International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.10, pp. 374-384
https://doi.org/10.1107/97809553602060000955

Chapter 3.10. Accuracy in Rietveld quantitative phase analysis with strictly monochromatic Mo and Cu radiations

L. León-Reina,a A. Cuesta,b M. García-Maté,c,d G. Álvarez-Pinazo,c,d I. Santacruz,c O. Vallcorba,b A. G. De la Torrec and M. A. G. Arandab,c*

aServicios Centrales de Apoyo a la Investigación, Universidad de Málaga, 29071 Málaga, Spain,bALBA Synchrotron, Carrer de la Llum 2–26, Cerdanyola, 08290 Barcelona, Spain,cDepartamento de Química Inorgánica, Cristalografía y Mineralogía, Universidad de Málaga, 29071 Málaga, Spain, and dX-Ray Data Services S.L., Edificio de institutos universitarios, c/ Severo Ochoa 4, Parque tecnológico de Andalucía, 29590 Málaga, Spain
Correspondence e-mail:  g_aranda@uma.es

This chapter is based on an article Accuracy in Rietveld quantitative phase analysis: a comparative study of strictly monochromatic Mo and Cu radiations by León-Reina et al. [(2016[link]), J. Appl. Cryst. 49, 722–735]. It reports Rietveld quantitative phase analyses using laboratory-based Mo and Cu radiations where synchrotron powder diffraction [λ = 0.77439 (2) Å] has been used to validate the most challenging analyses. From the results for three series with increasing contents of an analyte (inorganic crystalline phases, organic crystalline phases and a glass), it is inferred that Rietveld analyses from high-energy Mo Kα1 patterns have slightly better accuracies than those obtained from Cu Kα1 diffraction data. This behaviour was established from the results of calibration graphs obtained through the spiking method and also from Kullback–Leibler distance-statistic studies. The better accuracies achieved when using Mo Kα radiation can be attributed to the higher penetration of Mo K radiation compared with Cu radiation, and hence the larger number of crystallites that diffract with Mo radiation; the higher energy also allows the recording of patterns with fewer systematic errors, even though the diffraction power for Mo radiation is lower than for Cu radiation. Limits of detection (LoDs) and limits of quantification (LoQs) have also been established for the studied series. For similar recording times, LoDs in Cu patterns (∼0.2 wt%) are slightly lower than those derived from Mo patterns (∼0.3 wt%). The LoQ for a well crystallized inorganic phase using laboratory powder diffraction was established as close to 0.10 wt%, as stable fits were obtained. However, the accuracy of these analyses was very poor, with relative errors close to 100%. Only contents higher than 1.0 wt% yielded analyses with relative errors lower than 20%.

3.10.1. Introduction

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Most industrial materials are multiphase systems and the accurate determination of their phase assemblage is key to understanding their performances. There are different approaches to carrying out quantitative phase analysis (QPA; see Chapter 3.9[link] ); however, nowadays, the Rietveld method is the most widely employed methodology for QPA of crystalline materials (Madsen et al., 2001[link]; Scarlett et al., 2002[link]), including cements (Stutzman, 2005[link]; León-Reina et al., 2009[link]; Chapter 7.12[link] ).

The factors affecting the accuracy and precision of Rietveld quantitative phase analysis (RQPA) results can be gathered into three main groups: (i) instrument related, (ii) sample-preparation related and (iii) data-analysis protocol(s). The Rietveld method is a standardless methodology which uses the crystal-structure descriptions of each crystalline component to calculate its powder pattern. For this reason, the correct choice of crystal-structure description for each phase in multiphase materials is key (Zevin & Kimmel, 1995[link]; Madsen et al., 2001[link], 2011[link]). The influence of the instrument type on RQPA has previously been evaluated (Madsen et al., 2001[link]) and the main conclusion was that neutron and synchrotron powder diffraction yielded the best results owing to larger irradiated volumes and also to the minimization of microabsorption effects.

High-energy (short-wavelength) X-rays contribute (i) to minimize absorption and microabsorption effects, (ii) to the measurement of a higher number of Bragg peaks and (iii) to increase the irradiated volume of the specimen. Figs. 3.10.1[link](a) and 3.10.1[link](b) show the irradiated volumes bathed by X-rays when using flat samples for Mo and Cu radiations in transmission geometry, and Fig. 3.10.1[link](c) shows the irradiated volume for Cu in reflection mode (Cuesta et al., 2015[link]). Mo radiation combined with a flat sample in transmission geometry allows an irradiated volume of close to 100 mm3; meanwhile, for Cu radiation (flat samples in reflection and transmission geometries) the irradiated volumes are close to 5 mm3 (Cuesta et al., 2015[link]). In this context, it is worth mentioning that the absorption correction for flat-sample transmission geometry is conceptually similar to that for flat-plate reflection geometry, but the length of the scattered beam path has to be properly defined. The corresponding equation is given in section A5.2.5 of Egami & Billinge (2003[link]).

[Figure 3.10.1]

Figure 3.10.1 | top | pdf |

Irradiated volume for a flat sample holder in transmission mode using (a) Mo radiation and (b) Cu radiation, and (c) reflection mode using Cu radiation. Diffraction-geometry sketches: (d) transmission geometry with primary monochromator, (e) transmission geometry with focusing mirror and (f) reflection geometry with primary monochromator. [Reprinted from Cuesta et al. (2015[link]) with permission from Cambridge University Press.]

It must also be noted that Mo radiation has a major drawback when compared with Cu radiation. The λ3 dependence of diffraction intensity favours the use of Cu radiation by a factor of 10.2. Thus, a detector receives approximately ten times as many diffracted X-ray photons with Cu than with Mo (this calculation neglects the different fractions of photons lost in the diffractometer optical paths). This fact can be partially overcome in modern X-ray detectors by increasing the counting time for patterns collected with Mo radiation without reaching prohibitively long times.

As discussed in Chapter 3.9[link] , there are many factors that affect the accuracy and precision of QPA results. It must be recalled that accuracy is the agreement between the analytical result and the true value, and precision is the agreement between results for analyses repeated under the same conditions. Precision may be further divided into repeatability, the agreement between analyses derived from several measurements on the same specimen, and reproducibility, the agreement including re-preparation, re-measurement and data re-analysis of the same sample. Since the largest sources of errors in RQPA are experimental, sample preparation is key, as the reproducibility of peak-intensity measurements is mainly governed by particle statistics (Elton & Salt, 1996[link]). It is generally accepted that the diffraction intensities have to be collected with an accuracy close to ±1% to obtain patterns that are suitable for good RQPA procedures (Von Dreele & Rodriguez-Carvajal, 2008[link]). Milling the sample to reduce the particle size is an approach that should be exercised with care to avoid peak broadening or amorphization (Buhrke et al., 1998[link]). In order to improve particle statistics, a very common practice is to continuously spin the sample during data collection. A much less developed approach is to use high-energy, highly penetrating laboratory X-rays.

Another important issue in the QPA of mixtures is the limit of detection (LoD) and the limit of quantification (LoQ). In this context, the LoD can be defined as the minimal concentration of analyte that can be detected with acceptable reliability (Zevin & Kimmel, 1995[link]), i.e. for which its strongest (not overlapped) diffraction peak in the powder pattern has a signal-to-noise ratio larger than 3.0. The `reliability' criterion is flexible and may be defined by regulatory agencies, as is mainly the case for active pharmaceutical ingredients. Evidently, the LoD can be reduced (improved) by increasing the intensity of the X-ray source, for example using synchrotron radiation. In this context, the LoQ can be defined as the minimum content of an analyte that can be determined with a value at least three times larger than its standard deviation and determined to an acceptable reliability level. For RQPA, this type of approach can be straightforward, although the accuracy for minor phases may be quite poor.

The main aim of the study described here was to test whether the use of high-energy Mo radiation, combined with high-resolution X-ray optics, could yield more accurate RQPA than well established procedures using Cu radiation. In order to do so, three sets of mixtures with increasing amounts of a given phase (the spiking method) were prepared and the corresponding RQPA results were evaluated with calibration curves (least-squares fits) and quantitatively by statistical analysis based on the Kullback–Leibler distance (KLD; Kullback, 1968[link]). The three series were (i) crystalline inorganic phase mixtures with increasing amounts of an inorganic phase, (ii) crystalline organic phase mixtures with increasing amounts of an organic compound and (iii) a series with an increasing content of amorphous ground glass. This last series is the most challenging case because the amorphous content is derived from a small overestimation of the internal standard employed. Amorphous content determination is important for many industries, including cements, glasses, pharmaceuticals and alloys.

3.10.2. Compounds and series

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3.10.2.1. Single phases

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Table 3.10.1[link] provides information about the phases used in this work. Further details can be found in the original publication (León-Reina et al., 2016[link]). All of the mixtures were prepared by grinding the weighed phases by hand in an agate pestle and mortar for 20 min to ensure homogeneity.

Table 3.10.1| top | pdf |
Cambridge Structural Database (CSD)/Inorganic Crystal Structure Database (ICSD) reference codes for all phases used for Rietveld refinements in this work and the linear absorption coefficients for the wavelengths used

PhaseChemical formulaCSD/ICSD refcodeμ (cm−1), Cu Kα1, λ = 1.5406 Åμ (cm−1), Mo Kα1, λ = 0.7093 Åμ (cm−1), λ = 0.7744/0.4959 ÅReference
Glucose C6H12O6 Glucsa10 12 1 1.3/— Brown & Levy (1979[link])
Fructose C6H12O6 Fructo11 12 1 1.3/— Kanters et al. (1977[link])
α-Lactose monohydrate C12H22O11·H2O Lactos10 12 1 1.3/— Fries et al. (1971[link])
Xylose C5H10O5 Xylose 12 1 1.2/— Hordvik (1971[link])
Gypsum CaSO4·(H2O)2 151692 141 16 22/— De la Torre et al. (2004[link])
Quartz SiO2 41414 92 10 11/2.9 Will et al. (1988[link])
s-Anhydrite CaSO4 16382 219 24 31/— Kirfel & Will (1980[link])
i-Anhydrite CaSO4 79527 219 24 31/— Bezou et al. (1995[link])
Zincite ZnO 65120 285 244 —/89.1 Albertsson et al. (1989[link])
Calcite CaCO3 80869 194 22 27/7.3 Maslen et al. (1995[link])
SrSO4 SrSO4 22322 299 187 40/— Garske & Peacor (1965[link])

3.10.2.2. Crystalline inorganic series

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A constant matrix of calcite (C), gypsum (Gp) and quartz (Q) was prepared. Six samples with known increasing amounts of insoluble anhydrite (i-A) were then produced and were labelled CGpQ_xA, where x repesents the target i-A content: 0.00, 0.125, 0.25, 0.50, 1.0, 2.0 or 4.0 wt%.

3.10.2.3. Crystalline organic series

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A constant matrix of glucose (G), fructose (F) and lactose (L) was prepared. Six samples with known increasing amounts of xylose (X) were then produced and labelled GFL_xX, where x represents the target X content: 0.00, 0.125, 0.25, 0.50, 1.0, 2.0 or 4.0 wt%.

3.10.2.4. Variable amorphous content series

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A constant matrix of calcite (C) and zincite (Z) was prepared. Five samples with increasing contents of amorphous ground glass (Gl) were then prepared. The elemental composition of the ground glass is given in García-Maté et al. (2014)[link]. The mixtures were labelled CZQ_xGl, where x indicates 0, 2, 4, 8, 16 or 32 wt% Gl. The amorphous content was determined by adding ∼20 wt% quartz (Q) as an internal standard.

3.10.3. Analytical techniques

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All phases and mixtures were studied with Mo Kα1 (transmission geometry) and Cu Kα1 (reflection geometry) monochromatic radiation. Table 3.10.1[link] shows the X-ray linear absorption coefficients for all of the phases, as microabsorption is always a concern in RQPA. A microabsorption correction was not applied in this work, but readers must be aware that this effect, if relevant, is one of the greatest source of inaccuracy in RQPA (Madsen et al., 2001[link]; Scarlett et al., 2002[link]). All of the phases were also characterized by scanning electron microscopy (see Fig. 3.10.2[link]).

[Figure 3.10.2]

Figure 3.10.2 | top | pdf |

Scanning electron microscopy micrographs for the studied phases (×1000). The inset in the zincite micrograph shows the powder at higher magnification (×20 000).

3.10.3.1. Mo Kα1 laboratory X-ray powder diffraction (LXRPD)

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Mo Kα1 powder patterns were collected in transmission geometry in constant irradiated volume mode, in order to avoid any correction of the measured intensities, on a D8 ADVANCE (Bruker AXS) diffractometer (188.5 mm radius) equipped with a Ge(111) primary monochromator, which gives monochromatic Mo radiation (λ = 0.7093 Å). The X-ray tube operated at 50 kV and 50 mA. The optics configuration was a fixed divergence slit (2°) and a fixed diffracted anti-scatter slit (9°). A LYNXEYE XE 500 µm energy-dispersive linear detector, optimized for high-energy radiation, was used with the maximum opening angle. Using these conditions, the samples were measured between 3 and 35° 2θ with a step size of 0.006° and with a total measurement time of 3 h 5 min. The flat samples were placed into cylindrical holders between two Kapton foils (Cuesta et al., 2015[link]) and rotated at a rate of 10 revolutions per minute during data collection. Moreover, the absorption factor of each sample was experimentally measured by comparing the direct beam with and without the sample (Cuesta et al., 2015[link]). The amount of sample loaded (which determines the height of the cylinder) in the holders was adjusted to obtain a total absorption (μt) of ∼1, which corresponds to an absorption factor of ∼2.7 or 63% of direct-beam attenuation. For the organic samples this criterion was not followed as it would lead to very thick specimens. In this case, the maximum holder thickness was used (1.7 mm).

3.10.3.2. Cu Kα1 laboratory X-ray powder diffraction (LXRPD)

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Cu Kα1 powder patterns for exactly the same samples were recorded in reflection geometry (θ/2θ) on a X′Pert MPD PRO (PANalytical B.V.) diffractometer (240 mm radius) equipped with a Ge(111) primary monochromator, which gives monochromatic Cu radiation (λ = 1.54059 Å). The X-ray tube was operated at 45 kV and 40 mA. The optics configuration was a fixed divergence slit (0.5°), a fixed incident anti-scatter slit (1°), a fixed diffracted anti-scatter slit (0.5°) and an X′Celerator RTMS (real-time multiple strip) detector operating in scanning mode with the maximum active length. Using these conditions, the samples were measured between 6.5 and 81.5° 2θ with a step size of 0.0167° and a total measurement time of 2 h 36 min. The flat samples were prepared by rear charge of a flat sample holder in order to minimize preferred orientation and were rotated at a rate of 10 revolutions per minute.

The lowest analyte content samples, CGpQ_0.12A and GFL_0.12X, were measured three times using both radiations, Mo Kα1 and Cu Kα1, for a precision (reproducibility) assessment. Therefore, regrinding and reloading of the mixtures in the sample holder was carried out prior to every measurement.

3.10.3.3. Transmission synchrotron X-ray powder diffraction (SXRPD)

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Powder patterns for the lowest analyte content samples, CGpQ_0.12A and GFL_0.12X, were also measured using synchrotron radiation. SXRPD data were collected in Debye–Scherrer (transmission) mode using the powder diffractometer at the ALBA Light Source (Fauth et al., 2013[link]). The wavelength, λ = 0.77439 (2) Å, was selected with a double-crystal Si(111) monochromator and was determined using the NIST SRM640d Si standard (a = 5.43123 Å). The diffractometer is equipped with a MYTHEN-II detector system. The samples were loaded into glass capillaries 0.7 mm in diameter and were rapidly rotated during data collection to improve the diffracting-particle statistics. The data-acquisition time was 20 min per pattern to attain a very good signal-to-noise (S/N) ratio over the angular range 1–35° 2θ. Three patterns, taken at different positions along the capillaries, were collected for each sample.

SXRPD data for the amorphous content series, CZQ_xGl, were also measured at the ALBA Light Source. The experimental setup was the same as described above but the working wavelength was λ = 0.49591 (2) Å.

3.10.4. Powder-diffraction data analysis

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All powder patterns were analysed by the Rietveld method using the GSAS software package (Larson & Von Dreele, 2000[link]) with the pseudo-Voigt peak-shape function (Thompson et al., 1987[link]) for RQPA. The refined overall parameters were phase scale factors, background coefficients (linear interpolation function), unit-cell parameters, zero-shift error, peak-shape parameters and preferred-orientation coefficient, when needed. The March–Dollase preferred-orientation adjustment algorithm was employed (Dollase, 1986[link]). The modelling direction must be given as input for the calculations. In this case, the directions for the different phases were taken from previous studies. Alternatively, this direction can be extracted from the pattern from an analysis of the differences between observed and calculated intensities for non-overlapped diffraction peaks. The crystal structures used are reported in Table 3.10.1[link].

In order to provide a single numerical assessment of the performance of each analysis, a statistic based on the KLD distance was used (Kullback, 1968[link]). This approach was previously used to evaluate the accuracy of RQPA applied to standard mixtures (Madsen et al., 2001[link]; Scarlett et al., 2002[link]; León-Reina et al., 2009[link]). Both phase-related KLD distances and absolute values of the Kullback–Leibler distance (AKLD) were calculated. Accurate analyses are mirrored by low values of AKLD.

The overall amorphous content was determined from the internal standard methodology approach (De la Torre et al., 2001[link]; Aranda et al., 2012[link]) with quartz as an internal standard [using isotropic atomic displacement parameters (ADPs) of 0.045 and 0.0087 Å2 for Si and O, respectively]. If the original sample contains an amorphous phase, the amount of standard will be overestimated in RQPA. From the (slight) overestimation of the standard, the amorphous content of the investigated sample can be derived (De la Torre et al., 2001[link]). The important role of the values of the ADPs in the results of RQPA mainly in amorphous content determinations using the internal-standard method has been discussed previously (Madsen et al., 2011[link]).

3.10.5. Crystalline single phases

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All of the single phases were selected according to several parameters, such as relevance to selected applications, purity, particle size of the powder and preferred orientation. In order to check the suitability of the crystal structures used, all of the phases were first studied using powder diffraction with Mo Kα1 radiation. These preliminary studies were of special interest for organic phases, as the CIF files obtained from the Cambridge Structural Database (CSD) did not contain the atomic displacement parameters (ADPs). For lactose and fructose, the ADPs were obtained from the original publications and were introduced manually into the GSAS control file. For glucose and xylose, the ADP values were not reported in the original publications. Hence, they were obtained from the fits to the Mo Kα1 patterns for the single phases. Three groups of isotropic ADPs were refined: those for O, C and H atoms. The final ADP values are given in León-Reina et al. (2016[link]) as well as the RF values before and after optimization, showing the improvements in the fits. For RQPA of all of the mixtures the ADPs were kept fixed.

Preferred orientation was modelled by the March–Dollase algorithm along the [001] axis for both glucose and lactose. Since microparticle sizes and distributions for different phases may result in some sample-related effects, such as preferred orientation, microabsorption and `rock-in-the-dust/graininess' effects, all powders were characterized by scanning electron microscopy (SEM). Fig. 3.10.2[link] shows SEM micrographs for all of the phases. All inorganic samples were single phases except for gypsum and insoluble anhydrite. The impurity-phase contents for these two samples were reported in León-Reina et al. (2016[link]).

Both organic and inorganic phases were also measured using Cu Kα1 radiation in reflection mode. As expected, a transparency effect was observed in the Cu Kα1 patterns for organic samples (Buhrke et al., 1998[link]).

3.10.6. Limits of detection and quantification

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LoD and LoQ are two important quantities in the validation of any analytical method. LoD/LoQ are terms that are used to describe the smallest concentration of an analyte that can be reliably detected/assessed by an analytical procedure, as discussed in Section [link]3.10.1. In techniques such as Rietveld analysis, the approach of having a powder pattern with its strongest (not overlapped) diffraction peak with an S/N ratio of larger than 3.0 is not straightforward because the full powder pattern is evaluated.

Fig. 3.10.3[link] shows Mo Kα1 and Cu Kα1 raw patterns for the inorganic series with increasing amounts of insoluble anhydrite (labelled with solid squares) and Fig. 3.10.4[link] shows the strongest diffraction peak for i-A in the mixtures containing 0.123 wt% anhydrite (CGpQ_0.12A) and 0.25 wt% anhydrite (CGpQ_0.25A) to evaluate the limits of detection in the conditions reported in Section [link]3.10.5. For CGpQ_0.12A, both laboratory powder patterns yielded peaks with S/N ratios lower than 3.0 (top panels in Fig. 3.10.4[link]). For CGpQ_0.25A, the Cu Kα1 pattern yielded a clear peak with S/N = 4.1; therefore, it can be concluded that the LoD for insoluble anhydrite with this radiation in this mixture is slightly lower than 0.2 wt%. For Mo Kα1 radiation, the CGpQ_0.25A and CGpQ_0.50A samples yielded patterns with peaks with S/N ratios of 2.4 and 5.1, respectively. Hence, it can be concluded that the LoD for i-A with this radiation in this mixture is quite close to 0.3 wt%.

[Figure 3.10.3]

Figure 3.10.3 | top | pdf |

(a) Raw Mo Kα1 powder patterns for the inorganic series composed of a constant matrix of calcite, gypsum and quartz, and increasing amounts of insoluble anhydrite (peaks highlighted with a solid square). (b) Raw Cu Kα1 powder patterns for the same inorganic series. (c) Raw SXRPD patterns for CGpQ_0.12A collected at three different positions of the capillary (red, black and blue traces). The intensity values in (c) have been artificially offset to show the three different patterns.

[Figure 3.10.4]

Figure 3.10.4 | top | pdf |

Selected region of the powder patterns showing the main diffraction peak of insoluble anhydrite for the low-content samples to investigate the limit of detection. Top left: Cu Kα1 pattern for CGpQ_0.12A. Middle left, Cu Kα1 pattern for CGpQ_0.25A. Bottom left, SXRPD pattern for CGpQ_0.12A. Top right, Mo Kα1 pattern for CGpQ_0.12A. Middle right, Mo Kα1 pattern for CGpQ_0.25A. Bottom right, Mo Kα1 pattern for CGpQ_0.50A. The main peak of anhydrite, (θ)/λ = 0.143 Å−1, is located at 25.4, 11.6 and 12.7° 2θ for Cu Kα1, Mo Kα1 and synchrotron radiations, respectively. The peak at sin(θ)/λ = 0.1445 Å−1 is due to the soluble anhydrite from gypsum (constant content in all the samples). The very tiny peak at sin(θ)/λ = 0.1457 Å−1, which is slightly visible only in the SXRPD pattern, arises from SrSO4 (0.39 wt%) from gypsum.

The LoQ for i-A in this matrix was also studied. Three Mo Kα1 and Cu Kα1 patterns were collected for CGpQ_0.12A. For the three Mo Kα1 patterns, the average analysis result for i-A was 0.28 (2) wt%, but the accuracy of the obtained value is poor, as the expected value was 0.12 wt%. Similarly, the average value for the analyses of three Cu Kα1 patterns was 0.24 (2) wt%. The RQPA results are given as supporting information in León-Reina et al. (2016[link]). It was concluded that i-A can be quantified in this mixture at the level of 0.12 wt%, but with a relative error close to 100%. If the `acceptable reliability' criterion in the analysis is taken into consideration, the LoQ value would be close to 1.0 wt% in order to have a relative associated error lower than 20%.

CGpQ_0.12A was also studied by SXRPD. Fig. 3.10.3[link](c) shows the SXRPD patterns collected at three different positions of the capillary, which were almost identical, and Fig. 3.10.4[link] (bottom left) shows the main diffraction peak of anhydrite. The S/N ratio for the strongest diffraction peak of anhydrite was 12.8 and hence the limit of detection for i-A with synchrotron radiation in this matrix is below 0.10 wt%.

To quantify the accuracy of the analyses, the KLD methodology was used. The AKLD values for each analysis as well as the KLD values for i-A are reported in León-Reina et al. (2016[link]). The synchrotron analyses clearly had better accuracy than those using laboratory radiation. Moreover, the Mo Kα1 radiation analyses were slightly better than those obtained using Cu Kα1 radiation.

Fig. 3.10.5[link] shows Mo Kα1 and Cu Kα1 raw patterns of the organic mixtures with increasing amounts of xylose. The strongest powder-diffraction peak for xylose in the GFL_0.12X patterns (with both Mo and Cu radiations) was not observed. The corresponding peak was observed in the GFL_0.25X patterns. Therefore, the LoD can be established as close to 0.25 wt%. The analysis results for xylose in GFL_0.25X were reported in León-Reina et al. (2016[link]). These values showed that the results from Mo Kα1 powder diffraction were slightly more accurate.

[Figure 3.10.5]

Figure 3.10.5 | top | pdf |

(a) Raw Mo Kα1 powder patterns for the organic series composed of a constant matrix of glucose, fructose and lactose, and increasing amounts of xylose (peaks highlighted with an asterisk). (b) Raw Cu Kα1 powder patterns for the same organic series. (c) Raw SXRPD patterns for GFL_0.12X collected at three different positions of the capillary (as collected).

The LoQ for xylose was also studied. Once again, three Mo Kα1 and Cu Kα1 patterns were collected for GFL_0.12X. The average value for the analysis of the three Mo patterns was 0.18 (8) wt%. Similarly, the average result for the analyses of three Cu patterns was 0.34 (6) wt%. Full RQPA results are reported in the supporting information of León-Reina et al. (2016[link]). The LoQ for xylose in this mixture for the two radiations can be established as close to 0.12 wt%. Indeed, if one applies an `acceptable reliability' criterion, the LoQ would be much higher at above 1 wt%. The output of this study was that Cu Kα1 radiation yielded a slightly less accurate result than that obtained from the Mo Kα1 data.

GFL_0.12X was also studied by SXRPD in a rotating glass capillary in transmission mode. Fig. 3.10.5[link](c) shows SXRPD patterns for GFL_0.12X collected at three different positions of the same capillary. The powder patterns showed quite different peak ratios. It is important to bear in mind that filling a glass capillary with organic compounds is sometimes not easy due to electrostatic charge effects. For this reason, the phase ratio within the part of capillary bathed by the X-rays might not be the same as that of the sample under study. The behaviour observed in Fig. 3.10.5[link](c) could be explained by inhomogeneous capillary filling. Hence, in this case, the RQPA results are unreliable. Even in `well behaved' samples, inhomogeneous filling of small capillaries could result in problems. Readers should be aware of this, and the authors strongly recommend that at least three patterns should be collected along the capillary and superimposed. If there is inhomogeneous filling the patterns will differ, and extreme care has to be exercised when filling capillaries in order to minimize this problem.

3.10.7. Increasing inorganic crystalline phase content series

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Table 3.10.2[link] reports the RQPA results for six inorganic mixtures with increasing amounts of i-A measured with Mo Kα1 (transmission) and Cu Kα1 (reflection). The Rietveld plots of the mixture with 4 wt% i-A are shown in Fig. 3.10.6[link]. For most of the samples, the AKLD values (see Table 3.10.2[link]) for Mo Kα1 radiation are slightly smaller than the corresponding values obtained for Cu Kα1 radiation. For this reason, we can conclude that the Mo Kα1 analyses are slightly better than those derived using Cu Kα1 radiation.

Table 3.10.2| top | pdf |
Rietveld quantitative phase analyses for the crystalline inorganic mixtures measured with Cu Kα1 and Mo Kα1 radiations

Weighed amounts (wt%) are also shown for comparison. Absolute values of the Kullback–Liebler distance (AKLD) for each mixture and the KLD value for i-anhydrite are also included. Trm, transmission; rfl, reflection.

 CGpQ_0.0ACGpQ_0.25ACGpQ_0.50A
Phaseswt%Mo trmCu rflwt%Mo trmCu rflwt%Mo trmCu rfl
C 32.9 32.6 (1) 30.4 (2) 32.8 32.0 (1) 33.6 (1) 32.7 33.2 (1) 32.8 (1)
Gp 31.7 31.7 (1) 34.5 (1) 31.7 32.5 (1) 31.6 (1) 31.6 30.1 (1) 30.7 (1)
Q 34.2 34.6 (1) 33.7 (1) 34.1 33.9 (1) 33.0 (1) 34.0 34.6 (1) 34.2 (1)
s-A 0.8 0.66 (3) 0.76 (5) 0.8 0.77 (4) 0.78 (5) 0.8 0.97 (3) 1.15 (5)
SrSO4 0.4 0.44 (4) 0.70 (6) 0.4 0.44 (4) 0.67 (5) 0.4 0.39 (4) 0.56 (5)
i-A 0.28 0.42 (3) 0.42 (4) 0.52 0.71 (3) 0.71 (4)
                   
AKLD sum   0.0089 0.0605   0.0198 0.0235   0.0295 0.0180
(i-A) KLD         −0.001 −0.001   −0.002 −0.002

 CGpQ_1.0ACGpQ_2.0ACGpQ_4.0A
Phaseswt%Mo trmCu rflwt%Mo trmCu rflwt%Mo trmCu rfl
C 32.5 32.8 (1) 32.6 (2) 32.2 31.3 (1) 31.4 (1) 31.6 31.2 (1) 31.8 (1)
Gp 31.5 30.4 (1) 30.7 (1) 31.1 32.1 (1) 32.3 (1) 30.5 30.7 (1) 30.5 (1)
Q 33.8 34.1 (1) 33.8 (1) 33.5 33.5 (1) 32.6 (1) 32.8 32.8 (1) 32.0 (1)
s-A 0.8 1.03 (4) 1.11 (5) 0.7 0.54 (3) 0.58 (5) 0.7 0.67 (3) 0.77 (4)
SrSO4 0.4 0.43 (4) 0.68 (5) 0.4 0.48 (4) 0.68 (6) 0.4 0.45 (4) 0.63 (5)
i-A 1.02 1.23 (3) 1.17 (5) 2.02 2.05 (4) 2.38 (9) 4.02 4.30 (8) 4.33 (9)
                   
AKLD sum   0.0214 0.0152   0.0218 0.0358   0.0095 0.0156
(i-A) KLD   −0.002 −0.001   0.000 −0.003   −0.004 −0.003
[Figure 3.10.6]

Figure 3.10.6 | top | pdf |

Selected range of the Rietveld plots for CGpQ_4.0A: (a) Mo Kα1 and (b) Cu Kα1 patterns. The inset highlights the effect of preferred orientation for gypsum and calcite.

On the other hand, calcite and gypsum presented preferred orientations, with the axes being [104] and [010], respectively. This effect was modelled using the March–Dollase algorithm. Preferred orientation makes the 0l0 reflections for gypsum have higher intensities in the Cu Kα1 patterns, and smaller intensities in the Mo Kα1 patterns, than those calculated from the crystal structure (see insets in Fig. 3.10.6[link]). As a consequence, the refined values for flat samples in reflection and transmission geometries were smaller and larger than 1.0, respectively (Cuesta et al., 2015[link]). Although preferred orientation is present in all patterns, the Cu Kα1 patterns were recorded in reflection geometry (flat samples), while the Mo Kα1 measurements were collected in transmission (also flat samples). This results in opposite diffraction intensity changes and points towards another (possible) fruitful use: joint refinement of these two types of patterns to counterbalance the effects of preferred orientation in RQPA.

Fig. 3.10.7[link](a) shows the quantified i-A contents (wt%), as determined by the Rietveld methodology, as a function of the weighed i-A amount. The two R2 values for the fits are very close to 1.00, and the intercept values are very close to zero, showing the appropriateness of the Rietveld methodology for quantifying crystalline materials. Furthermore, the slopes of the calibration curves are also 1.00 in both cases. Consequently, this study allows it to be concluded that RQPA for crystalline inorganic phases using powder-diffraction patterns collected using Mo Kα1 radiation yields results that are as accurate as those obtained from the well established method using Cu Kα1.

[Figure 3.10.7]

Figure 3.10.7 | top | pdf |

Rietveld quantification results for (a) the insoluble anhydrite series (within an inorganic crystalline matrix), (b) the xylose series (within an organic crystalline matrix) and (c) the ground-glass series (within an inorganic crystalline matrix) as a function of the weighed amount of each phase. Open symbols represent the derived amorphous contents in the mixtures without any added glass. The results of the least-squares fits are also shown.

3.10.8. Increasing crystalline organic phase content series

| top | pdf |

Table 3.10.3[link] shows RQPA results for six mixtures prepared with G, F, L and an increasing amount of X measured with Mo Kα1 (transmission) and Cu Kα1 (reflection). In general, the values obtained using both radiations are quite similar to the weighed values. The AKLD values and the KLD values for the xylose phase are also reported in Table 3.10.3[link]. The AKLD values from Mo Kα1 and Cu Kα1 radiations are relatively similar. The main problem for RQPA of organic mixtures measured in reflection geometry is related to the low X-ray absorption of the samples and the transparency effects that lead to poor peak shapes and even some split peaks in the powder patterns, as discussed previously (León-Reina et al., 2016[link]).

Table 3.10.3| top | pdf |
RQPA for the crystalline organic mixtures measured with Cu Kα1 and Mo Kα1 radiations

Weighed amounts (wt%) are also shown for the sake of comparison. Absolute values of the Kullback–Liebler distance (AKLD) for each mixture and the KLD value for xylose are also included. Trm, transmission; rfl, reflection.

 GFL_0.0XGFL_0.25XGFL_0.50X
Phaseswt%Mo trmCu rflwt%Mo trmCu rflwt%Mo trmCu rfl
G 33.4 33.8 (1) 33.5 (3) 33.3 33.6 (1) 33.1 (2) 33.2 32.3 (2) 33.5 (2)
F 33.5 31.7 (1) 32.7 (3) 33.4 32.3 (1) 34.3 (2) 33.3 32.1 (2) 33.4 (2)
L 33.1 34.5 (1) 33.7 (3) 33.0 33.7 (1) 32.0 (2) 33.0 35.0 (3) 32.5 (2)
X   0.27 0.33 (4) 0.57 (9) 0.55 0.53 (8) 0.61 (9)
                   
AKLD sum   0.0362 0.0150   0.0216 0.0231   0.0410 0.0096
(X) KLD     −0.001 −0.002   0.000 −0.001

 GFL_1.0XGFL_2.0XGFL_4.0X
Phaseswt%Mo trmCu rflwt%Mo trmCu rflwt%Mo trmCu rfl
G 33.0 34.7 (1) 33.6 (2) 32.7 32.2 (1) 31.5 (2) 32.0 32.8 (1) 33.6 (2)
F 33.1 32.6 (1) 33.7 (2) 32.8 31.7 (1) 34.4 (2) 32.2 30.7 (1) 32.5 (2)
L 32.8 31.6 (2) 31.4 (2) 32.5 34.3 (1) 32.0 (2) 31.8 32.9 (1) 30.5 (2)
X 1.1 1.10 (5) 1.3 (1) 2.0 1.76 (5) 2.1 (1) 3.9 3.70 (5) 3.4 (2)
                   
AKLD sum   0.0338 0.0280   0.0363 0.0339   0.0361 0.0372
(X) KLD   0.000 −0.002   0.003 −0.001   0.002 0.005

Fig. 3.10.7[link](b) shows the quantified xylose contents (wt%) as determined by the Rietveld methodology as a function of the weighed amount of xylose added to the mixtures. The results were plotted to obtain the calibration lines with increasing content of the analyte. Both plots gave R2 values close to 1.0. However, the slope values were 0.92 and 0.82 for Mo Kα1 and Cu Kα1 radiations, respectively. Slope values close to 1.0 mirror accurate analyses. Furthermore, the y-intercept values were 0.04 and 0.30 for Mo Kα1 and Cu Kα1 radiations, respectively. A y-intercept value close to 0.0 mirrors accurate analyses. Hence, it can be concluded that slightly more accurate analyses are obtained for Mo Kα1 powder diffraction in transmission when compared with Cu Kα1 powder diffraction in reflection for organic crystalline samples.

3.10.9. Increasing amorphous content series within an inorganic crystalline phase matrix

| top | pdf |

Fig. 3.10.8[link] shows Mo Kα1 (transmission), Cu Kα1 (reflection) and SXRPD (transmission) raw patterns for the mixtures with increasing amounts of glass. It is important to highlight that the increase in the background due to the glass is very modest even for ∼32 wt% of glass. Table 3.10.4[link] shows the RQPA of these mixtures, prepared with C, Z and an increasing amount of Gl, for the three radiations. The glass-free sample may contain amorphous material from the employed phases. Hence, we used the SXRPD data to calculate a correction factor for quartz to yield zero amorphous content for the glass-free sample (León-Reina et al., 2016[link]).

Table 3.10.4| top | pdf |
Rietveld quantitative phase analyses of the CQZ_xGl mixture, where quartz (Q) is the internal standard, to derive amorphous content (am), obtained from SXRPD, Mo Kα1 and Cu Kα1 patterns

Absolute values of the Kullback–Liebler distance (AKLD) for each mixture and the KLD value for the amorphous content are also included. Trm, transmission; rfl, reflection.

MixtureWeighedSynchrotron trm
C wt%Z wt%Gl wt%C wt%Z wt%Am wt%AKLD sumAm KLD
CZQ_0Gl 50.01 49.99 0.00 49.9 (1) 49.6 (1) 0.4 (1) 0.0050
CZQ_2Gl 48.98 48.96 2.05 49.7 (1) 49.0 (1) 1.3 (1) 0.0169 0.009
CZQ_4Gl 47.93 47.91 4.17 47.9 (1) 47.6 (1) 4.5 (1) 0.0066 −0.003
CZQ_8Gl 46.00 46.00 7.99 46.6 (1) 45.9 (1) 7.5 (1) 0.0120 0.005
CZQ_16Gl 41.99 41.99 16.01 42.0 (1) 41.6 (1) 16.4 (1) 0.0079 −0.004
CZQ_32Gl 34.00 34.00 31.99 34.0( 1) 33.7 (1) 32.3 (1) 0.0061 −0.003

MixtureMo Kα1 trmCu Kα1 rfl
C wt%Z wt%Am wt%AKLD sumAm KLDC wt%Z wt%Am wt%AKLD sumAm KLD
CZQ_0Gl 47.5 (1) 49.0 (1) 3.5 (1) 0.0358 47.2 (1) 40.8 (1) 12.0 (1) 0.1305
CZQ_2Gl 45.9 (1) 47.7 (1) 6.4 (1) 0.0679 −0.023 47.4 (1) 40.6 (1) 12.0 (1) 0.1440 −0.036
CZQ_4Gl 46.5 (1) 47.0 (1) 6.5 (1) 0.0422 −0.019 45.8 (1) 39.7 (1) 14.6 (1) 0.1641 −0.052
CZQ_8Gl 42.6 (1) 44.8 (1) 12.5 (1) 0.0832 −0.036 45.3 (1) 38.1 (1) 16.6 (1) 0.1522 −0.058
CZQ_16Gl 39.9 (1) 41.7 (1) 18.5 (1) 0.0475 −0.023 40.9 (1) 35.8 (1) 23.4 (1) 0.1388 −0.061
CZQ_32Gl 31.7 (1) 33.1 (1) 35.2 (1) 0.0635 −0.031 32.2 (1) 28.7 (1) 39.1 (1) 0.1403 −0.064
[Figure 3.10.8]

Figure 3.10.8 | top | pdf |

Raw powder patterns for the amorphous-material-containing series composed of a constant matrix of calcite and zincite, and increasing amounts of ground glass. Quartz was added as internal standard. (a) Mo Kα1, (b) Cu Kα1 and (c) SXRPD radiations. The intensities of the patterns have been rescaled to highlight the contributions of the glass to the background.

The linear fit to the amorphous content values obtained using SXRPD was very good, R2 = 0.998, with the slope being 1.00 within the errors (see Fig. 3.10.7[link]c). This plot also shows the quantified amorphous contents, in weight percentage, as a function of the amount of added ground glass, measured with Mo Kα1 and Cu Kα1 radiations. Open symbols indicate the derived amorphous contents obtained with the internal-standard method in the mixture without any added glass, CZQ_0Gl. Both R2 values are quite close to 1.00, showing the consistency of the internal-standard methodology. However, the slope values were 0.98 and 0.89 for Mo Kα1 and Cu Kα1 radiations, respectively. Furthermore, the y-intercept values were 3.7 and 10.0 for Mo Kα1 and Cu Kα1 radiations, respectively. Again, slope values close to 1.0 and y intercepts close to 0.0 mirror accurate analyses. It must also be pointed out that for the Mo Kα1 analyses the value from the measurement of the Gl-free sample, 3.5 wt%, matches the value from the y intercept of the plot, 3.7 wt%, very well. Meanwhile, there is a much larger discrepancy for the similar Cu-based analyses, 12.0 and 10.0 wt%, respectively, which is quite far from zero. Hence, it is concluded that the amorphous contents derived from Mo Kα1 data are more accurate than those derived from Cu Kα1 data. However, it is not possible to reliably quantify amorphous contents below ∼8–10 wt% from Mo Kα1 and Cu Kα1 diffraction data (see Table 3.10.4[link]) with the internal-standard method.

On the contrary, SXRPD reliably allows quantification of amorphous contents down to ∼2 wt% for this relatively simple mixture. In addition, the AKLD and the KLD values reported in Table 3.10.4[link] demonstrate that the synchrotron analyses are indeed much better than the laboratory analyses.

3.10.10. Conclusions

| top | pdf |

  • (i) We have thoroughly studied the limit of detection for a well crystallized inorganic phase in an inorganic compound matrix. We have determined the following LoDs for insoluble anhydrite: ∼0.2 wt%, ∼0.3 wt% and lower than 0.1 wt% for Cu Kα1, Mo Kα1 and synchrotron radiations, respectively. We conclude that the LoD is slightly better for Cu Kα1 than for Mo Kα1 because the λ3 dependence of the diffraction intensity, with similar acquisition times, yielded slightly better signal-to-noise ratios in the Cu patterns. Of course, detector efficiencies also play a role in the measured signal-to-noise ratios.

  • (ii) We have also studied the limit of quantification for a well crystallized inorganic phase using laboratory X-ray powder diffraction. This phase could be quantified at the level of 0.12 wt% in stable fits with repeatable outputs and good precision. However, the accuracy of these analyses was quite poor, with relative errors close to 100%. Only contents higher than 1.0 wt% yielded analyses with relative errors lower than 20%.

  • (iii) The Rietveld quantitative phase analysis results from high-resolution Mo Kα1 powder diffraction (transmission geometry) and high-resolution Cu Kα1 powder diffraction (reflection geometry) were quite similar for a series of crystalline inorganic phase samples. We inferred the validation of the Mo-based analyses procedure from this initial study, as it yielded results very close to well established high-resolution Cu radiation analyses (see Fig. 3.10.7[link]a). From the comparison of the AKLD values for the two types of analyses, it was demonstrated that the Mo Kα1 analyses were slightly better than those using Cu Kα1.

  • (iv) Comparison of the results obtained from Mo-based and Cu-based patterns for a series of crystalline organic phase mixtures showed that the Mo Kα1 analyses gave slightly more accurate values. This conclusion was drawn because the calibration curve obtained from Mo patterns with increasing content of xylose gave an R2 value closer to 1.0, a slope closer to 1.0 and an intercept value close to 0.0 (see Fig. 3.10.7[link]b). The slightly poorer results from Cu Kα1 analyses are very likely to be due to the transparency effects in reflection geometry.

  • (v) Comparison of the results obtained from Mo Kα1 and Cu Kα1 patterns for a series containing increasing amounts of amorphous glass also indicated that the Mo-based analyses were slightly more accurate than the corresponding Cu Kα1 analyses. This conclusion was drawn because the obtained calibration curve from the Mo data has (1) a slope closer to 1.0, (2) a smaller amorphous value for the glass-free sample and (3) a closer agreement between the intercept from the least-squares fit and the determined amorphous value for the glass-free sample (see Fig. 3.10.7[link]c). The AKLD analysis confirmed this outcome. Furthermore, the results from synchrotron data have the best accuracy, as shown by the calibration plot and the AKLD analysis.

Finally, we conclude that for the challenging quantification analyses studied here, the results derived from high-energy Mo Kα1 patterns were slightly more accurate than those obtained from Cu Kα1 patterns. We justify this conclusion based on the larger tested volume for Mo Kα1 analyses, which led to better statistics/accuracy in the recorded powder-pattern intensities. The minimization of microabsorption in the Mo Kα1 transmission data is very likely to be an additional factor in the improved accuracy.

Acknowledgements

This chapter is based on an article Accuracy in Rietveld quantitative phase analysis: a comparative study of strictly monochromatic Mo and Cu radiations by León-Reina et al. [(2016[link]), J. Appl. Cryst. 49, 722–735]. The work was supported by Spanish MINECO through BIA2014-57658-C2-2-R, which is co-funded by FEDER, and BIA2014-57658-C2-1-R research grants. Funding from Junta de Andalucía (grant P11-FQM-07517) is also acknowledged. We thank CELLS-ALBA (Barcelona, Spain) for providing synchrotron beam time on the BL04-MSPD beamline. All raw powder-diffraction data files underlying this work can be accessed at Zenodo at https://doi.org/10.5281/zenodo.1291900 and used under the Creative Commons Attribution license.

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