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Journal of Petrology Volume 42 Number 4 Pages 673-683 2001
© Oxford University Press 2001
Calculation of Phase Relations Involving Haplogranitic Melts Using an Internally Consistent Thermodynamic Dataset
1DEPARTMENT OF EARTH SCIENCES, UNIVERSITY OF CAMBRIDGE, DOWNING STREET, CAMBRIDGE CB2 3EQ, UK
2SCHOOL OF EARTH SCIENCES, UNIVERSITY OF MELBOURNE, PARKVILLE, VIC. 3052, AUSTRALIA
Received October 28, 1999; Revised typescript accepted June 23, 2000
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHE THERMODYNAMIC MODELCALIBRATION OF THE MODELPREDICTIONS AND APPLICATIONS OF...DISCUSSIONREFERENCES |
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A simple thermodynamic model is developed for silicate melts in the system CaO–Na2O–K2O–Al2O3–SiO2–H2O (CNKASH). The Holland & Powell (Journal of Metamorphic Geology, 16, 289–302, 1998) internally consistent thermodynamic dataset is extended via the incorporation of the experimentally determined melting relationships in unary and binary subsystems of CNKASH. The predictive capability of the model is evaluated via the experimental data in ternary and quaternary subsystems. The resulting dataset, with the software THERMOCALC, is then used to calculate melting relationships for haplogranitic compositions. Predictions of the P–T stabilities of assemblages in water-saturated and -undersaturated bulk compositions are illustrated. It is now possible to make useful calculations of the melting behaviour of appropriate composition rocks under crustal conditions.
KEY WORDS: thermodynamics; melts; granite; dataset
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHE THERMODYNAMIC MODELCALIBRATION OF THE MODELPREDICTIONS AND APPLICATIONS OF...DISCUSSIONREFERENCES |
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The history of experimental investigation into the petrology of granites goes back to the days of N. L. Bowen and colleagues, summarized in the classic monograph of Tuttle & Bowen (1958)
on the origin of granitic rocks in the system NaAlSi3O8–KAlSi3O8–SiO2–H2O. Experimental and theoretical work continues, not only in the field of phase equilibrium studies but also on the structure of melts and their viscosities, densities and compositions as functions of pressure and temperature. The results of much of this endeavour have been collected in the book by Johannes & Holtz (1996). Thermodynamic approaches to calculating melting relations for basic magmas and wet granitic systems have also been made (e.g. Burnham, 1975; Nicholls, 1980; Berman & Brown, 1987; Stolper, 1989; Nekvasil, 1990; Ghiorso & Sack, 1995).
Metaluminous and peraluminous bulk compositions, primarily in the silica-saturated portion of the haplogranitic system, CaO–Na2O–K2O–Al2O3–SiO2–H2O (CNKASH), will be considered here. These include compositions made up of combinations of the end-member components albite (NaAlSi3O8), K-feldspar (KAlSi3O8), anorthite (CaAl2Si2O8), aluminosilicate (Al2SiO5), quartz (SiO2) and water (H2O). The purpose of this study is to extend the thermodynamic dataset of Holland & Powell (1998) to allow prediction of melting relations involving water-saturated and -undersaturated granitic melts. The effects of adding Fe and Mg will be addressed in a forthcoming publication on beginning of melting in pelitic and felsic rocks.
The approach followed in extending the thermodynamic dataset is to incorporate the experimentally determined melting curves for each of the individual minerals in CNKASH, dry and water saturated, as well as in other binary subsystems. In this way the melt model will be anchored in the overall system, particularly given that there is a large volume of data in the unary and binary subsystems. Also, the melt model will apply not only to ‘granite’ melts but also to melts in the broader CNKASH system. The model is validated by comparison with experimental results in larger subsystems. With an appropriate thermodynamic model, the advantage of being able to calculate melt equilibria is that the petrologist is not restricted to the P–T and composition conditions under which experiments are performed, but can consider any conditions within the range of applicability of the model.
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THE THERMODYNAMIC MODEL |
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TOPABSTRACTINTRODUCTIONTHE THERMODYNAMIC MODELCALIBRATION OF THE MODELPREDICTIONS AND APPLICATIONS OF...DISCUSSIONREFERENCES |
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The thermodynamic model adopted for the liquid is in the spirit of Nicholls (1980)
, involving the macroscopic mixing of a set of liquid end-members formulated, following Burnham (1975), on an eight-oxygen basis (with the exception of H2O) and assuming that non-ideality is accounted for by symmetrical interactions. The approach used is similar to that of Ghiorso & Carmichael (1980) and of Ghiorso & Sack (1995), although some assumptions are rather different; in particular: (1) the end-members in the melt used are simple mineral-like units such as albite (abL, NaAlSi3O8), K-feldspar (kspL, KAlSi3O8), anorthite (anL, CaAl2Si2O8), quartz (qL, 4SiO2) and sillimanite (silL, 8/5 Al2SiO5), rather than oxide and multi-oxide units on a variable oxygen basis; (2) volumetric properties use the Murnaghan equation of state; (3) partial molar volumes of H2O in the melt are modelled using the recent data of Ochs & Lange (1997); (4) the end-member liquid properties are calibrated entirely on melting experiments on subsystems in NCKASH. One of the principal goals of this model has been to describe and predict phase relations in unary, binary, ternary and quaternary subsystems as well as forming a basis for prediction of more complex natural granitic melts involving addition of Fe and Mg. Although water is known to exist as both hydroxyls and molecular water in silicate melts (e.g. Silver & Stolper, 1989; Novak & Behrens, 1995) and the distribution of these species depends on pressure and temperature, a macroscopic approach is here used that considers total H2O in the form of the single end-member, h2oL (H2O).
In using this model to calculate a feature such as the minimum melting curve K-feldspar + quartz + H2O = liquid in the system KAlSi3O8–SiO2–H2O, the melt phase is described by the three end-members kspL, qL and h2oL. Two compositional variables are required, and it is convenient to choose two of the three end-member proportions, denoted Xk (the third proportion being found by difference, 1 - 
Xk). There are three independent reactions between the end-members of the phases:
- 4q = qL;
- san = kspL;
- H2O = h2oL.
Writing the equilibrium relationships for each of these, 0 = 
G° + RTln K, the resulting set of non-linear equations may be solved iteratively at a chosen pressure, given suitable starting guesses for the compositions and temperature. The formulation of G° follows Holland & Powell (1998) whereas the equilibrium constants, K, use the activities of the end-members
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following Nicholls (1980), and the activity coefficients, 
k, are found from the regular solution model,
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In the expression above, 
l = 1 if l = k and l = 0 if l 
k, and Wij is the macroscopic interaction energy between end-members i and j (e.g. Powell & Holland, 1993). For more complex equilibria, for example in NKASH, the independent reactions between the phases will include ones to allow the equilibrium compositions of the solids to be determined.
Application of the model to the granite system requires calibration of the 15 binary regular solution energies amongst the six end-members in the melt as well as the thermodynamic properties of the six melt end-members themselves.
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CALIBRATION OF THE MODEL |
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The model requires values for the heat capacity, volume, thermal expansion, compressibility, entropy and enthalpy for each end-member in the melt.
The heat capacities, based on the work of Stebbins et al. (1984), of the melt end-members are taken from the compilation of Holland & Powell (1998). The volumes, thermal expansions and compressibilities were modified from the tables in Ochs & Lange (1997), via the use of a Murnaghan equation of state. with K' set to four so as to minimize the number of adjustable parameters required to satisfy the data. The properties for h2oL were also obtained in the same way.
The compressibilities so derived required some adjustment in the next stage of data retrieval during which the dry melting temperatures as a function of pressure were used to constrain the thermodynamic data for abL, kspL, anL, silL and qL. Fits to the melting experiments were made by non-linear regression to determine optimum enthalpy of formation, entropy and bulk modulus for each end-member to give best agreement between calculated melting curves and experimental brackets (Fig. 1). In addition, literature values for enthalpy of fusion (Lange & Carmichael, 1990) were used as further constraints in the least-squares analysis. The resulting set of thermodynamic data is listed in Table 1, and differs from that of Holland & Powell (1998) because (1) the volume behaviour of H2O is now based on Ochs & Lange (1997) rather than Burnham & Davis (1971), and (2) the enthalpies of fusion are now included as constraints. The data for sillimanite melting are provisional and are based on the suggested melting temperature (1200 ± 40°C) at ambient pressure from Cameron (1977) and an estimate of 1700–1750°C at 20 kbar from Holland & Carpenter (1986). The agreement of calculated curves and experimental brackets in Fig. 1 shows that the thermodynamic data and formulation used here provide an adequate description of the thermodynamics of melting of these mineral end-members.
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The thermodynamic properties of the h2oL end-member were derived simultaneously from (1) the solubility data for H2O in albitic melts, as measured by Goranson (1931, 1932), Burnham & Jahns (1962), Morey & Hesselgesser (in Clark, 1966) and Behrens (1995), and (2) the melting experiments of Goldsmith & Jenkins (1985). There are two equilibrium relations, ab (solid) = abL (liquid) and h2oL (liquid) = H2O (fluid). The H2O solubility experiments yield the entropy and enthalpy of the h2oL end-member from the second equilibrium relation, if the value for the regular solution parameter Wab h2o is known (in subscripts to W, the L part of the liquid end-member name will be omitted for ease of reading, so that Wab h2o replaces WabL h2oL). The temperatures of the wet melting of albite depend on the properties of the abL end-member (known from above) and the value of Wab h2o. Simultaneous fitting of the two equilibria yields the values for Wab h2o in Table 2 and the enthalpy and entropy of h2oL given in Table 1. It was found that a small pressure dependence was required for Wab h2o, a feature that probably incorporates the cumulative errors and simplifications in the models used here. Because the water contents of albitic melts are rather insensitive to temperature, calculated water solubilities (at 1100°C) may be compared with experimental values at a variety of temperatures in Fig. 2, where the agreement is seen to be good, given the scatter in the measurements. The predicted solubility of water in K-feldspar melts is a little lower than that in albite melts (Fig. 3), in agreement with the study of Behrens (1995).
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The interaction energies in the qL–h2oL and anL–h2oL binaries, Wq h2o and Wan h2o, were derived by adjusting their values to bring agreement with the wet melting curves of quartz (Fig. 4) and anorthite (Fig. 5), respectively. The value for Wksp h2o was derived from the wet melting experiments of Goldsmith & Peterson (1990) and, in the absence of any experimental constraint, the value for Wsil h2o has been estimated. The interaction energies among the components qL, abL, kspL and anL were determined by fitting the experimental eutectics for binary joins involving dry and wet melting; for example, the wet melting curves of albite and K-feldspar are shown in Figs 6 and 7. The calculated curves in Figs 2–7 show that a simple regular solution model is capable of reproducing the shape of the experimental data. All W values involving the silL end-member have been estimated, in the absence of appropriate experimental data to calibrate them, but this will introduce imperceptible errors in calculations because the proportion of silL in granitic melts is always small (e.g. Joyce & Voigt, 1994). Derived values of all of the interaction energies are given in Table 2.
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, Goranson (1931
, Shairer & Bowen (1947, 1955), Tuttle & Bowen (1958)
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The mixing properties of alkali feldspars in this study (albite–sanidine solutions) are taken directly from the subregular model of Thompson & Waldbaum (1969).
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PREDICTIONS AND APPLICATIONS OF THE THERMODYNAMIC MODEL |
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The calculated curves for wet melting are compared with experimental data for sanidine + quartz in Fig. 6, albite + quartz in Fig. 7, and albite + sanidine + quartz in Fig. 8. As these data were not used in the calibration for Tables 1 and 2, they provide a test of the predictive capability of the model. The agreement between calculated and experimental melting temperatures in the range 0–10 kbar is reasonable, especially when viewed in the light of the scatter present between the various experimental studies. The curves in Fig. 8 are simplified in that the change from alkali feldspar to albite + sanidine (solvus) and associated singularities are not shown here; however, these features are discussed below with respect to calculated phase diagrams such as in Fig. 11. An additional test is to compare the predicted water contents in a haplogranite melt of composition Qz28Ab38Or34 (weight percent basis) for which the experimental solubility data of Holtz et al. (1995)
are available. Figure 9 shows that the model used here, involving a macroscopic approach that ignores the details of the speciation of water, reproduces the measured solubilities adequately, particularly at lower pressures, but tends to underestimate the solubility at higher pressure. The change in temperature dependence of solubility, with isopleths of positive slope at low pressures and negative slopes at higher pressures (and high temperatures), is also predicted. It should be noted that different experimental studies (Goranson, 1931, 1932; Tuttle & Bowen, 1958; Luth et al., 1964; Steiner et al., 1975; Holtz et al., 1992, 1995) disagree by up to 1 wt % in absolute value for granitic melts.
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, Tuttle & Bowen (1958)
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Figure 10 shows a calculated P–T projection for the KASH system involving feldspar (ksp), muscovite (mu), quartz (q), sillimanite (sill), corundum (cor), liquid (liq) and H2O. The locations of the two invariant points are in very good agreement with those experimentally determined by Huang & Wyllie (1974, Fig. 3), and the quartz-absent curve linking them, ksp + sill + H2O = mu + liq, also has the same sense of curvature as suggested by Huang & Wyllie. The minimum melting curve in the peraluminous system ksp + q + sill + H2O = liq is lowered by about 25° relative to that in the haplogranite system (ksp + q + H2O = liq), in agreement with experimental evidence (Joyce & Voigt, 1994; Johannes & Holtz, 1996). The calculated position of the fluid-absent melting reaction mu + q = ksp + sill + liq, which would play a major role in melting of low-Na pelitic schists of low porosity, is also in excellent agreement with the experimental data of Huang & Wyllie (1974).
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Addition of Na to the KASH system dramatically lowers the calculated temperatures of melting, and Fig. 11 shows the P–T projection calculated for muscovite + albite + K-feldspar + quartz + sillimanite + corundum + liquid + H2O. The dashed lines show the reactions in the KASH system from Fig. 10 for comparison. Features to note are the azeotrope traces in the system at temperatures above the alkali feldspar solvus crest, the related singular points on the muscovite-absent reactions at temperatures below the solvus crest [the reader is referred to the discussion by Luth (1976, p. 339) or Morse (1980, p. 397) for further details] and the singular point on the albite-absent reaction (where quartz changes from product to reactant with rising temperature). Dehydration melting in pelitic rocks will begin at temperatures as low as 650°C at 4 kbar to 750°C at 10 kbar, heralding the change from muscovite schists to high-grade gneisses.
Figure 12 is a predicted ternary liquidus for ab–or–q at 5 kbar, showing the calculated contours and azeotrope traces. The experiments of Morse (1970) have the melting loop and solvus just meeting on the feldspar join at this pressure, whereas the thermodynamic model predicts that the solvus and the melting loop on the binary feldspar join have just separated. The singular (critical) point on the two-feldspar boundary at which the solvus meets the melting loop is shown as a simple open circle. The inset shows that adding excess alumina to the system at this pressure introduces a field of stability for muscovite + liquid.
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The simplest way to see phase relationships in complex systems is with pseudosections—phase diagrams showing just those phase relationships experienced by a particular bulk composition, or bulk composition line (e.g. Powell et al., 1998). The following pseudosections are based on a molar bulk composition: SiO2 = 75·00, Al2O3 = 18·32, K2O = 4·55, Na2O = 2·14. In the case of looking at melting relationships, choice of H2O content in the bulk composition is critical in that liquid has a strong affinity for H2O. The importance of H2O can be seen in the T–XH2O pseudosection at 5 kbar in Fig. 13a, and complemented by Fig. 13b, in which the H2O content in the bulk composition runs from zero through to just above H2O saturation at 850°C. The affinity of H2O for the liquid can be seen by the way the H2O saturation line behaves once the solidus is crossed. The kick-back of this line as muscovite melts out with rising temperature is an interesting (but geologically irrelevant) feature. It is interesting to note that the solidus temperature is not very composition dependent at this pressure, but the melt mode contours show that, as the amount of H2O in the bulk composition decreases, the amount of melt decreases. Nevertheless, this shows that heating a muscovite-bearing fluid-absent rock will cause it to melt at more or less the same temperature as an H2O-saturated rock at this pressure. Melting temperatures for fluid-present and fluid-absent melting diverge considerably at higher pressures (Fig. 11). The relative complexity of the pseudosection between 650 and 700°C at relatively high H2O contents relates to the H2O dependence of which of quartz and K-feldspar disappears first as melting proceeds.
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Considering a marginally H2O-saturated bulk composition heating up with no H2O infiltration (line A in Fig. 13b), the solidus is the [cor, ksp] univariant in Fig. 11, but substantial melting occurs across the [cor, H2O] univariant, and particularly the [cor, H2O, ab] divariant immediately above it, across which muscovite is lost. The solidus and the two dehydration-melting reactions referred to above for this bulk composition are shown rather more clearly in Fig. 14a. Such a migmatite may lose its melt, in which case the residue composition moves to the left in Fig. 13b as its H2O is removed with the melt. With no melt loss the rock returns down the line A. Significantly, much of the crystallization of the liquid, and consequent retrogression of such a migmatite, occurs across [cor, H2O, ab] and [cor, H2O] under H2O-undersaturated conditions.
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To see the P–T dependence of the equilibria in the context of the importance of H2O, P–T pseudosections corresponding to lines A and B in Fig. 13b are given in Fig. 14a and b complemented by Fig. 14c. The differences between Fig. 14a and b occur at temperatures above the wet melting curves, where K-feldspar is stabilized in melts to much higher temperatures (at higher pressure) in Fig. 14a. In this latter figure, the fluid-absent melting reaction mu + ab + q = ksp + sill + liq is responsible for the first appreciable melting with heating at pressures above 4 kbar. This is in contrast to the excess-H2O case, where the first appreciable melting occurs across the solidus reaction mu + ab + q + H2O = sill + liq at 650°C, at which point the albite is melted out. In Fig. 14a, for the geologically more appropriate situation of a fixed H2O content leading to H2O-undersaturated conditions for most super-solidus conditions, at moderate to higher pressures the melt modes and melt water contents mirror each other (Fig. 14c). However, at low pressure, the water contents cut through to the solidus, whereas the melt modes are closely spaced parallel to the solidus. This decrease of solubility of H2O in the melt not only gives the solidus its shape, but means that H2O saturation of melts formed under H2O-undersaturated conditions occurs at temperatures of some 20–70°C above the minimum melting at pressures below 4 kbar. This implies that such magmas will exsolve their H2O on decompression at greater depths (by some 2–3 km) than given by the minimum melting curve.
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DISCUSSION |
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The assumptions and simplifications made in the model are worth reiterating so that the limitations of the applicability of the thermodynamic model are appreciated:
- the melt model is developed from a simple macroscopic viewpoint, particularly in neglecting the speciation of water and the dissolved silicate in the accompanying aqueous fluid. No account is made of structural changes in melts with pressure and temperature, as reflected, for example, in viscosity changes, apart from letting interaction energies be a function of pressure. As such, there is an implicit assumption that the model applies to Na–K-dominated largely polymerized melts.
- The volume behaviour is assumed to obey the Murnaghan equation of state with an imposed constant value of K' = 4. Although this reproduces the silicate melt volumes reported by Ochs & Lange (1997) in the range 0–10 kbar, it may introduce error if extrapolated to very high pressures for hydrous melts. The derived values for the bulk modulus, as well as entropy and enthalpy, for melt end-members derived from the melting experiments may incorporate some errors introduced through this assumption.
- It is assumed that the fluid coexisting with the melt phase is pure H2O. This is a reasonable assumption, given that the silicate content of such fluids remains very small except close to the critical point for albite or quartz (Kennedy et al., 1962; Shen & Keppler, 1997). If such fluids were to be described by mixing models, the tendency to unmix to fluid + melt implies that activity is larger than mole fraction, causing water activities to remain close to unity in the aqueous component. However, the values for the regular solution parameters between silicate and h2oL end-members used in this model will probably reflect the effect of a slightly reduced water activity stemming from the pressure-induced solubility of silicate in the fluid.
The temperatures at which dry granitic rocks begin to melt at high pressures has become a matter of debate (Becker et al., 1998). The experimental results of Huang & Wyllie (1975) indicate melting at the solidus at 1050°C at 8 kbar, whereas more recent work (W. Johannes, personal communication, 1999) suggests that melting occurs at 
1050°C with 1 wt % water, and that extrapolation to dry conditions should yield a solidus nearly 90°C higher than this with a shallower dP/dT slope than that proposed by Huang & Wyllie. In addition, Becker et al. (1998) performed melting experiments at very low water activities (aH2O = 0·07) at 5 kbar and found melting temperatures (990°C) very similar to those encountered by Huang & Wyllie in nominally dry runs. This led them to suggest that small amounts of water-containing gel in the starting materials used by Huang & Wyllie might be the explanation of their low melting temperatures. Our calculations can shed further light in this debate. First, calculated melting temperatures for dry melting of quartz–albite and quartz–K-feldspar eutectics agree very closely with the experiments of Luth (1968) up to 20 kbar. Second, calculations for aH2O = 0·07 at 5 kbar reproduce the results of Becker et al. (1998) within experimental error, corroborating their inferences. Finally, the dry granite solidus calculated in this study occurs at temperatures of 1130°C at 8 kbar (Fig. 9). The slope as well as the position of our calculated granite solidus match the extrapolation of W. Johannes & F. Holtz (personal communication, 1999) remarkably well.
The model of melt thermodynamics presented here, although rudimentary and preliminary in nature, appears capable of reproducing the major features of the experimentally determined phase relations, and may be used to investigate some of the complexities seen in natural compositions. An extension of this model to include Fe and Mg and to predict the melting behaviour of pelitic schists and gneisses will be presented elsewhere.
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ACKNOWLEDGEMENTS |
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Alan Thompson, Francois Holtz and an anonymous reviewer are thanked for helpful comments and suggestions.
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FOOTNOTES |
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*Corresponding author. Email: tjbh{at}esc.cam.ac.uk
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