature and depth. Both of these facts can be considered as evidence that under low-temperature diagenesis and hydrothermal conditions, Al-rich cv 1M illite is more stable than its tv 1M polymorph.
According to the literature, structural mechanisms of formation of I-S containing interstratified tv and cv layers may be different. Cuadros and Altaner (1998b) showed that smectite illitization in the bentonite I-S samples proceeds through the solidstate transformation (SST) mechanism. In this case, transformation of cv into tv layers occurs within I-S crystallites preserving significant morphological changes of the I-S particles. Evolution of cv/tv ratios in I-S during hydrothermal transformation of rhyolitic volcanoclastic from Dolna Ves, Slovakia, showed that the reaction was a SST of smectite to illite layers up to 50% illite layers. At this stage, all 2:1 layers have cv sites, whereas further increase in the amount of illite layers up to 90% was accompanied by a dissolution-reprecipitation process, during which cv layers were replaced by tv 2:1 layers .
The I-S from a hydrothermally altered rhyolitic hyaloclastite from Ponza Island, Italy were represented by a full series from pure cv smectite to almost pure tv illite through intermediate phases consisting of interstratified cv and tv layers (Ylagan et al., 2000). On the basis of abrupt changes in morphology, smectite illitization on Ponza involved a dissolution and recrystallization mechanism with multiple stages of nucleation and crystal growth. This means that synthesis of the I-S included simultaneous growth of tv and cv layers in each individual I-S crystal.
It is well known that the cv smectites are primary products of transformed volcanoclastic rocks of rhyolitic composition. Therefore, as a rule, I-S consisting of tv and cv layers are associated with volcanic materials (Drits, 1987a,b;McCarty & Reynolds, 1995, 2001Drits et al., , 1998bDrits et al., , 2002aCuadros & Altaner, 1998b;Ylagan et al., 2000). Thus, one may assume that structures consisting of tv and cv layers originate from altered volcanic material (Drits et al., 1998b).
In contrast, I-S primarily formed from weathered illite typically consist of tv 2:1 layers independent of the content of I and S layers, such as found for I-S from the Upper Jurassic oil-source rocks . Investigation of I-S from black Cambrian shales (Baltic area) showed that samples having a small degree of diagenetic transformation consist of a physical mixture of detrital illite and I-S composed of tv and cv layers . It was concluded that the parent material of the I-S was volcanic.
Investigation of I-S in one Upper Jurassic Kimmeridgian mudstone core from East Greenland showed that the I-S samples located at different levels in the core differ from each other in the amounts of cv layers (Drits et al., 2002a). In particular, I-S at the bottom (37 m) and in the middle (13 m) of the core contained 60 -80% cv layers and 60% of expandable layers, whereas in I-S located at 12.4 m and 17.7 m depth, tv layers prevail (~80%) although content of expandable layers in their structure was still high (45%). It was calculated that I-S with a large cv layer content were formed from cv smectite which is probably of volcanic origin. Thus these findings of I-S reflect episodes of volcanic activity during Kimmeridgian time, as suggested by Lindgreen and Surlyk (2000).
Despite some progress in the study of I-S, further investigations are required in order to clearly determine the crystal chemical nature of I-S of different origins, the mechanisms and dynamics in their structural transformations and the factors controlling the coexistence of 2:1 layers differing in cation distribution. Reynolds (1992) showed that there is no threedimensional coherence across the expandable interfaces that separate stacks of thin illite fundamental particles (Nadeau et al., 1984). Therefore, a randomly oriented specimen of I-S diffracts in three dimensions like a randomly oriented set of thin illite crystals.

Diffraction methods for determination of tv and cv layers in illite and I-S. Generalization of Mé ring's rules
Different distributions of octahedral cations over transand cis-sites in 2:1 layers are accompanied by specific distortions of cisand trans-octahedra (Drits et al., , 1995. In a tv layer, the centre of the ditrigonal ring of the upper tetrahedral sheet is shifted from the centre of the lower tetrahedral sheet in projection on the ab plane by more than the ideal -a/3 value. As a result, in tv 1M illites, projection of the c axis on the ab plane (c cosb) is equal to T tv = -(0.38a70.40a) depending on the octahedral cation composition of the 2:1 layers. In particular, it was found that T tv values vary from -0.400a to -0.390a for almost Fe,Mg-free illites and from -0.386a to -0.383a for illites having phengite cation composition (Bailey, 1984;Drits et al., , 1993. In the cv layer this shift is < -a/3, and in cv 1M illites T cv = c cosb = -(0.30 -0.32)a (Drits et al., , 1993Zvyagin et al., 1985;Reynolds & Thomson, 1993). Thus, interstratification of tv and cv layers is associated with interstratification of two different Heterogeneity of layer silicates translations and is accompanied by significant variations of positions, intensities, and profiles of non-basal reflections in the powder XRD patterns.
There are two diffraction methods for the determination of coexistent tv and cv layers in mica structures which will be considered below in succession. One of them is based on calculation of XRD patterns from different structural models containing interstratified tv and cv layers, as well as rotational and translational stacking faults . Atomic coordinates for unit cells for tv and cv 1M illites used in these calculations were obtained by Drits et al. (1984). Examples of the application of this approach are given by , McCarty & Reynolds (1995, 2001, Ylagan et al. (2000).
The structural study of smectites is a difficult task because of their turbostratic structure. However, even for these minerals, essential information concerning the distribution of octahedral cations over trans and cis sites can be obtained by simulation of diffraction effects. Indeed,  showed that the intensity of the twodimensional 02-11 band for dioctahedral micas and dehydrated smectites is constant if their layers are either tv or cv. In contrast, if octahedral cations are randomly distributed over trans and cis sites within the same layer, the intensity of the 02-11 band decreases significantly. At the same time, the intensity of the 20-13 band does not depend on the cation distribution. Therefore the intensity ratio of the 02-11 and 20-13 bands is sensitive to the distribution of octahedral cations between cis and trans sites within the same octahedral sheet.
Taking into account this result, Manceau et al. (2000) calculated the intensity of the 02-11 and 20-13 bands for different occupancies of trans and cis sites and compared them with the experimental patterns of four different nontronites having turbostratic structures. Best fits were obtained by assuming 100% occupancy of cis-sites in the nontronite samples within the detection limit of 5% of total Fe.
The scientific significance of this result arises from the fact that the actual occupancy of trans and cis sites is important for reliable interpretation of magnetic properties of nontronites. As a matter of fact, oblique texture electron diffraction and XRD showed that nontronites are trans-vacant (Besson et al., 1983a;. However, the accuracy of these techniques was not sufficient to exclude completely the presence of some Fe 3+ in trans-octahedra. Therefore, to explain the non-ideal antiferromagnetic behaviour of SWa-1 nontronite at low temperature, Lear and Stucki (1990) assumed that~13% of trans-sites were occupied by Fe 3+ .
Another way to determine the distribution of octahedral cations over trans and cis sites in a smectite structure is related to an artificial increase of its 3D order. To do this, smectite samples are saturated with large anhydrous K or Cs cations, and subjected to wetting-and-drying cycles, and then dehydrated by heating and using vacuum (Besson et al., 1983b;Cuadros, 2002).
Powder XRD patterns of the treated smectites contain significant intensity modulation in the diagnostic region of 02l and 11l reflections. Recently, Cuadros (2002) studied Cs-saturated smectites of different cation compositions and using simulation of the experimental XRD patterns. He showed that the studied nontronite samples consist of tv layers whereas the montmorillonite samples are cis vacant.
The second method for the determination of coexistent tv and cv layer content is based on the ability of XRD to average the structural parameters of a defective sample (Drits & McCarty, 1996). For illites consisting of tv and cv layers, diffraction should average interstratified interlayer translations. This means that a given interstratified structure should be characterized by a statistically weighted interlayer displacement equal to T ef = W cv T cv + W tv T tv where W cv and W tv are the occurrence probabilities for cv and tv layers; T tv and T cv are the projections of the c axis on the ab plane for tv 1M and cv 1M illites, respectively. Taking into account that W cv + W tv = 1 the equation for T ef can be transformed to: W cv = (T tv -T ef )/(T tv -T cv ). Thus, the cv and tv layer contents can be calculated because T tv and T cv are known for illite of a given cation composition (Drits et al., 1993(Drits et al., , 1995 and T ef can be calculated using experimental values of d(11l) and the formulae given by Drits & McCarty (1996).
There is a simple and clear way to interpret the nature of diffraction effects from structures consisting of interstratified cv and tv layers. Figure 3 shows XRD patterns corresponding to periodic tv and cv 1M micas, respectively. In these patterns, reflections having the same hkl indices have different intensities and, in addition, 11l reflections having the same 1 values have different positions because of different c cosb values. The middle curve corresponds to a structure in which tv and cv layers are interstratified in equal proportions. One can see that 11l reflections are shifted and located between the 11l reflections corresponding to pure tv and pure cv micas whereas 02l reflections have the same positions for the three patterns because they do not depend on b. In general, the positions of 11l reflections depend on the content of tv and cv layers and will migrate within the interval restricted by the vertical lines in Fig. 3. This observation has a general meaning -interstratification of two interlayer translations shift hkl reflections sensitive to these translations and the positions of these reflections become irrational, i.e. they do not obey Bragg's law.
To account for the behaviour of non-basal reflections for any layer defective structures in which two translations are irregularly interstratified, Méring's rules can be generalized. According to these generalizations, the non-basal reflections are located between the neighbouring hkl reflections of the periodic phases whose elementary layer units are interstratified. The positions of these reflections depend on the relative proportions of the interstratified interlayer translations. These simple rules are very useful for the identification of the nature of the interstratified components and are effectively used for determination of tv and cv layer contents in I-S (Drits & McCarty, 1996;, as well as for the determination of stacking faults in various birnessite varieties (Drits et al., 1998c;Lanson et al., 2000Lanson et al., , 2002 and in layer double hydroxides (Drits & Bookin, 2001).

Determination of tv and cv layers in I-S having turbostrat ic structures
Neither of the diffraction techniques described above can be applied if powder XRD patterns do not contain three-dimensional diffraction maxima in the diagnostic region containing 11l and 02l reflections. In this case the proportions of tv and cv layers can be determined with the help of the third method based on the analysis of losses of structural water during dehydroxylation (Drits et al., 1998b). This technique employs DTA in combination with evolved-water analysis. As a matter of fact, for quite a long time it remained unclear why the dehydroxylation temperature of fine dispersed and highly disordered dioctahedral smectites is 150 -200ºC higher than that of illites which have very ordered structures. Moreover, I-S often have two dehydroxylation maxima at different temperatures. On the other hand, certain so called 'abnormal' illites have the same dehydroxylation t emperatures as that of t ypical smecti te. Interpretation of the observed thermal effects was given when different structural features of tv and cv layers, as well as different structural mechanisms of their dehydroxylation were taken into account (Drits et al., 1995;Muller et al., 2000a,b,c). For tv layers, dehydroxylation occurs in one stage, when each two adjacent hydroxyls are replaced by a residual oxygen atom, which leads to five-fold coordination for Al. In this case dehydroxylation occurs at temperatures below 600ºC, independent of whether these layers belong to an illite or smectite.
In cv structures, dehydroxylation occurs in two stages (Drits et al., 1995). At the first stage, each two adjacent hydroxyls are replaced by a residual oxygen atom, and the cations that originally occupied cisand trans-sites become 5-and 6-coordinated, respectively. The resulting structure, however, is unstable, and any movement of the cation within the 6-coordinated octahedron should be unstable to compensate for the charge imbalance. Therefore, when additional thermal energy is applied, each cation initially occupying a transsite migrates to the nearest five-fold coordination polyhedron corresponding to the initially vacant cissite, and the resulting structure is the same as that in the case of dehydroxylated tv layers. This complex structural rearrangment of cv layers during dehydroxylation reaction increases the dehydroxylation temperature of cv smectites and cv illite up to 650 -700ºC. It is likely that another major factor responsible for higher dehydroxylation temperature in cv layers is the considerable lengthening of the OH -OH edges in cv layers in comparison with the OH -OH edges in the tv layers: the longer the distances, the greater the thermal energy required to provide the opportunity for hydrogen to jump to the nearest OH group to form a water molecule (Drits et al., 1995).
Comparison of the results obtained by XRD and thermal analysis showed that losses of structural water above and below 600ºC occurred during dehydroxylation of smectites, I-S and illites are proportional to the cv and tv layer contents (Drits et al., 1998b(Drits et al., , 2002aLindgreen et al., , 2002. Zvyagin et al. (1985) were the first to describe a monomineral Al-rich 1M mica consisting of cv layers. Later Reynolds & Thomson (1993) and Drits et al. (1993) also described Al-rich cv 1M illites. They noted that the tv-3T and cv-1M polymorphs produce similar diffraction effects, i.e. in their patterns, hkl reflections have similar intensities and close positions. These features explain the reason for confusion with identification of tv-3T illite. Careful analysis of the experimental data published in the literature showed that illite varieties described as tv-3T (Warshaw, 1959;Ey, 1984;Halter, 1988;Horton, 1983) in fact correspond to cv-1M (Drits et al., 1993). Reliable identification of tv-3T and cv-1M illites is possible if their XRD patterns contain sharp diffraction maxima. However, the broader reflections from illite and I-S (due to rotational disorder) preclude the precise location of peak positions and make it difficult to determine their actual layer stacking.

Some problems in the identification of the actual structure of dioctahedral micas
An even more complex problem is to distinguish a periodic tv-3T mica polytype from mica structures in which tv and cv layers are interstratified in such a way that each layer, independent of octahedral cation distribution, is rotated with respect to the preceding one by 120º, as in the periodic 3T structure. In this case, positions of hkl reflections will be the same as those in the periodic 3T structure. Figure 4 shows that XRD patterns of the periodic 3T dioctahedral mica (solid line) and the interstratified structure almost coincide, when the cv layer content is 20% and are still very similar when the cv layer content reaches 40%. Formally, the quality of these patterns is quite acceptable for the Rietveld refinement, but the refined results would be wrong if the initial model consisted only of tv layers. This result is not surprising because XRD patterns of tv 3T and cv 3T have the same positions and shape of maxima and differ from each other mainly by distribution of 10l reflection intensities.
Diffraction effects from a tv 2M 1 structure are more sensitive to the presence of cv layers. However, the XRD patterns of a periodic tv 2M 1 structure and that containing 20% of cv layers are very similar (Fig. 5a).
Recently, Pavese et al. (2001) studied 3T and 2M 1 powder phengites using low-temperature neutron diffraction and the Rietveld technique. They found that occupancies of trans-octahedra of these micas are not zero but roughly equal to 0.18 and 0.12 atoms per octahedron. The title of their paper contains a question: are these occupancies real or artefact?
To explain the observed results the authors postulated a stacking disorder consisting of ±b/3 slips in the octahedral sheet of the 2:1 layers occurring in 2M 1 and 3T phengites. These slips move the upper plane of the apical oxygens atoms with respect to the lower plane in the octahedral sheet. In order to preserve the interlayer coordination for K cations, the ±b/3 shifts in the n th layer require the same shift for the layer located just above it. The presence of such defects is accompanied by displacements of some octahedral and tetrahedral cations as well as oxygen atoms from their standard positions to those which are unoccupied in the defect-free polytypes. The contribution of these 'shifted' atoms disturbs the reflections' intensities. Due to the ability of diffraction to average structural parameters, an XRD pattern from the ±b/3 disordered mica structure should be treated in terms of the unit cell in which occupancies can appear in positions generated by the structural disorder. In particular, the position of a filled cis-site comes to that corresponding to a vacant trans-site in the defectfree structure. Pavese et al. (2001) assumed that the scaling of the structural factors should greatly smooth the effect of the contribution of the shifted atoms to the reflections' intensities, but they did not estimate this effect quantitatively.
Analysis of the XRD patterns calculated from the ±b/3 disordered 2M 1 and 3T models shows that the main contribution to the modification of the patterns results not from the partial occupancy of trans-sites in the defective layers but from the disordered displacements of the tetrahedral sheets and oxygen atoms of the octahedral sheets in these layers by ±b/3. Indeed a partial occupancy of transsites redistributes intensities of some reflections but does not change the background. In contrast, as can be seen in Fig. 5b the ±b/3 layer displacements in the defective 3T phengite model decrease the intensities of all 10l reflections and dramatically increase the background. These features are typical of the XRD patterns of the ±b/3 disordered phyllosilicates (Drits & Tchoubar, 1990;. As a consequence, the background increases and the 10l reflection intensities decrease so dramatically that the scaling of the structural factors cannot smooth this effect. An alternative and crystallochemically more plausible explanation of the results obtained by Pavese et al. (2001) is associated with interstratification of tv and cv layers in the studied micas. Diffraction effects from tv/cv 2M 1 and tv/cv 3T models can also be treated in terms of the averaged unit cell in which both transand cis-sites should be partially occupied. It means that the main modification of the XRD patterns from the tv/cv 3T structures appears as a result of the partial occupancy of transsites mimicking the presence of cv layers in the actual structure. Moreover, the same reason might explain the uncertainty in the literature concerning the cation distribution over octahedral and tetrahedral sites in 3T phengites determined by the Rietveld technique (Pavese, 2002). An independent means by which to solve the problem is to use thermal analysis, taking into account that tv and cv layers have different dehydroxylation temperatures.

Simulation of two-dimensional distribut ions of isomorphous cations in tv and cv 2:1 layers
Spectroscopic methods are a very effective tool for determining isomorphous cation distribution, since they probe local cation environments and can detect short-range order in the cation arrangements. Information concerning the vacancy distribution over transand cis-sites in octahedral sheets of the 2:1 layers is one of the basic requirements for correct interpretation of experimental data obtained by spectroscopic methods.
FIG. 5. Comparison of XRD patterns calculated for the periodic tv-2M 1 (upper curve, solid line) and tv-3T (lower curve, solid line) structural models as well as for the cv/tv-2M 1 structure containing 20% of cv layers (upper curve, dotted line) and for the ±b/3 disorder tv-3T structure containing 20% of defective T u and T l layers (lower curve, dotted line).
To make this clear, one can recall the history of the interpretation of Mössbauer spectra of Fe 3+bearing dioctahedral 2:1 layer silicates. For a long time these spectra were decomposed into two main doublets. One of them was assigned to Fe 3+ in transsites whereas the other was assigned to Fe 3+ in cissites (Heller-Kallai & Rosenson, 1981;De-Grave et al., 1985). Application of diffraction methods has shown, however, that Fe-rich dioctahedral layer silicates such as glauconites, celadonites and nontronites consist only of tv layers (Besson et al., 1983a;Sakharov et al., 1990;Manceau et al., 2000). Therefore, the conventional interpretation of the Mössbauer spectra had to be revised and a new approach for their interpretation was developed (Dainyak et al., 1984a(Dainyak et al., ,b,c, 1992Dainyak & Drits, 1987;Drits et al., 1997c). Its main assumption is that the observed difference in quadrupole splittings of Fe 3+ cations is caused by the distortion of Fe 3+octahedra due to different local cation environments around these octahedra. However, in cv smectites, cv illites and in I-S containing interstratified tv and cv layers, Fe 3+ can occupy cisas well as transoctahedra.
Two alternatives should be considered. The first one assumes that the observed difference in the quadrupole splittings of Fe 3+ depends mainly on local cation arrangements around a central Fe 3+ and that the location of Fe 3+ in cisor trans-octahedra does not significantly influence the quadrupole splitting values. Similarly, frequencies of OH IR vibrations depend mainly on the nature of cations bonded to OH groups, and the distribution of the cations between cisand trans-octahedra has a smaller effect on the frequencies. The alternative assumption includes a decisive or at least significant role for mutual disposition of OH groups for spectroscopic characteristics of dioctahedral 2:1 layer silicates. In this case, for example, the location of Fe 3+ in cisand trans-octahedra should have a stronger influence on the quadrupole splitting than the local cation environment. The results obtained by IR and Mössbauer spectroscopy for I-S samples in which either cv or tv layers prevail support the first alternative because the IR and Mössbauer parameters are almost independent of the content of cv and tv layers in the studied minerals (Drits et al., 2002a).
If these conclusions are valid, then transand cisvacant 2:1 layers in illite or smectite having the same cation composition and the same local cation environments around Fe and OH groups may have different octahedral cation distributions. For example, Fig. 6a shows the two-dimensional distribution of Al, Mg and Fe 3+ cations in a tv octahedral sheet in which each Fe 3+ is surrounded by one Fe 3+ and two Al cations and OH groups are bonded to 20 Al -Al, 12 Al -Fe 3+ and 8 Al -Mg cationic pairs. As can be seen in Fig. 6a, Fe -Fe pairs are oriented along directions rotated by ±60º with respect to the b axis. The cation distribution in a cv sheet in Fig. 6b has the same local cation environments around Fe 3+ and OH groups as those in the tv sheet shown in Fig. 6a. However, to provide these local cation environments Fe -Fe pairs must be oriented along the b axis.
In contrast, tv and cv layers having identical or similar two-dimensional cation distributions should be characterized by different spectroscopic parameters. For example, Fig. 7a shows the twodimensional cation distribution in a tv octahedral sheet in which Fe cations are segregated into chains in such a way that there are no Fe 3+ -Fe 3+ cationic pairs bonded to OH groups. It is clear that IR spectroscopic characteristics in the OH-stretching and bending regions are not sensitive to such a distribution of Fe 3+ cations and should be the same if the nearest positions of Al and Fe 3+ cations bonded to OH groups are exchanged. In this case, each Fe 3+ cation will be surrounded only by Al cations. However, if each local cation environment around Fe and its proportion are known from the Mö ssbauer spectrum interpretation, then, in combination with the IR data, the actual two dimensional cation distribution shown in Fig. 7a can be simulated. In contrast, the same segregation of Fe in cv layers must be accompanied by formation of a significant amount of Fe -Fe pairs bonded to OH groups whereas local cation environments around Fe remain the same as in tv layers (Fig. 7b).
Thus these two models should form quite similar Mö ssbauer spectra but different IR spectra in the OH-stretching and bending regions. If, for example, information about local environments around OH groups and Fe 3+ obtained for cv layers by spectroscopic methods is applied to tv layers, the simulated cation distribution will be incorrect.
These examples demonstrate that in order to simulate the actual distribution of isomorphous cations in dioctahedral layer silicates of complex cation composition, at least four requirements should be met: (1) interpretation of spectroscopic data should be carried out in the light of diffraction data, i.e. average occupancy and composition of crystallographic sites should be determined for the sample under study; (2) at least two complementary spectroscopic methods should be applied to the same sample; (3) reliable interpretation of the experimental data obtained by each method should be provided; and (4) computer simulation of the two-dimensional cation distribution should satisfy the experimental data obtained by different methods equally well.
Simultaneous satisfaction of these requirements is a complex problem and at present, there are only a few publications devoted to simulation of octahedral cation distribution in dioctahedral layer silicates. Remarkably, most of them appeared in the last 5 -10 years (Palin et al., 2001;Cuadros et al., 1999;Manceau et al., 2000;Sainz-Diaz et al., 2001a;Drits et al., 1997c;Dainyak et al., 1992;Schroeder, 1993;Schroeder & Pruet, 1996;Müller et al., 1997). The reason is that during this period significant progress has been achieved in the empirical and theoretical methodological developments of spectroscopic methods. In particular, a new level in the determination of local structural environments around heavy metal cations in various layer minerals has been achieved due to development of polarized EXAFS spectroscopy effectively combined with the conventional powder EXAFS technique (Manceau et al., 1988(Manceau et al., , 1998(Manceau et al., , 1999. 29 Si MAS NMR has been shown to be a sensitive probe of tetrahedral Si and Al order-disorder in alumosilicate minerals and in layer silicates in particular (Herrero et al., 1987;Herrero & Sanz, 1991;Palin et al., 2001;Vinograd, 1995). Lausen et al. (1999) suggested a procedure for iterative fitting of Si NMR spectra of I-S to determine cation compositions of tetrahedral sheets forming illite and smectite interlayers. New opportunites for MAS NMR spectroscopy are associated with the fact that for layer silicates of known cation composition, the intensity of the Al signal can be used to determine short-range order in Fe-distribution (Schroeder, 1993;Schroeder & Pruett, 1996;Cuadros et al., 1999).
The reliable interpretation of Mössbauer spectra of tv 2:1 micaceous minerals of various cation composition has been simplified by the opportunity to predict quadrupole splitting for each local cation environment around Fe 3+ (Drits et al., 1997c). In IR spectroscopy, determination of the frequency for each pair of cations bonded to OH groups provides reliable interpretation of IR spectra of dioctahedral mica-like minerals in the OH-stretching vibration region (Besson & Drits, 1997a,b).
Ab initio quantum mechanical modelling of IR frequencies of the OH groups in dioctahedral 2:1 layer silicates using Hartree-Fock (HF) and Density Functional Theory (DFT) provides a new insight into the physical mechanisms that govern the OH vibrations observed experimentally (Kubicki et al., 1996;Sainz-Diaz et al., 2000;Martinez-Alonso et al., 2002a,b). In particular, the modelled frequencies in the bending vibration region reproduce the experimental ones since their values are mainly determined by the nature of the neighbouring octahedral cations. In contrast, noticeable discrepancies between calculated and experimental frequencies in the stretching vibration region were explained by the influence of factors other than the nature of the neighbouring octahedral cations bonded to OH groups. Sainz-Diaz et al. (2000) found correlations between the atomic weights and the Milliken charges of the cations with the experimental and theoretical OH vibration frequencies, whereas Martinez-Alonso et al. (2002a,b) found that the mass of the octahedral cations does not affect the OH vibrations. It follows from their calculations that undersaturated charges of apical O atoms of the tetrahedral sheet can explain the observed discrepancies between the experimental and calculated OH-stretching vibrations.
Obviously, the greater the number of methods involved, the more comprehensive the structural information obtained. Each method, however, has its own advantages and limitations and therefore provides only a partial solution to the general problem of cation order-disorder in 2:1 dioctahedral phyllosilicates. Therefore, even if unambiguous interpretation of experimental data is provided, the actual cation distribution pattern can hardly be achieved even using a combination of various diffraction and spectroscopic methods.
The solution to this problem proposed by Dainyak et al. (1992), Cuadros et al. (1999) and Sainz-Diaz et al. (2001a) was based on computer simulation of the two-dimensional distribution of isomorphous octahedral cations in dioctahedral 2:1 phyllosilicates. In fact, this approach can be considered as a 'bridge' between the data obtained by different methods as well as between short-range and long-range cation order. Application of different spectroscopic methods along with the new methodologies shows that along with random cation distribution, clustering of isomorphous cations is wide-spread. Dainyak et al. (1992), using IR and Mö ssbauer techniques in combination with computer simulation, showed that glauconite B.Patom is composed of small illite-like and celadonite-like clusters. In addition, in the celadonite-like clusters there is a tendency for each trivalent cation to be surrounded by divalent cations and vice versa, providing homogeneous anion negative charge compensation. Similar regularities were found in celadonites, glauconites and Fe-illites by combining XRD, EXAFS, IR, Mössbauer and computer simulation (Drits et al., 1997c). Müller et al. (1997) showed, using XRD, EXAFS and IR, that Fe and Mg are segregated in small clusters in the aluminous octahedral sheet of the Camp-Bertaux montmorillonite. Cuadros et al. (1999) and Sainz-Diaz et al. (2001a) studied cation distributions in the octahedral sheets of bentonitic I-S using 27 Al MAS NMR, IR and inverse Monte Carlo simulation techniques. They found that Fe cations in the octahedral sheet have a tendency to be segregated and Fe segregation increases with illite proportion in the I-S. In contrast, Mg-Mg pairs were not detected, i.e. Mg cations are dispersed among other cations.
Almost random distribution of Fe, Mg and Al cations was found in SWa-1 nontronites by EXAFS and IR spectroscopies, although (Al,Mg)- (Al,Mg) pairs are preferentially aligned along the b axis and Fe-(Al,Mg) pairs along the [310] and [310] directions (Manceau et al., 2000).
A theoretical study of octahedral cation distribution in dioctahedral 2:1 layer silicates using ab initio calculations has been made by Sainz-Diaz et al. (2002). In particular, they found that Mg cations have a tendency to be dispersed in the octahedral sheet, in contrast to Fe 3+ cations that have a tendency to segregation.
In spite of the essential progress in the study of short-range order in cation distribution, the reliability of some results remains questionable. The main reason is the existence of unsolved problems in reliable interpretation of experimental data obtained by different spectroscopic methods. For example, assignments of OH stretching vibrations for dioctahedral smectites are still debatable (Madejová et al., 1994;Fialips et al., 2002a,b;Zviagina et al., 2002). Similar problems exist in identification of individual OH-bending frequencies for dioctahedral 2:1 layer silicates (Fialips et al., 2002a,b;Martinez-Alonso et al., 2002a,b). Recently Cuadros & Altaner (1998a,b) decomposed the bending vibration regions of IR spectra for bentonitic I-S samples and found that the cation composition of the octahedral sheets calculated from the integrated intensities of the decomposed OH bands correlated with structural formulae based on chemical composition (Cuadros et al., 1999). It was assumed that as in the stretching vibration region (Besson & Drits, 1997a;Slonimskaya et al., 1986), the integrated intensity of the individual OHbending band is proportional to the occurrence probability of a specific cation pair. In addition, this domain is not affected by the presence of residual water molecules and it seemed that decomposition in this region was easier because band assignments were more straightforward (Vantelon et al., 2001). However, Fialips et al. (2002a), analysing the bending OH-vibration region of SWa-1 nontronite, argued that the absorptivity of the different cationic pairs bonded to OH groups in this region could be different.
Predictions following from the theoretical calculations do not always correspond to actual cation distribution. For example, according to calculations made for the Mg 2 Al 2 octahedral composition of Caand K-bearing mica-like structures, the most stable distribution corresponds to a model where Mg cations are completely dispersed among Al cations and Mg-Mg cationic pairs are absent. However, according to the IR experimental data, leucophyll i t e s h a v i n g i d e a l i z e d c o m p o s i t i o n KSi 4 Al 1 Mg 1 O 10 (OH) 2 , contain a significant amount of Mg-Mg pairs bonded to OH groups (Besson & Drits, 1997a). Indirect but strong theoretical and experimental evidence for the existence of Mg-OH-Mg cation arrangements in cv montmorillonite from Camp-Bertaux was obtained from Méring & Glaeser (1954). Further theoretical and empirical investigations are required to provide unambiguous interpretation for spectroscopic data and thus to provide unambiguous cation distribution determinations.

Q U A N T I T A T I V E S P E C I A T I O N O F H E A V Y M E T A L S I N S O I L S , S E D I M E N T S A N D S O L I D W A S T E
Owing to their unique redox and sorption properties, clay minerals, Fe (oxyhydr)oxides and Mn oxides play a pivotal role in the transport and fate of heavy metals and other pollutants contaminating the surface of the earth. Therefore, a comprehensive structural and crystal chemical study of these minerals is required in order to understand the atomic-scale bonding mechanisms of metals incorporated in a mineral structure, oxidation state and local structural environments of these metals, and thus provide a solid scientific base for modelling the extent and kinetics of chemical reactions in soils, sediments and solid waste. Because the toxicity of metals depends strongly on their oxidation states and their bonding with the mineral host, determination of various forms of toxic metals in natural systems is a fundamental task of the environmental science. Soils and sediments, however, are multicomponent and open systems in which different heavy metals can be distributed heterogeneously within the complex heterogeneous assemblage of minerals. Therefore, for a long time, structural and chemical forms of trace elements in heterogeneous natural matrices were questionable. A new level in the quantitative speciation of heavy metals has recently been achieved due to the development of new methodologies based on advanced synchrotron X-ray techniques. In a set of publications, Manceau et al. (2000Manceau et al. ( , 2002aManceau et al. ( , 2003 have shown that combination of scanning X-ray microfluorescence (mSXRF), microdiffraction (mSXRD) and microextended X-ray absorption fine structure (mEXAFS) spectroscopy provides a new insight into uptake mechanisms of heavy metals by soil constituents determining structural forms of heavy metals and their contents in each particular soil mineral. The first two techniques can identify the host phase by mapping the distribution of elements and solid species, respectively, whereas mEXAFS can determine local structure of metals in a particular mineral and the molecular-scale bonding mechanisms of transition elements by the host phase. The unique opportunities of these techniques have been illustrated by Manceau et al. (2000Manceau et al. ( , 2002aManceau et al. ( , 2003, Strawn et al. (2002) and Isaura et al. (2002) who studied quantitative speciation of Zn and Ni in soil ferromanganese nodules, smelter-contaminated soils and Zn-contaminated sediments. In particular, it was shown that: in smectites the local structure around Zn is trioctahedral; in birnessite Zn cations are located in the interlayers above or below the layer vacant octahedra and can have four-and/or six-fold coordination, whereas in Zn-bearing Fe hydroxides, Zn cations have a ferrihydrite local structure. The study of Ni-bearing soil Fe-Mn nodules showed that the substitution of Mn 3+ for Ni in the Mn layers of lithiophorite is characteristic for such micronodules. Manceau et al. (2002b) presented the state of the art in the environmental geochemistry of heavy metals. ACKNOWLEDGMENTS I am grateful to S. Hillier and two anonymous referees for their valuable comments and corrections to the English and to B.A. Sakharov and B.B. Zviagina for their kind help in the preparation of the manuscript. This work was supported by the Russian Science Foundation.