QUANTITATIVE PHASE ANALYSIS IN POWDER DIFFRACTION

Julius Schneider

Institut für Kristallographie und Angewandte Mineralogie Universität München, Theresienstrasse 41 D-80333 München, Germany,

Identification and determination of relative abundances of constituent components in polycristalline samples is of paramount importance in such diverse fields as crystallography, mineralogy, petrology, solid state chemistry, materials science, archeology, criminology, pharmacy, etc. Powder diffraction is one of the few techniques quich is truly phase sensitive rather than purely element sensitive like X-ray flourescence. It has proved to be an ideal tool for quantitative phase analysis (QPA) because of it's non-destructive nature and easy adaptability to a great number of problems.

Phase identification, which of course has to precede QPA, is nowadays almost exclusively performed with the aid of computerized databases, like the Powder Diffraction File (PDF, about 43000 entries) and the Inorganic Crystal Structure Database (ICSD, about 43000 entries). QPA is based on the fact that diffraction intensities of the individual phases are proportional to their relative amount within the mixture. However, the intensities are modified by absorption effects and are therefore not independent of the other phases present.

Essentially four ways of QPA have emerged to solve this problem:

1. Discrete Peak Methods are based on the possibility to isolate one or more diffraction peaks of each phase and on their subsquent comparison. The internal standard method, where a known fraction of standard powder is added to the sample, allows to eliminate the unknown matrix absorption of the mixture. Use of multiple lines improves the accuracy but greatly increases the number of required calibration constants. The use of relative intensities within the individual phases and of reference intensities (RIR's) with respect to some standard were introduced to circumvent this problem. Constraining the sum of the phase fractions to unity a QPA without standards is possible.
2. Whole Pattern Methods Much better averaging over systematic errors like preferred orientation etc. may be obtained by the use of full powder patterns. Based on fixed instrument conditions a database of observed whole powder patterns and RIR's is used to perform QPA by weighed summation of the patterns of the constitutents. With this method as well as with the following one the critical process of pattern decomposition is not needed. Materials of unknown structure, disordered materials, amorphous phases etc. may be treated also. However, much effort has to be put into the production of the standardized reference powder diagrams.
3. Multiphase Rietveld Analysis uses a completely different approach. Here the sum of calculated powder patterns, generated from crystallographic structure models and line profile models, is compared with the observed diffraction trace. By least square fitting the model parameters the weight fractions of the components are determined from optimized scale factors of the individual phases. Phase-pure standards are not needed, since the intensities are calculated on an absolute scale. Overlapped lines and patterns may be treated with no difficulty. Although the need for structure models is a drawback, great flexibility is provided in that the model parameters like cell parameters, atom species, atom coordinates, temperature factors, profile parameters etc. are adapted to the problem. The obtainable accuracy and precision issignificantly better then with conventional methods. Systematic errors, like preferred orientation etc., may be detected and possibly be included in the model calculations.
4. Combined Methods - Modified Rietveld methods allow the combination of calculated and observed whole powder patterns, which can releave method 3 from the structure model constraint. - Combinations with results from non-diffraction methods like element abundance obtained from X-ray fluorescence etc. may considerably improve accuracy.

The accuracy of QPA, be it discrete peak or whole pattern, may severly be hampered by sample induced aberations like particle size or particle statistics, sample preparation variability, preferred orientation, mikroabsorption, extinction, angle dependent absorption (thin samples), etc. Some of these systematic errors may be greatly reduced by experimental precautions like careful sample preparation, sample spinning, sample geometry etc. On the other hand, great effort has been put into the creation of mathematical models to simulate these errors and to integrate them into Rietveld systems. Being aware of pitfalls, verification of XRD-QPA is thus a very important step. First of all, checks for internal consistency like comparison of high and low angle data should be performed. Comparison with complementary diffraction information, like X-ray data at different wavelengths or neutron data can serve to detect and overcome absorption related errors. Possible common mode errors of Rietveld-based QPA, namely wrong structure models, may be cross-checked by non-diffraction methods like optical spectroscopy, calculations via normative analysis etc. Most recent applications include a chemometrics approach in which the whole diffraction pattern is used to derive information (QPA, plant operating conditions etc.) through the use of 'learning sets'. Many of the procedures decribed above have been integrated into software packages, the more prominent one's will be shortly discussed also. The review will be complemented by representative examples from diverse fields of application.