International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 3.9, pp. 366368
Section 3.9.10.3.3. Microabsorption^{a}CSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,^{b}TU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D09596, Germany, and ^{c}Bruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany 
The strongest ongoing impediment to accuracy in QPA using XRD data is microabsorption. The microabsorption effect occurs when a multiphase sample contains both low and highly absorbing phases. For the highly absorbing phases, the Xray beam is more likely to be absorbed in the surface layers of the grain; thus, the fraction of the grain contributing to the diffraction pattern will decrease as the size of the grain increases above the beampenetration depth. For the lowabsorbing phases, the beam penetrates further into the particle resulting in a greater likelihood of the desired `volume diffraction' occurring (Brindley, 1945). The overall effect is the observation of a disproportionate amount of observed intensity from individual grains relative to what would be expected for the average absorption of the sample; the highly absorbing phases are underrepresented relative to the lowabsorbing phases. There is extensive discussion of the microabsorption issue in Zevin & Kimmel (1995).
Brindley (1945) has described the particle absorption contrast factor τ_{α} aswhere V is the particle volume, and μ_{α} and are the linear absorption coefficients of phase α and the entire sample, respectively. While it is relatively easy to calculate the absorption coefficients, equation (3.9.48) implies knowledge of the particle size of each component; this information is only available through independent microscope or lightscattering characterization.
This correction term is commonly incorporated into QPA through a modification to equation (3.9.26) of the formBrindley has also devised criteria by which to assess whether a microabsorption problem is likely to be present or not. Calculation of μD (where μ is the linear absorption coefficient and D is the particle diameter) yields the following criteria:
It is difficult for the analyst encountering a new sample to determine whether a correction for microabsorption is required without first obtaining additional information. A minimum requirement should be to calculate μD for each phase present. However, this requires knowledge of the particle size which, in a multiphase sample, can be very difficult to obtain unambiguously. Even when the particle size is measured by, for example, dynamic light scattering or optical or SEM image analysis, the applicability of the correction can still be unclear. In addition, the correction factor embodied in equations (3.9.48) and (3.9.49) makes the assumption that the particles of the phase of interest are spherical and of uniform size. This assumption is unrealistic in almost all samples; in reality, each phase is likely to be present at a wide range of particle sizes and the particles are highly unlikely to be spherical.
Table 3.9.5 shows the calculated values of μD for Cu Kα radiation for some commonly encountered phases in mineralogical analysis. For the least absorbing phase (corundum), the upper range of applicability of the Brindley model (medium powder) is reached at about 5 µm; by 8 µm, the coarse powder criterion has been reached and the correction model is no longer applicable. For magnetite, these limits are reached an order of magnitude earlier at about 0.5 and 0.9 µm, respectively.

Fig. 3.9.19 shows an SEM image of a mixture of approximately equal amounts of corundum, magnetite and zircon. The individual components of the sample were weighed and the mixture ground in ethanol in a McCrone micronizing mill (McCrone Research Associates, London) for 10 min g^{−1}. This approach to sample preparation is generally accepted as best practice for powder XRD because it minimizes structural damage during grinding (Hill & Madsen, 2002). After decanting and drying, the sample was backpacked into a cavity sample holder for XRD data collection; the same sample was then used to obtain the SEM image in Fig. 3.9.19. Visual observation shows a wide range of particle sizes (from submicron to greater than 10 µm) and shapes that do not even approximate spheres. Even if this information is obtained, selection of a particle size that best represents each individual phase is a difficult task. In addition, in many sample suites, the component phases exhibit a range of hardness resulting in different rates of grinding and hence difference size ranges. Regrettably, what happens too often in practice is that analysts will micronize the sample and then select an arbitrary particle size in order to derive a `preferred' value for the final analysis. Therefore, caution is advised in the application of these correction models. The IUCr CPD round robin on QPA (Madsen et al., 2001; Scarlett et al., 2002) showed that many participants severely degraded their results by applying a correction when none was necessary.
Equation (3.9.48) shows that there two ways to minimize microabsorption. The first is to reduce the absorption contrast by, for example, changing the Xray wavelength. While corundum and magnetite have very different linear absorption coefficients for Cu Kα radiation (126 and 1123 cm^{−1}, respectively), the difference is reduced to 196 and 231 cm^{−1}, respectively, for Co Kα radiation. The second approach is to reduce the particle size in order to meet Brindley's fine or mediumpowder criteria.
However, even these steps may not be sufficient to eliminate the microabsorption effect. Slightly different absorption coefficients, or different particle sizes for phases with the same absorption coefficients, may still introduce a bias between expected and analysed concentrations. In this situation, it may be better to use a calibrated hkl_phase or peaks_phase (Section 3.9.6) instead of a Rietveld, structurebased phase. The calibration step involved in the generation of such a phase incorporates the microabsorption problem into the calibration constant.
Fig. 3.9.20 shows the bias between known concentrations (derived from chemical analysis) and QPAdetermined concentrations for a series of salt samples. The samples contain halite (NaCl), sylvite (KCl) and kieserite (MgSO_{4}·H_{2}O) as major phases and small amounts of anhydrite (CaSO_{4}), langbeinite [K_{2}Mg_{2}(SO_{4})_{3}] and carnallite [KMgCl_{3}·6(H_{2}O)]. The linear absorption coefficient of sylvite (254 cm^{−1}) is much higher than halite (165 cm^{−1}). Using crystalstructurebased analysis, there is a systematic deviation of up to 3% with an overestimation of the low absorber (halite) and an underestimation of the high absorber (sylvite). After replacing sylvite by a calibrated hkl_phase, the bias is reduced to about 1% and does not show systematic deviations.
It should be noted, however, that the phase constants developed using such a calibration approach will only be applicable to the sample suite and preparation conditions for which it was developed. The calibration process will need to be repeated if there are significant changes to the sample suite or samplepreparation conditions.
References
Brindley, G. W. (1945). The effect of grain or particle size on Xray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by Xray methods. London Edinb. Dubl. Philos. Mag. J. Sci. 36, 347–369.Google ScholarHill, R. J. & Madsen, I. C. (2002). Structure Determination from Powder Diffraction Data, edited by W. David, K. Shankland, L. McCusker & C. Baerlocher, ch. 6, pp. 98–116. Oxford University Press.Google Scholar
Madsen, I. C., Scarlett, N. V. Y., Cranswick, L. M. D. & Lwin, T. (2001). Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 1a to 1h. J. Appl. Cryst. 34, 409–426.Google Scholar
Scarlett, N. V. Y., Madsen, I. C., Cranswick, L. M. D., Lwin, T., Groleau, E., Stephenson, G., Aylmore, M. & AgronOlshina, N. (2002). Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 2, 3, 4, synthetic bauxite, natural granodiorite and pharmaceuticals. J. Appl. Cryst. 35, 383–400.Google Scholar
Zevin, L. S. & Kimmel, G. (1995). Quantitative Xray Diffractometry. SpringerVerlag New York, Inc.Google Scholar