Quantitative Phase Analysis of Skarn Rocks by the Rietveld Method Using X-ray Powder Di ﬀ raction Data

: The Rietveld method using X-ray powder di ﬀ raction data was applied to selected skarn samples for quantitative determination of the present minerals. The specimens include garnet, clinopyroxene–garnet, plagioclase–clinopyroxene–wollastonite–garnet, plagioclase–clinopyroxene– wollastonite, plagioclase–clinopyroxene–wollastonite–epidote, and plagioclase– clinopyroxene skarns. The rocks are coarse- to ﬁne-grained and characterized by an uneven distribution of the constituent minerals. The traditional methods for quantitative analysis (point-counting and norm calculations) are not applicable for such inhomogeneous samples containing minerals with highly variable chemical compositions. Up to eight individual mineral phases have been measured in each sample. To obtain the mineral quantities in the skarn rocks preliminary optical microscopy and chemical investigation by electron probe microanalysis (EPMA) were performed for the identiﬁcation of some starting components for the Rietveld analysis and to make comparison with the Rietveld X-ray powder di ﬀ raction results. All of the reﬁnements are acceptable, as can be judged by the standard indices of agreement and by the visual ﬁts of the observed and calculated di ﬀ raction proﬁles. A good correlation between the reﬁned mineral compositions and the data of the EPMA measurements was achieved. Mag—magnetite, Ti-Grt—Ti-rich garnet, Pl—plagioclase, An—anorthite.


Introduction
The estimation of relative quantities of the constituent minerals in rocks is essential for their classification and determination of paragenesis as well as for studies of their genesis, sequence of different mineralogical stages and geological events. The mineral modal analysis is also important for assessment of potentially valuable minerals and is applied in multi-component systems such as cements, ceramics, and archaeological materials.
The Rietveld method is one of the most appropriate techniques for modal analysis of rocks and synthetic polycrystalline materials based on X-ray powder diffraction (XRPD). This method was originally developed for the refinement of structures studied with neutron-diffraction [1,2] and subsequently was successfully applied to crystal-structure refinements of minerals and synthetic equivalents investigated with X-ray powder diffraction and synchrotron radiation [3][4][5][6]. This methodology and the possibilities for quantitative analysis were developed in detail in the works of: Albinati and Willis [7], Hill [8], Hill and Madsen [9], Hill and Howard [10], Bish and Howard [11], Post and Bish [12], Snyder and Bish [13], Hill [14], Young [15].
The Rietveld XRPD method is a full-profile approach for quantitative analysis based on a least squares fit between the calculated diffraction pattern and the observed experimental data of a measured Table 1. Sources of crystal structure data used to derive starting models of the minerals for the quantitative phase analysis.

Mineral
Sources of Crystal Structure Data  [54] The sequence of parameters, which were refined, is the following: scale factors for all phases, zero-shift parameter, background polynomial coefficients, unit-cell parameters for each phase, half-width parameters, atomic site occupancies, atomic coordinates, and preferred orientation. Details of the Rietveld refinement strategy and suggested guidelines are reported by Hill and Madsen [9,55] and Young et al. [56].
A correction for preferred orientation (exponential function implemented in the program Fullprof and preferred orientation spherical harmonics in Topas v4.2 software) was refined for minerals with marked grain-shape anisotropy (feldspars, pyroxene, wollastonite, calcite, epidote, chlorite, etc.). The effect of preferred orientation on the scale factors of the phases is minimized by Rietveld full profile fitting, especially for phases with a large number of XRPD reflexes [13]. The unit cell parameters and the atomic site occupancies of the major minerals with expressed isomorphism were set as variables in the final cycles of refinement (e.g., the Ca and Na content of the feldspars, the Mg and Fe content of the pyroxenes, Al and Fe content of the garnets). The chemical composition determined by EPMA was assigned to the minor minerals present.
The fit of the calculated profile with the measured XRPD pattern gives scale factors for each phase which are related to the weight fractions by Equation (1) [10]: where S, Z, M and V are, respectively, the Rietveld scale factor, the number of formula units per unit cell, the mass of the formula unit and the unit-cell volume.
The quality of the fit between the calculated and observed diffraction profiles obtained in a Rietveld refinement is usually given with the standard agreement indices Rp, Rwp, Rexp and the goodness of fit index (GofF), defined by Young et al. [15].

Garnet Skarn
The garnet zone is dominantly composed of garnets, calcite and quartz with minor prehnite and chlorite. Magnetite, apatite and titanite occur as accessory minerals. The grain-size of the rock varies in a very wide range-from a few microns to 4 mm. Coarse-grained euhedral garnet crystals occur in a fine-grained quartz-calcite-grossular matrix (Figure 1a,b). Due to the presence of fine-grained material, an optical mode was not applicable for this sample.
Based on optical characteristics and electron probe microanalyses, different garnet generations were identified. The earliest garnets are Ti-rich (TiO 2 from 5.30 wt% to 11.23 wt% and schorlomite component up to 31.8%) and under plane polarized light they appear as brown to red-brown isotropic crystals (Figure 1c,d). The second generation is represented by coarse-grained garnets, which belong to the grossular-andradite solid solution ranging in composition from Adr 8.5 Grs 88. 6 to almost pure andradite Adr 94.3 Grs 2.8 with spessartine, pyrope, almandine, uvarovite, schorlomite, and goldmanite collectively less than 3%. On a local scale, these garnets are continuously zoned with Fe-rich cores and Al-rich rims or exhibit oscillatory zoning, chemically expressed by variations in Al-content. The garnets from the third generation form fine-grained aggregates in the quartz-calcite matrix and are more grossular-rich (grossular component 60-85%).
Minerals 2020, 10, x FOR PEER REVIEW 5 of 27 in a very wide range-from a few microns to 4 mm. Coarse-grained euhedral garnet crystals occur in a fine-grained quartz-calcite-grossular matrix (Figure 1a,b). Due to the presence of fine-grained material, an optical mode was not applicable for this sample. Based on optical characteristics and electron probe microanalyses, different garnet generations were identified. The earliest garnets are Ti-rich (TiO2 from 5.30 wt% to 11.23 wt% and schorlomite component up to 31.8%) and under plane polarized light they appear as brown to red-brown isotropic crystals (Figure 1c,d). The second generation is represented by coarse-grained garnets, which belong to the grossular-andradite solid solution ranging in composition from Adr8.5Grs88.6 to almost pure andradite Adr94.3Grs2.8 with spessartine, pyrope, almandine, uvarovite, schorlomite, and goldmanite collectively less than 3%. On a local scale, these garnets are continuously zoned with Fe-rich cores and Al-rich rims or exhibit oscillatory zoning, chemically expressed by variations in Al-content. The garnets from the third generation form fine-grained aggregates in the quartz-calcite matrix and are more grossular-rich (grossular component 60-85%). The mineral composition of garnet skarn determined by XRPD is shown in Figure 2. The peaks of two type of garnets are well resolved in the patterns especially at d-spacing 2.99-2.97 Å, 2.67-2.65 Å. The grossular-andradite in this sample is modeled in the Rietveld analysis as a separate phase from grossular and both phases have unit cell dimensions that allow them to be distinguished in the diffraction patterns. Because of the relatively low abundance, Ti-rich andradite is not included in the refinements. The X-ray Rietveld refinements began with five major phases: quartz, calcite, grossular, grossular-andradite, and chlorite ( Figure 3). No traces of the other minor phases could be detected in the diffraction pattern. (d) backscattered electron images of garnet skarn showing a sharp contact between Ti-rich andradite and grossulare of later generation. Abbreviations: Ap-apatite, Grt-garnet, Prh-pehnite, Cal-calcite, Qz-quartz, Grs-grossular, Ti-Grt-Ti-rich garnet, Adr-andradite, Prp-pyrope, Sch-schorlomite, Sps-spessartine, Alm-almandine. The mineral composition of garnet skarn determined by XRPD is shown in Figure 2. The peaks of two type of garnets are well resolved in the patterns especially at d-spacing 2.99-2.97 Å, 2.67-2.65 Å. The grossular-andradite in this sample is modeled in the Rietveld analysis as a separate phase from grossular and both phases have unit cell dimensions that allow them to be distinguished in the diffraction patterns. Because of the relatively low abundance, Ti-rich andradite is not included in the refinements. The X-ray Rietveld refinements began with five major phases: quartz, calcite, grossular, grossular-andradite, and chlorite ( Figure 3). No traces of the other minor phases could be detected in the diffraction pattern. The sequence of parameters, which were refined, is the following: scale factors for quartz, calcite, grossular, grossular-andradite, chlorite (phases 1, 2, 3, 4 and 5); zero shift, background polynomial parameters (six coefficients); unit cell parameters for all phases; half-width parameters for all phases (U, V, W); atomic site occupancies of Al and Fe in garnets (phases 3 and 4); atomic coordinates of phases 3 and 4; preferred orientation correction for phases 1 and 2.
Visualization of the fit is given in a difference plot on  The sequence of parameters, which were refined, is the following: scale factors for quartz, calcite, grossular, grossular-andradite, chlorite (phases 1, 2, 3, 4 and 5); zero shift, background polynomial parameters (six coefficients); unit cell parameters for all phases; half-width parameters for all phases (U, V, W); atomic site occupancies of Al and Fe in garnets (phases 3 and 4); atomic coordinates of phases 3 and 4; preferred orientation correction for phases 1 and 2.

Clinopyroxene-Garnet Skarn
This is a fine-to coarse-grained rock containing garnets and clinopyroxene in quartz-calcite matrix with minor prehnite and chlorite. Accessory minerals include magnetite, apatite and titanite.
The garnets in this sample, like those in garnet skarn, are represented by at least three generations: Ti-rich garnets, zoned coarse-grained grossular-andradites (Figure 4a), and fine-grained grossular aggregates in quartz-calcite matrix. Grossular-andradites form poikiloblastic crystals that host relict minerals from the early high temperature contact metamorphic stage: melilite (Figure 4d), wollastonite−2M (I generation) ( Figure 4c) and "fassaitic" clinopyroxene (I generation). Clinopyroxene of the IInd generation commonly occurs as inclusions in garnet or as crystal aggregates in the quartz-calcite matrix, which exhibit strongly corroded margins. Optically, it shows strong dispersion of the optic axes, r > v, and displays anomalous blue and brown interference colors ( Figure 4b). The chemical compositions of the clinopyroxene from both generations are characterized by significant deficiency of SiO 2 , (Si 1.32-1.50 apfu) and high contents of esseneite (up to 46%) and Ca-Tschermak (up to 24%) components. Clinopyroxene of the I st generation has lower content of esseneite (up to 20%).
The mineral composition of clinopyroxene-garnet skarn determined by XRPD is shown in Figure 5. The grossular-andradite in this sample is modeled in the Rietveld analysis as separate phases from grossular and clinopyroxene as esseneite. The XRPD Rietveld refinements incorporate only the six phases quartz, calcite, grossular, grossular-andradite, chlorite and clinopyroxene. No traces of the other minor phases could be detected in the diffraction pattern. According to optical and SEM studies Ti-rich andradite, melilite and wollastonite occur in quantities that are too small to be detected as discrete phases in the X-ray pattern. Clinopyroxenes from both generations could not be distinguished crystallographically. The sequence of the refined parameters is: scale factors for quartz, calcite, grossular, grossular-andradite, chlorite, clinopyroxene (phases 1, 2, 3, 4, 5, 6); zero shift; background polynomial parameters (six coefficients); unit cell parameters for phases 1, 2, 3, 4, 5, 6; half-width parameters for all phases (U, V, W); atomic site occupancies of Al and Fe in garnets (phases 3 and 4); atomic coordinates of phases 3 and 4; atomic site occupancies of Mg and Fe in clinopyroxene (phase 6) atomic coordinates of phase 6; preferred orientation correction for phases 1, 2 and 6.
in the quartz-calcite matrix, which exhibit strongly corroded margins. Optically, it shows strong dispersion of the optic axes, r > v, and displays anomalous blue and brown interference colors ( Figure  4b). The chemical compositions of the clinopyroxene from both generations are characterized by significant deficiency of SiO2, (Si 1.32-1.50 apfu) and high contents of esseneite (up to 46%) and Ca-Tschermak (up to 24%) components. Clinopyroxene of the I st generation has lower content of esseneite (up to 20%). The mineral composition of clinopyroxene-garnet skarn determined by XRPD is shown in Figure 5. The grossular-andradite in this sample is modeled in the Rietveld analysis as separate phases from grossular and clinopyroxene as esseneite. The XRPD Rietveld refinements incorporate only the six phases quartz, calcite, grossular, grossular-andradite, chlorite and clinopyroxene. No traces of the other minor phases could be detected in the diffraction pattern. According to optical and SEM studies Ti-rich andradite, melilite and wollastonite occur in quantities that are too small to be detected as discrete phases in the X-ray pattern. Clinopyroxenes from both generations could not be distinguished crystallographically. The sequence of the refined parameters is: scale factors for quartz, calcite, grossular, grossular-andradite, chlorite, clinopyroxene (phases 1, 2, 3, 4, 5, 6); zero shift;

Plagioclase-Clinopyroxene-Wollastonite-Garnet Skarn
The rock is coarse-to fine-grained and contains garnet, wollastonite (II generation), plagioclase, clinopyroxene, and calcite. Minor or secondary minerals present are Ti-rich andradite (Figure 7a), quartz, prehnite, chlorite, and thaumasite. Accessory phases include titanite, apatite and magnetite. All of the constituent minerals are unevenly distributed within the zone. The rock is affected by intense retrograde processes represented by albitisation of former plagioclase, prehnitization of former wollastonite, and fracture-filling thaumasite (Figure 7b,d).
The garnets in this sample are grossular-andradites of intermediate composition (with andradite component from 42% to 50%) (Figure 7c,d), but later almost pure andradite is also found ( Figure 7d). EPMA analysis and optical studies identified two types of clinopyroxene: "fassaitic" clinopyroxene (esseneite component-up to 50% and Ca-Tschermak-up to 19%) with anomalous blue and brown interference colors and clinopyroxene of the diopside-hedenbergite series (mean Wo 51 En 41 Fs 8 ) (Figure 7a).

Plagioclase-clinopyroxene-wollastonite skarn
This skarn rock is characterized by significant difference in the grain size of the constituent minerals. Minerals identified in thin section include plagioclase, clinopyroxene, wollastonite ( Figure  10a,b) with minor or secondary Ti-rich andradite, calcite, quartz, prehnite, and chlorite. Titanite, apatite and magnetite occur as accessory phases. All of the minerals are unevenly distributed within the zone.

Plagioclase-Clinopyroxene-Wollastonite Skarn
This skarn rock is characterized by significant difference in the grain size of the constituent minerals. Minerals identified in thin section include plagioclase, clinopyroxene, wollastonite (Figure 10a,b) with minor or secondary Ti-rich andradite, calcite, quartz, prehnite, and chlorite. Titanite, apatite and magnetite occur as accessory phases. All of the minerals are unevenly distributed within the zone. The anorthite content in plagioclase in this sample (obtained from EPMA) shows large variations-from 1.9% to 76.4% because plagioclase is partially replaced by albite during the later retrograde processes (Figure 10c,d). The clinopyroxenes in this zone belong to the diopsidehedenbergite series (Wo50En36-39Fs11-14). Wollastonite forms coarse poikiloblastic crystals that host clinopyroxene and other rock-forming minerals (Figure 10a). Prehnite occurs in interstitial positions or replaces earlier formed minerals. The mineral composition of Pl-Cpx-Wo skarn determined by XRPD is shown in Figure 11. The XRPD Rietveld refinement began with five phases: plagioclase, clinopyroxene, wollastonite−2M, calcite and prehnite. According to optical and EPMA (Tables A3, A4, A5, and A7) studies, the plagioclase is modeled as labradorite and clinopyroxene as diopside-hedenbergite. None of the other minor phases could be detected above background. The sequence of refined parameters is the following: scale factors for plagioclase, wollastonite−2M, clinopyroxene, calcite, prehnite (phases 1, The anorthite content in plagioclase in this sample (obtained from EPMA) shows large variations-from 1.9% to 76.4% because plagioclase is partially replaced by albite during the later retrograde processes (Figure 10c,d). The clinopyroxenes in this zone belong to the diopside-hedenbergite series (Wo 50 En 36-39 Fs [11][12][13][14]. Wollastonite forms coarse poikiloblastic crystals that host clinopyroxene and other rock-forming minerals (Figure 10a). Prehnite occurs in interstitial positions or replaces earlier formed minerals.
Rietveld refinement plot of Pl-Cpx-Wo skarn is given in Figure 12.
According to EPMA and optical data, the mean content of the anorthite component in plagioclase is about 58%. Usually the primary plagioclase is partially replaced by albite. The clinopyroxenes in this zone belong to the diopside-hedenbergite series (Di36-80Hd21-63). The Fe 3+ /(Fe 3+ + Al) ratio in the epidote composition ranges from 0.16 to 0.32. Locally, titanite forms porphyroblasts up to 1 mm long (Figure 13c). Prehnite occurs in interstitial positions or replaces the earlier formed plagioclase and wollastonite.
The mineral composition of Pl-Cpx-Wo-Ep skarn determined by XRPD is presented in Figure  14. The XRPD Rietveld refinements incorporate eight phases: plagioclase, clinopyroxene, wollastonite−2M, calcite, epidote, prehnite, quartz, and chlorite. According to optical and EPMA (Tables A3-A7) studies the plagioclase is modeled as labradorite and clinopyroxene as diopsidehedenbergite. No traces of other minor phases could be detected in the diffraction pattern. The
According to EPMA and optical data, the mean content of the anorthite component in plagioclase is about 58%. Usually the primary plagioclase is partially replaced by albite. The clinopyroxenes in this zone belong to the diopside-hedenbergite series (Di 36-80 Hd 21-63 ). The Fe 3+ /(Fe 3+ + Al) ratio in the epidote composition ranges from 0.16 to 0.32. Locally, titanite forms porphyroblasts up to 1 mm long (Figure 13c). Prehnite occurs in interstitial positions or replaces the earlier formed plagioclase and wollastonite.

Plagioclase-Clinopyroxene Skarn
This skarn is present close to monzonitic rocks in contact with the skarn xenoliths. The rock is medium-grained and contains plagioclase and clinopyroxene (Figure 16a) with minor or secondary calcite, quartz, prehnite, and chlorite. Accessory minerals include titanite, apatite and magnetite.
Usually the primary plagioclase is partially replaced by albite or prehnite (Figure 16b-d). The mean plagioclase composition measured by EPMA is An60. The clinopyroxene in this zone belongs to the diopside-hedenbergite series (Di48-73Hd25-49Jhn0-2) and forms zoned crystals with Mg-rich cores and Fe-rich rims (Figure 16c,d). Prehnite occurs in interstitial positions or replaces the earlier formed plagioclase.
The mineral composition of Pl-Cpx skarn determined by XRPD is shown in Figure 17. The XRPD Rietveld refinements include five phases: plagioclase, clinopyroxene, calcite, quartz, and chlorite. Based on optical and EPMA (Tables A3 and A5) studies, the plagioclase is modeled as labradorite and clinopyroxene as diopside-hedenbergite. The minor phases are present only at the 0-2 vol.% level and could not be detected in the XRPD pattern. The obtained mean plagioclase composition is in agreement both with the mean of the optical determinations and with those measured by EPMA. The sequence of refined parameters is the following: scale factors for plagioclase, clinopyroxene, calcite, quartz, chlorite (phases 1, 2, 3, 4, 5); zero shift, background polynomial parameters (22 coefficients); unit cell parameters for phases 1, 2, 3, 4, 5; half-width parameters for all phases (U, V, W); atomic site occupancies of Ca and Na in plagioclase (phase 1); atomic coordinates of phase 1; atomic site occupancies of Mg and Fe in clinopyroxene (phase 2); atomic coordinates of phase 2; preferred orientation correction for phases 1, 2 and 3.

Plagioclase-Clinopyroxene Skarn
This skarn is present close to monzonitic rocks in contact with the skarn xenoliths. The rock is medium-grained and contains plagioclase and clinopyroxene (Figure 16a) with minor or secondary calcite, quartz, prehnite, and chlorite. Accessory minerals include titanite, apatite and magnetite.
Usually the primary plagioclase is partially replaced by albite or prehnite (Figure 16b-d). The mean plagioclase composition measured by EPMA is An 60 . The clinopyroxene in this zone belongs to the diopside-hedenbergite series (Di 48-73 Hd 25-49 Jhn 0-2 ) and forms zoned crystals with Mg-rich cores and Fe-rich rims (Figure 16c,d). Prehnite occurs in interstitial positions or replaces the earlier formed plagioclase.  The mineral composition of Pl-Cpx skarn determined by XRPD is shown in Figure 17. The XRPD Rietveld refinements include five phases: plagioclase, clinopyroxene, calcite, quartz, and chlorite. Based on optical and EPMA (Tables A3 and A5) studies, the plagioclase is modeled as labradorite and clinopyroxene as diopside-hedenbergite. The minor phases are present only at the 0-2 vol.% level and could not be detected in the XRPD pattern. The obtained mean plagioclase composition is in agreement both with the mean of the optical determinations and with those measured by EPMA. The sequence of refined parameters is the following: scale factors for plagioclase, clinopyroxene, calcite, quartz, chlorite (phases 1, 2, 3, 4, 5); zero shift, background polynomial parameters (22 coefficients); unit cell parameters for phases 1, 2, 3, 4, 5; half-width parameters for all phases (U, V, W); atomic site occupancies of Ca and Na in plagioclase (phase 1); atomic coordinates of phase 1; atomic site occupancies of Mg and Fe in clinopyroxene (phase 2); atomic coordinates of phase 2; preferred orientation correction for phases 1, 2 and 3.

Discussion
The quantitative analysis of the studied skarn samples showed that the Rietveld method combined with optical and EPMA studies is a suitable approach for quantification of mineral concentrations. This combination of methods was applied in previous studies to improve the quantitative determination of minerals [18][19][20]24,25].
The studied skarn rocks are inhomogeneous, locally very fine-zoned, containing mineral assemblages from different stages of the skarn processes. In addition, they consist of areas with significant differences in grain size-from fine-to very coarse-grained. The presence of minerals with highly variable mineralogical compositions and chemically zoned crystals further complicate the determination of phase quantities. In this case, the estimation of mineral content by routine methods such as point-counting in thin sections or whole-rock analyses are inappropriate.
When developing a Rietveld XRPD quantification method, a knowledge of mineralogy and crystal structure is important factor for the starting set of the phases. The refined crystal-structure parameters included the unit-cell parameters of each phase and element substitutions with pronounced influences on intensities and positions of the reflections. Atomic positions and site occupancies can be successfully varied for major phases to obtain accurate scale factors. For instance, the Ca 2+ and Na + occupancy in plagioclase structure; the Ca 2+ , Mg 2+ , Fe 2+ , Al 3+ Fe 3+ occupancy of the octahedral and tetrahedral positions in the clinopyroxene structure; the Ca 2+ , Fe 2+ , Fe 3+ , Al 3+ , and Ti 4+ occupancy of the eight-coordinated dodecahedral, 6-coordinated octahedral, and tetrahedral positions in garnet structure; the Mg 2+ and Fe 2+ occupancy of the octahedral positions in the chlorite structure; the Fe 3+ and Al 3+ occupancy in M3 octahedral position in epidote structure.
The results of some refined parameters of the minerals from the studied skarn samples are presented in Tables 2-7. The derived garnet compositions of grossular and grossular-andradite of the garnet skarn by Rietveld XRPD method are in good agreement with the data of the EPMA measurements (Tables 2 and A1).
The refined unit cell parameters of the "fassaitic" type clinopyroxene from the clinopyroxenegarnet skarn are very close to those in the starting model [48] and the compositions of the garnets and clinopyroxene are in accordance with the EPMA data (Tables 3, A1 and A2).
The comparison of the XRPD results with the EPMA analyses of the garnet, clinopyroxene and plagioclase compositions from Pl-Cpx-Wo-Grt skarn (Tables 4, A1, A3 and A5) shows good agreement, given the presence of diopside-hedenbergite and grossular-andradite solid solutions and albite, respectively. The refined unit cell parameters of the wollastonite−2M for all samples are very close to those reported by Hesse [49] (Tables 4-6). Table 5 presents refined parameters of the minerals from the Pl-Cpx-Wo skarn. The average plagioclase composition of An40 derived by XRPD is slightly deficient in Ca relative to the mean value obtained by EPMA (Table A5) but the pyroxene composition is well determined by XRPD, given the presence of cation isomorphic substitution in M1 site in the crystal structure (Tables 5 and A3). The

Discussion
The quantitative analysis of the studied skarn samples showed that the Rietveld method combined with optical and EPMA studies is a suitable approach for quantification of mineral concentrations. This combination of methods was applied in previous studies to improve the quantitative determination of minerals [18][19][20]24,25].
The studied skarn rocks are inhomogeneous, locally very fine-zoned, containing mineral assemblages from different stages of the skarn processes. In addition, they consist of areas with significant differences in grain size-from fine-to very coarse-grained. The presence of minerals with highly variable mineralogical compositions and chemically zoned crystals further complicate the determination of phase quantities. In this case, the estimation of mineral content by routine methods such as point-counting in thin sections or whole-rock analyses are inappropriate.
When developing a Rietveld XRPD quantification method, a knowledge of mineralogy and crystal structure is important factor for the starting set of the phases. The refined crystal-structure parameters included the unit-cell parameters of each phase and element substitutions with pronounced influences on intensities and positions of the reflections. Atomic positions and site occupancies can be successfully varied for major phases to obtain accurate scale factors. For instance, the Ca 2+ and Na + occupancy in plagioclase structure; the Ca 2+ , Mg 2+ , Fe 2+ , Al 3+ Fe 3+ occupancy of the octahedral and tetrahedral positions in the clinopyroxene structure; the Ca 2+ , Fe 2+ , Fe 3+ , Al 3+ , and Ti 4+ occupancy of the eight-coordinated dodecahedral, 6-coordinated octahedral, and tetrahedral positions in garnet structure; the Mg 2+ and Fe 2+ occupancy of the octahedral positions in the chlorite structure; the Fe 3+ and Al 3+ occupancy in M3 octahedral position in epidote structure.
The results of some refined parameters of the minerals from the studied skarn samples are presented in Tables 2-7. The derived garnet compositions of grossular and grossular-andradite of the garnet skarn by Rietveld XRPD method are in good agreement with the data of the EPMA measurements (Tables 2 and A1).
The refined unit cell parameters of the "fassaitic" type clinopyroxene from the clinopyroxenegarnet skarn are very close to those in the starting model [48] and the compositions of the garnets and clinopyroxene are in accordance with the EPMA data (Tables 3, A1 and A2).
The comparison of the XRPD results with the EPMA analyses of the garnet, clinopyroxene and plagioclase compositions from Pl-Cpx-Wo-Grt skarn (Tables 4, A1, A3 and A5) shows good agreement, given the presence of diopside-hedenbergite and grossular-andradite solid solutions and albite, respectively. The refined unit cell parameters of the wollastonite−2M for all samples are very close to those reported by Hesse [49] (Tables 4-6). Table 5 presents refined parameters of the minerals from the Pl-Cpx-Wo skarn. The average plagioclase composition of An 40 derived by XRPD is slightly deficient in Ca relative to the mean value obtained by EPMA (Table A5) but the pyroxene composition is well determined by XRPD, given the presence of cation isomorphic substitution in M1 site in the crystal structure (Tables 5 and A3). The refined unit cell parameters of prehnite in the Pl-Cpx-Wo and Pl-Cpx-Wo-Ep skarns are close to those in the starting model [51]. Table 2. Some refined parameters in the quantitative analysis of the garnet skarn and chemical compositions of garnets determined by X-ray powder diffraction (XRPD) and electron probe microanalysis (EPMA).  Table 3. Some refined parameters in the quantitative analysis of the clinopyroxene-garnet skarn and chemical compositions of the minerals determined by XRPD and EPMA.  The refined crystal chemical parameters of the minerals from the Pl-Cpx-Wo-Ep skarn are shown in Table 6. The derived by XRPD mean plagioclase composition after refinement is in agreement both with the mean of the optical determinations and with those measured by EPMA (Table A5). The compositions of the clinopyroxene and the epidote are in good accordance with those determined by EPMA analyses (Tables A3 and A6).

Minerals Unit Cell Parameters (Å) Chemical Compositions
The results for Pl-Cpx skarn after refinements are presented in Table 7. The variation of the Na/Ca content of the plagioclase during the XRPD refinement provided a composition of~An 50 , in agreement with optical and EPMA data (Table A5). The pyroxene composition is in agreement with the EPMA results (Table A5). Table 6. Some refined parameters in the quantitative analysis of the Pl-Cpx-Wo-Ep skarn and chemical compositions of the minerals determined by XRPD and EPMA.  The Rietveld refinement plots (Figures 3, 6 , 9, 12, 15 and 18) show the observed and calculated patterns for the studied samples. All of the refinements are acceptable, as can be judged by the standard indices of agreement and by the visual fits of the observed and calculated diffraction profiles, considering the complexity of the pattern and the large number of reflections.

Minerals
Each sample studied presents unique difficulties, as a result of its particular assemblages of minerals but the results show that the Rietveld XRPD method can be confidently applied to the wide range of rock types and phases under consideration here.

Conclusions
In the present study, we have shown that accurate mineral quantification may be obtained by the Rietveld XRPD method from inhomogeneous fine-zoned skarn rocks. The method is equally applicable to fine-grained and coarse-grained rocks, and is useful in distinguishing between the components in solid solutions phases.
The samples studied contain minerals with highly variable chemical compositions and involve up to eight major phases, causing significant overlap of peaks in the XRPD patterns. We demonstrate that the Rietveld XRPD method was able to refine these complex diffraction patterns. All of the refinements are acceptable, as can be judged by the standard indices of agreement and by the visual fit of the observed and calculated diffraction profiles.
Summarizing the discussed quantitative approach, we may claim that the considered technique is a serious tool for accurate and time saving quantitative determination of mineral contents in complex geological materials that may give important mineralogical information leading to additional genetic conclusions.
Appendix A Table A1. Representative EPMA analyses and structural formulae of garnets from skarn rocks in the Zvezdel-Pcheloyad ore deposit (Fe 2+ and Fe 3+ according to the charge balance).