algebraic analvsis of the biotite sillimanite ...classically been done by the construction of...

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


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


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


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


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-


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)


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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Received November 6, 1990, revised manuscriptoccepted February 14, 1991.

algebraic analvsis of the biotite sillimanite ...classically been done by the construction of...

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