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X-ray powder diffraction (XRPD) profiles provide valuable information on the geometrical shape of crystallites. This work delves into the complexity of determining the geometrical shape of crystallites, including plate-like and spherical TiO 2 specimens, and discusses crucial parameters of edge length, shape coefficient, and orientation vector. The analysis demonstrates the necessity of using both line-broadening analysis and preferred orientation to elucidate the underlying physics of XRPD analysis. The choice of empirical formula is illustrated in analyzing plate-like crystallites with the characteristic of flat morphology. For spherical crystallites, the March-Dollase formula is recommended. • X-ray diffraction powder profiles are used to determining crystallite morphology. • Crystallite shape with edge length (L), shape coefficient (η), and orientation [UVW]. • The integration of line-broadening analysis and preferred orientation provides insights into crystallite shape. [ABSTRACT FROM AUTHOR]

Specific layer-stacking irregularities have been identified in C36 (4H) Nb-Cr and Ti-Cr Laves phases on the basis of X-ray diffraction line-profile analysis and high-resolution transmission electron microscopy. Domain boundaries and transformation errors within domains could be distinguished. The layer-stacking irregularities in both C36-NbCr2 and C36-TiCr2 can be associated with a preceding C14 (2H) → C36 (4H) phase transformation carried out by glide of mobile synchro-Shockley partial dislocation dipoles in an ordered fashion. The stacking irregularities observed can be interpreted as deviations from such perfect “ordered glide”. The interpretation is supported by the observation that, in the case of C36-NbCo2, where no preceding C14 → C36 transformation occurs, different layer-stacking irregularities are observed. [ABSTRACT FROM AUTHOR]

Abstract: X-ray powder diffraction data of NiAs/Ni2In-type Ni1.60Sn and Ni1.63Sn alloys annealed at or below about 573K reveal the development of an incommensurately ordered phase called LT″. In this phase Ni(2) atoms occupy partially the trigonal–bipyramidal interstices formed by five Sn within an NiAs-type arrangement Ni(1)Sn. The modulated occupational ordering of Ni(2) in the LT″ phase can be described in the superspace group Cmcm(α00)0 s 0, and the parameters describing this occupational modulation were refined together with atomic displacement modulations using the Rietveld method. The structure parameters revealed close structural analogies of the LT″ phase with the previously reported commensurate LT-Ni1+ δ Sn and incommensurate LT′-Ni1+ δ Sn phases (A. Leineweber, J. Solid State Chem. 177 (2004) 1197–1212), which both occur for lower Ni contents than the LT″ phase. The 1st-order satellite reflections visible in the powder-diffraction patterns exhibit, with respect to the fundamental reflections, a considerable diffraction-line broadening, caused by a small size of the particularly ordered domains. This small-domain-size broadening was successfully described by a recently developed reflection-index (hklm) dependent (anisotropic) line-broadening model (A. Leineweber, V. Petricek, J. Appl. Crystallogr. 40 (2007) 1027–1034) designed to consider the effect of fluctuations of the lattice metrics on the peak widths in powder diffraction patterns of incommensurately modulated crystal structures. The small domain sizes encountered for the LT″ phase indicate that domain coarsening is much more difficult than for the LT and LT′ phases. This special feature of the LT″ phase goes along with a compared to the LT and LT′ phases absent orthorhombic distortion and the low ordering temperature, which are discussed as a consequence of the ordering patterns due to the Ni(2) atoms. [Copyright &y& Elsevier]

Abstract: We have developed a novel, analytical model for the determination of grain size distributions in nanocrystalline (nc) materials with low internal stresses by X-ray diffractometry (XRD). The model assumes explicitly that the XRD peaks are pseudo-Voigtian and that the grain size distributions are lognormal, both of which are assumptions amply supported by the experimental evidence. It was found analytically that the grain size dispersion depends on the shape of the XRD peaks only, whereas the grain size median depends on both the shape and width of the XRD peaks. In addition, the theoretical predictions resulting from the model were validated using standard XRD peaks obtained by computer simulation from first principles. Particular emphasis is given to the discussion of the validity limits of the model, and to the analysis of the influence of the characteristics of the grain size distributions on the nature of the XRD peaks. We then show how to calculate the average and apparent grain sizes from the grain size distribution determined with the model, and how this compares with the Scherrer method. Implications for the characterization of (undistorted and distorted) nc-materials are indicated, and a case study of an nc-powder of cubic ZrO2 is presented. The application of the model itself is simple, involving only the fit of a pseudo-Voigt function to a single XRD peak followed by the use of two equations. This suggests that the model may have an important role to play in the characterization of nc-materials. [Copyright &y& Elsevier]

Silicon carbide (SiC), owing to its unique combination of chemical and physical properties, is the material of choice for advanced engineering applications both at room and high temperatures. Due to the difficulty of SiC to be solid-state sintered, nowadays there has been growing interest in liquid-phase sintering (LPS) of SiC. Here, there is a critical need for developing methodologies for the accurate X-ray diffraction characterization of the microstructures of these ceramics as their microstructural evolution is not yet totally understood. Thus, the aim of the present work is to apply a procedure for the accurate line-broadening study of any given phase in LPS SiC ceramics. To that end, we have pressureless liquid-phase sintered (at 1950 °C for 0.5 h in flowing Ar atmosphere) a specimen starting from a commercially available β-SiC powder and using the simultaneous addition of Al2O3 and Y2O3. The resulting material was subsequently characterized using X-ray powder diffraction; more specifically, the Rietveld method was applied for the quantitative phase-composition analysis and the Warren–Averbach method for determining the mean crystallite size, the crystallite size distribution and the root-mean-square strain along the crystallographic direction <111> of the β-SiC polytype. The nanometric nature of the crystallite size distribution is finally related with the stacking faults in the SiC starting powder. [Copyright &y& Elsevier]

An alternative formulation of the variance method for the line-broadening analysis of polycrystalline materials is presented. It maintains the theoretical basis of the earlier formulations of the variance method, but differs in the manner of calculating the variance coefficients of the line profiles. In the proposed formulation, these are evaluated analytically in terms of the shape parameters of Voigt functions fitted to the X-ray diffraction data. Explicit expressions are thus derived for calculating the (surface-weighted) crystal sizes and (root-mean-square) lattice microstrains from the integral breadths of the Gauss and Lorentz components of the Voigt functions that model the experimental and instrumental line profiles. [ABSTRACT FROM AUTHOR]