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. 3.9, pp. 345-350

Section 3.9.3. QPA methodology

I. C. Madsen,a* N. V. Y. Scarlett,a R. Kleebergb and K. Knorrc

aCSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,bTU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D-09596, Germany, and cBruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany
Correspondence e-mail:  ian.madsen@csiro.au

3.9.3. QPA methodology

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The integrated intensity I(hkl of reflection hkl for phase α in a multiphase mixture, measured on a flat-plate sample of infinite thickness using a diffractometer with Bragg–Brentano geometry, is given by (Snyder & Bish, 1989[link]; Zevin & Kimmel, 1995[link]; Madsen et al., 2013[link])[\eqalignno{&I_{(hkl)\alpha}= \left [{{{{I_0}{\lambda ^3}} \over {32\pi r}}{{{e^4}} \over {m_e^2{c^4}}}} \right] &\cr&\quad\times \left [{{{{M_{hkl}}} \over {2V_\alpha ^2}}{{\left| {{F_{(hkl)\alpha }}} \right|}^2} \left({{{1 + {{\cos }^2}2\theta \, {{\cos }^2}2{\theta _m}} \over {{{\sin }^2}\theta \cos \theta }}} \right) \exp\left({ - 2B{{\left({{{\sin \theta }/ \lambda }} \right)}^2}} \right)} \right] &\cr&\quad\times\left [{{{{W_\alpha }} \over {{\rho _\alpha }\mu _m^*}}} \right],&(3.9.1)}]where I0 is the incident-beam intensity, λ is the wavelength, e is the charge on an electron, me is the mass of an electron, r is the distance from the scattering electron to the detector and c is the speed of light. Mhkl and Fhkl are the multiplicity and structure factor of the hkl reflection, respectively, Vα is the unit-cell volume of phase α, and θ and θm are the diffraction angles for the hkl reflection and the monochromator (if present), respectively. B is the mean atomic displacement parameter (ADP). Wα and ρα are the weight fraction and density of phase α respectively, while [\mu_m^*] is the mass absorption coefficient of the entire sample.

3.9.3.1. Absorption–diffraction method

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The various terms in equation (3.9.1)[link] are related to the (i) instrument configuration (first set of square brackets), (ii) crystal-structure-related parameters for reflection hkl of phase α (second set of square brackets), and (iii) phase-specific and whole-sample parameters including the weight fraction Wα for phase α (last set of square brackets).

The instrument-related and phase-dependent parameters, including phase density, can be grouped together and defined as a constant Ciα for the ith reflection of phase α for a specific set of measurement conditions. This greatly simplifies the relationship between reflection intensity and weight fraction to[{I_{i\alpha }} = {C_{i\alpha }}{{{W_\alpha }} \over {\mu _m^*}}. \eqno(3.9.2)]On rearrangement, the weight fraction can be derived from[{W_\alpha } = {{{I_{i\alpha }}\mu _m^*} \over {{C_{i\alpha }}}}. \eqno(3.9.3)]Application of equation (3.9.3)[link] (Klug & Alexander, 1974[link]), referred to as the absorption–diffraction method, requires:

  • (1) the determination of Ciα using a rearranged equation (3.9.3)[link] by (i) the preparation of standards with known additions W of phase αα, (ii) measurement of peak intensity Iiα for the standards, and (iii) estimation of the standard sample mass absorption coefficient [\mu_m^*];

  • (2) measurement of Iiα and estimation of [\mu _m^*] for the unknown samples; and

  • (3) calculation of Wα via equation (3.9.3)[link].

The value of [\mu _m^*] can be estimated by direct measurement of the beam intensity through a sample of known thickness t in a beam of the same wavelength as that used in the XRD data collection. Following measurement of the beam intensity with the sample in (I) and removed from (I0) the beam, [\mu _m^*] can be calculated using[{I \over {{I_0}}} = \exp \left({ - \mu _m^*{\rho _m}t} \right). \eqno(3.9.4)]However, this usually involves (i) the preparation of an additional, thinner, sample for presentation to the X-ray beam, (ii) in some cases, the addition of a diluent with a low mass absorption coefficient to produce I/I0 ratios in a range where reasonable accuracy can be achieved, and (iii) knowledge of the `mass thickness' ρmt. It should also be noted that there are few commercially available instruments that would be suitable for such measurements due to safety-related issues in accessing the X-ray beam path.

An alternative approach is to calculate [\mu _m^*] from the sum of the products of the theoretical mass absorption coefficient ([\mu_j^*]) of each element (or phase) and the weight fractions (Wj) of all n elements (or phases) in the sample. The elemental composition may be determined, for example, by X-ray fluorescence (XRF) measurement and its use is more accurate than the use of phase composition as it takes into account any amorphous material not represented by peaks in the diffraction pattern but which still contributes to [\mu _m^*],[\mu _m^* = \textstyle\sum\limits_{j = 1}^n {\mu _j^*} {W_j}. \eqno(3.9.5)]

3.9.3.2. Internal standard method

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A more general, and experimentally simpler, approach is to eliminate [\mu _m^*] from the analysis altogether via the inclusion of an internal standard s in known weight fraction Ws. Substitution of the measured intensity of the jth peak (or group of peaks) of the standard phase, Ijs, into equation (3.9.2)[link] yields[{I_{js}} = {C_{js}}{{{W_s}} \over {\mu _m^*}}. \eqno(3.9.6)]The ratio of equations (3.9.2)[link] and (3.9.6)[link] gives[{{{I_{i\alpha }}} \over {{I_{js}}}} = {{{C_{i\alpha }}} \over {{C_{js}}}}{{{W_\alpha }} \over {{W_s}}}. \eqno(3.9.7)]Since [\mu _m^*] now appears both in the numerator and denominator, its effect on the analysis, and hence the need to measure or calculate it, is removed from the calculation. Rearrangement of equation (3.9.7)[link] yields[{{{I_{i\alpha }}} \over {{I_{js}}}}{{{W_s}} \over {{W_\alpha }}} = {{{C_{i\alpha }}} \over {{C_{js}}}} = C_{js}^{i\alpha },\eqno(3.9.8)]where [C_{js}^{i\alpha }] is a calibration constant specific to the phase and internal standard used. Once [C_{js}^{i\alpha }] has been determined, the weight fraction of the unknown, Wα, can then be determined from[{W_\alpha } = {{{W_s}} \over {C_{\alpha s}^{ij}}} {{{I_{i\alpha }}} \over {{I_{js}}}}. \eqno(3.9.9)]This approach, referred to as the internal standard method, relies on the determination of [C_{\alpha s}^{ij}] using known mixtures of standard and analyte phases. The value of [C_{\alpha s}^{ij}] will be specific to the diffraction peaks used in its determination; if other lines are used in subsequent analysis, then an appropriate value of C will have to be redetermined.

It should be noted that the presence of systematic errors (such as preferred orientation and microabsorption) that influence the measurement of intensity and vary as a function of Wα will not be detected through application of equation (3.9.9)[link]. The use of consistent sample-preparation and presentation techniques is required to minimize the effect of these aberrations on the analysis (Zevin & Kimmel, 1995[link]).

3.9.3.2.1. Selection of an internal standard

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The selection of an appropriate material for use as an internal standard for QPA is not always straightforward. Ideally, the material selected should:

  • (1) Have a simple diffraction pattern resulting in minimal overlap with peaks of interest in the sample.

  • (2) Have a mass absorption coefficient similar to that of the sample to avoid introducing microabsorption effects and thus reducing accuracy.

  • (3) Have minimal sample-related aberrations that may affect observed intensities. For example, it should be fine-grained to ensure minimal grain-size effects on the observed intensities and not be subject to preferred orientation. Importantly, it should have 100% (or known) crystallinity.

  • (4) Be stable over an extended time and be unreactive, especially for in situ studies where it may be subjected to extreme conditions.

Some possibilities for use as internal standard include α-Al2O3 (corundum), TiO2 (rutile), ZnO (zincite), Cr2O3 (eskolaite), α-Fe2O3 (haematite), CeO2 (cerianite), CaF2 (fluorite) and C (diamond). Cline et al. (2011[link]) have described the certification of the standard reference material SRM 676a with accurately known amorphous content for use as an internal standard for QPA (see Chapter 3.1[link] ). Alternatively, it is possible to use an independent measure (e.g. chemical analysis) to derive the concentration of a phase already present in the sample and then to designate it as the internal standard.

Selection of the amount of internal standard to add is often based on folklore or local practices with reported additions ranging from 5 to 50 wt%. Westphal et al. (2009[link]) have described the mathematical basis for selecting the optimal internal standard addition in the context of amorphous phase determination. The amount of internal standard added has a strong influence on the precision of the determination of amorphous content and `a poor choice can make determination impossible, while a clever choice can enhance the precision'.

With the exception of diamond, all of the phases listed above tend to have absorption coefficients that are too high for use with organic materials. The development and verification of a suitable low-absorption-coefficient standard material that meets the criteria given above remains an important area of research.

3.9.3.3. Reference intensity ratio methods

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The reference intensity ratio (RIR) (Hubbard et al., 1976[link]; Hubbard & Snyder, 1988[link]) is an instrument-independent phase constant developed specifically for use in quantitative phase analysis and is defined as the ratio of strongest peak of phase α to the strongest peak of standard s. The RIR can be derived directly from equation (3.9.8)[link]:[{\rm RIR}_{\alpha s} = C_{js}^{i\alpha } = {{{I_{i\alpha }}} \over {{I_{js}}}}{{{W_s}} \over {{W_\alpha }}}. \eqno(3.9.10)]In some cases, the strongest lines of either the standard or phase of interest may not be accessible for measurement if, for example, they strongly overlap with peaks from another phase or if they are out of the 2θ range considered. Equation (3.9.10)[link] can be generalized (Hubbard & Snyder, 1988[link]) to use less intense peaks while keeping the same value of RIR:[{{{I_{i\alpha }}} \over {{I_{js}}}}{{I_{js}^{\rm rel}} \over {I_{i\alpha }^{\rm rel}}}{{{W_s}} \over {{W_\alpha }}} = {\rm RIR}_{\alpha s}, \eqno(3.9.11)]where Irel is the ratio of the intensity of the peak used for analysis to the most intense peak for the phase. RIRα,s is now the generalized reference intensity ratio for phase α with respect to standard s.

Quantification of the unknown phase in the presence of a known standard addition can be achieved by the rearrangement of equation (3.9.11)[link]:[{W_\alpha } = {I_{i\alpha } \over I_{js}}{I_{js}^{\rm rel} \over I_{i\alpha }^{\rm rel}}{W_s \over {\rm RIR}_{\alpha s}}. \eqno(3.9.12)]The generally accepted reference material for QPA via the RIR method is corundum because of its relatively simple diffraction pattern, stability and availability as a highly crystalline and pure single phase. If corundum is used, the RIR equates to I/Ic (or `I over I corundum') for the phase; these are the most commonly reported values in the literature.

RIRs can be determined either by (i) calculation using published crystal-structure information with Rietveld analysis software set to pattern-calculation mode, or (ii) direct measurement by taking the ratio of the strongest peak of the pattern to the intensity of the strongest peak of corundum in a 50/50 weight mixture [or through use of equation (3.9.11)[link] for non-equal proportions]. However, for some phases, there can be ambiguity about which peak is the most intense. For example, the 104 (2.551 Å) and 113 (2.085 Å) peaks of corundum have very similar observed intensities, as do the 111 (3.154 Å) and 022 (1.932 Å) peaks of fluorite. This may lead the analyst to select a peak different from that chosen for reported RIR values.

Collated lists of RIR values for frequently encountered phases can be found in the ICDD database (Fawcett et al., 2017[link]) and Smith et al. (1987[link]). It is important to note, however, that the user must be very careful when selecting an appropriate RIR value for their particular experiment. The values of RIR will depend upon the data-collection and measurement strategy employed (for example, peak height, integrated peak area, whole pattern, X-ray wavelength employed and so on) in their derivation. This must match the conditions used in the experiment to which the value is to be applied. In general, RIR values should be determined for the material currently being studied using the methodologies employed rather than relying on published values. Greater accuracy will be achieved if the relative intensities are determined as part of the calibration process using pure samples of the phase and standard or, preferably, samples in which the phases of interest have high and known concentration. If published values of RIR are used, then the determined phase abundances must be referred to as being only semi-quantitative.

3.9.3.4. Matrix-flushing method

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An important feature of RIR-based techniques is that, once the RIRs are determined for the analyte phases of interest, the standard phase does not need to be present in the sample. The effect of the sample mass absorption coefficient is also removed by taking the ratio of the intensity of phase α to another unknown phase β. Hence the ratio of the weight fractions of the two phases can be derived from[{{{W_\alpha }} \over {{W_\beta }}} = {{{I_{i\alpha }}} \over {{I_{j\beta }}}}{{I_{j\beta }^{\rm rel}} \over {I_{i\alpha }^{\rm rel}}}{{\rm RIR}_{\beta s} \over {\rm RIR}_{\alpha s}}. \eqno(3.9.13)]For a system comprising n phases, equation (3.9.13)[link] allows the derivation of n − 1 weight fraction ratios. Chung (1974a[link],b[link]) has demonstrated that, if all components are crystalline and included in the analysis, an additional constraint of the following form can be included:[\textstyle\sum\limits_{k = 1}^n {{W_k}} = 1.0. \eqno(3.9.14)]This forms a system of n linear equations which can be solved to give the weight fractions of all components in the analysis according to (Chung, 1974a[link],b[link]; Snyder & Bish, 1989[link])[{W_\alpha } = {I_\alpha \over {\rm RIR}_{\alpha s}I_\alpha ^{\rm rel}} \left(\sum\limits_{k = 1}^n {{I_k} \over {\rm RIR}_{ks}I_k^{\rm rel}} \right)^{ - 1}.\eqno(3.9.15)]The weight fractions analysed via this method are correct relative to each other but may not be correct in an absolute sense if unidentified or amorphous materials are present in the sample. In this case, the reported phase abundances will be overestimated. The addition of an internal standard to the system, or knowledge of the amount of a component phase determined by another technique, allows calculation of the absolute amount Wα(abs) of each phase [equation (3.9.16)[link]] and thus the derivation of the amount Wunk of unknown (amorphous and/or unidentified) components [equation (3.9.17)[link]].[{W_{\alpha ({\rm abs})}} = {W_\alpha } \times {{{W_{{\rm std(known)}}}} \over {{W_{\rm std(meas)}}}}, \eqno(3.9.16)][{W_{\rm unk}} = 1.0 - \textstyle\sum\limits_{k = 1}^n {{W_{k({\rm abs})}}}, \eqno(3.9.17)]where Wα(abs) is the absolute weight fraction of phase α, Wstd(known) is the known weight fraction of the standard added to the sample, Wstd(meas) is the weight fraction of the standard reported by equation (3.9.15)[link] and Wunk is the weight fraction of the unidentified and/or amorphous component in the mixture.

If an internal standard has been used, then initial calculation of its concentration via equation (3.9.15)[link] may be:

  • (i) the same as the weighed amount, indicating that there are unlikely to be amorphous or unidentified phases present;

  • (ii) greater than the weighed amount, indicating that amorphous or unidentified phases may be present; or

  • (iii) less than the weighed amount, indicating operator error or the use of invalid RIR or Irel values.

3.9.3.5. Full-pattern fitting methods

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The quantitative XRD techniques described above have traditionally been applied using phase intensity estimates derived from either single peaks or a small group of peaks. This approach can be effective when there is minimal peak overlap but becomes less useful in complex phase systems where it may be difficult to identify freestanding peaks in the pattern. In addition, the presence of sample-related effects such as preferred orientation skew the measured intensities from what would be expected from an ideal powder diffraction pattern, thus reducing the expected accuracy.

Some of these effects can be partially overcome by using full-pattern fitting methods (Smith et al., 1987[link]; Batchelder & Cressey, 1998[link]; Chipera & Bish, 2002[link], 2013[link]; Eberl, 2003[link]; Toraya & Tsusaka, 1995[link]; Cressey & Schofield, 1996[link]), where wide-range diffraction patterns of phases of interest are scaled, summed and compared with the observed diffraction data in a least-squares minimization process. The method relies on the generation of a library of standard patterns for each phase expected in the analysis collected under the same instrumental conditions as those used in subsequent analyses. The selection of standards that are a good match for the phases in the unknown sample is a critically important step. While the library will normally contain patterns of well ordered phases, it can also include patterns for less well ordered material such as glasses, polymers, clay minerals and gels, thus allowing their direct quantification. Where it is not possible to obtain a measured pattern, calculated patterns may also be included in the library.

Weight fractions are obtained by the solution of simultaneous equations that take into account the scale factors of the individual components and the mass absorption coefficients derived from knowledge of the elemental composition of each phase.

Alternatively, the contribution of library patterns to observed data can be normalized by scaling phases to an internal standard, typically corundum, using an RIR approach. Given the compositional and structural variability of some phases, especially in mineralogical applications, RIRs measured using the same minerals as those to be analysed are preferred to reported RIRs. Toraya (Toraya 2016a[link],b[link]) has devised a QPA method which uses observed integrated peaks intensities measured of a wide 2θ range. Phase calibration constants are calculated using only their chemical formula weight and the sum of the square of the number of electrons in the formula unit. While the method is effective for wide-range data, it cannot be applied to single-peak data or data that cover only a limited 2θ range.

The full-pattern fitting method is relatively easy to use and can be applied to difficult samples containing highly disordered materials. For some disordered phases where no crystal structure is available and where peak overlap means that individual peak intensities cannot be measured, full-pattern fitting may be the most appropriate approach to QPA. The major limitations of the method include the need (i) to define and subtract the pattern background, with a subsequent impact on QPA, and (ii) to obtain or generate a library of standard patterns of the phases of interest. The use of an internal standard is recommended and the method is best applied when all standard patterns have first been normalized to an internal standard intensity (Chipera & Bish, 2002[link], 2013[link]).

3.9.3.6. Rietveld-based QPA

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The advent of the Rietveld method (Rietveld, 1969[link]) and its extension into the field of QPA (Bish & Howard, 1988[link]; Hill, 1983[link]; Hill & Howard, 1987[link]; O'Connor & Raven, 1988[link]; Taylor, 1991[link]) has brought some significant benefits when compared with the conventional single-peak and pattern-addition methods. Recent surveys (Madsen et al., 2001[link]; Scarlett et al., 2002[link]) show that the majority of participants, greater than 75%, use a Rietveld-based approach for QPA. The benefits derive from (Hill, 1991[link]; Kaduk, 2000[link]):

  • (i) The use of the entire diffraction pattern. Depending on the 2θ range of the data and the crystallography of the component phases, this may involve hundreds or thousands of reflections rather than the few peaks in conventional methods. This helps to minimize the impact of some systematic sample-related effects such as preferred orientation and extinction.

  • (ii) The ability to accurately deconvolute overlapping peaks to extract the component intensities, thus allowing more complex patterns to be analysed. The development of fundamental-parameters models (Bergmann et al., 1998[link], 2000[link]; Cheary & Coelho, 1992[link]; Cheary et al., 2004[link]), which aim to distinguish instrument from sample contributions to the diffraction pattern, minimizes the number of profile parameters that need to be refined, further enhancing this profile-fitting step.

  • (iii) Refinement of the crystal structure, when supported by the data, to minimize differences between the intensities in the calculated and observed patterns. This brings additional information such as systematic changes in structure parameters from published data.

  • (iv) The ability to model some remaining systematic effects such as preferred orientation or anisotropic crystallite size/strain peak broadening.

The Rietveld method uses a least-squares procedure to minimize the difference between a calculated pattern and the measured data. The calculated pattern is derived from a model containing crystal-structure information for each phase included in the analysis, convoluted with expressions describing peak shape and width, along with functions to correct systematic variances such as preferred orientation. The calculated pattern is multiplied by an overall scaling factor which may be equated to the peak intensities (Iiα) considered by the single-peak methods. The Rietveld scale factor for phase α, Sα, can be defined as (Bish & Howard, 1988[link]; Hill, 1991[link]; Hill & Howard, 1987[link]; O'Connor & Raven, 1988[link])[{S_\alpha } = \left [{{K \over {V_\alpha ^2}}} \right] \left [{{{{W_\alpha }} \over {{\rho _\alpha }}}} \right] {1 \over {2\mu _m^*}}, \eqno(3.9.18)]where K is an 'experiment constant' used to put Wα on an absolute basis, and Vα, Wα and ρα are the volume of the unit cell, the weight fraction and the density for phase α, respectively.

Since equation (3.9.18)[link] inherently contains the weight-fraction information, it can be rearranged to derive Wα:[{W_\alpha } = {{{S_\alpha }{\rho _\alpha }V_\alpha ^2\mu _m^*} \over K}. \eqno(3.9.19)]O'Connor & Raven (1988[link]) and Bish & Howard (1988[link]) have shown that K is dependent only on the instrumental conditions and is independent of individual phase and overall sample-related parameters. Therefore a single measurement is sufficient to determine K for a given instrument configuration and set of data-collection conditions. Determination of K may be carried out by (i) a measurement of either a pure phase, or a phase of known proportion in a mixture, separately from the measurement of the actual unknown mixture, or (ii) using a phase that is present in the sample in a known amount. The value of K calculated in this way will be appropriate for the calibration of subsequent measurements as long as all instrumental and data-collection conditions remain the same as those used in its determination.

For each phase, the density ρα can be calculated from the published (or refined) crystal-structure information using[{\rho _\alpha } = {{Z{M_\alpha }} \over {{V_\alpha }}}, \eqno(3.9.20)]where ZM is the mass of the unit-cell contents (Z is the number of formula units in the unit cell and M is the molecular mass of the formula unit) and V is the unit-cell volume.2

Substitution of equation (3.9.20)[link] in equation (3.9.19)[link] shows that[{W_\alpha } = {{{S_\alpha }{{\left({ZMV} \right)}_\alpha }\mu _m^*} \over K} .\eqno(3.9.21)]In this context, (ZMV)α is the `phase constant' and can be calculated from published or refined crystal-structure information alone. It is worth noting that, if the crystal structure is refined as part of the analysis, ZMV is updated and hence becomes a dynamic phase constant.

The methodology embodied in equation (3.9.21)[link] is important in many applications in that it produces, within the limits of experimental error, absolute phase abundances and is referred to hereafter as the external standard approach. While the use of a phase that already exists within the sample to determine K may be considered as an internal standard approach, in some applications, including in situ studies, that phase may be removed from the system through, for example, decomposition or dissolution. However, the value of K remains valid for subsequent determination of phase abundances provided that the instrumental and data-collection conditions do not change.

Equation (3.9.21)[link] is directly applicable to the analysis of those phases for which detailed crystal-structure information is available. For phases where only a partial structure (for example, an indexed unit cell but no atom coordinates or site-occupation factors) is available, an empirical ZMV can be derived using mixtures of the phase of interest with known amounts of a well characterized standard (Scarlett & Madsen, 2006[link]). QPA of phases with partial structure is also possible through the use of equation (3.9.19)[link], but an estimate of the phase density, obtained through direct measurement, is required.

The limitations of the approach embodied in equations (3.9.19)[link] and (3.9.21)[link] derive from the need to (i) conduct a measurement of K and (ii) estimate the value of the mass absorption coefficient [\mu_m^*] for the sample(s) used to determine K, as well as for each sample of interest. However, similar to the earlier discussion about the single-peak methods, [\mu_m^*] can be determined by direct measurement or calculation using equations (3.9.4)[link] or (3.9.5)[link], respectively. The benefits that can be derived from the extraction of the absolute, rather than relative, phase abundances, make it worth pursuing in many analytical situations. For example, in time-resolved studies where phases transform and material is lost in the course of reaction, the calculation of relative abundances summed to 100% may give a misleading impression of increased amounts of the remaining phases.

In some diffraction instruments, there may be decay in the incident-beam intensity during the course of measurement. This may happen on the timescale of months for a laboratory-based instrument owing to X-ray tube ageing, or on the scale of minutes at a synchrotron instrument where the storage-ring current is only refreshed once or twice per day. In this case, the change in incident intensity can be taken into account by incorporating an additional term into equation (3.9.21)[link]:[{W_{\alpha i}} = {{{S_{\alpha i}}{{\left({ZMV} \right)}_\alpha }\mu _m^*} \over K} {{{I_0}} \over {{I_i}}} ,\eqno(3.9.22)]where I0 and Ii are the incident beam intensities present during the determination of K and the collection of data set i, respectively.

The need to measure K, and measure or calculate [\mu_m^*], serves to increase the overall experimental difficulty and can be eliminated in ways analogous to those used in the single-peak methodology described earlier. For a simple two-phase mixture where both phases, α and β, are 100% crystalline, the sum of their weight fractions Wα and Wβ equals unity and can be expressed as (Bish & Howard, 1988[link])[{W_\alpha } = {{{W_\alpha }} \over {{{W_\alpha } + {W_\beta }} }} .\eqno(3.9.23)]Substitution of equation (3.9.21)[link] for phases α and β in equation (3.9.23)[link] results in[{W_\alpha } = {{{S_\alpha }{{\left({ZMV} \right)}_\alpha }} \over {{S_\alpha }{{\left({ZMV} \right)}_\alpha } + {S_\beta }{{\left({ZMV} \right)}_\beta }}} .\eqno(3.9.24)]Alternatively, in a multiphase sample, the addition of an internal standard s in known amount Ws and taking the ratio of equation (3.9.21)[link] for analyte and standard phases provides the relationship[{W_\alpha } = {W_s} {{{S_\alpha }{{\left({ZMV} \right)}_\alpha }} \over {{S_s}{{\left({ZMV} \right)}_s}}}. \eqno(3.9.25)]The method embodied in equation (3.9.25)[link] is analogous to the internal standard approach in equation (3.9.9)[link] and also serves to produce absolute phase abundances Wα(abs). Once again, the benefit accruing from the use of absolute phase abundances is the ability to estimate the presence and amount of any amorphous and/or unidentified phases W(unk) through application of equations (3.9.16)[link] and (3.9.17)[link].

Hill & Howard (1987[link]) and Bish & Howard (1988[link]) have adapted the matrix-flushing method of Chung (1974a[link],b[link]) to the Rietveld analysis context. By constraining the sum of the analysed weight fractions to the assumed concentration of the crystalline components (usually unity), the weight fraction of phase α in an n-phase mixture is given by the relationship3[{W_\alpha } = {{{S_\alpha }{{(ZMV)}_\alpha }} \over {\textstyle\sum_{k = 1}^n {{S_k}{{(ZMV)}_k}} }}. \eqno(3.9.26)]The use of equation (3.9.26)[link] in QPA again eliminates the need to measure the instrument calibration constant and estimate the sample mass absorption coefficient. However, the necessity of normalizing the sum of the analysed weight fractions to unity only produces the correct relative phase abundances. This approach is the most widely used in Rietveld-based QPA and is almost universally coded into Rietveld analysis programs. If the sample contains amorphous phases and/or minor amounts of unidentified crystalline phases, the analysed weight fractions will be overestimated. Where absolute phase abundances are required in, for example, the derivation of reaction mechanisms in in situ studies, then one of the methods that produces absolute phase abundances must be used.

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