Clinopyroxene/Melt Trace Element Partitioning in Sodic Alkaline Magmas


 Clinopyroxene is a key fractionating phase in alkaline magmatic systems, but its impact on metal enrichment processes, and the formation of REE + HFSE mineralisation in particular, is not well understood. To constrain the control of clinopyroxene on REE + HFSE behaviour in sodic (per)alkaline magmas, a series of internally heated pressure vessel experiments was performed to determine clinopyroxene–melt element partitioning systematics. Synthetic tephriphonolite to phonolite compositions were run H2O-saturated at 200 MPa, 650–825°C with oxygen fugacity buffered to log f O2 ≈ ΔFMQ + 1 or log f O2 ≈ ΔFMQ +5. Clinopyroxene–glass pairs from basanitic to phonolitic fall deposits from Tenerife, Canary Islands, were also measured to complement our experimentally-derived data set. The REE partition coefficients are 0·3–53, typically 2–6, with minima for high-aegirine clinopyroxene. Diopside-rich clinopyroxene (Aeg5–25) prefer the MREE and have high REE partition coefficients (DEu up to 53, DSm up to 47). As clinopyroxene becomes more Na- and less Ca-rich (Aeg25–50), REE incorporation becomes less favourable, and both the VIM1 and VIIIM2 sites expand (to 0·79 Å and 1·12 Å), increasing DLREE/DMREE. Above Aeg50 both M sites shrink slightly and HREE (VIri ≤ 0·9 Å ≈ Y) partition strongly onto the VIM1 site, consistent with a reduced charge penalty for REE3+ ↔ Fe3+ substitution. Our data, complemented with an extensive literature database, constrain an empirical model that predicts trace element partition coefficients between clinopyroxene and silicate melt using only mineral major element compositions, temperature and pressure as input. The model is calibrated for use over a wide compositional range and can be used to interrogate clinopyroxene from a variety of natural systems to determine the trace element concentrations in their source melts, or to forward model the trace element evolution of tholeiitic mafic to evolved peralkaline magmatic systems.

Sodic clinopyroxene appear to more readily incorporate REE than their calcic equivalents (Marks et al., 2004), but despite fractionation of these minerals, melts in evolved (per)alkaline systems can attain high REE contents, even up to economic levels (Typically considered $1 wt % total Rare Earth Oxides; Kogarko, 1990;Downes et al., 2005;Marks et al., 2011;Sjö qvist et al., 2013;Goodenough et al., 2016;Mö ller & Williams-Jones, 2016;Borst et al., 2018). Precise knowledge of mineral/liquid partition coefficients for REE is required to forward model the effects of crystal fractionation on metal enrichment in residual liquids, and thus to understand the magmatic portion of the REE enrichment history of these systems. Furthermore, this facilitates interrogation of clinopyroxene from successive units of REE-mineralised nepheline syenite systems (cf. van Hinsberg et al., 2010;Slack & Trumbull, 2011;Dare et al., 2014;Vasyukova & Williams-Jones, 2019) and may reveal metal enrichment paths in ephemeral melts, providing insight into the relative importance of magmatic and hydrothermal processes for mineralisation.
Experimental investigations of element partitioning behaviour in alkali-enriched systems are limited in terms of composition (Wood & Trigila, 2001;Huang et al., 2006), and none yet have explored peralkaline conditions where molar (NaþK)/Al exceeds 1. Additional information has been obtained from natural samples by measuring the concentration ratios of phenocryst-glass pairs from volcanic and intrusive rocks (Larsen, 1979;Wö rner et al., 1983;Shearer & Larsen, 1994;Fedele et al., 2009;Severs et al., 2009;Mollo et al., 2016). However, these results may be biased by the presence of melt inclusions, mineral inclusions and mineral zoning, and their interpretation is commonly complicated by unknown or poorly constrained P-T-H 2 O-fO 2 conditions of equilibration and assumptions of closed-system behaviour.
In this contribution, we present trace element partition coefficients between sodic clinopyroxene and silicate melts of tephriphonolite to phonolite composition, as determined from internally heated pressure vessel experiments on synthetic and natural compositions. These are complemented by well-constrained natural volcanic phenocryst-glass pairs from Canary Islands pyroclastic fall deposits. We characterise the mineral compositional controls on element partitioning behaviour and present a predictive model for clinopyroxene/ melt element partitioning that can be used to generate clinopyroxene/melt partition coefficients using only clinopyroxene major element compositions (e.g. as measured by electron-microprobe). This approach permits forward modelling of element budgets during differentiation processes in magmatic systems, including evolved sodic alkaline systems, as well as providing a mineral-based tool that can be used to reconstruct parental melt compositions from clinopyroxene compositions in natural rocks.

METHODOLOGY Experimental starting materials
Clinopyroxene was synthesised in sodic alkaline silicate melts of varying composition to obtain a range of mineral compositions consistent with those in natural systems. Starting glass compositions are given in Table 1 and Supplementary Data Figs S1 and S2; supplementary data are available for downloading at http://www. petrology.oxfordjournals.org. Synthetic glasses L1 and L5 were prepared from reagent-grade oxide and carbonate powders, ground together in an agate mortar, decarbonated for 6 hrs at 450 C, and then homogenised in air for 3 hrs at 1400 C in a Pt crucible. Repeated fusion and grinding in agate ensured chemical homogeneity of the starting glasses, which was confirmed by electron microprobe analyses of the final fused glass. Finely ground Mud Tank zircon was added to the homogeneous major element glasses as a source of Zr, Hf, Nb and Ta; the glasses were then fused for a further three hours at 1400 C. The remaining trace elements were added as a cocktail of single element nitrate solutions (ICP-MS standards), dried onto the powdered glasses and then denitrified at 450 C for 30 minutes. The synthetic basanite L1 and phonolite L5 were mixed in varying proportions to make the low alkali (L) series of starting glasses, while dried NaOH and KOH powders were added to make the middle (M) and high (H) alkali starting glasses. Analyses of glass compositions L5 and H5, quenched from super-liquidus conditions, confirm that the melt was homogeneous at the start of the experiments (Supplementary Data Electronic Appendix 1).
A well-mixed rock powder from the Nechalacho Layered Suite in Canada (Mö ller & Williams-Jones, 2016, Unit NLS-9, sample L09-194-405Á5, alternative sample name VM 11-6) was used as experimental starting material for a more extreme peralkaline composition. This composition was not fused prior to loading into experiment capsules, so as to preserve the original volatile element concentrations. An experiment using this material that was quenched from super-liquidus conditions is homogeneous with respect to major elements, as determined from electron microprobe measurements.

Experimental equipment and procedures
In total, 36 partitioning experiments were conducted in a Harwood-type internally heated pressure vessel (IHPV) in the HP-GeoMatS laboratory at the Deutches GeoForschungsZentrum in Potsdam, Germany (Supplementary Data Electronic Appendix 1). Temperature was measured with Type-S thermocouples calibrated against the melting point of NaCl (Borchert et al., 2010, accuracy of 6 5 C at 200 MPa) at the top, middle and bottom of the furnace and showed there to be no thermal gradient over the capsule. Argon gas was used as the pressure medium, and pressure was measured with a strain gauge to an accuracy of 6 7 MPa. During the experiments, pressure was controlled automatically and held within 6 50 MPa of stated values.
Oxygen fugacity was buffered to the intrinsic redox conditions of the experimental setup, which corresponds to log f O 2 ca DFMQ þ1 in our H 2 O saturated charges (see Chou, 1986;Berndt et al., 2002;Jugo et al., 2010, FMQ is the fayelite-magnetite-quartz oxygen buffer). One experiment using the Nechalacho Layered Suite composition was performed in a double capsule with a haematite solid buffer, designed to increase f O 2 and promote crystallization of Fe 3þ -rich clinopyroxene (Eugster & Wones, 1962). This experiment ran at the Hm-Mt buffer, which corresponds to log f O 2 % DFMQ þ5, as confirmed by presence of both buffer minerals in the outer capsule following quench. Experimental temperatures were chosen to produce run products with a high proportion of melt relative to crystals for each of the compositions, and were found by trial and error to be 650 to 825 C, all at 200 MPa pressure (Table 2 and Supplementary Data Electronic Appendix 1).
Powdered starting glass and distilled, deionised water were welded inside 3Á0 or 3Á5 mm outer diameter platinum capsules of 20-25 mm length (see Supplementary Data Electronic Appendix 1). To verify that capsules were sealed, they were heated to 110 C and weighed before and after, as well as at the end of the run. The amount of water sealed inside the capsules varied with temperature and composition and was kept in excess of saturation (Carroll & Blank, 1997;Moore et al., 1998;Schmidt & Behrens, 2008). Water saturation was confirmed post-run by puncturing capsules and checking for water expulsion and mass loss on drying at 110 C.
The capsules were loaded into the vessel, pressurised to 200 MPa, then heated to super-liquidus temperatures for at least 16 hrs (Supplementary Data Electronic Appendix 1) to permit homogenisation of trace element concentrations, dissolution of water, and equilibration of f O 2 by exchange of H 2 through the capsule wall (Gaillard et al., 2002). Following homogenisation, temperature was lowered to run conditions. An initial set of experiments were cooled rapidly from homogenisation to run temperature at about 100 C/min. A second set of experiments were cooled to run temperature at 1 C/min to promote slow growth of crystals and to minimise the formation of compositional gradients in the melt. Vessel temperature was then cycled twice between run temperature (held for 30 min) and run temperature þ 10 C (held for 30 min, followed by cooling at 1 C/min) to promote dissolution of small crystals at the expense of larger grains, and to promote crystal growth close to the run temperature. Element partitioning results are consistent among experiments of different cooling paths. In all experiments, run temperature was maintained for >40 hrs to allow for chemical homogenisation of melt and growth of crystals via Ostwald ripening. Capsules were then quenched to room temperature.
To minimise the growth of groundmass crystals on cooling, a rapid-quench apparatus was used where possible (Table 2). We have not measured quench rates in this vessel, but the capsules probably cooled at rates of several hundred degrees Celsius per second (e.g. Berndt et al., 2002). For the rest of the experiments, quenching was achieved by cutting power to the furnace, which resulted in cooling to below the glasstransition temperature (<350 C; Giordano et al., 2005) in less than 150 seconds.

Sample preparation
Experiment capsules were torn open with pliers, and charges were split using a low-speed wafering saw.
Samples were mounted in epoxy resin and polished for in situ chemical analysis.
Natural pyroclastic rocks were rinsed in tap water and crushed with a hammer. Crystals with adhered glass, hand-picked from a sieved size fraction between 125 mm and 1Á18 mm, were used to make grain mounts containing 5-20 crystals of clinopyroxene per sample. Natural samples also contain biotite, sanidine, 6 spinel, amphibole, olivine, titanite and sodalite. Clinopyroxene mineral mounts were examined using backscattered electron imaging to select euhedral crystals that were free from melt inclusions and chemical zoning (n ¼ 6, Fig. 1).

ANALYTICAL TECHNIQUES
All experimental products were examined by reflected light optical microscopy and scanning electron microscopy, and all phases produced were identified by electron microprobe using an energy-dispersive spectrometer. Backscatter electron images were obtained to estimate modal proportions of phases, which were calculated using ImageJ freeware (Rasband, 2016; see Supplementary Data Electronic Appendix 1).

Electron microprobe analysis
Major element compositions of experimental products and natural minerals and glasses were measured with a JEOL 8900 EPMA instrument at McGill University and a JEOL 8230 EPMA instrument at the University of Ottawa, both using wavelength dispersive spectrometers (Table 3). An accelerating voltage of 15 kV was used with a 15 nA beam of 5 mm diameter for minerals, and a 4 nA beam of 50 mm diameter for glasses. Count times for all elements were 60 sec. Using the above routine, _RQ indicates use of a rapid-quench apparatus; (f) indicates failure of the rapid-quench apparatus, where capsules became jammed in the hot-zone of the IHPV (i.e. equiv. to regular quench); ** indicates use of a haematite double capsule, for run conditions at the haematite-magnetite fO 2 buffer (Eugster and Wones, 1962 we observed no sodium loss over the measurement time period. For the silicate minerals and glasses, both synthetic and natural minerals or oxides were used for calibration (see Supplementary Data Electronic Appendix 1). Analytical uncertainties were determined by multiple analyses of reference materials and duplicate analyses of samples. Element distribution maps were generated with a JEOL 8900 instrument at McGill University using an accelerating voltage of 15 kV, a focussed 40 nA beam, and a dwell time of 50 ms per pixel ( Fig. 3 and Supplementary Data Fig. S3). Fe, Ti, Al, Si and Na were measured with wavelength-dispersive detectors, while Ca and Mg were measured with an energy-dispersive spectrometer.
Transects of Ce, Mg and Fe concentrations across the experimental clinopyroxene were measured with a JEOL 8900 instrument at McGill University using a focussed beam of 50 nA with an accelerating voltage of 20 kV. Ce was counted with a wavelength-dispersive spectrometer with an LIFH crystal for 100 sec (MAC-Ce standard). Mg and Fe were counted for 20 sec, with TAP and LIF crystals respectively (diopside and olivine standards, respectively).

Laser-ablation ICP-MS
Trace element concentrations in clinopyroxene and glass were determined by laser-ablation ICP-MS. Analyses were performed at McGill University using a NewWave 213 nm Nd-YAG laser system coupled to a Thermo Finnigan iCAP-Qc quadrupole ICP-MS instrument. Typical fluence was 3-12 J/cm 2 (from 80 mm to 8 mm spot sizes), with a repetition rate of 10 Hz. Ablated material was transferred to the ICP-MS in a He flow of 800 mL/min and mixed with Ar prior to injection into the plasma. Instrumental drift was monitored by repeat analyses of the primary standard glass BCR-2G, with bias monitored by repeat analyses of secondary standards UTR-2 peralkaline rhyolite and USGS-RGM-1 rhyolite glasses (Supplementary Data Electronic Appendix 1). Sample surfaces were pre-ablated to remove residues from polishing materials and to improve ablation efficiency.
Primary and secondary standard glasses and an unzoned Canary Islands clinopyroxene were analysed with beam sizes of 8-80 mm to check for crater sizedependent element fractionation effects; none were found for the elements reported here. For minerals and glasses in the experimental charges, beam sizes of 8-40 mm and 16-80 mm were used, respectively. Depending on grain size and availability, 5-14 mineral grains and 3-10 areas of glass were analysed per charge. Natural clinopyroxene and rim glasses were analysed using a beam size of 12-80 mm and 12-30 mm, respectively. Wherever possible, mineral cores were avoided and ablation was performed along lines parallel to crystal rims (1-3 mm /sec).
Drift corrections and data reduction were conducted with the iolite v2.5 extension for Igor Pro (Paton et al., 2011). The total of major elements measured by LA-ICP-MS or, where available, the Al concentration from electron microprobe analyses, was used as an internal standard (Supplementary Data Electronic Appendix 1). For some experiments, ablation through the minerals was too rapid to generate a stable signal for data reduction purposes. In these cases, a mixing model was applied to estimate the composition of this clinopyroxene following Yang et al. (2018); see Supplementary Data Electronic Appendix 3. Trace element partition coefficients derived from these two data reduction methodologies are consistent to better than 2r.

Run product phase stability and crystallinity
Trace element data are reported for eleven experimental charges. 25 additional experiments were rejected as their run temperatures were super-liquidus or subsolidus, or because their growth textures were indicative of disequilibrium (e.g. Supplementary Data Fig. S3).

Phase proportions are given in Supplementary Data
Electronic Appendix 1.
All reported experimental runs were near-liquidus (3-22% crystals) and are characterised by a homogeneous distribution of phases ( Fig. 2 and Supplementary Data Fig. S3), except for experiments H5 3 and NLS-9 2À HM, in which sanidine crystals are concentrated at the centre. Glasses are clean, homogeneous and show a limited range of major and trace element compositions for each experiment (Supplementary Data Fig. S1 and S2). Clinopyroxene crystals are generally small, euhedral blades with a narrow range of sizes for a given experiment (<10 mm to 100 mm minor axis in cross section, Figs 2 and 3). In addition to clinopyroxene and glass, experiments on mafic to intermediate compositions produced magnetite, titanite 6 kaersutite amphibole, whereas some phonolitic experiments produced biotite, alkali feldspar 6 magnetite (Supplementary Data Electronic Appendix 1).
Constraining the full phase equilibria of all of the investigated compositions was beyond the scope of this study. However, in our experiments it can be seen that clinopyroxene has a wide stability field across the investigated physico-chemical conditions with only a single starting composition generating amphibole in place of pyroxene (experiments L3 and L3 2 ). Run conditions and run products given in Table 2.

Glass compositions
All run-product glasses are homogeneous across the length and width of the experimental capsules based on All values in wt %, except (Na þ K)/Al, which is a unitless molar ratio. LMB, the lower Montaña Blanca unit; UMB-II, the upper Montaña Blanca unit II; PV 2 Ka, the c.2 kyr basal pyroclastic fall deposit of Pico Viejo, all phonolites from Tenerife (Ablay et al., 1995). Montaña Samara is a monogenetic mafic cinder cone, also on Tenerife (Albert et al., 2015).
multiple electron microprobe and laser ablation ICP-MS analyses. The major element oxide compositions, as measured by electron microprobe, show relative standard deviations of <5% within experiment capsules, with minor element oxides (concentration <1%) showing greater variability, most with standard deviations between 5 and 10% relative (see Supplementary Data Electronic Appendix 1). Trace element compositions of the glass, as measured by laser-ablation ICP-MS, typically show time-weighted relative standard deviations of 1 to 8% within each capsule, with this variability depending on both the absolute concentration of that element and on the beam-size that was used for the analyses (Supplementary Data Electronic Appendix 1). Low totals of major element oxide concentrations in the electron microprobe analyses are a result of high dissolved water contents in the quenched melt. On a total-alkalis vs silica diagram the experiment glasses are predominantly phonolitic in composition, though span the trachyte-phonolite join (Supplementary Data Fig. S1). Their alkalinity index (molar (Na þ K)/ Al) is 0Á85 to 1Á40, crossing the alkaline-peralkaline join, and the Mg# of these quenched melts is 0 (Mg-free) to 22. The dissolved water content of these quenched melts is 8Á8 to 10Á7 wt % (by difference method from EPMA data, with the Fe oxidation state assigned following Kress & Carmichael, 1991) and their NBO/T ratios are 0Á98 to 1Á42 (Mysen et al., 1982(Mysen et al., , 1985; the ratio of non-bonding oxygen anions to tetrahedral cations in the melt). The halogen content of the synthetic experiments is nominally zero, whereas glasses from the Nechalacho Layered Suite composition experiments typically contain 0Á1% F and 0Á02% Cl by weight.
The Canary Islands glasses are mostly trachytic to phonolitic, with one basaltic trachyandesite (sample 17-12, Montaña Samara), and are alkaline to weakly peralkaline in composition (A.I. of 0Á78 to 1Á16) with Mg# of 8 to 39. Dissolved water content and NBO/T ratios are not reported for these glasses because of post-eruptive loss of volatiles. These Canary Islands glasses typically contain 0Á2% F and 0Á4% Cl by weight.

Compositions of the clinopyroxene and major element exchange
Experiments on the synthetic compositions produced clinopyroxene of diopside to aegirine-augite composition (Di 29-82 , Aeg 9-45 Hd 7-33 ), a subset of which overlap with the compositional space defined by the Canary  Islands clinopyroxene on a diopside-hedenbergiteaegirine ternary diagram (Fig. 4a, Table 3 and Supplementary Data Electronic Appendix 1). This lowaegirine group of synthetic clinopyroxene displays a positive correlation between IV Al þ IV Fe 3þ (in the IV T site) and VI Ti content (in the VI M1 site, red, low-aegirine, Aeg 5-25 , Fig. 4b), and notably contain both of these elements at elevated concentration relative to the Canary Islands clinopyroxene ( IV Al þ IV Fe 3þ ¼ 0Á20-0Á49 c.p.f.u., VI Ti ¼ 0Á07-0Á17 c.p.f.u.). A further subset of synthetic clinopyroxene have higher aegirine contents (blue, med-aegirine, Aeg 25-50 ) and show a similar range of Ti contents to the low-aegirine synthetic minerals (0Á05-0Á18 c.p.f.u.), but a limited range of lower IV Al þ IV Fe 3þ contents (0Á1-0Á25 c.p.f.u., Fig. 4b). Experiments performed on the Nechalacho Layered Suite composition (NLS-9, NLS-9 2-HM , green, high-aegirine, Aeg 55-95 ) reproduced the aegirine-augite to aegirine clinopyroxene from the natural system (Mö ller & Williams-Jones, 2016) that contain low concentrations of IV Al þ IV Fe 3þ (up to 0Á14 c.p.f.u.) and almost no Ti or Mg.
The six Canary Islands clinopyroxenes that were selected for determination of mineral-melt partition coefficients are diopside in composition (Di 65-71 , Aeg 3-10 Hd 21-28 ), with most crystals containing about 0Á05 c.p.f.u. IV Al þ IV Fe 3þ and 0Á02 c.p.f.u. VI Ti (Fig. 4b). These minerals are free from melt inclusions, suggesting slow rates of crystal growth (Kennedy et al., 1993, Fig . 1). Chemical zonation, if present, is of a similar magnitude to the analytical precision of our electron microprobe, indicating that pressure, temperature and melt composition remained stable during crystal growth (Fig. 1). Trace element abundances within these Canary Islands clinopyroxenes show core-to-rim variations of about 1-5% relative, with larger variations in element abundance associated with sector zoning than with concentric growth zoning (Supplementary Data Electronic Appendix 1).
In contrast our experimental clinopyroxene displays systematic, albeit minor compositional variation between their cores and rims; this variation recording changes to melt composition during growth of these  (Akasaka & Onuma, 1980). Fe 2þ /Fe 3þ within the clinopyroxene was assigned following Droop (1987), then site occupancies and end-member proportions were allocated following Morimoto (1989). End-member proportion calculations are given in Supplementary Data Electronic Appendix 6. The range of clinopyroxene compositions within each experiment record both sector zoning and changes to the composition of the melt during crystal growth, whereby the rims (highest Fe, Na) are in equilibrium with the quenched melt. The rims of this clinopyroxene were preferentially sampled in trace element analyses by LA-ICP-MS. crystals (e.g. Fig. 3; cf. Adam et al., 1993;Blundy & Dalton, 2000;Dygert et al., 2014). The systematics of crystal-chemical zonation both within individual experimental charges and within groups of experiments reveal major element exchange vectors between the clinopyroxene and the melt. Three major element exchange mechanisms correspond to the low-(Aeg 5-25 ), medium-(Aeg 25-50 ) and high-(Aeg 55-95 ) aegirine compositional domains defined above (Figs 4b, 5 and Supplementary Data Fig. S4).
For the low-aegirine series; 13ð IV Al; IV Fe 3þ Þ þ 4 VI Ti þ 3 VI Fe 3þ þ2 VIII Ca 13 IV Si þ 7 VI Fe 2þ þ2 VIII Na (1) For the medium-aegirine series; ð IV Al; IV Fe 3þ Þþ VI Ti þ 10 VI Fe 2þ þ10 VIII Ca IV Si þ 11 VI Fe 3þ þ10 VIII Na (2) and for the high-aegirine series; 3 IV Al þ 7 VI Fe 2þ þ10 VIII Ca 3 IV Si þ 7 VI Fe 3þ þ10 VIII Na (3) As the aegirine content of clinopyroxene increases, substitutions at the IV T site become relatively less important than exchanges at the VI M1 and VIII M2 sites (Fig. 5). At the VI M1 site, the importance of Ti substitution for ions of 2þ and 3þ valence also decreases with increasing aegirine content in the clinopyroxene. In low-aegirine clinopyroxene, the concentration of 3þ ions at the VI M1 site is negatively correlated with X M2 Na , whereas in medium-and high-aegirine clinopyroxene, the VI M1 site takes progressively more 3þ ions as X M2 Na increases. Substitutions of VIII Ca for VIII Na are relatively unimportant in low-aegirine clinopyroxene, but play a large role in medium-and high-aegirine exchange vectors.
The interdependent nature of clinopyroxene compositional terms provides a link between established compositional influences on trace element partitioning and those that are not traditionally considered (e.g. the IV Al content of low-aegirine cpx will be positively correlated with fO 2 due to VI Fe 3þ interdependency; essenite substitution; Eqn 1; cf. Mollo & Vona, 2014). Furthermore, not all of the VIII Na in the clinopyroxene is charge balanced by VI Fe 3þ ; some is charge balanced by VI Ti 4þ (Eqns 1, 2; cf. Flower, 1974). The latter effect correlates positively with the alkalinity of the starting compositions and may explain why D values for HFSE do not vary consistently with IV Al (Fig. 7).

DISCUSSION
The effect of chemical heterogeneity on apparent partition-coefficients During our experiments, crystals of clinopyroxene were grown directly from unseeded glasses and the run temperature was approached from super-liquidus conditions. No attempts were made to reverse these experiments by re-equilibrating crystals and liquids with deliberately mismatched trace element concentrations because of the slow diffusion of most elements through the clinopyroxene structure (Van Orman et al., 2001;Cherniak & Dimanov, 2010). As a result of this slow diffusion all experimental clinopyroxene preserves subtle concentric compositional growth zonation, generated as the cores of the grains become chemically . Sector zonation is also observed for some clinopyroxene and is associated with larger compositional differences; otherwise the major and trace element concentrations are generally uniform in the quenched melt and mineral phases (Supplementary Data Electronic Appendix 1).
Given the presence of growth zoning, only the crystal rims record true chemical equilibrium with the adjacent melt, and partition coefficients should, therefore, be determined from rim compositions. Unfortunately, in experimental studies, trace element analyses of crystal rims are rarely possible because the mineral grains are commonly comparable in size to the effective beam size of laser-ablation or SIMS systems ($ 10 À 20 mm), as is the case here. We quantify the departure of our apparent partition coefficients, calculated using bulk crystal compositions, from the true values based on the equilibrium clinopyroxene rim compositions using highresolution Ce-transects measured by EPMA (Supplementary Data Electronic Appendix 4). The transects cross seven experimental clinopyroxene from three experiments, each crystallized to varying degrees, with Ce as a proxy for the compatible elements. Based on these transects, apparent clinopyroxene-melt partition coefficients for compatible elements are offset from equilibrium (rim Ce) to higher values by 4-12% (apparent D values are calculated using bulk crystal Ce contents, expressed as the transect median Ce, see Supplementary Data Electronic Appendix 4). These values are not correlated with the degree of crystallization in the three experiments, however the highest values are found for the largest clinopyroxene crystals (c.100 mm) and in these, because of their size, we were able to avoid the cores in the analyses by LA-ICP-MS. The 12% deviation thus represents a maximum estimate of the bias from equilibrium in the partition coefficients. Moreover, at the low degrees of crystallization of our experiments median incompatible element concentrations in clinopyroxene are more subtly offset from equilibrium values than compatible elements because their concentration in the melt varies less during crystallization (see Supplementary Data Electronic Appendix 7).
Most importantly, the systematic positive bias in compatible D values observed here is small relative to the variation in partition coefficients for our complete sample set, as well as that of literature values (e.g. Fig. 7 below). Our experimentally-derived apparent partition coefficients plot on trends with clinopyroxene major element composition defined by literature data (Fig. 7). Because the potential bias in our apparent partition coefficients is not significant relative to the variation in the full dataset, no bias correction factor has been applied to our data. A limited deviation from equilibrium is further evidenced by a lack of correlation between the composition and the size of the analysed clinopyroxene, and the small variability in clinopyroxene and melt compositions within each experiment.

Attainment of equilibrium in the Canary Islands rocks
The Canary Islands trace element partition coefficients presented here were determined from euhedral, bladeshaped crystals free of melt inclusions and chemical zonation. The corresponding quenched melt was in direct contact with these crystals and shows no zonation in backscattered electron images (Fig. 1). While equilibrium conditions are challenging to confirm for a natural volcanic system, the euhedral forms, chemical homogeneity of crystals, and congruency between samples from separate eruptions suggest that the crystals grew in a stable environment, and were not subject to chemical or physical perturbations during growth (Figs 1 and  4). In the case of these Canary Islands clinopyroxene, Fig. 6. Diagram of trace element partition-coefficients between clinopyroxene and silicate melt, as determined from internally heated pressure vessel experiments (n ¼ 11; low-, medium-and high-aegirine types) and from clinopyroxene-glass pairs from Tenerife, Canary Islands (n ¼ 6; low-aegirine type). Uncertainties on the partition-coefficients are at the 1r level. the entire mineral is interpreted to be in chemical equilibrium with the adhered quenched melt.

Trace element partitioning
Apparent Nernst partition coefficients (D values) and their uncertainties were calculated as mass concentration ratios between clinopyroxene and coexisting glass (Table 4 and Supplementary Data Electronic Appendix 1). Three markedly different behaviours of rare earth element partitioning are observed in the experiments (Fig. 6). These depend on the aegirine concentration in the clinopyroxene and match the major element exchange vector domains discussed above. Low-aegirine experimental clinopyroxene (Aeg 5-25 ) prefer the MREE; medium-aegirine clinopyroxene (Aeg 25-50 ) shows a similar behaviour, save for higher LREE partition coefficients, whereas high-aegirine clinopyroxene (Aeg 55-95 ) strongly prefer HREE and show incompatible behaviour for the light and middle REE. The experimental REE partition coefficients are 0Á3-53, typically 2-6, with minima for LREE and MREE in high-aegirine clinopyroxene (Fig. 6). Apparent REE partition coefficients determined from our experiments are positively correlated with the IV Al content of the clinopyroxene and are an order of magnitude higher than most literature values, the majority of which were determined for more mafic compositions (e.g. Fig. 7e). The Canary Islands clinopyroxene and low-aegirine experiments show similar interelement partitioning systematics for the REE, with absolute values for the Canary Islands partition coefficients of about one order of magnitude lower (Fig. 6). The high field-strength elements (HFSE) Ti, Zr, Hf, Nb and Ta are compatible to slightly incompatible in the experimental clinopyroxene, and typically 1-2 orders of magnitude less compatible in the Canary Islands clinopyroxene ( Fig. 7a-c). D HFSE for the low-aegirine experiments and Canary Islands rocks plot on trends with the IV Al content of clinopyroxene, as defined by literature data from Italian volcanoes (Wood & Trigila, 2001;Fedele et al., 2009;Mollo et al., 2013Mollo et al., , 2016. In our medium-and high-aegirine experiments D HFSE values are not correlated with the IV Al (or VI Ti) content of clinopyroxene, consistent with a distinct incorporation mechanism for these elements relative to the low-aegirine experiments. Major element exchange mechanisms derived from our experiments suggest that HFSE (e.g. Ti) incorporation is charge balanced by substitution of VIII Ca for VIII Na, as well as by IV Si for IV Al, the relative importance of the former mechanism increasing with the alkali content of the system (cf. Eqn. 1-2). A weak positive correlation between X M2 Na and D i for the HFSE Ti, Zr and Nb is consistent with this mechanism (Fig. 7 &  Supplementary Data Fig. S5; cf. Flower, 1974).
Partition coefficients for the large-ion lithophile elements K, Sr, Pb are positively correlated with X M2 Na in the low-and medium-aegirine clinopyroxene, but are lower in high-aegirine clinopyroxene ( Fig. 6; Supplementary Data Electronic Appendix 1). The Rb, Cs and Ba partition coefficients have a high uncertainty and are maximum estimates owing to concentrations of these elements in the clinopyroxene being close to the LA-ICP-MS detection limit. Lithium is incompatible (D Li ¼ 0Á1-0Á4) in both Canary Islands and experimental clinopyroxenes and, like Sr and Pb, becomes more compatible with increasing aegirine content in the clinopyroxene, plateauing at X M2 Na ¼ 0Á4 and decreasing thereafter ( Supplementary  Data Fig. S5g). The actinides U and Th show contrasting partitioning behaviour; the former showing no correlation with aegirine content in the clinopyroxene, the latter becoming more incompatible with increasing aegirine content (Fig. 7h). The U and Th partition coefficients for our Canary Islands samples are similar to those for the Italian volcanoes, and are 1-2 orders of magnitude more incompatible relative to the experimental clinopyroxene.
In polymerised silicate melts (NBO/T < 0Á49), the melt structure can impart a significant influence on trace element partitioning behaviour (Gaetani, 2004;Huang et al., 2006;Schmidt et al., 2006). However, in the depolymerised peralkaline melts studied here apparent partition coefficients are not correlated with NBO/T, except for D Sr that shows a weak positive correlation (Fig. 8). NBO/T could not be calculated for our Canary Islands compositions because the water content of the melt prior to quench is not known.

Fits to the lattice-strain model
The equilibrium partitioning of trace elements between minerals and melts is largely controlled by the structure of the crystal lattice, its elasticity (Onuma et al., 1968;Kumazawa, 1969;Weidner & Vaughan, 1982) and its ability to accommodate an excess or deficit in charge (Blundy et al., 1998;Hanchar et al., 2001;Wood & Blundy, 2001;Corgne & Wood, 2005). The lattice-strain model provides a framework in which the influence of these variables on partitioning behaviour can be quantified, and thus predicted under conditions bracketed by a calibrating data set (Onuma et al., 1968;Blundy & Wood, 1994;Wood & Blundy, 2014).
To investigate systematics in D i values and the mechanisms by which trace elements are incorporated into clinopyroxene, element partitioning behaviour was explored in the light of lattice-strain theory, quantitatively described by the lattice-strain equation: where r 0 is the ideal radius for the lattice site, E s is the Young's modulus (i.e. the lattice site stiffness in GPa), D 0 is the strain-free partition coefficient, N a is Avogadro's number, R is the gas constant, T is temperature in Kelvin, and r i is the ionic radius of the element in question, all radii in Å . We focused on 3þ ions that cover a wide range of radii and fitted lattice-strain parameters for both the VI M1 and VIII M2 sites of clinopyroxene ( Fig. 9): Parabolae for 3þ ions were fitted for the VI M1 and VIII M2 sites using the REE, Ga and Al assigned to the VI M1 site of clinopyroxene (Fig. 9a). Fits are weighted based on uncertainties for the element partition coefficients. HREE have higher element partition coefficients than can be predicted by substitution into the VIII M2 site, hence were fitted with ionic radii for six fold coordination into the VI M1 site (Olin & Wolff, 2010;Reguir et al., 2012;Bé dard, 2014). Lattice-strain parameters as obtained from fits to the data are shown in Figs. 9-12 and Supplementary Data Electronic Appendix 1.
In some low-aegirine experiments and the Canary Islands rocks, lattice-strain fitting for 3þ ions at the VI M1 site was not possible, because too few HREE partitioned onto the VI M1 site of this clinopyroxene. Here, we chose to fit only lattice-strain parameters for the VIII M2 site, or fix D 3þ 0 values for the VI M1 site to match those for the VIII M2 site, and fit only the r 0 and E s parameters for the VI M1 site (Supplementary Data Electronic Appendix 1). Fitting of element partitioning data for 1þ, 2þ and 4þ ions was less successful owing to sparse coverage of suitable radii and detection limit issues for some elements. Partition coefficients for 1þ, 2þ, and 4þ elements follow radius-and charge-dependent trends consistent with lattice-strain theory and reported effects of charge on lattice-strain parameters ( Fig. 9b; e.g. Hazen & Finger, 1979;Law et al., 2000;Adam & Green, 2006).
Effects of composition on ideal site size, r 0 As the composition of clinopyroxene shifts from augite toward aegirine, the size of the VI M1 and VIII M2 sites, or strain-free radii (r 0 ), should diverge following the sizes of the major element cations on these sites. Lattice-strain fits for 3þ cations indicate expansion of the VIII M2 site between low and medium-aegirine clinopyroxene, with r 3þ 0; M2 correlating well with Na replacing Ca (Figs 9 and  10). Expansion of the VIII M2 site stalls at r 3þ 0;M2 % 1:12 Å and X M2 Na % 0Á4, changing little in size between medium and high-aegirine clinopyroxene. We suggest that this is a 'saturation effect', whereby the smaller ions in the IV T and VI M1 sites prevent further expansion of the VIII M2 site as additional R þ M2 (any cation of 1 þ valence) is added to the clinopyroxene. For the VI M1 site of clinopyroxene, strain free radii for R 3þ cations indicate expansion between low and medium-aegirine clinopyroxene and contraction between medium and high-aegirine clinopyroxene (Figs 9 and 10). These trends broadly follow the substitution of Mg 2þ for Fe 2þ , then Fe 2þ for Fe 3þ with increasing aegirine content in the clinopyroxene.
The effect of cation charge on the D 0 parameter The D 0 parameter of the lattice-strain model describes ideal, strain-free partitioning and tracks the solubility of an ideal cation in the mineral with changing pressure, temperature and the bulk composition of the system (Wood & Blundy, 2014). D 0 , therefore, correlates with the major element composition of the clinopyroxene, as well as that of the melt (e.g. Jd-melt, Jd-DiHd and CaTS-DiHd exchanges have been shown to control REE incorporation in cpx; Putirka, 2008;Wood & Blundy, 2014;Mollo et al., 2017). Incorporation of trace elements of a different charge introduces an electrostatic penalty that leads to a lower D 0 for that charge (Wood & Blundy, 2001. The average charge of major elements on the VIII M2 site of clinopyroxene decreases from 2þ to 1þ on the compositional join between Ca-rich diopside and (a) High-aegirine Medium-aegirine Low-aegirine Fig. 8. Diagrams of clinopyroxene-melt trace element partition coefficients for the IHPV experiments as a function of NBO/T of the quenched melt. NBO/T was calculated following Mysen et al. (1985) with melt Fe oxidation state assigned following Kress & Carmichael (1991) and the water content of the melt estimated by difference from 100% major element oxides.
Na-rich aegirine. Consequently, the electrostatic penalty for substituting a REE 3þ cation into the clinopyroxene VIII M2 site is increased (Fig. 11c). Conversely, as the average charge on the VI M1 site of clinopyroxene increases from 2þ toward 3þ in end-member aegirine, the electrostatic penalty incurred when substituting REE 3þ cations onto the VI M1 site is reduced (Fig. 11d) D 3þ 0;M1 consequently increases by an order of magnitude between our medium-aegirine and high-aegirine experimental clinopyroxene, an effect that when combined with the shrinking VI M1 site size, leads to strong fractionation of the HREE (Figs 9 and 11d).
A positive correlation between IV Al and partition coefficients for highly charged trace elements has been extensively documented in studies on clinopyroxene (Lundstrom et al., 1994;Gaetani & Grove, 1995;Blundy et al., 1998;Francis & Minarik, 2008;Hill et al., 2011;Mollo et al., 2016). The low-aegirine experimental clinopyroxene and most of the Canary Islands rocks extend trends defined by clinopyroxene from mafic systems (Figs 7 and 11a), whereas the remainder of the experimental data set and Canary Islands rocks show element partitioning behaviour similar to Italian volcanoes (Wood & Trigila, 2001;Fedele et al., 2009;Mollo et al., 2016), confirming that an IV Al-controlled substitution mechanism for REE extends to peralkaline conditions (Figs 7 and 11a).
D 3þ 0 parameters for the VI M1 site are strongly correlated with those for the VIII M2 site, except at aegirine (a) ( b) Fig. 9. Onuma diagrams for clinopyroxene-melt partitioning of R 3þ cations, as determined from internally-heated pressure vessel experiments. Parabolae are non-linear, weighted least-squares fits to the element partitioning data following the lattice-strain model of Blundy & Wood (1994). Shaded uncertainty envelopes, determined via bootstrapping, are at the 2r level. (a) Representative fits to 3þ ion partitioning behaviour with examples for low-(red), medium-(blue) and high-aegirine (green) clinopyroxene experiments. Vertical coloured lines indicate ideal ionic radii (r 3þ 0 ) of VI M1 and VIII M2 sites. (b) Measured partition-coefficients for ions of 1þ, 2þ, 4þ and 5þ charges that are consistent with the lattice-strain model. Uncertainties are shown at the 1r level. R 3þ ions were assigned to 6 or 8 fold coordination (Shannon, 1976), based on minimal residuals in the fit of parabolae (cf. Olin & Wolff, 2010). Y was not included in the fitting routine for R 3þ ions because of mass fractionation effects (ibid.). Fitted latticestrain parameters are given in Table 4.

High-aegirine
Medium-aegirine Low-aegirine with XNa for the VI M1 and VIII M2 sites of clinopyroxene. Shown for comparison are single crystal x-ray diffraction data from the hedenbergite-aegirine compositional join (heavy solid lines, from Redhammer et al., 2006). Shaded boxes represent the range of ionic radii for rare earth elements in VI and VIII coordination, with labelled fine dashed lines indicating selected ionic radii (Shannon, 1976). Literature data for Italian volcanoes are from Wood & Trigila (2001); Fedele et al. (2009); Mollo et al. (2013Mollo et al. ( , 2016 and for mafic systems from Hill et al. (2000Hill et al. ( , 2011; Gaetani (2004); Huang et al. (2006). concentrations exceeding 50 mol %. Similarities to VIII M2 partitioning behaviour likely reflect the dominance of IV T site substitution mechanisms in augite clinopyroxene. In the high-aegirine clinopyroxene, IV T site substitutions become less important as the IV T sites become saturated with Si 4þ (Fig. 5). Here, the dominant ion on the VI M1 site is Fe 3þ , the replacement of which by R 3þ trace elements does not introduce a charge penalty. Consequently D 3þ 0; M1 values for aegirine are higher than for more magnesian clinopyroxene.

An element partitioning model extending to aegirine clinopyroxene
Partition coefficients vary systematically with the physicochemical conditions of natural and synthetic magmas (cf. Wood & Blundy, 2003). A number of models have been presented that describe these systematics for clinopyroxene-melt partitioning (Wood & Blundy, 1997Hill et al., 2011;Yao et al., 2012;Bé dard, 2014;Dygert et al., 2014;Mollo et al., 2016Mollo et al., , 2018. The majority of these models are based on lattice-strain theory and predict how the lattice parameters r 0 , E s , and D 0 vary with composition, temperature and pressure. This semi-thermodynamic approach theoretically permits calculation of partition coefficients for any trace element, at any set of P ÀT ÀX conditions. In reality, all models have a limited applicable range, as restricted by the input data set. Because existing partitioning models do not reproduce the high r 3þ 0; M2 values for clinopyroxene with aegirine content that exceed Fig. 11. Diagrams of strain-free partition-coefficients (D 0 ) for 3þ ions into clinopyroxene vs clinopyroxene major element composition. (a, c) are for the VIII M2 site, and (b, d) are for the VI M1 site. The diagrams show that variability in partitioning behaviour is highly dependent on mineral composition, and that incorporation of trace elements into aegirine-rich clinopyroxene cannot be explained well by the same mechanisms as for mafic systems (c). Literature data for element partitioning in mafic þ meteorite and Italian volcano compositions are from the compilations of Bé dard (2014) and Mollo et al. (2016). 1r uncertainties on the fitted lattice strain parameters are smaller than the symbol sizes.  Table 5. 50 mol % (Fig. 12a), they cannot accurately predict REE partitioning behaviour for strongly peralkaline systems. We here present a new empirical model calibrated on both our work (n ¼ 16) and existing partitioning data from the literature (n ¼ 75, compilation of Bé dard, 2014 and Mollo et al., 2016;Fig. 12; Supplementary Data Electronic Appendix 1) to extend the applicable compositional range for our model. Our calibration database covers a wide range of composition, pressure, temperature and oxygen fugacity (0Á0001-3Á5 GPa, 650-1345 C, log f O 2 ¼ IW to MH % DFMQ -5 to þ5). Clinopyroxene compositions are XMg 0Á031-1, X Na M2 0-0Á84 and IV Al 0-0Á49 c.p.f.u. and melt compositions are Mg# (0-100) and XH 2 O (0-0Á38). REE partitioncoefficients are D La 0Á01-4Á79; D Sm 0Á02-47Á24, and D Yb 0Á11-8Á00. The majority of partition coefficients in the data set were measured via SIMS or LA-ICP-MS, minimising analytical uncertainty (e.g. from analyses by electron microprobe). An empirical partitioning model that includes both clinopyroxene and melt parameters has the highest potential for accuracy (e.g. Wood & Blundy, 2014). However, such models cannot be applied in geological scenarios where melt composition cannot be directly measured, for example to the cores of zoned phenocrysts in tephra, or to cumulate systems. Therefore we restrict our model to clinopyroxene compositional terms alone, an approach that is valid because crystallization is a thermodynamically-controlled process and the composition of the melt, and thus its effects on element partitioning are, at least in part, recorded by the major element composition of the clinopyroxene.

The clinopyroxene M2 site
To determine the principal physico-chemical factors that affect element partitioning at the VIII M2 site of clinopyroxene, a stepwise least-squares multiple linear regression analysis was performed using the lattice-strain parameters r 3þ 0 , D 3þ 0 and E 3þ s , temperature, pressure and clinopyroxene composition as inputs. Input parameters were initially examined in binary scatter diagrams to ascertain whether correlations with lattice-strain parameters were linear. If not, interaction compositional terms were added to the initial set of possible fitting parameters that had linear correlations with latticestrain parameters (e.g. X T AlþFe 3þ ). Intensive variables for multiple regression models for r 0 , D 0 and E s were introduced following a hierarchical forward selection criterion with switching. Only parameters that were statistically significant at the 95% confidence level were retained, with the largest number of significant terms being eight for E M2 (cf. Supplementary Data Electronic Appendix 5). The resultant empirical model accounts well for changes in lattice-strain parameters over a range of compositions from basalt to peralkaline phonolite, reproducing large r M2 0 values typical for sodic clinopyroxene ( Fig. 12a; model coefficients in Table 5). Student t-tests show that all of the independent variables included in the models are significant at the 95% confidence level and PRESS R 2 values obtained by repeated random sub-sampling of the dataset (Stevens, 1996) are close to R 2 values calculated by regular methods, indicating that the models are robust and have high predictive power. Full multiple regression reports are available in Supplementary Data Electronic Appendix 5. Equations generated by the multiple linear regression calculations are given below for the VIII M2 site, where a i are the regression coefficients for the respective variables: The model for r M2 0 incorporates compositional controls from the VI M1 and VIII M2 sites, as well as temperature. Elevated concentrations of large VIII M2 cations Ca 2þ and Na þ are correlated with large VIII M2 sites. Ti 4þ cations in the neighbouring VI M1 site are also correlated with expansion of the VIII M2 site, and the concentration of small Al 3þ minus larger Fe 3þ on the VI M1 site is negatively correlated with r M2 0 . The negative correlation between r M2 0 and temperature reflects the sum of changes to major element composition that lead to smaller clinopyroxene VIII M2 sites at higher temperatures. This compositional effect swamps the minor influence of thermal expansion.
The model for D M2 0 incorporates compositional terms from all three sites in clinopyroxene, and temperature. The positive effect of tetrahedral R 3þ on D M2 0 is the largest contribution to the model, which is consistent with published studies (see above). The relationship between clinopyroxene compositional terms on the VI M1 and VIII M2 sites and D 0 are indirect and are tied to the solubility of the mineral in the melt (Wood & Blundy, 2003), which in turn is tied to the physico-chemical conditions of the system (largely melt composition). The model for D M2 0 is less robust that that for r M2 0 , largely because there are melt compositional effects that are not recorded in the composition of the clinopyroxene. We tested the Mg# and XH 2 O of the melt, neither of which are significant predictors for D M2 0 (95% confidence interval).
The model for E M2 is less well constrained than for the other two VIII M2 lattice-strain parameters, suggesting that VIII M2 site stiffness is not tied strongly to clinopyroxene composition, temperature or pressure. Despite a significantly lower predictive power, this model still has physical grounding. Stiffness of the VIII M2 site is positively correlated with pressure, as might be expected following a simple Hooke's law relationship, and there are some subtle compositional controls imparted by the IV T and VI M1 sites. The poor correlation between E M2 , clinopyroxene composition, temperature and pressure is also evident in published element partitioning models, where E M2 is either poorly predicted (Fig. 12c), or set to a fixed value (e.g. Dygert et al., 2014).
Diagrams of measured vs predicted D values for R 3þ cations are given in Fig. 13a, showing the predictive power of the models over a compositional range between basalt and peralkaline phonolite. For the VIII M2 site, 95% of the measured R 3þ partition coefficients are reproduced within a factor of þ2:5 À2:9 (hard dashed lines), and in extreme cases, the model still reproduces D values within an order of magnitude, sufficient for the prediction of element partitioning trends over a wide range of P À T À X. D MREE , including Sm, are reproduced more faithfully than D LREE , because their radius is closer to r M2 0 (Fig. 13c, d), and, therefore, prediction of their partitioning behaviour is affected less strongly by inaccuracies in predicted E M2 values.

The clinopyroxene M1 site
Using a methodology similar to the VIII M2 site, we fitted a predictive model for partitioning of R 3þ cations onto the smaller VI M1 site of clinopyroxene. Lattice-strain parabola were constrained by partitioning data for Cr, Ga, Sc, and, where suitable, the HREE Tm, Yb and Lu (Our IHPV experiments plus Hill et al., 2000;Fedele et al., 2009;Mollo et al., 2013;Dygert et al., 2014). The data set for the VI M1 site partitioning model is small relative to that for the VIII M2 site (n ¼ 18), and because it is strongly skewed towards alkaline compositions, it has lower predictive power and is not recommended for application to mafic magmatic systems. Equations for the VI M1 site lattice-strain parameters, as generated by multiple linear least squares regression, are given below and shown in Fig. 12 where b i are the regression coefficients (Table 5) for the respective variables: The model for r M1 0 is robust and accurately reproduces the input data set. A negative pressure term may reflect compressional strain on the crystal lattice. Fe 3þ cations have a positive effect on the size of the VI M1 site, while E M1 is predicted more accurately than E M2 and is largely described by variations in temperature and pressure. Much like the VIII M2 site, the stiffness of the VI M1 site appears to be controlled dominantly by physico-chemical factors that are not recorded in the composition of the clinopyroxene.
The model for D M1 0 contains compositional terms from all three crystallographic sites in clinopyroxene. X T AI has a strong positive correlation with D M1 0 , consistent with a charge compensation mechanism that aids incorporation of R 3þ cations, while terms for VI M1 and VIII M2 site cations may be indirectly recording melt compositional effects. Because D M1 0 is unusually high for our high-aegirine experiments, these had to be excluded from the fitting procedure to permit model convergence. Therefore, while the models for r shows a comparison between measured partition coefficients and model-derived values for the VIII M2 site of clinopyroxene. Hard dashed lines represent 95% (2r) confidence intervals of the model, and correspond to maximum uncertainties of factor þ2:5 À2:9 . Fine dashed lines represent 1 order of magnitude uncertainty (extreme outliers for VIII M2 model). Partition coefficients in this diagram are the REE La to Er for our IHPV experiments, Canary Islands rocks, and literature data from the Italian volcanoes (Wood & Trigila, 2001;Fedele et al., 2009;Mollo et al., 2013Mollo et al., , 2016, and all the REE plus Y for the rest of the data compilation (Bé dard, 2014), which is split by analytical methodology. (b) shows performance of the predictive model for the VI M1 site that is calibrated for alkaline magmatic systems, and includes data from our IHPV experiments and the Italian volcanoes Mollo et al., 2013Mollo et al., , 2016. Maximum uncertainties at the 95% (2r) confidence interval are a factor of þ7 À11 , higher than for the VIII M2 site because of the smaller calibrating data set. (c) performance of the VIII M2 site model for La, and (d) for Sm. between those that generated our medium and highaegirine clinopyroxene would be required to better constrain the clinopyroxene compositional record of D M1 0 in strongly peralkaline Fe-rich magmas.
When applied to our experimental data, and the compilation of partition coefficients from Italian volcanoes Mollo et al., 2013Mollo et al., , 2016, the VI M1 stepwise model reproduces element partitioning data to a factor of þ7 À11 at the 95% confidence interval (Fig. 13b). Full regression reports are provided in Supplementary Data Electronic Appendix 5.
To assess the utility of the partitioning models and to monitor for potential introduction of radius-dependent bias, we show predicted REE patterns normalised to measured ratios for some literature data and our internally heated pressure vessel experiments (Fig. 14)

Implications for formation of REE deposits in evolved alkaline intrusions
The solubility of REE and HFSE minerals is strongly enhanced in peralkaline melts (Watson, 1979;Linnen & Keppler, 1997;Boehnke et al., 2013;Aseri et al., 2015), and thus the high concentration of these elements in peralkaline systems may (partially) reflect this fact (Dostal, 2017). Melts containing high concentrations of REE and HFSE are thought to be generated through low degrees of partial melting in the mantle, followed by residual enrichment during protracted fractional crystallization (Marks & Markl, 2017). The budget of REE and HFSE in a fractionating magma is influenced by the mineralogy of the crystallizing assemblage, and the extent to which these elements are incorporated at minor or trace concentrations.
Clinopyroxene is a major ferromagnesian phase that is commonly saturated throughout the entire differentiation histories of peralkaline magmatic systems (Ablay et al., 1998;Marks & Markl, 2001;Mö ller & Williams-Jones, 2016). The composition of the fractionating   Marks & Markl, 2003). C/C 0 is the modelled concentration of La or Sm divided by the initial concentration of this element in the melt. Phase relations and clinopyroxene compositions were determined experimentally by Giehl et al. (2013) and pertain to both oxidising and reducing conditions (log f O 2 ¼ DFMQ þ1 and -3), nominally dry to water bearing (to 3 wt % H 2 O at 100 MPa). Colour shading indicates the weight fraction of clinopyroxene within the crystallizing assemblage. Bold arrows indicate residual enrichment pathways for the REE in the melt for Rayleigh fractionation (no incorporation into crystals), reduced dry conditions, and oxidised dry conditions (the latter two are least-squares power law fits to the data). For simplicity, this model does not consider REE incorporation into magnetite, alkali-feldspar, olivine, nepheline or aenigmatite (see Larsen, 1979;Kovalenko et al., 1988;Mahood & Stimac, 1990;Fedele et al., 2015), all of which are phases generated in the experiments of Giehl et al. (2013). Calculations follow White (2013). clinopyroxene has a strong impact on the absolute REE concentrations and REE pattern of the residual melt, and ultimately on the ability of a system to develop economic concentrations of the REE (e.g. Kogarko, 1990;Sørensen, 1992;Marks et al., 2011). Clinopyroxene in alkaline magmatic systems is initially calcic for mafic melts, and becomes increasingly sodic as crystal fractionation proceeds (Marks et al., 2004). Although the REE are compatible in the majority of our experimentally generated clinopyroxene, those approaching aegirine end-member composition, as found in evolved alkaline magmatic systems, have the lowest D REE values (Fig. 6). Strongly alkaline magmatic systems are thought to crystallize abundant Ca-clinopyroxene early in their evolution that may deplete residual liquids with respect to REE. Consequently, even though crystallization of Na-clinopyroxene could enrich residual liquids with REE, the resultant concentration of these metals in the melt would remain low. However, clinopyroxene is not the only phase to crystallize from alkaline magmas, and the majority of additional silicate phases, such as olivine, biotite and feldspar have DREE ( 1, typically 1-4 orders of magnitude lower than clinopyroxene (Larsen, 1979;Kovalenko et al., 1988;Mahood & Stimac, 1990;Fedele et al., 2015). Consequently, if the mode of clinopyroxene is low enough, the bulk D REE of the crystallizing assemblage would remain below unity, allowing the REE to become enriched in the residual silicate melt.
To give insight into the optimum conditions for residual magmatic enrichment of the REE in (per)alkaline systems we forward modelled the evolution of REE concentrations in the melt during fractional crystallization of a nepheline syenite body ( Fig. 15; Supplementary Data Electronic Appendix 8). Phase relation data and clinopyroxene compositions are from the experimental study of Giehl et al. (2013). Their starting composition, based on the MiKa dyke, from the Gardar Province, Greenland, is already extremely evolved, with Mg# ¼ 2, (Na þ K)/Al ¼ 1Á44 and FeO* ¼ 12 wt %. In these models, crystallization under water-bearing, oxidising conditions produces a high fraction of clinopyroxene that depletes residual melts with respect to Sm, while subtly enriching La. Dry conditions promote abundant alkali feldspar (Afs) crystallization, which effectively enriches the REE content of residual melts. Under oxidising, dry conditions, the La/Sm ratio of the residual melt increases with fractionation, because Sm is more effectively incorporated into clinopyroxene. Residual enrichment is most effective under dry, reducing conditions because of a relatively lower fraction of clinopyroxene within the crystallizing assemblage. Because of this, the REE enrichment path of the residual melt is close to that of ideal Rayleigh fractionation. Under these reducing, dry conditions and at a temperature of 750 C, the experiments of Giehl et al. (2013) attained a crystal fraction of 0Á8. Here, residual melts would have 3Á2 times the La concentration and 2 times the Sm concentration relative to their starting composition (C/C 0 ). Clinopyroxene generated in the higher fO 2 experiments of Giehl et al., (2013) did not display systematic increases to VI Fe 3þ and IV Al, nor higher modelled D M1 0 and D M2 0 values relative to their reduced experimental clinopyroxene (Supplementary Data Electronic Appendix 8). The enhanced residual enrichment of the REE under reduced conditions here reflect lower proportions of clinopyroxene in the crystallizing assemblage, rather than a clinopyroxene compositional control.
Considering the modelling results above alongside our experimental and Canary Islands data, the best systems to develop high REE concentrations are those that would produce small proportions of Ca-clinopyroxene early in their crystallization histories, quickly evolving to more sodic compositions that crystallize aegirine clinopyroxene. Cooling under low-pressure, dry, reducing conditions produces abundant alkali feldspar that in the case of a peralkaline composition, would serve to further increase the alkalinity of the residual melt. Lowdegrees of source melting would produce primary melts with (1) high REE concentrations and (2) low melt Mg þ Fe, and low modal abundance of clinopyroxene, which would aid residual enrichment in silicate melt via fractional crystallization.
The HREE-rich nature of peralkaline silicate magmatic systems, both granites and nepheline syenites, is compatible with fractionation of moderately sodic clinopyroxene that has high D LREE /D HREE (e.g. Mö ller & Williams-Jones, 2016;Dostal, 2017). As crystal fractionation progresses and clinopyroxene composition evolves toward the aegirine end-member composition, D LREE /D HREE decreases (Fig. 6). This systematic change in element partitioning behaviour would result in strong HREE enrichment in aegirine-clinopyroxene cumulates, and would enrich the residual melt with respect to LREE-MREE.

CONCLUSIONS
Eleven IHPV crystallization experiments and six natural clinopyroxene-glass pairs from Tenerife, Canary Islands were investigated to determine the clinopyroxene-melt partitioning behaviour of trace elements in sodic alkali-rich magmas and to extend empirical partitioning models toward the aegirine end-member clinopyroxene composition. Our experiments reveal three distinct element partitioning behaviours for Na-rich clinopyroxene that depend on the aegirine content of clinopyroxene. Each of these is associated with a distinct major element exchange vector and it is unclear at present if the transition between these behaviours is smooth or step-like. For all investigated compositions, apparent REE partition coefficients are correlated with the IV Al content of clinopyroxene, confirming that a REE incorporation mechanism coupled with IV Al-Si substitution extends to peralkaline conditions. HFSE 4þ and HFSE 5þ partition coefficients determined from lowaegirine experiments (Aeg 5-25 ) and Canary Islands pyroclasts are also correlated with the IV Al content of clinopyroxene. No such correlation was found at aegirine contents exceeding Aeg 26 , indicating a distinct incorporation mechanism for HFSE in alkali-rich systems that is correlated with the VIII Na content of clinopyroxene.
Fits of R 3þ partition-coefficients to the lattice-strain model of Blundy & Wood (1994) indicate expansion of the VIII M2 site with increasing Na þ M2 , to a maximum r 3þ 0; M2 of 1Á12 Å at Na þ M2 ¼ 0Á4 c.p.f.u. Further expansion did not occur at higher Na contents, rather both the VI M1 and VIII M2 sites of clinopyroxene shrink in response to increasing P R 3þ M1 . Charge effects lead to a progressive increase in D M1 0 at the expense of D M2 0 , following the exchanges VIII Ca 2þ for Na þ and VI M 2þ for Fe 3þ .
We have calibrated a new empirical model that predicts partition coefficients for R 3þ ions between silicate melt and the VI M1 and VIII M2 sites of clinopyroxene. This model requires as input clinopyroxene major element composition, temperature and pressure and is calibrated for use on compositions between tholeiitic basalt and peralkaline phonolite. For convenience we provide an EXCEL spreadsheet of this model (Supplementary Data Electronic Appendix 6). Application of our empirical partitioning model to a peralkaline-silicate composition from the Gardar Province, Greenland, indicates that fractional crystallization under dry, low-pressure, low-fO 2 conditions permits efficient orthomagmatic enrichment of the REE. Under these conditions abundant Ca-Mg rich clinopyroxene crystallization is supressed in favour of an alkalifeldspar dominated assemblage with lower bulk D REE values. Here, clinopyroxene mode represents the dominant control on the REE budget of the residual melt, whereas clinopyroxene composition determines its REE pattern. The modelling further suggests that nepheline syenites which host REE deposits originate from lowdegree melts with sufficient alkali enrichment to saturate clinopyroxene similar to our medium-aegirine clinopyroxene (Aeg 25-50 ). Fractionation of such clinopyroxene enriches residual melts with respect to the HREE, in accord with the composition of REEmineralised nepheline syenite systems.