X-Ray Diffraction by Disordered 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.