Subsolidus Evolution of the Magnetite-Spinel-UlvöSpinel Solid Solutions in the Kovdor Phoscorite-Carbonatite Complex, NW Russia

: The Kovdor phoscorite-carbonatite ore-pipe rocks form a natural series, where apatite and magnetite ﬁrst gradually increase due to the presence of earlier crystallizing forsterite in the pipe marginal zone and then decrease as a result of carbonate development in the axial zone. In all lithologies, magnetite grains contain (oxy)exsolution inclusions of comparatively earlier ilmenite group minerals and/or later spinel, and their relationship reﬂects the concentric zonation of the pipe. The temperature and oxygen fugacity of titanomagnetite oxy-exsolution decreases in the natural rock sequence from about 500 ◦ C to about 300 ◦ C and from NNO + 1 to NNO − 3 (NNO is Ni-NiO oxygen fugacity buffer), with a secondary positive maximum for vein calcite carbonatite. Exsolution spinel forms spherical grains, octahedral crystals, six-beam and eight-beam skeletal crystals co-oriented with host magnetite. The ilmenite group minerals occur as lamellae oriented along {111} and {100} planes of oxy-exsolved magnetite. The kinetics of inclusion growth depends mainly on the diffusivity of cations in magnetite: their comparatively low diffusivities in phoscorite and carbonatites of the ore-pipe internal part cause size-independent growth of exsolution inclusions; while higher diffusivities of cations in surrounding rocks, marginal forsterite-rich phoscorite and vein calcite carbonatite result in size-dependent growth of inclusions.

The results of experimental and theoretical studies [4,5,38,[43][44][45][46][47][48][49][50] allowed us to estimate the equilibration temperature and oxygen fugacity of coexisting magnetite and its exsolution products based on their chemical composition. Unfortunately, we can perform only a very approximate estimation of PT-conditions of Al-rich magnetite exsolutions based on the models of magnetite-spinel miscibility [5,45]; however, Fe-Ti oxide geothermobarometers are much more informative. Most realistic results can be obtained using the model [49] calibrated with the cation-ordering data [51,52], and experimental data on Fe 2+ Ti ↔ 2Fe 3+ exchange between minerals of the ilmenite and spinel groups [26,53]. The model [44] can also be used for comparison.
The main objectives of the paper include studies of: (i) exsolution conditions (temperature, oxygen fugacity, cation diffusivities, etc.) of magnetite-spinel-ulvöspinel solid solution; and (ii) their dependence on rock type and spatial position within the Kovdor phoscorite-carbonatite pipe. These objectives were achieved by development and analysis of 3D-models of compositions, grain size and thermodynamic parameters of exsolution products formation.

Geological Setting
The Kovdor alkali-ultrabasic massif is situated in the SW part of the Murmansk Region, Russia (Figure 1a). It is a central-type multiphase volcano-plutonic complex of peridotite, rimmed by foidolite, melilitolite and related metasomatic rocks (diopsidite, phlogopitite, skarn-like rock and fenite) that intruded into Archean granite-gneiss 380 Ma ago. At the western contact of the diopsidized/phlogopitized peridotite core with a foidolite-melilitolite rim, a concentrically zoned phoscorite-carbonatite ore-pipe (0.8 × 1.3 × > 6 km) intrudes into the massif, forming several explosive funnels (up to 100 m in diameter) on the present-day surface (Figure 1b). Due to a steep dip of the ore-pipe wall towards its axis located in the center, the pipe diameter reduced by 40% at 600 m depth [16,54,55].
The rocks of the Kovdor phoscorite-carbonatite ore-pipe form a natural series [16,40,56], where content of apatite and magnetite first gradually increase at the expense of earliest forsterite and then decrease as a result of carbonate development in the axial zone ( Figure 2). Within this series, the rocks with carbonate content over 50 modal% are called "forsterite-related carbonatite", and the remainder of the magnetite-forsterite-apatite-carbonate rocks are called "phoscorite". Phoscorite varieties were designated according to the main minerals [16]: C, carbonate (mainly calcite); M, magnetite; A, apatite; and F, forsterite (the respective abbreviator of a rock-forming mineral is included in denomination of a rock if the content of this mineral exceeds 10 modal %), e.g., AF (apatite-forsterite phoscorite) etc.
It is obvious that magnetite composition determines both the modal and chemical composition of (oxy)exsolution inclusions. This fact enables us to complement the schemas of spatial distribution of the ilmenite-group minerals and spinel (see Figure 2) with similar schemas of their chemical composition and temperature of magnetite (oxy)exsolution. In addition, the shape and grain size of the inclusions will be estimated and correlated with temporal/spatial distribution of (oxy)exsolution temperature and oxygen fugacity. As was shown in [16,40], magnesium is the main subordinate component in magnetite of the Kovdor phoscorite-carbonatite complex. There is a minor increase in MgO in magnetite from earlier (apatite)-forsterite phoscorite (4 ± 2 wt %) to intermediate low-carbonate magnetite-rich phoscorite (5 ± 2 wt %), and then MgO in magnetite decreases in late carbonate-rich phoscorite and carbonatite (4 ± 2 wt %). Magnetite with a higher Al 2 O 3 content occurs in the intermediate low-carbonate magnetite-rich phoscorite (0.7 ± 0.5 wt %), while earlier (apatite)-forsterite phoscorite and late carbonate-rich phoscorite and carbonatite contain magnetite with lower content of Al 2 O 3 (0.5 ± 0.5 and 0.6 ± 0.9 wt % correspondingly). The TiO 2 content decreases from earlier (apatite)-forsterite phoscorite (2 ± 1 wt %) to intermediate low-carbonate magnetite-rich phoscorite (1.0 ± 0.6 wt %), and then slightly increases in late carbonate-rich phoscorite and carbonatite (1.1 ± 0.9 wt %). MnO content in magnetite is constant through all the pipe volume (0.5 ± 0.2 wt %).
It is obvious that magnetite composition determines both the modal and chemical composition of (oxy)exsolution inclusions. This fact enables us to complement the schemas of spatial distribution of the ilmenite-group minerals and spinel (see Figure 2) with similar schemas of their chemical composition and temperature of magnetite (oxy)exsolution. In addition, the shape and grain size of the inclusions will be estimated and correlated with temporal/spatial distribution of (oxy)exsolution temperature and oxygen fugacity.

Materials and Methods
For this study, we used 540 polished thin sections of phoscorite (mainly), carbonatites and host rocks from 108 exploration holes drilled within the Kovdor phoscorite-carbonatite ore-pipe [16]. The polished thin sections were analyzed using the scanning electron microscope LEO-1450 (Carl Zeiss Microscopy, Oberkochen, Germany) with an energy-dispersive X-ray analytical attachment (EDS) Röntek to obtain back-scattered electron (BSE) images of important regions and pre-analyze all minerals found in the samples under analysis. The Image Tool 3.04 (The University of Texas Health Science Center, San Antonio, TX, USA) was used to generate digital images from the BSE-images, and determine inclusions/magnetite area ratios and inclusions size (equivalent circular diameter of spinel grains, and thickness of ilmenite-geikielite lamellae).
The chemical composition of magnetite-magnesioferrite crystals and (oxy)exsolution inclusions (>10 µm thick/diameter) was determined using the electron probe microanalyzer (EPMA) Cameca MS-46 (Cameca, Gennevilliers, France) operating in wavelength-dispersive mode at 20 kV and 20-30 nA. Grains were analyzed using the beam size of 5 µm and the counting time of about 20 and 10 s on peaks and background respectively. The standards used, and limits of accuracy are given in Table 3. Cation and end-member contents were calculated using the MINAL program of Dmitry D. Dolivo-Dobrovolsky [58]. Equilibrium temperatures and oxygen fugacities of magnetite exsolution were estimated using Fe-Ti two-oxide geothermometers and oxygen-barometers of Andersen and Lindsley [44] and Ghiorso and Evans [49], with corresponding interactive programs ILMAT [59] and MELTS [60]. Statistical analyses were carried out using the STATIATICA 8.0 (StatSoft) and TableCurve 2.0 (The University of Texas Health Science Center, San Antonio, TX, USA) programs. For the statistics, resulting values of the analyses below the limit of accuracy (see Table 3) were considered to be ten times lower than the limit. Geostatistical studies and 3D modeling were conducted with the MICROMINE 16 program. Interpolation was performed with ordinary kriging. The automatic 3D geological mapping (see Figure 2b) was performed by means of chemistry-to-mineral conversion [61].

Spinel
According to the earlier evidence of Rimskaya-Korsakova [41], exsolution spinel forms spherical grains (up to 20 µm in diameter), well-shaped octahedral crystals (up to 200 µm in diameter) as well as six-beam (on [100]) and eight-beam (on [111]) skeletal crystals (up to 300 µm in diameter) structurally co-oriented with host magnetite crystals. In different sections of host magnetite crystals (Figure 4), these inclusions appear as circles (any sections of magnetite crystals); squares and crosses (on (100) planes); triangles, hexagons and three-beam stars (on (111) planes); rhombs, hexagons, crosses and lamellae (on (110) planes). The content of spinel inclusions in magnetite reaches 27 modal %, and their median content in spinel-containing grains is 9 modal %. Spinel inclusions are mainly concentrated in the cores of magnetite crystals, where their size increases significantly (see Figure 3). Comparatively large crystals of spinel (>20 µm in diameter) are usually rimmed by an inclusion-free magnetite aureole of twice the diameter (see Figure 4).
(on (100) planes); triangles, hexagons and three-beam stars (on (111) planes); rhombs, hexagons, crosses and lamellae (on (110) planes). The content of spinel inclusions in magnetite reaches 27 modal %, and their median content in spinel-containing grains is 9 modal %. Spinel inclusions are mainly concentrated in the cores of magnetite crystals, where their size increases significantly (see Figure 3). Comparatively large crystals of spinel (>20 µm in diameter) are usually rimmed by an inclusion-free magnetite aureole of twice the diameter (see Figure 4).
Spinel crystals commonly contain prismatic inclusions of baddeleyite as a co-product of magnetite exsolution, and sometimes most of the spinel crystals carry such inclusions. During the last stages of magnetite exsolution in the carbonate-rich rocks, quintinite-2H formed after, or instead of, spinel [   Spinel crystals commonly contain prismatic inclusions of baddeleyite as a co-product of magnetite exsolution, and sometimes most of the spinel crystals carry such inclusions. During the last stages of magnetite exsolution in the carbonate-rich rocks, quintinite-2H formed after, or instead of, spinel [ The distribution of the grain size of spinel inclusions in magnetite can be of two types ( Figure 5): (1) negative-exponential distributions (about 30% of investigated specimens), when cumulative frequencies are concave down in log-log space and linear in semilog space; and (2) power-law distributions (about 70% of investigated specimens), when cumulative frequencies are linear in log-log space and concave up in semilog space. Within different magnetite grains in the same specimen, the size of inclusions is distributed according to the same law. The simplified explanation of the difference between these types of distributions of spinel grain size is that the negative-exponential distribution reflects size-independent crystal growth, and the power-law distribution results from a positive-feedback process in which the time-averaged crystal growth rate is proportional to the crystal size [63][64][65]. Size-independent growth of spinel occurs in about 30% of low-carbonate magnetite-rich phosphorite of the intermediate zone, and carbonate-rich phoscorite and phoscorite-related carbonatite of the ore-pipe axial zone; however, the size-dependent growth of spinel grains is dominant everywhere ( Figure 6). The distribution of the grain size of spinel inclusions in magnetite can be of two types ( Figure 5): (1) negative-exponential distributions (about 30% of investigated specimens), when cumulative frequencies are concave down in log-log space and linear in semilog space; and (2) power-law distributions (about 70% of investigated specimens), when cumulative frequencies are linear in log-log space and concave up in semilog space. Within different magnetite grains in the same specimen, the size of inclusions is distributed according to the same law. The simplified explanation of the difference between these types of distributions of spinel grain size is that the negative-exponential distribution reflects size-independent crystal growth, and the power-law distribution results from a positive-feedback process in which the time-averaged crystal growth rate is proportional to the crystal size [63][64][65]. Size-independent growth of spinel occurs in about 30% of low-carbonate magnetite-rich phosphorite of the intermediate zone, and carbonate-rich phoscorite and phoscorite-related carbonatite of the ore-pipe axial zone; however, the size-dependent growth of spinel grains is dominant everywhere (Figure 6). The typical size of spinel inclusions Dchar (average equivalent circular diameter of a grain fraction with the largest summary area [40,66]) usually lies within the interval of 1-4 µm; however, in size-dependent sets, an additional maximum appears at 12-50 µm (Figure 7). For this reason, Dchar has a bimodal distribution with maxima at these intervals.   Typical chemical compositions of magnetite-spinel pairs, selected on the basis of median contents of Mg in spinel, is presented in Table 4. Spinel contains insignificant amounts of chemical impurities, and, on average, its composition corresponds to the formula (Mg0.88Fe 2+ 0.10 Zn0.02)Σ1.00(Al1.86Fe 3+ 0.13 Ti0.01)Σ2.00O4 [40]. The main compositional impurity is iron that replaces both Mg and Al during exsolution of Mg-Al-rich magnetite (Figure 8a). This process is also accompanied by the differentiation of Mn that remains in the magnetite structure (Figure 8b), and Zn that goes from magnetite into spinel. As a result, in the forsterite-rich phoscorite of marginal zone of the ore-pipe, the exsolved spinel is comparatively enriched both in Fe 2+ and Fe 3+ , and contains higher amounts of Fe 3+ in the carbonate-rich phoscorite and carbonatite of the axial zone (compare Figures 9 and 2). The typical size of spinel inclusions D char (average equivalent circular diameter of a grain fraction with the largest summary area [40,66]) usually lies within the interval of 1-4 µm; however, in size-dependent sets, an additional maximum appears at 12-50 µm (Figure 7). For this reason, D char has a bimodal distribution with maxima at these intervals.  Typical chemical compositions of magnetite-spinel pairs, selected on the basis of median contents of Mg in spinel, is presented in Table 4. Spinel contains insignificant amounts of chemical impurities, and, on average, its composition corresponds to the formula (Mg0.88Fe 2+ 0.10 Zn0.02)Σ1.00(Al1.86Fe 3+ 0.13 Ti0.01)Σ2.00O4 [40]. The main compositional impurity is iron that replaces both Mg and Al during exsolution of Mg-Al-rich magnetite (Figure 8a). This process is also accompanied by the differentiation of Mn that remains in the magnetite structure (Figure 8b), and Zn that goes from magnetite into spinel. As a result, in the forsterite-rich phoscorite of marginal zone of the ore-pipe, the exsolved spinel is comparatively enriched both in Fe 2+ and Fe 3+ , and contains higher amounts of Fe 3+ in the carbonate-rich phoscorite and carbonatite of the axial zone (compare Figures 9 and 2).   [40]. The main compositional impurity is iron that replaces both Mg and Al during exsolution of Mg-Al-rich magnetite (Figure 8a). This process is also accompanied by the differentiation of Mn that remains in the magnetite structure (Figure 8b), and Zn that goes from magnetite into spinel. As a result, in the forsterite-rich phoscorite of marginal zone of the ore-pipe, the exsolved spinel is comparatively enriched both in Fe 2+ and Fe 3+ , and contains higher amounts of Fe 3+ in the carbonate-rich phoscorite and carbonatite of the axial zone (compare Figures 2 and 9). gaps and equilibrium tie lines in the system of spinels-(Fe 2+ , Mg)Cl2 aqueous solution at 800 °C and 4 kbars [45]. Solid tie lines, connecting compositions of coexisting magnetite and spinel, are nearly parallel to dotted tie lines calculated with the Lehmann-Roux model [45]. The orientation of the tie lines in this model is almost independent of temperature; but pairs of low-temperature origin have much lower mutual solubility. This means that exsolution of Mg-Al-rich magnetite mostly took place at temperatures much lower than 800 °C, and only a few specimens of magnetite-bearing carbonatite showed temperatures of magnetite exsolution between 800 °C and 1000 °C.    The diagram in Figure 8a shows compositions of spinel-magnetite pairs in relation to miscibility gaps and equilibrium tie lines in the system of spinels-(Fe 2+ , Mg)Cl 2 aqueous solution at 800 • C and 4 kbars [45]. Solid tie lines, connecting compositions of coexisting magnetite and spinel, are nearly parallel to dotted tie lines calculated with the Lehmann-Roux model [45]. The orientation of the tie lines in this model is almost independent of temperature; but pairs of low-temperature origin have much lower mutual solubility. This means that exsolution of Mg-Al-rich magnetite mostly took place at temperatures much lower than 800 • C, and only a few specimens of magnetite-bearing carbonatite showed temperatures of magnetite exsolution between 800 • C and 1000 • C.

Ilmenite Group Minerals
Ilmenite, geikielite and pyrophanite (generic term ilmenite) usually occur as thin (up to 50 µm thick) lamellae oriented along the {100} and {111} planes of host magnetite [40,42,54,67], and form characteristic trellises on all other planes (see Figure 3b,c). Lamellae oriented along the {111} planes of magnetite are predominant and originate from reaction (2), while their orientation along the {100} planes of magnetite occurs much rarer as a probable result of consecutive reactions (1) and (3) [39,68]. The cores of magnetite crystals contain comparatively thick ilmenite lamellae, while outer zones are often free of inclusions or include the thinnest lamellae of "ilmenite". The content of "ilmenite" inclusions in magnetite reaches 26 vol %, and their median content in "ilmenite"-containing grains of magnetite is 10 vol %. Sometimes, ilmenite lamellae contain co-oriented inclusions of baddeleyite, and coexist with quintinite in magnetite crystals with spinel-impregnated cores and ilmenite-quintinite bearing marginal zones of titanomagnetite grains.
The distribution of ilmenite lamellae thickness is similar to the distribution of spinel grain diameter ( Figure 10): power-law distributions are predominant and negative-exponential distributions occur much more rarely (correspondingly, about 80% and 20% of investigated samples). Just as with spinel, size-independent growth of ilmenite lamellae prevails, and size-dependent growth occurs in magnetite-rich phoscorite and phoscorite-related carbonatite of the ore-pipe intermediate and axial zones, as well as in the latest magnetite-dolomite-(phlogopite)-serpentine rock and vein dolomite carbonatite (see Figure 6). Comparatively large ilmenite lamellae are always rimmed by an aureole of inclusion-free magnetite (see Figure 10). diameter ( Figure 10): power-law distributions are predominant and negative-exponential distributions occur much more rarely (correspondingly, about 80% and 20% of investigated samples). Just as with spinel, size-independent growth of ilmenite lamellae prevails, and size-dependent growth occurs in magnetite-rich phoscorite and phoscorite-related carbonatite of the ore-pipe intermediate and axial zones, as well as in the latest magnetite-dolomite-(phlogopite)-serpentine rock and vein dolomite carbonatite (see Figure 6). Comparatively large ilmenite lamellae are always rimmed by an aureole of inclusion-free magnetite (see Figure 10). The representative compositions of ilmenite-magnetite pairs from different rocks of the Kovdor massif are shown in Table 5 in accordance with the median content of Ti in ilmenite. On the ternary diagram (Fe, Mg, Mn)O-(Fe, Al)2O3-TiO2 (Figure 11a), compositions of co-existing magnetite and oxy-exsolution ilmenite are located along the magnetite-ulvöspinel (titanomagnetite) and hematiteilmenite (titanohematite) lines, correspondingly. As for divalent cation relations, the ilmenite The representative compositions of ilmenite-magnetite pairs from different rocks of the Kovdor massif are shown in Table 5 in accordance with the median content of Ti in ilmenite. On the ternary diagram (Fe, Mg, Mn)O-(Fe, Al) 2 O 3 -TiO 2 (Figure 11a), compositions of co-existing magnetite and oxy-exsolution ilmenite are located along the magnetite-ulvöspinel (titanomagnetite) and hematite-ilmenite (titanohematite) lines, correspondingly. As for divalent cation relations, the ilmenite compositions range widely from ilmenite-poor to geikielite-and pyrophanite-poor, and correspond to average formula of Mg-rich ilmenite: ( [40]. Fractionation of Mg and Mn between coexisting titanomagnetite and ilmenite (Figure 11b) has resulted in significant losses of Mg in host magnetite in comparison with fresh volcanic rocks [47] due to spinel exsolution following ilmenite oxy-exsolution [39,40].
As a result, the spatial distribution of ilmenite composition reflects the total zonation of the Kovdor phoscorite-carbonatite pipe (compare Figures 2 and 12). In particular, higher content of Fe 2+ characterizes oxy-exsolution ilmenite from low-carbonate magnetite-rich phoscorite of the ore-pipe intermediate zone, and from neighboring silicate rocks. Conversely, oxy-exsolution ilmenite with higher content of Mg (up to pure geikielite) is spread over the forsterite-rich marginal zone and carbonate-rich axial zone of the ore-pipe. Manganese-rich oxy-exsolution ilmenite (up to pyrophanite) occurs in marginal forsterite-rich phoscorite, and ilmenite with higher content of Fe 3+ (up to Ti-rich hematite) is spread over (apatite)-forsterite phoscorite of the ore-pipe marginal zone, carbonate-rich phoscorite and carbonatite of the axial zone, and neighboring diopsidite.   compositions range widely from ilmenite-poor to geikielite-and pyrophanite-poor, and correspond to average formula of Mg-rich ilmenite: (Fe 2+ 0.50Mg0.36Mn0.13)Σ0.99(Ti0.96Fe 3+ 0.05Nb0.01)Σ1.02O3 [40]. Fractionation of Mg and Mn between coexisting titanomagnetite and ilmenite (Figure 11b) has resulted in significant losses of Mg in host magnetite in comparison with fresh volcanic rocks [47] due to spinel exsolution following ilmenite oxy-exsolution [39,40]. As a result, the spatial distribution of ilmenite composition reflects the total zonation of the Kovdor phoscorite-carbonatite pipe (compare Figures 12 and 2). In particular, higher content of Fe 2+ characterizes oxy-exsolution ilmenite from low-carbonate magnetite-rich phoscorite of the ore-pipe intermediate zone, and from neighboring silicate rocks. Conversely, oxy-exsolution ilmenite with higher content of Mg (up to pure geikielite) is spread over the forsterite-rich marginal zone and carbonate-rich axial zone of the ore-pipe. Manganese-rich oxy-exsolution ilmenite (up to pyrophanite) occurs in marginal forsterite-rich phoscorite, and ilmenite with higher content of Fe 3+ (up to Ti-rich hematite) is spread over (apatite)-forsterite phoscorite of the ore-pipe marginal zone, carbonate-rich phoscorite and carbonatite of the axial zone, and neighboring diopsidite.
Temperatures of titanomagnetite oxy-exsolution, T1, and oxygen fugacities, log  (Figure 13a). However, as was shown [49], this model often gives a temperature and oxygen fugacity that is too high, especially when the oxidation state is estimated under relatively oxidized conditions.  Temperatures of titanomagnetite oxy-exsolution, T 1 , and oxygen fugacities, log f O 2 , for 371 ilmenite-magnetite pairs (177 samples) were determined first using the Fe 2+ Ti-Fe 3+ 2 exchange geothermometer/oxometer of Andersen and Lindsley [44], and the model of Stormer [69] for calculation based on molecular fractions. Obtained values of log f O 2 and T 1 increase from −55 at 246 • C to −9 at 1043 • C in accordance with the Ni-NiO oxygen fugacity buffer (Figure 13a). However, as was shown [49], this model often gives a temperature and oxygen fugacity that is too high, especially when the oxidation state is estimated under relatively oxidized conditions. For this reason, more accurate estimation of equilibration temperature, T2, and oxygen fugacity deviation from the Ni-NiO oxygen fugacity buffer at 200 MPa, ΔNNO, was performed on 94 samples using the Fe 2+ Ti-Fe 3+ 2 exchange geothermometer/oxometer of Ghiorso and Evans [49]. This produced estimated values in the range from 230 °C to 756 °C, and from NNO − 6.4 to NNO + 3 (Figure 13b), with difference between T1 and T2 up to 200 °C. Unfortunately, almost half of ilmenite-magnetite pairs cannot be estimated with the last model due to unsuitable chemical composition (higher content of Mn, Nb, Sc, etc.), which markedly constrains perspectives of 3D modeling. Nevertheless, there are good regressions between temperatures and oxygen fugacities obtained with both geothermometers/oxometers ( Figure 14) that enables to estimate equilibration temperatures and oxygen fugacity values for the rest of the 83 samples using the corresponding equations: T2' ≈ 198.46 + 3.58 exp(T1/153.61); (5) ΔNNO2' ≈ -1.24 + 0.80 ΔNNO1 Figure 14. Relations between equilibration temperatures and the oxygen fugacity values calculated using magnetite-ilmenite geothermometers of Andersen and Lindsley [44] and Ghiorso and Evans [49]. Dashed lines limit 95% prediction intervals. For this reason, more accurate estimation of equilibration temperature, T 2 , and oxygen fugacity deviation from the Ni-NiO oxygen fugacity buffer at 200 MPa, ∆NNO, was performed on 94 samples using the Fe 2+ Ti-Fe 3+ 2 exchange geothermometer/oxometer of Ghiorso and Evans [49]. This produced estimated values in the range from 230 • C to 756 • C, and from NNO − 6.4 to NNO + 3 (Figure 13b), with difference between T 1 and T 2 up to 200 • C. Unfortunately, almost half of ilmenite-magnetite pairs cannot be estimated with the last model due to unsuitable chemical composition (higher content of Mn, Nb, Sc, etc.), which markedly constrains perspectives of 3D modeling. Nevertheless, there are good regressions between temperatures and oxygen fugacities obtained with both geothermometers/oxometers ( Figure 14) that enables to estimate equilibration temperatures and oxygen fugacity values for the rest of the 83 samples using the corresponding equations: Values of T 2 ' and ∆NNO' obtained this way were added to T 2 and ∆NNO values, and then the results were used for statistics and 3D modeling. In the natural sequence of the Kovdor's rock formation, the oxygen fugacity and temperature of titanomagnetite oxy-exsolution sharply decreased ( Figure 15) from host foidolite (on average, NNO + 0.7 at 517 • C) to diopsidite and phlogopitite (NNO − 0.8 at 388 • C), (apatite)-forsterite phoscorite (NNO − 1.4 at 347 • C) and low-carbonate magnetite-rich phoscorite (NNO − 1.8 at 337 • C); then again the values increased in carbonate-rich phoscorite, phoscorite-related carbonatite (NNO − 0.6 at 343 • C) and vein calcite carbonatite (NNO − 0.9 at 361 • C); and decreased to a minimum (NNO − 1.9 at 316 • C) in magnetite-dolomite-(phlogopite)-serpentine rock and vein dolomite carbonatite. It is necessary to note that application of geothermometers-geooxometers to magnetite-ilmenite pairs in intrusive rocks does not correspond the parameters of the rocks formation, but rather reflects the closure of exchange of components between coexisting phases. estimated values in the range from 230 °C to 756 °C, and from NNO − 6.4 to NNO + 3 (Figure 13b), with difference between T1 and T2 up to 200 °C. Unfortunately, almost half of ilmenite-magnetite pairs cannot be estimated with the last model due to unsuitable chemical composition (higher content of Mn, Nb, Sc, etc.), which markedly constrains perspectives of 3D modeling. Nevertheless, there are good regressions between temperatures and oxygen fugacities obtained with both geothermometers/oxometers ( Figure 14) that enables to estimate equilibration temperatures and oxygen fugacity values for the rest of the 83 samples using the corresponding equations: T2' ≈ 198.46 + 3.58 exp(T1/153.61); (5) ΔNNO2' ≈ -1.24 + 0.80 ΔNNO1 Figure 14. Relations between equilibration temperatures and the oxygen fugacity values calculated using magnetite-ilmenite geothermometers of Andersen and Lindsley [44] and Ghiorso and Evans [49]. Dashed lines limit 95% prediction intervals.

Figure 14.
Relations between equilibration temperatures and the oxygen fugacity values calculated using magnetite-ilmenite geothermometers of Andersen and Lindsley [44] and Ghiorso and Evans [49]. Dashed lines limit 95% prediction intervals.  It was not a surprise when 3D-modeling showed a sharp decrease of both temperature and oxygen fugacity from host silicate rocks and (Ap)-Fo phoscorite of the ore-pipe marginal zone towards magnetite-rich phoscorite of the intermediate zone, with a secondary maximum in carbonate-rich phoscorite and carbonatite of the ore-pipe axial zone (compare Figures 16 and 2). It was not a surprise when 3D-modeling showed a sharp decrease of both temperature and oxygen fugacity from host silicate rocks and (Ap)-Fo phoscorite of the ore-pipe marginal zone towards magnetite-rich phoscorite of the intermediate zone, with a secondary maximum in carbonate-rich phoscorite and carbonatite of the ore-pipe axial zone (compare Figures 2 and 16).

Discussion
As was shown in [16,40,72], the properties and composition of all economic minerals, including magnetite, vary within the phoscorite-carbonatite complex following regular petrographic zonation of the ore-pipe. In particular, Mn-Ti-rich magnetite (with oxy-exsolution inclusions of ilmenite-pyrophanite) of the marginal (apatite)-forsterite phoscorite were replaced by Mg-Al-rich magnetite (with exsolution inclusions of spinel) of the intermediate low-carbonate magnetite-rich

Discussion
As was shown in [16,40,72], the properties and composition of all economic minerals, including magnetite, vary within the phoscorite-carbonatite complex following regular petrographic zonation of the ore-pipe. In particular, Mn-Ti-rich magnetite (with oxy-exsolution inclusions of ilmenite-pyrophanite) of the marginal (apatite)-forsterite phoscorite were replaced by Mg-Al-rich magnetite (with exsolution inclusions of spinel) of the intermediate low-carbonate magnetite-rich phoscorite, and then by Ti-V-rich magnetite (with oxy-exsolution inclusions of geikielite and Mg-rich ilmenite) of carbonate-rich phoscorite and carbonatite of the ore-pipe axial zone. New results allowed us to associate this zonation with thermodynamic conditions of the Kovdor phoscorite-carbonatite complex formation.
Since magnetite (oxy)exsolution is a subsolidus process, estimated temperatures of the magnetite-ilmenite equilibration can differ significantly from the temperatures of phoscorite-carbonatite melt crystallization. However, the estimated temperature sequence is well complemented by our results from calcite-dolomite geothermometry (using the formulation proposed by Anovitz and Essene [71]) that form an almost linear trend of temperature decrease from foidolite to dolomite carbonatite (see Figure 15). The figure shows that titanomagnetite oxy-exsolution occurs at a temperature that is lower than the temperature for the exsolution of carbonates by 250 • C.
It is generally recognized that intercrystalline diffusion, nucleation and growth of (oxy)exsolution inclusions are temperature-dependent processes. Therefore, the probability of critical nucleus formation is proportional to exp(−∆G*/kT), where ∆G* is a nucleation barrier corresponding to the critical radius of a nucleus, and k is the Boltzmann constant. The probability increases with temperature decrease because the latter causes an increase in the degree of supersaturation. However, when a stable nucleus is formed, its further growth is significantly constrained by the diffusivities of components [73,74].
The diffusion of cations through the magnetite structure depends on both temperature and oxygen fugacity [75][76][77]: where D 0 is the temperature independent diffusion coefficient, H the activation enthalpy for vacancy [V] and interstitial [I] regimes, f O 2 the oxygen fugacity in bars, R the gas constant and T the temperature in Kelvin. As it follows from this equation, a vacancy mechanism of diffusion prevails under oxidizing conditions (∆NNO > 0), and decreases with temperature increase; while interstitial diffusion occurs under reducing conditions (∆NNO < 0) and increases with temperature growth. Besides, the increase of vacancy concentration due to the substitution 2Fe 2+ ↔ V + Ti 4+ decreases the concentration of interstitial cations, and inhibits diffusion of divalent cations by the more "rapid" interstitial mechanism [78]. Estimation of the diffusion coefficients D* of Al and Ti in magnetite for the Kovdor phoscorite-carbonatite complex using this equation, with D 0 and H values given in [77], has shown ( Figure 17) that lower cation diffusivities occur in the rocks where size-independent growth of (oxy)exsolution inclusions was found (see Figure 6). Size-independent (constant) crystal growth occurs when equivalent faces on similar crystals grow at the same rate, i.e., increment of the crystal diameter dD is a constant k for all crystals, regardless of their size, in each time interval dt: dD/dt = k and D j+1 = D j + k j [79]. We believe that the constant growth of (oxy)exsolution inclusions in magnetite is caused by slower diffusion rates of cations, which is considered to be the main factor constraining the growth.
Size-dependent (proportional) crystal growth can be mathematically described by the equation D j+1 = D j + ε j D j , where ε j is a small random number within a narrow range, which differs for each crystal and for each growth cycle [79]. In this case, the increment of each crystal diameter dD in a certain time interval dt is proportional to the crystal initial diameter D: dD/dt = kD. There are four main explanations of the size-dependent crystal growth [79][80][81][82]. Firstly, due to the Gibbs-Thomson effect, equilibrium solubility of fine grains (<1 µm) decreases with their size growth, consequently, smaller grains will have lower supersaturation, and grow slower. Secondly, concerning the crystals coarser than 1 µm, the probability of dislocation occurrence increases with their surface growth, which in turn causes faster growth of such crystals. Thirdly, this can result from surface-controlled growth, and the volume of reactants during each cycle is actually unlimited. Fourthly, it occurs when crystals of the same size grow at different rates.
Although all these mechanisms can cause size-independent growth of (oxy)exsolution inclusions in magnetite, the first of them seems more important. It is necessary to note also that the removal of titanium from the magnetite matrix to a growing ilmenite inclusion causes interstitial diffusion increase, and thus accelerates inclusion growth. Besides, a growing inclusion increases stress in the magnetite matrix, which results in additional dislocations, and activates cation redistribution. Although all these mechanisms can cause size-independent growth of (oxy)exsolution inclusions in magnetite, the first of them seems more important. It is necessary to note also that the removal of titanium from the magnetite matrix to a growing ilmenite inclusion causes interstitial diffusion increase, and thus accelerates inclusion growth. Besides, a growing inclusion increases stress in the magnetite matrix, which results in additional dislocations, and activates cation redistribution.

Conclusions
Complex (oxy)exsolution of Mn-Mn-Al-Ti-rich magnetite in the Kovdor phoscorite-carbonatite pipe has formed concentric alternating zones of spinel-and ilmenite-impregnated magnetite within the pipe: (apatite)-forsterite phoscorite of the marginal zone and carbonate-rich phoscorite and carbonatite of the axial zone predominantly contain magnetite with exsolution lamellae of ilmenite-geikielite, while low-carbonate magnetite-rich phoscorite of the intermediate zone predominantly includes exsolved magnetite with spinel impregnation. Each of these zones (rock types) has certain features in terms of exsolution processes and products: (1) Exsolution spinel forms spherical grains, octahedral crystals, six-beam (along [100]) and eight-beam (along [111]) skeletal crystals co-oriented with host magnetite and having maximal morphological diversity in magnetite-rich phoscorites of the ore-pipe inner part. The ilmenite group minerals occur usually as thin lamellae on the (111) and (100) planes of host magnetite (respectively, due to direct oxy-exsolution of titanomagnetite and with intermediate ulvöspinel).
In accordance with the lower diffusivity of Al than Ti in studied magnetites, spinel crystallizes after the ilmenite-group minerals, which is emphasized by the formation of zoned magnetite crystals with spinel-impregnated core, ilmenite-impregnated intermediate zone and inclusion-free marginal zone; (2) The kinetics of inclusion nucleation and growth depends mainly on the diffusivity of cations in magnetite: comparatively low diffusivities of Al 3+ and Ti 4+ cations in magnetite-and/or carbonate-rich phoscorite and carbonatite cause size-independent growth of both spinel and ilmenite-group minerals, while higher diffusivities of these cations in surrounding rocks, marginal forsterite-rich phoscorite and vein calcite carbonatite lead to size-dependent growth of corresponding inclusions; (3) Three-dimensional mineralogical mapping of the Kovdor phoscorite-carbonatite pipe has shown its concentric (nested) zonation in regard to granulometry, shape, modal and chemical compositions of (oxy)exsolution inclusions in magnetite. In general, this zonation reflects concentric spatial change of host magnetite composition, corresponding in turn to the rock Figure 17. Diffusivities of Al and Ti in magnetite during its exsolution in rocks of the Kovdor massif (mean ± 95% confidence interval).

Conclusions
Complex (oxy)exsolution of Mn-Mn-Al-Ti-rich magnetite in the Kovdor phoscorite-carbonatite pipe has formed concentric alternating zones of spinel-and ilmenite-impregnated magnetite within the pipe: (apatite)-forsterite phoscorite of the marginal zone and carbonate-rich phoscorite and carbonatite of the axial zone predominantly contain magnetite with exsolution lamellae of ilmenite-geikielite, while low-carbonate magnetite-rich phoscorite of the intermediate zone predominantly includes exsolved magnetite with spinel impregnation. Each of these zones (rock types) has certain features in terms of exsolution processes and products: (1) Exsolution spinel forms spherical grains, octahedral crystals, six-beam (along [100]) and eight-beam (along [111]) skeletal crystals co-oriented with host magnetite and having maximal morphological diversity in magnetite-rich phoscorites of the ore-pipe inner part. The ilmenite group minerals occur usually as thin lamellae on the (111) and (100) planes of host magnetite (respectively, due to direct oxy-exsolution of titanomagnetite and with intermediate ulvöspinel).
In accordance with the lower diffusivity of Al than Ti in studied magnetites, spinel crystallizes after the ilmenite-group minerals, which is emphasized by the formation of zoned magnetite crystals with spinel-impregnated core, ilmenite-impregnated intermediate zone and inclusion-free marginal zone; (2) The kinetics of inclusion nucleation and growth depends mainly on the diffusivity of cations in magnetite: comparatively low diffusivities of Al 3+ and Ti 4+ cations in magnetite-and/or carbonate-rich phoscorite and carbonatite cause size-independent growth of both spinel and ilmenite-group minerals, while higher diffusivities of these cations in surrounding rocks, marginal forsterite-rich phoscorite and vein calcite carbonatite lead to size-dependent growth of corresponding inclusions; (3) Three-dimensional mineralogical mapping of the Kovdor phoscorite-carbonatite pipe has shown its concentric (nested) zonation in regard to granulometry, shape, modal and chemical compositions of (oxy)exsolution inclusions in magnetite. In general, this zonation reflects concentric spatial change of host magnetite composition, corresponding in turn to the rock crystallization sequence: surrounding silicate rocks-earlier forsterite-rich phoscorite-intermediate low-carbonate magnetite-rich phoscorite-late carbonate-rich phoscorite and carbonatite; (4) Temperature and oxygen fugacity of titanomagnetite exsolution decreases in this sequence from about 500 • C to about 300 • C and from NNO + 1 to NNO − 3, with local positive maximums in calcite carbonatite. The temperature of magnetite oxy-exsolution in phoscorite and carbonatites is about 250 • C below the temperature of equilibration of coexisting carbonates; (5) The intermediate low-carbonate magnetite-rich phoscorite was crystallized under oxidizing conditions resulting in the presence of Fe 3+ instead of Fe 2+ in melt/fluid. Therefore, oxy-exsolution of titanomagnetite finished here at lower temperature, oxygen fugacity and titanium diffusivity than in marginal and axial zones of the ore-pipe.