P. Scardi

Rietveld refinement guidelines

A set of general guidelines for structure refinement using the Rietveld (whole-profile) method has been formulated by the International Union of Crystallography Commission on Powder Diffraction. The practical rather than the theoretical aspects of each step in a typical Rietveld refinement are discussed with a view to guiding newcomers in the field. The focus is on X-ray powder diffraction data collected on a laboratory instrument, but features specific to data from neutron (both constant-wavelength and time-of-flight) and synchrotron radiation sources are also addressed. The topics covered include (i) data collection, (ii) background contribution, (iii) peak-shape function, (iv) refinement of profile parameters, (v) Fourier analysis with powder diffraction data, (vi) refinement of structural parameters, (vii) use of geometric restraints, (viii) calculation of e.s.d.s, (ix) interpretation of R values and (x) some common problems and possible solutions.

Simultaneous structure and size-strain refinement by the Rietveld method

A new procedure for simultaneous refinement of structural and microstructural disorder parameters for polycrystalline materials is proposed. It is based on the Rietveld method combined with Fourier analysis for broadened peaks. Crystallite size and shape and r.m.s. microstrain are regarded as fitting parameters, replacing the well known formula of Caglioti, Paoletti & Ricci [Nucl. Instrum. Methods (1958), 3, 223–228] for the angular dependence of the peak width. In particular, from these microstructural disorder parameters, by inverting the Warren–Averbach procedure [Warren & Averbach (1950). J. Appl. Phys. 21, 595–599; (1952), 23, 1059] for a single peak, it is possible to obtain the parameters of the pseudo-Voigt (pV) functions employed to fit the experimental data. The anisotropy of the crystallite size and microstrain is also taken into account. The method has been tested on three materials with different degrees of crystallization: tetragonal ZrO2 (P42/nmc, a = 3.5961, c = 5.1770 A, Vc = 66.95 A3, Z = 2, Rwp = 0.077, M ≃100 A, 〈∊2〉1/2 ≃ 3 × 10−3); tetragonal Zr0.82Ce0.18O2 (P42/nmc, a = 3.6419, c = 5.2440 A, Vc = 69.556 A3, Z = 2, Rwp = 0.0654, M ≃1000 A, 〈∊2〉1/2 ≃8 × 10−4); α-Al2O3 (R{\bar 3}c, a = 4.7605, c = 12.9956 A, Vc = 255.05 A3, Z = 6, Rwp = 0.0684, M ≃ 1400 A, 〈∊2〉1/2 ≃7 × 10−4).

Whole powder pattern modelling

A new approach for the modelling of diffraction patterns without using analytical profile functions is described and tested on ball milled f.c.c. Ni powder samples. The proposed whole powder pattern modelling (WPPM) procedure allows a one-step refinement of microstructure parameters by a direct modelling of the experimental pattern. Lattice parameter and defect content, expressed as dislocation density, outer cut-off radius, contrast factor, twin and deformation fault probabilities), can be refined together with the parameters (mean and variance) of a grain-size distribution. Different models for lattice distortions and domain size and shape can be tested to simulate or model diffraction data for systems as different as plastically deformed metals or finely dispersed crystalline powders. TEM pictures support the conclusions obtained by WPPM and confirm the validity of the proposed procedure.

Diffraction analysis of the microstructure of materials

1 Line Profile Analysis: A Historical Overview.- 2 Convolution Based Profile Fitting.- 3 Whole Powder Pattern Modelling: Theory and Applications.- 4 Full Profile Analysis of X-ray Diffraction Patterns for Investigation of Nanocrystalline Systems.- 5 Crystallite Size and Residual Strain/Stress Modeling in Rietveld Refinement.- 6 The Quantitative Determination of the Crystalline and the Amorphous Content by the Rietveld Method: Application to Glass Ceramics with Different Absorption Coefficients.- 7 Quantitative Analysis of Amorphous Fraction in the Study of the Microstructure of Semi-crystalline Materials.- 8 A Bayesian/Maximum Entropy Method for the Certification of a Nanocrystallite-Size NIST Standard Reference Material.- 9 Study of Submicrocrystalline Materials by Diffuse Scattering in Transmitted Wave.- 10 Determining the Dislocation Contrast Factor for X-ray Line Profile Analysis.- 11 X-ray Peak Broadening Due to Inhomogeneous Dislocation Distributions.- 12 Determination of Non-uniform Dislocation Distributions in Polycrystalline Materials.- 13 Line Profile Fitting: The Case of fcc Crystals Containing Stacking Faults.- 14 Diffraction Elastic Constants and Stress Factors Grain Interaction and Stress in Macroscopically Elastically Anisotropic Solids The Case of Thin Films.- 15 Interaction between Phases in Co-deforming Two-Phase Materials: The Role of Dislocation Arrangements.- 16 Grain Surface Relaxation Effects in Powder Diffraction.- 17 Interface Stress in Polycrystalline Materials.- 18 Problems Related to X-Ray Stress Analysis in Thin Films in the Presence of Gradients and Texture.- 19 Two-Dimensional XRD Profile Modelling in Imperfect Epitaxial Layers.- 20 Three-Dimensional Reciprocal Space Mapping: Application to Polycrystalline CVD Diamond.

Line broadening analysis using integral breadth methods: a critical review

Integral breadth methods for line profile analysis are reviewed, including modifications of the Williamson–Hall method recently proposed for the specific case of dislocation strain broadening. Two cases of study, supported by the results of a TEM investigation, are considered in detail: nanocrystalline ceria crystallized from amorphous precursors and highly deformed nickel powder produced by extensive ball milling. A further application concerns a series of Fe–Mo powder specimens that were ball milled for increasing time. Traditional and modified Williamson–Hall methods confirm their merits for a rapid overview of the line broadening effects and possible understanding of the main causes. However, quantitative results are generally not reliable. Limits in the applicability of integral breadth methods and reliability of the results are discussed in detail.

LSI - a computer program for simultaneous refinement of material structure and microstructure

A computer program has been written to introduce profile analysis into the Rietveld method. The devised algorithm simultaneously refines both structural and microstructural parameters, also accounting for anisotropy in crystallite size and microstrain. Instead of using phenomenological relations to describe the trend of profile width and shape as a function of diffraction angle, a model based on the Warren–Averbach approach has been developed that permits extraction of more information from data, also achieving faster convergence.

Diffraction line profiles from polydisperse crystalline systems

Diffraction patterns for polydisperse systems of crystalline grains of cubic materials were calculated considering some common grain shapes: sphere, cube, tetrahedron and octahedron. Analytical expressions for the Fourier transforms and corresponding column-length distributions were calculated for the various crystal shapes considering two representative examples of size-distribution functions: lognormal and Poisson. Results are illustrated by means of pattern simulations for a f.c.c. material. Line-broadening anisotropy owing to the different crystal shapes is discussed. The proposed approach is quite general and can be used for any given crystallite shape and different distribution functions; moreover, the Fourier transform formalism allows the introduction in the line-profile expression of other contributions to line broadening in a relatively easy and straightforward way.

Line profile analysis: pattern modelling versus profile fitting

Powder diffraction data collected on a nanocrystalline ceria sample within a round robin conducted by the IUCr Commission on Powder Diffraction were analysed by two alternative approaches: (i) whole-powder-pattern modelling based upon a fundamental microstructural parameters approach, and (ii) a traditional whole-powder-pattern fitting followed by Williamson-Hall and Warren-Averbach analysis. While the former gives results in close agreement with those of transmission electron microscopy, the latter tends to overestimate the domain size effect, providing size values about 20% smaller. The origin of the discrepancy can be traced back to a substantial inadequacy of profile fitting with Voigt profiles, which leads to systematic errors in the following line profile analysis by traditional methods. However, independently of the model, those systematic errors seem to have little effect on the volume-weighted mean size.

Thermal diffusivity/microstructure relationship in Y-PSZ thermal barrier coatings

A set of yttria partially stabilized zirconia coatings with different thickness was deposited on flat nickel-base alloy coupons by air plasma spray (APS) under uncontrolled temperature conditions. In this way, the length of the spraying process (and consequently the coating thickness) had a direct effect on phase composition as well as on the thermal properties of the material. In particular, both the monoclinic phase percentage and thermal diffusivity increased considerably with the thickness. Because this trend was observed together with a slight but clearly visible increase in the total porosity, the interpretation of the results was not straightforward, but required a detailed discussion of the thermal transport mechanism. Considering the complex microstructure typical of APS coatings and the relevant role of porosity, it was shown how a modest reduction in the fraction of closed pores can account for the observed increase in diffusivity. It was then proposed that the volume change associated with the progressive tetragonal to monoclinic phase transformation can be responsible for the reduction of the closed porosity of lenticular shape oriented parallel to the surface, in spite of the observed increase in the total porosity.

Fourier modelling of the anisotropic line broadening of X-ray diffraction profiles due to line and plane lattice defects

A new model of line-profile broadening due to the effect of linear and planar lattice defects has been incorporated into the conventional Rietveld algorithm for the structural refinement and whole-pattern fitting of powder data. The proposed procedure, applied to face-centred cubic (f.c.c.) structure materials, permits better modelling, even in the case of anisotropic line broadening and other hkl-dependent effects that can be related to the presence of dislocations and planar defects (stacking faults and twinning). Besides better quality of the profile fitting, detailed information on the defect structure can be produced: dislocation density and cut-off radius, stacking- and twin-fault probabilities can be refined together with the structural parameters. For each phase (in different samples or in multi-phase samples) the appropriate size–strain model can be selected. The Fourier formalism, which is the basis of the line-profile modelling, also permits an easy adaptation to different lattice-defect models. New approaches can be easily introduced and tested against or together with the existing ones. Finally, the devised program can also be used for the simulation of powder patterns for materials with different types and amounts of line and plane lattice defects.