J. metamorphic Geol., 1998, 16, 577–588
Calculating phase diagrams involving solid solutions via non-linear
equations, with examples using THERMOCALC
R . P OWE L L ( e -ma i l: rp @ e ar t hs c i. u n ime l b. e d u. a u )1 , T . H O L L AN D2 A N D B . W O RL E Y 1
1 School of Earth Sciences, The University of Melbourne, Victoria 3052, Australia
2 Department of Earth Sciences, University of Cambridge, Cambridge CB2 3EQ, UK
A B S TR A C T
Phase diagrams involving solid solutions are calculated by solving sets of non-linear equations. In
calculating P–T projections and compatibility diagrams, the equations used for each equilibrium are the
equilibrium relationships for an independent set of reactions between the end-members of the phases in
the equilibrium. Invariant points and univariant lines in P–T projections can be calculated directly, as
can coordinates in compatibility diagrams. In calculating P–T and T –x/P–x pseudosections – diagrams
drawn for particular bulk compositions – the equilibrium relationship equations are augmented by mass
balance equations. Lines in pseudosections, where the mode of one phase in the lower variance equilibrium
is zero, and points, where the modes of two phases are zero, can then be calculated directly. The software,
THERMOCALC, allows the calculation of these and a range of other types of phase diagram. Examples
of phase diagrams and phase diagram movies, with instructions for their production, along with the
THERMOCALC input and output files, and the MathematicaTM functions for assembling them,
are presented in this paper, partly in hard copy and partly on the JMG web sites
(http://www.gly.bris.ac.uk/www/jmg/jmg.html, or equivalent Australian or USA sites).
I NTR ODU CT I ON
The advent of large internally-consistent thermodynamic datasets (Helgeson et al., 1978; Holland &
Powell, 1985; Powell & Holland, 1985; Berman, 1988;
Holland & Powell, 1990; Gottschalk, 1997) has meant
that it has become possible to calculate mineral
equilibria involving solid solutions in complex systems
(e.g. Spear & Cheney, 1989; Powell & Holland, 1990;
Symmes & Ferry, 1992; Mahar et al., 1997). In
principle such calculations allow phase diagrams to be
constructed which closely approximate the phase
relationships in rocks.
Several different ways of approaching the calculation
of mineral equilibria involving solid solutions have
evolved out of those advocated in the chemical and
metallurgy literature (e.g. van Zeggeren & Storey,
1970): the Gibbs method (Spear & Cheney, 1989);
direct minimization of Gibbs energy (Wood, 1987;
Connolly, 1990; de Capitani & Brown, 1987) and the
solution of simultaneous non-linear equations (Powell
& Holland 1990). This last approach is the one
followed in the software, THERMOCALC, Powell &
Holland (1988, and succeeding upgrades), which uses
the internally-consistent thermodynamic dataset of
Holland & Powell (1985, 1990, 1998) (e.g. Guiraud
et al., 1990; Baker et al., 1994; Xu et al., 1994; Worley
& Powell, 1998).
The purpose of this paper is to consider the
calculation of phase diagrams involving solid solutions
by solving sets of non-linear equations. Examples of
the application of this approach are calculated with
© Blackwell Science Inc., 0263-4929/98/$14.00
Journal of Metamorphic Geology, Volume 16, Number 4, 1998, 577–588
the software THERMOCALC v2.6 using the April 20,
1996 version of the thermodynamic dataset. The
version of THERMOCALC, current at acceptance of
this paper for publication, this thermodynamic dataset,
datafiles, output files and documentation for the
examples are available from the JMG web sites.
THERMOCALC is available to be downloaded in
DOS, PowerMac and other Mac versions. The material
summarized in Table 1 and the most recent published
dataset are also available from the JMG sites. Current
versions of the software and dataset, as well as
Table 1. Summary of web-based material that is part of this
paper, located at the JMG web site. See also the material
at: http://rubens.unimelb.edu.au/ ~rpowell/THERMOCALC/
THERMOCALC.html for, for example, documentation of
datafile construction and THERMOCALC scripts, as well as
the most recent software and dataset.
Category
Software
Dataset
Documentation
Mathematica functions
Phase diagram movies
Contents
THERMOCALC v2.6 for DOS
THERMOCALC v2.6 for PowerMac
THERMOCALC v2.6 for other Macs (68k Macs)
TH.PD (‘20apr96’) for DOS
TH PDATA1 and THPDATA2 (‘20apr96’) for Mac
getting started
running THERMOCALC
annotated KFMASH datafile
annotated runs of THERMOCALC (logfiles)
assembling phase diagrams with Mathematica
low-level functions (drawline etc)
high-level functions for compatibility diagrams
high-level functions for T –x and P–x pseudosections
full examples of using functions
AFM compatibility diagram at 3 kbar, for 500 to 580 °C
T –x psuedosection for 510–650 °C, for 5 to 10.5 kbar
577
578
R. P OW E L L E T A L .
documentation of other THERMOCALC facilities,
are
at:
http://rubens.its.unimelb.edu.au/~rpowell/
THERMOCALC/THERMOCALC.html
C A L C U LAT IN G P H A S E DI AG RA M S
Calculating phase diagrams involves a number of steps:
1 Choose a model system in which to do the calculations.
A model system is just the chemical system, usually
specified in terms of oxides, in which the equilibria to
be calculated can be represented. This specifies which
phases, and the substitutions within them, that will be
able to be considered. Obviously these phases can only
involve those end-members that occur in the thermodynamic dataset used (or linear combinations of
them). For example, the model system normally used
to consider metapelites is K O–FeO–MgO–Al O –
2
2 3
SiO –H O (or KFMASH) (Thompson, 1957), allowing
2 2
most of the critical minerals, and the FeMg
and
−1
(Fe,Mg)SiAl Al
(Tschermak’s) substitutions in
−1 −1
them, to be considered. This model system is the one
used for the examples in this paper.
2 Formulate the thermodynamics of the phases in the
system. Given that a central part of calculations on
assemblages involving solid solutions is the calculation
of the equilibrium compositions of the phases, the
activity-composition (a–x) relationships of the phases
are needed in algebraic form, in terms of the composi-
tional variables to be calculated. The first stage of this
is matching the required substitutions in each phase
with end-members in the thermodynamic dataset, and
assigning composition variables to the substitutions.
In the case of chlorite in KFMASH, for example, the
FeMg
and (Fe,Mg)SiAl Al
substitutions are
−1
−1 −1
represented by clinochlore [clin, Mg Al Si O (OH) ],
5 2 3 10
8
daphnite [daph, Fe Al Si O (OH) ] and amesite
5 2 3 10
8
[ames, Mg Al Si O (OH) )] end-members. Two
4 4 2 10
8
composition variables are needed, and one choice is
x=Fe/Fe+Mg, reflecting the extent of FeMg , and
−1
y=Al/4, reflecting the extent of (Fe,Mg)SiAl Al
−1 −1
(see Appendix). For each phase, a mixing model needs
to be chosen. The approach followed in formulating
the a–x relationships of KFMASH chlorite in terms
of x and y, in this case in terms of the ideal-mixingon-sites model, is illustrated in Appendix 2. The a–x
relationships for the other phases in the examples are
also summarized in Appendix 2.
3 Decide on which phase diagrams are to be constructed.
This decision will depend mainly on what geological
problem is being addressed. Important types of
diagrams are:
$ A P–T projection, usually a key phase diagram,
shows the stable invariant points and univariant (or
reaction) lines for all of the bulk compositions in the
system. A P–T projection for KFMASH, constrained
Fig. 1. P–T projection for KFMASH
(+mu+q+H O); the in-excess phases
2 in the reactions labelling
are not included
the univariant lines, as is usual for such
diagrams. See Table 3 for the
abbreviations used.
C ALC ULATING PHA SE DIA GRA MS
Fig. 2. AFM compatibility diagram for KFMASH
(+mu+q+H O) at P=6 kbar and T =560 °C.
2
by stipulating the presence of muscovite, quartz and
H O (i.e. with mu+q+H O ‘in excess’), is shown in
2
2
Fig. 1. Such diagrams are the familiar petrogenetic
grids of the literature.
$ Compatibility diagrams show the mineral assemblages, and ranges of mineral solid solutions, at
specified P–T , for all of the bulk compositions in the
model system. A compatibility diagram for AFM, with
mu+q+H O in excess, is shown in Fig. 2.
2
$ P–T pseudosections show just those phase relationships for a particular bulk composition. A P–T
pseudosection for a model pelite composition in AFM,
with mu+q+H O in excess, is shown in Fig. 3.
2
$ T –x or P–x pseudosections show the phase relationships for a particular bulk composition line, at specified
P or T respectively. A T –x pseudosection for AFM,
with mu+q+H O in excess, is shown in Fig. 4.
2
Pseudosections are important because in systems with
solid solutions, it is not usually obvious which parts
of the equilibria in a P–T projection will be seen by a
particular bulk composition, given that the compositions of the phases vary along the univariant lines
(compare Figs 3 & 4 and Fig. 1).
The diagrams in Figs 1–4 were calculated with the
program THERMOCALC, using the a–x relationships
in the Appendix (see Table 3 for the phase and endmember abbreviations used). Although there are other
types of phase diagrams that can be drawn, for example
with activities or chemical potentials on the axes, it is
the calculation of the types of diagram in Figs 1–4
that will be the focus in this paper.
4 Build up the phase diagram via calculations on the
equilibria involved. Each mineral equilibrium calculation involves setting up and solving a mathematical
problem, as outlined in the next section for calculation
579
Fig. 3. P–T pseudosection in KFMASH (+mu+q+H O) for a
2
‘common’ pelite composition: Al O =41.89, MgO=18.19,
3
FeO=27.29, and K O=12.63 (in2 mol%).
2
Fig. 4. A T –x pseudosection in KFMASH (+mu+q+H O),
for a composition line along which FeO:MgO varies, with2 x=
FeO/(FeO+MgO), and Al O =41.89, FeO+MgO=45.48, and
2 3composition line goes through
K O=12.63 (in mol%). This
the2 composition used in Fig. 3.
in terms of sets of non-linear equations. Generating a
phase diagram usually involves many such calculations.
The calculations are computationally intensive so they
are usually carried out by computer: THERMOCALC
580
R. P OW E L L E T A L .
is software that performs such calculations. Phase
diagrams can be drawn from THERMOCALC output
using MathematicaTM functions which are made available here. These functions allow single phase diagrams
to be drawn, as well as phase diagram movies to be
constructed. THERMOCALC can also produce tabdelimited tables for importing into other graphical
software for manipulation and plotting.
C O N S T R U C TI O N A L F E AT U R E S OF P H A S E
D I A G R AM S
In the calculation of phase diagrams such as Figs 1–4,
it is necessary to understand their constructional
features. Such features of P–T projections, involving
invariant1 points and univariant lines, are generally
familiar. The constructional features of compatibility
diagrams are illustrated in Fig. 5. The tie triangles
(divariant equilibria) in the full system (within the
triangle) as well as the divariant equilibria in the
subsystems (edges of the triangle) specify much of the
geometry, including the apices of the one-phase fields
(quadrivariant equilibria). The edges of the one phase
fields can be determined via the adjacent trivariant
equilibria. In cases such as this, in which Fe–Mg is the
dominant substitution, the edges are close to being
straight; in other systems, such edges can be strongly
curved.
Pseudosections involve the invariant and univariant
equilibria they inherit from P–T projections, as well as
additional boundary lines and points. In a P–T
pseudosection these inherited equilibria are just those
parts of the P–T projection ‘seen’ by the bulk
composition being considered (compare Figs 1 & 3).
In a T –x pseudosection these inherited equilibria occur
as horizontal lines, spanning the range of bulk
compositions that ‘sees’ an equilibrium. The boundary
lines and points separate fields of different variance;
across lines the variance changes by one, and through
points by two, e.g. Fig. 6(iii), 7(iv). Divariant fields
adjoin univariant lines, separated by trivariant fields
that emanate from points along the univariant lines,
e.g. Fig. 6(i), 7(ii). The boundary points and lines can
be most easily discussed with a new notation, which
takes advantage of the fact that at lines one mode has
gone to zero, and at points two modes have gone to
zero. Labelling is carried out in terms of the lower
variance assemblage involved, with the names of phases
with zero modes given in brackets, see Figs 6 & 7. So
for example in Fig. 6(iii), going from the g–chl–bi
divariant field, through the point at 8 kbar and 580 °C,
into the chl quadrivariant field, the modes of bi and g
go to zero at the point. The point is therefore labelled
‘chl ( bi g)’. Similarly, in going down T at 10 kbar from
the g–chl–bi divariant field into the g–chl trivariant
field into the chl quadrivariant field, the two lines
crossed are labelled ‘chl g ( bi)’ and ‘chl (g)’. The zero
mode aspect of the boundary lines and points also
provides a means of calculating them directly, as
shown below.
H O W C A L C U LAT IO N S A R E S ET UP WI T H
NON- LINEAR EQUATIONS
The organization of phase diagram calculations is
predicted on the way the calculations are to be carried
out. Setup of calculations for use in a non-linear
equations solver, rather than, for example, a Gibbs
energy minimizer, is outlined here.
Fig. 5. The constructional features of the AFM compatibility
diagram in Fig. 2. The numbers give the variance of the
corresponding field. Part (i) focusses on a full-system divariant
field; (ii) focusses on the chlorite one-phase field; its apices are
defined by the corresponding apices of the full and subsystem
divariant fields; (iii) focusses on the KFASH divariant fields
that define the left hand edge of the diagram, (iv) focusses on
the KMASH divariant fields that define the right hand edge.
1for an n component system, involving p phases, the variance v
is equal to n−p+2, otherwise known as the phase rule.
Invariant: v=0; univariant: v=1; divariant: v=2; and so on.
Calculating P–T projections and compatibility diagrams
A non-linear equations approach is suited to calculating equilibria of a specified variance. Thus, the
calculation of P–T projections and compatibility
diagrams is straightforward. So, if the position of the
[bi,cd] invariant point in Fig. 1 is to be calculated,
the equilibrium ctd–st–chl–g–mu–q–H O is addressed.
2
Similarly, if the coordinates of the corners of
the g–chl–bi divariant triangle in Fig. 2 are to be
C ALC ULATING PHA SE DIA GRA MS
581
Calculating an invariant point, say [bi, cd] at about
P=12 kbar and T =595 °C on Fig. 1, will involve
solving a set of non-linear equations consisting of the
equilibrium relationships for reactions, forming an
independent set, written between the end-members of
the phases in the equilibrium being considered
0=DG°+RT ln K
1
1
0=DG°+RT ln K
2
2
e
Whereas DG° is just a function of P–T , the equilibrium
k
constant, K , is, in general, a function of P–T and the
k
compositions of all the phases whose end-members are
involved in the reaction. The P–T and these compositions are the unknowns that are solved for. For [bi,cd]
on Fig. 1 an independent set of reactions is:
(1) 72mctd+11py+3ames=8mst+17clin ;
(2) 8mst+17clin+6ky=84mctd+11py ;
(3) 10mst+19clin+3q=96mctd+13py ;
(4) 39mctd+4py=4mst+7clin+3H O ;
2
(5) 8fst+17daph+6ky=84fctd+11alm ;
(6) 10fst+19daph+3q=96fctd+13alm ;
(7) 39fctd+4alm=4fst+7daph+3H O ;
2
(8) 8mst+14clin+3cel=72mctd+11py+3mu ;
(9) 8fst+14daph+3fcel=72fctd+11alm+3mu ;
Fig. 6. The constructional features of part of the P–T
pseudosection in Fig. 3, indicating the necessary calculations to
define the lines and points. The numbers give the variance of
the corresponding field. In parts (i–iii), points and lines are
labelled in terms of the lower variance adjacent assemblage,
with the phase(s) whose modes are zero there being given in
brackets. Part (i) focusses on the g+chl=st+bi univariant
line, showing its relationship to adjoining trivariant fields that
terminate on it; (ii) focusses on the g–chl–bi divariant field, its
boundary lines at higher P being with trivariant fields, whereas
the lower P boundary is the univariant; and (iii) focusses on
the chl quadrivariant field.
calculated, the equilibrium g–chl–bi–mu–q–H O is
2
addressed. The approach can be extended to the
calculation of all of the equilibria of a particular
variance in a system. Calculating all of the equilibria
of variance, v, involves looking at all of the different
combinations of n−v+2 phases chosen from the p
phases in the system. There will be
p( p−1) ... (n−v+4)(n−v+3)
p
=
Cn−v+2
(n−v+2)(n−v+1) ... (2)(1)
of these (in the absence of composition degeneracies
amongst the phases).
using the abbreviations in Table 3, and the corresponding set of equilibrium relationships can be written out.
For example, the first one is
A
B
(ast )8(achl )17
mst clin
0=DG°+RT ln
1
(actd )72(ag )11(achl )3
mctd
py
ames
which becomes, on substituting for the activity
expressions in the Appendix
0=DG°+RT
1
{(1−x )4}8{16(1−x )5(1−y )2y2 }17
st
chl
chl chl
×ln
{(1−x )}72{(1−x )3}11{(1−x )4y4 }3
ctd
g
chl chl
Given that each DGo is a function of P and T , these
nine equations are in nine unknowns: P, T , x , x ,
st ctd
x, x , y , x
and y . Therefore they can be
g chl chl mu
mu
solved, giving the P–T coordinate of the invariant
point on the P–T projection, as well as the equilibrium
compositions of the coexisting minerals there.
(THERMOCALC gives P=12.2 kbar, T =595 °C,
x =0.865, x =0.662, x =0.830, x =0.436, y =
st
ctd
g
chl
chl
0.556, x =0.701 and y =0.793). See also Table 4
mu
mu
for the results of calculations around the [ctd,cd]
invariant point in Fig. 1, that terminates st+bi stability
to higher pressure at P=10 kbar and T =623 °C.
Calculating the g–chl–bi–mu–q–H O divariant equi2
A
B
582
R. P OW E L L E T A L .
Fig. 7. The constructional features of part of the T –x pseudosection in Fig. 4, indicating the necessary calculations to define the lines
and points. The numbers give the variance of the corresponding field. In parts (ii–iv), points and lines are labelled as in Fig. 3. Part
(i) shows the AFM diagram at the temperature of the g+chl=st+bi univariant equilibrium, showing where the bulk composition line
of the T –x diagram plots at this temperature, in projection from the calculated compositions of muscovite; (ii) focusses on the g+chl=
st+bi univariant line; (iii) focusses on the g–chl–bi divariant field, with the apex at x=1 given by the corresponding KFASH
univariant equilibrium; (iv) focusses on the chl quadrivariant field.
librium as a tie triangle on a compatibility diagram
(e.g. Fig. 2), a set of seven independent reactions is:
(1) clin+ames+4q=3py+8H O ;
2
(2) 5py+3daph=5alm+3clin ;
(3) clin+east=ames+phl ;
(4) ames+cel=clin+mu ;
(5) 3ames+2phl+6q=py+3clin+2mu ;
(6) 2ann+mu+6q=alm+3fcel ;
(7) 6daph+mu+3fcel=7alm+4ann+24H O .
2
giving seven equilibrium relationships in terms of nine
variables: P, T , and the compositions x , x , y , x ,
g chl chl bi
y , x and y . To calculate this equilibrium, two
bi mu
mu
variables must be fixed. Fixing P and T , as appropriate
for a compatibility diagram (P=6 kbar, T =560 °C in
Fig. 2), the compositions of the coexisting phases in
the equilibrium can be calculated by solving the
equilibrium relationships. The coordinates of the calculated compositions of the phases can then be plotted
on a compatibility diagram. The coordinates of biotite,
the only phase in this equilibrium which is not already
in the plane of the compatibility diagram, is found by
projecting from the calculated composition of the
muscovite. (THERMOCALC at P=6 kbar and T =
560 °C gives the {A,F,M} coordinates to be g: {0.250,
0.704, 0.046}, chl: {0.190, 0.558, 0.251} and bi: {−0.228,
Table 2. Notation (units used are kJ, K and kbar and
temperatures are quoted in °C).
Symbol
P
T
a
x
aj
i
x
i
y
i
R
DG°
k
n
p
s
c
v
p
j
xj
i
cj
i
x
i,j
s
[k]
x^n
Meaning
pressure
temperature
activity
composition
activity of end-member i in phase j
composition variable, Fe/Fe+Mg, for phase i
composition variable relating to Tschermak’s substitution for phase i
gas constant (0.0083144 kJ K−1)
Gibbs energy of reaction for reaction k
number of components in a model system
number of phases in an equilibrium
Total number of end-members in the phases in an equilibrium
number of composition variables set (i.e. given values) in an equilibrium
variance
modal proportion of phase j in an equilibrium
proportion of end-member i in phase j
number of molecules of component i in end-member j
site fraction of element i on site j
standard deviation
reaction not involving phase k (i.e. k-out)
in Table 8, means x to the power of n
0.871, 0.356}. The calculated AFM coordinates of the
stable divariant fields at P=6 kbar and T =560 °C, as
shown in Fig. 2, are summarized in Table 5.
To generalize, considering an n-component model
system, if phase k involves e end-members, then it
k
involves e −1 composition variables. For p phases in
k
the equilibrium, there will be Sp e ¬s end-members
k=1 k
of phases, and Sp
(e −1)=s−n composition
k=1
k
C ALC ULATING PHA SE DIA GRA MS
Table 3. Abbreviations and full names of the phases and endmembers, and the formulae of the end-members, used in the
KFMASH examples. The end-member abbreviations are those
understood by THERMOCALC.
Phase
abbrev.
Phase
full name
mu
muscovite
bi
biotite
chl
chlorite
cd
cordierite
ctd
chloritoid
st
staurolite
g
garnet
and
ky
sill
q
HO
2
andalusite
kyanite
sillimanite
quartz
fluid
End-member
abbrev.
End-member
full name
Formula
Table 5. Summary of the THERMOCALC output for the
stable KFMASH divariant triangles at P=6 kbar and
T =560 °C, giving the AFM coordinates on the compatibility
diagram in Fig. 2. The coordinates of biotite are calculated by
projection from the calculated composition of muscovite in this
divariant equilibrium.
Divariant
mu
cel
fcel
phl
ann
east
clin
ames
daph
crd
hcrd
fcrd
mctd
fctd
mst
fst
py
alm
gr
and
ky
sill
q
HO
2
muscovite
celadonite
Fe celadonite
phlogopite
annite
eastonite
clinochlore
amesite
daphnite
cordierite
hydrous cordierite
Fe cordierite
Mg chloritoid
Fe chloritoid
Mg staurolite
Fe staurolite
pyrope
almandine
grossular
andalusite
kyanite
sillimanite
quartz
water fluid
KAl Si O H
3 3 12 2
KMgAlSi O H
4 12 2
KFeAlSi O H
4 12 2
KMg AlSi O H
3
3 12 2
KFe AlSi O H
3
3 12 2
KMg Al Si O H
2 3 2 12 2
Mg Al Si O H
5 2 3 18 8
Mg Al Si O H
4 4 2 18 8
Fe Al Si O H
5 2 3 18 8
Mg Al Si O
2 4 5 18
Mg Al Si O H
2 4 5 19 2
Fe Al Si O
2 4 5 18
MgAl SiO H
2
7 2
FeAl SiO H
2
7 2
Mg Al Si O H
4 18 7.5 48 4
Fe Al Si O H
4 18 7.5 48 4
Mg Al Si O
3 2 3 12
Fe Al Si O
3 2 3 12
Ca Al Si O
3 2 3 12
Al SiO
2
5
Al SiO
2
5
Al SiO
2
5
SiO
2
HO
2
variables. The number of reactions between the
end-members that make up an independent set is the
number of end-members minus the number of components, s-n (Powell & Holland, 1988). Given that there
is a non-linear equation for each reaction in the
independent set, these relationships indicate how many
583
Phase
st–g–chl
st
g
chl
st
chl
ky
g
chl
bi
st–chl–ky
g–chl–bi
A
F
M
0.692
0.250
0.199
0.692
0.207
1.000
0.250
0.190
−0.228
0.294
0.702
0.542
0.274
0.356
0.014
0.048
0.259
0.033
0.437
0.704
0.559
0.871
0.046
0.251
0.356
unknowns can be solved for, and therefore how many
must be set, because the s-n equations can only be
solved for s–n unknowns. The number of things that
have to be set in order for an equilibrium to be
calculated can be represented in terms of degrees of
freedom, equal to the number of unknowns, (s–p)+2,
minus the number of equations, s–n, giving n–p+2. So
the number of degrees of freedom is just the variance.
Setting unknowns may involve setting P and/or T , or
setting compositional variables, as would be done, for
example, in calculating composition isopleths on a
P–T diagram. Therefore, by variance, v:
v=0. For an invariant equilibrium, a point on a P–T
diagram, all the unknowns can be solved for. With
no composition variables set, the equilibrium
Table 4. Summary of the THERMOCALC output for the equilibria around the [cd, ctd] invariant point at P=10 kbar and T =
623 °C in the P–T projection shown in Fig. 3. First is the calculated coordinates of the invariant point itself, with the uncertainties
(1s) arising from the uncertainties on the thermodynamic data, followed by the calculated coordinates of the univariant lines. The
P and T of the invariant point are only slightly correlated (r =−0.133).
PT
(a)
s
[ky]
[bi ]
[chl]
[g]
[st]
P ( kbar)
T (°C)
x
10.0
0.8
6.0
8.0
10.0
12.0
14.0
6.0
8.0
10.0
12.0
14.0
6.0
8.0
10.0
12.0
14.0
6.0
8.0
10.0
12.0
14.0
6.0
8.0
10.0
12.0
14.0
623
7
572
600
623
644
662
649
640
623
599
569
670
650
623
589
549
585
605
623
639
654
583
605
623
639
653
0.818
0.0203
0.937
0.884
0.817
0.737
0.642
0.752
0.782
0.818
0.859
0.902
0.895
0.855
0.818
0.786
0.761
0.859
0.838
0.818
0.797
0.776
st
x
g
0.771
0.0192
0.911
0.846
0.770
0.685
0.590
0.686
0.725
0.771
0.823
0.877
0.862
0.813
0.771
0.735
0.708
0.885
0.830
0.770
0.705
0.635
x
chl
0.362
0.0258
0.604
0.466
0.361
0.277
0.209
0.279
0.315
0.362
0.427
0.515
0.394
0.377
0.362
0.347
0.333
0.542
0.442
0.361
0.295
0.239
y
chl
0.565
0.0091
0.593
0.579
0.565
0.552
0.539
0.580
0.573
0.565
0.557
0.549
0.609
0.587
0.565
0.544
0.523
0.610
0.587
0.565
0.544
0.523
x
bi
y
0.384
0.0250
0.638
0.496
0.383
0.292
0.217
0.356
0.0483
0.417
0.386
0.355
0.326
0.297
0.574
0.468
0.383
0.315
0.259
0.429
0.406
0.384
0.363
0.343
0.578
0.471
0.383
0.310
0.247
0.493
0.425
0.355
0.284
0.213
0.448
0.400
0.355
0.313
0.274
0.446
0.399
0.355
0.314
0.275
bi
x
mu
0.620
0.0253
0.812
0.713
0.619
0.527
0.437
0.509
0.559
0.620
0.691
0.772
0.752
0.680
0.619
0.569
0.528
0.646
0.632
0.620
0.608
0.596
0.768
0.692
0.619
0.550
0.482
y
mu
0.822
0.0161
0.860
0.841
0.821
0.801
0.780
0.863
0.843
0.821
0.796
0.767
0.879
0.852
0.821
0.784
0.739
0.889
0.857
0.821
0.782
0.741
0.877
0.850
0.821
0.791
0.759
584
R. P OW E L L E T A L .
involves n+2 phases; the P–T of the point, and the
compositions of all the phases in the equilibrium,
can be solved for. For c composition variables set,
the equilibrium involves n+2−c phases. For
example, a divariant assemblage involving n phases,
with two composition variables set, corresponding
to the intersection of two isopleths, is (effectively)
invariant, and the P–T and remaining composition
variables of the equilibrium can be solved for.
v=1. For a univariant equilibrium, a line on a P–T
diagram, if one of the unknowns is set (i.e. one of P,
T and the composition variables), then the remaining
unknowns can be solved for. With no composition
variables set, the equilibrium involves n+1 phases,
and, given, say, P, the T and the compositions of all
of the phases can be solved for. For c composition
variables set, the equilibrium involves n+1−c
phases. For example, a divariant assemblage involving n phases with one composition variable set, is a
line on a P–T diagram (an isopleth), the equilibrium
is (effectively) univariant and, given, say, P, the T
and the remaining compositions of the phases can
be solved for.
v=v. For a v-variant equilibrium, if v of the unknowns
are set (i.e. v of P, T and the composition variables),
then the remaining unknowns can be solved for. For
v≤2, and no composition variables set, the equations
cannot be solved. With no composition variables
set, the equilibrium involves n+2−v phases; for c
composition variables set, the equilibrium involves
n+2−v−c phases.
Calculating pseudosections
The calculation of P–T and T –x/P–X pseudosections
(Figs 3 & 4) with the non-linear equation approach is
greatly aided by augmenting the non-linear equations
formed by the set of 0=DGo+RT lnK equations with
a set of equations derived from mass balance constraints. For the specified bulk composition that is
being used for the calculation, these additional equations relate the mole proportion of each component in
the bulk composition with the sum of the calculated
mineral compositions multiplied by their modal proportions. In other words, the bulk composition must
be able to be made up of an assemblage of the phases
of interest. Of course the phases in the assemblage
must each have non-negative modal proportions.
Situations where one or two phases have zero modes
are particularly relevant, as they correspond to
the lines and points in pseudosections, as discussed
above.
As can be seen in the following, the zero mode
character of the lines and points in pseudosections can
be utilized in calculations. For the n-component system
considered above, there will be a mass balance equation
for each component, giving n equations additional to
the equilibrium relationships involved. There are also
p additional variables, the modes of the phases. Then
the total number of equations is s-n+n, i.e. s equations,
in s-n+p+2, i.e. s+2 unknowns. So, if two things are
specified, the number of equations equals the number
of unknowns regardless of the number of phases involved.
This means that equilibria of any variance can be
calculated, once a bulk composition is specified (see
also Spear, 1986). The value of including the mass
balance constraints in calculations comes in setting
one or two modal proportions to zero. On a P–T
pseudosection, with one mode set to zero, an equilibrium (involving any number of phases) defines a P–T
line (i.e. is effectively univariant). With two modes set
to zero, an equilibrium defines a P–T point (i.e. is
effectively invariant). On a T –x pseudosection (at fixed
P ), with one mode set to zero, an equilibrium
(involving any number of phases) defines a T –x line,
and with two modes set to zero, an equilibrium
defines a T –x point. This means that the lines and
points that make up pseudosections can be calculated
directly.
The above approach can be seen in calculating the
divariantquadrivariant point, ‘chl ( bi g)’ on a P–T
pseudosection, occurring at 8 kbar and 580 °C on
Fig. 3. The divariant equilibrium, chl–bi–g, involves
the same set of seven independent reactions between
the end-members of the phases as listed above in the
compatibility diagram example. So there are seven
equilibrium relationships, in the nine unknowns, P, T ,
x , x , y , x , y , x and y . Rather than fixing P
g chl chl bi bi mu
mu
and T as in the above, the equilibrium relationships
are augmented with mass balance constraints. With
constraints for K O, FeO, MgO, Al O , and the
2
2 3
additional unknowns, the modal proportions, p , p ,
chl bi
p and p , the required P–T point can be calculated
g
mu
directly, once p and p are both set to zero. Mass
bi
g
balance constraints are not needed for SiO and H O
2
2
because quartz and H O2 correspond to these compo2
nents. The mass balance constraints that are needed
involve combinations of the composition and mode
variables. For example, for MgO, using the information
in Appendix 2 for converting composition variables
into mineral compositions
MgO=p {cclin xchl +cames xchl +cdaph xchl }
chl MgO clin MgO ames MgO daph
+p {cphl xbi +ceast xbi +cann xbi }
bi MgO phl MgO east MgO ann
+p {cpy xg +calm xg }
g MgO py MgO alm
+p {cmu xmu+ccel xmu+cfcel xmu }
mu MgO mu MgO cel
MgO fcel
In this, p , for example, is the modal proportion of
bi
biotite in the equilibrium, xbi , is the proportion of the
phl
end-member phlogopite in the biotite, and cphl is the
MgO
number of MgO in the formula of phlogopite. On
substituting the expressions for the proportions of the
end-members in the phases in terms of the composition
2H O is used here as a component and as a phase, with the
2
meaning
given by the context.
C ALC ULATING PHA SE DIA GRA MS
variables this constraint becomes
GA
B
2
MgO=p
5 1− x (3−y )−(2y −1)
chl
chl
chl
5 chl
H
+4(2y −1)
chl
GA
A
A B
H
1
+p 3 1−x 1− y
−y +2( y )
bi
bi
bi
bi
3 bi
+p {3((1−x ))}
g
g
+p {x (1−y )}
mu mu
mu
The coordinates of ‘chl ( bi g)’ on the P–T and T –x
pseudosections as calculated by THERMOCALC are
given in Table 6, section (ii) and Table 7, section (i).
In the latter case, the coordinates are determined via
the locus of the P–T point with bulk composition, with
the T –x position of the point at the fixed pressure of
the pseudosection found by interpolation, or by
running THERMOCALC over an appropriately
narrow composition range.
The calculation of a line on a P–T or T –x
pseudosection is illustrated with the trivariant
quadrivariant line, ‘chl ( bi)’. This runs down pressure
from the ‘chl ( bi g)’ point on Fig. 3, and to more
magnesium bulk compositions from this same point
on Fig. 4. The trivariant equilibrium, chl–bi, is represented by an independent set of five reactions:
(1) ames+cel=clin+mu ;
(2) clin+east=ames+phl ;
Table 6. Summary of THERMOCALC output that relate to the
calculation of the P–T pseudosection in Fig. 3, the p being the
k
modal proportions of the phases in the assemblage. The
mineral
composition information is omitted. (i) the g–chl–bi–st
univariant assemblage calculated with the modes of st and g,
and g and chl, set to zero, so determining the terminations of
the corresponding trivariant fields on the univariant line; (ii) the
bi–chl–g divariant field calculated with the modes of bi and g
set to zero, so determining the position of the apex of the chl
quadrivariant; (iii) the bi–chl–g divariant field calculated with
the mode of bi set to zero, so determining the position of the
boundary between the bi–chl–g divariant field and the g–chl
trivariant field; and (iv) the bi–chl trivariant field calculated with
the mode of bi set to zero, so determining the position of the
boundary between the bi–chl trivariant and the chl
quadrivariant fields. For (i) and (ii) the calculated uncertainties
(1s) on the positions of the P–T points are also given.
P (kbar)
i
s
s
ii
s
iii
iv
7.3
0.8
6.3
0.9
8.0
0.6
9.0
10.0
11.0
12.0
5.0
6.0
7.0
8.0
T (°C)
590
5
577
6
580
6
590
600
609
609
521
541
561
581
p
st
p
g
p
p
chl
bi
p
mu
0.207
0
0
0.575
0.217
0
0
0.444
0.102
0.454
0
0
0.495
0
0.505
0.088
0.158
0.219
0.273
0.407
0.337
0.276
0.222
0.495
0.495
0.495
0.495
0
0
0
0
0
0
0
0
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
585
Table 7. Summary of THERMOCALC output relating to
generating the T –x pseudosection in Fig. 4, with the mineral
composition information omitted: (i) the bi–chl–g divariant
field calculated with the modes of bi and g set to zero, so
determining by interpolation the position of the apex of the chl
quadrivariant field; (ii) the bi–chl–g divariant field calculated
with the mode of chl set to zero, so determining the position of
the boundary between the bi–chl–g divariant and the bi–g
trivariant fields; and (iii) bi–chl trivariant field calculated with
the mode of bi set to zero, so determining the position of the
boundary between the bi–chl trivariant and the chl
quadrivariant fields. The prop column refers to the
proportional position along the bulk composition line.
i
ii
iii
P (kbar)
T (°C)
prop
p
g
5.1
5.5
5.8
5.8
6.0
6.6
7.0
7.4
6.0
6.0
6.0
6.0
6.0
6.0
6.0
6.0
6.0
6.0
528
535
541
541
545
556
563
572
541
554
567
545
543
542
540
538
535
532
0.050
0.100
0.150
0.150
0.179
0.250
0.300
0.350
0.050
0.100
0.150
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0
0
0
0
0
0
0
0
0.430
0.413
0.399
p
chl
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0.495
0
0
0
0.495
0.495
0.495
0.495
0.495
0.495
0.495
p
bi
0
0
0
0
0
0
0
0
0.130
0.164
0.192
0
0
0
0
0
0
0
p
mu
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.505
0.440
0.423
0.409
0.505
0.505
0.505
0.505
0.505
0.505
0.505
(3) 5clin+3mu=4ames+3phl+7q+4H O ;
2
(4) daph+4fcel=3ann+mu+7q+4H O ;
2
(5) 5ames+5fcel+4clin+daph+5mu .
So there are five equilibrium relationships, in the eight
unknowns, P, T , x , y , x , y , x
and y .
chl chl bi bi mu
mu
Augmenting with mass balance constraints for K O,
2
FeO, MgO, Al O , with the additional unknowns, p ,
2 3
chl
p , and p , there are nine equations in 11 unknowns.
bi
mu
On setting p to zero, the set of equations defines a
bi
P–T line for a particular bulk composition, as in Fig. 3,
as well as a T –x line at fixed pressure across a bulk
composition range, as in Fig. 4. The corresponding
calculated coordinates are given in Table 6, section (iv),
and in Table 7, section (iii).
Determining the stability of calculated equilibria
Not addressed in the above development is the
determination of the stability of equilibria. When all
the invariant and univariant equilibria in a P–T
projection are calculated, many or most will be
metastable. The stability of the equilibria can be
determined by applying Schreinemakers Rule3 to the
univariant lines around each invariant point, then
putting the invariant points together. With the resulting
P–T projection, determining the stable equilibria in
compatibility diagrams and pseudosections is generally
straightforward.
3the metastable extension of the reaction not involving phase i
(i-out or [i]) lies between i-producing reactions (e.g. Zen, 1966).
586
R. P OW E L L E T A L .
In the context of determining which assemblage is
the most stable one at a particular P–T , for a particular
bulk composition, the minimization of the Gibbs
energy of the system can be utilized. From the last
subsection, with the mass balance constraints included,
equilibria of any variance can be calculated if two
unknowns are fixed. Choosing these to be P–T , it is
straightforward, if laborious, to loop through all the
possible divariant and higher equilibria in the system,
determine the Gibbs energy of each assemblage if it
can be calculated, and identify the one with the
smallest Gibbs energy. Such an approach, applied at a
series of P–T points along a P–T line, is very effective
in getting started in constructing pseudosections, in
identifying (stable) lines that may then be calculated
by the method of the last subsection.
In conclusion, the software THERMOCALC is built
around a non-linear equation solver, and it involves
an implementation of the material covered above. As
a consequence, the features of P–T projections, compatibility diagrams, P–T pseudosections and T –x/P–x
pseudosections can be calculated directly. Aspects of
this are discussed in Appendix 1.
Table 8. (Continued)
cd 3
g2
chl 3
2
fcrd
1
2
hcrd
bulk 1 1
1
2
bulk 1 1
x(cd) 0.45
h(cd) 0.7
py
1
1
check 0
alm
1
1
bulk 1 1
check 1
x(g) 0.9
clin
16
3
ames
east
mu 3
Table 8. Main part of a THERMOCALC datafile for
KFMASH, showing the coding of the a–x relationships etc
used in the examples, using the a–x relationships from the
Appendix. Everything after a % on a line is treated as a
comment and is ignored by THERMOCALC, allowing the
annotation of datafiles. See the documentation on the web for
details of datafile construction. See Table 3 for the
abbreviations of the phases and the end-members.
ctd 2
1
1
check 0
fst
1
1
bulk 1 1
check 1
x(st) 0.92
mctd 1
1
check 0
fctd
1
1
bulk 1 1
check 1
x(ctd) 0.91
1
1
−1
1
4
% a(mst)=(1−x)^4
0
0
1
1
1
1
1
1
4
%
bulk 1 1
check 0 1
x( bi) 0.5
y( bi) 0.49
mu
1/4 3
fcel
1
1
−1
1
1
1
1
1
1
1
1
1
% a(fctd)=x
%
x=Fe/(Fe+Mg)
check 0 1
1/4 4
check 0 0
1/4 4
bulk 1 2
ky q
0
0
check 1 0
1/4 4
bulk 1 2
a(fst)=x^4
% x=Fe/(Fe+Mg)
% a(mctd)=(1−x)
check 0 0
1/4 4
bulk 1/3 2
cel
mst
check 1 1/2
1
2
bulk 1 1
check 0 1
x(chl) 0.5
y(chl) 0.6
phl
1/4 4
ann
A P P EN D I X 1 : U SI N G THE R M O C A L C
st 2
check 0 1/2
16
3
bulk 1/5 2
We thank F. Spear, D. Pattison and J. Brady for their
helpful comments. We also thank all those friends and
others who have laboured with THERMOCALC, who
have helped in its evolution over the years, including
J. Baker, M. Guiraud, T. Will, G. Xu, K. Stuwe,
J. Arnold, E. Maher, C. Carson, among others.
The software THERMOCALC allows phase diagrams to be
calculated, as in Tables 4–7. In using THERMOCALC, provision
of the a–x relationships for the minerals is done via a datafile, of
which for example Table 8 is a part. Although for a few solid
solutions the mixing model to use is fixed by the way the
thermodynamic data for the end-members were extracted (see
Holland & Powell, 1990), in general the user can code whichever
mixing model is thought to be appropriate (within the Margules
1
daph
bi 3
ACKNOWLEDGEMEN TS
crd
check 1 0
x(mu) 0.75
y(mu) 0.95
HO
2
1
1
0
1
0
1
0
0
1
1
1
1
1
1
1
1
−1
−1
1
−1
1
−1
1
1
1
2
1
2
1
1
2
2
2
1
2
1
% a(crd)=(1−x)^2 *
%
(1−h)
% a(fcrd)=x^2 *
%
(1−h)
2
1
% a(crd)=(1−x)^2 *
%
h
% x=Fe/(Fe+Mg)
% h=H O pfu
2
% a(py)=(1−x)^3
1
1
−1
1
3
0
1
0
1
1
1
1
3
1
1
1
0
1
1
1
−1
−1
1
1
2
2
5
2
2
% x=Fe/(Fe+Mg)
% a(clin)=16 (1−x)^5 *
%
(1−y)^2 *
%
y^2
0
1
0
0
3
1
1
1
1
1
1
−1
1
2
−1
1
2
2
1
2
5
2
2
% a(daph)=16 x^5 *
%
(1−y)^2 *
%
y^2
1
0
−1
1
1
1
−1
1
2
1
2
2
4
4
% a(ames)=(1−x)^4 *
%
y^4
% a(alm)=x^3
1
2
1
1
1
1
1
1
−1
−1
1
−1
1
2
2
2
3
2
1
1
% x=Fe/(Fe+Mg)
% y=oct A1/2
% a(phl)=1/4 (1−x)^3 *
%
(2−y)^2 *
%
(1+y) *
%
(1−y)
0
2
1
1
0
3
1
1
1
1
1
1
1
−1
1
−1
1
−1
1
2
2
2
1
2
3
2
1
1
% a(ann)=1/4 x^3 *
%
(2−y)^2 *
%
(1+y) *
%
(1−y)
1
0
1
2
0
1
1
1
1
1
−1
1
1
−1
1
1
2
2
2
2
2
1
2
1
% a(east)=1/4 (1−x)^2 *
%
y*
%
(1+y)^2 *
%
(2−y)
1
2
0
1
1
1
1
−1
1
2
2
2
2
1
1
% x=Fe/(Fe+Mg)
% y=oct Al
% a(mu)=1/4 (1+y)^2 *
%
(2−y) *
%
y
1
1
1
2
1
1
1
1
1
1
1
1
−1
−1
1
−1
−1
−1
1
2
2
2
1
2
1
1
1
2
% a(cel)=1/4 (1−x) *
%
(1−y) *
%
(1+y) *
%
(2−y)^2
0
1
1
2
0
1
1
1
1
1
1
1
1
−1
1
−1
1
−1
1
2
2
2
1
2
1
1
1
2
% a(fcel )=1/4 x *
%
(1−y) *
%
(1+y) *
%
(2−y)^2
% x=Fe/(Fe+Mg)
% y=oct Al−1
family). Output from THERMOCALC, processed by Mathematica
functions made available here, can be used for assembling phase
diagrams and phase diagram movies. The mechanics of datafile
construction, running THERMOCALC, and assembling phase
C ALC ULATING PHA SE DIA GRA MS
587
diagrams, is covered in detail on the web. Additionally on the web
are QuickTime compatibility diagram movies for AFM, running
from 500 to 580 °C at 3 kbar, and a T –x diagram movie for AFM,
running from P=5 to 10.5 kbar, and the information needed to
construct them.
One aspect of using THERMOCALC to perform mineral
equilibrium calculations that is worth emphasizing is its ability to
propagate uncertainties in the thermodynamic data of the endmembers of the phases through calculations. These data uncertainties
are generated along with the thermodynamic data themselves as a
consequence of the way the data are generated (see below; also
Powell & Holland, 1993). The magnitude of the uncertainties on the
thermodynamic data reflect how well they are constrained by the
various experimental data used to calculate them. It is always
worthwhile calculating the uncertainties on equilibria, e.g. Tables 4(i),
6(i,ii), to give an indication of how well known the equilibria are.
Such an error propagation calculation as done by THERMOCALC
omits any contribution from errors in the formulation of a–x
relationships. However, in the absence of gross a–x errors, generally,
if uncertainties arising from the thermodynamic data are small, they
are likely to remain small overall. Of course, if uncertainties arising
from the thermodynamic data are large, they can only get larger if
other sources of uncertainty are also included. On this basis, the
position of the [ctd,cd] invariant point in Fig. 1, Table 4(i), is
relatively well known, with an uncertainty of about 1.5 kbar (2s),
whilst the pressure of [cd,as] is much less well-known (±>4 kbar).
The position of this point is obviously sensitive to the data used in
its calculation. In considering a whole phase diagram, the uncertainties on the positions of points and lines will usually be strongly
correlated, so care is required in developing on overall picture of
uncertainties. For example it can be asked if different topologies in
a P–T projection can be stable within error: can [cd,as] be at a
lower pressure than [g,cd] in Fig. 1 (so making both invariant
points metastable, and a different topology apply)? To answer this
requires knowing the correlations, because it is not sufficient to say
that the uncertainties on the two points overlap.
THERMOCALC can be used to perform calculations additional
to those addressed here. These include the geothermometry/
geobarometry calculations called average P–T methods (Powell &
Holland, 1994), as well as phase diagram calculations not involving
solid solutions (e.g. Powell & Holland, 1988). These latter calculations are simple to perform as the datafile needed requires just a
list of the phases (end-members) involved, and the results include
not only the positions of the equilibria but also the stability
relationships (see Powell & Holland, 1988).
amesite
[Mg ]M1[Al ]M2[Si ]T1[Al ]T2O (OH) .
4
2
2
2
10
8
The composition variables to use are not prescribed, although
some may be easier to use in calculations than others. A reasonable
choice is one that brings out the substitutions involved in the phase,
so define:
APPENDI X 2 : ACTI V IT Y- COMPOSI T ION A ND
B U L K C O M P O S IT I ON R E L AT I O N S H IP S F O R
M I N ER A L S I N K F M A S H
From the expression for y
chl
p
=2y −1
ames
chl
and substituting this into the expression for x
chl
2
p
= x (3−y )
chl
daph 5 chl
In any mineral equilibrium calculation involving solid solutions, the
thermodynamics of the phases need to be expressed in terms of their
compositional variables. Then, once a calculation is done, and the
equilibrium values of the compositional variables are found, the
bulk composition of the phase must be constructed, for example for
writing reactions or plotting the compositions on a compatibility
diagram. These processes are illustrated for chlorite in KFMASH.
The corresponding expressions for the other minerals are also
given below.
The starting point is deciding which end-members to include and
what a–x model to use. For chlorite in KFMASH, clinochlore (clin),
daphnite (daph) and amesite (ames), constitute an independent set
of end-members, if the FeMg and (Fe,Mg)SiAl Al substitutions
−1
−1 −1
are to be considered. The a–x relationships will be modelled using
ideal-mixing-on-sites, with Al mixing on two octahedral sites
(designated M2) and two tetrahedral sites (designated T2) (Holland
& Powell, 1990). Thus the end-members can be written:
clinochlore
[Mg ]M1[MgAl ]M2[Si ]T1[SiAl ]T2O (OH) ;
4
2
10
8
daphnite
[Fe ]M1[MgAl]M2[Si ]T1[SiAl ]T2O (OH) ;
4
2
10
8
Aloct
Fe
and y =x
=
x =
chl
Al,M2
chl Fe+Mg
2
Then the site fractions can be written in terms of x
and y :
chl
chl
=1−x
x
=x
Mg,M1
chl
Fe,M1
chl
x
=(1−x )(1−y ) x
=x (1−y ) x
=y
Mg,M2
chl
chl
Fe,M2
chl
chl Al,M2
chl
x
=0
x
=1
Al,T1
Si,T1
x
=y
x
=1−y
Al,T2 chl
Si,T2
chl
Then, for clinochlore using ideal mixing on sites:
x
a =16x4
x
x
x2 x
x
clin
Mg,M1 Mg,M2 Al,M2 Si,T1 Al,T2 Si,T2
in which the 16 is a normalization constant forcing a =1 for pure
clin
clinochlore (e.g. Powell, 1978, p. 70). This is necessary because both
the M2 and T2 sites have more than one element on them in pure
clinochlore. Substituting the site fractions into the ideal mixing
activity expression gives
a =16(1−x )5(1−y )2y2
clin
chl
chl chl
The same logic for daph and ames gives
a
=16x5 (1−y )2y2
daph
chl
chl chl
a
=(1−x )4y4
ames
chl chl
Following an equilibrium calculation, there will be values of x
chl
and y which can then be used to reconstitute the chlorite. This is
chl
done via expressions for the proportions of the end-members, p ,
clin
p
and p
, in the chlorite in terms of x and y . A start is
daph
ames
chl
chl
made by writing the definitions for x and y in terms of the p’s:
chl
chl
5p
5p
Fe
daph
daph
=
=
x =
chl Fe+Mg
5p
+5p +4p
5−p
daph
clin
ames
ames
in which the multiplier on p comes from the number of atoms of
k
Fe or Mg in the formula unit of k. The last step comes from the fact
that p +p
+p
=1. Similarly
clin
daph ames
p +p
+2p
1+p
Aloct
daph
ames=
ames
= clin
y =
chl
2
2
2
and, by difference
2
p =1− x (3−y )−(2y −1)
clin
chl
chl
5 chl
The a–x relationships of staurolite, chloritoid and garnet are
particularly simple to represent in terms of their composition
variables, in each case x is Fe/Fe+Mg. Thus, for their end-members:
=(1−x )4 a =x4 ;
mst
st
fst
st
=1−x ) a =x ;
mctd
ctd
fctd
ctd
a =(1−x )3
a =x3 .
py
g
alm
g
In considering biotite, using the compositional variables
a
a
Fe
and y =2x
=Aloct
x =
bi
Al,M2
bi Fe+Mg
with Al mixing with the Fe+Mg on two octahedral sites (M2), and
588
R. P OW E L L E T A L .
Al with the Si on two tetrahedral sites (T2), then
x
=1−x
Mg,M1
bi
x
A
y
x
=(1−x ) 1− bi
Mg,M2
bi
2
x
=
Al,T2
B
Fe,M1
x
1+y
bi
2
=x
bi
A
B
y
y
=x 1− bi x
= bi
Fe,M2
bi
Al,M2
2
2
x
=
Si,T2
1−y
bi
2
and the ideal mixing on sites expressions for the biotite end-members
are:
1
a = (1−x )3(2−y )2(1+y )(1−y );
phl 4
bi
bi
bi
bi
1
a = x3 (2−y )2(1+y )(1−y );
ann 4 bi
bi
bi
bi
1
a = (1−x )2y (1−y )2(2−y ).
east 4
bi bi
bi
bi
In considering muscovite, using the compositional variables
Fe
and y =x
=Aloct−1
x =
mu
Al,M2
mu Fe+Mg
with Fe+Mg mixing with the Al on two octahedral sites (M2), and
Al with the Si on two tetrahedral sites (T2), then
x
x
Mg,M2
=(1−x )
mu
y
= mu
Al,T2
2
A B
1−y
bi
2
x
x
=x
Fe,M2
Mu
Si,T2
A B
1−y
1+y
bi x
bi
=
Al,M2
2
2
y
=1− mu
2
and the ideal mixing on sites expressions for the muscovite
end-members are:
1
a = (1+y )2(2−y )y ;
mu 4
mu
mu mu
1
a = (1−x )(1−y )(1+y )(2−y )2;
mu
mu
mu
mu
cel 4
1
a = x (1−y )(1+y )(2−y )2.
fcel 4 mu
mu
mu
mu
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Received 3 March 1997; revision accepted 20 January 1998.