Quantitative X-Ray Diffractometry

1 Introduction.- 1.1 Phase analysis—when and why.- 1.2 Phase analysis as an analytical method.- 1.2.1 Direct methods of phase analysis.- 1.2.2 Indirect methods of phase analysis.- 1.2.3 General approach to phase quantification.- 1.3 History of quantitative X-ray phase analysis.- 1.3.1 Quantification techniques.- 1.3.2 Instrumentation.- 2 Physical basis.- 2.1 Interaction of X-rays with material.- 2.1.1 Scattering of X-rays.- 2.1.2 Absorption of X-rays.- 2.1.3 Various forms of absorption coefficients.- 2.2 Intensity in powder diffraction.- 2.2.1 General expression derived from the kinematic theory of diffraction.- 2.2.2 Polarization factor in X-ray diffractometry.- 2.2.3 Extinction of X-rays.- 2.3 Background—angle variation and intensity.- 2.3.1 Introduction.- 2.3.2 Scattering of continuous radiation.- 2.3.3 Scattering of characteristic radiation.- 2.4 X-ray diffraction by nonhomogeneous polycrystalline materials.- 2.4.1 Introduction.- 2.4.2 Two extreme cases: homogeneous and nonhomogeneous specimens.- 2.4.3 Particle absorption factor.- 2.4.4 Absorption by a nonhomogeneous system.- 2.4.5 Intensity correction for nonhomogeneous specimens.- 2.4.6 Application of the theory.- 2.4.7 Surface roughness.- 2.5 Orientation of reflecting particles.- 3 Geometric aspects of X-ray diffractometry.- 3.1 Geometric schemes in X-ray diffractometry.- 3.2 Diffractometers with Bragg-Brentano reflection focusing.- 3.2.1 General geometric features.- 3.2.2 Absorption correction.- 3.2.3 Real irradiated volume in the Bragg-Brentano scheme.- 3.2.4 Instrumental aberrations.- 3.2.5 Diffraction by a thin-layer specimen.- 3.3 Diffractometers with Seeman-Bohlin reflection focusing.- 3.3.1 Absorption correction.- 3.3.2 Instrumental aberrations.- 3.3.3 Comparison with the Bragg-Brentano scheme.- 3.4 Transmission technique with constant specimen-detector distance (Bragg-Brentano transmission analog).- 3.4.1 Geometric scheme.- 3.4.2 Absorption correction.- 3.4.3 Instrumental aberrations.- 3.4.4 Comparison between reflection and transmission Bragg-Brentano geometries.- 3.5 Transmission technique with an invariant focusing circle (Guinier diffractometer or the Seeman-Bohlin transmission analog).- 3.5.1 General description.- 3.5.2 Absorption correction.- 3.5.3 Chromatic aberration.- 3.5.4 Comparison of symmetric and asymmetric transmission geometries.- 3.6 Debye-Sherrer geometry.- 3.6.1 General view.- 3.6.2 Absorption correction.- 3.6.3 Comparison with former methods.- 3.7 Powder diffractometry with synchrotron radiation.- 3.7.1 General description.- 3.7.2 Instrumental aberrations.- 3.7.3 Diffractometric modes with synchrotron radiation.- 3.8 Position-sensitive detectors in powder diffractometry.- 4 Methodology of quantitative phase analysis.- 4.1 Introduction.- 4.2 Analysis of samples with a known mass absorption coefficient (diffraction-absorption technique).- 4.2.1 Multiphase system with a constant absorption coefficient.- 4.2.2 Two-phase system with a variable absorption coefficient.- 4.2.3 General case: multiphase systems with a variable absorption coefficient.- 4.2.4 Examples.- 4.3 Internal standard method.- 4.3.1 Basic principles.- 4.3.2 Evaluation of calibration constants.- 4.3.3 Precision of the internal standard method.- 4.3.4 Comparison of the internal standard method with the diffraction-absorption technique.- 4.3.5 Internal standard—materials and optimal amount.- 4.3.6 Application of the internal standard method.- 4.4 Doping method in quantitative X-ray diffractometry.- 4.4.1 Introduction.- 4.4.2 Constant absorption approach.- 4.4.3 General case.- 4.4.4 Two-phase approach.- 4.4.5 Precision of the analysis and optimal doping.- 4.4.6 Implementation of the doping method.- 4.5 Dilution Method.- 4.5.1 Basic aspects.- 4.5.2 Optimal dilution.- 4.5.3 Potential diluents.- 4.5.4 Use of a heavy absorber—constant-absorption approach.- 4.5.5 Implementation of the dilution method.- 4.6 Full-phase analysis of the n-phase sample.- 4.6.1 Basic equations.- 4.6.2 Application of the method to two-phase mixtures.- 4.6.3 Evaluation of the calibration coefficient ?rj.- 4.6.4 Presence of undetected phases in the analytical sample.- 4.6.5 Comparison of equations (4.62) and (4.77).- 4.6.6 Precision of the analysis.- 4.6.7 Invariance of ?rj values.- 4.6.8 Examples.- 4.7 Reference intensity ratios in quantitative analysis.- 4.7.1 Introduction.- 4.7.2 Definition and acquisition of reference intensity ratios.- 4.7.3 Implementation of reference intensity ratios.- 4.7.4 Invariance of reference intensity ratios.- 4.8 Diffraction patterns with overlapping peaks—full diffraction pattern approach.- 4.8.1 Introduction.- 4.8.2 Basic principles.- 4.8.3 Evaluation of phase abundances.- 4.8.4 Evaluation of the constants ?ij.- 4.8.5 Precision of the analysis.- 4.8.6 Internal standard technique in the case of overlapping peaks.- 4.8.7 Continuous pattern approach.- 4.8.8 Examples.- 4.9 Implementation of calculated powder patterns in QXRD.- 4.9.1 Introduction.- 4.9.2 Calculation of calibration constants.- 4.9.3 Full-pattern approach.- 4.9.4 Simultaneous QXRD and structure refinement by the pattern-fitting method.- 4.9.5 Sources of structural data.- 4.9.6 Quality of structural data.- 4.9.7 Comparison of calculated and experimental powder data.- 4.9.8 Summary.- 4.10 Standardless Methods.- 4.10.1 Introduction.- 4.10.2 Derivation of phase abundances and calibration constants.- 4.10.3 Full pattern approach.- 4.10.4 Number of phases in analyzed samples.- 4.10.5 Precision of the standardless techniques—optimal sample set.- 4.11 Combination of X-ray diffraction and chemical data.- 4.11.1 Phase analysis based on pure chemical data.- 4.11.2 Auxiliary X-ray diffraction data.- 4.11.3 Basic application of X-ray diffraction.- 4.12 Crystallinity of polymers.- 4.13 Analysis of low-mass samples.- 4.13.1 Critical sample masses.- 4.13.2 Analytical equations.- 4.13.3 Analytical techniques.- 5 Practical aspects of quantitative phase analysis.- 5.1 Introduction.- 5.2 Instrumentation.- 5.2.1 Introduction to instrumentation.- 5.2.2 Geometric alignment of the diffractometer.- 5.2.3 Geometric aberrations.- 5.2.4 Variable instrument parameters.- 5.2.5 Detectors and monochromators.- 5.2.6 Fixed divergence versus fixed irradiated area geometries.- 5.3 Specimen preparation.- 5.3.1 Preparation of bulk specimens.- 5.3.2 Sampling of powders.- 5.3.3 Fine grinding of powders.- 5.3.4 Powder aggregation and mixing.- 5.3.5 Powder mounts.- 5.3.6 Pressing.- 5.3.7 Dusting of loose powder on a plate.- 5.3.8 Powder deposition on a membrane filter.- 5.3.9 Preparation of oriented mounts.- 5.3.10 Handling of reactive samples.- 5.3.11 Handling of low-mass samples.- 5.4 Analytical standards.- 5.4.1 Effects of solid solution on diffracted intensity.- 5.4.2 Linear and planar structural imperfections.- 5.4.3 Particle morphology.- 5.4.4 Selection of analytical standards.- 5.5 Intensity measurements.- 5.5.1 Counting statistics.- 5.5.2 Difference and sum of intensities—counting strategy.- 5.5.3 Ratio of intensities.- 5.6 Definition and subtraction of background.- 5.6.1 Definition and subtraction of the background for a single diffraction peak.- 5.6.2 Full-pattern approach.- 5.6.3 Subtraction of incoherent scattering.- 5.7 Determination of sample absorption.- 5.7.1 Calculation of mass absorption.- 5.7.2 Measurements of absorption coefficients by transmission methods.- 5.7.3 Attenuation of reflection from a crystalline substrate.- 5.7.4 Determination of the absorption coefficient by diffraction techniques.- 5.7.5 Determination of the mass absorption coefficient by means of Compton scattering.- 5.8 Pattern decomposition and simulation.- 5.8.1 Introduction.- 5.8.2 Profile-fitting functions.- 5.8.3 Fitting of asymmetrical profiles.- 5.8.4 Adjustable parameters and constraints.- 5.8.5 Number of separated peaks and angular resolution.- 5.9 Methodology of corrections for preferred orientation.- 5.9.1 Introduction.- 5.9.2 Presentation of the pole distribution.- 5.9.3 Orientation distribution function.- 5.9.4 Determination of the orientation distribution function W(?, ?).- 5.9.5 Crystallite orientation distribution—the general case of axial texture.- 5.9.6 Approximation of the orientation distribution function W(?,?).- 5.9.7 W(?) of diffraction peaks not dependent on preferred orientation.- 5.10 Estimation of analysis errors.- 5.10.1 Introduction.- 5.10.2 Calculation of reproducibility.- 5.10.3 Scattering range of results.- 5.10.4 Comparison of observed and known phase abundances.- 5.10.5 Inter- and intralaboratory precision.- 5.10.6 Parameters of the calibration graph—estimation of standard deviation.- 5.11 Detection limit.- 5.11.1 Basic formulation.- 5.11.2 Reduction of the detection limit.- 5.11.3 Detection limit in the sample mass approach.- 5.12 Crystallite statistics.- 5.12.1 Intensity error.- 5.12.2 Methods of decreasing intensity fluctuations other than sample diminution.- 6 Industrial applications.- 6.1 Ceramics and glass ceramics.- 6.2 Naturally occurring (geologic) samples.- 6.2.1 Modal analysis of geologic samples.- 6.2.2 Analysis of bauxites.- 6.2.3 Mineralogical analysis of coal and coal ash.- 6.3 Analysis of Portland cement.- 6.4 Metallurgy.- 6.5 Thin films and coatings.- 6.6 Air pollution (aerosols and airborne dusts).- 6.7 Pharmaceuticals.- References.