X-ray diffraction by disordered lamellar structures. Theory and applications to microdivided silicates and carbons.

New methods for the determination of the nature, proportion, and distribution of structural defects in microcrystallized lamellar systems are of utmost importance not only to experimentalists but also to theoreticians. Mathematical formalism - indispensable for such analyses - is well-illustrated by various examples, allowing this method to be easily adopted and even to be applied to other solids with lamellar or pseudo-lamellar structures.

1 Overall Description of Imperfect Lamellar Crystals.- 1.1 Some Reminders on the Specific Characteristics of Crystals with a Triperiodic Structure.- 1.2 Range of Validity of the Direct Methods of Structural Analysis.- 1.2.1 Crystals with Point Defects.- 1.2.2 Crystals with Planar Defects.- 1.3 Indirect Structural Analysis of Partially Disordered Lamellar Systems. Principles of Their Modelization.- 1.4 Determination of the Structural Characteristics of the Layers.- 1.5 General Characteristics of Triperiodic Layer Stackings.- 1.5.1 Characteristic Translations of Layer Stackings.- 1.5.2 Polytypic Modifications.- 1.5.3 Triperiodic Lamellar Structures with Layers Containing Isomorphic Substitutions or with Different Types of Layers.- 1.6 Principal Characteristics of Lamellar Structures with Stacking Faults.- 1.6.1 Translation Stacking Faults.- 1.6.2 Rotation Stacking Faults.- 1.6.3 Stacking Faults Due to Enantiomorphism.- 1.6.4 Weil-Defined Stacking Faults.- 1.6.5 Random Stacking Faults.- 1.6.6 Stacking Faults Due to Fluctuations in the Position of the Layers. Disorders of the First and Second Types.- 1.6.7 Particles, Crystallites and Interferential Coherence Domains.- 1.7 Principal Characteristics of Interstratified Minerals.- 1.7.1 Interstratified System Characterization by the Stacking Mode of the Layers.- 1.7.2 Order-Disorder in the Sequence of Layers of Different Types.- 1.8 Commensurate and Incommensurate Structures in Interstratified Systems.- References.- 2 Theory of the Diffraction Phenomenon Produced by Powders of Microcrystals with a Lamellar Structure.- 2.1 Diffraction from an Isolated Layer of Finite Extent.- 2.2 Diffraction from a Defect-Free Stack of Identical Layers.- 2.2.1 General Description of the Diffraction.- 2.2.2 Effect of the Thinness of the Interferential Coherence Domains on the Intensity Distribution. Apparent Irrationality of the 00l Reflections.- 2.3 Diffraction by a Powder of Particles with Totally Random Orientation.- 2.3.1 General Expression for the Intensity of the Wave Diffracted by an Isotropic Powder.- 2.3.2 The Tangent Cylinder Approximation.- 2.3.3 Physical Significance of and General Expression for T(U).- 2.3.4 Computation of T(U) for Rectangular Interferential Coherence Domains.- 2.4 Diffraction from a Powder of Partially Oriented Particles.- 2.4.1 Definition of the Spatial Distribution of the Particles in a Powder.- 2.4.2 Diffraction from a Partially Oriented Powder in a Symmetrical ?-2? Transmission Mounting.- 2.4.3 Diffraction from a Partially Oriented Powder in an Asymmetrical Transmission Mounting or in a Reflection Arrangement.- 2.4.4 Diffraction from a Partially Oriented Powder in the Particular Case of the (00) Rod.- References.- 3 Diffraction from Lamellar Crystals with Stacking Faults.- 3.1 General Expression for the Diffraction Produced by Stacks of Layers with Position Defects.- 3.1.1 Mathematical Description of the Diffraction.- 3.1.2 The Matrix Formalism.- 3.2 Diffraction Produced by Stacks Containing Rotation or Translation Faults Without Mutual Interaction.- 3.2.1 Effects of Random Rotation or Translation Stacking Defects on the Diffraction.- 3.2.2 Effect of Well-Defined Translation Defects on the Diffraction.- 3.2.3 Effect of Well-Defined Rotation Defects on the Diffraction.- 3.3 Diffraction Produced by Stacks with Defects Due to Fluctuations in the Positions of the Layers.- 3.3.1 Position Fluctuations Leading to a Disorder of the First Type.- 3.3.2 Position Fluctuations Leading to a Disorder of the Second Type.- 3.3.3 Determination of the Mean Standard Deviation of the Fluctuations Affecting the Interlayer Distances by Direct Profile Analysis of the 00l Reflections.- 3.3.4 Comparison of the Effects of Random Defects and of Position Fluctuations on the Diffraction.- 3.3.5 Comparison of the Physical Significances Attached to the Concepts of Random Defects and of Position Fluctuation Defects.- References.- 4 Statistical Models and Parameters Used to Describe Interstratified Lamellar Systems.- 4.1 General Parameters Characterizing the Stacking of Different Layers in Interstratified Structures.- 4.2 Interstratified Structures with S = 0.- 4.3 Interstratified Structures with S = 1.- 4.3.1 Determination of the Independent Parameters Characterizing Two-Component Structures.- 4.3.2 Classification of Two-Component Structures as a Function of the Degree of Order in the Sequence of Layers.- 4.3.3 Interstratified Structures with Three Types of Layers.- 4.4 Interstratified Structures with S = 2.- 4.4.1 Relationships Between the Proportions of Different Types of Layers and the Conditional Probabilities.- 4.4.2 Choice of the Independent Parameters.- 4.4.3 Classification of Structures with S = 2 as a Function of the Degree of Order in the Sequence of Layers.- 4.4.4 Interstratified Structures with S = 2 and g Types of Layers.- 4.5 Interstratified Structures with S = 3.- 4.6 Degree of Homogeneity for Powders of Thin Particles with Markovian Interstratification (Quasi-Homogeneous System).- 4.7 Parameters for the Characterization of Homogeneous Interstratified Systems.- 4.7.1 Homogeneous Two-Component (A and B) Systems with S = 0.- 4.7.2 Homogeneous Two-Component Systems with S ? 0 and Restrictive Conditions for the Sequence of Layers.- References.- 5 Diffraction Methods Adapted to the Structural Analysis of Interstratified Systems.- 5.1 Direct Methods of Structural Analysis.- 5.1.1 The Method of D’yakonov.- 5.1.2 Computation of the Function ??(z).- 5.1.3 Comparison of the Mac Ewan and D’yakonov Direct Methods of Structural Analysis.- 5.2 Indirect Methods of Structural Analysis Based on the Computation of the Intensities of Basal Reflections.- 5.2.1 Calculation of an Interference Function Using a Single Structure Factor.- 5.2.2 Methods of Intensity Calculation Using Different Structure Factors.- 5.3 Diffraction by Systems with g Types of Layers, with a Specific Translation r Between the Adjacent i-Type and j-Type Layers, for any Given Value of S.- 5.3.1 Expressions for the Matrices [W], [?], and [Q], when S = 0 or 1.- 5.3.2 Expressions for [W], [?], and [Q] when S = 2.- 5.3.3 Expressions for [W], [?], and [Q] when S = 3.- 5.3.4 The Matrices [W], [?] and [Q] in the Case of Interstratified Systems with g Components, for any Given Value of S.- 5.4 Intensity of the Wave Diffracted by Systems with g Types of Layers, for any Value of S and R.- 5.4.1 Matrix Formalism for Systems with Identical Layers in the Same Azimuthal Orientation, with Translational Defects and an Interaction Parameter R ? 1.- 5.4.2 Matrix Formalism for Interstratified Structures with any Number of Translations Without Mutual Interaction (R = 0).- 5.4.3 Matrix Formalism in the General Case of Interstratified Systems.- 5.5 General Remarks.- References.- 6 Experimental Techniques Adapted to the Study of Microdivided Lamellar Systems.- 6.1 Survey of the Techniques Most Frequently Used in Powder Diffractometry.- 6.1.1 The Powder Diagram.- 6.1.2 The Debye-Scherrer-Hull Mountings.- 6.1.3 Use of a Recording Counter and of a Monochromator.- 6.1.4 Advantages and Drawbacks of the Reflection and Transmission Mountings.- 6.2 Adaptation of Transmission Techniques to the Study of Microdivided Lamellar Systems.- 6.2.1 The X-Ray Source.- 6.2.2 The Monochromator.- 6.2.3 Particular Features of the Specimen.- 6.2.4 The Goniometer.- 6.2.5 The Detector and Counting Equipment.- 6.3 Perturbing Factors which can be Minimized.- 6.3.1 Choice of the Slit-Widths in the Path of the X-Ray Beam.- 6.3.2 Optimal Slit Height.- 6.3.3 Choice of Sample Thickness.- 6.4 Principal Corrections on the Diffraction Patterns.- 6.4.1 Correction of Effects Due to Polarization of the X-ray Beams.- 6.4.2 Correction of Effects Due to Sample Absorption.- 6.4.3 Correction for the Nonlinear Response of the Localization Detector.- 6.5 Perturbing Factors Introduced in the Computation of the Theoretical Diffractograms.- 6.5.1 Lorentz Factor.- 6.5.2 Orientation Function for the Particles in a Powder.- 6.6 Determination of the Absolute Intensity Scale.- 6.6.1 Definition of the Absolute Scale.- 6.6.2 Determination of the Absolute Scale.- 6.6.3 Examples and Applications.- References.- 7 Structural Characteristics of Carbons.- 7.1 General Characteristics of Carbon Materials.- 7.1.1 General Description.- 7.1.2 Basic Features of the Graphitization Process.- 7.2 Structural Characteristics of the Graphitization Process.- 7.2.1 Structural Study of the Carbon Layers.- 7.2.2 Examples of Structural Evolution in the Carbon Layer as a Function of the Thermal Treatment.- 7.3 Organization of the Stacks.- 7.3.1 Structure of the Stacks in the two Graphite Polytypes.- 7.3.2 Structure of the Stacks in a Carbon Undergoing Graphitization.- References.- 8 The Modelization Method in the Determination of the Structural Characteristics of Some Layer Silicates: Internal Structure of the Layers, Nature and Distribution of the Stacking Faults.- 8.1 Structural Defects in Kaolinite.- 8.1.1 Common Features of the Layers in Kaolin Minerals.- 8.1.2 Common Features of the 1:1 Layers in Dickite and Nacrite.- 8.1.3 Characteristics of the 1:1 Layer in Kaolinite.- 8.1.4 Comparison of the Kaolinite and Dickite Unit Cells.- 8.1.5 Models for the Stacking Faults in Kaolinite.- 8.1.6 Comparison Between Calculated and Experimental XRD Patterns.- 8.2 Distribution of the Cations in the cis and trans Octahedral Sites of Dioctahedral Smectites.- 8.2.1 Preparation Techniques for Smectite Samples Used in the Diffractometric Determination of the Distribution of Cations in Octahedral Sites.- 8.2.2 Determination of the Octahedral Cation Distribution in K-Smectites by Oblique Texture Electron Diffraction.- 8.2.3 Determination of the Distribution of Octahedral Cations in K+-Nontronites by XRD.- 8.2.4 Analysis of the XRD Powder Patterns from Dioctahedral Cs-Smectites.- 8.3 Determination of the Distribution of Cations and Water Molecules in the Interlamellar Spaces of Dioctahedral Smectites.- 8.3.1 Experimental Conditions.- 8.3.2 Analysis of the Profile of the 00l Reflections from a Two-Water-Layer Na-Beidellite (Sample E2).- 8.3.3 Qualitative Description of the (02, 11), (20, 13) and (04, 22) Bands Given by Two-Water-Layer Na+-Beidellite.- 8.3.4 Determination of the x, y Coordinates of the Sites Occupied by Water Molecules.- 8.3.5 The Different Possible Stackings of Layers in Two-Water-Layer Na-Beidellite.- 8.3.6 Determination of the Structural Characteristics of the Two-Water-Layer Na+-Beidellite by Fitting the Calculated Pattern to the Experimental XRD Data.- 8.3.7 Structural Characteristics of One-Water-Layer Na+-Beidellite. Comparison with the Two-Water-Layer Hydrate.- 8.4 Structural Defects in Glauconites.- 8.4.1 Structure of the Glauconites.- 8.4.2 Choice of the Samples, Experimental Conditions and Description of the Experimental Diffractograms.- 8.4.3 Determination of the Unit Cell Parameters and of the Atomic Coordinates.- 8.4.4 Structural Models for Glauconites Devoid of Stacking Defects.- 8.4.5 Models with Well-Defined ±120° Rotational Stacking Faults.- 8.4.6 Structural Models with Enantiomorphic Layers.- 8.4.7 Structural Model with n 60° Rotational Stacking Faults and R = 0.- 8.4.8 Structural Model with n 60° Rotational Stacking Faults and R = 1.- 8.4.9 Determination of the Structural Parameters Characteristic of Glauconites.- References.- 9 Determination of the Structural Characteristics of Mixed-Layer Minerals.- 9.1 The Method of D’yakonov.- 9.1.1 Practical Example of the Use of the D’yakonov Method.- 9.1.2 Appraisal of the D’yakonov Method.- 9.2 General Guidelines for the Use of Modelization of X-Ray Diffractograms in the Study of Mixed-Layer Minerals.- 9.2.1 Determination of the Nature of the Layer Types.- 9.2.2 Chemical Composition and Structure of the Layers and of the Interlayer Spaces.- 9.2.3 Choice of the Origin for the z Ordinates of Atoms in the Scattering Units.- 9.3 Calculation of the Reference X-Ray Diffractogram for Quasi-Homogeneous Interstratified Minerals.- 9.3.1 Specific Features of the X-Ray Diffractograms Given by Interstratified i-m Systems with S = 0 or 1.- 9.3.2 Characteristics of the X-Ray Diffractograms Given by i-m Systems with S = 2 and Wi > Wm.- 9.3.3 Specific Features of the X-Ray Diffractograms from Interstratified i-m Structures with S = 3.- 9.3.4 Comparison of the Specific Features of the Diagrams Given by Systems with S = 0, 1, 2, 3.- 9.4 Parameters Other than W and S which Influence the Profile of the Calculated X-Ray Diagrams of Two-Component Interstratified Systems.- 9.4.1 Influence of the Thickness of the Scattering Units.- 9.4.2 Influence of the Thickness of the Interferential Coherence Domains.- 9.4.3 Physical Mixtures of Quasi-Homogeneous Mixed-Layer Systems.- 9.4.4 Homogeneous Mixed-Layer Models.- 9.5 Quantitative Determination of the Structural Characteristics of Interstratified Dioctahedral Mica-Smectite Minerals.- 9.5.1 Two-Component Interstratified Minerals: Celadonite-Nontronite.- 9.5.2 Three-Component Interstratified Minerals: Leucophyllite-Montmorillonite-“Vermiculite” with S=1.- 9.5.3 Two-Component Interstratified Minerals: Leucophyllite-Montmorillonite, with S = 3.- 9.6 Semi-Quantitative Determinations of the Structural Characteristics of Interstratified Minerals.- 9.6.1 Two-Component Interstratified Minerals: Illite-Montmorillonite.- 9.6.2 Interstratified Minerals with Kaolinite 1:1 Layers.- 9.6.3 Study of Hydrated Talcs.- References.- Author Index.