The Effect of Stoichiometry, Mg-Ca Distribution, and Iron, Manganese, and Zinc Impurities on the Dolomite Order Degree: A Theoretical Study

: The determination of the degree of Mg-Ca order in the dolomite structure is crucial to better understand the process or processes leading to the formation of this mineral in nature. I 01.5 / I 11.0 intensity ratios in the X-ray powder diffractograms are frequently measured to quantify dolomite cation order in dolomites. However, the intensity of diffraction peaks can be affected by factors other than the Mg-Ca distribution in the dolomite structure. The most relevant among these factors are (i) deviations from the ideal dolomite stoichiometry, and (ii) the partial substitution of Mg and Ca atoms by Fe, Mn, and Zn impurities. Using the VESTA software, we have constructed crystal structures and calculated I 01.5 / I 11.0 ratios for dolomites with Mg:Ca ratios ranging from 0.5 to 1.5, and with Fe, Mn, and Zn contents up to 30%. Our results show that both deviations from dolomite ideal stoichiometry and the presence of impurities in its structure lead to a signiﬁcant decrease in I 01.5 / I 11.0 intensity ratios, an effect which must be considered when cation orders of natural dolomites from different origins are compared.


Introduction
Dolomite, CaMg(CO 3 ) 2 , is an abundant carbonate mineral with a structure characterized by a highly ordered alternation of magnesium and calcium layers along the c axis. However, at temperatures below~100 • C, the formation of dolomites with such a high cation order cannot occur by direct precipitation from aqueous solutions, but it requires a Mg-Ca ordering process or processes after first crystallisation [1][2][3]. The investigation of both the mechanisms and kinetics of dolomite ordering process in nature first requires to accurately quantify the degree of Mg-Ca order within the dolomite structure. Since the dolomite structural order depends on the distribution of Mg and Ca cations in their corresponding crystallographic positions, the following order parameter can be defined: where X Mg is the occupancy factor of Mg atoms in the magnesium sublattice in the dolomite structure [4][5][6][7]. Although the s order parameter is suitable to describe cation order in dolomite, its direct experimental determination is not possible. However, given that changes in the X Mg in the dolomite structure have a clear effect on diffraction patterns [8], the degree of the cationic order of dolomites can be conveniently evaluated from X-ray powder diffractograms. One method to proceed is to calculate the ratio of the intensities of a selected ordering peak (i.e., a Bragg reflection which only appears in the dolomite diffractograms when Mg-rich and Ca-rich alternate along the dolomite c-axis) and the nearest non-ordering peak to the selected ordering peak (i.e., a dolomite Bragg reflection not related to the Mg-Ca order) [8][9][10][11]. Ordering peaks, also known as superstructure peaks or superstructure reflections, must fulfill the following condition: h0.l and 0k.l with l odd (e.g., 00.3, 10.1, and 01.5 peaks) [12].
By far, the I 01.5 /I 11.0 ratio is the most frequently intensity ratio used to quantify dolomite cation order (e.g., [9,10,13,14] and references therein). Using the I 01.5 /I 11.0 ratio, researchers have not only characterized some natural dolomites, but also monitored crystallisation processes of dolomites under controlled experimental conditions [1,2,15]. Nevertheless, it is important to take into account that two main factors can affect I 01.5 /I 11.0 ratios other than Ca-Mg distribution in dolomite structure: (i) the dolomite stoichiometry, i.e., deviations from Mg:Ca = 1 ratio, and (ii) the presence of impurities, which can partially substitute Mg atoms. Therefore, an adequate use of the I 01.5 /I 11.0 ratio as dolomite order index requires to precisely quantity the effects of these factors on the intensities of dolomite diffraction peaks. Otherwise, the degree of cation order in the dolomite structure might be underestimated.
The aim of this paper is to provide a first insight into the effects of Mg-Ca distribution, slight deviations from ideal dolomite stoichiometry and chemical composition (i.e., dolomites with Mg or Ca excess, and with Fe, Mn, and Zn impurities) on the index most commonly used to quantify the degree of cation order in the structure of dolomite. Our results can be useful for future crystallochemical characterizations of natural and synthetic dolomites.

Calculation Procedure
Dolomite structures with different Mg-Ca distributions and Mg:Ca stoichiometries have been constructed and depicted using the VESTA software 3.5 [16]. In a number of cases, dolomite structures also included small amounts of iron, manganese, and zinc atoms as impurities. In these dolomites, these cation impurities were only added in the Mg-sites of stoichiometric dolomites, i.e., with (Mg + M):Ca = 1, where M = Fe, Mn, or Zn. It is important to note here that the VESTA software does not use atomistic simulations or quantum-mechanical methods. Therefore, no optimization of the geometry of dolomite structures prior to the calculation of diffractograms was performed. Accordingly, the shifting of diffraction peaks was not calculated, and only changes in the intensities of these peaks were considered in this work.
Both dolomite and calcite structures were used as starting structures for calculations. Dolomite and calcite crystallographic data were taken from Miser et al. [17] and Maslen et al. [18], respectively (see Table 1). In the case of calcite, after generating all fractional coordinates of the atoms corresponding to the space group R3c, the s.g. P1 was assigned while keeping all fractional coordinates unvaried. By doing that, magnesium atoms can be added in alternating layers to create a "dolomite structure" in which the cation disorder can be subsequently modified. Once dolomite structures with different Mg-Ca arrangements and impurity contents were constructed, diffractograms were calculated with the RIETAN-FP code [19] included in VESTA software [16]. Then, I 01.5 /I 11.0 ratios were measured in the calculated diffractograms using a standard protocol that can be found elsewhere (e.g., [9,10,13,14,20,21]). This method consists in calculating the intensity I 01.5 /I 11.0 ratios by measuring peak intensities in counts, and without conducting any previous treatment of the diffractograms (e.g., Rietveld method).

Stoichiometric Dolomites
Stoichiometric dolomites have a Mg:Ca ratio equal to unity. Therefore, the relative amount of Ca and Mg cations must be changed in the Mg and Ca crystallographic sites to generate variations in cation order within these stoichiometric dolomites ( Figure 1). Thus, a maximum order is reached when both calcium and magnesium sites are fully occupied by the corresponding cation, i.e., magnesium in the Mg-site and calcium in the Ca-site, and, therefore, X Mg = 1. In turn, the order degree reaches a minimum when both sites are occupied by 50% of magnesium cations and 50% of calcium cations, i.e., X Mg = 0.   Figure 2 shows the calculated diffractograms for both a fully ordered and a fully disordered stoichiometric dolomite structure. As can be seen in this figure, the intensities of superstructure peaks (i.e., diffraction peaks which result from cation order) are different. In principle, superstructure peaks in the diffractogram of a fully disordered dolomite should be absent. Nevertheless, in Figure 2b, superstructure peaks are still visible. This is because calculations shown in Figure 2 were performed using crystallographic data for a dolomite structure [17]. In the dolomite 3 structure, the Ca-and Mg-sites are crystallographically distinct sites. As a consequence, diffraction peaks for superstructure planes remain even when the Mg-Ca distribution within them is random (i.e., XMg = 0.5). In a different way, when the calcite 3 structure is used for the calculations [18], both crys- anionic groups in black (carbon atoms) and red (oxygen atoms). The Mg:Ca ratio in each site is represented by spherical sectors with radii proportional to occupancy factors. Figure 2 shows the calculated diffractograms for both a fully ordered and a fully disordered stoichiometric dolomite structure. As can be seen in this figure, the intensities of superstructure peaks (i.e., diffraction peaks which result from cation order) are different. In principle, superstructure peaks in the diffractogram of a fully disordered dolomite should be absent. Nevertheless, in Figure 2b, superstructure peaks are still visible. This is because calculations shown in Figure 2 were performed using crystallographic data for a dolomite structure [17]. In the dolomite R3 structure, the Ca-and Mg-sites are crystallographically distinct sites. As a consequence, diffraction peaks for superstructure planes remain even when the Mg-Ca distribution within them is random (i.e., X Mg = 0.5). In a different way, when the calcite R3c structure is used for the calculations [18], both crystallographic sites are identical, and superstructure peaks vanish for X Mg = 0.5 (see Figure S1 in the Supporting Information). The results presented here were obtained by considering the structure of dolomite only. As explained in the introductory section, diffractograms of dolomites can show very low I01.5/I11.0 ratios, indicating a very low cation order [14,20]. However, the minimum order to distinguish between a dolomite and a VHMC, i.e., when its structure changes from an 3 to an 3 , has not yet been clearly defined. Taking this into account, we consider that a significant overestimation of the order indexes observed for a completely disordered dolomite will only occur for dolomites with an Mg mole fraction close to 0.5. Above that value, all dolomites will present a structure with 3 space group and, therefore, superstructure peaks will be present in the diffractograms.
From diffractograms like those shown in Figure 2 and calculated for XMg ranging from 0.5 to 1.0, I01.5/I11.0, ratios were systematically measured. These intensity ratios were found to increase nonlinearly from 0.13 to 0.53 as the XMg increases ( Figure 3a). Since intensity ratios were measured from diffractograms simulated using a dolomite structure [17], an I01.5/I11.0 value of 0.13 was obtained for a completely disordered dolomite (i.e., for xMg = 0.5). As explained above, this "residual order" for a fully disordered dolomite structure results from the symmetry constraints of the 3 space group. When the distribution of Mg and Ca is varied in the structure of a stoichiometric dolomite, the changes in I01.5/I11.0 ratios can be also related to the order parameter, s, using Equation (1). Figure 3b shows the non-linear increase of I01.5/I11.0 ratio as the s parameter The results presented here were obtained by considering the structure of dolomite only. As explained in the introductory section, diffractograms of dolomites can show very low I 01.5 /I 11.0 ratios, indicating a very low cation order [14,20]. However, the minimum order to distinguish between a dolomite and a VHMC, i.e., when its structure changes from an R3 to an R3c, has not yet been clearly defined. Taking this into account, we consider that a significant overestimation of the order indexes observed for a completely disordered dolomite will only occur for dolomites with an Mg mole fraction close to 0.5. Above that value, all dolomites will present a structure with R3 space group and, therefore, superstructure peaks will be present in the diffractograms.
From diffractograms like those shown in Figure 2 and calculated for X Mg ranging from 0.5 to 1.0, I 01.5 /I 11.0 , ratios were systematically measured. These intensity ratios were found to increase nonlinearly from 0.13 to 0.53 as the X Mg increases (Figure 3a). Since intensity ratios were measured from diffractograms simulated using a dolomite structure [17], an I 01.5 /I 11.0 value of 0.13 was obtained for a completely disordered dolomite (i.e., for x Mg = 0.5). As explained above, this "residual order" for a fully disordered dolomite structure results from the symmetry constraints of the R3 space group. The results presented here were obtained by considering the structure of dolomite only. As explained in the introductory section, diffractograms of dolomites can show very low I01.5/I11.0 ratios, indicating a very low cation order [14,20]. However, the minimum order to distinguish between a dolomite and a VHMC, i.e., when its structure changes from an 3 to an 3 , has not yet been clearly defined. Taking this into account, we consider that a significant overestimation of the order indexes observed for a completely disordered dolomite will only occur for dolomites with an Mg mole fraction close to 0.5. Above that value, all dolomites will present a structure with 3 space group and, therefore, superstructure peaks will be present in the diffractograms.
From diffractograms like those shown in Figure 2 and calculated for XMg ranging from 0.5 to 1.0, I01.5/I11.0, ratios were systematically measured. These intensity ratios were found to increase nonlinearly from 0.13 to 0.53 as the XMg increases (Figure 3a). Since intensity ratios were measured from diffractograms simulated using a dolomite structure [17], an I01.5/I11.0 value of 0.13 was obtained for a completely disordered dolomite (i.e., for xMg = 0.5). As explained above, this "residual order" for a fully disordered dolomite structure results from the symmetry constraints of the 3 space group. When the distribution of Mg and Ca is varied in the structure of a stoichiometric dolomite, the changes in I01.5/I11.0 ratios can be also related to the order parameter, s, using Equation (1). Figure 3b shows the non-linear increase of I01.5/I11.0 ratio as the s parameter increases from s = 0 to s = 1. Noticeably, for the maximum value of the order parameter (s = 1), a superstructure/non-superstructure intensity ratio is not equal to one, as previously When the distribution of Mg and Ca is varied in the structure of a stoichiometric dolomite, the changes in I 01.5 /I 11.0 ratios can be also related to the order parameter, s, using Equation (1). Figure 3b shows the non-linear increase of I 01.5 /I 11.0 ratio as the s parameter increases from s = 0 to s = 1. Noticeably, for the maximum value of the order parameter (s = 1), a superstructure/non-superstructure intensity ratio is not equal to one, as previously assumed in the analysis by Luth [5] of the Schultz-Güttler [11] method for quantifying the cation degree of order in dolomites. From this assumption, Luth [5] concluded that measurements of dolomite superstructure/non-superstructure intensity ratios might lead to a systematic underestimation of the s parameter. Our calculations do not support this conclusion, but they instead indicate that a precise correlation between intensity ratios and s requires a previous calibration with the experimental dolomite diffractograms.

Non-Stoichiometric Dolomites
For the non-stoichiometric dolomites considered in this work, Mg:Ca ratios were varied from 0.5 to 1.5, i.e., their compositions were gradually changed from Ca 0.5 Mg 1.5 (CO 3 ) 2 to Ca 1.5 Mg 0.5 (CO 3 ) 2 . Structurally, the cation in excess can fill its corresponding sites and a fraction of the sites of the minority cation (see Figure 4a), or alternatively, Mg:Ca ratios can vary in both cation sites, while keeping the same Mg:Ca ratio of the given stoichiometry (see Figure 4b). Dolomites of both types have been found in nature, i.e., dolomites with excess of Ca atoms which fill completely its own site and a fraction of the magnesium sites [22], and dolomites with an excess of Ca atoms which occupy both magnesium and calcium sites [23].
Minerals 2021, 11, 702 6 of 10 cation degree of order in dolomites. From this assumption, Luth [5] concluded that measurements of dolomite superstructure/non-superstructure intensity ratios might lead to a systematic underestimation of the s parameter. Our calculations do not support this conclusion, but they instead indicate that a precise correlation between intensity ratios and s requires a previous calibration with the experimental dolomite diffractograms.

Non-Stoichiometric Dolomites
For the non-stoichiometric dolomites considered in this work, Mg:Ca ratios were varied from 0.5 to 1.5, i.e., their compositions were gradually changed from Ca0.5 Mg1.5(CO3)2 to Ca1.5Mg0.5(CO3)2. Structurally, the cation in excess can fill its corresponding sites and a fraction of the sites of the minority cation (see Figure 4a), or alternatively, Mg:Ca ratios can vary in both cation sites, while keeping the same Mg:Ca ratio of the given stoichiometry (see Figure 4b). Dolomites of both types have been found in nature, i.e., dolomites with excess of Ca atoms which fill completely its own site and a fraction of the magnesium sites [22], and dolomites with an excess of Ca atoms which occupy both magnesium and calcium sites [23].  Figure 5a shows the variation of I01.5/I11.0 ratios in the diffractograms calculated for non-stoichiometric dolomites with x ranging from 0.5 to 1.5 in the Ca2−xMgx(CO3)2 formula, and for which one of both cation sites is always fully occupied by the corresponding cation. As can be seen in this figure, deviations from the dolomite stoichiometry towards an  Figure 5a shows the variation of I 01.5 /I 11.0 ratios in the diffractograms calculated for non-stoichiometric dolomites with x ranging from 0.5 to 1.5 in the Ca 2−x Mg x (CO 3 ) 2 formula, and for which one of both cation sites is always fully occupied by the corresponding cation. As can be seen in this figure, deviations from the dolomite stoichiometry towards an excess of calcium have a more pronounced effect on the I 01.5 /I 11.0 ratios than in the case of an excess of magnesium in the dolomite structure. The reason for this asymmetry lies in the larger atomic scattering factor of calcium compared to that of magnesium. Thus, the intensity of the I 01.5 peak is more effectively reduced when calcium atoms increasingly occupy Mg-sites than in the case of magnesium atoms occupying Ca-sites. In other words, dolomite order quantified by I 01.5 /I 11.0 ratios will be more underestimated for Ca-rich dolomites than for Mg-rich dolomites.
Minerals 2021, 11, 702 7 of 10 excess of calcium have a more pronounced effect on the I01.5/I11.0 ratios than in the case of an excess of magnesium in the dolomite structure. The reason for this asymmetry lies in the larger atomic scattering factor of calcium compared to that of magnesium. Thus, the intensity of the I01.5 peak is more effectively reduced when calcium atoms increasingly occupy Mg-sites than in the case of magnesium atoms occupying Ca-sites. In other words, dolomite order quantified by I01.5/I11.0 ratios will be more underestimated for Ca-rich dolomites than for Mg-rich dolomites. For a given dolomite stoichiometry, I01.5/I11.0 ratios can be further reduced by allowing Mg and Ca atoms to occupy both sites in different ratios, i.e., by removing the previous constraint of filling first the layer corresponding to the atom in excess. Figure 5b illustrates the effect on the I01.5/I11.0 ratios when both the molar ratio of Mg atoms in Ca-sites (Mg excess region in the plot) and the molar ratio of Ca atoms in Mg-sites (Ca excess region in the plot) increase. As in the case shown in Figure 5a, the decrease in I01.5/I11.0 ratios is more pronounced in the Ca-excess region of the plot.
The results shown in Figure 5a,b demonstrate that, in the case of non-stoichiometric dolomites, the degree of order directly derived from in I01.5/I11.0 ratios will be always underestimated, i.e., the experimental value of I01.5/I11.0 of a non-stoichiometric dolomite is always lower than that of a stoichiometric dolomite. Therefore, in order to precisely quantify the order of a dolomite crystal it is necessary to know first the maximum theoretical I01.5/I11.0 ratio for its particular stoichiometry and Ca-Mg distribution in the two cation sites.

Dolomites with Iron, Manganese, and Zinc Impurities
Frequently, natural dolomites contain variable amounts of Fe, Mn, and Zn impurities in their structures. It is obvious that the presence of these impurities will have an effect on dolomite cation order and, therefore, on I01.5/I11.0 ratios. Although cation impurities can be accommodated in both Ca-and Mg-sites, in this work, we will only analyse this effect in the case of the partial substitution of Mg 2+ by Fe 2+ , Mn 2+ , or Zn 2+ cations in the Mg-site of dolomites. Moreover, we have assumed that these dolomites fulfill the condition: (Mg+M):Ca = 1 (where M = Fe, Mn, or Zn atoms). Therefore, the general formula of the dolomites considered is Ca(Mg1-xMx)(CO3)2. As an example, Figure 6 shows a dolomite structure in which a 20% of the Mg-sites are occupied by Fe 2+ . For a given dolomite stoichiometry, I 01.5 /I 11.0 ratios can be further reduced by allowing Mg and Ca atoms to occupy both sites in different ratios, i.e., by removing the previous constraint of filling first the layer corresponding to the atom in excess. Figure 5b illustrates the effect on the I 01.5 /I 11.0 ratios when both the molar ratio of Mg atoms in Ca-sites (Mg excess region in the plot) and the molar ratio of Ca atoms in Mg-sites (Ca excess region in the plot) increase. As in the case shown in Figure 5a, the decrease in I 01.5 /I 11.0 ratios is more pronounced in the Ca-excess region of the plot.
The results shown in Figure 5a,b demonstrate that, in the case of non-stoichiometric dolomites, the degree of order directly derived from in I 01.5 /I 11.0 ratios will be always underestimated, i.e., the experimental value of I 01.5 /I 11.0 of a non-stoichiometric dolomite is always lower than that of a stoichiometric dolomite. Therefore, in order to precisely quantify the order of a dolomite crystal it is necessary to know first the maximum theoretical I 01.5 /I 11.0 ratio for its particular stoichiometry and Ca-Mg distribution in the two cation sites.

Dolomites with Iron, Manganese, and Zinc Impurities
Frequently, natural dolomites contain variable amounts of Fe, Mn, and Zn impurities in their structures. It is obvious that the presence of these impurities will have an effect on dolomite cation order and, therefore, on I 01.5 /I 11.0 ratios. Although cation impurities can be accommodated in both Ca-and Mg-sites, in this work, we will only analyse this effect in the case of the partial substitution of Mg 2+ by Fe 2+ , Mn 2+ , or Zn 2+ cations in the Mg-site of dolomites. Moreover, we have assumed that these dolomites fulfill the condition: (Mg+M):Ca = 1 (where M = Fe, Mn, or Zn atoms). Therefore, the general formula of the dolomites considered is Ca(Mg 1−x M x )(CO 3 ) 2 . As an example, Figure 6 shows a dolomite structure in which a 20% of the Mg-sites are occupied by Fe 2+ . Measurements conducted on diffractograms of dolomites in which the Mg-site was progressively occupied by Fe, Mn, and Zn show that dolomite cation order decreases as the impurity content increases. Figure 7 shows the variation of I01.5/I11.0 when Fe was progressively substituted in the Mg-site up to 30%. In this compositional range, I01.5/I11.0 ratios decrease from 0.53 (dolomite without iron impurities) to 0.21 (dolomite with XFe = 0.3 in the Mg-site). A similar decrease in the dolomite cation order also occurs when Mg atoms are partially substituted by Mn and Zn impurities in the Mg-site (see Figure S2 in the Supporting Information). However, the significantly larger atomic scattering factor of the Zn atoms compare to those of Fe and Mn atoms results in a more pronounced decrease of I01.5/I11.0 ratios when magnesium is gradually substituted by Zn. Thus, for XZn = 0.3 in the Mg-site, I01.5/I11.0 is only 0.14. Measurements conducted on diffractograms of dolomites in which the Mg-site was progressively occupied by Fe, Mn, and Zn show that dolomite cation order decreases as the impurity content increases. Figure 7 shows the variation of I 01.5 /I 11.0 when Fe was progressively substituted in the Mg-site up to 30%. In this compositional range, I 01.5 /I 11.0 ratios decrease from 0.53 (dolomite without iron impurities) to 0.21 (dolomite with X Fe = 0.3 in the Mg-site). A similar decrease in the dolomite cation order also occurs when Mg atoms are partially substituted by Mn and Zn impurities in the Mg-site (see Figure S2 in the Supporting Information). However, the significantly larger atomic scattering factor of the Zn atoms compare to those of Fe and Mn atoms results in a more pronounced decrease of I 01.5 /I 11.0 ratios when magnesium is gradually substituted by Zn. Thus, for X Zn = 0.3 in the Mg-site, I 01.5 /I 11.0 is only 0.14.

Conclusions
Our calculations using the VESTA software [16] allowed us to quantify the decrease in I01.5/I11.0 intensity ratios when the amount of calcium and magnesium was varied in the Mg-and Ca-sites of a dolomite structure with Mg:Ca = 1 (i.e., an ideally stoichiometric dolomite). Similarly, when non-stoichiometric dolomites were considered (i.e., dolomites with Mg:Ca ≠1), I01.5/I11.0 values decreased significantly when x was varied in the Ca2−xMgx(CO3)2 dolomite formula. We found that such a decrease is more pronounced for dolomites with an excess of calcium than for dolomites with and excess of magnesium. Finally, we calculate the decrease in I01.5/I11.0 intensity ratios as the Mg-site was progressively occupied by Fe, Mn, or Zn impurities up to 30%. Since both the deviations from stoichiometry and the presence of impurities result in a significant decrease in the I01.5/I11.0 intensity ratios, these factors should be considered to quantify the degree of cation order of natural and experimentally produced dolomites.