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Journal of Petrology Volume 42 Number 10 Pages 1869-1885 2001
© Oxford University Press 2001
Minor Phases as Carriers of Trace Elements in Non-Modal Crystal–Liquid Separation Processes I: Basic Relationships
DEPARTMENT OF EARTH SCIENCES, CARDIFF UNIVERSITY, PO BOX 914, CARDIFF CF10 3YE, UK
Received September 10, 1998; Revised typescript accepted March 16, 2001
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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Some trace elements have the property that, although they are incompatible with most mineral phases in magmatic systems, they are strongly concentrated in certain minor mineral phases. These minor phases, termed here ‘carrier-phases’, and their associated trace elements include platinum group elements in base metal sulphide and chromite; chromium and vanadium in magnetite; uranium group metals in zircon and monazite; and rare earth elements in monazite and xenotime. Carrier-phases may form only a small fraction of a source rock undergoing partial melting and tend to be eliminated from the residue at an intermediate point in the partial melting history; conversely, those same minor carrier-phases tend to precipitate late during fractional crystallization of a liquid produced in the above manner, but may constitute a high proportion of the cumulate then forming. This paper explores the phase equilibria aspects of such processes in a simple system, outlining a nomenclature which is then used in a mathematical treatment applicable to non-modal melting and crystallization processes involving several crystal species. The treatment at this stage assumes constant individual crystal–liquid distribution coefficients. Equations are developed, which are applied in a companion paper to illustrate the behaviour that can be anticipated when carrier-phases play a significant role in trace element location during melting and crystallization.
KEY WORDS: uranium; thorium; platinum group elements; carrier-phase; trace element
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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What is a trace element carrier-phase?
Here we consider aspects of the behaviour of those trace elements which are generally incompatible in mineral–melt equilibria, but which are compatible in a mineral species, the carrier-phase, which in general does not survive in the system when the melt fraction is high. This carrier-phase will often be a minor phase in the total crystal assemblage, but this is not a requirement. Throughout most of the discussion in this paper this carrier-phase is treated as though it is a crystalline solid, but the discussion is equally valid if the carrier-phase is an immiscible liquid (e.g. sulphide melt) as may well be the case in some of the most important potential applications of this study.
As examples, platinum group elements (PGE) are extremely compatible in the metal, chromite and sulphide phases. U and Th are extremely compatible in the trace minerals zircon and monazite, respectively. Such trace elements are highly incompatible in all the major phases of most rocks and for this reason are apparently difficult to separate from one another by melting or crystallization processes.
Less striking examples are provided by the behaviour of Ni in olivine, Sr in plagioclase, Cr in clinopyroxene and V in titanomagnetite during the separate or cotectic crystallization of these phases from basic magmas, and by the behaviour of Ba in potassium feldspar in the partial melting of crustal rocks or the crystallization of felsic magmas.
Petrological relevance of these considerations
Carrier-phases in which U, Th or the PGE are highly compatible may be present during partial melting of mantle peridotite and subsequent crystallization of the resultant basic magmas (provided that the mass fraction of liquid in the system is low); during incipient partial melting of crustal rocks; and during the later stages of the crystallization of granitic magmas. When these minor minerals compete for such elements in strongly contrasted ways, or when previously partially melted residues are being remelted, substantial variations in the concentrations and ratios of the PGE or in the U/Th ratio are predicted in the solid and liquid products. The extent to which these predictions are fulfilled in natural rocks may be used to constrain and quantify the processes involved, assist in ore prospecting and improve understanding of the distribution of the heat-producing elements within the crust.
Other fields where the considerations in this paper may prove important are in the separation during crystallization, or resorption during melting, of a vapour which acts as a carrier-phase for potentially volatile trace elements, and the associated separation or resorption of a metal phase in lunar or achondrite parent planet magmas as a result of the movement of oxygen into or out of the vapour phase where it is combined with C or S, which would otherwise be combined with iron as carbide or sulphide in the liquid.
Previous studies
The topic of non-modal melting and to a lesser extent non-modal crystallization, which forms the background to this paper, attracted much attention between 1967 and 1976 and there have been many subsequent applications, chiefly in the context of simulations of partial melting behaviour. Phase equilibria and major component aspects have been addressed by Presnall (1969)
. The general principles and specific formulations of trace element behaviour in non-modal melting have been addressed by Schilling & Winchester (1967), Gast (1968), Shaw (1970), Schilling (1971), O’Nions & Clarke (1972) and Hertogen & Gijbels (1976). Hertogen & Gijbels (1976) addressed the interfacing of the relationships derived by Shaw (1970) to extend them to cases where there are several stages of melting marked by the presence of different mineral assemblages, always with the restriction that the crystal–liquid distribution coefficients remain constant throughout each melting interval, and the modal melting proportions (i.e. the coefficients in the equation which represents the melting reaction) remain constant through each melting interval characterized by a particular coexisting mineral assemblage.
Hertogen & Gijbels (1976) extended the formulation to cover cases where neither of the above assumptions remains valid during a single melting interval. They pointed to the way in which these relationships might also be interfaced to deal with melting through a series of intervals with changing mineral assemblages, including the complications introduced by incongruent melting with or without the appearance of a new crystal species in the assemblage. They considered further complications which might be introduced by a change in melting style from equilibrium (batch) melting at low mass fractions of liquid development to fractional melting once melt connectivity had been established. They observed that the differential equations which they derived did not allow any general analytical solutions; pointed to the way in which the equations might be solved precisely, and suggested that stepwise calculation with adjustment of the parameters would be a practical way forward in many cases—a method which has in fact been implemented in several applications. Caution is required in this latter procedure, however, because any departure from infinitesimal steps has disproportionate effects on the predicted or modelled behaviour of highly incompatible elements during fractional melting, and on highly compatible elements during fractional crystallization (O’Hara, 1993).
Little attention has been paid previously to the formulation of relationships in non-modal crystallization, except for that implicit in equilibrium partial melting, which is the precise converse of equilibrium partial crystallization. Nevertheless, the effects of such crystallization have dominated the literature of layered mafic complexes for more than half a century.
Non-modal melting and the modal melting or crystallizing assemblage
In a simple (e.g. three- or four-component) system without crystalline solutions, when real multiphase mineral assemblages undergo partial melting, the minerals present do not, in general, contribute to the liquid in the same proportions as those in which they are found in the original rock. This is non-modal melting—the melt produced will not, on crystallization, yield solids having the same mode as the source. The liquid composition can be expressed in a simple equation as a mixture of the mineral phases present (involving negative coefficients if reaction relationships exist). The assemblage of minerals in the proportions in which they contribute to the formation of the liquid (the modal melting proportions) constitutes the modal melting assemblage. Similarly, in non-modal crystallization the minerals are, in the general case, precipitated in proportions (the modal crystallizing proportions) which are different from those in which they will be present in the final solid assemblage. Non-modal equilibrium crystallization is the precise reverse of non-modal equilibrium melting; in general, non-modal fractional crystallization is not the precise reverse of non-modal fractional melting, and neither is the same as the relevant equilibrium process.
The non-modal concept outlined above becomes less clear when there are major component crystalline solutions in the solid phases—the equation to express the liquid composition must now be written in terms of each end-member component of the crystalline solutions. The modal melting proportions of each mineral phase are then formed from the sum of the appropriate end-member components. Such equations may even be written for each trace element and the concept of distribution coefficients is inherent in such equations. This latter complexity is avoided in this treatment. The phase equilibria and the modal melting proportions are assumed to be defined by the major and minor chemical components in the system. Trace elements are assumed to dissolve into and partition between the liquid and solid phases with constant distribution coefficients between any pair of phases. Presence and variation in concentration of the trace elements is assumed to cause no significant modification of the phase equilibria.
Bulk and modal distribution coefficients
The bulk distribution coefficient, much used in the context of trace element modelling, is defined as the average concentration of an element in the total assemblage of solids present, divided by its concentration in the coexisting liquid. It is calculated by obtaining the products of the crystal–liquid distribution coefficient for each phase multiplied by the mass fraction of the solid assemblage which is composed of that phase. These quantities are then summed across all the phases present in the solid assemblage [as defined by equation (4) in a later section].
The bulk modal distribution coefficient emerges from the treatment as an important concept, not new to this paper. It is defined [see equation (5) in a later section] by combining the product of the crystal–liquid distribution coefficient and the mass fraction of each mineral in the proportions in which those minerals melt to contribute to that liquid, or precipitate simultaneously from that liquid during crystallization. Both distribution coefficients are specific properties of a given element in the system, but the bulk distribution coefficients are further specific to a given bulk composition within that system. The same formulation may be retained when one or more of the phases in the ‘solid’ assemblage is an immiscible liquid, a circumstance which probably arises when sulphides are involved.
As a rule of thumb derivable from examination of the figures or the equations in a later section (which are intended to be general in their application), significant effects of the type specifically addressed in this paper will be observed whenever the product of the crystal–liquid distribution coefficient for an element between an individual carrier-phase and the liquid, multiplied by the mass fraction of that carrier-phase in the solid mineral assemblage, yields a number significantly greater than unity. Evidently, when the crystal–liquid distribution coefficient is very high (e.g. 
1000–10 000), only trace amounts of the carrier-phase need to be present to produce significant effects. It is important to recognize that an element may be highly compatible in one of the mineral phases present, yet be at least mildly incompatible with respect to the solid assemblage as a whole.
Case of Ba and Rb in granitoid magmas
The broad outlines of what will happen to the concentrations of previously highly incompatible elements at the entry of a carrier-phase into the crystallizing assemblage were established by work on trace elements in differentiated igneous systems (Nockolds & Mitchell, 1947; Nockolds & Allen, 1953, 1954, 1956). The behaviour of Ba in granitoid magma systems provides an example of a nearly ideal trace element, which is incompatible in most of the early crystallizing minerals but is relatively highly compatible in potassium feldspar once that mineral becomes saturated at the liquidus. Barium concentrations in the residual liquids increase during the earlier stages of partial crystallization, peak at the point where K-feldspar starts to crystallize, and then decline as Ba is extracted into the potassium feldspar. Rb, on the other hand, remains incompatible throughout, even with respect to K-feldspar, although it is ultimately substantially hosted in this mineral. Consequently, Ba/Rb ratios in the residual liquids remain fairly constant during the early stages of crystallization, then decline once K-feldspar starts to crystallize. This sequence of events would be reversed during partial melting of suitable protoliths to produce granitoid magmas. In a simple and qualitative manner, this behaviour illustrates the way in which the phase equilibria behaviour of a carrier-phase can control the build-up or decline of some trace element concentrations and ratios.
Case of V in titanomagnetite in the Skaergaard intrusion
The basic geochemical principle identified by Nockolds & Allen (1953, 1954, 1956) is clearly illustrated in the Skaergaard intrusion (Wager & Brown 1967), which provides the classic case of nearly closed system near-perfect non-modal fractional crystallization of basaltic magma. The cumulus sequence is marked by the progressive entry of new, relatively minor minerals to the precipitating assemblage. V behaves as a mildly incompatible element in the magma up to the point where a mineral in which the element is relatively highly compatible begins to precipitate. This carrier-phase mineral is titano-magnetite in the case of V. The ratios of V to other elements of generally similar chemical character are little modified before the entry of the carrier-phase but thereafter are subject to rapid and extreme modification.
New approaches in this study
This study focuses on the behaviour of trace elements which are highly incompatible in most crystalline phases but are extremely compatible in one or two carrier-phases which are eliminated relatively early during partial melting, or which appear relatively late during crystallization. A simplified model is developed to illustrate their behaviour in terms of phase equilibria in a simple ternary system using largely graphical methods (Figs 1–4 ). Following this, a new approach to the theory of the behaviour of ideally behaved trace elements is presented, which extends previous treatments to multistage non-modal melting, to processes of fractional non-modal crystallization and to imperfect fractional non-modal melting and crystallization.
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The model employed in this paper recognizes, but ignores, all complications arising from variations in the crystal–liquid distribution coefficients with changes in the liquid composition or the concentration of the element. Elsewhere, Zou (2000) has addressed some aspects of the probability of variations in the distribution coefficients as they might affect the detection of a ‘garnet’ signal in erupted basalts. The model used here further assumes that there are no variations in the proportions in which the mineral phases enter or are extracted from the liquid within a melting or crystallization interval. It ignores all effects associated with solid solutions of major components within the solid phases, incongruent melting phenomena and the possible appearance of immiscible liquids, except insofar as these can be treated as simply another phase competing for elements of interest. Throughout this treatment complications such as large variations in the relevant distribution coefficients, which might arise from changes in the speciation of the trace elements in response, for example, to variations in oxygen fugacity, have not been considered. However, in this study some major geochemical effects are identified which are likely to survive, in principle if not in detail, the introduction of these additional factors. The model does not directly address the more sophisticated melting and crystallization models explored by O’Hara (1977, 1985, 1995), O’Hara & Mathews (1981), Langmuir (1989) and O’Hara & Fry (1996a, 1996b, 1997), although the potential effects of these upon the observable phenomena are surveyed qualitatively.
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TRACE ELEMENT CARRIER-PHASES IN A HYPOTHETICAL THREE-COMPONENT SYSTEM |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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The nature of the phenomena explored in this paper and the terminology necessary for the formulation used in a later section can be illustrated using simple phase equilibria diagrams. The equilibrium, integrated and small packet partial crystallization of bulk composition O (Fig. 1) in a hypothetical system analogous to a simplified Diopside–Forsterite–Anorthite was described by O’Hara & Fry (1996b
, fig. 3). In the postulated absence of solid solutions in the crystalline phases, the major element behaviour during perfect fractional crystallization of composition O will be the same as that described for its equilibrium partial crystallization, whereas equilibrium partial melting of composition O will simply be the reverse of the equilibrium partial crystallization sequence. The perfect fractional melting of bulk composition O is significantly different from equilibrium melting and is described below.
Equilibrium non-modal partial melting (ENMPM)
Major element behaviour
Partial melting of composition O begins with the appearance of a drop of liquid of major element composition E in equilibrium with all three solid phases. Further melting of composition O proceeds with the eventual production of 0·5 mass fraction of liquid with major element composition E, at which point the first solid phase ‘Anorthite’ has just been totally consumed.
During this stage of partial melting the proportions in which the three solid phases combine to form the liquid are fixed at the ratios specified by the composition E. These are the modally melting mass fractions, ti, of the three solid phases, symbolized as tFo3, tDi3 and tAn3, respectively, in the equilibrium between three solid phases and liquid. The subscripts identify which phase is involved and the appended superscripts convey the stage in the crystal–liquid history, counting from the first appearance of a crystal species downwards to the stage in which solidification occurs. This choice, which facilitates the handling of the equations, means that this superscript also, in this simple system, indicates the number of solid phases involved in this stage of melting; but it would not necessarily have to do so in more complex systems where discontinuous reaction relationships are involved.
The initial mass fractions of these solid phases in bulk composition O, symbolized as 0sFo3, 0sDi3 and 0sAn3, become modified during the melting process according to the relationship 0sq3 – tq3f = fsq3(1 - f), where the term fsq3 is the mass fraction of the residue which is still composed of phase q after mass fraction f of melt has formed in the equilibrium between three solid phases and liquid. It should be noted that this quantity must be multiplied by (1 - f) to obtain the mass fraction of the new state of the system which is composed of phase q.
At a critical value of the mass fraction of melting the system passes from having three solid phases in equilibrium with liquid to having only two. This mass fraction of melting is symbolized as fc2 and in this particular case is equal to 0·5. Here the first of the fsq3 terms, that for ‘Anorthite’, has been reduced to zero and ‘Anorthite’ has been totally consumed. Further partial melting of bulk composition O then proceeds with the development of liquids whose major element compositions evolve along E–L in Fig. 1 until another critical value of the mass fraction of melting, fc1 = 0·67, has been achieved. At this stage the liquid composition has evolved to L, and the second solid phase, ‘Diopside’, has now been totally consumed. As drawn in Fig. 1, the locus of the liquids E–L is a straight line (special case) and consequently the modal melting mass fractions of ‘Diopside’ and ‘Forsterite’ are given by the composition LE' throughout. Although the absolute magnitude of tFo2 and tDi2 have increased significantly relative to tFo3 and tDi3, their ratio has changed little in this particular case and their values remain approximately constant throughout the equilibrium involving these two solid phases and liquid.
In the meantime, the bulk composition of the residual solid assemblage evolves from O to C at fc2 = 0·5, always in equilibrium with liquid E. Then it evolves from C to ‘Forsterite’ at fc1 = 0·67 in equilibrium with liquids between E and L. The progressive evolution of the composition of this residual assemblage may be followed in the first melting interval by calculating the successive values of fsq3 at all values of f up to fc2 = 0·5. At the start of the second interval the values of fsq3 for the two remaining minerals are those to which the values of 0sq2 at fc2 = 0·5 are set. Then it is possible to follow the evolving composition of the residue by calculating the successive values of fsq2 as f increases further, using the appropriate values of tq2. At f greater than the value of fc1 (0·67) melting will proceed to eventual completion in equilibrium with just one solid phase and with liquid compositions evolving along L–O. In this stage of melting it is obvious that fsFo1 and tq1 must always have the value 1·0.
Trace element behaviour
Now we consider the behaviour during partial melting of bulk composition O of three trace elements, X, Y and Z, which are ideally behaved in the sense that their crystal–liquid distribution coefficients remain constant for each mineral species as the melting process proceeds. We suppose, however, that they display very different preferences for the available host minerals. These elements are assumed to be present in amounts so small that their presence does not significantly alter the phase equilibria. The three trace element concentrations are assumed to be normalized in such a way that they each have equal unit fractional concentrations in the bulk system and hence that their concentrations and concentration ratios taken in pairs are all 1·0 in the final liquid at f = 1·0 (Fig. 2) and when the individual concentrations of these trace elements are normalized to their sum, the bulk system composition falls at O (Fig. 3).
Let element X be highly incompatible in all the solid phases. The bulk distribution coefficient for X between any residual solid assemblage and the liquid must in consequence always be very low. The distribution coefficients between each mineral in the modally melting solid assemblage and the liquid are very low, hence all of the bulk modal distribution coefficients must also be low. Almost all of element X will enter the first drop of liquid to form with composition E, and thereafter it will be simply diluted by melting of more of the solid phases, which now contain very small amounts of element X, whose concentration is declining proportional to the value of 1/f as f increases (Fig. 2).
Let element Y be highly incompatible with respect to ‘Forsterite’, highly compatible with respect to ‘Diopside’ and neutral with respect to ‘Anorthite’. The bulk distribution coefficient for element Y at the start of melting of bulk composition O will be moderately high because of the substantial mass fraction of ‘Diopside’ in the assemblage. The contribution from ‘Diopside’ dominates the calculation for the bulk distribution coefficient of element Y in composition O. The modal melting distribution coefficient will be similar to that for the modal melting solid assemblage E because it contains a comparable mass fraction of ‘Diopside’. When ‘Anorthite’ has just been eliminated from the residue, the modal melting assemblage jumps to LE' (Fig. 1). The bulk modal distribution coefficient for element Y in the equilibrium between the two solid phases and liquid will now be very high because of the high proportion of the carrier-phase, ‘Diopside’, in the modal melting assemblage. The bulk distribution coefficient for the total residual solids in equilibrium with liquids along E–L, however, will be much lower than the modal distribution coefficient and it will decline steadily as melting proceeds because of the lower and declining mass fraction of ‘Diopside’ in the residual assemblages evolving between composition C and ‘Forsterite’.
Partial melting in the second stage with ‘Diopside’ and ‘Forsterite’ in equilibrium with a liquid between E and L proceeds with element Y becoming progressively more concentrated in the ever smaller amount of the carrier-phase, ‘Diopside’. However, the concentration of Y in the liquid is also increasing (Fig. 2). The total budget of Y in the system is coming to reside increasingly in the liquid phase—the concentration of element Y in the liquid phase will be at its maximum when the liquid composition is L and ‘Diopside’ has just been eliminated—virtually all of element Y is then in the 0·67 mass fraction of liquid present and further melting merely dilutes that concentration (Fig. 2). Element Y remains compatible in the modally melting mineral assemblage relative to the liquid throughout this stage, but the bulk distribution coefficient between total residue and the liquid must decrease below unity as liquid composition L is approached because of the interplay of the distribution coefficients and the ever declining mass fraction of the carrier-phase.
The partial melting history of bulk composition O, as it concerns the behaviour of the element Y in the liquid, begins with an interval covering the first two stages of melting (f = 0–0·50 and 0·50–0·67). Through most of this stage there is plenty of the carrier-phase, ‘Diopside’, present and the concentrations of Y in the liquid and the residual carrier-phase increase only slowly. This is followed by a narrow critical melting interval close to f = 0·67, where the last of the carrier-phase is being eliminated. Concentrations of Y in both liquid and residual carrier-phase are maximized at this point, at values which approximate to C0Y/fc and C0YdDiY/fc, respectively, where C0Y is the initial concentration of element Y in the bulk system and dDiY is the crystal liquid distribution coefficient for Y between ‘Diopside’ and the liquid. It should be noted that the concentration of the element Y in the bulk residue must decline to a very low value in the critical melting interval, despite the increase of the concentration of Y in the carrier-phase, because of the decline in the amount of that carrier-phase. There then follows a dilution interval as f increases between 0·67 and 1·0, from the elimination of the carrier-phase up to total melting (Fig. 2).
Let trace element Z be highly compatible in ‘Anorthite’, its carrier-phase, but neutral in ‘Diopside’ and highly incompatible in ‘Forsterite’. The initial bulk distribution coefficient for Z is high, but it decreases to a low value as the first stage of partial melting proceeds and its carrier-phase, ‘Anorthite’, is progressively eliminated. The behaviour of Z as f approaches 0·5 and ‘Anorthite’ is eliminated is analogous to that described above for element Y, but is now related to the elimination of the appropriate carrier-phase ‘Anorthite’. When element Z is considered, partial melting intervals of slow concentration, critical melting and dilution as defined above in connection with element Y may again be recognized. The boundaries between these intervals are, however, displaced to the lower values of f associated with elimination of the carrier-phase for element Z. Clearly, these terms, like compatible and incompatible, are not absolute features descriptive of the element or the system, but terms which are descriptive of the behaviour of a specific element in a specific bulk composition in that system.
Ratios of the trace elements
Next we consider the evolving concentrations of, and ratios between, the three trace elements in the liquid and solid assemblages as melting proceeds (Figs 2 and 3). Caution needs to be exercised in approaching Fig. 3, which is a schematic representation, not a precise plot. The concentrations of the three trace elements have been normalized to their total concentration in making this diagram, a procedure which leaves binary mixing relationships as straight lines but destroys the simple proportional relationship between the two parts of a mixing line and the mass fractions of the two components being mixed, i.e. the mixture is collinear with the components but the Lever Rule no longer applies.
Points E0 and E0·5 in Fig. 3 represent the ratios of the three trace elements in the liquid E of Fig. 1 at the start of melting and at the total consumption of the first crystalline phase ‘Anorthite’ at f = 0·5, respectively. The ratios in the liquid L0·67 at f = 0·67 are virtually identical to those in the bulk system O. Locus M3 represents the evolving ratios in the three-phase modal melting solid assemblage; locus M2 represents the evolving ratios in the two-phase modal melting solid assemblage between E and L of Fig. 1. Points on the dashed curve representing the bulk residues of O during the progress of melting are related to their coexisting liquids by a tie-line through composition O. The concentration of X in the liquid starts high and decreases steadily throughout the melting history; in the bulk residual solid it decreases to nearly zero at the start of melting. The concentration of Y in the liquid is very low at the start of melting and increases slowly as melting proceeds through the elimination of ‘Anorthite’; then it increases more rapidly as the elimination of ‘Diopside’ is approached, after which almost all of Y will be in the liquid phase. The ratio of Y/X (not plotted in Fig. 2, readable from Fig. 3) in consequence is very low at the onset of melting (E0 in Fig. 3), increases relatively slowly up to the elimination of ‘Anorthite’ (E0 to E0·5 in Fig. 3) and then increases sharply as the carrier-phase for Y is eliminated (E0·5 to L0·67 in Fig. 3) and the liquid composition attains approximately the same Y/X ratio as the bulk system because all three elements are incompatible in ‘Forsterite’.
The concentration of Z and the ratio Z/X also commence very low in the initial liquid but both quantities increase relatively rapidly as ‘Anorthite’, the carrier-phase for Z, is progressively eliminated in the first stage of melting (E0 to E0·5 in Fig. 3). Once this carrier-phase has been eliminated the concentration of Z is merely diluted by further melting and the ratio Z/X (not plotted in Fig. 2 but readable from Fig. 3) changes little because both elements are already largely contained in the liquid. The Z/Y ratio in the liquid (plotted in Fig. 2 and readable from Fig. 3) has increased somewhat during the first stage of melting, but decreases sharply as the carrier-phase for element Y nears elimination (E0·5 to L0·67 in Fig. 3) attaining a value close to 1·0 in liquid L of Fig. 1.
Perfect non-modal fractional melting (PNMFM)
Presnall (1969) addressed this specific problem with respect to the major element behaviour of the liquids and residua. Given the assumed absence of major component crystalline solutions in the hypothetical system of Fig. 1, the PNMFM history for bulk composition O is simple. Only three liquid compositions (E, LE' and ‘Forsterite’) are produced.
Initially, and for the first 0·5 mass fraction of total melt extraction, each infinitesimal melt increment will have composition E, just as in the case of EPM. The behaviour of the trace elements will be very different one from another, however. Almost all of the highly incompatible trace element, X, is removed into the first liquid increments. The concentration of element Z, which is highly compatible in ‘Anorthite’, rises even more sharply as carrier-phase ‘Anorthite’ nears exhaustion than it does at equivalent values of f in the ENMPM case. Element Y is highly compatible in ‘Diopside’, whose mass fraction in the residuum is changing little during this part of the melting process—concentrations of Y in the liquid remain low throughout. During this process the residual bulk composition migrates from O to C, directly away from the composition of the liquid E, which is being removed.
After exhaustion of ‘Anorthite’ no further liquid can form until the temperature appropriate for the formation of liquid at LE' is reached, in equilibrium with residual ‘Diopside’ and ‘Forsterite’. The last drop of liquid extracted with composition E will contain a very high concentration of the trace element Z, which was highly concentrated in the carrier-phase which has just disappeared. In this respect, the behaviour of Z is similar to that in the liquid developed at the same value of f in ENMPM, but the concentrations achieved are higher. Thereafter, as f increases element Z behaves as a highly incompatible element, which will be rapidly eliminated in the next few drops of liquid to form. This liquid has major element composition LE' (a major departure from the behaviour in ENMPM), which will be formed until ‘Diopside’ is exhausted, at which point the total mass fraction of the system which has been melted is 0·67. The bulk residue migrates from C to pure ‘Forsterite’ during the second stage, after which no further liquid can be produced until pure ‘Forsterite’ itself begins to melt, producing a liquid of its own bulk composition.
Accumulated non-modal perfect fractional melting (APNMFM)
Were the melts to be extracted, accumulated and well mixed to homogenize the trace element contents during the above process, the average composition of the aggregated melt would remain at E up to 0·5 mass fraction melting. During the second stage this average liquid composition migrates along the straight line from E towards LE', because of a growing contribution from liquid of composition LE'. This line will in the general case be close to, but not coincident with, the liquidus boundary E–L–LE' (it is coincident in Fig. 1 because the boundary has been arbitrarily depicted as a straight line). When the residue composition reaches pure ‘Forsterite’ the fractionally melted system consists of this mineral plus an average liquid composition close to L and lying exactly at the intersection of the lines E–LE' and ‘Forsterite’–O projected (it falls at L in Fig. 1 for the reason already given). The behaviour of trace element Y as ‘Diopside’ nears and passes exhaustion will be similar to that described above for element Z at the exhaustion of ‘Anorthite’.
During this evolution of the average liquid composition in APNMFM, the bulk modal distribution coefficients for the trace elements will be the same as in the equilibrium non-modal partial melting case (at comparable stages of the melting history the same mineral assemblages are being melted). The bulk distribution coefficients at the start of each stage will be approximately the same as before, as will the bulk distribution coefficient between the residue and the liquid at any intermediate stage of melting, given the assumptions made concerning the system of Fig. 1. In real systems there may be some departure from this simple picture, e.g. to the extent that the liquidus phase boundaries are curved, a complication which is ignored throughout the rest of this treatment on the qualitative assessment that in most cases such departures will give rise to effects that are small when compared with those which are the focus of this study.
The principal points to note about the PNMFM process are the stepwise jumps in the major element composition of the individual liquid increments which are being produced; the smooth evolution of the average liquid composition produced in APNMFM and its small divergence from that of the ENMPM liquid; and the ‘spike’ of concentration of a highly compatible element, which is produced in the infinitesimal liquid increments around the point in the melting where the relevant carrier-phase is eliminated.
Equilibrium non-modal partial crystallization (ENMPC)
This process must display the exact reverse of the major component and trace element features seen during ENMPM. For a given bulk composition it is easiest to calculate phase proportions in the final solid assemblage and then to calculate trace element concentrations at each appropriate mass fraction of melt from the equations.
Perfect non-modal fractional crystallization (PNMFC)
In the special case chosen in Fig. 1 the major element behaviour during perfect fractional crystallization happens to be the same as that in ENMPC and the exact reverse of that during ENMPM. In systems involving continuous reaction relationships between crystals and liquids (i.e. there are crystalline solutions between the major components in the system) these liquid evolution paths will in general be different, and given only the initial liquid composition it will be necessary to have more information about the phase equilibria in the system than is necessary to handle the equilibrium cases, a point further addressed in the next section.
It was noted above that the major component behaviour during PNMFC in the simple case assumed in Fig. 1 is identical to that in ENMPC. The behaviour of highly compatible trace elements will, however, be significantly different in PNMFC from that observed in ENMPC, ENMPM and PNMPM, although the overall pattern of events is recognizable in relation to the results for liquids and solids in ENMPM and ENMPC. When each new crystalline phase begins to precipitate, the bulk distribution coefficient immediately assumes the value of the bulk modal distribution coefficient for this new equilibrium, and retains this value throughout the ensuing stage of the crystallization. If the bulk modal distribution coefficient for an element, following the first appearance of its carrier-phase, is very large then the concentrations in the residual liquids decline sharply as the melt fraction falls below that of the appearance of the carrier-phase. This decline with decreasing values of f is approximately as 1/(1 - f) in PNMFM and in the two equilibrium processes (the fall in PNMFM is slightly greater). However, the concentrations fall as f d - 1 in the residual liquids of PNMFC, leading to a more rapid decline in highly compatible element concentrations immediately after the first appearance of a carrier-phase. Extraction of the appropriate elements into the crystallized products will tend to peak very sharply at and immediately after the first appearance of a carrier-phase in the crystallization sequence, because it is immediately being extracted in response to the bulk modal distribution coefficient. This peaking of concentration in real systems is, however, vulnerable to changes in the crystal–liquid distribution coefficients if the concentration of the trace element increases dramatically.
In ENMPC, on the other hand, the liquid is continuously re-equilibrating with the solids at the current bulk distribution coefficient, which must initially be low because the mass fraction of the carrier-phase in the bulk solids during ENMPC must initially be very low—in fact, the element of interest must pass through an interval of crystallization within which it is still incompatible with respect to the bulk solids and during which its concentration in the average solids must rise. Concentrations in ENMPC will peak as more and more of the element is transferred to the solids, and then decline as the mass fraction of solids increases with most of the element of interest already contained in the solids.
Reaction relationships and crystalline solutions
Discontinuous (incongruent) reaction relationships between crystals and liquids in the various equilibria encountered during non-modal melting and crystallization may be addressed through the introduction of negative values for the mass fractions of relevant phases in the modal melting or crystallization relationships (the mass balance equation involves the reacting solid on the same side as the liquid and its coefficient appears with a negative sign when the liquid composition is expressed wholly in terms of changes in the solids). The matter has not been pursued in this contribution because the basic principles of interest for the purpose are illustrated without introduction of this complication.
By inspection it may be appreciated that it would become a particularly important factor in the detail of the effects if a carrier-phase were to appear and then disappear entirely within the melting or crystallization history as a consequence of such liquid–crystal reaction relationships. Examples may be found in the behaviour of sapphirine in the system MgO–Al2O3–SiO2 (Schreyer & Schairer, 1961) and in the behaviour of spinel in some basalt- and peridotite-like compositions within the system CaO–MgO–Al2O3–SiO2 (Anderson, 1915; Osborn & Tait, 1952; O’Hara, 1969; Biggar et al., 1972). Indeed, if Fig. 1 were drawn for the true system Diopside–Forsterite–Anorthite, then a field of liquidus spinel would appear along the Forsterite–Anorthite join and appropriate compositions within the system (not composition O) would display the transient appearance of spinel during their evolution.
Transient appearance of such a carrier-phase would lead to major differences in the evolution of the concentration of trace elements highly compatible in this carrier-phase as between the non-modal equilibrium, perfect fractional melting and perfect fractional crystallization processes. The removal of the liquid before development of the carrier-phase during fractional melting would deprive that carrier-phase and the residuum of the opportunity, which they have during the equilibrium process, to glean the elements of interest—they would largely have gone during the earlier interval during which they were incompatible. The removal of the crystals precipitated during fractional crystallization would deplete the liquid in those elements concentrated into the carrier-phase and deprive the later residual liquids of the opportunity, which they have during equilibrium processes, to regain them when the carrier-phase reacts out. There is a possibility that monazite may behave in this way during partial melting of crustal rocks and the crystallization of granitic magmas.
The existence of significant crystalline solutions between the major components of the system considered is a further complication because continuous reaction relationships between crystals and liquids during melting and crystallization then become a factor. The most prominent effect is that the PNMFC path will no longer coincide exactly with the ENMPC path even in the absence of discontinuous reaction relationships. The PNMFM stepwise path will be progressively blurred as crystalline solutions in the major phases become more important. This matter also has not been pursued here because it is our assessment that it will introduce effects which will be of minor geochemical importance relative to those discussed here.
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TRACE ELEMENT CONTROLLED CARRIER-PHASES |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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In the simple example used above the carrier-phases are also major minerals in the assemblage and their controlling elements or components are major features in the bulk composition, O, which is considered. This choice was made in the interests of clarity in the graphical treatment. A consequence of this choice is that the distribution coefficients of the controlling elements or components between the carrier-phases and the relevant liquids are always relatively low (and variable). The distribution coefficient for ‘Diopside’ in Fig. 1 (calculated as concentration of diopside in crystals
1·0 divided by concentration of diopside in liquid 0·28–0·4) cannot be greater than 3·5 and may be as low as 2·5 in liquid L; that for ‘Anorthite’ cannot exceed 1·5 in the compositions discussed. The distribution coefficients envisaged for the trace elements Y and Z in their respective carrier-phases are much greater than this. There is, however, no bar to the existence of trace elements which are incompatible with respect to the other two minerals and are carried in the third, but whose crystal–liquid distribution coefficients are less than that of the controlling component and possibly less than 1·0. Rubidium displays such behaviour relative to potassium in K-feldspar in an example cited in the Introduction. In such a case the concentration of the element in the carrier-phase will peak at the low-f end of that mineral’s coexistence with liquid. As melt fraction increases, the ratio of that element to a truly incompatible element such as X in the liquid will increase only slightly from its initially relatively high value. Its ratio in the liquid phase to the component which controls the appearance of the carrier-phase will decrease as melt fraction increases.
Behaviour of this type is increasingly likely to be encountered when the distribution coefficient of the component that controls the appearance of the carrier-phase is itself high. Such is frequently the case where a trace component of the system gives rise to a mineral very rich in that particular component. The distribution coefficients for Zr between zircon and silicate melts, and of Ce between monazite and silicate melts must both be 3000. Furthermore, the trace amounts of these controlling elements ensure that the carrier-phase so stabilized can only be present in very small amounts. The principles of behaviour illustrated in connection with Fig. 1 and in this section remain valid, however. The matter is illustrated in detail in a companion paper (O’Hara et al., 2001).
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A SIMPLIFIED TREATMENT OF NON-MODAL MELTING AND CRYSTALLIZATION FOR TRACE ELEMENTS OF CONSTANT CRYSTAL–LIQUID DISTRIBUTION COEFFICIENT |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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Scope and limitations
The mathematical relationships presented in this section are those used in the computation of the figures in the companion paper (O’Hara et al., 2001
). Cases involving two or more immiscible liquids are not at this time addressed. Nor is the case of complications arising from reaction relationships which can cause a new phase, one not present in the original solidus assemblage, to appear during the partial melting history; or conversely during crystallization, a phase which will not be part of the solidus assemblage to crystallize at an earlier stage before its partial or complete resorption. The sections on fractional and imperfect fractional partial melting implicitly assume that there is no significant difference between the residual solid phases generated during equilibrium and fractional partial melting. As argued above, the effects of the breakdown of these assumptions (other than that concerning transient appearance of a carrier-phase) are likely to give rise to only second-order effects with respect to the gross behaviour of the elements concentrated into a carrier-phase.
Implementing the calculations
The strategy necessary to implement the calculations requires values to be given for the mass fractions of all the solid phases present in the solidus assemblage and values for mass fractions of each solid phase in each one of all the modal melting assemblages in which it might be involved. Also required are the crystal–liquid distribution coefficients (assumed constant) for each element of interest in each of the solid phases. The various critical values of the mass fraction of melt may then be calculated, together with the appropriate values of the mass fractions of each solid phase, sqp; the bulk distribution coefficient at the start of each interval of melting, 0dbp; the modal distribution coefficient within each interval of melting, dmp; and the bulk distribution coefficient, fdbp, at any value of f. These data are also used to determine the sequence of solid phase assemblages which will be encountered during the partial melting of that particular bulk composition. Then it is possible to determine an appropriate set of relationships for the bulk distribution coefficient as a function of the mass fraction of liquid which has developed in the system, of the type shown in equation (6) below. These are then used to calculate the concentrations of trace elements in the liquids, and residues if so desired. In the course of these calculations it is necessary to determine which of the phases will be next to be totally consumed, and also to cater for the possibility of simultaneous consumption of two or more solid phases together.
For a system of two solid phases this is relatively easy, with only three possibilities in the two-major-component system, two of them general cases, and with a further two very special cases (only one solid phase present in the bulk composition) to be considered. For a system of three major components it is more complicated, with 13 possibilities for compositions involving all three solid phases, six of them general cases, and a further 12 more special cases for initial bulk compositions involving two or only one of the solid phases—a total of 25 cases to be considered. This is illustrated in Fig. 4. Phase assemblage relationships encountered during equilibrium partial melting by all possible bulk compositions within the three-component system A–B–C are each represented along some radius of the figure read from the centre towards the circumference. Different radii represent different possible ratios of the solid phases in the initial solid composition. The relative diameters of the three types of circle may vary provided they satisfy the requirement that the radii through D, E and F are tangent to the mid-sized circles (which therefore touch but do not intersect) at D, E and F, and that the mid-sized circles touch but do not cut the outer circle at G, H and I. Critical values of the mass fraction of melting are represented where a chosen radius cuts one of the circles, and are labelled appropriately in Fig. 4. The mass fraction of melt in the system increases towards the outer circumference of the figure. The six general cases (each applying to an arc of radii), and six of the special cases (applying to the radii through D–J, respectively) are immediately obvious; the seventh is encountered when circle DEF coincides with GHJ (melting of the eutectic bulk composition). Representation of phase assemblage changes during equilibrium partial crystallization merely requires the phase assemblages to be read from circumference to centre.
For systems with more solid phases the decision-taking process remains simple in principle but becomes rapidly more complicated in practice—a four-solid-phase system has 68 possibilities, 24 of which are general cases. There are in addition a further 74 special cases arising from initial bulk compositions falling in the four bounding ternary joins, six binary joins and at the four pure phases.
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SIMPLIFIED NON-MODAL MELTING AND CRYSTALLIZATION FOR IDEAL TRACE ELEMENTS |
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Symbols and nomenclature
Stages in the crystal–liquid history are identified by the number of solid phases present in equilibrium with the liquid. This is adequate in this simplified system in which the possibility of discontinuous crystal–liquid reaction relationships is ignored. The possible appearance of an immiscible liquid phase, which might be the carrier-phase, is not directly addressed but can be accommodated within the algorithms developed. A more satisfactory and general way to designate these stages might be to number them consecutively from zero, designating the stage commencing with zero per cent melt present, up to whatever number is reached at the liquidus—bearing in mind that closely adjacent starting compositions might reach the liquidus in different numbers of stages when reaction relationships are encountered by one and not by the other. Alternatively, the stages could be designated by abbreviations indicating the actual solid phases present, requiring independent knowledge of the phase equilibria before the melting or crystallization history can be symbolized. Neither method, however, is attractive in the context of this simplified treatment.
Stage p is the stage in which p solid phases are in equilibrium with the melt; in a system with n components and n solid phases at the solidus, p = n marks the first phase of partial melting or the last phase of partial crystallization. Each phase in the melting or crystallization history is bounded by two critical values of the mass fraction of melting, designated as fcp at the lower limit of f at which the p phases are in equilibrium with the liquid and fcp - 1 at the upper limit of f at which the p phases are still present. It follows that fcn = 0·0 in a system of n components and n solid phases, because this represents the onset of partial melting or the end of partial crystallization; and fc0 = 1·00 in all systems because this is always the stage of completion of partial melting or start of partial crystallization.
Treating the case solely from the point of view of equilibrium partial melting for the time being, a system of n solid phases in a system of n components begins melting at fcp= fcn = 0·0 and continues with p = n solid phases present, defining stage n, until f = fcn - 1, when one of the solid phases has been totally consumed and p becomes indeterminate at the discontinuity between stages n and n - 1 when (n - 1) phases first coexist with the liquid. Melting then continues through stages (n - 1), (n - 2) ... to stage 1, which terminates at f = fc0 =1·0 or total melting.
Table 1 illustrates relationships between the various symbols used, and their scope relative to the phase equilibria and the melting stages. With this nomenclature there is no ambiguity but some superabundance of symbols. Specifically, the zero-order continuous, first-order discontinuous function which is the bulk distribution coefficient, fdb, and which, within intervals, it is convenient to label fdbp, will, at the discontinuity at critical melt fractions, fcp, only, be the limiting value of both adjacent intervals such that fdbp = fdbp + 1, symbolized as 0dbp in this paper.
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Simple mass balance approach
The general expression for the mass balance for any ideally behaved trace element is then
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The following relationship exists between the mass fraction of a phase q in the solid at the start of melting stage p and the mass fraction of that phase present at previous critical values of the melt fraction:
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Bulk distribution coefficients
It is convenient to define some bulk distribution coefficients at this point, always bearing in mind that the bulk distribution coefficient is defined as the concentration in the bulk solids divided by that in the coexisting liquid. This is the same as the fractional mass of the element in the solids divided by the mass fraction of the solids in the whole system. Then the bulk distribution coefficient, 0dbp, at the onset of melting in stage p, i.e. at f = fcp, is given by
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We note also the successive replacements for 0dbp which may be derived from equation (2) above,
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Equilibrium non-modal partial melting (ENMPM)—first approach
Returning to equation (1), this may also be presented as
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,
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Alternative approach through displacement of mass balance
An alternative approach that is easier to extend to the cases of perfect and imperfect fractional melting may be developed from consideration of the mass balance involved in a small increase in the mass fraction of liquid in an already partially melted system. The basic mass balance equation expressing the change, after mass fraction, f, of melt has already formed in stage p of partial melting, involved in producing a further increment of liquid from unit mass of residue is
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Equilibrium non-modal partial melting (ENMPM)—alternative approach
If f in equation (12) is set equal to 0·0 and 
f is made equal to f, the total mass fraction of melt developed since the onset of melting, then,
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) when p = n and fcp = 0·0 as discussed above.
Perfect non-modal fractional melting (PNMFM)
Unlike the situation in equilibrium partial melting, fractional processes preserve a memory of the path by which the present state of the system has been reached. The solution required will be of the form
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f and CL are allowed to tend to zero in equation (15), the resulting differential equation may be integrated between the limits f = fcp and f = f to yield the change in composition relative to the liquid composition at the last critical point passed, or, with superscript p changed to p + 1 throughout, between the limits f = fcp+1 and f = fcp to obtain the change in composition between the last two critical points relative to the liquid composition at the last but one critical point; thus in stage p, up to the current value of f:
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When one is dealing with the first stage of melting this expression reduces to
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. Equation (18) further reduces to the standard expression for simple perfect fractional (modal) partial melting when the initial bulk and modal distribution coefficients are made equal. In the simple system envisaged here, the composition of the residue is given by multiplying the expression in equation (18) by the bulk distribution coefficient at the appropriate mass fraction of melting, f. In the more general case there will generally be a difference between the value of the bulk distribution coefficient in ENMPM and PNMFM at a given value of f.
The concentration in the average liquid composition is given by the fractional mass of the element which is not in the residue, divided by the fractional mass of liquid in the whole system, or
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Imperfect (finite increment) non-modal fractional melting (INMFM)
In imperfect non-modal fractional melting the concentration of an ideally behaved element in the jth batch of melt, mass fraction f, extracted will relate to that of the first batch of liquid extracted by
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Then, from equation (12), substituting for f = ( j - 1)f and f + f = jf in the jth liquid increment, and letting CL and CL + CL represent the concentration of an element in successive liquid increments,
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Perfect non-modal fractional crystallization (PNMFC)
The residual liquid composition will relate to the source liquid composition by
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f, C tend to zero, to
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The solid being precipitated and removed at any stage may be related to the source liquid composition by multiplying the current residual liquid composition by the appropriate value of the bulk modal distribution coefficient.
The average concentration of an element in the solids precipitated thus far at any stage of partial crystallization is obtained simply by determining the fraction of the mass of the element which is not in the residual liquid at that stage, and dividing it by the mass fraction of the system which has been crystallized,
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Imperfect (finite increment) non-modal fractional crystallization (INMFC)
At the start of crystallization the mass fraction of melt, f, is unity. Then, for increment j which takes the solid fraction crystallized from (1 - f) to (1 - f - f ), let f = 1 - ( j - 1)f, and (f - f ) = (1 - jf ). In imperfect non-modal fractional crystallization the concentration of an ideally behaved element in the jth batch of melt, mass fraction f, extracted (which has the same concentration as the entire residual liquid) will relate to that of the first batch of liquid extracted by
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SUMMARY |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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The basic principles which govern the location of trace elements which are highly compatible in minor carrier-phases have been explored. The effects of the relationships described on the concentration and dispersal of scarce elements, some of which have considerable economic importance, will be pursued in a companion paper.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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ACKNOWLEDGEMENTS |
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We wish to thank J. R. Cann, K. G. Cox, C. T. Herzberg, R. K. O’Nions, D. Presnall, D. M. Shaw and M. Wilson for their efforts as readers of an earlier version of this paper, which led to significant improvements in substance and presentation.
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FOOTNOTES |
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*Corresponding author. E-mail: sglmjo{at}cardiff.ac.uk
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REFERENCES |
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TOPABSTRACTINTRODUCTIONTRACE ELEMENT CARRIER-PHASES IN...TRACE ELEMENT CONTROLLED CARRIER...A SIMPLIFIED TREATMENT OF...SIMPLIFIED NON-MODAL MELTING AND...SUMMARYAPPENDIXREFERENCES |
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