JOURNAL OF PETROLOGY
VOLUME 42
NUMBER 10
PAGES 1869–1885
2001
Minor Phases as Carriers of Trace Elements
in Non-Modal Crystal–Liquid Separation
Processes I: Basic Relationships
M. J. O’HARA∗, N. FRY AND H. M. PRICHARD
DEPARTMENT OF EARTH SCIENCES, CARDIFF UNIVERSITY, PO BOX 914, CARDIFF CF10 3YE, UK
RECEIVED SEPTEMBER 10, 1998; REVISED TYPESCRIPT ACCEPTED MARCH 16, 2001
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
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.
INTRODUCTION
What is a trace element carrier-phase?
Petrological relevance of these
considerations
Here we consider aspects of the behaviour of those trace
elements which are generally incompatible in mineral–
Carrier-phases in which U, Th or the PGE are highly
compatible may be present during partial melting of
∗Corresponding author. E-mail: sglmjo@cardiff.ac.uk
 Oxford University Press 2001
JOURNAL OF PETROLOGY
VOLUME 42
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
NUMBER 10
OCTOBER 2001
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
1870
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
equilibrium crystallization is the precise reverse of nonmodal equilibrium melting; in general, non-modal fractional crystallization is not the precise reverse of nonmodal 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 endmember 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 nearperfect non-modal fractional crystallization of basaltic
magma. The cumulus sequence is marked by the
1871
JOURNAL OF PETROLOGY
VOLUME 42
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 carrierphase 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 carrierphases 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 nonmodal crystallization and to imperfect fractional nonmodal melting and crystallization.
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)
NUMBER 10
OCTOBER 2001
and O’Hara & Fry (1996a, 1996b, 1997), although the
potential effects of these upon the observable phenomena
are surveyed qualitatively.
TRACE ELEMENT CARRIER-PHASES
IN A HYPOTHETICAL THREECOMPONENT SYSTEM
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.
1872
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
Fig. 1. Part of a hypothetical pseudo-ternary system not unlike Anorthite–Diopside–Forsterite at atmospheric pressure (e.g. O’Hara & Fry,
1996b). The relative height scale at the left is intended to assist estimation
of phase proportions. Further discussion in the text.
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
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
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
1873
JOURNAL OF PETROLOGY
VOLUME 42
Fig. 2. Plot of relative concentrations of hypothetical trace elements
X, Y and Z and of the Z/Y ratio in the liquid phase which is produced
by equilibrium partial melting of composition O in Figs 1 and 3, plotted
as a function of the mass fraction of melt formed. The vertical numerical
scale shows the concentration of each trace element in the liquid
relative to its concentration in the bulk composition (continuous curves),
and the consequent ratio of the concentrations of elements Z/Y in that
liquid phase as melting proceeds. With the assumptions about initial
trace element concentrations made in the text, the three trace element
concentrations in the liquid are identical at f >0·67. Further description
in the text.
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
NUMBER 10
OCTOBER 2001
L proceeds with element Y becoming progressively more
concentrated in the ever smaller amount of the carrierphase, ‘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,
1874
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
Fig. 3. Relative proportions of hypothetical trace elements X, Y, Z
in the liquids (continuous lines and symbols), modal melting solid
assemblages (open symbols) and bulk residues (dashed curve) of equilibrium partial melting of bulk composition O of Fig. 1. Further
description in the text.
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 carrierphase 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 carrierphase 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 carrierphase 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
1875
JOURNAL OF PETROLOGY
VOLUME 42
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
NUMBER 10
OCTOBER 2001
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
1876
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
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
1877
JOURNAL OF PETROLOGY
VOLUME 42
which will be of minor geochemical importance relative
to those discussed here.
TRACE ELEMENT CONTROLLED
CARRIER-PHASES
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).
NUMBER 10
OCTOBER 2001
A SIMPLIFIED TREATMENT OF NONMODAL MELTING AND
CRYSTALLIZATION FOR TRACE
ELEMENTS OF CONSTANT
CRYSTAL–LIQUID DISTRIBUTION
COEFFICIENT
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
1878
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
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.
Fig. 4. Diagrammatic representation of all possible melting sequences
for a hypothetical ternary system A–B–C illustrating the relationships
between the critical values of f. Further discussion in the text.
(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 threecomponent 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
SIMPLIFIED NON-MODAL MELTING
AND CRYSTALLIZATION FOR IDEAL
TRACE ELEMENTS
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 carrierphase, 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
1879
JOURNAL OF PETROLOGY
VOLUME 42
NUMBER 10
OCTOBER 2001
Table 1: Relationship between melting interval during non-modal melting, value
of the index p, location and nomenclature of critical values of f, s and their
relationship to the ranges of modal and bulk distribution coefficients, which are
assumed throughout the treatment; items vertically beneath each other refer to the
same mass fraction of melt present
INTERVAL
|
3
|
2
|
1
|
1
|
(Number of crystalline phases present)
p
|
f
0
3
|
2
|
1
> MELTING
<
Critical f
fc3
Mineral proportions
0
0
>
CRYSTALLIZATION <
fc2
sq . . . . . . tq . . .
0
db3 . . . . . . dm3 . . .
0
3
3
fc1
fc0
sq . . . . . . tq . . .
0
sq . . . . . . tq . . .
db2 . . . . . . dm2 . . .
0
db1 . . . . . . dm1 . . .
2
2
1
1
Modal distribution coefficients and special
cases of bulk coefficient
Bulk distribution coefficients
f
db
3
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.
Simple mass balance approach
The general expression for the mass balance for any
ideally behaved trace element is then
f
db
2
f
db1
q
C 0S (1·0) = C Lp { f + [(1 − f cp )0s qp dq − t qp dq( f − f cp )]}
1
(a)
(b)
(c)
(d)
(1)
= C Lp [ f + fd qp (1 − f )]
where CS0 is the concentration of the element in the
initial source system (which may be entirely solid or
entirely liquid according to the current sense of usage),
p is the stage identifier, q identifies the solid phase
concerned, 0sqp is the mass fraction of the solid assemblage
which is composed of phase q at f = fcp the last critical
mass fraction of liquid passed, and tqp is mass fraction of
phase q in the modal melting assemblage. More generally,
f p
sq is the mass fraction of q in the residual solid when
mass fraction of melt f has formed, at which point the
fraction of the whole initial system which is still composed
of solid q is given by 0sqp(1 − f ). dq is the crystal–liquid
distribution coefficient for a particular element in phase
q. The second version of equation (1) incorporates the
bulk distribution coefficient, fdqp, at this value of f and is
self evident. When multiplied out, term (a) gives the total
mass of the element, term (b) expresses the mass in the
liquid phase when mass fraction f of the original system
has become liquid, term (c) expresses how much of the
element was still contained in solid phase q at the start
of the current stage of partial melting and term (d) adjusts
term (c) for the amount of that element which has been
transferred to the liquid by partial melting since the start
of the current stage of partial melting.
The following relationship exists between the mass
fraction of a phase q in the solid at the start of melting
1880
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
stage p and the mass fraction of that phase present at
previous critical values of the melt fraction:
s = [(1 − f pc + 1)0s pq + 1 − t pq + 1 ( f cp − f pc + 1))]/(1 − f cp)
0 p
q
= [(1 − f pc + 2)0s pq + 2 − t pq + 2( f pc + 1 − f pC+ 2) −
t
p+1
q
(f −f
p
c
p+1
c
(2)
)]/(1 − f )
p
c
q
d = 0s qp .dq.
0 p
b
(4)
1
The bulk modal distribution coefficient dmp (assumed
constant for each modal melting assemblage) in stage p
is given by
q
where the left-hand side multiplied by the denominator
of the right-hand side is the mass fraction of phase q in
the whole system at f = fcp. The numerator of the righthand side gives the mass fraction of the whole system
which is phase q at the start of the previous stage of
partial melting reduced by the mass fraction of that phase
which melts during the previous stage. Replacements of
this type can be pursued in equation (2) until p = (n −
1), when
d mp = t qp .dq.
(5)
1
After mass fraction ( f − fcp) of liquid has formed in stage
p the fraction of the total system that is phase q is fsqp(1
− f ), which is equal to 0sqp(1 − fcp) − tcp( f − fcp). Then
the bulk distribution coefficient is
q
d = fs qp dq =
f p
b
1
d (1 − f cp) − d mp ( f − f cp)
(1 − f )
0 p
b
(6)
Then all quantities can be calculated from a knowledge
of the initial mass fractions of the solid phases in the
system and values or assumptions for all values of tqp, dfq.
More generally,
where 0dbp is the bulk distribution coefficient for the
combined solid phases at f = fcp (start of stage p) and dmp
is the modal melting bulk distribution coefficient during
stage p. This relationship is also implicit from equation
(1) above.
We note also the successive replacements for 0dbp which
may be derived from equation (2) above,
s = [(1 − f cp)0s qp − t qp ( f − f cp)]/(1 − f )
0s qp dq = 0d bp = [(1 − f pc + 1) 0d pb + 1 −
0 n−1
q
s
= [(1 − f cn)0s qn − t qn( f nc − 1 − f cn)]/(1 − f nc − 1)
= (0s qn − t qn f nc − 1)/(1 − f nc − 1).
(3)
q
f p
q
= [(1 − f pc + 1)0spq + 1 − t pq + 1( f cp − f pc + 1) −
1
(2a)
(7)
( f cp − f pc + 1)d pm+ 1]/(1 − f cp)
t qp ( f − f cp)]/(1 − f )
It should be noted that 0sq1 and tq1 must always be equal
to 1·0, regardless of which phase is designated as q,
because at this stage there is only one solid phase remaining. In consequence, in the final stage of equilibrium
non-modal melting or the first stage of equilibrium nonmodal crystallization,
and which can in turn be used to expand equation (6)
and others herein when required.
C 0S (1·0) = C 1L { f + [(1 − f 1c)0s 1q − t 1q ( f − f 1c)]dq}
Returning to equation (1), this may also be presented as
= C 1L [dq + f (1 − dq)]
(1a)
regardless of the identity of the solid phase q. This is the
standard equation for simple equilibrium partial melting
or crystallization which implicitly assumes modal melting
with a single residual solid phase throughout.
Equilibrium non-modal partial melting
(ENMPM)—first approach
C 0S (1·0) = C Lp [ f + (1 − f cp )0d bp − ( f − f cp )d mp ]
which may be rearranged to obtain the general expression
for the liquid composition relative to the initial solid
composition in equilibrium non-modal partial melting:
C Lp
1
=
C 0S
[ f + 0d bp (1 − f cp ) − d mp ( f − f cp )]
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
(8)
=
1
.
[ f + fd bp (1 − f )]
(9)
In the special case where p = n and consequently fcp =
0·0, and valid only through the first melting stage where
all initial phases in the solid assemblage are still present,
equation (9) reduces to the relationship derived by Shaw
(1970),
1
C Lp
= 0 p
.
C 0S
[ d b + f(1 − d mp )]
1881
(10)
JOURNAL OF PETROLOGY
VOLUME 42
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
q
(1 − f ) s .dq .C
f p
q
p
L
1
(11)
(a)
NUMBER 10
OCTOBER 2001
where CL∗ is the concentration in the first drop of liquid
to form, CS0 is the concentration in the initial solid,
f
CLp is the concentration in the liquid and fCRp is the
concentration in the solid residue when mass fraction f
has melted. Whence,
f
C Lp
1
=
C 0S
f + 0d bp (1 − f cp ) − d mp ( f − f cp )
(13a)
0 p
Cp
d b (1 − f cp ) − d mp ( f − f cp )
and 0R =
CS
f + [ 0d bp (1 − f cp ) − d mp ( f − f cp )]
f
which reduce to the equations for liquid and residue
compositions during non-modal equilibrium partial melting of a two-solid phase assemblage (Shaw, 1970) when
p = n and fcp = 0·0 as discussed above.
q
= [ f + fs qp .dq (1 − f − f ) + f ](C Lp + C Lp )
Perfect non-modal fractional melting
(PNMFM)
1
(b) (c)
(d)
where (a) is the mass in the residual solid related to the
appropriate liquid composition, (b) is the old mass, less
(c) the newly melted mass but plus (d) the new increment
of liquid, all related to the new liquid composition.
q
Replacing fs qp dq by a unique value of the distribution
1
coefficient, this mass balance equation can be processed
and integrated to yield the familiar standard expressions
for the concentrations of an ideal trace element during
perfect modal fractional partial melting. It follows, using
equation (6) above, that
q
C + C
=
C Lp
p
L
fs qp .dq (1 − f )
p
L
f + fs qp .dq (1 − f − f ) + f
(12)
d (1 − f cp ) − d mp ( f − f cp )
.
[ d (1 − f cp ) − d mp ( f + f − f cp )] + f
0 p
b
0 p
b
Equilibrium non-modal partial melting
(ENMPM)—alternative approach
Then, from equations (6) and (11) above,
C Lp
= (d mp − 1)
C Lp
1
C Lp
p
fc p =
CL
f
f
f p
C Rp
db
0 =
f p
CS
f + d b (1 − f )
(15)
If 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:
f
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,
0 p
f p
1
C Lp
db
CL
=
or
∗
f p
0 =
f p
CL
f + d b (1 − f ) C S
f + d b (1 − f )
(14)
because f cn =0.
1
f p
b
and
p
q
d (1 − f )
= f + f p
d b(1 − f f ) + f
n−1
fc p
fc
0 n
CL
C nL − 1
1
d
C Lp
...
and f nc n = 0b
=
f nc n
f pc p
f pc + 1 p + 1
CL
CL
CS
CL
CL
f
f
.
{f + [ 0d bp (1 − f cp ) − d mp ( f + f − f cp )]}
1
=
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
=
(13)
[ 0d bp (1 − f cp ) − d mp ( f − f cp )]d
1
[0d bp (1 − f cp )]d
[ fd bp (1 − f)]
p
m
p
m
−1
−1
(16)
1
−1
d pm
1
[0d bp (1 − f cp )]d
p
m
−1
and, in stage (p + 1) between critical values fcp + 1 and
fcp,
1882
O’HARA et al.
[ f cd bp (1 − f cp )]d
p
f pc
C Lp
=
p
+
1
fc
C pL + 1
1
p+1
m
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
−1
1
−1
.
(17)
[0d pb + 1 (1 − f pc + 1 )]d
Linking these replacements together and relating the
initial liquid to the initial solid via the initial bulk distribution coefficient,
p+1
m
f
1
1
1 f p
C Lp
−1 0 p
d
.[ d b (1 − f cp )]d
0 = 0 p .[ d b (1 − f )]
CS
db
p+1 −
p
m
m
[0d pb + 1 (1 − f pc + 1 )]d
1
p+2
m
−
1
d pm+ 1
... (0d bn )
−
1
d pm
1
.
(18)
d pm+ 2
where CS0 is the concentration in the source solid, 0dbn is
the bulk distribution coefficient at the onset of melting
of the system, i.e. when p = n in the terms of an earlier
part of the discussion, and dmn is the modal distribution
coefficient in the first stage of melting.
When one is dealing with the first stage of melting this
expression reduces to
1
C Lp
1 f n
−1 0 n −
d
.( d b )
0 = 0 n [ d b (1 − f )]
CS
db
f
n
m
dn
1
= 0 n 1 − 0 mn f
db
db
1
−1
d mn
(19)
1
−1
d mn
which is the expression derived by Shaw (1970).
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
1 − (1 − f )(fd bp fC Lp /C 0S )
C Lp
.
0 =
CS
f
f
av
and these further reduce to the standard expressions for
products of modal fractional melting when the initial
bulk and modal distribution coefficients are equal.
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
n − 1f
jc
j cf p
CL
C Lp
...
p
+
1
C Lp j c fC pL + 1
p
jf
j pcf
f
C Rp
=
C 0S
(1 − f )
1− 1−
f
av
C Lp
=
C 0S
(22)
because j cn = 0
where jcp is the number of increments required to reach
fcp (in real systems with finite increment sizes the evaluation will be complicated by this number being a real,
not an integer). n is the number of solid phases present
at the solidus and 0dbn is the bulk distribution coefficient
for the system at the start of melting.
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,
( j − 1) f p
d b [1 − ( j − 1)f ]
C Lp + C Lp
=
jf p
C Lp
d b (1 − jf ) + f
d (1 − j cp f ) − d mp [( j − 1)f − j cp f ]
=
d (1 − j cp f ) − d mp [( jf − j cp f )] + f
0 p
b
0 p
b
(23)
whence,
j ( j − 1) f p
C Lp
d b[1 − ( j − 1)f ]
=
“
p
jf p
C L j pc + 1
d b[1 − jf ] + f
jf
j pc f
j
=“
j pc + 1
j
(20)
=“
j pc + 1
When one is dealing with the first stage of melting these
expressions reduce to
dn
1 − 0 mn f
db
0 n
C nL − 1
1
db
and
=
n
n
j cf n
CL
CL
C 0S
j ncf
d (1 − j cp f ) − d mp ( j − 1 − j cp )f
d (1 − j cp f ) − d mp ( j − j cp )f + f
0 p
b
0 p
b
(24)
d (1 − j cp f ) − d mp ( j − j cp − 1)f
0 p
b
d (1 − j cp f ) − d mp j − j cp −
0 p
b
1
f
d mp
and
1
d mn
j cp f
;
n
m
0 n
b
d
f
d
p
jc
( j − 1) f p + 1
C Lp
d b [1 − ( j − 1)f ]
=
“
p+1
j f p + 1
CL
d b [1 − jf ] + f
j pc + 1+ 1
j pc + 1 f
(21)
1
d mn
(1 − j pc + 1 f ) − d pm+ 1 ( j − j pc + 1 − 1)f
.
= “
1
j pc + 1+ 1 0 p + 1
d b (1 − j pc + 1 f ) − d pm+ 1 ( j − j pc + 1 − p + 1 )f
dm
j cp
f
1883
0 p+1
b
d
(25)
JOURNAL OF PETROLOGY
VOLUME 42
p − 1 (d pm− 1)
f
f 0c
The residual liquid composition will relate to the source
liquid composition by
p−1
OCTOBER 2001
p
f p
C Sav
C Sav 1 − f.( f/ f c
=
1 =
CL
C 0S
Perfect non-modal fractional crystallization
(PNMFC)
)
p
p−1
( f pc − 1 / f pc − 2)(d m
(1 − f )
p−1
= (1−f )− 1.[1−f d m.( f pc − 1 )d m
fc
fc 1
C pL − 1
CL
1
1
C Lp
because f 0c = 1·0.
f pc − 1 p − 1 . f pc − 2 p − 2 .... f 0c 0 and f 0c 0 =
CL
CL
CL
CL
C 0L
f
NUMBER 10
p−2
.( f pc − 2 )d m
p
− dm
− 1)
...
− d pm− 1
(28a)
1
( f 1c )d m − d m ... ( f 0c )1 − d m
1
2
1
and the final term in fc0 = 1·0 as before.
Commencing with the appropriate mass balance equation,
q
C f = ( C − C )( f − f ) + t d f ( C − C )
f
p
L
f
p
L
f
p
L
p
q q
f
p
L
f
p
L
(26)
1
which leads, when f, C tend to zero, to
fC Lp
f
p
f p = (d m – 1)
CL
f
and, after integration between the appropriate limits,
to
C Lp
p−1
= ( f/f pc − 1 )(d m ).
C pL − 1
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
f
(27)
f pc − 1
f pc − 2
(1 −
1
j pc − 1f )
(29)
1
1
because j 0c = 0.
(1 − j 0cf ) 0 =
C L C 0S
Now
f pc − 1
p−1
(1 − j c
f ) p − 1
(1 − j c f ) 1
CL
CL
C Lp
=
...
p
−
2
(1 − j 0cf ) 0 and
p−1
(1 − j c
f ) p − 2
CL
CL
CL
(1 − jf )
C pL − 1
p−1
p−1
/f pc − 2 )(d m − 1)
p−2 = ( f c
CL
From equation (26) and the replacements indicated
above,
hence
f
C Lp
p
p−1
1
= ( f/f pc − 1 )(d m − 1) ( f pc − 1/f pc − 2 )(d m − 1) ... ( f 1c / f 0c )d m − 1
C 0S
p−1
p − 1d p
m
= f d m .( f pc − 1 )d m
p−2
.( f pc − 2 )d m
− d pm− 1
...
(28)
f
( fC Lp − fC Lp )
=
f p
CL
f − f + d mp f
Hence it may be written,
(1 − jf ) p
L
[1 − ( j − 1)f ] p
L
C
1
c
(f )
d 1m
−
d 2m
0 1−
c
.( f )
C
d 1m
(30)
1 − ( j − 1)f
.
=
1 − ( j − d mp )f
=
1 − ( j − 1)f
1 − ( j − d mp )f
(31)
from which it follows that
with the final term in fc0 equal to unity because fc0 itself
equals 1·0.
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,
(1 − jf )
j
C Lp
1 − ( j − 1)f
, and
p−1 = “
CL
1
− ( j − d mp )f
p
−
1
jc
+1
(1 − j pc − 1f )
(1 − j pc − 1f )
p−1
jc
C pL − 1
1 − ( j − 1)f
=
“
.
p
−
2
(1 − j c
f ) p − 2
p−1
CL
)f
j pc − 2+ 1 1 − ( j − d m
(32)
SUMMARY
The basic principles which govern the location of trace
elements which are highly compatible in minor carrierphases have been explored. The effects of the relationships
1884
O’HARA et al.
MINOR PHASES AS CARRIERS OF TRACE ELEMENTS, I
described on the concentration and dispersal of scarce
elements, some of which have considerable economic
importance, will be pursued in a companion paper.
ACKNOWLEDGEMENTS
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.
REFERENCES
Anderson, O. (1915). The system anorthite–forsterite–silica. American
Journal of Science, 4th Series 39, 407–454.
Biggar, G. M., Humphries, D. J. & O’Hara, M. J. (1972). Flow sheet
for CaO–MgO–Al2O3–SiO2 at atmospheric pressure. In: Progress in
Experimental Petrology: Second Progress Report of Research Supported by NERC
at Edinburgh and Manchester Universities, 1969–1971. NERC Publication
Series D 2, 99–102.
Gast, P. W. (1968). Trace element fractionation and the origin of
tholeiitic and alkaline magma types. Geochimica et Cosmochimica Acta
32, 1057–1086.
Hertogen, J. & Gijbels, R. (1976). Calculation of trace element fractionation during partial melting. Geochimica et Cosmochimica Acta 40,
313–322.
Langmuir, C. H. (1989). Geochemical consequences of in situ crystallization. Nature 340, 199–205.
Nockolds, S. R. & Allen, R. (1953). The geochemistry of some igneous
rock series. Part 1. Geochimica et Cosmochimica Acta 4, 105–142.
Nockolds, S. R. & Allen, R. (1954). The geochemistry of some igneous
rock series. Part 2. Geochimica et Cosmochimica Acta 5, 245–285.
Nockolds, S. R. & Allen, R. (1956). The geochemistry of some igneous
rock series. Part 3. Geochimica et Cosmochimica Acta 9, 34–77.
Nockolds, S. R. & Mitchell, R. L. (1947). The geochemistry of some
Caledonian plutonic rocks: a study of the relationship between the
major and trace elements of igneous rocks. Transactions of the Royal
Society of Edinburgh 61, 533–575.
O’Hara, M. J. (1969). The atmospheric pressure equilibrium and
fractional crystallization of basalt-like mixtures in the MgSiO3-rich
part of the plane CaSiO3–MgSiO3–Al2O3 and the nature of thermal
divides. In: Progress in Experimental Petrology First Report NERC Supported
Research Units in British Universities 1965–1968, 129–152.
O’Hara, M. J. (1977). Geochemical evolution during fractional crystallization of a periodically refilled magma chamber. Nature 266,
503–507.
O’Hara, M. J. (1985). Importance of the ‘shape’ of the melting regime
during partial melting of the mantle. Nature 314, 58–62.
O’Hara, M. J. (1993). Trace element geochemical effects of imperfect
crystal–liquid separation. In: Prichard, H. M., Alabaster, T., Harris,
N. W. B. & Neary, C. R. (eds) Magmatic Processes and Plate Tectonics.
Geological Society, London, Special Publications 76, 39–59.
O’Hara, M. J. (1995). Trace element geochemical effects of integrated
melt extraction and ‘shaped’ melting regimes. Journal of Petrology 36,
1111–1132.
O’Hara, M. J. & Fry, N. (1996a). The highly compatible trace element
paradox—fractional crystallization revisited. Journal of Petrology 37,
859–890.
O’Hara, M. J. & Fry, N. (1996b). Geochemical effects of small
packet crystallization in large magma chambers—further resolution
of the highly compatible element paradox. Journal of Petrology 37,
891–925.
O’Hara, M. J. & Fry, N. (1997). Integrated compositions of perfect
fractional liquid and solid products are identical, with respect to
ideal trace element behaviour, to simple equilibrium (batch) products
in very special cases. Journal of Petrology 38, 425–430.
O’Hara, M. J. & Mathews, R. E. (1981). Geochemical evolution in an
advancing, periodically replenished, periodically tapped, continuously fractionated magma chamber. Journal of the Geological Society,
London 138, 237–277.
O’Hara, M. J., Fry, N. & Prichard, H. M. (2001). Minor phases as
carriers of trace elements in non-modal crystal–liquid separation
processes II: illustrations and bearing on behaviour of REE, U, Th
and the PGE in igneous processes. Journal of Petrology 42, 1887–1910.
O’Nions, R. K. & Clarke, D. B. (1972). Comparative trace element
geochemistry of Tertiary basalts from Baffin Bay. Earth and Planetary
Science Letters 15, 436–446.
Osborn, E. F. & Tait, D. B. (1952). The system diopside–
forsterite–anorthite. American Journal of Science (Bowen Volume) 252A,
413–433.
Presnall, D. C. (1969). Geometrical analysis of partial fusion. American
Journal of Science 267, 1178–1194.
Schilling, J.-G. (1971). Sea-floor evolution: rare earth evidence. Philosophical Transactions of the Royal Society of London, Series A 268, 663–706.
Schilling, J.-G. & Winchester, J. W. (1967). Rare-earth fractionation
and magmatic processes. In: Runcorn, S. K. (ed.) Mantles of the Earth
and Terrestrial Planets. New York: Interscience, pp. 267–283.
Schreyer, W. & Schairer, J. F. (1961). Compositions and structural
states of anhydrous Mg-cordierites: a reinvestigation of the central
part of the system MgO–Al2O3–SiO2. Journal of Petrology 2, 324–406.
Shaw, D. M. (1970). Trace element fractionation during anatexis.
Geochimica et Cosmochimica Acta 34, 237–243.
Wager, L. R. & Brown, G. M. (1967). Layered Igneous Rocks. San
Francisco, CA: W. H. Freeman.
Zou, H. (2000). Modeling of trace element fractionation during nonmodal dynamic melting with linear variations in mineral/melt distribution coefficients. Geochimica et Cosmochimica Acta 64, 1095–1102.
1885