Mineral Quantification with Simultaneous Refinement of Ca-Mg Carbonates Non-Stoichiometry by X-ray Diffraction, Rietveld Method

Quantitative phase analyses of carbonate rocks containing Mg-rich calcite and non-stoichiometric dolomite by the Rietveld method yielded improved results when the substitutions are refined for either minerals. The refinement is constrained by the c-axis of the lattice for both minerals using the formula c = −1.8603 nMg + 17.061 for calcite, where nMg is the molar fraction of Mg replacing Ca, and c = 16.0032 + 0.8632ΔnCa for dolomite, with ΔnCa being the excess Ca in its B site. The one-step procedure was implemented into the Topas software and tested on twenty-two carbonate rock samples from diverse geological settings, considered analogues to petroleum system lithotypes of the pre-evaporite deposits of Southeastern Brazil. The case study spans over a wide range of calcite and dolomite compositions: up to 0.287 apfu Mg in magnesian calcite, and Ca in excess of up to 0.25 apfu in non-stoichiometric dolomite, which are maximum substitutions the formulas support. The method overcomes the limitations for the quantification of minerals by stoichiometry based on whole-rock chemical analysis for complex mineralogy and can be employed for multiple generations of either carbonate. It returns the mineral quantification with unprecedented detailing of the carbonates’ composition, which compares very well to spot analysis (both SEM-EDS and EMPA) if those cover the full range of compositions. The conciliation of the quantification results based on the XRD is also excellent against chemical analysis, thermogravimetry, and carbon elemental analysis.


Introduction
Calcite and dolomite, the most common carbonate minerals, are ubiquitous in a great variety of sedimentary, metamorphic, and igneous rocks. Along the calcite-magnesite compositional series, dolomite, as the intermediary mineral equivalent to the reaction CaCO 3 + MgCO 3 = CaMg(CO 3 ) 2 boasts a cell volume decrease [1] of up to 12%, as compared to calcite [2]. Besides end-members and the intermediary phase, the continuous exchange of cations produces magnesian calcite (Ca 1−x Mg x CO 3 ) and non-stoichiometric dolomite as the most important minerals.
Research on the determination of seawater composition over geological time depends on the correct quantification of the Mg/Ca ratio in calcite. The percentage of Mg incorporated by marine

Results
The experimental data of Goldsmith et al. [23] and Zhang et al. [38] related the Mg for Ca substitution in magnesian calcite to the lattice parameters a and c. Both relations are linear up to 0.287 Mg apfu, but the linearity does not extend to higher substitutions ( Figure 1). The c axis correlates better, through the Equation: c = −1.8603 nMg + 17.061 (1) where c is the length of the crystallographic axis c (in Å), and nMg is the molar fraction or number of atoms per formula unit(apfu) of Mg that replace Ca in calcite. Restraining the application of this equation for the nMg range from 0.000 to 0.287 corresponds to c ranging from 17.061 to 16.528 Å, respectively (Table 1).  (1) where c is the length of the crystallographic axis c (in Å), and nMg is the molar fraction or number of atoms per formula unit(apfu) of Mg that replace Ca in calcite. Restraining the application of this equation for the nMg range from 0.000 to 0.287 corresponds to c ranging from 17.061 to 16.528 Å, respectively (Table 1).  McCarthy et al. [39] present a similar approach for non-stoichiometric dolomite. The relationship between both of the crystallographic axes a and c of dolomite and the molar fraction of excess Ca, allocated on the B site of the mineral, is presented in Figure 2. Only Mg occupies the B site when the dolomite is stoichiometric. As for calcite, the c axis achieves a better correlation (factor of 0.9983), and relates to the excess Ca in the B site ∆n Ca by Equation (2): where c is the length of the crystallographic axis c and c o is the length of the crystallographic axis when the dolomite is stoichiometric (∆n Ca = 0.00). The lattice parameters a and c of McCarty et al. [39] (a = 4.8071 Å, c = 16.0032 Å), coincident with those calculated by Reeder and Sheppard [40], allow for Equation (2) to be rewritten as: c = 16.0032 + 0.8632∆n Ca McCarthy et al. [39] discovered that, when excess Ca (∆n Ca ) in the dolomite exceed 0.07 (apfu), a second dolomite phase could be detected, and two well-delimited groups of dolomite might be observed ( Figure 2). The authors defined the first, low excess-Ca dolomite, to occur in the range of the number of total Ca atoms from 1.00 to 1.10, high excess-Ca dolomite from 1.10 to 1.15 apfu, although their data supports the latter up to 1.25 apfu. With these well-established relationships, two different structures of non-stoichiometric dolomite were created, with the limit lattice parameters for the two phases as in Table 2.  McCarthy et al. [39] present a similar approach for non-stoichiometric dolomite. The relationship between both of the crystallographic axes a and c of dolomite and the molar fraction of excess Ca, allocated on the B site of the mineral, is presented in Figure 2. Only Mg occupies the B site when the dolomite is stoichiometric. As for calcite, the c axis achieves a better correlation (factor of 0.9983), and relates to the excess Ca in the B site ΔnCa by Equation (2): where c is the length of the crystallographic axis c and co is the length of the crystallographic axis when the dolomite is stoichiometric (ΔnCa = 0.00). The lattice parameters a and c of McCarty et al. [39] (a = 4.8071 Å, c = 16.0032 Å), coincident with those calculated by Reeder and Sheppard [40], allow for Equation (2) to be rewritten as: McCarthy et al. [39] discovered that, when excess Ca (ΔnCa) in the dolomite exceed 0.07 (apfu), a second dolomite phase could be detected, and two well-delimited groups of dolomite might be observed ( Figure 2). The authors defined the first, low excess-Ca dolomite, to occur in the range of the number of total Ca atoms from 1.00 to 1.10, high excess-Ca dolomite from 1.10 to 1.15 apfu, although their data supports the latter up to 1.25 apfu. With these well-established relationships, two different structures of non-stoichiometric dolomite were created, with the limit lattice parameters for the two phases as in Table 2.   The Equations (1) and (3) were implemented in the TOPAS software as a restriction to the XRD refinement of the cationic site occupancy by the Rietveld method, respectively, for the Mg-for-Ca isomorphic substitution in magnesian calcite and excess Ca in non-stoichiometric dolomite.
The results of the Rietveld method for all of the samples with the mineral quantification (% by mass), MgCO 3 molar fraction in magnesian calcite, molar fraction of excess CaCO 3 in non-stoichiometric dolomite (dolomite I and dolomite II), as well as the lattice parameters a and c, the position of the reflection d(104), and the cell volume, are presented in Table 3. The quality of the refinement is exemplified in Figure 3, choosing samples with large amounts of magnesian calcite and non-stoichiometric dolomite.  The Equations (1) and (3) were implemented in the TOPAS software as a restriction to the XRD refinement of the cationic site occupancy by the Rietveld method, respectively, for the Mg-for-Ca isomorphic substitution in magnesian calcite and excess Ca in non-stoichiometric dolomite.
The results of the Rietveld method for all of the samples with the mineral quantification (% by mass), MgCO3 molar fraction in magnesian calcite, molar fraction of excess CaCO3 in nonstoichiometric dolomite (dolomite I and dolomite II), as well as the lattice parameters a and c, the position of the reflection d(104), and the cell volume, are presented in Table 3. The quality of the refinement is exemplified in Figure 3, choosing samples with large amounts of magnesian calcite and non-stoichiometric dolomite.     The conciliation of the chemical composition calculated from the XRD results against X-ray fluorescence analyses (XRF) is graphically shown in Figure 4A, while Figure 4B emphasizes Ca and Mg, the elements calculated directly from the structure of the carbonates. It is assumed that some deviation in the conciliation is due to several other minerals besides the carbonates, while the calculated Mg values, whose bearers are exclusively the fitted carbonates, match the chemical analyses very well. The conciliation of the chemical composition calculated from the XRD results against X-ray fluorescence analyses (XRF) is graphically shown in Figure 4A, while Figure 4B emphasizes Ca and Mg, the elements calculated directly from the structure of the carbonates. It is assumed that some deviation in the conciliation is due to several other minerals besides the carbonates, while the calculated Mg values, whose bearers are exclusively the fitted carbonates, match the chemical analyses very well. The results of the Rietveld method refinement were further checked against C measured by the elemental analysis and CO2 determined by thermogravimetry under CO2 atmosphere, which separates the thermal events of mass loss of the carbonate from the magnesite molecule due to dolomite from calcite [41], in Figure 5.  Table S4.
Other analytical techniques were also used to compare the results and validate the proposed method. Analyzes by EDS and WDS demonstrate that the carbonates' compositional range for each sample might be very large.  The results of the Rietveld method refinement were further checked against C measured by the elemental analysis and CO 2 determined by thermogravimetry under CO 2 atmosphere, which separates the thermal events of mass loss of the carbonate from the magnesite molecule due to dolomite from calcite [41], in Figure 5. The conciliation of the chemical composition calculated from the XRD results against X-ray fluorescence analyses (XRF) is graphically shown in Figure 4A, while Figure 4B emphasizes Ca and Mg, the elements calculated directly from the structure of the carbonates. It is assumed that some deviation in the conciliation is due to several other minerals besides the carbonates, while the calculated Mg values, whose bearers are exclusively the fitted carbonates, match the chemical analyses very well. The results of the Rietveld method refinement were further checked against C measured by the elemental analysis and CO2 determined by thermogravimetry under CO2 atmosphere, which separates the thermal events of mass loss of the carbonate from the magnesite molecule due to dolomite from calcite [41], in Figure 5.  Table S4.
Other analytical techniques were also used to compare the results and validate the proposed method. Analyzes by EDS and WDS demonstrate that the carbonates' compositional range for each sample might be very large.   Table S4.
Other analytical techniques were also used to compare the results and validate the proposed method. Analyzes by EDS and WDS demonstrate that the carbonates' compositional range for each sample might be very large.   Table S5.
The general agreement of the refined results with the point analysis is excellent. XRD is a bulk method, and therefore is able to accommodate for the compositional variations while refining. It will retrieve the mean composition for the carbonates for every compositional domain. Point analysis, on the other hand, is spatially limited to a few micrometers of the volume excited by the electron beam, and a larger number of analyses are required to encompass the whole range. Almost all compositions refined from the XRD plot in the range, as determined by EDS. A few exceptions occur, however. Examples are given in Figure 6B,F, where non-stoichiometric dolomite II, with 0.25 apfu excess of Ca, was refined by the Rietveld method for samples, respectively, LV07 and Am2a, but was not detected by EDS; this is due to the low grade of the mineral in the sample, only 2.3% and 3.1%, respectively (Table  3), but within the detection limit of XRD. The same happens to sample PA13 ( Figure 6A), where dolomite I could not be detected by EDS. Careful reanalysis of the sample (Figure 7) showed the fine intergrowth with dominant (58.4%) quartz (Table 3). Intergrowth and fine particle size is not limiting XRD-based analysis.  Table S5.
The general agreement of the refined results with the point analysis is excellent. XRD is a bulk method, and therefore is able to accommodate for the compositional variations while refining. It will retrieve the mean composition for the carbonates for every compositional domain. Point analysis, on the other hand, is spatially limited to a few micrometers of the volume excited by the electron beam, and a larger number of analyses are required to encompass the whole range. Almost all compositions refined from the XRD plot in the range, as determined by EDS. A few exceptions occur, however. Examples are given in Figure 6B,F, where non-stoichiometric dolomite II, with 0.25 apfu excess of Ca, was refined by the Rietveld method for samples, respectively, LV07 and Am2a, but was not detected by EDS; this is due to the low grade of the mineral in the sample, only 2.3% and 3.1%, respectively (Table 3), but within the detection limit of XRD. The same happens to sample PA13 ( Figure 6A), where dolomite I could not be detected by EDS. Careful reanalysis of the sample (Figure 7) showed the fine intergrowth with dominant (58.4%) quartz (Table 3). Intergrowth and fine particle size is not limiting XRD-based analysis.
Some samples (EC01, PS01, 04, 10, Lithot, C12, C14, AC0370, AC3623, Am2a, PA13, and PA2B) were selected for point chemical analysis by electron microprobe analysis (EMPA). Figure 8 shows the composition of the samples, again comparing it with the refinement result by the Rietveld method. As for the EDS analysis, no dolomite II could be found in sample Am2a ( Figure 8C), due to the low content of the phase. The significantly smaller number of analyzed points (163), to some extent, reduces the range of sampled compositions, but mostly the Rietveld refinement results also plot among the chemical analyses for each sample. Some samples (EC01, PS01, 04, 10, Lithot, C12, C14, AC0370, AC3623, Am2a, PA13, and PA2B) were selected for point chemical analysis by electron microprobe analysis (EMPA). Figure 8 shows the composition of the samples, again comparing it with the refinement result by the Rietveld method. As for the EDS analysis, no dolomite II could be found in sample Am2a ( Figure 8C), due to the low content of the phase. The significantly smaller number of analyzed points (163), to some extent, reduces the range of sampled compositions, but mostly the Rietveld refinement results also plot among the chemical analyses for each sample.   Some samples (EC01, PS01, 04, 10, Lithot, C12, C14, AC0370, AC3623, Am2a, PA13, and PA2B) were selected for point chemical analysis by electron microprobe analysis (EMPA). Figure 8 shows the composition of the samples, again comparing it with the refinement result by the Rietveld method. As for the EDS analysis, no dolomite II could be found in sample Am2a ( Figure 8C), due to the low content of the phase. The significantly smaller number of analyzed points (163), to some extent, reduces the range of sampled compositions, but mostly the Rietveld refinement results also plot among the chemical analyses for each sample.   Table S6.
It should be stressed that the perfect alignment of the analyses on the theoretical lines in Figures 6 and 8 is due to the calculation X cal Ca + X cal Mg = 1 for calcite, where the total atoms for the A site is equal to 1, and X dol Ca + X dol Mg = 2 for 2 atoms at sites A + B in dolomite.

Discussion
Titschack et al. [32] also used a fundamental parameters approach to the Rietveld method for magnesian calcite quantification, starting refinement with the structure defined by Paquette & Reeder [42], with molar fraction of 0.129 MgCO 3 . After retrieving the refined lattice parameters and plotting them against the chemical analyses of his samples, they used a calibration curve based on cell volume to calculate the Mg and Ca site occupancy, adjusting these values in the software for phase quantification. The results obtained by Titschack et al. [32] are compared to those of the present work in Figure 9. Both of these studies display very close results, with similar correlation coefficients. Rather than dynamically constraining the Mg-for-Ca substitution, Titschack et al. [32] applied a two-step refining routine, i.e., refining lattice parameters, using them to determine substitution, and then adjusting the occupancy in the structure for further operations. The authors also discuss the possibility of multiple calcite compositions in the samples, and use their method to quantify the (simulated) binary mixtures. It was a very fortuitous case study, as the echinoid remains (coronas and spines) contained exclusively magnesian calcite of uniform composition, allowing them to use the whole-sample ICP-AES chemical analysis after full dissolution to correlate them to the lattice parameters. These are high-quality data on magnesian calcite. It should be stressed that the perfect alignment of the analyses on the theoretical lines in Figures  6 and 8 is due to the calculation + = 1 for calcite, where the total atoms for the A site is equal to 1, and ( + = 2) for 2 atoms at sites A + B in dolomite.

Discussion
Titschack et al. [32] also used a fundamental parameters approach to the Rietveld method for magnesian calcite quantification, starting refinement with the structure defined by Paquette & Reeder [42], with molar fraction of 0.129 MgCO3. After retrieving the refined lattice parameters and plotting them against the chemical analyses of his samples, they used a calibration curve based on cell volume to calculate the Mg and Ca site occupancy, adjusting these values in the software for phase quantification. The results obtained by Titschack et al. [32] are compared to those of the present work in Figure 9. Both of these studies display very close results, with similar correlation coefficients. Rather than dynamically constraining the Mg-for-Ca substitution, Titschack et al. [32] applied a twostep refining routine, i.e., refining lattice parameters, using them to determine substitution, and then adjusting the occupancy in the structure for further operations. The authors also discuss the possibility of multiple calcite compositions in the samples, and use their method to quantify the (simulated) binary mixtures. It was a very fortuitous case study, as the echinoid remains (coronas and spines) contained exclusively magnesian calcite of uniform composition, allowing them to use the whole-sample ICP-AES chemical analysis after full dissolution to correlate them to the lattice parameters. These are high-quality data on magnesian calcite. Comparison of the results of our study with different calibrations [22][23][24]26,27,30,32,42] for determination of the molar fraction of MgCO3 in calcite shows that the new calibration is in agreement with the trend of the former publications (Figure 10), except for the c/a relation, as already observed by Titschack et al. [32]. Some authors validate this linear relationship only up to 0.18 apfu of Mg [20,27], stating that the relation is not linear any more above this content. Our data, based mostly on data of Goldsmith et al. [23] and Zhang et al. [38], extends the linear relation continuously up to 0.287 apfu of Mg. Comparison of the results of our study with different calibrations [22][23][24]26,27,30,32,42] for determination of the molar fraction of MgCO 3 in calcite shows that the new calibration is in agreement with the trend of the former publications (Figure 10), except for the c/a relation, as already observed by Titschack et al. [32]. Some authors validate this linear relationship only up to 0.18 apfu of Mg [20,27], stating that the relation is not linear any more above this content. Our data, based mostly on data of Goldsmith et al. [23] and Zhang et al. [38], extends the linear relation continuously up to 0.287 apfu of Mg.
The method for quantifying the substitution of Mg for Ca at the A site of magnesian calcite (up to 0.287 apfu) and the excess of Ca, located at the B site in non-stoichiometric dolomite (up to 0.25 apfu) was applied and validated in different rocks samples from diverse geological environments with varied mineralogy.
In addition to having obtained a high degree of data correlation, the implemented method is fast, efficient, and refines more than one generation for calcite and dolomite, considering all of the reflections of each mineral phase. The new method convolutes the entire X-ray diffraction pattern, making quantification possible when the mineralogy is complex, even with peaks overlapping. In addition to having obtained a high degree of data correlation, the implemented method is fast, efficient, and refines more than one generation for calcite and dolomite, considering all of the reflections of each mineral phase. The new method convolutes the entire X-ray diffraction pattern, making quantification possible when the mineralogy is complex, even with peaks overlapping.
As always when using the Rietveld method, a comprehensive knowledge of the mineralogy of the sample prior to quantification is a requirement. Another paramount necessity is using the structure files that really describe the refined minerals. For mixed crystals, as in samples studied here, choosing adequate structures was not an easy task, and was used to require a multistep approach as the one adopted by Titschack et al. [32] for magnesian calcite. If solid solutions are a part of samples with complex mineralogy, the initial cell parameters refinement without constraints on occupancy can easily retrieve approximate values, leading to wrong occupancy. Refining occupation while constraining it to a lattice parameter allows a one-step procedure for the fundamental parameters of the Rietveld method refinement. Refining the occupancy provides a much better structure for the method to find real minima, and this gives improved results for the overall mineral quantification, as the effect of different scattering powers of Mg and Ca on intensities, i.e., scale factors, is taken into consideration. Titschack et al. [32] had already drawn attention to the possible bias of not considering the effect of scattering powers, which can be circumvented by our method. Exactly the same approach discussed for calcite has also been adopted for dolomite in this study.
By applying self-contained structures, which refine to its correct parameters, it is also possible to use it for more than one compositional variety in a sample, by setting precise lattice parameter limits. Here, we defined only one magnesian calcite variety with MgCO3 allowed to refine between As always when using the Rietveld method, a comprehensive knowledge of the mineralogy of the sample prior to quantification is a requirement. Another paramount necessity is using the structure files that really describe the refined minerals. For mixed crystals, as in samples studied here, choosing adequate structures was not an easy task, and was used to require a multistep approach as the one adopted by Titschack et al. [32] for magnesian calcite. If solid solutions are a part of samples with complex mineralogy, the initial cell parameters refinement without constraints on occupancy can easily retrieve approximate values, leading to wrong occupancy. Refining occupation while constraining it to a lattice parameter allows a one-step procedure for the fundamental parameters of the Rietveld method refinement. Refining the occupancy provides a much better structure for the method to find real minima, and this gives improved results for the overall mineral quantification, as the effect of different scattering powers of Mg and Ca on intensities, i.e., scale factors, is taken into consideration. Titschack et al. [32] had already drawn attention to the possible bias of not considering the effect of scattering powers, which can be circumvented by our method. Exactly the same approach discussed for calcite has also been adopted for dolomite in this study.
By applying self-contained structures, which refine to its correct parameters, it is also possible to use it for more than one compositional variety in a sample, by setting precise lattice parameter limits. Here, we defined only one magnesian calcite variety with MgCO 3 allowed to refine between >0.000 to 0.287 apfu following the compositional range as defined by Goldsmith et al. [23] and Zhang et al. [38], but it is usually refined together with stoichiometric calcite. For dolomite, we have defined two varieties, low calcium-excess (LCD, >0.00 to 0.10) and high calcium-excess (HCD, >0.10 to 0.25) dolomite, as adopted by McCarthy et al. [39], usually refined together with stoichiometric dolomite. The limits of the varieties could be shifted, as the equations are valid throughout the entire compositional range, and the inclusion of another variety could be considered, if the limits are well defined.
While potentially providing more information than competing techniques, X-ray diffraction with the Rietveld method analysis is still cost-effective and competitive. As discussed before, spot chemical analysis by EMPA or EDS might require a large number of analyzed points for statistical reasons, which increase both time and costs. Bulk sample chemical analysis is cheaper, but the reach of the results is impaired by complex mineralogy. Nevertheless, it is recommended to combine chemical and XRD analysis, as the conciliation of results greatly adds to its reliability.

Conclusions
We implemented a method for quantifying the substitution of Mg for Ca in magnesian calcite and the excess of Ca in non-stoichiometric dolomite by the constrained Rietveld method refinement of X-ray diffraction spectra, while performing quantitative phase analysis.
The tool developed in this work allows for the generation of high quality quantitative mineralogical data, with unprecedented detail on the composition of Ca and Mg carbonates. Despite the restrictions imposed onto carbonate composition, X Mg ≤ 0.287 for high-Mg calcite and X Mg ≥ 0.75 for high-Ca dolomite, most samples could be refined within these limits. The original data we used to define the tool, however, precludes its application to extreme substitutions.
Samples derived from the petroleum systems show the most significant Mg substitutions for Ca in calcite and Ca excess replacing Mg at the B site of dolomite. Validation analyses of the method with samples of petroleum systems constitute an excellent case study. As pointed out by Wright [17], a large range of carbonates with Mg in the crystal structure might be produced in lakes with volcanic influence, especially in rift settings as those of the pre-salt rocks in Southeastern Brazil. The capability of determining the complete mineralogy by a fast and reliable method allows for the generation of the great amount of data needed to adequately interpret the deposition environment and further evolution, thus fostering better prospective models for exploration.

Supplementary Materials:
The following are available online at www.mdpi.com/2075-163X/7/9/164/s1, Table S1: Analytical parameters for EMPA (WDS), Table S2: Data as plotted in Figure 4: chemical composition as calculated from mineral quantification by the Rietveld method (wt %), Table S3: Data as plotted in Figure 4: results from XRF analysis in wt %, Table S4: Data as plotted in Figure 5: CO 2 content from different methods, Table S5: Data as plotted in Figure 6: EDS analysis of calcite and dolomite and structural formula, Table S6: Data as plotted in Figure 8: WDS analysis of calcite and dolomite and structural formula.