-
Canadian MineralogistVol. 29. pp. 673-686 (1991)
ABSTRACT
Mineral compositions from pelitic assemblages thalstraddle the
sillimanite - biotite isograd in the File Lakearea, Manitoba, have
been examined using singular valuedecomposition and linear
programming. If the rocks aremodeled in a ten-component system, the
mineralogy ofindividual assemblages can be interpreted as
representingat least divariant equilibrium. A modification
ofCreenwood's (1967) technique of linear programmingpermits the
identification of mass balances (crossingtielines in n-dimensional
space) and also identifies theconstituents that fail to balance
where no mass balanceexists. The isograd can be approximated by the
reaction:Sr + Ms + Chl + Crr + Ilm = Pl + Qtz + Bt + Sil+ H2O.
Metamorphic conditions are inferred to havebeen near 540'C and 330
MPa.
Keywords: electron-microprobe analyses, garnet,staurolite,
biotite, muscovite, chlorite, metamor-phism, algebraic analysis,
File Lake Formation,Manitoba.
Sovnaernr
La composition des min6raux des assemblages pdliti-ques qui
chevauchent l'isograde sillimanite - biotite dansla r6gion du lac
File, au Manitoba, a fait l'objet d'une6tude par d6composition de
valeurs singulibres et parprogr,tmmation lin6aire. Si les roches
correspondent ir dessystemes d dix composantes, la min6ralogie des
assem-blages individuels peut r€sulter d'€quilibres au
moinsbivariants. Une modification de la technique de program-mation
lineaire de Greenwood (1967) permet l'identifica-tion des
6quivalences des masses (le croisement de lignesd'attache dans un
espace i n dimensions) et, en plus, les6l6ments qui sont en
violation de l'6quivalence des masses.La r€action isogradique
serait: St + Ms + Chl + Gfi+ Ilm = Pl + Qtz + Bt + Sil + H2O. Les
conditionsdu mdtamorphisme auraient 6t6 prbs de 540'C et
330MPa.
(Traduit par la R6daction)
Mots-clds; analyses ir la microsonde 6lectronique,
grenat,staurotide, biotite, muscovite, chlorite, m6tamor-phisme,
analyse alg6brique, Formalion de File Lake,Manitoba.
ALGEBRAIC ANALVSIS OF THE BIOTITE_SILLIMANITE ISOGRADIN THE FILE
LAKE AREA, MANITOBA
TERENCE M. GORDON, EDWARD D. GHENT AND MAVIS Z. STOUTDepartment
oj Geotog! and Geophysics, University of Calgary, Calgary, Alberta
T2N IN4
INTRoDUCTtoN
A fundamental problem in metamorphic petrol-ogy is the
identification of the underlying changesin phase compatibilities
that result in the develop-ment of isograds (Thompson 1957). This
hasclassically been done by the construction ofpetrogenetic grids
using generalizations aboutassemblages and mineral compositions
(e.g., com-pilations in Philpotts 1990, Figure l7-11; Yardley1989,
Figures 3.11, 4.7). To be applicable to anumber of field areas and
simple enough to beuseable, such grids usually model nature in
termsof a small number of components. Rocks that don'tquite "fit"
these models are common, however,which leads to speculation that
such rocks eitherdo not represent equilibrium assemblages, or
thatadditional components are required to adequatelydescribe the
natural occurrences. Following thepioneering work of Greenwood
(1967, 1968),several investigators have attempted to resolve
thesequestions for specific isograds using a variety ofalgebraic
techniques (Fletcher 1971, Fletcher &Greenwood 1979,Pigage
1976, 1982, Lang & Rice1985a, Fisher 1989).
This paper uses the SVD procedure suggested byFisher (1989) and
a new linear programmingformulation of the r-dimensional tieJine
problemthat provides several advantages in investigationsof this
type. The results obtained in the study ofthe sillimanite - biotite
isograd in pelitic rocks fromthe File Lake area in northern
Manitoba indicatethat: a) the assemblages above and below
theisograd can be interpreted as divariant, and b) theisograd can
be modeled by a discontinuous reactionin the ten-component system
SiO2-TiO2-Al2O3-FeO-MnO-MgO-CaO-KzO-Na2O-HrO. Pressure- temperature
estimates show that this transitiontakes place near 540oC and 300
MPa, in concor-dance with the results of others on rocks
fromdifferent settings.
Ggoloctcel SetttNc
The File Lake area in northern Manitoba (Fig.
673
-
5OO metres#
File Lake
674 THE CANADIAN MINERALOGIST
PernocnapHv aNo Mtmnnr- CHrnatsrnv
The assemblages are shown in Table l. Chloriteand muscovite,
although ubiquitous, are notabundant. Trace amounts of apatite and
tour-maline occur in all samples, and graphite is foundin most.
The rocks contain a weak to moderatelywell-developed
schistosity, which developed priorto and during the highest grade
of metamorphism(Bailes 1980). In thin section, sparse flakes of
IABLE 2. IIEMON-UIEOPROBA DAIA ON PGCIOCNSE
1 0 0 1
s 1 0 2 6 1 . 9 64L203 23.65C a O 5 . 3 4N a 2 O a . 7 8K 2 0 0
. 0 5fotaf, 99.7a
s i 2 . 7 5 4Al L.239c a 0 , 2 5 4Xa 0.757K 0 . 0 0 3
h 2 6 . 4 9 2 5 . 5 0s 7 4 . 2 9 7 4 . 7 26 t 0 . 3 0 0 . 5
9Totat xo1.oa 100.41
5 1 . 8 9 5 0 . 6 0 5 9 . 5 2 6 2 . 2 7 5 6 . 5 52 a . 6 0 2 4 .
9 1 2 5 . 2 L 2 3 . 7 5 2 7 . 6 r
5 . 1 4 6 . 3 6 6 . 9 8 5 . 1 2 9 . 3 38 . a 3 a . 1 1 7 . 7 a 4
. 9 2 6 . 1 90 . 1 0 0 . o 5 0 . 0 4 0 . 0 6 0 . 1 8
9 9 . 5 6 1 0 0 . 0 3 9 9 . 5 3 1 0 0 . 1 2 9 9 . 8 6
2 . 7 5 6 2 . 6 9 4 2 , 6 6 5 2 . 7 5 7 2 . 5 4 LL . 2 ? 9 X . 3
0 5 1 . 3 3 0 1 . 2 3 9 L . 4 6 20 . 2 4 5 0 . 3 0 3 0 . 3 3 5 0 .
2 4 3 0 . 4 4 90 . 7 6 2 0 . 6 9 9 n . 6 7 5 0 . 7 6 6 0 . 5 3 90 .
0 0 5 0 . 0 0 3 0 . 0 0 2 0 . 0 0 3 0 . 0 1 0
wolgbt percent end Denbers
NuEb€r of lons on th6 baalB of, x5 poaitlve charg€s
Ftc. 1. Sample localities. Biotite-sillimanite isograd fromthis
study and rhat of Bailes (1980).
l) lies within the Early Proterozoic Flin Flonmetavolcanic belt.
The area is partly underlain bythe File Lake Formation of pelitic
metasedimentsthat preserve a prograde metamorphic sequenceranging
from the chlorite - muscovite zone to thegarnet - sillimanite -
biotite zone (Bailes 1980).Regional metamorphic studies in the area
have beensummarized by Bailes & McRitchie (1978) andGordon
(1989).
The prograde appearance of aluminosilicate plusbiotite in
muscovite-bearing rocks has been mappedfor 75 kilometers along
strike east of File Lake(Froese & Gasparrini 1975, Froese &
Moore 1980,Bailes 1980, 1985, Gordon & Gall 1982, Zaleski1989).
Pressure - temperature calibration of theisograd will therefore
provide an important datumfor reconstruction of the regional
thermal history.This study also complements the derailed study
ofreaction mechanisms of Briggs & Foster (1989) andFoster
(1991) in rocks from the File Lake area.
3 1 . 5 5 3 4 . 6 3 2 5 . 4 0 4 6 . 2 46 4 . 6 2 6 5 . 8 3 7 5 .
4 5 5 2 . 3 4
0 . 3 0 0 . 2 4 0 . 3 5 1 . 0 61 0 0 . 4 7 1 0 0 . 7 0 1 0 1 . 2
3 9 9 . 7 2
llole trDrcent €nd dedb€rs
h 2 5 . 0 8A ! 7 4 . 6 4o r 0 . 2 8
2 4 . 2 0 3 0 . 1 57 5 . 2 4 6 9 . 5 7
0 . 5 6 0 . 2 4
3 3 . 0 7 2 4 . O O 4 4 . 9 16 6 . 7 0 7 5 . 6 7 5 3 . 9 9
o . 2 3 0 . 3 3 1 . 0 3
All analyaea aie averagea of, riEs ln reight 8.
TABTA 3. MCROI{-UI&OPrcBE DAIA ON BIO1ITE
1001 20254
s 1 0 2 3 5 . 5 1 3 5 . 1 3a 1 0 2 L . 6 2 L . 6 2AL203 49.74
19.54n6o 20.47 21.6LM n O 0 . 0 5 < 0 . 0 4x E o 9 . 2 1 9 . 6
5e o 0 . 0 3 0 . 0 2B a O 0 . 2 3 O . 2 4N a 2 O o . 2 4 O , 2 5K 2
0 S . 5 5 8 . 6 0F n . d . n . d .8 2 0 3 . 7 0 3 . O afoial 99.a1
99.74rot-F 99.41 99.74
s L 5 . 3 7 0A l ( i v ) 2 . 5 3 0
4 . 0 0 0
203a 2026-2 2027 2040-2
3 5 . 0 0 3 7 . 1 1 3 5 . 1 4 3 4 . 9 71 . 7 5 1 . 4 5 1 . 4 3 1
. 4 2
2 0 . 0 1 1 9 . 0 4 1 9 . 4 5 1 9 . 4 02 0 . 1 2 1 5 . 6 9 ! 9 .
2 4 1 4 . 5 3< o . 0 4 < 0 . 0 4 0 . 0 6 0 . 0 5
9 . 5 9 7 2 . 7 9 1 0 . 0 7 1 0 . 3 2o . 0 3 < 0 . 0 1 < 0
. 0 1 0 . 0 70 . 2 7 0 . 3 0 0 . 3 4 0 . 3 4o . 2 4 0 . 3 0 0 . 3 0
0 . 2 7a . 6 7 8 . 4 4 S . 9 5 9 . X 9o . 2 3 n . a l . o . 2 5 o .
2 r3 . 8 8 3 . 5 3 3 . 6 4 4 . 4 7
9 9 . 7 9 9 9 . 7 5 9 9 . 7 5 9 9 . 8 59 9 . 6 9 9 9 . 7 5 9 9 .
6 4 9 9 . 7 6
!{eb€r of i.on6 on tbo badl.6 of 44 trDsltlve charg€6, 4 OH
TABIf I. IIIf,ERAL ASSEUBI"ASES
r t 0 . 1 8 4 0 . 1 8 4 0 . 2 0 0 0 . 1 6 1 0 . 2 0 4 0 . 2 0 9A
t 0 . 8 8 8 0 . 7 9 4 0 . 8 8 7 0 . a 0 0 0 . € 5 s 0 . 8 1 4F e z
2 . € 3 9 2 . 7 3 L 2 . 5 5 2 2 . o 6 a 2 . 4 3 4 2 . 3 6 2l l n 0
. 0 0 6 0 . 0 0 0 0 . O O 0 0 . 0 0 0 0 . 0 0 4 0 . 0 0 8v s 2 . 0
9 0 2 . L 1 3 2 . 1 6 8 2 . a L 7 2 . 2 7 1 2 . 3 4 4
5 . 8 0 7 5 . 8 4 2 5 . 8 0 7 5 . 4 4 1 5 . 7 7 6 5 . 7 3 7
5 . 3 0 8 5 . 3 0 9 5 . 4 8 42 . 6 9 2 2 . 6 9 L 2 . 5 1 6a . 0
0 0 8 . 0 0 0 8 . 0 0 0
0 . 0 0 5 0 . 0 0 30 . 0 1 4 0 . 0 1 40 . 0 7 0 0 . 0 ? 37 . 6 4
9 1 . 6 5 4L . 7 3 8 4 . 1 4 4
0 . 0 9 64 . 0 0 0 4 . 0 0 0 3 . a 9 0 4 . 0 0 04 . 0 0 0 4 . o
o o 4 . 0 0 0 4 . 0 0 0
5 . 3 1 6 5 . 3 2 92 . 6 A 4 2 , 6 7 L8 . O 0 0 4 . 0 0 0
0 . 1 2 0 0 . 1 0 13 . a a o 3 . 8 9 94 . O O 0 4 . O O O
Sanple
tool
e023!
ao3a
2026-2
2027
aoao-2
PI
xxxxx
Bt
xxxxxx
xxxx
x
crt
xx
x
chl
x
x
xx
Ma
xxx
ITDxxxxxx
Qtzxxxxxx
BANa
FOE
0 . 0 0 5 0 . 0 0 0 0 . 0 0 0 0 . 0 1 x0 . 0 1 6 0 . 0 1 7 0 . 0
2 3 0 . 0 2 00 . 0 7 1 0 . 0 4 6 0 . 0 4 4 0 . 0 4 0L . 6 7 A 1 . 5
9 1 ! . 7 2 7 1 . 4 2 5L . 7 7 0 1 . 6 9 4 1 . 4 3 4 1 . 9 3 6
All aLa1y6€s arE av€tagea of ri.DB ln rElghi 8.n.d. - not
detealleal.Welght perc€nt f2o ostldat€d andt lterated.Abbroviat
iona u€ froE K!6tz (1983).
-
ANALYSIS OF THE BIOTITE-SILLIMANITE ISOGRAD 675
rsr 4. rumotr-E&oPrcBE DATA ON SrAmOnrE TABLE 7.
ELECIRON-MICROPROBE DATA ON I'TUSCOVITE
sio2TiO24L203FSholtgo820fotal
1001 20254 203S 2026-2
2 6 . 6 1 2 6 . 5 2 2 6 . 9 0 2 7 , 0 40 . 4 0 0 . 4 4 0 . 5 5 0
. 6 4
5 3 . 1 6 5 2 . 8 S 5 3 . A 3 5 2 . 2 7L4.42 L4.44 14.14
13.72
0 . 1 1 0 . 1 1 0 . O 8 < O . O 51 . 4 5 1 . 5 0 L . 6 7 2 .
t 22 . 0 0 2 . 0 0 2 . o o 2 . o o
9 4 . 1 5 9 7 . 4 9 9 9 , L 7 9 7 . 7 9
3 . 4 5 7 3 . 8 5 0 3 . 9 2 00 . 0 4 8 0 . 0 5 9 0 . 0 ? 09 . 0
6 4 9 . 0 S 1 8 . 9 3 2L . 7 5 6 1 . 6 9 3 1 . 6 6 40 . 0 1 4 0 . 0
1 0 0 . 0 0 00 . 3 2 5 0 . 3 5 5 0 . 4 5 42 . 0 0 0 2 . 0 0 0 2 , 0
0 0
2027 2040-2
2 6 . A 6 2 6 . 8 00 . 6 1 0 . 5 5
5 3 . 7 0 5 3 . 0 71 3 . 9 9 1 4 . 4 6
o . 1 5 0 . x 21 . 7 9 2 . 0 02 . O O 2 . 0 0
9 9 . 1 1 9 9 . 2 0
3 , 8 4 7 3 . 4 5 0o . 0 6 6 0 . 0 5 99 . 0 6 5 S . 9 6 5L . 6 7
6 4 . 7 3 70 . 0 1 9 0 . 0 3 90 . 3 s 2 0 . 4 2 42 . 0 0 0 2 . 0 0
0
1001
s io2 46 .1ar i o 2 0 . 3 34L203 36 .35FeO 1 .OgMgO 0.48cao
-
676 THE CANADIAN MINERALOGIST
phyllosilicate cut the dominant foliation, but areidentical in
composition to the majority of grainsthat are parallel to the
foliation. There is no texturalevidence of disequilibrium in these
samples.
Analyses were carried out on the ARL SEMQelectron microprobe at
the University of Calgary.Within each polished section, the
smallest areacontaining all minerals was selected for analysis.The
individual mineral grains were examined forhomogeneity, but only
analyses from withinapproximately ten pm of grain boundaries
wereused for this study. Data-reduction proceduresfollowed Nicholls
& Stout (1988). The results ofthe analyses are presented in
Tables 2 to 8. Table9 gives the one-sigma error determined by
propaga-tion of counting errors in measurements of thecalibration
standards, background standards andthe unknown.
Individual mineral grains do not showpronounced zoning.
Plagioclase cores are generallyless calcic than rims, but the
compositionaldifference is usually less than I mol Vo An.
Garnetrims are from zero to 3 mol 7o richer in almandinethan cores,
whereas spessartine, pyrope andgrossular components show even less
variability.
INrrral Susrgclvr DEcrsroNSAND ASSUMPTIONS
The conclusions reached in this and similarstudies are almost
entirely dependent on threeunavoidable and necessarily subjective
decisionsthat must be made at the outset. Although thesechoices may
appear obvious, it is important tounderstand their implications
when interpreting theresults of the mathematical manipulations.A)
The systems assumed to be in equilibrium arethe rims of mineral
grains within the smallest areacontaining all minerals in each
polished thinsection. This portion of each rock, i.e., that
portionof the minerals within approximately l0 pm ofgrain
boundaries, is assumed to have retainedequilibrium compositions.
Microprobe-determinedcompositions of individual spots in this
region areall within the one-sigma analytical errors tabulatedin
Table 9 and are hence indistinguishable.
The decision not to use the means of rimcompositions from a
number of areas within eachpolished section is important. In the
context of thisstudy, such averaged compositions cannot
besatisfactorily interpreted in terms of ther-modynamic
equilibrium, which requires uniformityof compositions of phases.
Using analyticallyindistinguishable compositions from a single
smallregion maximizes the probability that assumption(A) is valid.
The exceptions to this restriction are
the ilmemite compositions in Table 8, which resultfrom analyses
averaged over several grains.
The algebraic tests described below attempt tofind mass balances
within and between the mineralcompositions of individual
assemblages. The inter-pretation of the results is based entirely
on whetheror not such mass balances can be found
withincompositional uncertainty. The larger uncertaintiesthat would
result from averaging mineral composi-tions over larger volumes of
rock would make iteasier to obtain mass balances; hence this
definitionof the equilibrium system has a major impact onthe
interpretations.B) Phoses in the model system are
plagtoclase,quartz, biotite, staurolite, garnet, chlorite,
mus-covite, ilmenite and sillimanite, plus a fluid phaseof unknown
composition, taken to be H2O. Theseminerals account for over 9990
of the modes of therocks. Whole-rock analyses of the File
LakeFormation show an average of only 0.24wt.0/o CO2and 0.20 wt.9o
P2O5 (Bailes 1980). The mass-balance calculations described below
do not includeB, C or P, so that the omission of
tourmaline,graphite and apatite will not affect the results.C)
Compositionol variability of the phases maybe described in terms of
SiOz, TiO2, Al2O3, totaliron as FeO, MnO, MgO, CaO, K2O, Na2O
andHrO, present in the phases for which there areanalytical data
reported in Tables 2 to 8.Microprobe analyses give satisfactory
totals whenrecalculated in terms of these oxides, and alsoproduce
satisfactory structural formulae. This is aminimum number of
constituents. The less abun-dant cations Mn and Ti are known to
affect mineralstabilities and hence cannot be excluded
fromconsideration.
Determinations of Fe2O3 are not available forthese samples. The
calculations that follow thuscontain the implicit assumption that
the rocks wereopen to Or, i.e., any imbalance in O, betweenmineral
compositions was compensated exactly byan appropriate gain or loss
of 02 to the coexistingfluid phase (Greenwood 1967, p. 480).
The ten oxides are the initial constituents fromwhich the number
of components may be deter-mined numerically. Note that the number
ofcomponents in the model system can be no greaterthan the lesser
of a) the number of phases asdetermined by assumption (B) above, or
b) thenumber of constituents, as determined by (C). Theoutcome of
the phase-rule-based calculations beloware thus largely constrained
by these choices.
VARIANCE OF INDIVIDUAL ASSEMBLAGES
AcconorNc ro rHE PHASE RULE
A common assumption in metamorphic studies
-
ANALYSIS OF THE BIOTITE-SILLIMANITE ISOGRAD 677
is that the minerals in an assemblage attainedequilibrium under
externally imposed, arbitrarypressure and temperature, and hence
shouldrepresent divariant equilibrium. A permissive testof this
assumption is the determination of thevariance of an
assemblage.
Brinkley (1946a, b) showed that the number ofcomponents, c, in a
thermodynamic system is equalto the rank of the matrix of vectors
that express- the compositions of the p phases in terms of
aninitial set of constituents, in this paper, oxides. Thephase-rule
variance of a single assemblage, lf = s+ 2 - p, follows directly.
As pointed out byThompson (1988) and Fisher (1989), a variance
ofless than two is equivalent to a statement that amass balance or
mass balances can be writtenamong some or all of the phase
compositions ofthe assemblage.
For each assemblage studied here, there is thusa composition
matrix formed of columns cor-responding to the compositions of
minerals in thatassemblage, with the addition of columns
cor-responding to the ideal compositions of quartz,water, and
sillimanite where appropriate. Deter-mination of the ranks of these
l0 by l0 or l0 by9 matrices will therefore determine the number
ofindependent components in each assemblage.
In this type of study, the initial subjective choicesof
constituenls and phases determine the numberof rows and columns in
each composition matrix.The number of components in a model
assemblagetherefore cannot exceed the lesser of a) the numberof
rows (constituents); or b) the number of columns(phases) in that
assemblage. The subjective choiceof a smaller number of
constituents or largernumber of phases would result in a
smallerphase-rule variance.
Compositions of minerals are not known per-fectly, so that the
problem becomes that ofdetermining the rank of a matrix where its
elementsare subject to uncertainty. Least-squares (LSQ)techniques
have commonly been used for thiscomputation (Greenwood 1968,
Fletcher 1971,Fletcher & Greenwood 1979, Pigage 1976,1982,Lang
& Rice 1985a). The LSQ approach attemptsto find a linear
dependency among the mineralcompositions. If at least one such
dependencyexists, then the composition matrix cannot be offull
rank.
A more direct technique is the use of singularvalue
decomposition (SVD) to determine the rankof the matrix directly
(e.9., Noble & Daniel 1988,p.342-343, Fisher 1989). SVD
routines are readilyavailable to perform the computation (e.g.,
Presset al. 1989, Fisher 1989, The Mathworks 1989).
The application to petrological problems wasintroduced and
clearly elucidated by Fisher (1989).
Briefly, the procedure is to express a compositionmatrix as the
product of three other matrices:
/ = Qclthl/r (1)
where U and Z have orthogonal columns, and Wis diagonal with
nonnegative elements. U and Wmay be rectangular or square'
depending onparticular implementations of the SVD algorithm'but }/
always contains positive numbers known asthe singular values of
matrix A. They appear inorder of decreasing magnitude. The rank of
.4 isequal to the number of nonzero singular values.Matrices of
lower rank that form the "best"approximations to I can be formed by
setting thesmaller nonzero singular values in Z to 0.0, whichleads
to malrix 14/, then performing the multiplica-tion:
A* = IJoW*cVr (2)
Matrices A. and A can then be compared todetermine whether the
matrix of reduced rankapproximates the original matrix within
analyticaluncertainty. If this is the case, then for theassemblages
studied here, the number of com-ponents of the modeled system would
be less thanthe number of phases, and the phase-rule variance,less
than 2.
Several investigators who have used LSQmethods to determine rank
have weighted thecomposition matrices by multiplying by rhe
inverseof the measurement-error covariance matrix (e.g.'Reid el al.
1979, Pieaee 1982). This procedurehomogenizes the influences of
components withvery different abundances. It should be
noted,however, that a) there is no unanimity on thevalidity of
applying statistical weighting topetrological models (see
discussion in Le Maitre1982, p. 103-105), and b) general
weighting(scaling) strategies for linear problems are unreli-able,
hence "It is best to scale (if at all) on the basisof what the
source problem proclaims about thesignificance of each e1" (Golub
& Van Loan 1989,p. r25).
One argument against this type of weighting isthat it is the
major constituents that determine thestable assemblage in an
equilibrium system.Weighting drastically increases the influence
ofminor constituents, although their abundances donot determine the
presence and amounts of majorphases. In this study, unweigftted
matrices wereused in the numerical determination of rank.
The assemblages consist of either nine or tenphases described in
terms of ten constituents. Theapproximate rank of each unweighted
compositionmatrix was tested by SVD techniques as described
-
678 THE CANADIAN MINERALOGIST
IIABI;E IO. UISTI! BEtrTEEN ONICINA! COUPOAIIION UATRICESAND
}IATRICES OF REDUCED RANK
guishable from the corresponding matrices A, andthe conclusion
would be that all assemblages aremonovariant. In this case, the
rightmost column ofeach matrix Z would represent the
one-dimensionalnull space of the corresponding A", and could
beinterpreted as a mass balance corresponding to aunivariant
equilibrium.
toot!{nO
Na20
2023tlnoCaONa2o
2 o 3 altnoCaONa2O
2026-2CaO
2027caoNa20
204O-2!. lno 2.96cao -0.16 -14.: . tN a 2 O o . ! 2 1 0 . 7
6
St elt gbl liB
5 . a 6 - 0 . 5 1 - 3 . 9L . 2 L 6 . t 6 e . 7 2
- 1 . 6 4
5 . 6 9 - 0 . 5 3 - 4 . 3 3L . 0 3 5 . 6 5 7 . 5 1
3 . 5 9 - 0 . 3 4 - 2 . 6 5L . O 7 8 . 4 7 7 . 7 7
- 2 . 0 5
- 3 . 5 4 - 0 . 9 9
1 2 . 5 { 1 9 . 9 9- 7 . 2 4 - 3 . 7 ?
- 0 . 2 - 0 . 0 1 - r . 1 60 . 0 7 a . 2 5 1 t . o 4
- 6 . 6 3
Pl At
a . 8 3- o . 3 3 - 9 . 3
0 . 1 2 . 6 9
9 . 3 5- 0 . 2 8 - 4 . 1 5
0 . 0 8 2 . 2 L
6 . 0 3- 0 . 2 5 - 4 . 4 6
0 . 1 3 . 3 9
o . 2 9 0 . o 2
- o . 7 - 1 9 . t 5o . 2 5 . 7 6
llE
- O . 2 4
Reaults obtaLned by gVD calculations rl.thout !@ o!
qoller€tghtlng. UnLts are outtLlrtEa of one-sLg@ ilalyttcatenols in
Table 9.
by Noble & Daniel (1988) and Fisher (1989), usingroutines in
PC-MATLAB (The Mathworks 1989).For each assemblage, the difference
between 1",the matrix of rank one less than initial matrix A,was
computed and compared on a term-by-termbasis with the uncertainties
shown in Table 9.Because ideal compositions were used for
quartz,sillimanite and water, and because all minerals werenot
analyzed for all oxides, comparisons wererestricted to matrix
entries for which analyticaluncertainty was measured.
For each assemblage, rows containing values ofA" - A greater
than one sigma are shown in unitsof one sigma in Table 10. In every
case, the rowscorresponding to CaO and either Na2O or MnOcontain
one or more terms greater than three timesthe corresponding
standard deviation (shown inboldface).
The values in Table l0 are much larger than theanalytical
uncertainties; hence the model matricesare not adequate
representations of the composi-tion matrices, and the composition
matrices haveranks equal to the number of minerals in
thecorresponding assemblages. This means that thephase-rule
variance of each model assemblage mustbe at least two, consistent
with the usual assump-tion that mineral assemblages in
regionallymetamorphosed rocks represent divariant equi-librium.
The dependence of this conclusion on the initialsubjectively
chosen constituents in each mineral isprofound. If the initial
subjective constraints weremodified such that MnO, CaO, and Na2O
werc notincluded in the model phases where they occur insmall
quantities, matrices I' would be indistin-
We believe that our choice of constituents issound, As a matter
of interest, however, the mass
o.a2 balances that would result from the modifiedproblem are
presented below, using mineralabbreviations from Kretz (1983). For
samples 1001,2025 and 2038 from below the isograd, the massbalance
is:
o . 4 4
cn + Chl + Ms + Ilm = Pl + Qrz + Br + St + H2O
Samples above the isograd have differentassemblages, and hence
the mass balances differ.They are:for sample 2-026-2:
Pl + Qtz + Bt + St = Crt + Chl + Ilm + Sil + H2O,
for sample 2027:
Ms + St + Chl + Ilm = Pl + Qtz + 81 + Sil + H2O,
and for sample 2040-2:
N{s + S + Grt + Chl + Ihn : Pl + QE + B + Sil + H2O.
DIFFERENCES IN EXTERNALLY IMPOSEDCoNrrrroNs AcRoss rHE
IsocRAD
Greenwood (1967) provided the analytical for-mulation for
"deciding whether the mineralogy ofone rock differs from another
because of somedifference in their bulk compositions or because
ofsome difference in the physical conditions of theirformation."
The solution to this problem dependson the premise that at given
values of two externallyimposed variables (such as pressure and
tempera-ture), any particular bulk composition has a uniquestate at
equilibrium (1.e., the amounts and com-positions of phases). This
is not universally true(e.g., Berry 1990), but for the macroscopic
systemsencountered in geology, an exception is virtuallyimpossible.
Zen (1966, p. 7-9), for example, usedan equivalent statement as a
necessary "fundamen-tal axiom" to establish geometric rules for
construc-tion of phase diagrams.
The test thus reduces to the search for ahypothetical bulk
composition that can be ex-pressed as a positive combination of the
mineralcompositions in each assemblage. This is equivalentto the
search for a mass balance between the twoassemblages. If any such
mass balance is found to
-
ANALYSIS OF THE BIOTITE-SILLIMANITE ISOCRAD 679
exist, then a bulk composition exists that can berepresented by
two distinct sets of phase composi-tions. The postulate of
uniqueness then requiresthat the two assemblages must have
equilibratedunder different externally imposed conditions.Note that
the bulk compositions, and hence modesof the assemblages, are not
important in thesecalculations; only the mineral compositions
are.
Although this procedure is commonly describedas the search for a
balanced "reaction" betweenmineral assemblages, it is important to
note thatthe successful identification of such a
mass-balancerelationship does not demonstrate that a
particularchemical process acted, nor does it imply that themass
balance corresponds to a univariant equi-librium relationship
between the two assemblagesin the model system. Note also that
failure to finda mass-balance relationship implies only that thetwo
assemblages do not share any bulk composi-tions. It is permissive,
not diagnostic, of equilibra-tion under the same conditions,
The simplest mathematical statement of themass-balance condition
is:
fi"o*, ,fr,*r*, . o t.r,z, ..,c
xr > 0 J=L'2" "ntn (3)
i ' , - tJ.\
where ci; is the weight 9o of constituent i in phasej, x; is the
unknown weight fraction of phase 7 inthe-sought-after mass balance,
c is the number ofconstituents in the system, and m, n are
thenumber of phases in the respective assemblages.
The inequalities in (3) express the requirementthat the unknown
weight fractions have nonnega-tive values, which will ensure that
if the massbalance can be found, it is also physicallyattainable.
In order to exclude the trivial solution(all xr = 0) and to specify
a unique set ofmass-6alance coefficients, an additional
restriction,such as the equation normalizing the values of x;for
one assemblage, is required.
Following Greenwood's (1968) paper, mostinvestigators have used
LSQ approaches to thisproblem (Fletcher 1971, Fletcher &
Greenwood1979, Pigage 1976, 1982, Lang & Rice 1985a).Standard
LSQ algorithms do not permit inclusionof the inequality and
normalization equations in(3); hence LSQ methods commonly include
aniterative step to determine whether a solution canbe found with
correct signs for each set of .1values.The linear programming (LP)
approach originallyproposed by Greenwood (1967) avoids
theseproblems, however; furthermore, the recent in-crease in
availability of inexpensive spreadsheetprograms and computer codes
for the solution of
such problems (Press et ol. 1989), make the use oflinear
programming techniques relatively simple'
As with the determination of the variance of asingle assemblage,
uncertainties in mineral com-positions must be taken into account.
Greenwood's(1967) formulation included the uncertainty
termsexplicitly in the inequalities. His inequalities (20)and
equations (22) can be rearranged as:
a b a
p (aJJ-0!J) xr;!, (as+6s) xj : o L'L,2, ",c
f rrrr 'arrt*r-f (as-Err)x, > o t--L,2,, "c
\ > o
E * t = 't-\
&a
E t r - rJea
where the notation is the same as (3), and 6;; is theuncertainty
in the analysis of constituent I in phasei.
This form of Greenwood's statement of theproblem can be easily
compared with (3) above.The nominal compositions of minerals are
per-mitted to vary within their known uncertainties,thus expanding
the equation into two inequalities.If there are nontrivial
solutions to (4), they willdefine an infinite set of x7 values
satisfying theinequalities.
A system of inequalities cannot be "solved" inthe sense of
giving a unique solution, but LPalgorithms can be used to (a)
determine whetherthe inequalities are consistent or feasible' and
(b)maximize or minimize a linear function (obiective
function) of the unknown values of xr. Using thisformulation, a
number of objective functions,usually the weight fractions of each
phase, aremaximized in turn. If a feasible solution is found,then
an infinite number of hypothetical mass-balances exist, and the
solution to any particularproblem provides one of them.
A disadvantage of this approach is that failureto find a
feasible solution causes most algorithmsto return an "infeasible"
flag without providinginformation as to which constituents are
failing tobalance. An alternative formulation that avoids
thisproblem is presented here:
a r yi ^u * r | a ;A . ' x t
+ s ; - s i ' o l e l ' z " " cJ.l 7-@t
x t > o
e i > o
s ; > 0
F x . = 1r.1
(5)
Jg r . 2 , . . , n+4 )
l - L ' 2 ' . . ' c
l o r , 2 , , . , c
-
680 THE CANADIAN MINERALOGIST
where,s,+ and E are new variables that account forany imbalance
in constituent i. Note that linearprogramming algorithms ensure
that at most onlyone ofs/ or q will appear in the solution for
eachi.
This statement of the problem is a simpleextension to (3),
forcing each of the i equations tobe satisfied exactly. For each
constituent i, the valueof either s/ or .g provides the amount by
whichthe mineral compositions fail to satisfy the massbalance.
If the objective function is chosen as:
tila!'nl.ze i ttl*";114' ' - (6)
then linear programming algorithms will return asingle solution
that may include nonzero values ofsome sf / s;and as well as
x;.
For any particular /, nonzero s1 or s; values inthe solution
will give the amounts by which themineral compositions fail to
balance for particularconstituents. These values can then be
comparedwith the imbalance permitted by the propagationof
analytical error for constituent i predicted by thevalues of x; and
the known values of theuncertainties in the analytical results:
l - L , z , . . ' c 7 )
where e,7 is the known one-sigma analyticaluncertainty of
constituent i in phase 7.
If s;+ or q is less than (7), then the equation forconstituent i
balances within the predicted uncer-tainty. If, on the other hand,
s1* or si is greaterthan (7), then there is no permissible
mass-balance.In contrast to the other methods, this formulationof
the problem results in the identification of theconstituent(s) that
fail to balance.
There are nine possible combinations of as-semblages formed by
pairing each sample frombelow with each sample from above the
isograd.Each of these pairs was tested using measuredcompositions
and the formulation given by (5) and(6), and the simplex routine in
Press et al. (1989).Because the compositional data have only
twosignificant digits following the decimal place, anequation was
considered to balance if the right-hand side was less than 0.005
weight go. Quartzand a fluid phase containing H2O are assumed tobe
present in both assemblages, hence neither SiO2nor H2O was included
in the mass-balanceequations. The solutions that minimize the
im-balances are presented in Table 11.
Only 20254 = 2O26-2 and 2038 : 2026-2
require an additional term in order to altain a massbalance. In
both cases, CaO fails to balance. Theamounts of CaO required are
much higher than theerror permitted by (7), even if e6ue; is taken
asthree times the one-sigma errors in Table 8. Thesepairs of
assemblages clearly differ in bulk composi-tion sufficiently that
no conclusion can be drawnabout differences in their conditions of
formation.
The remaining pairs do show mass-balancerelationships, however,
which demonstrates thatthe isograd is due to a change in externally
imposedconditions, pressure or temperature (or borh).
UNpenlvrNc ReecrroNs AND EeurLrBRrA
Although it is tempting to equate mass balancesobtained between
different assemblages to reac-tions or phase equilibria, such
correlations can bemisleading. This can be illustrated by the
simplifiedsituation shown in Figure 2. A three-componentsystem has
four phases of interest, all of whichshow solid solution. Under
some initial conditionsof pressure and temperature, phases a, b,
and dare stable in assemblages abc and abd; at differentpressure
and temperature, the equilibrium composi-tions of a, b, c and d
have shifted to A, B, C andD, respectively, and the stable
assemblages havebecome ACD and BCD. At some intermediateconditions,
a univariant reaction has caused the
Frc. 2. Assemblages from a hypothetical three-componentsystem
with solid solurion and discontinuous equi-librium. Only one of the
algebraically determinedfour-phase mass balances between ensemble
abcd andensemble ABCD corresponds to the discontinuousequilibrium.
Numbers refer to the mass balances: 1)a + b = A + C , 2 ) a + b + d
= A , 3 ) b + d =A + D , 4 ) b + d = C + D , a n d S ) a + b = C +D
.
fr txrrrrr
-
ANALYSIS OF THE BIOTITE.SILLIMANITE ISOGRAD
TABIJE 11. RESUIJTS OF I'INEAR PROGRAMUING CAI]CUI,ATIONS TO
DRTERI'TINE
UASS BAI,ANCES B TWAEN ASSB'TBI,AGES ABOVE AND BELOW TIIE
ISOGRAD
1001 = 2026-20 .0143 P1 + 0 .0244 ChL + 0 .1072 Gr t + 0 .0057 S
t + 0 .8388 I 1 t00 .0161 P l + 0 .1469 Gr t + 0 .8369 I l t o
LoOt a 20270.3079 Pl + O.22L3 Chl + 0.0437 Grt + 0.2927 t ' tE +
0.1855 l ln0 .3397 F l + 0 .2970 B t + 0 .1877 s t + 0 .U55 I
Ln
1001 = 2010-2O.O35O P l + 0 .2374 Gr t + O .00OB ME + 0 .7525 l
l t u0 .0499 F l + 0 .0235 Ch1 + 0 .0489 Gr t + 0 .8314 l l n + 0
.0463 S i l
20254 = 2026-20.0511 ChL + 0.6393 St + 0.2974 I I ] l + O.7O73
CaOo .OOo7 B t + 0 .0571 Gr t + 0 .6465 s t + 0 .2956 I 1 : r
2025A = 20270 .1335 Ch l + 0 .0133 Gr t + 0 .8520 Us + 0 .0497 I
l r 0O .OO48 P l + 0 .1836 B t + 0 .6878 l t s + 0 .0813 S t + 0
.0425 I L l r
2025A = 2040-2O.O ] -27 P l + 0 .2416 Gr t + 0 .0002 Us + 0
.7765 l l n0 .0181 PL + O .O145 Ch l + 0 .0531 Gr t + 0 .8621 I 1n
+ 0 .0522 S11
2038 = 2026-20.0367 ChI + 0.7872 st + 0.1560 ILe + O.05as
caoO.OOO2 B t + 0 .0288 Gr t + 0 .8158 S t + 0 .1553 I l n
2OtB = 20270 .2644 BX + 0 . ! 924 Ch l + 0 .0375 Gr t + 0 .5481
! ' l s + 0 .0093 I 1 t00 .0124 P l + 0 .5241 B t + 0 .3040 l { s +
0 .1594 S t
2038 = 2010-2O.OO45 P l + 0 .1791 Gr t + 0 .3961 1 .16 + 0 .4540
l l n0 .0253 PL + 0 .0338 B t + 0 .3358 l . I s + 0 .1006 S t + 0
.5044 I 1n
68r
l.Ieasured conpositLon6 fron Tables 2 to 8 and lnequal.ities.
(5),degcrlbed in the text, rtere used to find rnaEs balances
betlteenassenblages. Coeffl-cLents are r,telght fractlons.
IrnbaLances occuronly for cao (bold ital.lcs) and can be cornpared
wlth the errorperrnltted by the analytl-cal uncertalntLes ln Table
9 and thecoefficlents ln the equatlons.
A-B tieline to become unstable with respect to theC-D
tie-line.
Consider two samples consisting of an as-semblage with bulk
composition x, with assemblageabd, and one with bulk composition X,
withassemblage ACD. The two phase diagrams clearlyintersect, and an
infinite number of mass balancesbetween the assemblages can be
obtained fromwithin the shaded area. Because this is a
three-com-ponent system, the usual algebraic techniqueswould
essentially examine all four-phase combina-tions selected from a,
b, c, d, A, B, C and D. Themass balances obtained would correspond
tohypothetical equilibria:
l ) a + b : A + C2 ) a + b * d = A3 ) b + d = A + D4 ) b + d = C
+ D5 ) a + b = C + D
Ofthese, only the last bears any relationship to theactual
univariant equilibrium responsible for thetopological differences
between the phasediagrams.
The situation in ten dimensions is very likelymuch worse. It is
not difficult to imagine acircumstance in which tieline C-D would
fallcompletely outside of abcd, in which case, none ofthe
algebraically obtained mass-balances wouldcorrespond to the
underlying univariant equi-librium.
Whereas algebraic techniques may not identifyan actual reaction
or equilibrium, they may be usedto test reactions proposed on other
grounds. In theFile Lake area, field and petrographic
observations(Bailes & McRitchie 1978, Bailes 1980, this
study)indicate that the prograde appearance and increasein
abundance of sillimanite coincide with a decreasein modal chlorite,
muscovite and staurolite. This iscommonly observed in metamorphosed
pelitic
-
682 THE CANADIAN MINERALOGIST
rocks (e.g., Guidotti 1974,Lang & Rice 1985a, b).In the
model KFMASH system, this reaction isattributed to the univariant
equilibrium:
C h l + M s + S t = Q t z + S i l + B r + H 2 O ( 8 )
To extend this model reaction to the File Lakeassemblages, it is
necessary to consider theadditional phases plagioclase, garnet, and
ilmenite.The problem is to attempt to find mass balancesamong
analyzed minerals that preserve the six-com-ponent equilibrium
above and contain the addition-al constituents and phases.
For this aspect of the study, Greenwood'sformulation of the
problem, given in (4) above,was used. To ensure that the mass
balances wouldresult in sillimanite as a product phase, a
linearprogramming problem was set up for each of thenine pairs of
assemblages with rhe objectivefunction:
Maximize xr;111,,,on;," (9)
Although all pairs produced feasible mass-balan-ces, only two
provided solutions that containequilibrium (8) as a subset. The
results were:
1001 = 2040-20.5569 Ilm + 0.0128 Chl + 0.0659 Grt + 0.2828sr +
0.0825 Ms =0.6208 Ilm + 0.0147 Pl + 0.0788 Br + 0.2788 Sil+ 0.0069
H2o
2037 = 20380.1301 Pl + 0.3363 Chl + 0.0241 St + 0.4966 Ms+
0.0128 llm : 0.1658 Pl + 0.0048 Qtz + 0.5152w. + 4.2709 sil +
0.0433 H2o
lf these mass balances have any relationship toactual
"reactions", plagioclase is a product andgarnet a reactant. The
role of ilmenite is unclear.This result is in accordance with the
processesinferred by Bailes (1980) for this area, by Guidotti(1974)
for pelitic schists in the Rangely area, Maine,and by Lang &
Rice (1985a) for assemblages arSnow Peak, northern ldaho.
PRESSURE-TEMpERATURE CoNorrroNsOF THE ISOCRAD
It is not our intention in this study to evaluatethe numerous
calibrations of exchange equilibriarelevant to pelitic rocks. We
decided to estimatepressure-temperature conditions using the
publiclyavailable PTAXSS program and data-base (Berman& Brown
1988) that is parr of rhe GEO-,CALCsystem (Berman et al. 1987).
Estimates of pressure and temperature werebased on the
simultaneous solution of the Gibbsfree energy equations
corresponding to H2O-freeequilibria among appropriate end-members
in thedata-base. This can be done with GE@CALC byexamining
intersections of the pressure-tempera-ture curves corresponding to
equilibria betweenmineral end-members for which there are
reference-state data. The end members used were: almandine(alm),
annite (ann), anorthite (an), clinochlore(chl), grossular (grs),
muscovite (ms), phlogopite(phl), pyrope (prp), B-quartz(pqtz),
sillimanite (sil)and staurolite (st).
The ver$ion of the GE@CALC data-base utilizedwas NOV89.RGB.
which uses the thermochemicaldata of Berman (1988, 1990), the
activity modelfor garnet of Berman (1990), and the activity
modelfor biotite of Indares & Martignole (1985). Thesedata are
consistent with the experiments of Ferry& Spear (1978). The
muscovite-paragonite activitymodel of Chatterjee & Froese
(1975) and theplagioclase mixing model of Furhman &
Lindsley(1988) also were adopted. The formula forend-member
clinochlore used in the data-base isMg5Al2Si3O16(OH)6; hence we
adopted a single-site,coupled-substitution model (Chernosky el
a/.1988). The activity of clinochlore was estimated by:
-cnraurrl - l xYagr, , slt. I x(Nl-j-) , r ltqo . rdE tcp^ .E -L
5 , 61 [ 6
,CJ
The activity of lhe iron end-member of staurolite,computed
as
a#Wr* = [ x (Fez " ) ] 6 ,
from the staurolite formula in the data-base,
gaveFeaAl13Si7.5Oaa(OHJ.
Although in the assemblages studied here thereare as many as 32
possible equilibria between thechosen end-members, the number of
linearlyindependent equilibria is much smaller. All equi-librium
curves can be generated from linearcombinations of a particular set
of independentequilibria. For each assemblage, the number
oflinearly independent equilibria is equal to the rankof the matrix
formed of all possible equilibria(Jouguet l92l), or equivalently,
the rank of thenull space of the composition matrix of end-mem-ber
compositions (Aris 1965, Thompson 1982,1988).
The stoichiometry of the end members used inthis study is such
that three independent equilibriasuffice to generate all equilibria
in assemblages1001, 20254,2038, and 2026-2, and four inde-
-
ANALYSIS OF THE BIOTITE-SILLIMANITE ISOCRAD 683
pendent equilibria suffice for assemblage 2040-2.Because
assemblage 2027 does not contain garnet,it defines only one
equilibrium, hence was excludedfrom further consideration.
In general, the computed equilibria for any oneassemblage are
inconsistent, i.e., they do not definea single pressure and
temperature. Examination ofthe results indicates that the problem
arises frominconsistency between equilibria involving mus-covite on
the one hand and those involvingclinochlore and staurolite on the
other. The onlyequilibria that do not involve
muscovite,clinochlore, or staurolite are the garnet - biotiteFe-Mg
exchange:
alm + phl : prp + ann (A)
and the sillimanite - qvartz - grossular -
anorthiteequilibrium:
si l + pqtz + grs = 3an
Equilibrium (A) can be determined in five of thesamples, hence
it was chosen as the first inde-pendent equil ibrium to
examine.
In samples from below the isograd (1001,20254.,and 2038),
equlibrium (A) and all chlorite- andstaurolite-free equilibria are
linearly dependent andhence intersect at a single point. One of
these:
a l m + g r s + m s = 3 a n + a n n ( C )
was thus chosen as the second of the threeindependent
equilibria. Similarly, a number ofmuscovite-free equilibria
intersect (A) at a uniquepressure, and
6st + 45 Bqtz + Sprp + 24grs =S a l m + 7 2 a n + 3 c h l
was chosen as the third independent equilibrium.Assemblages from
above the isograd include
equilibrium (B), as well as a single muscovite-free,biotite-free
equilibrium involving the minimumnumber of end members:
6st + 21 9qtz + 5FrF = 8alm + 3 chl + 48sil (E)
hence this also was used in the calculations.The sensitivity of
the results to uncertainties in
mineral compositions was estimated by repeatedcalculations with
the weight 9o of each constituentvarying by two standard deviations
from itsnominal value. The maximum uncertainty intemperature of
equilibration of a single assemblage,as determined from (A), was +
l5oc, whereas themaximum uncertainty in pressure of a single
assemblage, as determined from (B), was + 30MPa.
The pressures and temperatures defined by theseequilibria are
plotted in Figure 3. The temperaturesdetermined for samples 2038
and 2026-2 areinconsistent with the assumed increase of
tempera-ture across the isograd, but lie within theuncertainty from
analytical measurement, giving amedian of 540oC. Pressures
determined by chlorite-and staurolite-bearing equilibria are
uniformly inthe andalusite field and hence suspect, not asurprising
result, given the approximations used foractivities of these end
members. Muscovite-bearingequilibria are most consistent with
equilibrium (B),and the two give a median pressure of 330 MPa.
These results are comparable with the work ofother investigators
on similar assemblages fromrocks metamorphosed at higher pressures.
Forexampleo in the study of Lang & Rice (1985b),assemblages
containing plagioclase + quartz +biotite + staurolite + garnet +
muscovite +chlorite + kyanite. were estimated to have ex-perienced
pressures near 600 MPa at temperaturesnear 525oC. The conditions
determined at File Lakeare consistent with their results and
suggest thattemperatures recorded by the biotite -
sillimaniteisograd are between 500"C and 550oC throughoutthe
pressure range 300 to 600 MPa.
CoNCLUStoNS
Algebraic techniques have been applied tomineral compositions
obtained from assemblagesthat straddle the sillimanite - biotite
isograd in theFile Lake area. If the rocks are modeled in
aten-component system, the mineralogy of in-dividual samples can be
interpreted as representingat least divariant equilibrium. The
isograd can beapproximated by the reaction:
S t + M s + C h l + G r t + I l m =P l + Q t z + B t + S i l + H
r O
Metamorphic conditions are inferred to have beennear 540oC and
330 MPa.
ACKNOWLEDGEMENTS
It will be obvious to the reader that the approachtaken in this
paper owes its origin to HughGreenwood's innovations in the
quantitative studyof metamorphic rocks. We express our
profoundthanks to Hugh, not only for his excellentcontributions to
science, but also for his kindnessand generosity as a teacher,
colleague, and friend.
Alan Bailes kindly furnished air photographsand led several
field trips to the File Lake area.Steve Jackson, Eva Zaleski, and
Julie Jacksonprovided able field assistance during sample
(B)
(D)
-
684 THE CANADIAN MINERALOGIST
250
2005 1 0 520 530 540 550 560
Temperature (oC)Ftc. 3. Pressure-temperature conditions for
samples (numbers) obtained from
GE@-CALC @ermanet al. 1987). Letters refer to cquilibria (A) to
@) discussedin text. For each sample, pressure-temperature
conditions from equilibrium (A),alm + phl = prp + ann, are the
near-vertical lines. For clarity,pressure-temperature conditions
obtained from the equilibria: B) sil + pqtz +grs = 3an, C)alm + grs
+ ms = 3 an + ann, D)6st + 45 $qtz + 5 prp+ 24 grs = 8 alm + 72 an
+ 3 chl, and E) 6 st + 21 9qtz + 5 prp = 8 alm+ 3 chl + 48 sil, are
shown as short lines where they intersect the
correspondingequilibrium (A).
400
350
ctTL
o 300trlooo(L
collection. Rob Berman generously supplied betaversions of
GEGCALC software and date-base.Many enjoyable discussions with Jim
Nicholls, RobBerman and Eva Zaleski helped clarify the
ideasexpressed in this paper. We are very appreciativeof the
perceptive and constructive reviews byHoward Day, George Fisher,
and Charles Guidotti.The research was supported by a University
ofCalgary Research Grant, a Lithoprobe grant andNSERC grant
GP0046219 to TMG, and NSERCgrant 44379 to EDG. This is Lithoprobe
Publica-tion 214.
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ANALYSIS OF THE BIOTITE-SILLIMANITE ISOCRAD 685
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