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Journal of Petrology | Volume 40 | Number 4 | Pages 629-652 | 1999
© Oxford University Press 1999

The Stability of Hydrous Potassic Phases in Lherzolitic Mantle—an Experimental Study to 9.5 GPa in Simplified and Natural Bulk Compositions

J. Konzett* and P. Ulmer

Institut Für Mineralogie Und Petrographie, Eth-Zentrum Sonneggstr. 5, Ch-8092 Zürich, Switzerland

Received January 30, 1998; Revised typescript accepted September 30, 1998


    ABSTRACT
 TOP
 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
To investigate the pressure stability limit of phlogopite and the pressure–temperature stability field of its breakdown product K-richterite, experiments were conducted from 4.0 to 9.5 GPa and between 800°C and 1400°C in a subalkaline system K2o–Na2O–CaO–MgO–Al2O3–SiO2–H2O (KNCMASH) and in natural phlogopite and K-richterite-doped lherzolite systems. In KNCMASH, phlogopite breaks down between 6.0 and 6.5 GPa at 800°C and between 6.5 and 7.0 GPa at 1100°C to form potassic amphibole by the reaction phlogopite + clinopyroxene + orthopyroxene = K-richterite + garnet + olivine + H2O. In the natural system, the stability field of amphibole is shifted towards lower pressures by ~0.5 GPa. The high-temperature stability limit of K-richterite in KNCMASH was located between 1300 and 1400°C at 8.0 GPa and at <1300°C at 7.0 GPa. Thus, K-richterite can be stable in the mantle wedge above subduction zones below a depth of ~180–200 km. Because of the small difference in K/OH ratios between phlogopite and K-richterite, only a small amount of aqueous fluid is likely to be produced during phlogopite breakdown in an average mantle lherzolite bulk composition. This fluid might be trapped by nominally anhydrous minerals before it can migrate to hotter portions of the mantle wedge. Phlogopite breakdowntherefore is unlikely to be a factor in inducing significant melting of the wedge leading to arc magmatism.

KEY WORDS: experimental study; high pressure; K-richterite; lherzolite; phlogopite


    Introduction
 TOP
 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
Hydrous potassic phases are potential hosts for alkalis and H2O in metasomatized lherzolitic mantle to depths near the transition zone (Trønnes et al., 1988Go; Trønnes 1990Go) and are likely to be source components for deep-seated generation of potassium-rich magmas (e.g. Wendlandt & Eggler, 1980Go; Lloyd et al., 1985Go; Foley, 1992Go; Thibault et al., 1992Go; Mitchell, 1995Go; Luhr, 1997Go). The hydrous phases phlogopite and K-rich richteritic amphibole are potentially stable in natural peridotitic bulk compositions and carry significant amounts of potassium and water. Phlogopite–spinel peridotites are common in xenolith suites of metasomatized shallow upper mantle sampled by basaltic volcanism (Nixon, 1987Go, and references therein), whereas phlogopite–garnet peridotites are typically part of xenolith suites from deep continental lithosphere. In rare cases K-richterite-bearing peridotites are found as xenoliths in kimberlites (e.g. Erlank et al., 1987Go; Tallarico & Leonardos, 1995Go). Although phlogopite peridotites typically show well-equilibrated textures, textures of K-richterite-bearing peridotites indicate that the amphibole is not in equilibrium with the host lherzolite (Erlank et al., 1987Go). The occurrence of diamond-bearing phlogopite peridotites (Dawson & Smith, 1975Go; Waters & Erlank, 1988Go) testifies that phlogopite is stable above 4 GPa in peridotitic bulk compositions under cool lithospheric PT conditions. With respect to T, phlogopite is stable in lherzolite to <1250°C at 3.0 GPa under water-absent conditions and to <1150°C at the same pressure under water-saturated conditions (Wendlandt & Eggler, 1980Go), consistent with results of Mengel & Green (1989)Go.

Experimental work in the KCMASH system has shown that phlogopite + clinopyroxene ± orthopyroxene assemblages break down to form KK-richterite1 + garnet + olivine at P ≥; 6–7 GPa or ~180–210 km, thus making phlogopite-bearing peridotites potential host rocks for potassium amphibole (Sudo & Tatsumi, 1990Go; Luth, 1997Go). This suggests that in a subalkaline2 bulk composition K-amphibole can be stable in mantle regions only where K–OH metasomatism is active within the diamond stability field or where metasomatized phlogopite-bearing mantle can be transported to a depth exceeding the upper-pressure stability limit of phlogopite. At present, no K-richterite-bearing peridotites have been found in which the amphibole is in textural equilibrium with the lherzolitic host assemblage. Nevertheless, the widespread occurrence of K-richterite in equilibrium with a garnet lherzolite assemblage in subduction zone peridotites has been proposed by Tatsumi (1989)Go, Tatsumi et al. (1991)Go and Tatsumi & Eggins(1995Go), on the basis of the stability of phlogopite in the peridotitic mantle wedge above subducting slabs and experimental results of Sudo & Tatsumi (1990)Go. Tatsumi & Eggins (1995)Go proposed that fluids released by the phlogopite-to-potassium amphibole reaction can trigger the mantle melting responsible for arc volcanism above subduction zones. Only reconnaissance studies have been available to date on the stability field of K-richterite in subalkaline bulk compositions relevant to normal lherzolitic mantle compositions (Trønnes et al., 1988Go; Trønnes, 1990Go; Harlow, 1995Go). The principal objectives of this study therefore were to locate the PT conditions of the reaction by which phlogopite breaks down to form amphibole precisely and determine the phase compositions across the phlogopite–amphibole transition so as to assess the amount of fluid released during phlogopite breakdown. The KNCMASH system was studied first to avoid the complications arising from heterovalent cation substitutions and bulk compositional change from reaction of the experimental charge with the noble metal capsule. In a second step, experiments were performed using a natural lherzolite composition doped with phlogopite or K-richterite.


    Previous Experimental Work
 TOP
 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
Kushiro & Erlank (1970)Go first reported K-richterite breakdown to phlogopite in the presence of garnet at 1000°C at 2.0 GPa and proposed a reaction

Formula 1

Formula 2
This reaction accounts for the P-induced breakdown of phlogopite in lherzolites or websterites into which phlogopite was introduced by metasomatic melts or fluids. Subsequent experimental studies on the stability of pure K-richterite (Trønnes, 1990Go; Foley, 1991Go) constrained the maximum stability field of amphibole and thus are of restricted relevance to real mantle rocks. Sudo & Tatsumi(1990Go) and Luth (1997)Go investigated the stability field of KK-richterite in a KCMASH system using phlogopite + diopside or phlogopite + diopside + enstatite mixes as starting materials. Sudo & Tatsumi (1990)Go proposed that a reaction of the form

Formula 3
governs the stability of phlogopite in the upper mantle. Harlow (1995)Go used an Na-free system to investigate the influence of halogens on K-amphibole stability to 11 GPa. Thibault (1993)Go used phlogopite + diopside mixes in a KNCMASH(F) system. This system considers jadeite–diopside and K-richterite–richterite solid solution but rules out orthopyroxene as a potential reactant for the formation of amphibole.


    Composition and Preparation of the Starting Material
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 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
Our starting composition in the KNCMASH system contains an excess of phlogopite relative to orthopyroxene and thus can be used to locate the upper-pressure stability limit of phlogopite in the presence of K-richterite (Table 1, Fig. 1). A spinel lherzolite from Mont Briançon, French Massif Central, was used as natural starting material (see Downes, 1987Go) because it represents a moderately fertile bulk composition slightly higher in modal clinopyroxene than average spinel lherzolite but less fertile than KLB-1 (see Table 1, Fig. 1). This composition was modified by subtracting 30 wt % olivine (Fo91) to increase the amount of pyroxenes relative to olivine and by adding 5 wt % synthetic phlogopite and 0.4 wt % Na2O. For two runs within the K-richterite stability field (PU667, PU698; see Table 2) the lherzolite was doped with 10 wt % synthetic K-richterite. Phlogopite and K-richterite were used as carriers for both potassium and water, with 5 and 10 wt % representing a minimum amount that contains sufficient potassium to produce detectable quantities of potassic phases in the experiments. Although in the natural system an excess of orthopyroxene with respect to phlogopite is present, phlogopite could still ‘survive’ the amphibole formation and coexist with K-richterite as a result of exhaustion of jadeite component in clinopyroxene. To avoid this less realistic scenario, 0.4 wt % Na2O was added to the natural starting material.


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Table 1: Composition of synthetic and natural starting materials used in this study compared with starting materials of previous experimental studies and of natural metasomatized peridotites from the Kaapvaal Craton

 

Figure 01
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Fig. 1. Projection from the SiO2 apex onto the Al2O3–(Na2O + K2O)–Mg2SiO4 plane of the KNMAS tetrahedron (Sweeney et al., 1993Go) (inset upper right). {blacktriangleup}, simplified KNCMASH bulk composition; *, spinel lherzolite BRIAN 2; {star}: modified BRIAN 2 (30 wt % olivine subtracted, 0.4 wt % Na2O and 5 wt % phlogopite added). bullet, compositions of pure K-richterite, phlogopite and pyrope. {square}, bulk compositions used by Kushiro & Erlank (1970)Go (KE70), Sudo & Tatsumi (1990)Go (ST90), Luth (1997)Go (L97) and Trønnes et al. (1988)Go (T88). Shaded areas: compositional range of MARID (mica–amphibole–rutile–ilmenite–phlogopite) xenoliths (Waters, 1987aGo),PKP (phlogopite–K–richterite–peridotite) and GPP (garnet–phlogopite–peridotite) xenoliths (Erlank et al., 1987Go). The inset upper left triangle shows the modal composition of BRIAN 2 in wt % of olivine, orthopyroxene and clinopyroxene compared with the composition of average spinel lherzolite (Maaløe & Aoki, 1977Go) and of fertile spinel lherzolite KLB-1 (Takahashi, 1986Go). (For abbreviations, see Table 9, below.)

 

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Table 2: Summary of experimental run conditions and run products

 
Preparation of the starting materials
For the simplified KNCMASH bulk composition, a mix of synthetic oxides and carbonate was used. H2O was added to the anhydrous oxide mix as Mg(OH)2 and Na2O as sodium metasilicate (Na2SiO3). Details of the preparation procedure have been given by Konzett et al. (1997)Go. The natural starting composition (modified BRIAN 2) used synthetic oxides and carbonates, with FeO added as synthetic fayalite and Na2O as sodium metasilicate. Phlogopite and K-richterite were synthesized at 0.3 GPa and 680°C, and 3.0 GPa and 1000°C, respectively, from oxide–carbonate mixes. Phlogopite turned out to be too fine grained for reliable microprobe analysis; X-ray diffraction showed the presence of well-crystallized phlogopite without additional diffraction lines. K-richterite was large enough (up to 15 µm) to be analysed with the electron microprobe; an averaged composition is given in Table 1. After mixing, the starting materials were stored at 250°C to minimize adsorption of moisture.


    Experimental and Analytical Techniques
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 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
Experiments were performed in a Walker-type two-stage 6/8 multi-anvil module (Walker et al., 1990Go; Walker, 1991Go) in a 1000 ton press. The pressure-transmitting octahedra and gasket fins are fabricated from an MgO-based castable ceramic. For runs to 9.5 GPa (including the high-temperature calibration runs), 3.5 mm outer diameter (o.d.) stepped graphite furnace assemblies, in combination with WC cubes with a truncated edge length (TEL) of 12 mm were used. The run at 9.0 GPa (Ma99s) was conducted with 8 mm TEL WC cubes and a 3.5 mm LaCrO3 stepped heater assembly. Temperatures were measured with one thermocouple at the bottom of the capsule and in some experiments with a second thermocouple attached to the capsule wall (see fig. 2 of Konzett et al., 1997Go). Temperature gradients measured with this arrangement do not exceed 30°C/mm; the vertical length of the experimental charge is ≤ 1.5 mm. All temperatures quoted are thermocouple readings from the Eurotherm controller uncorrected for the pressure effect on e.m.f. During the runs temperatures were kept constant to within ± 2°C and the applied load to within ±1 ton (strain-gauge reading; corresponding to ≤ 0.02 GPa) and recorded during the entire duration of the experiment. Details of the calibration procedure have been given by Konzett et al. (1997)Go. For runs in the simplified KNCMASH system, 1.6 mm o.d. Pt100 capsules were used. Except for three runs (Table 2), experiments using the lherzolite starting composition were conducted in 1.6 mm o.d. Pt100 capsules, with an inner graphite capsule to reduce f(O2) withi n the experimental charge relative to f(O2) in Pt100 capsules and to avoid loss of Fe to the capsule wall.

Analytical technique
After each experiment, capsules were embedded longitudinally in epoxy resin and ground to expose the centre of the charge. During opening of the KNCMASH charges small amounts of water extruded from the capsules, which was taken as an indication of water-saturated conditions at subsolidus temperatures and of integrity of the capsule during the run. Run products were analysed with a Cameca SX50 electron microprobe at ETH Zürich and with a JEOL Superprobe at the Geophysical Laboratory. The following standards were used: Si, Ti, Cr, Al, Mg: synthetic oxides; Ca, Fe, Mn, Na, K: natural wollastonite, haematite, tephroite, aegirine or omphacite, and orthoclase. Correction of the raw data was performed on-line with the PAP (ETH) or the PRZ (Geophysical Laboratory) correction procedure. Analytical conditions were 15 kV acceleration voltage and 20 nA sample current; the beam size was minimized (~1 µm) for most analyses, except for analyses of phlogopite and amphibole if the grain size was sufficiently large. Counting times of 20 s on peaks and 10 s on backgrounds were used. Ni in the experimental charges was analysed using synthetic NiO as a standard and counting times of 50 and 25 s on peak and background, respectively.

Recalculation of mineral formulae
K-richterite analyses from both simplified and natural bulk compositions were recalculated from the oxide wt % assuming 23 oxygens, stoichiometric OH and Fetot = Fe2+. In this case cation sums are usually very close to 16.00. Because of the uncertainty of individual microprobe analyses, cation sums up to 16.05 were accepted as representative. Cations were assigned to structural positions as recommended by Leake (1978)Go [revised Leake et al. (1997)Go]. In case of {Sigma}(Mg + Ca + Na) on M(4)< 2.0, K was assigned to M(4) to make KK-richterite component. Phlogopite was recalculated to 11 oxygens and stoichiometric OH. In the case where {Sigma}(Si + Al)< 4, Fe3+ was added to the tetrahedral cations to bring the sums up to 4.0.


    Results of the Experiments
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 Results of the Experiments
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Petrography
In both the simplified and the natural lherzolite system, mineral phases form euhedral to subhedral grains ≥ 10–20 µm in diameter except for grains of clinopyroxene and phlogopite, which rarely exceed 5–10 µm in size (Fig. 2). At temperatures <1100°C, the average grain size decreases rapidly and clinopyroxene grains may become too small (<5 µm) to be analysed with the microprobe. Garnets are typically poikiloblastic and contain numerous inclusions of clinopyroxene and/or olivine. At run durations >24 h, olivine often forms grain aggregates with perfect 120° triple point junctions. In run Ma80s orthopyroxene is absent and instead, coexisting omphacitic and diopsidic clinopyroxene can be texturally distinguished. The Al-poo clinopyroxene grains form aggregates of tiny, needle-like crystals, whereas the Al-rich ones are blocky isolated crystals that do not exceed 5 µm in length. Although the textures of the Al-poor clinopyroxenes would be consistent with a quench origin, these clinopyroxenes are not associated with K-richphases and, therefore, are considered a stable phase coexisting with omphacite. In cases where quench could be identified, the quench phases were always K rich.


Figure 02
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Fig. 2. Back-scattered electron photomicrographs of experimental charges from simplified and natural bulk compositions. (a) Ma62s (8 GPa, 1100°C), KNCMASH above the K-richterite-in reaction; (b) Ma85s (5 GPa, 1100°C), KNCMASH below the K-richterite-in reaction; (c) PU649 (8 GPa, 1200°C), Pt100 + inner graphite capsule; (d) PU633 (6.5 GPa, 1100°C), Pt100 capsule with coexisting K-richterite + phlogopite. (For abbreviations, see Table 9, below.)

 
The phase distribution within individual charges is often inhomogeneous. Quenched fluid is typically concentrated at the capsule bottom along with olivine in low-temperature runs (e.g. Ma76, Ma86). Possible explanations for the concentration of olivine would be incongruent dissolution of phlogopite at subsolidus conditions or grain coarsening in a T gradient. Modal amounts of garnet may also strongly increase towards the capsule bottom. A similar, but less pronounced increase can be observed for clinopyroxene. Unlike the H2O-rich peralkaline KNCMASH system investigated by Konzett et al. (1997)Go, the high-temperature breakdown of K-richterite in the H2O-poorer subalkaline KNCMASH system does not produce easily recognizable segregated melt. Instead, melt forms numerous small pools that contain K-rich quench phases, dispersed in the matrix. This feature makes the modal amounts of quench phases impossible to estimate in runs Ma100s and Ma101s. In the natural lherzolite system the breakdown of K-richterite occurs without formation of any detectable quench. Instead, X-ray mapping reveals that K, possibly deposited as extremely fine-grained quench, is dispersed along the interface between the experimental charge and the graphite capsule.

Chemical homogeneity of the phases
In both the simplified and the natural systems there is no systematic difference in the composition of phases analysed at the bottom and the top of individual experimental charges. With the exception of large (~20 µm or larger) garnet and K-richterite grains, no zoning within individual crystals can be observed. Garnets and clinopyroxenes, however, in part show considerable scatter in their Cr contents, which is ascribed to random inclusions of Cr2O3 remnants from the starting material. Most large garnet grains in both the simplified and thenatural systems show Al- and Ca-rich cores. In the natural system some amphibole grains show cores distinctly lower in Al and Ca compared with rims. Both garnets and amphiboles show 5–10 µm wide rims of constant composition, which are thought to represent the equilibrium garnet and amphibole composition at prevailing P and T. Average temperatures derived from diopside solid solution in enstatite as calibrated by Brey & Köhler, (1990)Go are in good agreement with actual run temperatures (see Table 2), indicating that exchange reactions as fast as CaMg–1 in pyroxene reached equilibrium.

Phase relations in KNCMASH
In the subalkaline KNCMASH system K-richterite is stable at pressures >6.0 GPa (Figs 2a and 3; Table 2). Below the K-richterite-in curve, the stable assemblage is phlogopite + clinopyroxene + orthopyroxene + garnet + fluid (Figs 2b and 3). Olivine joins the assemblage at pressures close to the K-richterite-in curve. The position of the K-richterite-in reaction curve was located between 6.0 and 6.5 GPa at 800°C, and between 6.5 and 7.0 GPa at 1100°C. Hence, the reaction has a positive slope with dP/dT < 3.3x 10–3 GPa/K.


Figure 03
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Fig. 3. PT diagram of experimental results in the KNCMASH system (Table 1). Phases present in the experimental charges are represented by black sectors within the run symbol; phases not detected are denoted by white sectors (inset lower right). The dashed line is a 40 mW/m2 continental geotherm (Pollack & Chapman, 1977Go). The numbers next to the run symbols are run numbers (Table 2) and observed approximate amounts of quench products. Vertical bars denote the P range within which Sudo & Tatsumi, (1990)Go located the K-amphibole-in-reaction (ST90) (see their fig. 1).

 
In the presence of orthopyroxene as a possible reactant phase, the changes of mineral assemblage associated with the appearance of K-richterite are consistent with a reaction phlogopite + clinopyroxene + orthopyroxene = amphibole + garnet + olivine + fluid as initially proposed by Sudo & Tatsumi (1990)Go. The high-temperature stability limit of K-richterite is reached between 1300 and 1400°C at 8.0 GPa, and at <1300°C at 7.0 GPa amphibole breakdown produces an anhydrous lherzolite assemblage garnet + olivine + clinopyroxene + orthopyroxene. The fine-grained interstitial quench material found in runs Ma100s and Ma101s is interpreted as quenched melt. Because of the surplus of phlogopite with respect to orthopyroxene, phlogopite coexists with amphibole between 7.0 and 8.0 GPa at 1100°C. Phlogopite finally breaks down in the absence of orthopyroxene between 8.0 and 9.0 GPa at 1100°C. This is accompanied by an abrupt increase of K p.f.u. in the amphibole, with a resulting KK-richterite component. The schematic reaction is

Formula 4
Phlogopite breakdown between 1100 and 1200°C at 8.0 GPa again produces KK-richterite component in amphibole and is also responsible for the reappearance of orthopyroxene in subsolidus run Ma96s (see Fig. 3). The upper T stability of K-richterite with KK-richterite solid solution at 8.0 GPa is reached between 1300 and 1400°C. Under these conditions amphibole breaks down to form a lherzolitic assemblage plus small amounts of fine-grained quench products.

Phase relations in the natural lherzolite system
As opposed to KNCMASH, no free fluid phase is present in the natural lherzolite system below the PT conditions of the K-richterite-in curve, and the influence of Fe has to be considered. The stability of K-richterite in the lherzolite system is shifted slightly towards lower pressures compared with KNCMASH. K-richterite first appears and coexists with phlogopite at 6.5 GPa. At 1100°C, the pressure interval of coexisting phlogopite + amphibole is <1.0 GPa (Fig. 4). The shift of the K-richterite-in curve probably reflects partitioning of Fe2+ into garnet which is part of the product assemblage of phlogopite breakdown. The persistence of phlogopite to 6.5 GPa is ascribed to a high Fe3+/Fe2+ ratio in the Pt capsules, which stabilizes phlogopite with respect to amphibole. This assumption is supported by unusually high forsterite contents of coexisting olivine and pronounced tetrahedral cation deficits in phlogopite analyses when normalized to 11 oxygens and stoichiometric OH. To reduce Fe3+ in the experimental charges, experiments were rerun at 1100°C using double capsules with an inner graphite liner. This led to the disappearance of phlogopite at 6.5 GPa and 1100°C along with a significant change in the mineral compositions. At 8.0 GPa K-richterite disappears between 1200 and 1300°C. The K-richterite breakdown does not produce optically [back-scattered electron (BSE) and secondary electron (SE) imaging] detectable quench phases nor is there any significant change in the composition of lherzolite phases between 1100 and 1300°C. At 9.5 GPa no solid hydrous potassic phase was found at 1200 and 1300°C regardless of whether phlogopite or K-richterite-bearing starting material was used. X-ra mapping revealed that within the experimental charge K is concentrated in clinopyroxene and to a smaller degree dispersed along grain boundaries. The highest K concentration was found along grain boundaries of graphite grains within an ~20 µm wide zone at the interface between graphite capsule and experimental charge. The complete absence of K-richterite in runs at 9.5 GPa would imply a negative slope of the reaction that defines the high-temperature breakdown of K-richterite, opposite to the curvature observed in the synthetic KNCMASH system. A possible explanation for this discrepancy is that, in the natural system, increasing oxidation of graphite forms CO2 with increasing pressure and temperature. This would reduce the activity of H2O in the fluid and destabilize K-richterite, and possibly form trace amounts of alkaline CO2-rich fluids or melts (e.g. Sweeney, 1994Go).


Figure 04
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Fig. 4. PT diagram summarizing experimental results using a natural lherzolite starting composition (Table 1). Symbols and 40 mW/m2 geotherm are as in Fig. 3. ACMA is an average current mantle adiabat for a surface potential temperature of 1280°C (Thompson, 1992Go). Numbers next to run symbols are run numbers (Table 2). All runs except PU667 and PU698 had 5 wt % phlogopite added; PU667 and PU698 had 10 wt % K-richterite added.

 
Mineral chemistry
Amphibole
Amphiboles in both the KNCMASH and the lherzolite systems are K-richterites (see Leake, 1978Go; revised Leake et al., 1997Go). The major chemical variations are an increase in K and K/(K + Na) and a decrease in Na with increasing pressure at constant temperature (Fig. 5, Table 3). These effects are more pronounced in KNCMASH than in the natural system. Maximum observed K contents in the KNCMASH system are 1.25 ± 0.02 (n = 4) K p.f.u. at 9.0 GPa and 1100°C, equivalent to 27 mol % of KKCaMg5Si8O22(OH)2 component (Fig. 6). In the natural system K contents are lower, with a maximum of 1.10 ± 0.02 (n = 3) K p.f.u. at 8.0 GPa and 1200°C, as a result of the considerably lower bulk K content of the lherzolite. All analysed K-richterite grains contain small amounts of Al(IV) and Al(VI), both of which decrease with increasing P at constant T in both chemical systems (Fig. 7). This decrease can be explained by a reaction

Formula 5
causing clinopyroxene and garnet growth at the expense of amphibole. Fe contents of amphiboles in the lherzolite system range between 2.1 and 2.6 wt % FeO, corresponding to XMg of 0.94–0.95 on the basis of Fetot = FeO. Total cation numbers of lherzolitic amphiboles in Pt capsule experiments are slightly higher than those in the KNCMASH and the lherzolite system experiments performed in graphite capsules (see Table 3), indicating the presence of some Fe3+. All lherzolitic amphiboles contain minor Ti, Cr and Ni. TiO2 and Cr2O3 contents are ≤ 0.20 and 0.43 wt %, respectively, whereas Ni contents range between 640 and 1110 ppm. The significance of Ni contents in phlogopite and K-richterite will be discussed below.


Figure 05
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Fig. 5. K/(K + Na) ratios of phlogopite and K-richterite as a function of P at a constant temperature of 1100°C for runs in the KNCMASH and the natural lherzolite system (inset lower left). Shaded bands denote the pressure of K-richterite-in and phlogopite-out reactions in KNCMASH and the natural lherzolite system. Individual data points represent an average of several analyses (see Tables 3 and 4). Error bars are less than or equal to the size of plot symbols.

 

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Table 3: Average analyses of K-richterite

 

Figure 06
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Fig. 6. Composition of amphibole in KNCMASH in terms of K-edenite, K-richterite and KK-richterite endmembers. Arrows denote exchange vectors for transformation of the amphibole endmembers. Mol % KK-richterite and K-richterite based on K(M4) and Na(M4), respectively; mol % K-edenite based on remaining Ca(M4).

 

Figure 07
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Fig. 7. Al contents of K-richterite as a function of P at constant T in KNCMASH (filled symbols) and in the natural lherzolite system using graphite capsules (open symbols). Individual data points represent an average of several analyses per run (Table 3).

 
Phlogopite
Phlogopite in KNCMASH shows only minor deviations from the ideal endmember composition (Table 4). The principal chemical variations are an increase in K and K/(K + Na), and a decrease in Al and Na p.f.u. with pressure at constant temperature (Figs 5 and 8). These effects duplicate the behaviour of amphibole. The decrease in Al can be attributed to garnet- or clinopyroxene-forming reactions. Modal amounts of garnet and clinopyroxene were determined in runs at 4.0 GPa and 1100°C, and 6.0 GPa and 1100°C using image analysis in three areas for garnet and six areas for clinopyroxene at the bottom, the centre and the tip of the capsules. The selected areas have an edge length of 400–500 µm. Although garnet distribution is inhomogeneous, the modal amount strongly increases, from 7 to 26 vol %, from 4 to 9 vol % and from 4 to 7 vol % at the bottom, centre, and tip of the capsules between 4.0 and 6.0 GPa. The modal amount of clinopyroxene is much less variable, increasing from 21 to 25 vol %, from 19 to 25 vol % and from 18 to 26 vol % at the bottom, centre, and tip of the capsules within the same P range. This would be consistent with the assumption that garnet-forming reactions are mainly responsible for the Al decrease in phlogopite. For phlogopites with Al(IV) > 1.0 p.f.u. and in the presence of orthopyroxene, the relevant equilibrium is

Formula 6
which is a combination of two equilibria involvingTschermak exchange component (Thompson, 1982Go)

Formula 7

Formula 8
At P ≥ 6.0 GPa, phlogopite has <1.0 Al p.f.u. In this case, garnet formation can involve montdorite component KMg2.5[Si4O10](OH)2 (Seifert & Schreyer, 1971Go) in phlogopite (see Konzett et al., 1997Go). Phlogopites formed in lherzolite runs using Pt capsules show tetrahedral cation deficits when normalized to 11 oxygens + stoichiometric OH. In runs PU624 and PU633 {Sigma}(Si + Al) is 3.96 and 3.92, respectively. Fe3+ was used in these cases to fill the tetrahedral sites with a resulting Fe3+/(Fe3+ + Fe2+) of 0.16 and 0.31 (Table 4). In run PU654 (6.0 GPa and 1100°C, graphite capsule) no tetrahedral deficits are present and the mica composition is very similar to that in the KNCMASH system at identical P and T (see Table 4).


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Table 4: Average analyses of phlogopite

 

Figure 08
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Fig. 8. Al and Na contents of phlogopite as a function of P at constant T in KNCMASH (filled symbols) and in the natural lherzolite system using graphite capsules (open symbols). Shaded bands denote the P range of K-richterite-in in KNCMASH and the natural lherzolite system. Individual data points represent an average of several analyses per run (Table 4).

 
Clinopyroxene
Clinopyroxenes in KNCMASH are jadeite–diopside–enstatite solid solutions with a small Ca-Tschermak component. The latter rapidly decreases from 0.035 Al(IV) p.f.u. at 4.0 GPa and 1100°C to values well within the count statistical error of a microprobe analysis (Fig. 9, Table 5). Enstatite solid solution is constrained by the cpx–opx solvus (Brey et al., 1990Go) and varies little with pressure between 10 and 14 mol % at 1100°C. At higher temperatures, enstatite component increases, reaching 27 mol % at 8µ0 GPa and 1400°C. Na in clinopyroxene increases with pressure to a maximum averaged value of 19 mol % jadeite at 6µ5 GPa. At higher pressures, jadeite component is consumed by the amphibole-forming reaction and the re-partitioning of Na between amphibole and clinopyroxene decreases Na contents of clinopyroxene formed at pressures between 7.0 and 8.0 GPa. At 6.0 GPa and 1100°C two coexisting clinopyroxenes—diopside and omphacite—occur instead of orthopyroxene in the KNCMASH system. The diopside ranges in composition from jad7di91cats2 to jad21di71en7cats1, whereas omphacite ranges from jad52di41en3cats4 to jad67di29en2cats2 (Table 5). The system jadeite–augite shows complex subsolidus exsolution behaviour at low T–high P with two immiscible regions across which omphacite can coexist with sodic diopside or calcic jadeite (Rossi, 1988Go, and references therein). Coexisting clinopyroxenes have been reported from natural rocks (Droop et al., 1990Go) and are known from experimental studies. Schmidt (1992)Go reported coexisting omphacite and sodic omphacite–augitic omphacite in a tonalitic bulk system between 1.7 and 3.6 GPa and 650°C. The presence of coexisting clinopyroxenes at temperatures as high as 800°C in this study probably reflects the high experimental pressure, which might shift the critical point of the immiscible region to higher temperatures compared with natural rocks (Carpenter & Smith, 1981Go) and/or the lack of Fe, and particularly Fe3+ in the subalkaline KNCMASH system.


Figure 09
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Fig. 9. Composition of clinopyroxene (a) in KNCMASH and (b) in the natural lherzolite system. Open and filled symbols in (b) denote clinopyroxene compositions from experiments performed in Pt and Pt + graphite capsules, respectively. Jadeite and jadeite + acmite compositions plot along the dashed lines. (c) and (d) K2O contents of individual clinopyroxenes in KNCMASH across K-richterite-in and phlogopite-out reactions. Error bars denote 2{sigma} error of an individual microprobe analysis.

 

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Table 5: Average and representative analyses of clinopyroxene

 
Clinopyroxene in lherzolite experiments between 6.0 and 8.0 GPa show jadeite contents of 6–10 mol % and an acmite component from 4 to 9 mol % in graphite capsules and from 11 to 16 mol % in Pt capsules. The formation of amphibole decreases the jadeite component from 10 to 7 mol % between 6.0 and 6.5 GPa. Potassium contents of KNCMASH clinopyroxene in the presence of phlogopite are small with values <0.2 wt % K2O (see Fig. 9). The breakdown of phlogopite at high pressures (phl -> Kr) and high temperatures (phl -> Kr + en) is accompanied by a significant increase of K2O in the clinopyroxene that coexists with either K-richterite or melt-fluid. Such clinopyroxene displays up to 0.45 wt % K2O (Fig. 9). Under very high pressure conditions, clinopyroxene in the lherzolite system contains between 0.05 and 0.10 wt % K2O as a result of the much lower bulk contents of the natural starting material.

Garnet, olivine, and orthopyroxene
Garnet in KNCMASH is a binary pyrope–grossular solid solution with Ca/(Ca + Mg) decreasing with increasing pressure (Brey et al., 1990Go) to 6.5 GPa. Within the K-richterite stability field, XCa remains constant to the highest pressure we studied. At 8.0 and 9.0 GPa individual garnet analyses typically show >3.0 Si p.f.u., with a maximum of 3.06 ± 0.02 Si p.f.u. at 9.0 GPa. Garnet compositions in the lherzolite system are constant between 6.0 and 8.0 GPa, with Mg/(Mg + Fetot) and XCa between 0.82 and 0.84, and 0.09 and 0.11, respectively (Table 6).


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Table 6: Average and representative analyses of garnet

 
Olivine in the lherzolite system experiments with graphite capsules ranges between Fo91 and Fo92, compositions very similar to those of olivine in BRIAN 2 (Table 7). Olivine formed in Pt capsules displays forsterite contents close to Fo93. Ni contents of olivine are between 2000 and 3500 ppm, which is within the expected range for primary mantle olivines (Griffin et al., 1989Go).


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Table 7: Average analyses of olivine

 
Orthopyroxene shows decreasing Mg-Tschermak and diopside components along with increasing Na with pressure and temperature (Table 8), constrained by the coexistence with garnet and clinopyroxene (Brey & Köhler, 1990Go; Brey et al., 1990Go, and references therein).


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Table 8: Average analyses of orthopyroxene

 

    Discussion
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 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
Amphibole-forming reactions in Na-free and Na-bearing systems
Sudo & Tatsumi (1990)Go and Luth (1997)Go used an Na-free KCMASH system to assess the potential stability range of KK-richterite under upper-mantle PT conditions. The KCMASH system is relevant to this study because K-richterite in the KNCMASH system contains a significant KK-richterite component at very high pressures.

Phase relations and mineral assemblages found in the KNCMASH and KCMASH systems are shown in Fig. 10. Because endmember phlogopite and K-richterite have identical K/OH ratios, reactions in KNCMASH do not produce a free fluid (Fig. 10a). The grossular-absent reaction (Fig. 10a) corresponds to the K-richterite-forming reaction proposed by Luth et al. (1993)Go. In Fig. 10a, H2O is a phase component but not a phase, and a(H2O) defined by equilibria involving hydrous phases is less than a(H2O) of a pure water phase at the same P and T. According to Thompson (1983)Go, H2O-conserving reactions generally cannot be located in PT space by direct experiments involving a free fluid, because the presence of H2O as a phase shifts the bulk composition to a region in composition space not accessible to H2O-conserving equilibria. In the KCMASH system the amphibole endmember is KKCaMg5Si8O22(OH)2, which is very close to K1.9Ca1.1Mg5[Si7.9Al0.1]O22(OH)2 reported by Sudo & Tatsumi, (1990)Go. The higher K/OH ratio of KK-richterite compared with that of phlogopite requires that all phlogopite breakdown reactions that form KK-richterite must release H2O (see Fig. 10b). If phlogopite and amphibole deviate from their endmember stoichiometry, the number of moles of H2O produced during formation of 1 mol amphibole by phlogopite breakdown is (Kamphibole p.f.u./Kphlogopite p.f.u.) – 1. In Fig. 10b, the enstatite- and grossular-absent reactions are equivalents to the amphibole-forming reactions of Sudo & Tatsumi (1990)Go.


Figure 10
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Fig. 10. Schematic phase relations in PT space of clinopyroxene, orthopyroxene, garnet, olivine, phlogopite and amphibole using endmember mineral compositions (a) in KNCMASH and (b) in KCMASH. No CaMg solid solution was considered for garnet and no CaMg–NaAl solid solution was considered for clinopyroxene. The reactions with endmember garnet and clinopyroxene ([jad, Kr, phl] and [Kr, phl, ol, H2O], respectively) are single isopleths that represent a particular garnet and clinopyroxene composition. (For abbreviations, see Table 9.) Arrows in (a) denote observed reactions towards high pressures (K-richterite formation) and high temperatures (K-richterite + orthopyroxene formation) observed in the experiments. Reactions are arranged to be consistent with a positive slope of the grossular-absent reaction and balanced using the REACTION program of Finger & Burt (1972)Go.

 
The high Na content of K-richterite close to endmember composition and the resulting small difference in K/OH between K-richterite and phlogopite compared with KK-richterite and phlogopite require that only a small amount of fluid is produced during K-richterite formation. The reaction cannot be strictly H2O conserving because it is observed in the experiments (Thompson, 1983Go). The positive slope of the K-richterite-forming reaction in KNCMASH implies that H2O is on the low-entropy-low-temperature side of the reaction. Although this is not a common feature of water-releasing reactions, it is typical of reactions that involve glaucophane (Holland, 1988Go). The negative slope of the KK-richterite-forming reaction, as suggested by Sudo & Tatsumi (1990)Go for the KCMASH system, resembles the slopes of calcic amphibole dehydration reactions, but it is not well constrained within the spacing of the experimental data points of Sudo & Tatsumi (1990)Go (see Fig. 3).

The stability of potassium amphibole in lherzolitic mantle
Subcontinental lithosphere
The PT conditions defined by an average 40 mW/m2conductive continental shield geotherm (Pollack &Chapman, 1977Go) are outside the K-richterite stability field for the natural lherzolite system (Fig. 3). In regions of subcontinental mantle with a heat flow <40 mW/m2, PT conditions straddle the high-temperature corner of this amphibole stability field. Unless it is stabilized by F, K-richterite is probably not present in subcontinental mantle lherzolite. Foley (1991)Go showed that at 5.0 GPa, the T stability limit of K–F-richterite is increased by ~200°C compared with that of K–OH–richterite. However, phlogopites from metasomatized peridotites generally contain <1.0 wt % F (Delaney et al., 1980Go; Smith et al., 1980Go).

Subduction zone settings
Because subducting slabs are heat sinks, temperatures in the mantle wedge above a descending slab are anomalously cold compared with other mantle regions. Thermal models for subduction zones (e.g. Davis & Stevenson, 1992Go; Furukawa, 1993Go) predict steady-state temperatures of 800–1000°C at the slab–mantle interface, with isotherms parallel to the subducting slab. PT conditions in part of the mantle wedge are thus within the K-richterite stability field below a depth of ~180 km or 6 GPa. The mantle volume in which K-richterite is potentially stable varies with the spacing of isotherms. According to the theoretical model of Davis & Stevenson, (1992)Go, increasing subduction velocity leads to a hotter wedge with a tighter spacing of the isotherms. Therefore, fast subduction should reduce the mantle volume in which K-richterite is stable (Fig. 11). Different subduction angles and varying thickness of the slab, on the other hand, have little influence on the T distribution in the wedge in the model of Davis & Stevenson (1992)Go.


Figure 11
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Fig. 11. Temperature distribution in the mantle wedge above a subducting slab according to the model of Davis & Stevenson (1992)Go for a subduction angle of 60° and three different subduction velocities. The shaded area in the wedge represents an approximate potential stability field of K-richterite as deduced from experiments of the present study. The upper T stability limit of K-richterite is thought to be at T < 1400°C as determined in the KNCMASH system and was (arbitrarily) placed at 1300°C in the absence of unambiguous evidence for K-richterite melting in the lherzolitic system (see text for discussion). The dashed line connecting dots is an approximate upper T stability limit for phlogopite in lherzolite extrapolated from experimental data of Wendlandt & Eggler, (1980)Go. Lines with numbers refer to isotherms and corresponding temperatures.

 
Both phlogopite breakdown in peridotites dragged down along the slab–mantle interface and interaction between mantle peridotites and K-rich fluids or melts at pressures above the stability limit of phlogopite can form K-richterite in subduction zones. Mechanisms for the generation of K-enriched fluids from subducting slabs within the phlogopite stability field include: (1) breakdown of Ca-amphibole in the subducted slab (e.g. Tatsumi & Eggins, 1995Go); (2) decomposition of K-feldspar + phlogopite (Massonne, 1992Go); (3) continuous phengite dehydration or K-rich mid-ocean ridge basalt (MORB) melting (Schmidt, 1996Go; Domanik & Holloway, 1996Go); (4) hybridization between hydrous silicic melts and peridotitic mantle (Sekine & Wyllie, 1982Go; Wyllie & Sekine, 1982Go). Together, these mechanisms can form phlogopite in a large P(–T) interval within the mantle wedge. The presence of phlogopite in arc magma sources is supported by the occurrence of K-rich lamprophyric rocks in some young subduction zones (Esperança & Holloway, 1987Go; Hochstaedter et al., 1996Go; Luhr, 1997Go). Moreover, some arc-related ophiolites contain phlogopite–calcic amphibole peridotites (Arai & Takahashi, 1990Go). Because phlogopite is stable to higher temperatures than is pargasite (see Wendlandt & Eggler, 1980Go; Mengel & Green, 1989Go), phlogopite can survive low-P partial melting of hydrated peridotite that extracts the Ca-amphibole. A possible source for K-rich fluids or hydrous melts at pressures exceeding the P stability of phlogopite is phengite breakdown to K-hollandite or melting of K-rich MORB, the latter occurring over a large P interval as a result of the low-angle intersection between typical subduction PT paths and the solidus of K-rich MORBs (Domanik & Holloway, 1996Go; Schmidt, 1996Go; Sorensen et al., 1997Go).

Fluid production during pressure-induced phlogopite breakdown
Tatsumi (1989)Go, Tatsumi et al. (1991)Go and Tatsumi & Eggins, (1995)Go, on the basis of available experimental data (Sudo & Tatsumi, 1990Go), proposed that hydrous K-rich fluids are released during phlogopite breakdown to form potassium amphibole at a depth of ~180 km. These fluids are thought to trigger large-scale melting in the overlying mantle and be responsible for back-arc volcanism. Tatsumi & Eggins (1995)Go assumed that pure KK-richterite is formed from phlogopite breakdown, both phases having the maximum possible difference in their K/OH ratios. This assumption would yield the maximum amount of fluid that can be released during the phlogopite-to-amphibole reaction [reaction (2)]. A natural mantle environment that would approach the KCMASH system studied by Sudo & Tatsumi (1990)Go is a phlogopite–garnet peridotite, in which the grossular component of reactant garnet could provide Ca for the formation of KK-richterite (the diopside-absent reaction in Fig. 10b). In an Na-bearing system, the amount of fluid produced during phlogopite breakdown will be reduced because our experiments show that the first K-richterite stable in a subalkaline lherzolite composition contains significant amounts of Na and has a K/OH ratio very similar to that of phlogopite. Because nominally anhydrous minerals (NAMs) can accommodate significant amounts of water under high P and T conditions (e.g. Bell & Rossman, 1992Go; Young et al., 1993Go; Kohlstedt et al., 1996Go) it is uncertain whether small amounts of aqueous fluid can migrate through a mantle peridotite without being trapped by NAMs.

Experiments in the KNCMASH system indicate that phlogopite could persist to pressures as high as 8–9 GPa at 1100°C in peridotites with modal phlogopite > orthopyroxene, or in a relatively refractory peridotite bulk composition with very small amounts of Na-poor clinopyroxene. In such bulk compositions, phlogopite + K-richterite would coexist in a pressure interval of 2–3 GPa. In non-peridotitic, orthopyroxene-absent environments (phlogopite clinopyroxenites) the first K-richterite would form by clinopyroxene + phlogopite breakdown between 8 and 9 GPa.

Hydrous potassic phase composition from lherzolitic and MARID-type sources
Experimental K-richterite and phlogopite in equilibrium with a lherzolitic assemblage are similar in major element composition to their natural counterparts from metasomatized peridotites (PPs and PKPs; see Erlank et al., 1987Go). This is not surprising, as PKP amphiboles have probably formed by metasomatic replacement of peridotite phases. Although experimentally grown K-richterite shows significantly higher Al contents at its lower P stability limit compared with PKP amphibole [see table I of Erlank et al., (1987)Go], this difference vanishes at very high pressures because of transfer of Al from phlogopite to garnet and clinopyroxene through Tschermak or combined Tschermak + plagioclase exchange. MARID-suite K-richterite and phlogopite are higher in Fe and Ti compared with their peridotitic counterparts (Dawson & Smith, 1977Go; Erlank et al., 1987Go; Waters, 1987bGo), which testifies to their origin from a different bulk composition. K-richterite and phlogopite from our experiments show Ni contents in the range of 640–1110 ppm and 1180–1730 ppm, respectively, values very similar to those reported for calcic amphibole and phlogopite in equilibrium with primary undifferentiated olivine (Griffin et al., 1989Go) from various peridotitic mantle sources (Delaney et al., 1980Go; O'Reilly et al., 1991Go). For example, experimental K-richterite–olivine pairs from our experiments at 6.5 GPa and 1100°C and 8.0 GPa and 1100°C yield DNiKr/ol = 0.30 ± 0.08 (six pairs) and 0.31 ± 0.07 (eight pairs), respectively. These values are identical to the DNiCa-amph/ol of 0.28 ± 0.04 for coexisting pargasite and olivine from spinel peridotites reported by O'Reilly et al. (1991)Go in a T range between 825 and 1036°C, as well as to a value of 0.27 for the same partition coefficient determined by Mysen (1978)Go for coexisting pargasite + olivine + water-saturated liquid at 1.5 GPa and 1000°C. DNiamph/ol is insensitive to P and T (and amphibole composition) because Ni contents of peridotitic amphibole and phlogopite are buffered by coexisting olivine. On the other hand, MARID K-richterite and phlogopite are significantly lower in Ni, with values less than 570 and 860 ppm, respectively (Konzett, 1996Go) (Fig. 12). Assuming a DNiKr/ol of 0.30, then olivine coexisting with MARID K-richterites must have had Ni contents of 1230–1440 ppm. This suggests that MARID amphibole and phlogopite did not crystallize in equilibrium with primary mantle olivine, but rather from a differentiated (presumably kimberlitic) melt that has undergone olivine fractionation with an attendant decrease of Ni.


Figure 12
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Fig. 12. Ni contents of amphibole, phlogopite and olivine from various mantle sources compared with experimentally produced Ni contents of K-richterite, phlogopite and coexisting olivine in the natural lherzolite system (PU numbers; see Table 2). G89, average Ni contents in olivine from diamond inclusions (bullet) and from garnet peridotites ({circ}) reported by Griffin et al. (1989)Go; O91, pargasitic amphiboles and coexisting phlogopite and olivine from spinel lherzolites (O'Reilly et al., 1991Go); D80, primary textured phlogopites and coexisting olivine from harzburgitic and lherzolitic xenoliths (Delaney et al., 1980Go); M78, pargasite and coexisting olivine in equilibrium with water-saturated liquid at 1.5 GPa and 1000°C (Mysen, 1978Go). K-richterite and phlogopite from MARID xenoliths were analysed by PIXE, all other analyses are by electron microprobe (Konzett, 1996Go). Error bars represent errors of individual microprobe analyses. Errors for PIXE analyses are smaller than the plotting symbols (Sweeney et al., 1995Go).

 


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Table 9: List of mineral endmembers, abbreviations and formulae used in this study

 


    Acknowledgements
 
We are indebted to Russell Sweeney and Alan Thompson for stimulating discussions. Sincere thanks are due also to Russell Sweeney for providing PIXE analyses of MARID phases. Careful and constructive reviews by George Harlow, Robert Luth and Gautam Sen are gratefully acknowledged. This study was financially supported by the Swiss National Science Foundation.


    FOOTNOTES
 
1 Throughout this study, the terms ‘KK-richterite’ and ‘K-richterite’ will be used to refer to amphiboles with the idealized composition KKCaMg5Si8O22(OH)2 and KNaCaMg5Si8O22(OH)2, respectively. Back

2 Throughout this study, the term subalkaline will be used to refer to bulk compositions with molar (K2O + Na2O)/Al2O3 < 1.0 as opposed to peralkaline compositions with molar (K2O + Na2O)/Al2O3 > 1.0. Back


* Corresponding author. Present address: Geophysical Laboratory, 5251 Broad Branch Road, N.W., Washington, DC 20015-1305, USA. Telephone: 202-686-2410, ext. 2443. Fax: 202-686-2419. e-mail: konzett{at}gl.ciw.edu


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 TOP
 ABSTRACT
 Introduction
 Previous Experimental Work
 Composition and Preparation of...
 Experimental and Analytical...
 Results of the Experiments
 Discussion
 References
 
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