Modern powder diffraction employing computer-controlled diffractometers now allows quantitative analytical methods to use the whole diffraction trace rather then only individual peaks. Two such methods are in common use the Rietveld method which refines the crystal structures of the component phases as part of the matching calculation, and the patternfitting method which uses reference patterns from a database. Potential accuracies of these methods seems to be around 1 % absolute. The most severe limitation on the potential accuracy of these methods is particle statistics, which has been reviewed in considerable detail.

Calculated patterns play an essential role in X-ray powder diffraction analysis. This paper gives examples of their use in qualitative analysis for evaluating and supplementing reference patterns in the ICDD Powder Diffraction File (PDF), in quantitative analysis for calculating Reference Intensity Ratios (RIRs), in ceil parameter refinements for indexing of low-symmetry/large unit cell diffractograms, in powder pattern determination for validating intensities and recognizing preferred orientation, in new materials synthesis for verification of structure type and phase purity, and for modeling the effects of solid solution substitution.

The utilization and optimization of the properties of materials follows most effectively from a detailed knowledge and understanding of the positions and energetics of their constituent atoms, generally obtained from scattering/diffraction experiments involving electrons, neutrons or electromagnetic radiation. For the most part, these experiments are undertaken on individual crystals of the material, thereby preserving the resolution (and information content) of the three-dimensional reciprocal lattice. However, many of the substances of greatest academic and technical importance either do not crystallize with dimensions large enough for single-crystal studies, or display the properties of maximal interest only when present in finely-divided (powdered) form. In a diffraction experiment, the reciprocal lattice is then collapsed on to the single dimension of the 2θ scale.

The magnitudes of the peak intensities in a powder diffraction pattern are not uniformly distributed as a function of diffraction angle. The Lorentz-polarization factor, X-ray form-factor and the thermal vibration parameters all conspire to progressively decrease the intensities of the peaks as sinθ/λ increases. In spite of this, diffraction data for Rietveld analysis is universally collected using the same counting time for each step in the pattern. As a result, peaks at high angles are collected with lower counting precision than those at low angles, despite the fact that the high-angle region has a higher density of peaks and therefore contains more information than the low-angle part of the pattern. Indeed, the intensity, position and profile of reflections at medium and high sinθ/λ are often more easily modelled since they are subject to less interference from systematic effects such as extinction, specimen transparency and α1/α2 overlap errors. In the present work a novel variable step-counting-time regime has been devised that increases the counting times with diffraction angle in a manner inversely proportional to the intensity fall-off calculated from the Lorentz-polarization, form-factor and thermal parameters. The effect of this new data collection regime on the results of Rietveld refinement of a number of materials of varying structural complexity is discussed.

SIROQUANT is a suite of personal ccmputer programs which runs on an IBM-compatible 386 or 486 personal computer, and gives phase quantification of mineral samples hy Rietveld refinement using the full XRD profile. Eight phases can be quantified in a mineral with the 640 K RAM version and 11 in the EM version.

New features have been added since the original description was written (J.C. Taylor, Powder Diffraction, 6, 2, 1991) including calculations of amorphous percentages using spiked samples, particle size calculations, use of “measured” as well as calculated patterns as standards, pre-programmed refinement “Stages” with different variables in each stage, and improved menus.

Some advanced applications of the system using these new features are described.

The effect of sample granularity on diffracted x-ray intensity was evaluated by measuring the 2θ dependence of x-ray fluorescence from various samples. Measurements were made in the symmetric geometry on samples ranging from single crystals to highly absorbing coarse powders. A characteristic shape for the absorption correction was observed. A demonstration of the sensitivity of Rietveld refined site occupation parameters is made on CuAu and Cu50Au44Ni6 alloys refined with and without granularity corrections. These alloys provide a good example of the effect of granularity due to their large linear x-ray absorption coefficients. Sample granularity and refined thermal parameters obtained from the Rietveld analysis were found to be correlated. Without a granularity correction, the refined thermal parameters are too low and can actually become negative in an attempt to compensate for granularity, A general shape for granularity correction can be included in refinement procedures. If no granularity correction is included, data should be restricted to above 30° 2θ, and thermal parameters should be ignored unless extreme precautions are taken to produce <5 (μm particles and high packing densities.

Rietveld Analysis on X-Ray powder diffraction data was used to build a comprehensive model of the structures in gallmm rich R-Ga systems (R is a rare earth between La and Gd). The ability to refine occupation factors as well as atomic positions allowed for the analysis of the e solid solution, its structural relation to the ordered ∊′ phase, and the unique role of the Ga-Ga pairs in these systems.

A modified version of the Rietveld method has been developed for quantitative phase analysis in conjunction with crystallite size and microstrain determination.

Besides the information on the microstructural disorder in the different phases present in the sample, accurate phase analysis can be performed, even in presence of anisotropic crystallites, taking into account the non-instrumental peak broadening effect in a better way compared to conventional methods.

The method has been tested on binary mixtures of corundum, ceria stabilized zirconia and on a polyphase stabilized zirconia sample.

The electron density distribution maps for CaF2 and TiO2 (rutile) were obtained from profile fitting of powder diffraction data by a Maximum Entropy Method (MEM) analysis. The resultant electron density maps show clearly the nature of the chemical bonding. In order to interpret the results, the nuclear density distribution was also obtained for rutile from powder neutron diffraction data. In the electron density map for rutile obtained by HEM analysis from the X-ray data, both apical and equatorial bonding can be seen. On the other hand, the nuclear density of rutile Is very simple and shows the thermal vibration of nuclei.

A Debye-Scherrer camera (radius 572 mm) was designed for the use at the Photon Factory BL-6A2. It uses 4 pieces of the Imaging Plate to cover 0° ∼ 160° in 2θ. The profiles measured by the camera had 0.083° full width at half maximum (FWHM) and were rather symmetric compared with conventional X-ray source. The observed profiles were well represented by either the split Pearson VII or pseudo-Voigt function. In order to examine the performance of the new camera, the whole powder data of LiF was analyzed by the Maximum Entropy Method(MEM). The MEM analysis of LiF was successfully accomplished using 20 measured structure factors and gave R(Bragg) 0.25%. The density distribution obtained represented a characteristic feature of ionic crystals and was consistent with the theoretical result of the self-consistent LCAO method.

This paper investigates the contributions of sample preparation to the analytical lower limits of detection and errors for the quantitative X-ray diffraction analysis of a simple, binary system of quartz and calcite. This study further demonstrates the danger of using only statistical and instrumental contributions to detection limits and analytical errors.

Quantitative phase analysis (QPA) methods which employ only one diffraction line for each phase arc generally unreliable due to the influence of systematic errors such as preferred orientation and extinction. Gazzara and Messier (1977) proposed a multi-line QPA method using normalised line intensities for line j of phase i,

where Iij = measured Bragg intensity, m = multiplicity, Lp = Lorentz-polarisation factor, |F| = structure amplitude and Vi = cell volume. The method proposed here is a refinement of the Gazzara-Messier procedure. Phase compositions are calculated directly from the mean normalised Intensity for resolved lines which span the 2θ range of the instrument. The method makes use of the variance in to estimate standard deviations for the phase compositions. Test results are given for binaries of corundum-quartz and corundum-ceria.

A menu-driven IBM-PC/MS-DOS software system for quantitative XRF analysis was designed combining elements of pattern recognition (spectral “fingerprinting”) with mathematical correction models. The following topics are covered:

1. - pattern recognition method; N - dimension representation

2. - local empirical Lachance-Traill or Lucas-Tooth & Price models

3. - EGA/VGA intelligent driver for a calibration display

4. - dSASE III Plus “data files - accessed from dBXL/Quicksifver”, dBASE III Plus or from compiled BASIC equipped with “call subroutines”

5. - Examples of applications (stainless steels and ceramics).

In this approach, experimental alphas are computed utilizing the closest automatically preselected standards from available data files. Alpha values generated in this fashion we term “local alphas”. An intelligent software program is employed which automatically recognizes the resolution of tiie CRT display and delivers the best possible display for the available equipment. This new design also permits the utilization of dBXL/Quicksitver, dBASE III Plus or compiled BASIC resources for additional on site system customization programming. A dBASE III Plus file format is used for XRF data storage. This permits fast data exchange with a local ASCII database and also with all popular spreadsheet formats. Additionally, there are utility subroutines available which allow direct and rapid manipuiation of dBASE III Plus files in compiled BASIC. Application of this software system and graphics to stainless steels and engineered ceramic materials are demonstrated. RIGAKU Smax/PDP-11/73 and TRACOR X-RAY SPECTRACE 5000/Compaq 386 spectrometers are used in these studies.

A new approach to quantitative XRD by Partial Least Squares (PLS) used region(s) or the entirety of the diffraction pattern of calibration standards (also called a training set) in the model. The basic concept of this approach states that the information in many observed variables, expressed as matrix I = (i1, i2 … , ik,) is concentrated onto a few underlying latent variables, called factors, by the process of data compression. In XRD, the data points of the diffraction pattern are compressed to few factors T, computed according to their ability to explain the variation in the diffraction pattern or matrix I. The procedure incorporates into the model that part of I that is correlated to C concentrations. Data compression preserves the redundancy between variables due to collinearity and stabilizies the predictions against noise in I. The resulting calibration model allows for detection of outliers. Another important effect of data reduction is the ability to analyze muticomponent systems even when lines of the components are overlapped, Examples of quantitative analysis by PLS are demonstrated in the analysis of a commercial product.

At grazing incidence all X-ray techniques become surface sensitive. Far below the angle of total reflection the X-rays penetrate only 20 to 70 Å into condensed matter. X-ray reflectivity measurements give the density and the thickness of layers on substrates and the roughness of external and internal interfaces. The diffuse scattering in the vicinity of the specular reflection gives, besides the interface roughness, also the height-height correlation of the surface. From the angular dependence of the fluorescence intensity, emitted as a consequence of X-ray absorption, the depth profile of the absorbing species can be deduced. Artificial periodic multilayers give rise to Bragg reflexes and to standing X-ray wave fields, the angular dependence of which give the distribution of atomic species in the periodic stack. A number of examples illustrate the possibilities of the technique.

Precision X-ray reflectivity data were obtained with a high-resolution reflectometer equipped with a rotating anode X-ray source and Ge 220 channel monochromators (one placed before and the other after the specimen). The surfaces and buried interfaces of thin films were characterized by ieast-squares refinement of experimental data. Values of thickness, density, and/or roughness of Pt “single-layer” and Pt/Co based multiple-layer films were determined.