American Mineralogist, Volume 73, pages 201-215, 1988

Ternary-feldspar modeling and thermometry

Mrnr.ll.r L. FunnvreN, DoNlr,o H. LrNosr.nvDepartment of Earth and Space Sciences, State University of New York, Stony Brook, New York ll'194-2100, U.S.A.

AssrRAcr

A revised thermodynamic model for ternary feldspars-an extension of the approach ofGhiorso (1984)-was made possible by (a) inclusion of volume data for ternary feldsparsand (b) adjustment of the compositions (within analytical error) of some of the experi-mental data points (Seck, l97la). The excess free-energy expressions of Newton et al.(1980) are used for the plagioclase binary join and those of Haselton et al. (1983) for alkalifeldspar. Excess terms for the An-Or join and for ternary interaction were obtained fromSeck's experiments by the method of linear least squares. The new model is consistentwith most occurrences of natural ternary feldspars and with experiments on strongly ter-nary feldspars. Most ternary feldspars have negative excess volumes-an important con-straint on pressure corrections in geothermometry.

The activity expressions of this model permit yet another formulation of the two-feld-spar geothermometer that fully takes ternary solution into account; it yields three tem-peratures, one each for Ab, Or, and An equilibria. Few published analyses of feldspar pairsgive concordant temperatures, probably because small errors in composition strongly affectthe calculated temperatures. However, if the compositions of coexisting feldspars are al-lowed to vary within expected analytical error, three concordant temperatures can becalculated for feldspar pairs that were close to being in equilibrium. The program presentedwill also indicate pairs that could not have been in equilibrium. This thermometer is moreuseful than previous two-feldspar thermometers because it (l) fully accounts for ternarysolid solution, (2) indicates whether the two feldspars could have been in equilibrium, and(3) gives further information about how the feldspar compositions may have been changedby such postcrystallization processes as alkali exchange and subsolidus exsolution. Ourthermometer yields temperatures similar to those obtained using the formulation of Has-elton et al. (1983) for pairs of feldspars close to binary compositions, and much morereasonable temperatures for strongly ternary compositions than any previous thermome-ter'

INtnonucrroN AND pREvrous woRx double-binary thermometers do not utilize the fact that

' two feldspars coexisting in equilibrium must have theThe ternary-feldspar system has been given much at- activities of all their components equal, not just those of

tention over the past few decades, mainly for purposes of Ab. What is needed is a description of the system in ter-geothermometry. In the ternary-feldspar system, Na- nary terms: ternary mixing parameters can be modeledAlSirO8-KAlSirOr-CaAlrSirO, (for this paper, Ab, Or, and from existing temary experimental data and then used toAn components, respectively), there is complete solid so- predict temperatures of equilibrium of coexisting naturallution at high temperatures on the Ab-Or and Ab-An feldspars. A thermometer based on ternary expressions isbinaries, but limited solid solution between An and Or not limited to compositions close to the feldspar binaries;and between plagioclase and alkali feldspars within the it eliminates arbitrary projection of ternary compositionsternary. The compositions of the coexisting feldspars are to the binaries; and, most important, it provides valuablestrong functions of temperature; effects of pressure must tests for equilibrium between coexisting feldspars. It isalso be taken into account. Most formulations of the feld- incumbent on any petrologist, when applying any ther-spar thermometer concentrate on two feldspar binaries, mometer, to offer evidence that the phases involved wereAb-An and Ab-Or, and do not fully account for ternary at some time in equilibrium; it may be necessary to re-compositions (Barth, l95l; Stormer, 1975; Powell and construct original compositions of exsolved phases, or toPowell, 1977; Whitney and Stormer, 19771- Brown and decide which portions of zoned feldspars may have beenParsons, l98l; Haselton et al., 1983). Thermodynami- inequilibrium.Agreatadvantageofafeldsparthermom-cally based thermometers should be designed to account eter based on a ternary model is that it allows an addi-for ternary solid solutionin both the plagioclase and al- tional test of equilibrium-provided, of course, that onekali feldspar phases (Brown and Parsons, 1985). These believes the model to be correct-because the three si-

0003-004x/88/0304-020 l $02.00 20r

202 FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

TABLE 1. Excess terms for ternary feldspars (in joules)

Ghiorso (1984) Green and Usdansky (1986) This study

wsWHwvwsWHwswH

ttt +

Wonoo

W^^o,

Wo,o^

309781706228226

84712798367469

- 1 3869

21.4

-20.211 1.06

- 14.63

1881027320282308473

-6540765305

- 1 1 4 . 1 0 412.537

2.1121

1881027320282268471

s2468(497)47396(23s)8700(897)

0.394(0.017)0.394(0.017)

-0.120(0.04)

- 1.094(0.1 25)

0.3610.361

1 0 31 0 3

10.3 0.36410.3 0 364

Note: Wn : WH - TWs + PyVy. Absence oJ an entry means that the parameter was set or assumed to be zero Boldface values are results of least-squares fits to the data of Seck (1971a, 1971b). Values in ( ) are uncertainties (+1o) in parameters

' Excess volumes for this study were fitted to the data of Newton et al. (1980) and Kroll et al. (1986)..' lryterms for Ab-Or taken from Thompson and Hovis (1979) [Ghiorso] and from Haselton et al. (1983) [Green and Usdansky, this study].

f ytl terms for An-Ab taken from Newton et al. (1980).

multaneous equilibria (among Ab, An, and Or compo-nents) provide redundant information on temperature.

Thus far, the only thermodynamic models that utilizeboth existing experimental data and explicit formulationsfor ternary mixing parameters are those of Ghiorso (1984)and of Green and Usdansky (1986). Both of these models,though successful for some feldspars, have certain draw-backs: (l) they are based on synthesis experiments, (2)they use a single equation of state for both monoclinicand triclinic feldspars, (3) they lack volume data for ter-nary feldspars, and (4) they predict unrealistic tempera-tures for many natural and synthetic ternary feldspars.

Because we needed to apply a solution model to strong-ly ternary natural and synthetic feldspars (Fuhrman, 1986;Fuhrman et al., unpub. ms.), we have devised yet anothermodel-one that also includes volume data for ternaryfeldspars (Kroll et al., 1986). Our model shares the firsttwo drawbacks of the earlier models-use of synthesisdata and of a single equation of state. Because there mustbe a first-order thermodynamic difference between feld-spars with the two different structures, at least two equa-tions are required for a rigorous formulation. Althoughwe recognize that feldspar models must ultimately use atleast two separate equations, we believe that the data nowavailable-both for phase equilibria and for volumes-do not justify such a level of sophistication. For example,it is not always clear which structure a given feldspar hadwhen it was in equilibrium, yet such information wouldbe absolutely essential ifone were to try to fit a separateequation to each feldspar structure. Furthermore, sepa-rate equations would more than double the number ofunknowns to be fitted to the data: additional Margulesparameters would be required, and we would also needfree-energy terms for the monoclinic-to-triclinic reactionsfor end-member Ab, An, and Or. Accordingly, we haveretained the single-equation model despite its shortcom-ings, with the full realization that as well-reversed exper-imental data on ternary feldspars become available, twoor more equations of state will be more appropriate.

Ternary solution model

A ternary solution model involves expressions for con-figurational entropy (ideal part ofactivity terms) and for

excess free energy (activity coefficients). Ghiorso (1984)extended the Al-avoidance model for the entropy of pla-gioclase (Newton et al., 1980; Kerrick and Darken, 1975)to ternary feldspars; for pure ternary feldspars, his expres-sions are equivalent to the site-mixing terms (Price, 1985)that were adopted by Green and Usdansky (1986). Bothof these models employ ternary Margules expressions forexcess energy (Wohl, 1953; Andersen and Lindsley, 1981),which involve parameters for each of the three binaries,plus-in Ghiorso's model-a ternary excess term. Bothmodels used published values for the Ab-An and theAb-Or joins (Table l). Because there are essentially noexperimental data for the third binary (An-Or), they cal-culated asymmetric Margules parameters for it by mod-eling the experimental data of Seck (l97la, l97lb) forternary parrs.

The results of these earlier calibrations are shown inTable l. The conventions for labeling the parameters inTable 1 are those of Andersen and Lindsley (1981),Ghiorso (1984), and Green and Usdansky (1986); theyare opposite to those used by Haselton et al. (1983).Ghiorso concluded that the ternary term is significantonly at low L His method yields a temperature depen-dence for each unknown, so his model uses six new pa-rameters to describe the system: three excess enthalpies(Wr) and three excess entropies (2"). Green and Usdan-sky likewise calculated six new parameters; however, theyconcluded that the ternary terms were unnecessary, andinstead they extracted large positive excess volume termsfor the An-Or binary on the basis of Seck's experimentsat 1,5, and l0 kbar ( l97 la,197lb) .

The approach adopted by Ghiorso and by Green andUsdansky provides a definite improvement on previoustwo-feldspar geothermometers by using explicit ternarymixing parameters to describe the feldspar system. How-ever, both models have problems in predicting some feld-spar relationships: they predict less miscibility in the ter-nary feldspars than is suggested by many natural feldspars.Among these are ternary mesoperthites from the SybilleMonzosyenite (Fuhrman, 1986) and from the KlokkenSyenogabbro (Parsons and Brown, 1983). Independentthermometers suggest that these feldspars crystallized at950-1020 'C, whereas Ghiorso's model would require

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY 203

temperatures above 1200 oC and the thermometer ofGreen and Usdansky gives strongly disparate tempera-tures for each ofthe three equilibria. It is likely that thedeterminations of Seck (197 la, I 97 I b) underestimate Anin alkali feldspar, possibly as the result ofanalytical erroror of incomplete equilibrium in Seck's experiments orboth. Seck's analytical method, X-ray determinationscombined with geometric constraints, has a probable un-certainly of up to 2 molo/o (and even more for stronglyternary compositions) and allows considerable latitude inthe choice of "best" values. He reported very low solu-bility of An in alkali feldspar and, more important, verylittle increase in that solubility with increasing tempera-ture. By adopting Seck's values, the earlier modelers wereforced to include a negative excess entropy term for oneof the An-Or Margules parameters, a highly unusualproperty for silicates if real. As we show in a later section,very small increases (well within the analytical uncertain-ty) in the An content of alkali feldspar for Seck's higher-temperature experiments can eliminate the need for thisnegatrve excess entropy.

PnBsrNr sruDY

We have followed the approach of Ghiorso (1984) withthe following exceptions: (1) we fitted the volume data ofKroll et al. (1986) and Newton et al. (1980) to obtainexcess volumes for the An-Or join and for the ternary;(2) like Green and Usdansky (1986) we used the alkalifeldspar model of Haselton et al. (1983) for the Ab-Orjoin; (3) we used Seck's data but (for reasons explainedin a later section) omitted the 650 "C experiments; (4) weadjusted the compositions of some of Seck's experimentswithin his analytical uncertainty to give slightly more ter-nary solution at the higher temperatures,

Modeling of volumes

The molar volume of a ternary feldspar can be ex-pressed as

v: xo,vo, + xAbvob + x^"v^" + w, (l)

where, from the expression for G.' from Andersen andLindsley (1981),

v*: wr.o,oo(Xo{o)(Xoo + X^"/2)+ wv^bo,(xo,X"o)(x", + x""/2)+ wv.o,A"(xoxi(x"" + x^J2)+ wv.^^o,(xo,xo)(x", + xAb/2)+ Wv.^b^,(X^rX""XX^, + Xo,/2)+ wv.A"Ab(x^oXo")(xoo + xo,/2)+ wv.o,^b^^(xoxooXo"). (2)

Because excess volume parameters for the plagioclasebinary join are not significant (Newton et al., 1980), theyare set to zero. In addition, we treat excess volume foralkali feldspar as symmetric (Hovis, 1977; Kroll et al.,I 986) :

Equation I therefore becomes

v: Xo,vo, + x^hv^b + x^nv^n + wv.o,^b(xo.x^b)+ w y.o,^"(xo,x A")(x ""+

x ^J 2)+ wvo"o,(xox"")(x", + x^"/2)+ wvo,^b^"(xo1"oxo,). (4)

We used volume data from Kroll et al. (1986) andNewton et al. (1980) in a least-squares refinement ofEquation 4 to yield values for Wr.o,oo, W.o,o' W.ono,,artd W,o,ooon fiable l). (Attempts to use other combi-nations of parameters-for example W,*o" and W.o"*without the ternary term-did not fit the data nearly sowell; the refined value for W.o,on was found to be zerowithin error and was therefore set to zero in the finalfitting.) The value for W.o^o is similar to such valuesfound by Thompson and Hovis (1979) and Kroll et al.(1986), but is about l0o/o larger, a result ofour fitting datafor binary and ternary feldspar simultaneously. The datayield significant negative excess volumes for the ternaryand for one of the An-Or binary terms, an importantresult because it implies that as feldspars become morestrongly ternary, the effect of pressure on the miscibilitygap becomes the opposite of that on the alkali feldsparbinary: at constant temperature, the ternary-feldspar mis-cibility gap becomes narrower with increasing pressure.This effect is suggested independently by experiments onternary feldspars, where at similar temperatures, higherpressures (5 kbar) produced more strongly ternary feld-spars than did low pressures (1 atm) (Fuhrman, 1986).As in the case for the free energies, rigorous treatment ofthe volumes would require two equations, one each fortriclinic and monoclinic feldspars. Nevertheless, the vol-ume data clearly show that a single-phase ternary feldsparhas a lower molar volume than do coexisting plagioclaseand alkali feldspar (lying either on or close to the binaryjoins) of equivalent bulk composition. The small negativeexcess volume for the An-Or join derived from the vol-ume data of Kroll et al. contrasts strongly with the largepositive excess volume calculated by Green and Usdan-sky (1986) from Seck's tie line data. If Seck's polybaricexperiments at 650 'C had been reversed and the com-positions rigorously determined, we would accept thevolumes extracted by Green and Usdansky; since neitheris the case, we consider it preferable to use the measuredvolumes. That choice is being confirmed by continuingexperiments on feldspar equilibria at l0 kbar (Hadji-georgiou et al., 1987). The implications for the pressureeffect on two-feldspar thermometers are profound.

We do not have a physical explanation for these neg-ative volumes of mixing, which are unusual for silicates.If they were releated to the Al-Si substitution, the Ab-Anjoin should be similarly affected, but it is not. We canonly surmise they result from an unexpected interactionbetween K and Ca in the feldspar structure.

Activity expressions

The Al-avoidance model (Ghiorso, 1984) results inexpressions for ideal activities of pure ternary feldsparsWr.*oo: Wr,ooo.. (3)

204 FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

identical to those given by the site-mixing model of Price(1985). Thus the site-mixing expressions used by Greenand Usdanskv (1986) are identical to Ghiorso's and ourown for ternary compositions. Combination of the con-figurational terms with the expression for excess free en-ergy of mixing yields

@mix: - fsconi + Gs. (5)

G^^: RT{X"oln[X"o(1 - f?Jl+ X".ln[X..(l - XL)]+ X^.lnlx^"(l - X^")'z/4lj+ wo,Ab(xo,x^)(x^b + x^"/2)+ w^bo,(xo,x^J(xo, + x^"/2)+ wo,^"(xo,xA)(x^" + x^b/2)+ W^"o,(Xo,X^")(Xo, + X^b/z)+ WobA,(X^bX^")(X^" + Xo,/2)+ wA^^b(x^bx^")(x^b + xo,/2)+ wo,^b^,(xoxooXo). (6)

If we accept previously determined values for the Ab-An and Ab-Or joins, there are three unknown parametersin the expression above: Wo,An, WAno,, and Zo.ooo"-orsix if each of these Wo terms includes a temperature-dependent term (Wr).

At a given P and T, two coexisting feldspars must meetthe equilibrium conditions:

+ w""o,l2xo,x""(l - xo,)I XooXo"(Yz - Xo,)l

+ W^b^nlX^,.X^"(Yz - X", - 2X"")l

+ W^^^blX^bX^"(Yz - X", - 2X'Jl

+ wo,Ab^,lx^,x""(1 - 2x;l)/RT Q2)

ao^: lX'^(l + X^")'z/41

. exp{(Wo^o[X ooXo,(Vz - X ̂ " - 2X ̂ J]+ w^bo,lxAbxo,(h - x"" - 2x",)l

+ wo,^^l2xo,x""(1 - &")* XooXo,(I/z - X^")1

+ w""",\K",(l - 2x^")I XpXo,(Vz - X"")1

+ w^bA^l2x^bx^,(l - xo^)

I XooXo,(I/z - X"")1

+ w"""olxoo(l - 2x^")

* X^oXo,(t/z - X"")]

+ wo,^b^,lxo,Xoo(r - 2x^")l)/R7\ (13)

These activity expressions are identical to those ofGhiorso (1984), except that we include excess volumeterms in W,, ar;,d we include the ternary terms that wereaccidentally omitted from his published equations.

By substituting Equations ll-13 into Equations 7-9and gathering unknown parameters together, we obtainexpressions with a constant term on one side (all knownparameters) and the three unknown W,, lerms with theircoemcients on the other. Like Ghiorso and Green andUsdansky, we then employ the method of least squaresto solve for these unknown parameters using 3n equa-tions derived from the n experimentally determined tielines.

Experimental data base

Seck (l97la, 1971b) has presented the most extensiveexperimental data on coexisting ternary feldspars. Therehas been some question as to the attainment of equilib-rium in these experiments (e.g., Johannes, 1979; Brownand Parsons, l98l) both because of the use of gels asstarting materials and because of the general problem ofslow reaction rates in feldspar experiments. Johannes(1979) determined two tie lines at 800'C and one at 650oC using crystalline starting materials; his experimentsmay have approached equilibrium more closely than didthose of Seck. It is not clear a priori which data set is"better." but we have chosen to use Seck's data for ourmodel because (l) Brown and Parsons (1981) noted thatthe regularity of Seck's results suggests an approach toequilibrium; (2) the number of Johannes's tie lines is toosmall to permit modeling whereas Seck's data base is moreextensive; and (3) the previous workers had modeled thesystem using Seck's data, and we wanted to compare theresulting models.

^alk - ^Dlasu Ab

- qA6-

a$l: aulz

a" | \ : 6o ; "c '

Equation 6 is combined with the relationship

RTln a,: Gmtx + (I - X)(dG/dX,),,o,*,.,0 (10)

and, after differentiation and some simplification, we ob-tain the activities:

aoo: Xoo(l - X7"). exp {(W",ool2 X ̂ bxo,(l - X oo)

t Xo,Xo^(Vz - XoJl

+ w"o",lY",(I - 2X") r Xo,Xo^(t/z - &o)l+ Wo,^,lxo,X^"(t/z - Xoo - 2X"")l

+ W^"o,lxo,X^"(t/z - Xoo - 2X")l

+ W"oo"[P""(l - 2X^) -t Xo,Xn(Vt - X"o)]

+ w^^Abf2x^x""(l - &o)I Xo,Xo^(Vz - X"o)]

+ wo,^b^"lxo,Xo"(l - 2x^b)l)/RT\ (11)

ao,: Xoll - Xtr")

. exp{(W",ooffib(l - 2Xo,) I X ooX n"(t/z - X"Jl

+ wobo,l2xArx".(1 - x"JI XooXo"(t/z - X"Jl

+ w",o"lx?""(l - 2xo)I X^oX^^(t/z - X".)l

(7)(8)(e)

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY 205

The most important difference between our approachand those of Ghiorso and of Green and Usdansky lies inthe use of the experimental data base. We omitted Seck'sdata at 650 "C because (a) they had the largest residualsin our initial least-squares refinement, (b) they are leastlikely fo have approached equilibrium, and (c) it is notclear that high-temperature parameters would be appro-priate for temperatures below 700'C. Ghiorso used Seck'sdata as tabulated, an approach that would avoid bias ifthe errors in composition were randomly distributed. Butit is clear from Seck's text and the smooth variation ofcomposition in his figures that his analytical method-combined d spacings and geometric constraints-did notproduce random errors! For each isotherm, one coulddraw slightly different tie lines and solvi that are equallycompatible with the experiments. The X-ray data pro-vide good determinations for Or but only poor con-straints for An in alkali feldspars. Particularly striking areSeck's reported contents ofAn in alkali feldspars at 900oC-values that seem suspiciously low when compared tomany sanidines and that led to a negative W" for one ofthe An-Or terms in the models both of Ghiorso and ofGreen and Usdansky. That negative I/", in turn, is largelyresponsible for the unrealistic temperatures these modelspredict for many ternary feldspars, both natural and syn-thetic. Accordingly we attempted to improve the modelby allowing small changes in Seck's published composi-tions subject to the following constraints: (l) no pointswere changed by more than 1.5 mo10/o; (2) new pointsremained consistent with Seck's auxiliary experiments thatdetermine the extent of the two-feldspar field; (3) if onefeldspar composition was changed by a large amount (> Imolo/o), the composition of the coexisting feldspar wasalways moved in a compensating fashion to maintain bulkcomposition; (4) data points were changed in directionsthat reduced the least-squares residual values and thatwere also compatible with the probable directions of an-alytical error. We emphasize that all compositions thatwe used as input to our least-squares refinement are com-patible with Seck's experiments and thus are just as "val-id" as his published values. The final data points used tocalibrate the model, together with Seck's original data,are given in Fuhrman (1986). In general, the changes inSeck's published data involve small counterclockwise ro-tations for some tie lines and slightly more ternary solu-tion for most of the 900 "C experiments; they also happento make the revised tie lines more consistent with thoseofJohannes (1979) and with the calculated tie lines givenby Ghiorso (1984).

Our method further departed from that of Ghiorso(1984) at this stage. We used a polythermal data base(7 50, 825 , and 900 "C experiments of Seck) to solve oursystem of equations simultaneously for both Wo and W,portions of each Wo. When we use the adjusted compo-sitions as outlined above, there is no significant temper-ature dependence of the thermodynamic parameters; weobtained a good fit using only three W, parameters (Ta-ble l).

r200

rooo

800

600

400

I kbor

---Green- Usdonsky

- Fuhrmo n- Li ndsley

Two Feldspors

mole froclion

A n 1 2 . 3 4 5 6 7 8 9 O r

Fig. l. Calculated solvi for the join An-Or based on themodels of Green and Usdansky (1986) and this study. The solvicalculated by Ghiorso (written comm., 1987) are closely similarto ours. The apparent decreasing solubility ofAn in potassiumfeldspar with increasing temperature for the Green-Usdanskymodel reflects their negative values for W"(AnOr) and for,V"(AnOr). Above approximately 1030 'C, the relations shownhere are metastable with respect to assemblages containing leu-cite or liquid or both.

Green and Usdansky (1986) also fitted Seck's datapolythermally, \Mith the important diference that they alsoaccepted his 650 oC experiments at l, 5, and l0 kbar.Like Ghiorso, they used six adjustable parameters; how-ever, they omitted ternary terms and instead calculatedasymmetric Margules parameters for enthalpy, entropy,and volume on the An-Or join. We consider the directvolume measurements of Kroll et al. (1986, which pre-sumably were not available to Green and Usdansky) tobe strongly preferable to volumes extracted from the ex-perimental tie lines-especially in view of our doubts asto whether Seck's 650'C experiments reached equilibri-um. Green and Usdansky clearly recognized the difficul-ties with Seck's compositions. They tried to avoid theproblems of the uncertain An content of the alkali feld-spars by using only the activity expressions for ao. (ourEq. 8) to solve for the unknown Margules parameters.That approach is clever and indeed elegant, but as wewill show below, it unfortunately results in some unde-sirable features of their model. We believe that our meth-od of adjusting some of Seck's compositions, though lesselegant, is preferable.

Calculated solubilities for the An-Or join. The Green-Usdansky model and our own predict similar solubilitiesbetween An and Or at 650-750'C, but there are strongdisagreements at higher and lower temperatures (Fig. l).In particular, the strongly negative values for Wo^^o, andW, ono, in the Green-Usdansky model predict decreasingsolubility of An in Or with increasing temperature-amost unlikely result. This feature carries over into theternary system and seriously impairs temperatures cal-culated for any feldspar pairs that have more than min-imal solution of An in alkali feldspar. The solubilitiescalculated by Ghiorso for his model (written comm., 1987)are similiar to those for our model up to approximately

l-\-Nl - \-N.\\.\ 'r \

sNl\..\\r,,'.

206 ruHRMAN AND LINDSLEY: TERNARY.FELDSPAR MODELING AND THERMOMETRY

Fig. 2. Comparison of isotherms calculated from our model with the experimental data of Seck (197la). (a) 750 "C, 1 kbar; (b)825 "C, I kbar; (c) 900'C, 0.5 kbar. In Fig. 2c, * marks the bulk composition of a mesoperthite homogenized by Morse (1969, p.t2s).

1030 'C; above that temperature, An-Or relations be-come metastable with respect to assemblages involvingleucite and liquid.

Comparison of isotherrns

One way to test our model is to calculate isothermalsections of the ternary-feldspar miscibility gap and com-pare these with the data used to produce the model. Tielines were calculated using a computer program that si-multaneously matches activities of the three feldsparcomponents. These tie lines and the isotherms they gen-erate are compared with Seck's original data in Figures2a-2c. ln each figure, the dashed tie lines connect coex-isting feldspars as predicted by our model; Seck's data areshown connected by solid lines. There is generally goodagreement except at low An contents where our modelpredicts /ess miscibility at 7 50 'C and more misclbilityat 900'C than reported by Seck. Seck's compositionaldeterminations for plagioclase should have the greatesterror in this region, so the disagreement may be moreapparent than real. Johannes's (1979) dala at 800'C alsodescribe tie lines less steep than Seck's, but are subpar-allel to tie lines given by our model. Johannes's data sug-

gest lower solubility of An in alkali feldspar than do Seck'sdata or our model.

Also shown in Figure 2c is the bulk composition of amesoperthite that Morse (1969), on the basis of homog-enization experiments and geometric constraints, consid-ered to lie on the consolute curve for ternary feldspars at0.5 kbar and 925 + 5 "C. The agreement with our pre-dicted consolute point for 0.5 kbar and 900 oC is remark-able, especially in view of the fact that Morse's experi-ments, which were of the dissolving type only, wouldpermit a consolute point at a temperature somewhat low-er than 925'C.

We have also calculated isotherms for 1000 'C andI 100 "C at I kbar and 5 kbar (Fig. 3) in order to test thecompatibility of the model with a number of natural ter-nary feldspars. These feldspars are estimated to havecrystallized at temperatures of 950-1 100'C and pressuresof l-7 kbar. Our model is compatible with most of theseestimates. The model of Ghiorso would require consid-erably higher temperatures for most of these feldspars

[his 1200'C isotherm (1984, Fig. 4) virtually coincideswith our 1000 .C isotherm in much of the region Ab >501, whereas the thermometer of Green and Usdansky

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

An

b825"C

20'7

I - - - I M O D E LH SECK

r - - - I MODELo-o sEcK

,.':\'t\. \ \. \

' \

r t t

Fig. 2-Continued.

208 FUHRMAN AND LINDSLEY: TERNARY.FELDSPAR MODELING AND THERMOMETRY

An

- - - 5 kbo r- I kbor

Fig. 3. Isotherms calculated by our model for 1000'C and 1 100'C at 1 kbar and at 5 kbar; the corresponding consolute pointsare shown by *. Also plotted are compositions of strongly ternary natural feldspars: L: Sybille Monzosyenite, Fuhrman (1986);PB : Klokken Syenogabbro, Parsons and Brown (1983); OA: Oaxacan Complex, Mora and Valley (1985); SC: Scourie Complex,Rollinson (1982) and O'Hara and Yarwood (1978); TIG : Tigalak Intrusion, Wiebe and Wild (1983); WHC : Wyatt HarborComplex, Huntington (l 980).

(1986) gives divergent temperatures for each ofthe equi-libria.

Two-rnr,ospAR THERMoMETRy

Most of the two-feldspar geothermometers developedto date are based solely on the distribution of Ab com-ponent between plagioclase and alkali feldspars-theequilibrium a:it : aolt". However, as pointed out byStormer (1975), Ghiorso (1984), and Green and Usdan-sky (1986), similar equations can be written for the Orand An components in two coexisting feldspaars. Substi-tution ofthe activity expressions (Eqs. I l-13) into Equa-tions 7-9 permits calculation of three temperatures, eachone expressing the equilibrium temperature based on theactivity of that component. Ideally, for two feldspars inequilibrium, these temperatures will all be the same; inreality they rarely are, because of(l) analytical difficultiesin determining exact feldspar compositions, (2) difficul-ties in estimating amounts of exsolution and re-equili-bration that occurred after each feldspar crystallized orin identifying coexisting pairs when one or both feldsparsare zoned, (3) uncertainties in the estimation of pressure

(pressure coemcients of the three thermometers are dif-ferent), and (4) errors associated with the model. Themole fractions of An in alkali feldspar and Or in plagio-clase are particularly sensitive to the first two sources oferror; this drastically affects the activity expressions forthese two components. Ghiorso (1984) acknowledgedthese problems and developed a technique utilizing a least-squares method to find the one temperatufe that best sat-isfies all three activity equations. However, his methodloses some important information, particularly a simpletest for equilibrium: whether the three temperatures de-termined from the three equilibria (Ab, An, and Or com-ponents in the two feldspars) are close to each other orcan be made close by modifying the feldspar composi-tions within their estimated analytical error.

Green and Usdansky ( I 986) used the different pressurecoefficients for each equilibrium as a barometer by de-termining the pressure at which the differences among thetemperatures are minimized. The approach is intriguing,but unfortunately errors of 0.5 molo/o for Or in plagioclaseor An in alkali feldspar can affect the calculated optimumpressure by 2 kbar or more. Thus we believe that it ismore valid to use independent estimates of pressure and

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

to use the redundant temperature information to evaluate tered temperatures, and (3) comparison of temperatures

the effects of analytical uncertainty on the thermometer. for natural two-feldspar pairs with independent temper-ature estrmates.

209

The thermometer program

Our thermometer utilizes all the information given bythe coexisting feldspars and provides insight into howand why different feldspar pairs give various kinds ofresults. It is based on three major assumptions: (l) theternary-feldspar model presented above describes the sys-tem well; (2) feldspar can be analyzed with an overallaccuracy of +2 molo/o; (3) the samples to which it is ap-plied represent equilibrium compositions-or nearly so.Given these assumptions, an equilibrium two-feldspar pairshould yield three congruent temperatures (here definedas being within -40'C of each other) if the compositionsare allowed to vary up to 2 molo/0. The program firstcalculates equilibrium temperatures based on the initialfeldspar compositions and then systematically searchesat successively larger increments around each feldsparcomposition for a pair of compositions that will mini-mize the sum of dffirences function

f (Xo,, X oo, X o, in 2 feldspars) :

Comparing results of various feldspar thermometers

At present there are three diferent solution models for

ternary feldspars and three quite different approaches for

applying these in thermometry. Various combinations of

the models and approaches could yield a total of nine"feldspar thermometers"l Possibly one of the six remain-ing combinations will be found to be the "best" ther-mometer, but we leave that project to others; in the dis-

cussion that follows, unless otherwise noted, we applyeach thermometer in the way the original author(s) pro-

posed.

Temperature recovery from experimental feldspar pairs

Table 2 shows the results of applying our feldspargeothermometer to the experiments of Seck (1971a) and

Johannes (1979). Seck's 750 oC compositions yield tem-peratures that average about 30'C high, the 825 "C sam-ples yield temperatures very close to the experimentaltemperature, and the pairs from the 900'C experimentsgenerate temperatures l0 to 40 'C lower than the exper-imental temperatures. We compare these results with thetemperature recovery from the preferred model of Greenand Usdansky (1986; the "orthoclase calibration" of theirTable l). Their recovery of temperatures for the Ab andOr equilibria is in all cases as good as-in several casesbetter than-ours, but the An temperatures are a differentstory. For all but the data set at 650 oC and I kbar, theirpublished An temperatures average 80 to 270 "C belowthe actual temperatures. In one sense, this comparison isunfair, for the Green-Usdansky temperatures are for Seck'scompositions as originally reported, whereas ours reflectthe best fits for compositions within the estimated ana-lytical error, i.e., +2 molo/o of the original compositions.However, the intent here is to compare the approachesto thermometry as well as the models; we suggest that theimproved overall fit for our model shows that our ap-proach is the better one. The poor fit of the Green-Us-dansky An temperatures is especially noteworthy becauseit is precisely the negative pressure coefficient for the Anequilibrium that dominates the pressure calculation usingtheir thermobarometer.

When our geothermometer is applied to the 800 'C

pairs reported by Johannes (1979), it yields concordanttemperatures that average 734 "C for one pair and 770'C for the other (Table 2). The Green-Usdansky ther-mometer yields nearly concordant temperatures that av-erage 801 "C for the first pair and somewhat discordanttemperatures that average 836'C for the other' Johan-nes's experiment at 650'C yields temperatures that av-eruge 639'C for our thermometer, but rather discordanttemperatures by that of Green and Usdansky. For Seck's650 'C experiments (which were omitted from our mod-eling) the original compositions used with our model yieldZoo and ?"o, very close to 650'C, with Io" much higher-

I lZ "o l - lZ " . l I+ l l 7 o . l - l l " " l l+ | 17" "1 - l r "o l l . (14)

In other words, the program finds two compositions closeto the original values that yield the smallest differencesamong the three temperatures given by the three activityequations. The details of this iteration and search methodare given in Appendix l. This concept of searching foran optimum tie line arbitrarily close to the analyzed com-positions is similar but not identical to the proceduredeveloped for pyroxene pairs by Davidson (Davidson andLindsley, 1985, p. 402). The error in temperature, in-cluding analytical uncertainty of the compositions, is ap-proximately +40'C.

Although our solution model was developed using theAl-avoidance model for configurational entropy (Ghior-so, 1984), we can substitute site-mixing expressions forthe ideal portions of the activity terms because the site-mixing and Al-avoidance models are equivalent for pureNa-Ca-K feldspars (Price, 1985). It is especially useful tobe able to include the effects of celsian component(BaAlrSirOr, Cn) for some volcanic feldspar pairs in whichCn equals or even exceeds An in the alkali feldspars. Wehave therefore included Cn in the configurational (idealactivity) terms even though we lack expressions for thecontribution of Cn to the activity coefficients. Use of Cnin our program is optional; 0.01 to 0.02 mole fraction Cnin alkali feldspar raises the apparent temperature by l0-20"c.

We have evaluated our two-feldspar thermometer bythree kinds of tests: (l) its ability to yield correct tem-peratures for the experiments used to calibrate the model,(2) its ability to recover original compositions and tem-peratures for feldspar pairs that have been deliberatelydisplaced from compositions that give three tightly clus-

2r0 FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

Trere 2. Temperature recovery from experiments on ternary feldspars

This study Green and Usdansky (1986)

rfc) Ioo

P(kbar) r""L"To,r".

(1) (2\Too I^o

Seck (1 971 a)

Johannes (1 979) 805819

699 + 14 696 + 11 691 + 11780 + 19 779 + 18 779 + 19825 + 28 825 + 23 824 + 23863 + 26 863 + 27 861 + 23

733 730 740769 768 772621 641 656

650 1750 1825 1900 0.5

800 1800 1650 1

665 + 21 638 + 64760 + 26 670 + 22811 + 22 651 + 17885 + 29 695 + 18

770. 818-840. 865.586' 731"

638 + 18736 + 17826 + 28917 + 23

813" 783-803- 862'674' 565'

Note: (1) : Hasetron et at. (1983); (2): chiorso (1984).- Calculated by us. Green and Usdansky temperatures for Seck's data are from their paper

the last being just the opposite of the results from theGreen-Usdansky thermometer. For these feldspar pairs,our thermometer eventually converged on congruenttemperatures that average 30-40 'C too high. Possibleexplanations for the relatively poor recovery of the 650'C temperatures by our thermometer include (l) our modelis inaccurate in this region; or (2) Seck's 650 .C experi-ments did not reach equilibrium. On the basis of the dis-cussion of volumes above, Seck's 650'C experiments atl, 5, and l0 kbar cannot c// represent equilibrium; per-haps none ofthem do.

Overall, the ability of our thermometer to recover theexperimental temperatures is good.

Recovery of original temperatures from randomlydisplaced compositions

We have tested the ability of our thermometer programto converge on consistent temperatures by taking pairs ofcompositions that yield closely similar temperatures(within 5 'C), randomly displacing one or both by 2 molo/o,and allowing the program to search for the best answer.In almost all cases the program found the original com-positions or pairs whose temperatures differ by only a fewdegrees from those indicated by the original ones. Theone type of displacement for which the original compo-sitions and temperatures (understandably!) could not berecovered is that which closely corresponds to an equilib-rium pair for a lower or higher temperature. With thatexception, the excellent recovery of the original temper-atures shows the power of using the three thermometryequations simultaneously to search for equilibrium com-posrtrons.

Furthermore, the deviations between the original tem-peratures and those initially given by the displaced com-positions provide insight into reasons why some naturalpairs fail to yield concordant temperatures, because re-setting of one or both compositions by geologic processesmay also mimic our displacements of the compositions.For almost all random displacements from an equilibri-um pair of compositions, ?nAb and Zo. change in the samedirection while Zo, changes, in many cases strongly, inthe opposite direction. Thus suspiciously high or low an-orthite temperatures commonly reflect overestimation orunderestimation of An in alkali feldspar. Underestima-

tion of Or in plagioclase (as for example when exsolvedalkali feldspar is not included in the analysis) typicallyyields 2". very much lower than the other two tempera-tures.

Results from natural feldspars

Feldspar pairs from volcanic rocks. Two-feldspar tem-peratures from volcanic rocks are an important test ofathermometer because it is probable that equilibriumcompositions-if init ially present-have been main-tained. Analytical errors are more likely to be random,and thus adjustments to the compositions by our ther-mometer should be randomly distributed. Table 3 showstemperatures given by our thermometer and by those ofGhiorso (1984), Haselton et al. (1983), and Green andUsdansky (1986) for a number of volcanic feldspar pairs,along with independent estimates of temperature givenby Fe-Ti oxide pairs where available. In most cases, thetemperatures given by our thermometer compare favor-ably with those given by the other thermometers. Has-elton et al. (1983) recommended against using their ther-mometer for feldspars that formed above 850 "C, wherethere may be a large ternary component in one or bothof the feldspars. The results in Table 3 support their rec-ommendation.

Feldspar pairs from the Fish Canyon Tuffare especiallyinteresting because Stormer and Whitney (1985) havesuggested a depth of origin corresponding to approxi-mately 9 kbar, the pressure needed to produce agreementbetween two-oxide and two-feldspar temperatures (asbased on thermometers available at that time) for theserocks. We do not take sides on this controversial issue,but we note that our thermometer makes smaller andmore nearly random adjustments to the compositions foran assumed pressure of 3 kbar as compared to either Ior 6 kbar. However, the 6-kbar to 8-kbar temperaturesindicated by our thermometer still provide the closestmatch to revised oxide temperatures, so the general re-lationship noted by Stormer and Whitney still holds,whatever its cause. Inclusion of celsian components (whichare available for these samples) raises the feldspar tem-peratures by about 20 {-approximately equal to theeffect of I kbar. If the Stormer and Whitnev barometer

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

TneLe 3. Comparison of calculated temperatures for igneous ternaryJeldspar pairs

Plag. Alk. This study Green-Usdansky (1) (21roo To, ron roo To, rnn roo roo

2rl

OrP

(kbar)

Carmichael (1965)Trachyte

Ewart (1 965)lgnimbrite [750'C]

Hildreth (1979)Bishop Tuff, early [720'C]

late [780'C]Baldridge et al. (1981)

Leucite basanitePhonolite-tephrite

W34-8-1 monzonite(LAC)SR-6A monzosyenite (LAC)Klokken Syenogabbro;

Parsons and Brown (1983)

0.654 0.073 0.273

0.632 0 041 0.326

0.798 0.061 0.1410.702 0 076 0.222

0.253 0.029 0.7180.477 0.072 0.451

Volcanic oairs0379 0.603 0.018 849 821 832

0.319 0.653 0028 807 756 776

933 799

810 631

698 603789 730

774 7791011 1343

890 857 993

708 689 871

622 708761 812

817817

588o /o

676 676 671770 772 779

909 892 902905 868 903

0.100 984 984 9890.047 884 885 9210.060 932 932 970

0.544 0.286 0.1700.553 0.278 0.1810 520 0.260 0 220

11

0.0010.001

31

0.362 0.6300.337 0.650

0.168 0.7910.293 0.675

Plutonic pairs0.417 0 4840342 0.6120.330 0.610

0.0080 013

0.0410.033

789 1't46 444863 1755 562880 3163 608

856 10869 1 0 1 1 9 5

>1200. 2010>1200. 1298>1200' 1356

Note: Temperatures from method of Green and Usdansky (1986) were calculated by us. (1) : Ghiorso (198a); (2) : Haselton et al. (1983)' [ ] Fe-Ti

oxide temperatures Laramie (LAC) data from Fuhrman (1986).* Temperature estimated from position of isotherms.

(1985) is used, it is clearly essential that the effects ofCnbe taken into account.

Adjustments made by our thermometer to the com-positions of volcanic feldspars are semirandom (Fuhr-man, 1986, and unpub. data); in no case tested by us isthe Or content of plagioclase, and only rarely is the Ancontent of alkali feldspar, decreased. With these excep-tions, other adjustments are about equally likely. It isentirely possible that our model slightly overestimates Orin plagioclase and An in alkali feldspar; conceivably thosephases are able partially to purge themselves of thosecomponents even in quickly cooled rocks.

Feldspar pairs from plutonic igneous rocks. When ourthermometer is applied to strongly ternary feldspar pairs(Table 3), it yields nearly concordant temperatures of 90G-1000 'C. The pair from the Laramie monzonite gives atemperature of 985 + 50 'C, which is consistent withtwo-pyroxene thermometry (980-1020 "C). These pairs

consist of highly exsolved feldspars whose integratedcompositions have large uncertainties (+4 molo/o). The

large uncertainty in composition is partially compensatedfor by the gentler slopes of the solvus for these compo-sitions. The feldspar pair given by Parsons and Brown(1983) generates a temperature of -940"C, close to their

estimate of 950-1000 "C for crystallization temperaturesfor the rocks in which the pair occurs'

The Green-Usdansky thermometer generally gives

strongly divergent temperatures for these plutonic ternaryfeldspars, with Ab tempertures being closest to the in-

dependent estimates and An and Or temperatures typi-cally several hundreds of degrees higher and lower, re-

spectively.Metamorphic feldspar pairs. Table 4 shows calculated

temperatures using several thermometers for metamor-phic feldspar pairs from Adirondack granulite-facies rocks.

Very few metamorphic pairs give concordant tempera-

TneLe 4. Calculated temperatures for some feldspar pairs from granuliteJacies metamorphic rocks

Plag. Atk. This study Green-Usdansky

(2)( 1 )IonTo,To,Io.To,Ioo

BM-15 0.743tN-11 0.785MM-2 0.762sR-31 0713sc-2 0722xY-g 0.796xY-12 0.808

HA-243 0.802HA-260 0.697HA-309 0.728HA-349 0 73sHA-390 0 751

0.014 0 2420.017 0.1 930.030 0.2050.010 0.2730.008 0.2700.014 0 1830.013 0.178

0.012 0 1860.008 0.2950.013 0.2590.009 0.2560.013 0.236

Bohlsn and Essene (1977); calculated for I kbar0.235 0.751 0 014 707 612 7080.348 0.639 0.006 764 604 7s00.256 0.700 0.036 723 665 7250.314 0 638 0.031 763 598 7660.196 0.800 0 004 675 582 6260.355 0.605 0 037 770 583 7040.38s 0.586 0.026 778 590 779

Ehrhard (1 986); calculated lor 7.2 kbar0.317 0.656 0.027 736 5800.192 0.798 0.010 648 5860 235 0.751 0.014 703 6130.223 0.760 0.017 695 5850.191 0.796 0.013 643 593

Notej IAb, Io,, Ia" from method of Green and usdansky (1986) calculated by us using their thermometer. (1) : Haselton et al. (1983); (2): Stormer(1 975).

693 504 460791 510 707635 623 336730 464 438682 428 630678 461 331720 456 405

668 677 441 389655 668 425 532699 696 487 495681 667 428 460649 624 472 468

711 676804 750670 687797 776685 644741 747774 758

714 704678 642714 674685 658632 616

212

PLAG ALKFig. 4. Distribution of adjustments in compositions made by

the thermometer program for feldspar pairs from Adirondacksgranulite-facies metamorphic rocks. The innermost hexagonrepresents adjustments of 0.5 mo10/o; outward from this the hex-agons represent adjustments of 1.0, 1.5, and 2.0 mol0/0. Adjust-ments are made while always holding one mole fraction con-stant. Numbers indicate frequency of samples adjusted.

tures with our thermometer; fo, tends to be lower thanthe other temperatures, although Zoo and ?"o" are gener-ally similar. We assume that loo reflects an equilibriumtemperature but that Zo. may have been affected by lossof Or from plagioclase in these slowly cooled rocks. ?"Abis close to that given by Haselton et al. (1983), hardlysurprising inasmuch as the two thermometers are basedon the same binary parameters. Temperatures for thesemetamorphic rocks using the Green-Usdansky thermom-eter are less satisfactoryi Tet is generally 20-40 "C lowerthan for the other thermometers (hence within uncertain-ty), but Zo" and 2". both tend to be much lower, by 100-200.c.

We emphasize that it is essential to reintegrate thecompositions of plagioclases as well as those of the alkatifeldspars in order to obtain consistent temperatures forthese and other deep-seated rocks. For many such pairs,it is likely that there has been some re-equilibration orother modification of compositions after peak tempera-tures were reached. Possible changes in composition couldbe due to internal and/or granule exsolution or perhapsto alkali exchange with a fluid phase. Such changes shouldbe reflected in the directions that the compositions tendto be adjusted using our thermometer. For alkali feld-spars, the major process is the exsolution of Ab-rich pla-gioclase. It is now an accepted practice to reintegrateexsolved perthites in order to estimate original compo-sitions (Bohlen and Essene, 1977). However, some work-ers appear to ignore exsolved K-rich feldspar in plagio-clase even in cases where it can still be recognized. Quiteclearly such obviously exsolved plagioclases should bereintegrated, but users of our thermometer (or any ther-mometer based on a ternary model) should also look forevidence of external granule exsolution ofalkali feldsparfrom plagioclase and take that into account as well. Fig-ure 4 shows the directions and amounts that composi-tions have been adjusted by our thermometer for 25metamorphic pairs from the Adirondacks (Bohlen andEssene, 1977;,Ehrhard, 1986; the original data are sum-

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

marized in Fuhrman, 1986). The perthites of these pairshave been integrated, and there is a fairly regular distri-bution of adjustments around the original compositionfor alkali feldspars. For plagioclase, however, all the com-positions "require" increased Or component, implyingthat some regular process or processes may have affectedtheir compositions.

Appr,rcarroN, DISCUSSIoN, AND coMpARrsoN oFTHERMOMETERS

Choosing a two-feldspar thermometer

Potential users of two-feldspar thermometry are facedwith a number of possible thermometers calibrated inrecent years: Haselton et al. (1983); Ghiorso (1984); Greenand Usdansky (1986); and the one presented here. Thethermometers based on ternary formulations should beinherently preferable to the double-binary formulation ofHaselton et al. for feldspars that have more than trivialamounts of the third component. For several reasons, webelieve that our ternary formulation is superior to thoseof Ghiorso and of Green and Usdansky. By accepting theexperimental data of Seck as tabulated (Seck, 1971a,l97lb), those workers were forced to employ a negativeterm for excess entropy on the An-Or join. That negativeterm greatly restricts the amount of ternary solution cal-culated for temperatures at and above 1000 "C, whereboth natural and synthetic systems show extensive solu-tion. Our model incorporates the extensive volume dataof Kroll et al. (1986), which show distinct negative vol-umes of mixing for both the An-Or join and ternary feld-spars. The large positive volumes calculated for the An-Or join from Seck's polybaric experiments (Green andUsdansky, 1986) are clearly inappropriate and should notbe used. The poor recovery of anorthite temperatures bythe Green-Usdansky model (7"" : 531 'C at 5 kbar, 380"C at l0 kbar) for the 650 "C experiments used to calibratetheir model (Green and Usdansky, 1986; their Table l)probably reflects the deleterious effect of these volumeterms.

Thermobarometry using two feldspars?

Although each ofthe three equilibria for coexisting feld-spars has a different pressure coefficient, we strongly cau-tion against using two feldspars for simultaneous ther-mometry and barometry (Green and Usdansky, 1986),because small errors in composition have a very largeeffect in the apparent pressure. Furthermore, most of thepotential barometric information in the Green-Usdanskythermobarometer lies in the negative pressure coefrcientof ?"o". As outlined above, however, the ability of theirmodel to recover Zo" values for the S-kbar and lO-kbarexperiments used to calibrate the model is distinctly poor.Thus, apparent pressures calculated for the calibratingexperiments themselves would be seriously in error.

The examples of thermobarometry presented by Greenand Usdansky (1986) are very striking. Their examplefrom Little Glass Mountain Rhyolite (their Fig. l) shows

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY 213

the three feldspar temperatures converging at about 2.2kbar to a temperature very close to that indicated by Fe-Ti oxides. At first glance, this result provides support forthe idea of using their model for thermobarometry, be-cause independent estimates of pressure are around 3.2kbar (Mertzman and Williams, 198 l). However, the feld-spar compositions used by Green and Usdansky to pro-duce their Figure I are evidently the representative anal-yses of Mertzman and Williams (1981;sample 33P, Table3). The plagioclase composition appears to be that of arim, but the sanidine plots well within a cluster of points(Fig. 4 of Merlzman and Williams) and is most unlikelyto have been in equilibrium with the plagioclase. Use ofthe probable rim composition (the most Ab-rich sanidine)in the Green-Usdansky thermobarometer yields a poorconvergence at approximately 725 "C and - I kbar, muchless satisfactory than those workers obtained using theapparently nonequilibrium pair (1986; their Fig. l). Incontrast, it is interesting to note that when the composi-tions used by Green and Usdansky are allowed to vary(that is, using their model but our approach to thermo-metry), the sanidine composition moves toward the in-ferred rim composition, and the resultant temperature(calculated for 3.2 kbar) is in excellent agreement with thetemperature independently estimated by Mertzman andWilliams (1981). Clearly the Green-Usdansky model is abetter thermometer than barometer for this example!

The agreement between temperatures inferred for theBishop Tuff using two-oxide thermometry by Green andUsdansky using only their two-feldspar thermobarometer(1986, Fig. 2a) and those from two-oxide thermometry(Hildreth, 1977) is impressive. However, their inferredrange of pressure (2.5 to 6.5 kbar; their Fig. 2b) is lesswell constrained and is marginally higher than the pressurerange estimated by Hildreth (1.5 to 5 kbar). It is true thatStormer and Whitney (1985) inferred pressures of 4 to 6kbar for the Bishop Tuff using their two-oxide, two-feld-sparbarometer, but those pressures are controversial. Fur-thermore, application of the Stormer-Whitney techniqueusing our feldspar thermometer, together with an im-proved oxide thermometer, yields pressures of 2 to 3 kbar(Grunder et al., 1987), much closer to Hildreth's originalestrmates.

Use of our two-feldspar thermometer

We emphasize that the thermometer presented here (orany ternary-feldspar thermometer) should not be usedwithout an understanding of the method and the impli-cations of the results. The thermometer yields three gen-eral classes of results, designated by the relationship in-ferred for the original feldspars: ( I ) equilibrium pair, Z^o =

To" = Toi (2) close-to-equilibrium pair, T, = T, + To;and (3) nonequilibrium pair, TAb + T^n + 7o.. In thediscussion that follows, the reader should bear in mindthat Z^o is controlled mainly by the orientation (rotation)of the tie line, whereas Io" and To, are dominated by thewidth and position of the two-feldspar field.

Equilibrium pair : concordant temperatures. If, within

the +2 molo/o analytical error, the geothermometer pro-gram finds two feldspar compositions that yield temper-atures whose sum of differences VV) of Eq. l4l is lessthan 80,-that is, the highest and lowest temperatwes arewithin -40 "C of each other, well within the estimateduncertainties of the thermometer and model-it is rea-sonable to conclude that the feldspar compositions rep-resent equilibrium. In this case, the temperature acceptedfor the rock should be the mean of the three tempera-tures, with an error of 30-50 "C that will include all threetemperatures.

Close-to-equilibrium pair: two concordant tempera-tures, one different by -100 "C. In many cases, two ofthe activity equations yield concordant temperatures,while the third yields a temperature about 100 "C higheror lower. As mentioned above, Zo, is quite sensitive toAn content of alkali feldspar and Io. to Or content inplagioclase. Usually it is one of these temperatures thatis discordant, indicating a probable error in the deter-mination of the "minor" ternary component in eitherplagioclase or alkali feldspar. A common example fromplutonic rocks is the case of perthite (whose compositionhas been carefully reintegrated) that is said to coexist withplagioclase having a small reported Or content (l-2mol0/o). Such pairs invariably give either two concordanttemperatures (if the plagioclase is not too far from itsequilibrium composition) or three discordant tempera-tures (if even the 2 molo/o variation allowed does not yieldan equilibrium composition for the plagioclase). For thecase where two temperatures are approximately equal andthe third is -100 "C ofl the temperature taken for therock should be the mean of the two concordant temper-atures, but the feldspar pair should be re-examined care-fully in an attempt to explain the discordance of the third.Another possible cause of discordant temperatures is aserious error in estimating pressure, but the error wouldhave to be larger than several kilobars to have much ef-fect.

Nonequilibrium pair : discordant temperatures. Thereare three general cases where the three temperatures areall significantly different. The first is where Zoo lies be-tween Io" and Io. and the difference between the lattertwo is less than 500 "C. In this case, the orientation ofthe tie line (which dominates ZoJ may be close to equi-librium; however, the more sensitive Zon and To, atethrown offby relatively small errors in composition. Herethe temperature Z^o can be taken as that of the rock, butthe user should try to understand why the other two tem-peratures are offand, for plutonic feldspars, should prob-ably try estimating possible granule exsolution or alkaliexchange and input new compositions reflecting these es-timates to see if they give more reasonable and concor-dant temperatures.

In the second case, the range between the highest andlowest temperatures is also less than 500 "C, but 7"oo doesnot lie between Zo" and Zo.. In many such cases, Zoo isthe highest and probably reflects a close-to-equilibriumorientation of the tie line; but as described above, 7o.

2t4

and ?"A, are low because compositions may not quite beon the solvus at Zoo. When Zoo is the lowest temperature,it is probable that the orientation ofthe tie line has beenstrongly affected by some postcrystallization process andIoo does not reflect an equilibrium temperature. In thesecases, the thermometer should not be used without anattempt to determine why or how the tie line has beenrotated and which temperature is closest to representingthe equilibrium value.

The third case of three discordant temperatures is wherethe difference between the highest and lowest tempera-tures is greater than 500'C. Such a feldspar pair is no-where near being in equilibrium-most likely the orien-tation of the tie lines, which dominates Zoo, is inconsistentwith the model at geologically reasonable temperatures.Wildly discordant temperatures will also be generated byfeldspar pairs that plot on the same side of the consolutecurve, as is the case for some pairs given by Mora andValley (1985) where plagioclase having very little Or co-occurs with ternary feldspars. These pairs are stronglyincompatible with the model and therefore give unlikelyand discordant temperatures. For pairs that give very dis-cordant temperatures, the user should compare them withthe isotherms to verify that they lie on opposite sides ofthe consolute curve. Ifthey do, but still give very discor-dant temperatures, one should consider the possibility ofmagma mixing or incorporation of xenoliths in igneousrocks and should look for evidence ofdisequilibrium suchas resorbed rims or overgrowths.

CoNcr,usroi.,rs

The model presented here represents an improvementover existing solution models for ternary feldspars. Whenused for two-feldspar thermometry, it gives consistent andreasonable temperatures both for feldspar pairs with nearlybinary compositions and for feldspars with substantialternary component. Perhaps one of the most practicaluses of this model and thermometer is as a test for equi-librium between co-occurring feldspars. The relative suc-cess of this model is permissive evidence that the 750-900 'C experiments of Seck (l97la) closely approachedequilibrium. Our model should also be useful in model-ing the crystallization of ternary feldspars from hot, drymelts; it is much more successful for such feldspars thanare the models of Ghiorso (1984) and of Green and Us-dansky (1986). The thermometer of Green and Usdanskygenerally gives good temperatures in its range of calibra-tion (650-900 "C) and at relatively low pressures. It is notappropriate for strongly ternary feldspars, and-becauseit uses volume terms now known to be in conflict withmeasured volumes-it should not be used as a barometer.

AcrNIowt,rlcMENTS

We thank S. R. Bohlen, R. T. Dodd, G. N Hanson, and S. A. Morsefor their constructive comments on an earlier version of this paper. Weare also most grateful for helpful reviews of the current version by M.Ghiorso, N Green, and S A Morse (again); they may well not agree withall views expressed here, however This work was supported by NSFgrants EAR-84 I 6254 and EAR-86 I 8480.

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY

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MaNuscnlm RECEIvED M.r.rcH 13. 1987MlNuscn.Ipt ACCEmED Noveuser 13. 1987

2t5

AppnNltx 1. DnscnrprroN oF THERMoMETRY

PROGRAM MTHERM3

The thermometer program calculates three temperatures' one

for each activity equation (Eqs. 7-9) for any given two co-oc-

curring feldspars. The program searches in hexagons of increas-

ing radius around the original compositions until either the sum

ofdifferences (see Eq. 14) is smaller than that requested, or until

the limits of analytical uncertainty have been reached (2 mo10/o)'

The hexagons are made by holding one component constant while

the other two components are changed (see program lines 250

to 3 1 0 and 700 to 800). The program flags the compositions that

give temperatures with the smallest sum of diferences and up-

dates these compositions as smaller sums are encountered' The

order of search is as follows: plagioclase composition is changed

by 0.5 molo/0, then by 1.0 mol0/0, then by 1.5 mol0/0, and finally

by 2.0 mol0/0, while alkali feldspar is first unchanged and then

successively varied in steps of 0.5, 1.0, 1.5, and 2.0 molo/o' This

process continues until every combination of plagioclase and

alkali feldspar compositions from the original ones to 2 molo/o

away from the original compositions has been searched through

or until the lower limit ofsum ofdifferences requested is reached.

When the search is completed, the program prints out the most

concordant temperatures, and the compositions used to calculate

these temperatures.Where celsiarr contents of the feldspars are known' that infor-

mation should be used in the thermometer; our program has an

option for including Cn. Although our model has no information

on the effect of Cn on the activity coefrcients of the other com-ponents, we have used the site-mixing approach ofPrice (1985)

to express the contribution of Cn to the ideal activities of those

components. Cn is excepted from the permutations of compo-

sition described above.We will provide listings of BASrc program MTHERM3. Those

who would like copies of the program (for IBM-PC and com-patible computers) should send a formatted 5.25-in. diskette.

The compiled program is approximately 9 kilobytes long; it re-

quires approximately the same amount of memory to run' All

requests should be addressed to Lindsley.

FUHRMAN AND LINDSLEY: TERNARY-FELDSPAR MODELING AND THERMOMETRY