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. 353356
Section 3.9.5. Alternative methods for determination of calibration constants^{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 
In order to determine the phase calibration constant C, it is common to obtain (i) a pure sample of the phase of interest that accurately reflects the form of the phase in the samples to be analysed, or (ii) a multiphase sample in which the phase concentration is known by other means (for example, chemical analysis or point counting). In some systems, there may be insufficient sample available to risk `contaminating' it with an internal standard, especially if the material needs to be analysed using other techniques. The addition of an internal standard may also introduce microabsorption problems or increase the complexity of patterns that are already highly overlapped. For other situations, the time frame demanded for the analysis may prohibit the timeconsuming procedures of standard addition, data collection and separate determination of the phase calibration constant.
Zevin & Kimmel (1995) have described an approach to the derivation of phase constants which relies on having a suite of samples to be analysed that (i) have the same phases present in all samples and (ii) exhibit a wide range of composition of these phases in various samples in order to stabilize the analysis. If we reconsider the relationship between the weight fraction W_{α} and the observed intensity [equation (3.9.3)],and assume that all phases in the system are known and included in the analysis, we can introduce the additional constraint that the sum of all W_{α}'s is unity (or at least a known value):In a system of n samples containing m phases, we can explicitly write the relationships expressed in equations (3.9.34) and (3.9.35) as a set of simultaneous equations:where is the mass absorption coefficient for the nth sample.
Knudsen (1981) has described a modification to this approach by including an internal standard in each of the samples to be analysed and using the ratio of intensities of the analyte and internal standard phases in place of the I_{nm} in equation (3.9.36). While this eliminates the need to determine and use the mass absorption coefficient, the tedious procedure of adding and mixing an internal standard is required for each sample and for reasons described above may not be appropriate.
The relationships embodied in equations (3.9.36) can be expressed more simply in matrix notation aswhere is a column vector (dimensions 1 × n) containing the known (or assumed) sum of weight fractions for each sample (unity in this case), is a column vector (dimensions 1 × m) containing the calibration constants for each phase and is a rectangular matrix (dimensions n rows × m columns) containing the measured peak intensities (or scale factors) for each phase multiplied by the sample mass absorption coefficient.
A leastsquares solution of equation (3.9.37) to derive the value for C for each phase can be calculated using matrixmanipulation methods (Knudsen, 1981):where the superscripts T and −1 represent the transpose and inverse matrix functions, respectively.
Accuracy in the calculation of the individual values of C is improved by having (i) phases of the same or similar composition in all samples and (ii) a wide range of concentrations of each phase across the sample suite. These conditions may be met in, for example, mineral exploration samples where a limited number of phases are present in a drill core but their abundance varies as a function of depth. In mineral processing or industrially based material manufacture, the goal is usually to control the system to minimize compositional variation in the product. The side effect of this is that the values of intensity in matrix I have too little variation, resulting in large errors in the derived values for C. In the limiting case, the system may become indeterminate with no unique solution available. To overcome this, forced or accidental changes to processing conditions may introduce sufficient compositional variation to stabilize the determination of the C values through equation (3.9.38). Alternatively, physical or chemical separation of selected components may be sufficient to provide the required compositional variation. Knudsen (1981) provides a detailed statistical analysis used in the determination of the errors in the phase constants.
While Zevin (Zevin & Kimmel, 1995) and Knudsen (1981) have demonstrated the application of this approach for singlepeak methods, it is equally applicable if scale factors derived from wholepattern fitting or Rietveldbased methods are used.
The sample 1 suite from the IUCr CPD round robin on QPA again provides an ideal platform for demonstrating the applicability of this method due to the wide variation of concentration of the constituent phases. A measure of intensity was derived using an hkl_phase (see Section 3.9.6) in which the peak positions were constrained to the space group and unitcell parameters but the individual peak intensities were refined to empirical values using a pure subsample of each of the three phases. For the analysis of the samples, the relative peak intensities were fixed and an overall scale factor S for each phase in each sample (eight samples, three replicates, three phases), multiplied by the mass absorption coefficient calculated from the XRFdetermined composition, was used as the measure of intensity. These values then formed the intensity matrix I in equations (3.9.37) and (3.9.38) while all values in the vector L were assumed to be 1.0 (i.e. all samples were assumed to be fully crystalline). Microsoft Excel provides a useful platform for these calculations since it contains all of the matrixmanipulation functions required by equation (3.9.38). The determined values for C for the three phases are given in Table 3.9.3. The values in the C/C_{corundum} column should be compared with the values derived in Section 3.9.4.3 above.

Application of these C values to the analysis of all samples via equation (3.9.34) yields the results given in Fig. 3.9.9. The results, displayed as bias from the known values, show that at all concentration ranges the analyses are within about ±1% of the weighed values. The important point to note here is that there has been no prior calibration conducted to obtain this result; the system is selfcalibrating and has only relied on having a wide range of concentrations of the three phases in the sample suite. The only prior knowledge used in the analysis is (i) a measure of peak intensity embodied in the empirical phase scale factor and (ii) an estimate of for each sample calculated from the elemental composition.
Earlier discussion has noted that the experiment constant K used in equation (3.9.21) can be determined using (i) a standard pure phase or mixture measured separately from the measurement of the actual unknown mixture being analysed, or (ii) using a phase that is present in the sample in a known amount. However, in some cases, these approaches are not always effective in producing reliable values of K because the methodology assumes that the mass of sample contributing to the diffraction process is constant. While this condition is true for infinitely thick samples in Bragg–Brentano geometry, it is unlikely to be true for capillary or flatplate samples in transmission geometry. In these cases, the sample thickness and packing density will have a significant influence on the amount of sample contributing to the diffraction process and hence on the observed intensity and the derived values of K. Therefore, a K value determined from one capillary sample is unlikely to be applicable to another capillary even though all other instrumental conditions remain the same. However, for in situ studies, a K value determined at the start of an experiment should remain valid as the analysis proceeds.
K can also be determined using the whole sample, rather than an individual phase. Since the determined value of K then applies equally to all phases in the sample, equation (3.9.21) can be summed over all analysed components thus:If the crystallinity of the sample is known (or can safely be assumed), then individual phase abundances are not required and K can be calculated fromwhere is the assumed crystallinity of the entire sample.
For a sample that is 100% crystalline and all components included in the analysis, then the denominator is unity and K is simply the sum of the product of the scale factors and their respective ZMV's multiplied by the mass absorption coefficient of the entire sample.
For in situ studies where a reaction or process is examined dynamically, sealed capillary sample geometry is frequently used. In this environment, the chemical composition of the capillary contents will not change during the course of the reaction even though individual phases may be undergoing transformation. Equation (3.9.40) can be further simplified since the overall sample mass absorption coefficient remains constant throughout the reaction and can therefore be deleted and its effect incorporated into K.
This wholesample approach to the determination of K is also useful in systems where there are residual errors that may not be evident when equation (3.9.21) is used with the concentration of a single phase. By way of demonstration, the sample 1 suite from the IUCr CPD round robin on QPA has been used to calculate K in two distinct ways:
Fig. 3.9.10 shows the 72 individual determinations of K from the phasespecific method as a function of known phase concentration. At high concentrations, the values for K derived from each of the three phases are similar indicating that, for effectively pure phase samples, the approach embodied in equation (3.9.21) is valid. However, if K is determined using the known concentration of a single phase at a lower concentration in a multiphase sample, then residual errors in the measurement of pattern intensity serve to reduce its accuracy. At lower concentrations of corundum, there is a systematic increase in the determined value of K resulting from a small microabsorption effect present in these samples. Since corundum has the lowest mass absorption coefficient of the three phases in this system its intensity, and hence Rietveld scale factor, is slightly overestimated relative to the fluorite and zincite. This results in an overestimation of the value of K relative to an ideal sample; the magnitude of this difference is about 5% relative. Use of these values for subsequent analysis will result in an underestimation of phase concentrations using equation (3.9.21). The converse is true if fluorite or zincite is used to determine K.
However, if the wholesample approach embodied in equation (3.9.40) is used for the determination of K, these residual samplerelated aberrations can be eliminated; the results of the determination of 24 values of K using this approach are also included in Fig. 3.9.10. The mean of all 24 determinations is 427.6 (3.7) representing a relative error of <0.8%. The important point to note here is that knowledge of the individual phase concentrations is not needed; the only assumption needed relates to the total crystallinity of each sample.
For in situ studies, using equation (3.9.40) to calculate K at each step i in the reaction (defined as K_{i}) can be useful in deriving details of the reaction mechanism. If K_{i} increases as the reaction progresses, this may be indicative of increasing crystallinity in the sample. Reductions in K_{i} during the reaction may point to the formation of intermediate amorphous material or unidentified crystalline components, the total concentration of which can be readily calculated usingApplication of this can be demonstrated using sample 3 from the QPA round robin (Scarlett et al., 2002), as it contains the same three crystalline phases as the sample 1 suite with the addition of 29.47 wt% amorphous silica flour. Calculation of K_{i} for sample 3, based only on the three crystalline phases, results in a value of 301.8. Substituting this into equation (3.9.41) along with the previously determined value of K (427.6) gives a measured amorphous content of 29.42 wt% – this is in good agreement with the known weighed amount. The important point to note here is that the data for sample 3 were collected at the same time, and under the same instrumental conditions, as for sample 1, which ensured that the true value of K was the same for all data.
References
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