Manganese, ferric iron, and the equilibrium between garnet ...888 WILLIAMS AND GRAMBLING:...

American Mineralogist, Volume 7 5, pages 886-908, 1990

Manganese, ferric iron, and the equilibrium between garnet and biotite

Mrcn,c.nr- L. Wrr,r,ravrsDepartment of Geology and Geography, University of Massachusetts, Amherst, Massachusetts 01003, U.S.A.

Jnrrnnv A. Guvrnr,rNcDepartment of Geology, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A.

Ansrnacr

Garnets from the Proterozoic rocks of northern New Mexico vary widely in Mn contentbut contain very little Ca. In specimens collected from three small areas, X.o, in garnetvaries from nearly zero to 0.76. Biotite and Fe-Ti oxides occur in each specimen; theircompositions change systematically with X"o". After corrections are made for Fe3* in bio-tite, garnet and biotite show several compositional relationships critical to understandingtheir equilibrium. Ko (Fe-Mg, garnet-biotite) varies as a function X"o. in garnet, but thisrelationship alone does not constrain the mixing behavior of Mn in garnet because thedata show strong linear dependencies among most compositional variables. The mostimportant covariances occur among X"ou (X^^ - Xr*), (Xr" - X-")t', -Xli, and (Fe3*/totalFe)B'. This multicolinearity seriously obscures the mixing properties of any one compo-nent.

Stepwise least-squares regression and stepwise ridge regression analyses, carried out onsubsets of the data population, suggest that Ca mixing into iron-magnesium garnet issignificantly nonideal even at low X," (AWE1: 12 550 J/mol), consistent with most pre-vious results. The effects of Ti, Fe3t, Mn, and Al mixing in biotite are not significant inthis study because of the small or constant amounts of these components in the analyzedbiotite. The magnitude of Mn-Mg and Mg-Fe nonideality in garnet cannot be determinedfrom any one data set because of the nearly perfect correlation between X"o" and (X"'- -

Xo.o). Instead, an array of paired values is permitted. For example, if WW"is near 0 J/mol,AW1A : 5 500 J/mol; 1f WW. : 8000 J / mol, AWffi : I 9000 J/mol. Simultaneous solutionof the three data sets tentatively suggests that the larger values for the two interactionparameters may be more realistic.

Regression results allow a number of models that can successfully describe mixing effectsin the analyzed garnets. Two end-member models are proposed. The simplest, permittedbecause of the strong correlation between X"o" and most other components, artificiallycombines all nonideal mixing in both garnet and biotite into a single X.o"-dependent pa-rameter. The small magnitude, approximately 5500 J/mol, of this composite interactionparameter emphasizes that the nonideal mixing effects of all components tend to cancel,resulting in an overall behavior that mimics ideality. A more rigorous approach treats Mn,Fe, and Mg mixing in garnet independently, using an independently constrained Mg-Femixing model for garnet. For this study, two recently proposed models from the literaturewere employed. The simplified and the more rigorous gamet mixing models result inalternative calibrations of the garnet-biotite geothermometer that yield consistent andreasonable temperatures for garnet with X"," between 0 and 0.5.

INrnooucrroN This study was undertaken to evaluate the mixingproperties of Fe-Mg-Mn garnet with the goal of deriving

Garnet has complex nonideal solution properties. Mn a garnet-biotite geothermometer applicable to Mn-richand Ca apparently do not mix ideally with Fe and Mg, rocks. It is based on 7 7 garnet-biotite specimens collectedand Fe-Mg mixing itself may not be ideal (Sen and Chak- from the Proterozoic rocks of north-central New Mexico.raborty, 1968; Hodges and Spear, 1982; Ganguly and Although these specimens generally have X^ < 0.10, theySaxena, 1984; Geiger et al., 1987; Hackler and Wood, have X"o" ranging from 0 to 0.75.1989). Numerical expressions representing departures For this report, suites of 18-30 specimens each werefrom ideality are not well constrained, especially for Mn selected from three small areas (Fig. l): (l) west of Pecosmixing. These problems limit garnet thermobarometry, Baldy in the Truchas Range, a region of I km2 whereincluding garnet-biotite geothermometry, to rocks low in peak metamorphic conditions were near 500 oC, 4 kbarMn and Ca. (Grambling, 1986; Grambling et al., 1989); (2) the Rio

0003404x/90/0708-o886$02.00 886

WILLIAMS AND GRAMBLING: EQUILIBRIUM BETWEEN GARNET AND BIOTITE 887

Mora uplift, an area of l0 km, straddling the kyanite-sillimanite isograd, where peak conditions were near 550"C,4.5 kbar (Grambling and Williams, 1985; Gramblinget al., 1989); and (3) Cerro Colorado, an area of I km, inthe southern Tusas Mountains, where peak metamorphicconditions were 550-600'C, 4-5 kbar (Williams, 1987).

Because of the small size of each area, as well as theP-?" constraints cited in the references above, we haveassumed that peak metamorphic conditions were con-stant within each study area. Most specimens containstaurolite or aluminum silicate. and as a result. most bio-tites are saturated in the aluminum-biotite component.Chemical variation in garnet and biotite is mainly limitedto the Fe2+, Fe3*, Mg, and Mn components.

Atrt,c.LI"TrcA,r, TEcHNIeuES

Microprobe analyses were obtained with a JEOL 733electron microprobe at the Department of Geology, Uni-versity of New Mexico, using a consistent set of silicateand oxide standards. Operating conditions were 15 kV,20 nA, with a beam diameter of I prm for garnet, 3-10pm for biotite, and 2-10 pm for Fe-Ti oxides. Coexistinggarnet, biotite, and Fe-Ti oxide were analyzed during thesame operating session. We examined analytical preci-sion by standard statistical methods and also by reana-lyzing selected specimens several months after the origi-nal analysis. Precision was better than +0.2 wtolo for allmajor elements, and a l0lo of the amount present for ele-ments less abundant than 5 wto/o.

Zoned garnets were analyzed by multiple microprobetraverses with step increments less than l0 pm. Detailedtwo-dimensional maps of garnet zoning were constructedfrom several samples. Numerous biotite grains were ana-lyzed in each specimen; over 20 and up to 50 spot anal-yses were obtained from biotite in each thin section.

Fe3+/Fe2+ ratios of Fe-Ti oxides were determined byrecalculation from microprobe analyses following themethod of Rumble (1973). Fe3+/Fe2+ ratios of garnet andbiotite separates were determined on six samples byMdssbauer spectroscopy by M. Darby Dyar and RogerBurns at the Massachusetts Institute of Technology. Splitsfrom each of the biotite separates, along with five addi-tional biotite separates, were analyzed by wet chemistryat the University of New Mexico. In most cases whereboth methods were used on the same specimen, M6ss-bauer spectroscopy and wet-chemical analysis gave sim-ilar results. For the few exceptions, Milssbauer data wereaccepted in preference to wet-chemical data on the as-sumption that local patches of alteration or fine mineralinclusions, observed in some biotites, biased the wet-chemical results.

D,l,r,L

Garnet

Garnet typically preserved compositional zoning char-acterized by continuous declines in X.o" and consistentchanges in Xr" and Xil^/(X^h t Xr*) from cores to pointsnear garnet rims. However, the outermost 5 to 50 rrm of

many grains showed sharp reversals in the zoning trends,particularly for X"o", X^r^, and Xo* (see Grambling et al.,1989, Figs. l0 and I l). The thicknesses and magnitudesof these reversals varied even along the rims of singleeuhedral garnets. Typically, reverse-zoned rims werethickest and most pronounced where garnet was in directcontact with biotite or chlorite. Rims were absent or leastpronounced where garnet directly contacted quartz orfeldspar. Based on these observations, we have interpret-ed the reverse zoning as a retrograde feature. For thisstudy, garnet compositions were taken from the edges oftraverses not showing the apparent retrograde rims, or atthe point ofreversal along the zoning profile, with anal-yses accepted only from samples where the volume ofretrograde garnet was insignificant with respect to the vol-ume of biotite. These constraints resulted in composi-tions that were constant for all garnets in a thin sectionor even a small outcrop.

It should be noted that the conectness ofthe interpre-tation that the zoning reversals are retrograde is not crit-ical to our conclusions. Regardless ofthe interpretation,the reversals provide a consistent point of comparisonfor the garnets within a single data set, characterized bya consistent phase assemblage.

Mcissbauer analyses showed thal 95o/o of the Fe in gar-net is Fe2+. For this study, all Fe was treated as Fe2*. Thecompositions of 29 low-calcium garnets from Pecos Baldyare in Table l. X"o" varies from 0.00 to 0.32. In 35 spec-imens collected at Rio Mora, X.o" varies from 0.00 to 0.75(Table 2). From Cerro Colorado, X.o. in low-calcium gar-nets range from 0.01 to 0.47 (Table 3).

Biotite

Biotite is fresh in most samples, but grains in somerocks showed evidence of retrograde or deuteric altera-tion. The partial alteration included thin streaks of low-interference color parallel to (00 I ), rare stripes of chloriteparallel to (001), and less commonly, small patches ofextremely fine-grained chlorite, clay minerals, or epidote.Special care was taken to avoid partial alteration. All op-tically fresh biotite grains in a thin section varied by lessthan I wto/o in all analyzed elements. However, partiallyaltered grains showed variations in KrO and total weightpercent that corresponded to systematic changes in FeOand MgO. Analyses of some altered biotite had no KrOand gave analytical totals approaching 89 wto/0, suggestingthat small amounts of clay or chlorite were mixed withthe altered biotite.

Microprobe analyses accepted for this study were ob-tained by analyzing at least 20 optically fresh grains in asingle thin section, then averaging the ten analyses withhighest KrO and total weight percant. This technique dra-matically reduced the scatter in biotite analyses from asingle section. Biotite compositions from the three studyareas, screened using these criteria, are in Tables 4, 5,and 6.

Biotite from Cerro Colorado shows somewhat moreextensive compositional variation, even within a single

888 WILLIAMS AND GRAMBLING: EQUILIBRIUM BETWEEN GARNET AND BIOTITE

TABLE 1, Microprobe analyses of garnet, Pecos Baldy, New Mexico

77-858 76-420 77]125 77-15 77-81E 77-103 76-4008 7741 76-446 77-39

Welght percenttFeO 39.97 39.28MgO 1.75 2. ' t4MnO 0 .13 0 .14CaO 1.14 0.86Tio, 0.03 o.o2At,o3 21.00 20.69sio, 37.41 36.75Torat 101.43 99.88

37.24 39.232.O7 1.510.43 0.472.21 1.510.00 0.00

20.75 20.3737.19 36.7599.89 99.84

2.525 2.6860.250 0.1840.030 0.0330.192 0.1320.000 0.0001 .983 1.9653.015 3.008

0.843 0.8850.083 0.0610.010 0.01 10.064 0.043

39.92 35.771.55 2.080.69 0.840.91 2j60.03 0.01

20.74 20.6737.28 37.02

101.12 98.55

2.697 2.4480j87 0.2540.047 0.0580.079 0.1890.002 0.0011 .975 1.9943.012 3.030

0.896 0.8300.062 0.0860.016 0.0200.026 0.064

38.43 38.201.98 1 .881 .00 1 .151.88 1 .500.01 0.01

20.90 20.8337.08 37.01

101 .28 100.58Cations.'

FeMgMnCaTiAISi

Mole fractionstalmprpspsgrs

39.051.940.601 . 1 90.04

21.0837.97

101 .87

2.6020.2300.0400.1020.0021.9803.026

0.875o.0770.0130.034

38.151.930.881.870.03

20.7837.45

101 .09

2.5660.2310.0600.1610.0021.9703.012

0.8500.0770.0200.053

2.686 2.6780.210 0.2600.009 0.0100.098 0.0750.002 0.0011.989 1 .9883.006 2.996

0.894 0.8860.070 0.0860.003 0.0030.033 0.025

2.5870.2380.0680.1620.0011.9832.985

0.8470.0780.o220.053

2.5870.2270.0790.1300.0011.9882.997

0.8560.0750.0260.043

77-6 77-42 76-445 78-22A 77-73 77-23 8345 77-46D 84-1632 84-1631

Weight percent'FeO 38.94 37.66MgO 1.86 1.86MnO 1.29 1 .83CaO 1.49 '1.74

Tio, 0.05 0.00Al,o3 20.74 20.95sio, 37.13 37.11Toral 101 .50 101 .15

38.24 36.26't.32 2.023.51 3.540.34 0.880.00 0.03

20.94 21.1636.98 37.52

101 .33 101 .41

2.588 2.4300.159 0.2410.241 0.2400.029 0.0760.000 0.0021.997 1.9982.993 3.006

0.8580.0530.0800.010

0.8140.0810.0800.025

37.00 35.250.95 1.523.80 4.511 .15 1 .090.01 0.01

21.08 20.3736.54 36.59

100.53 99.34

2.523 2.4240.1 15 0 .1860.262 0.3140.100 0.0960.001 0.0012.026 1.9742.979 3.009

0.841 0.8030.038 0.0620.087 0.1040.033 0.032

28.35 28.672.98 3.108.96 9.201 .86 1.660.04 0.00

20.78 20.8037.06 37.37

r 00.03 100.80

1 .914 1.9220.359 0.3700.613 0.6250.161 0.1430.002 0.0001.977 1 .9652.992 2.996

0.6280.1 180.2010.053

0.6280.1210.2040.047

Cations..FeMgMnCat l

AIJ I


39.471.402.34o.420.01

20.9438.43

103.01

2.6160.1650.1570.0360.0011.9563.046

0.8800.0550.0530.012

28.262.948.691.790.01

21.0337.64

100.36

1.8940.3510.5900.1540.0011.9863.016

0.634o.1170.1970.052

2.6220.2230.0880.1290.0031.9682.990

0.8560.0730.0290.042

2.5380.223o.1250.1 500.0001.9892.990

0.8360.0730.0410.049

84-1692 76-437 84-169A 78-2576-400 76-379 77-45 84-1622 84-1621

Weight percent'27.962.919.322.220.07

20.9037.06

Total 100.44

FeOMgoMnOCaOTio,Alro3sio,

1.8810.3490.6350.1910.0041.9822.981

0.6160 . 1 1 40.2080.063

30.201.449.291.470.01

20.7336.3599.49

2.0730.1 760.6460.1290.0012.0052.983

0.6860.0580.2140.043

26.972.859.882.130.00

21.2136.9399.97

1 .8190.3430.6750.1840.0002.0162.978

0.6020.1 14o.2230.061

26.582.92

10.521.910.03

21.O137.99

100.96

1.770o.3470.7100.1630.0021.9723.025

0.5920.1 160.2370.055

26.342.90

11 .042.000.03

21.0938.36

101 .76

1.7400.3410.7390.1690.0021.9643.031

0.5820.1 140.2470.057

25.762.98

11.571.850.02

21.0737.94

1 01 .19

1.7130.353o.7790.1580.0011.9743.017

0.5700 . 1 1 80.2590.053

25.322.83

11.732.180.02

20.8137.91

100.80

1.6910.3370.7930.1860.0011.9583.027

0.5620.1120.2640.062

21.44 21.773.03 3.32

14.03 14.513.29 2.790.05 0.08

21.25 21.0037.98 37.89

1 01 .07 101 .36

1.421 1.4420.358 0.3920.942 0.9740.279 0.2370.003 0.0051.984 1 .9613.009 3.002

0.4740 . 1 1 90.3140.093

0.4740.1290.3200.078

Cations"FeMgMnCaTiAISi


. Weight percont TiO, is below detection in all garnets." Bas€d on 12 O atoms.t alm : cations Fe/cations (Fe + Mg + Mn + Ca).

Fig. l. Generalized geologic map showing exposures of Pro-terozoic rocks in a part of north-central New Mexico. Boxesoutline sampling areas.

thin section, than that from the other two study areas.Although we have eliminated all samples with obviouslyaltered biotite, we suspect that the greater scatter ob-served in these data may reflect greater degrees of re-trogression (enhanced by the higher peak temperature)and minor amounts of alteration. This region, unlike theother two, was not recently glaciated.

Fe3* in biotite. The Fe3*/Fe2* ratio of biotite is expectedto vary with 6r, an expectation supported by Figure 2.Fe3+/(Fe3+ + Fe2*) in biotite, calculated on a cation basis,increases from 0.13 in rocks with .fa;S#f' : 0 to 0.26 inrocks with .lTUd"- : 0.86, with an expected complicationinduced by the ilmenite-hematite phase transition. Figure2 provides a useful way to estimate Fe3* in biotites thathave not been analyzed directly for Fe3*. Fe3+/Fe2+ ratios,determined directly for eight samples from Rio Mora andestimated using Figure 2 for the remaining samples, areincluded in Table 5.

Although Fe3t/(Fe3t + Fe2*) in biotite increases with-XTU$"'(i.e., with/6r), the absolute number of Fe3* cationsin biotite varies little, decreasing slightly as/o, increases.Calculated on the basis of I I O per formula unit, biotitehas 0.20 cations Fe3* at relatively low fo, but only 0.12cations Fe3* in samples with hematite. This is not asanomalous as it might seem, because total Fe in biotitefalls sharply asf, increases. Such behavior is consistentwith wet-chemical data presented in previous studies

889

Xr",o" in Hemoii te- l lmenite

Fig. 2. Fe3*/(Fe3* + Fe'*) in biotite plotted against molefraction of hematite in hematite-ilmenite. Fe3*/Fe'?* ratios arefrom wet-chemical and Mrissbauer analyses of biotite separates.Oxides were analyzed with the electron microprobe (Table 7).All samples are from the Rio Mora map area (Table 5).

(Chinner, 1960; Phinney, 1963 Hounslow and Moore,1967).

For this study, mole fractions for octahedral cations inbiotite were calculated using two techniques. The firstassigned AI cations to all tetrahedral sites not occupiedby Si. The remaining Al and all Fe, Fe3*, Ti, Mg, and Mnwere assumed to be in octahedral sites. The second tech-nique assumed that some Fe3* was tetrahedrally coordi-nated in biotite (as suggested by M. Darby Dyer, personalcommunication). In all samples, Fe3t exceeded l09o oftotal Fe. Therefore, for the purpose of comparison, weassumed that 100/o of the total Fe was tetrahedrally co-ordinated. Vacant tetrahedral sites were then assumed tobe filled with Al and all remaining cations were consid-ered to be in octahedral sites. These two techniques re-sulted in two distinct values for Xfi and Xf5-, and thusmay have an effect on regression analyses that investigatemixing of these phase components.

Ilmenite-hematite

Ilmenite and hematite showed some retrograde or su-pergene alteration. Exsolution lamellae were evident inmost grains, and some ilmenite crystals were altered to amixture of geothite and leucoxene. Altered crystals wereavoided during microprobe analysis.

Several techniques were tested to overcome the prob-lem of exsolution. These included (l) averaging numer-ous broad-beam microprobe analyses from single exsolvedgrains, (2) determining compositions of subdomainswithin exsolved crystals and then, using back-scatter elec-tron imaging, calculating the homogenized oxide com-position, and (3) measuring compositions of oxides foundas small inclusions in other metamorphic minerals. Themost consistent and reproducible oxide analyses were ob-tained from small oxide inclusions (2-30 y.m in diameter)found within matrix grains of qvafiz. These inclusions,

WILLIAMS AND GRAMBLING: EQUILIBRIUM BETWEEN GARNET AND BIOTITE

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f-'l Gornet - biotile[- l som0le oreo

TRUCHAS

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Tmu 2. Microprobe analyses of garnet, Rio Mora, New Mexico

81-23a 80-142 80-46b C81-174 80-187 80-42 80-51 C81-18180-46 80-30 80-2080-43

Weight percent'FeOMgoMnOCaOTio,Al,o3sio,Total '101.27

39.19 40.162.06 2.040.34 0.430.45 0.540.00 0.02

20.96 20.9636.31 37.2799.31 101.42

2.688 2.7010.252 0.2450.024 0.0290.040 0.0470.000 0.0012.026 1.9872.978 2.998

0.895 0.8940.084 0.0810.008 0.0100.013 0.016

39.70 39.632.00 1.550.44 0.470.55 1.750.00 0.02

20.62 20.9537.31 37.30

1 00.62 101 .67

2.688 2.6620.241 0.1860.030 0.0320.048 0.1510.000 0.0011 .968 1.9833.021 2.996

0.894 0.8780.080 0.0610.010 0.0110.016 0.050

39.64 39.872.30 2.200.53 0.540.75 0.650.00 0.00

20,91 20.9237.62 37.47

101 .75 101 .6s

2.651 2.6730.274 0.2630.036 0.0370.064 0.0560.000 0.0001.971 1.9763.009 3.004

0.876 0.8820.091 0.087o.o12 0.0'120.021 0.018

38.01 40.232.32 2.220.55 0.611.32 0.430.04 0.00

21 .03 21.0537.42 37.88

100.69 102.42

2.557 2.6750.278 0.2630.037 0.0410.1 14 0.0370.002 0.0001.993 1 .9733.01 0 3.012

0.856 0.8870.093 0.0870.012 0.0140.038 0.012

39.69 37.39 36.881.72 2.23 3.060.85 0.86 1.230.57 1.58 1.540.01 0.00 0.00

20.73 20.80 20.9037.18 37 .41 36.05

100.75 100.27 99.66

2.689 2.525 2.5150.208 0.268 0.3720.058 0.059 0.0850.049 0.137 0.1350.001 0.000 0.0001 .980 1.980 2.0093.012 3.021 2.940

0.895 0.845 0.8090.069 0.090 0.'t200.019 0.020 0.0270.016 0.046 0.043

39.102.510.231.03o.02

20.8537.53

Cations"FeMgMnCaTiAI5 l


2.6210.3000.0160.0880.0011.9703.009

0.8660.0990.0050.029

80-178a 8'l-225 8018a 80-192-2 81-234 C81-148 C81-105 C81-213 81-13a C81-310 80-178 81-145

Weight percent-FeOM9oMnOCaOTio,Al,o3sio,

37.1'1 31 .642.36 5.701.62 1 .731.07 2.420.00 0.00

20.95 21.2137.80 37.49

100.91 100.19

2.488 2.0960.282 0.6730 . 1 1 0 0 . 1 1 60.092 0.2050.000 0.0001.979 1 .9803.030 2.970

0.837 0.6780.095 0.2180.037 0.0380.031 0.066

36.06 35.55 30.722.30 2.33 4.591.67 1.87 3.002.05 2.59 2j40.01 0.00 0.00

20.93 20.90 20.8737.39 37.54 38.23

100.41 100.78 99.55

2.429 2.385 2.0450.276 0.279 0.5450.114 0.127 0.2020.177 0.223 0.1830.001 0.000 0.0001 .987 1.976 1.9583.01 1 3.011 3.044

0.811 0.791 0.6870.092 0.093 0.1830.038 0.042 0.0680.059 0.074 0.062

34.82 32.652.32 1.874.06 5.401.81 2.930.02 0.00

20.76 20.3636.38 36.86

100.17 100.07

2.370 2.2210.281 0.2270.280 0.3720.158 0.2550.001 0.0001 .991 1.9522.961 2.998

0.767 0.7220.091 0.0740.091 0.1210.051 0.083

27.41 25.253.29 2.76

10.23 10.521.67 3.640.00 0.04

20.97 20.6337.48 37.19

101 .05 1 00.03

1 .830 1.7010.391 0.3310.692 0.7180.143 0.3140.000 0.0021.973 1.9592.992 2.996

0.599 0.5550.128 0.1080.226 0.234o.o47 0.102

26.57 26.49 20.153.35 2.96 5.48

10.68 10.58 12.431.44 ',t.73 3.400.03 0.00 0.03

21.05 21.00 21.0737.62 37.24 38.55

100.74 100.00 101 .1 1

1.774 1.784 ',t.317

0.399 0.35s 0.6380.722 0.722 0.8230.123 0.149 0.2850.002 0.000 0.0021 .981 1 .994 1.9403.004 2.999 3.012

0.588 0.593 0.4300.132 0.1 18 0.2080.239 0.240 0.2690.041 0.050 0.093

TotalCations..

FeMgMn

TiAIJ I

Mole fractionsfalmprpspsgrs

81-70a 80-218 82-62

High-Mn garnets, excluded from regression analysis

C81-244 82-12 84Jt47e 84-147d 81-221 81-175a 81-139 81139a

Weight percent'FeOMgoMnOCaOTio,Al,o3sio,Total

Cations--FeMgMnCaTiAIsi

Mole fractionsfalmprpspsgrs

25.603.31

12.261 . 1 40.03

20.9737.63

100.94

1.7090.3940.8290.0980.0021.9733.004

0.5640.1300.2740.032

1 . 1 8 10.4941.2480.1 390.0051.9363.011

1.1120.5621.2520.1240.0021.9692.997

22.46 17.173.66 4.22

14.38 '17.3'l1.70 2.890.08 0.03

20.89 21.0937.88 38.07

101 .05 100.78

1.492 1 .1350.433 0.4970.968 1 .1580.145 0.2450.005 0.0021.956 1 .9643.009 3.008

0.491 0.3740.143 0.1640.319 0.3820.048 0.081

17.58 16.834.13 4.77

18.35 18.721 .61 1 .470.09 0.03

20.45 21.1537.49 37.9499.70 100.91

9.98 5.4s4.06 1.96

22.94 28.793.34 5.670.04 0.04

2',t.12 20.2638.49 37.0699.97 99.23

0.660 0.3700.478 0.2371.536 1 .9780.283 0.4930.002 0.0021.968 1 .9373.043 3.006

0.223 0.1200.162 0.0770.519 0.6430.096 0.160

0.66 0.554.11 4 .14

33.68 34.062.49 2.430.08 0.02

20.76 21.1036.93 37.5498.71 99.84

0.108 0.045 0.0370.373 0.495 0.4922.'t91 2.304 2.3000.364 0.216 0.2080.003 0.00s 0.0011 .959 1.976 1.9833.010 2.983 2.993

0.036 0.015 0.0120.123 0.162 0.1620.722 0.753 0.7570.120 0.071 0.068

0.386 0.3650.161 0.1840.408 0.4100.045 0.041

9.514.42

22.974.000.04

21.O837.7199.73

0.6320.5231.5460.3410.0021.9742.996

0.2080.1720.508o.112

1.613.12

32.224.230.05

20.7037.4999.42

'Weight percent TiO2 is below detection in all garnets.-t Based on 12 O atoms.f alm : cations Felcations (Fe + Mg + Mn + Ca).

WILLIAMS AND GRAMBLING: EQUILIBRIUM BETWEEN GARNET AND BIOTITE 8 9 1

which typically lacked exsolution textures, were inter-preted to have been trapped as quartz annealed near thepeak of metamorphism and were accepted as oxide com-positions at peak metamorphic conditions. The armoringquartz may have protected the inclusions from subse-quent retrograde or deuteric alteration. Typically, at leastsome larger matrix oxide grains have compositions iden-tical to the small inclusions. Table 7 presents analyses ofpeak oxide minerals.

TrrpruronyNAMrc ExpR.ESSroNS

The systematics of Fe-Mg partitioning between garnetand biotite have been presented by Ganguly and Ken-nedy (1974), and Ganguly and Saxena (1984), and Gan-guly and Saxena (l 987). The Fe-Mg exchange reaction canbe expressed as

%FerAlrSirO,, * YrKMg.AlSirO,o(OH), :almmdine pNogopite

%MgrAlrSirO,, * %KFerAlSi3Oro(OH)r. (l)pyrope mnite

At equilibrium

K : (Xo*/X^^) ,

(Yo./'Yn) ()\--eq (XM,/ XF.)B' (7""/7..)t,

where the first term (K*) is the equilibrium constant forEquation 1, the second, compositional, term (K") is thedistribution coefficient, and the third term (K,) treatsnonideal mixing.

Following Ganguly and Saxena (1984, 1987), we as-sume that mixing in garnet and biotite can be approxi-mated by a simple mixture model (Guggenheim, 1967)within the compositional range of our isothermal-isobar-ic suites of samples, and we ignore multicomponent in-teractions. With these assumptions, Equation 2 can berewritten as

-RZln K": -RZln K.o+ lwMEF"(x^* _ Xr*) + awc^(x#)

+ azM"(x"oJlc"- lw*"o"(Xr, - xr*) - awM"(xM")

- Lwri(xr) - AW^|(X^)- AWF&-(XF?3,)IB' (3)

where W,, represents a binary simple mixture interactionparameter and AW,: (W,*" - W,o.).All biotite mixingterms refer exclusively to cations in octahedral sites. TheW,., terms can be modeled as constant over a limitedcompositional range (symmetric mixing) or as a variablefunction of composition (asymmetric mixing). Equation3 assumes that, along with Fe and Mg, the octahedralsubstituents in biotite are Mn, Fe3*, Al, and Ti and theeightfold substituents in garnet are Mn and Ca.

Constraints on garnet and biotite mixing parameters

Garnet. Most early investigators assumed that Mg-Femixing in garnet is nearly ideal, and Ferry and Spear

(1978) found no evidence in their experiments for non-ideality. However, based on an analysis of experimentaland natural data, Ganguly and Saxena (1984) suggestedthat Mg-Fe mixing is significantly nonideal and that non-ideality is asymmetric, reaching a maximum in Fe-richgarnets. This suggestion has been supported by experi-mental data (Geiger et al., 1987). Ganguly and Saxena(1984) proposed a symmetric approximation for asym-metric Mg-Fe mixing where

WffiF.: Wo.MeIX,*/(Xr* + X,'-)]+ WME-F"lXut^/(Xr* + X^r^)l (4)

with W-"-o.: 10460 J/mol and Wr".-":835 J/mol (one-site basis). Recently, Hackler and Wood (1989) conclud-ed that this model is incompatible with experimental par-titioning data, especially at high X"r-. Using the new andexisting experimental data, they concluded that Mg-Femixing behavior is much more nearly ideal, W-"-r": 2100J/mol and Wu.-*": 700 J/mol.

Most workers have agreed that Ca does not mix ideallyin magnesium-iron garnet (cf. Hodges and Spear, 1982;Ganguly and Saxena, 1984; Hodges and Crowley, 1985;Anovitz and Essene, 1987; Geiger et al., 1987). Gangulyand Saxena suggest a value of 12550 J/mol for AW."forXr." below 0.10, a value supported by the experiments ofGeiger et al. (1987).

Ganguly and Kennedy (1974) suggest that Mn-Fe mix-ing is nearly ideal in garnet, but Mn-Mg mixing may besignificantly nonideal (Ganguly and Kennedy, 1974;Ganguly and Saxena, 1984). Using partitioning of Mnbetween garnet and ilmenite, O'Neill et al. (1989) con-cluded thal. W*^u. is approximately 1800 J/mol, but untilpresent, W-^-" has been largely unconstrained. Manyworkers have observed a negative correlation betweenX.o" and KB-c" (Ganguly and Kennedy,1974 Dahl, 1980;Newton and Haselton, 1981; St. Onge, 1984; and thisstudy), and using regression analyses of natural data,Ganguly and Saxena (1984) proposed that AW-^ is ap-proximately 12550 J/mol. However, as discussed below,the negative correlation may result from multicolinearityin the compositional data and thus may not directly con-strain Mn mixing behavior.

Biotite. Following Mueller (1972), most workers haveassumed that Fe and Mg mix ideally in biotite. There arefew data to contradict this assumption, although Gangulyand Saxena (1984, p. 92) allowed that Fe-Mg mixing mayshow some positive departure from ideality. NonidealMn mixing is unlikely to be important in biotite becauseof its vanishingly small amounts of Mn, even where bio-tite coexists with spessartitic garnet.

Several other components may mix nonideally in mag-nesium-iron biotite. Dallmeyer (1974) and Indares andMartignole (1985) have shown that Ko correlates posi-tively with Fe3*, Al, and Ti in biotite. The effects of non-ideal mixing of these components are probably most im-portant in the upper amphibolite to granulite facies, whereAl and Ti become maior substituents. As noted above,

br"o o o o f

I


0.8

0 . 6

o

t nz -x - "

o.2

0.0

Fig. 3. X.o. plotted against (X.,- - Xo*) for garnets from RioMora (squares), Pecos Baldy (circles), and Cerro Colorado (tri-angles), New Mexico (slope : -0.70, "r-intercept: 0.77, and 12: 0.e4).

the mole fraction of the aluminum biotite component isrelativley constant in biotite from New Mexico.

RncnnssroN ANALYSIS

Multicolinearity of compositional variables

For an isothermal-isobaric set of garnet-biotite pairs,such as any one ofthe three sets presented here, statisticalregression of ln K" against the compositional variables(X,) in Equation 3 should constrain the values of the bi-nary interaction parameters (Ganguly and Saxena, 1987).However, the results of the regression analysis may becompromised when two or more compositional variablesare highly correlated, meaning that at least one variablecan be predicted from the others (Devore, 1982; Belsley

et al.. 1980). Correlation coefficients for the variables inEquation 3, based upon analyses of the New Mexico rocks,are shown in Table 8. Strong correlations exist amongmany of the variables. Most irnportantly for Mn-bearinggarnets, X"o" is correlated to some degree with most othercompositional variables, and it is strongly correlated with(X.,- - Xo-) (correlation coefficient >0.9). This last cor-relation is extremely important because X"o" and (X.'- -

X,*) are probably the two most important nonideallymixing components in Equation 3. These correlationshave an important efect on the magnitude of the regres-sion coefficients (interaction parameters), and, therefore,on any conclusions about garnet mixing behavior and thegarnet-biotite Fe-Mg exchange equilibrium. One addi-tional correlation, between X.o.andfo, must also be con-sidered. Several of the strongest linear relationships arediscussed individually in the following section.

X"- vs. (X",. - X"."). A negative linear correlation be-tween X"o" and (X",. - Xo,) characterizes all 85 analyzedsamples (Fig. 3). The correlation coefficient for a linearfit to all data is high (r'z:0.91). Separate linear regres-sions of specimens from each individual study area yieldslightly different slopes, -0.64, -0.74, and -0.70 forPecos Baldy, Rio Mora, and Cerro Colorado, respective-ly. These slopes are statistically distinct at the 980/o con-fldence level.

This correlation is a consequence ofphase equilibriumas shown on an MgO-FeO-MnO projection in Figure 4.For amphibolite-facies samples, all garnets coexisting withstaurolite, aluminum silicate, or chlorite must lie along alinear trend nearly parallel to the FeO-MnO side of thetriangle. Because lines of constant (X"'- - Xo.o) are per-pendicular to the MgO-FeO side, Mn and (Fe-Mg) willtend to be negatively correlated. Further, a negative X.o"vs. (X",. - Xo*) correlation would be expected to char-

TABLE 3, Microprobe analyses of garnet, Cerro Colorado, New Mexico

w83-302 w84-117 w84-130 w83-300 w84-145

Weight percenttFeOMgoMnOCaOAlro3sio,Total

Cations-.FEMgMnCaAISi

Mole fractionsfatmprpspsgrs

35.504.240.581.75

20.6936.7299.48

2.4000.5110.0400.1521.9712.969

0.7730.1 650.0130.049

36.001.613.381.78

20.7837.2'l

100.76

2.4330.1940.2310.1 541.9803.008

28.394.244.643.19

20.6437.4098.50

1.9150.5100.3170.2761.9633.017

0.6350.1690.1050.091

32.641.665.452.33

20.6936.9699.73

2.2220.2010.3760.2031.9853 009

o.7400.067o.1250.068

29.992.635.803.67

20.9837.60

1 00.67

2.0040.3130.3930.3141.9763.004

0.6630.1040.1300.104

30.232.307.292.22

20.7337.33

100.10

2.042o.2770.4990.1 921.9733.015

27.073.087.713.58

21.O737.2399.74

1 .8190.3690.5250.3081.9952.991

0.6020.122o.1740.1 02

29.472 . 1 89.',t71 .53

20.7937.38

100.52

1.9870.2620.6260.1 321.9763.014

0.8080.064o.o770.051

0,6780.0920.1660.064

0.6610.0870.2080.044

'Weight percent TiO, is below detection in all garnets.*'Based on 12 O atoms.f alm : cations Felcations (Fe + Mg + Mn + Ca).


FeO Ch lo r i t e FeO

Fig. 4. (A) Generalized FeO-MgO-MnO projection from quartz, muscovite, HrO, and biotite (modified after Spear, 1988). (B)FeO-MgO-MnO plot for samples from Rio Mora (squares), Pecos Baldy (circles), and Cerro Colorado (triangles), New Mexico.Following Spear (1988), molar FeO-MgO-MnO ratios of the garnets have been plotted. Note that lines of constant (FeO-MgO) areperpendicular to the FeO-MgO side of the triangle. Thus, for staurolite-, chlorite-, or aluminum-silicate-bearing samples, a linearrelationship between X"o" and (X^^ - Xo*) is predicted.

Mgo

aateize pelitic garnet-bearing assemblages elsewhere.Figure 5 illustrates this based on the garnet- and staurol-ite-zone samples analyzed, by Ferry (1980). Similar cor-relations can be seen in most other compilations ofgarnetanalyses (cf. Phinney, 1963; Hodges and Spear, 1982;Hudson, 1985). A consequence ofthis correlation is thatall garnet specimens reported in Tables l-3, that have(X"o, + X,.") less than 0.04, have nearly identical X",-, 0.89+ 0.02, regardless of the specific assemblage in which

Ttete 3.-Continued

they occur. The same X'- is derived by projecting Ferry's(1980) garnet analyses (Fig. 5) to X.o" : 0. This helps toexplain the widespread success of the Ferry and Spear(1978) garnet-biotite calibration applied to garnets low inMn and Ca: their experiments were calTied out on garnetof virtually the same X",- as those most common in na-ture.

A similar negative linear correlation couples X"o" to (XF"- Xr")"' or (Xo./X*")Bt. This is required by chemical equi-

Sfourol i te

w85-195 w84-58 w84-153 w85-253 w84-75 w83-242 w84-133 w85-254

28.212.379272.2s

2't .4537.85

101 .40

1.8750.2810.6240.1922.0093.008

0.6310.0950.2100.065

27.032.63

1 0 . 1 82.O5

20.75J / . C b

100.20

1 .8190.3150.694o.1771.9683.022

0.6050.1050.2310.0s9

25.513.23

10.801.83

21.0637.77

100.20

1.7060.3850.7320.1 571.9853.021

U . J I Z

0.1290.2460.053

26.4'l2.50

10.741.88

20.9637.3499.83

1.7830.3010.7340.1 631.9943.014

0.5980.101o.2460.055

23.493.69

11.352.37

20.8638.0899.84

1 5680.439o.7670.2031.9623.039

0.5270.1470.2580.068

25.743.11

1 1 .901.53

20.8337 18

100.29

1.7330.3730.811o 1321.9762.992

0.5680.1220.2660.043

25.032.74

11.971.90

20.5738.00

1 00.21

1.6800.3280.8140.1631.9453.049

0.5630 . 1 1 00.2730.055

24.622.87

13.041.35

20.9536.8899.71

1.6670.3460.8940.1 171.9992.987

0.5510 . 1 1 40.2960.039

17.273.68

19.581.23

20.9037.76

100.42

1 1530.4381.3240.1051.9663.014

0.3820.1450.4380.035

TABLE 4. Microprobe analyses of biotite, Pecos Baldy, New Mexico

77-858 76-420 77-125 77-15 77-81E 77-103 76-4008 77-41 76-446 77-39

Weight percent'FeOMgoMnOTio,Alro3sio,KrONaroCaOTotalYo Fe3*

Cations".Fer*Fe3*MgMnTiAIsiKNaCa

23.706.700.001.59

20.4334.408.740.320.001

95.880.12

1.3310.1810.7620.0000.0911.8372.6240.851o.0470.000

21.557.810.011.57

20.3134.498.910.250.060

94.960.13

1.1970.1790.8890.0010.0901.8282.6330.8680.0370.005

20.638.500.001.50

20.4435.468.540.320.040

95.430.14

1 . 1 1 60.1 820.9530.0000.085't.8122.6670.819o.o470.003

24.686.300.001.56

19.9433.418.180.160.030

94.26o.12

1.4160.1 930.7320.0000.0911.8322.6050.8140.0240.003

22.757.070.001.58

20.1934.168.620.310.040

94.720.13

1.2730.1900.8100.0000.0911.8302.6270.8460.0460.003

23.226.210.001.48

20.5634.148.700.210.020

94.54o.12

1 .3190.1800.7150.0000.0861.8712.6360.8570.0310.002

19.888.760.001.43

19.9533.978.640.330.000

92.960 1 5

1.0940.1 931 . 0 1 10.0000.0831.8202.6290.8530.0500.000

21.388.300.011.37

20.1 935.448.340.270.020

95.320.13

'1.175

0.1760.9340.0010.0781.7972.6760.8030.0400.002

1.2160.'t820.9060.0000.0831.8312.6280.8200.0520.001

22.26 21.898.50 7.960.00 0.001.46 1.45

19.97 20.3435.24 34.428.69 8.420.33 0.350.030 0.007

96.48 94.840.14 0 .13

1.2020.1960.9520.0000.0821.7682.6470.8330.0480.002

77-6 7742 76-445 77-73 77-23 83-4578-224 77-46D 84-1632 84-1631

Weight percent-FeOMgoMnOTio,Al,o3sio,K"oNaroCaOTotalo/o Fe3+

Cations..Fe2*Fe3*MgMnTiAISiKNaCa

23.117.100.001.58

20.3234.159.270.190.040

95.760.13

1.2860.1920.8090.0000.0911.8312.6110.9040.0280.003

22.867.950.011.45

20.6534.928.550.330.025

96.750.13

1.2490.1860.8890.0010.0821.8272.6210.8190.0480.002

25.096.340.011.45

21.3434.068.910.110.000

97.310.13

1.3790.206o,7140.0010.0821.9002.5730.8590.0160.000

24.855.940.041.48

20.5533.918.840.180.016

95.800.12

1.4060.1910.6800.0030.0861.8622.6060.8670.o270.001

21.888.860.001.63

20.1334.957.820.170.060

95.500.14

1 .1860.1930.99s0.0000.0921.7882.634o.7520.0250.005

26.665.120.061.69

20.4933.897.380.180.050

95.520.16

1.438o.2730.5860.0040.0981.8552.6030.723o.o270.004

22.467.420.051.60

20.0934.948.800 . 1 00.003

95.460.14

1.2280.2000.8410.0030.0911.8002.6560.8530.0150.000

16.2912.150.101.37

19.4736.219.050.430.015

95.090.19

0.8190.1921.3440.0060.0761.7032.6870.8570.0620.001

16.68 16.6012.62 12.730.15 0 .141.34 1 .34

19.45 19.6735.91 36.209.19 9.280.49 0.470.000 0.000

95.83 96.430.21 0.21

0.8140.2161.3900.0090.0741.6932.6530.8660.0700.000

0.8040.2141.3920.0090.0741.7002.6550.8680.0670.000

84-1 692 76-437 84-1694 76-400 78-25 76-379 7745 84-1622 84-1621

Wdght percent.FeOMgoMnOTio,Alro3sio,K.ONaroCaOTotalo/" Fe3*

Cations".Fe-Fe,3-MgMnTiAISiKNaCa

16.2912.790.141.32

19.5935.959.220.400.005

95.71o.22

0.784o.2211.4060.0090.0731.7032.6520.8680.0570.000

21.597.910 .161 . 8 1

19 .1334.288.760 .180.00s

93.830.12

1.2310.1680.9140.0110.1 051.7472.6570.8660.o270.000

16.5512.600.121.36

1 9 . 1 336.879.070.440.002

96.14o.22

0.791o.2231.3770.0070.0751.6532.7030.8480.0630.000

15.5912.600 .151 Q F

19.4136.328.910.450.002

94.780.20

o.7730.1931.3930.0090.0751.6962.6930.8430.0650.000

15.6512.230.181.41

19.4337.068.650.520.000

95.130.22

0.7510.2111.3410.0110.0781.6852.7270.812o.o740.000

14.5813.480.171.36

19.2136.949.060.450.000

95.250.20

0.7160.1791.4760.0110.0751.6632.7140.8490.0640.000

15.1612.420.131.42

19.2236.109.090.430.009

93.970.22

0.7390.2081.3830.0080.0801.6932.6970.8660.0620.001

12.72 12.7514.91 15.640.15 0.201.19 1 .04

18.80 18.4737.44 37.399.10 9 .150.35 0.300.000 0.000

94.66 94.940.25 0.25

0.584 0.5840.194 0.1951.626 1.7030.009 0.0120.065 0.0571.621 1 .5902.739 2.7310.849 0.8530.050 0.0420.000 0.000

' o/o Fe3+ : Fe3+/(Fe3+ + FeF*), estimated based on composition of coexisting oxides and Figure 3.'" Based on 11 O atoms.

Tlale 5. Microprobe analyses of biotite, Rio Mora, New Mexico

Sample 81-23a 80-142- 80-46- 80-46b C81-174 80-30- 80-43 80-187 80-42 80-20 80-51 C81-181

Weight percent"FeOMgoMnOTio,Al,o3sio,K.oNaroCaO

21.66 23.128.24 6.250.00 0.001.62 2.08

20.54 20.3734.99 33.878.66 9.080.42 0.360.02 0.00

96.1 5 95.130.14 0 .14

1 . 1 7 1 . 2 80.19 0.210.92 0.720.00 0.000.09 0.121.82 1 .852.63 2.610.83 0.890.06 0.050.00 0.00

23.27 23.066.89 6.660.00 0.011.91 2.00

20.29 20.4034.25 34.438.96 8.690.34 0.310.00 0.00

95.91 95.560.13 0 .12

22.77 23.757.29 6.850.01 0.002.25 1.98

20.37 20.3533.43 34.318.99 9.030.23 0.300.00 0.00

95.34 96.570.13 0 .12

1.27 1 .330.19 0 .180.83 0.780.00 0.000 . 1 3 0 . 1 11.84 1.822.57 2.610.88 0.880.03 0.040.00 0.00

2034 23.308.30 6.710.00 0.001.75 1 .84

20.34 20.4735.02 34.468.93 9.13o.29 0.150.00 0.00

95.57 96.060.13 0 .12

1 . 1 5 1 . 3 10.17 0 .180.93 0.760.00 0.000.10 0 .111 .81 1.842.65 2.620.86 0.890.04 0.020.00 0.00

24.92 21.47 20.785.63 8.05 9.710.00 0.00 0.002.10 1.76 1 .55

20.14 20.17 19.2933.86 34.66 35.028.91 9.06 9.060.42 0.36 0.170.01 0.00 0.00

95.99 95.53 95.580.13 0 .15 0 .15

1.39 1 .16 1 .12o.21 0.20 0.200.65 0.91 1.090.00 0.00 0.000 j2 0 .10 0 .091.82 1.80 1.722.60 2.63 2.650.87 0.88 0.870.06 0.05 0.030.00 0.00 0.00

25.056.130.002.03

19.6534.169.220.210.01

96.46o . 1 2

Totalo/" Fe3*

Cations.'Fe2*Fe3*MgMnTiAISiKNaCa

1 .29 1 .30 1.410.19 0 .18 0 .190.78 0.76 0.700.00 0.00 0.000.11 0 j2 0 .121 .82 1 .84 1.782.61 2.63 2.620.87 0.85 0.900.05 0.05 0.030.00 0.00 0.00

Sample 80-178a 81-225 80-18a 80-192-2 81-234 C81-148' C81-105 81-213 81-13a C81-310 80-178 81-145

Weight percent'FeOMgoMnOTio,Al,o3sio,K.oNaroCaO

21.06 14.278.81 15.580.01 0.011.71 1 .30

20.04 18.4434.95 37.528.80 8.360.34 0.450.01 0.01

95.73 95.940.15 0 .16

1 .13 0.730.20 0.140.99 1.690.00 0.000.10 0.071.78 1.582.64 2.720.85 0.770.05 0.060.00 0.00

20.03 20.04 15.539.09 9.06 12.960.01 0.04 0.051.58 1 .64 1 .39

19.98 20.2't 18.5934.95 35.68 36.928.89 8.87 9.030.35 0.30 0.440.01 0.00 0.01

94.89 95.84 94.92o j4 0 .15 0 .16

1 .09 1.07 0.810.18 0 .19 0 .151 .03 1.01 1 .430.00 0.00 0.000.09 0.09 0.081.79 1.78 ' t .622.65 2.67 2.740.86 0.85 0.850.05 0.04 0.060.00 0.00 0.00

22.09 21.998.15 8.250.05 0.141.63 2.41

20.04 19.1834.56 35.019.26 9.230.25 0.180.00 0.01

96.03 96.400.17 0 .15

1 . 1 6 1 . 1 80.24 0.210.92 0.930.00 0.010.09 0.141.79 1 .712.62 2.640.89 0.890.04 0.030.00 0.00

1 6.20 16.0612.91 13.420.13 0 .141.69 1.72

19.34 18.2335.87 36.649.21 9.280.30 0.240.00 0.01

95.65 95.740.21 0.20

0.79 0.790.21 0.201.42 1 .480.01 0.010.09 0.101.68 1 .592.65 2.700.87 0.870.04 0.030.00 0.00

1 6.32 1 6.45 10.9812.51 11 .37 18.350.17 0.15 0.221.73 1.58 1.29

19.57 19.15 17.7335.77 35.87 37.809.15 9.57 8.510.34 0.25 0.370.01 0.00 0.03

95.57 94.39 95.280.20 0.19 0.23

0.81 0.84 0.510.20 0.20 0.151 .38 1.27 1.970.01 0.01 0.010.10 0.09 0.071.71 1 .70 1 .512.65 2.69 2.730.86 0.92 0.780.05 0.04 0.050.00 0.00 0.00

Totalo/o Fe3+

Cations-'Fe2*Fe3'MgMnTiAISiKNaCa

Sample 81-70a' 80-218" 82-62 C81-244 82-12 U-147d 84-147e. 81-221' 81-175a 81-139 81-139a

Weight percent'FeOMgo

15.0812.740.171.54

19.3935.939.270.340.00

94.460.20

0.750.191.410.010.091.702.680.880.050.00

13.59 9.9414.42 16.020.23 0.291.09 0.60

19.72 17.7036.73 36.609.03 9.160.33 0.310.01 0.05

95.15 90.670.19 0.25

0.67 0.470.16 0 .161.57 1 .810.01 0.020.06 0.031.70 1 .582.69 2.780.84 0.890.05 0.050.00 0.00

9.73 9.26 7.2717 .78 1 7.80 18.540.28 0.32 0.510.61 0.72 1.01

19.13 19.89 17.6437.70 37.93 38.868.92 8.86 8.880.33 0.34 0.220.03 0.05 0.00

94.51 95.17 92.93o.24 0.26 0.25

0.45 0.41 0.330.14 0 .14 0 .111.92 1 .90 2.010.02 0.02 0.030.03 0.04 0.061.63 1 .68 1.512.73 2.72 2.830.82 0.81 0.820.05 0.05 0.030.00 0.00 0.00

6.78 8.4820.61 17.060.52 1.210.96 1.95

17.27 17.4838.97 37.839.25 10.220.19 0.050.01 0.01

94.56 94.290.25 0.29

0.31 0.370.10 0 .152.20 1.860.03 0.080.05 0.111.46 1 .502.79 2.760.85 0.950.03 0.010.00 0.00

2.22 1.52 1 .4322.96 22.62 22.810.81 0.76 0.770.88 0.60 0.56

18.87 18.32 18.4637.95 39.20 39.268.21 9.31 9.480.22 0.30 0.320.00 0.00 0.00

92.12 92.63 93.090.33 0.35 0.35

0.09 0.06 0.060.04 0.03 0.032.46 2.41 2.420.05 0.05 0.050.05 0.03 0.031.60 1 .55 1 .552.73 2.80 2.800.75 0.85 0.860.03 0.04 0.040.00 0.00 0.00

MnOTio,Alr03sio,KrONaroCaOTotalVo Fe3*

Cations.'Fe2*Fe3*MgMnTiAISiKNACa

* o/o Fe3+ : Fe,3r(Fep. + Fe'?*), estimated based on composition of coexisting oxides and Figure 3. Ratios for starred samples (') were determineddirectly by M6ssbauer spectroscopy and wet-chemical analysis (see text).

'" Based on 11 O atoms.


TABLE 6. Microprobe analyses of biotite, Cerro Colorado, New Mexico

Sample w83302 w84-35 w84-86 w84-94 w84-117 W84-130 W83-300 W84-145

Weight percent-FeOMgoMnOTio,Al203sio,KrONaroCaOTotalVo Fe3*

Cations--te.'Fe3*MgMnTiAIJ I

KNaCa

21.00 20.098.86 9.260.07 0.121.55 1 89

18.84 19.2334.76 35.169.36 9.'t7o.24 0.160.04 0.01

94.72 95.090.15 0 .16

1.144 1 .0690.202 0.203't.o12 1.0450.00s 0.0080 089 0.1081.702 1 .7172.664 2.6630.915 0.8860.036 0.0230.003 0.001

19.2110.430.021.56

18.9935.259.250.230.04

94.980 . 1 6

1.0210.1941.1750.0010.0891.6922.6650.8920.0340.003

22.728.530.101.61

19.5434.759.380.310.01

96.950.15

1 . 2 1 70.2140.9580.0060.0911.7352.6190.9020.0450.001

13.7514.340.091.20

18.3636.448.860.280.01

93.330.19

0.6970.1 631.5980.0060.0671 .6182.7250.8450.0410.001

18.6810.52o.171.70

19.2835.719.400.290.02

95.770.16

0.9820.1871.1730.0110.0961.7002.6710.897o.0420.002

17.8710.37o.171.87

19.4035.109.290.280.03

94.380.'17

0.9390.1921.1700.0110.1061.7302.6560.8970.0410.002

18.1610.720.131.34

19.0235.579.370.310.01

94.630.16

0.9640.1841.2080.0080.0761.6942.6880.9030.04s0.001

' "/o Fe3* : FeC-/(Fe* + Fe'*), estimated based on composition of coexisting oxides and Figure 3.

.. Based on 11 O atoms.

librium between garnet and biotite, given the relationshipshown in Figure 3.X"^ vs. Xlf. A strong negative covariance exists betweenX,o. and -$j (conelation coefrcient >-0.6). This is notsurprising considering the correlations discussed above

TABLE 7. Selected ilmenite-hematite analyses and assemblages'

and well-documented positive correlation between Ti andFel(Mg + Fe) in biotite (Guidotti, 1984). This relation-ship suggests that it may be difficult to separate the effectsof Mn mixing in garnet and those of Ti mixing in biotite.

X"* "s./o"

Chinner (1960) and Guidotti (1984) noted

Pecos Baldy, New Mexico77-858 76-420 77-81E 77-4'1 78-224 77-73 77-23 83-45 7745 77-46D 76-437 77-45 84-1622 84-162

X(Fe,Oi) 0.00X(FeTiO3) 1.00X(MnTiOs) 0.00RutileMagnetite

Rio Mora, New Mexico8046

0.04 0.030.93 0.970.07 0.00

X X

80-30 80-187

0.o20.960.02

X

0.020.970.01

X

80-46b

0.00 0.06 0.00 0.67 0.180.98 0.86 0.98 0.32 0.790.o2 0.13 0.02 0.01 0.03

_1

0.20 0.72 0.670.77 0.27 0.33o.o2 0.01 0.01

1_81-213 C81-310 80-218

0.000.880. ' t2

80-20 81-225 81-225 80-1922 81-234s c81-148 C81-105

X(FerO3)X(FeTiO3)X(MnTiOs)RutileMagnetite

Rio Mora, New Mexico (continued)82-62 C81-244 82-12 84-147e 84-147d 81-221 81-175a 81-139 81-139a

0.016 0.009 0.000 0.0000.983 0.986 0.960 0.9330.001 0.005 0.040 0.017

X X

0.028 0.109 0.7320.967 0.883 0.2680.005 0.008 0.000

X -X X

0.000 0.123 0.0730.969 0.859 0.9150.031 0.018 0.012

X -X X

0.636 0.422 0.7160.352 0.567 0.2800.012 0.011 0.004

X X

0.0670.8630.071

X

X(FerO3)X(FeTi03)X(MnTiOs)RutileMagnetite

0.850 0.832 0.8440.146 0.165 0.1460.003 0.003 0.010

X -

0.8020.1910.007

X

0.7960.1980.006

0.872 0.959 0.978 0.9790.122 0.035 0.003 0.0090.006 0.006 0.019 0.012

x -x -

Cerro Colorado, New Mexicow83- W84- W84- W84-302 86 86 94b

w84- W8+ W84- W84- W84-129 130 130 145 145

w84-7 ^

X(FerO3)X(FeTi03)X(MnTiO3)RutileMagnetite

0.06 0.65 0.110.93 0.34 0.870.01 0.01 0.02

XX X

0.09 0.110.89 0.870.o2 0.01

X X

o.77 0.10 0.750.23 0.89 0.250.01 0.01 0.00

X X X

0.680.310.01

x

o.140.840.01

X

Note.'X indicates the presence of rutile or magnetite in the assemblage.* Mole fractions calculated according to the method of Rumble (1973).


f ABLE 6.-Continued

897

w85-195 W84-58 w84-153 w84-87 w85-253 w84-75 w83-242 w84-133 w85-254

17.5211.370.141.42

19.5935.159.220.240.01

94.660.17

0.9150.1871.2760.0090.0801.7382.6450.8850.0350.001

17.2011.270.181.44

19.7035.669.310.260.00

95.020.17

0.8930.1831.2560.0110.0811.7372.6670.8880.0380.000

14.7012.960.191.37

19.3036.068.940.330.03

93.880.20

0.7340.183't.442

0.0120.o771.6982.6920.8510.0480.002

16.631 1 . 6 10.191.79

19.3035.769.56o.230.00

95.07o.17

0.862o.1771.2930.0120.1011.7002.6720.9110.0330.000

14.3812.860 .171.42

19.4936.649.3s0.300.01

94.620.20

0.7120.'1781.4180.0110.0791.7002.7110.8830.0430.001

13.7513.710.151.27

19.0836.279.200.310.01

93.75o.20

0.6860.' t711.5230.0090.0711.6762.7040.8750.0450.001

15.7312.09o.221.31

19.7535.769.360.260.01

94.49o.20

0.7860.1961.3460.0140.0741.7382.6700.8920.0380.001

13.7514.010. ' t7't.34

19.2336.289.450.340.01

94.580.20

0.6810.1 701.5460.0110.0751.6782.6850.8920.0490.001

10.2216.440.311.O2

18.9937.009 . 1 10.360.02

93.470.23

o.4840.1441.8020.0190.0561.6462.7210.8550.0510.002

that the manganese content of gamet increases with Or.The samples from New Mexico show the same behavior,as illustrated by the correlation between Mn in garnet andX."ro, in ilmenite-hematite solid solution (Fig. 6). Thiscorrelation is also a result of partitioning relationships.An increase in/", is associated with a decrease inFe2,/

(Fe':* * Mg) of the coexisting silicate phases, includingbiotite and garnet, which in turn is coupled to increasesin X.o. (Figs. 3 and 4).

It should be stressed that the linear relationship be-tween f, and X.o, would not be expected to characterizeall regions or even all samples from New Mexico. To a

TABLE 8. Correlation coefficients for compositional variables in Equation 3

Pecos Baldy, New MexicoX"o" 4- (Xn^- Xwl )el xii (Xr" - Xrn)'')(EF5'

xsXn(X"'^ - X*)FriX'Fgr

)qr(X* - X"s)"'In Kgn'"'

1 . 00.592

-0.964- 0.618

0.250-0.852-0.876-0.828

1 . 0-0.729-0.665

0.058-0.636-o.776-0.744

1 . 0o.737

- 0.1 910.8890.9700.841

1 . 0-0.079

0.6630.8190.719

1 . 0-o.270-0 .113-0.258

1 . 00.861o.727

1 . 00.823

Rio Mora, New Mexico )e#.4"Xo" (X"'. - X",") xgt )qi (X* - X"J"'

X"*Xn.(X"'. - X*))qXBFS-XFI-(Xr" - X"J"'In Kg''"'

1.00.423

-0.952- 0.780-0.640-0.601-0.903-0.680

1 . 0-0.541-0.280-0.271-o.784-0.553-0.741

1 . 00.844o.7280.7410.9860.706

1 . 0o.7440.4390.8740.580

1 . 00.7520.646

1.00.734

1 . 00.431o.7450.355

Ceno Colorado, New Mexicoj=p" )qi4- (X"^ - \,1 xpl )GFi; (X.. - XrJ"'

X*"4.(X"- - X-)nxBF!-.xFt'(X." - Xd'ln Kgn-e'

1 . 0-0.449-o.827-0.682-0.542

0.359-o.754-0.480

1 . 00.1660.3390.5620.0710.3370.573

1 . 00.8380.580

-0.255o.9240.084

1 . 00.7200.1 950.9320.313

1.00.1390.7730.515

1 . 0-0.001

0.2011 . 00.300

- Octahedral cations onlv.


oco

><

0.6

o.4

o.2

0.0 0 .2 0 .4 0 .6 0 .8

(Xo,, Xo.o)

Fig. 5. X,o, us. (X,,- - X".) for garnets from the WatervilleFormation, Maine (Ferry, 1980) (slope : -0.89, x-intercept :0.77, 12: 0.83). Note the similar linear trend but different slopefrom data from New Mexico (Fig. 3).

large extent, the correlation may reflect the generally ox-idized nature of the Mn-rich sedimentary protoliths.However, where present, this relationship can seriouslyobscure the mixing behavior of other components. In mostsamples from New Mexico, as X"e" increases, an increas-ing percentage ofFe in biotite is Fe3* (Fig. 7). Ifthis Fe3*is not subtracted from total Fe, as in typical microprobeanalyses, K$-cn utrd estimated temperature will be erro-neously high. The effect is opposite ofthat predicted fornonideal mixing of Mn in magnesium-iron garnet andthus partially cancels the effect of Mn on Ko, creating theerroneous impression that Mn mixing is more nearlyideal (Williams and Grambling, 1984).

E

l H e m o t i t e - | "j l lmen i te ij S o l v u s i o B

r!o

| | o E a o

o l l o

- E l l o a

" f l g Io a - I I

t rA; a | |- o

o

0 .6

0.5

o.4oF 0.3><

o.2

0 .1

0 .00.0 0. ' l o.2 0.3 0.4

Fe3* /(Fe2*+Fer*) in Biotite

Frg. 7. Mole-fraction spessartine in garnet vs. Fest/(Fe3* *p"z*) in biotite. Fe3*/Fe2* ratios are from wet-chemical andMiissbauer analyses of biotite separates. Garnets were analyzedwith the electron microprobe (Table 2). All samples are fromthe Rio Mora area.

Finally, for the regression analyses that follow, it isimportant to note that, although Fe3+/(Fe3+ * Fe2*) inbiotite is positively correlated \Mith X,e" in the New Mex-ico rocks, -X?:3- does not show the same relationship. Asnoted above, the number of Fe3* cations is nearly con-stant, decreasing slightly with increasing X"0".

X"o" vs. garnet-biotite K". Table 8 and Figure 8 docu-ment a negative correlation between X"o, and K$"-n'. tn'tcorrelation suggests that Mn mixes nonideally in garnet.However, it is difrcult to use the data in Table 8 directlyto calculate the magnitude of the Mn-Mg interaction pa-rameter because most other compositional variables arecorrelated to some degree with X"o". Therefore, the neg-ative slopes in Figure 8 incorporate the combined non-ideal mixing effects of a number of components in garnetand biotite.

Six specimens from Rio Mora have X.o, between 0.50and 0.76. These samples, not shown in Figure 8, displaya dramatic reversal of the X,o, vS. K$n'at trend; as X"o. risesfrom 0.50 to 0.76, Ko rises. These samples are shown onFigure 9, but because several ofthe garnets contain slight-ly higher Xo., It Ko has been corrected for the effects ofnonideal Ca mixing (Wg : 12550 J/mol). The reasonfor this reversal is unclear to us. Although a number ofthese samples have relatively high Xr,,, no direct relation-ship exists between Xr" and ln K". As expected, thesesamples have garnets and biotites characterized by highMe/(Mg + Fe) and are associated with ilmenite-hematiteoxides rich in the hematite component. However, the InKo reversal cannot be explained by asymmetric mixingbehavior ofmagrresium garnet according to any recentlyproposed model. Further, although it is difficult to esti-

a l ' l

0 .6

& 0.4o

><

u.z

0.00.0 0.2 0.4 0.6 0.8 1.0

Xp""e, in Hemoti te- l lmenite

Fig. 6. Mole-fraction spessartine in garnet vs. mole-fractionhematite in hematite-ilmenite. Constructed from recalculatedmicroprobe analyses in Tables l-3 and 7. Note that the plot isnot a rigorous representation of an isothermal solvus becausethe three study areas crystallized at different peak metamorphictemperatures.

- Carnat , Taaa

o Stour. /And. Zone

WILLIAMS AND GRAMBLING: EQUILIBR.IUM BETWEEN GARNET AND BIOTITE 899

mate the amount of Fe3+ in these low-Fe biotites, and KDis extremely sensitive to errors in the estimate, Fest wasdetermined directly in two of these biotite samples, andso errors in the Fe3*/Fe2* ratios do not seem to be re-sponsible for the reversals. Also, because ofthe low totalFe, ,l}5- is extremely small in these biotites, so nonidealFe3* mixing would also not seem to be responsible forthe reversal. At present, we suspect that the reversals mayrelate to an important change in the mixing behavior ofgarnet or biotite at high X,o" and high f"r.Because of thedramatic nature of the reversal and because most of thesesamples have slightly elevated X",", we have elected toexclude them from our regression analyses and to restrictour conclusions to garnets with X-e" less than 0.50.

Regression techniques for multicolinear data sets

Statistical regression of multicolinear data can lead tolarge standard deviations for the estimated coefftcientsbecause variation produced by one coefficient is, to vary-ing degrees, attributed to another coefficient. In the caseof garnet, some variation in ln K" due to nonideal Mnmixing might be attributed erroneously to another coef-ficient, such as (X"r^ - Xo.). Further, nonideality may beunrecognized because the effects of one coefficient maycancel the effects of another. For example, the predictedeffects of nonideal mixing of Ti in biotite and Mn ingarnet are opposite in sign (Dallmeyer, 1974; Indares andMartignole, 1985). Because of the correlation between X.*and ,$1, these effects will tend to cancel and obscure thetrue mixing behavior of both components.

Traditional least-squares multiple regression analysismay not reveal the presence or effects of multicolinearity.Hence, two alternate approaches have been used here.The first involves elimination of the least important coef-ficients from the model, fixing the value of coefficientsthat are reasonably well known, or both. The second in-volves the use ofa regression technique that yields biasedrather than unbiased estimates of the model coefrcients(ridge regression, Devore, 1982). A biased regtession mayreduce the standard deviation of regression coefficientsand highlight those coefficients not important for the re-gression model.

For this study, multiple regressions (least-squares andbiased regressions) were carried out in a backward step-wise manner: regression coefficients were subtracted fromthe model one at a time in order to evaluate the influenceof any one on the regression model. The least significantcoefrcients were eliminated from the model one at a timeand the remaining coefficients re-regressed. Coefficientswere evaluated on the basis of their t ratio. For any coef-ficient i, t,: 0,/o, where B is the value of the estimatedcoefficient and o is the standard deviation of that coefr-

-+

Fig. 8. Mole-fraction spessartine in garnet vs. ln K$"-B'whereFeB' includes only Fe'?*. Fe3* contents of biotite were estimatedfrom Figure 2.Each point represents one sample.

' - i . e ' - i . + ' i . z ' - i . o -1 .8 -1 .

Inxo Gornet-Biot i te

0.5

0.5

o

6>< o.2

0.1

0.5

0.3

oeo

x o.2

0.1

o.4

0.0 t-2.4

0.0-2.6 -2.4 -2.2 -2.O -1.8 -1.6 -1.4

A Rio Moro

\ t r\ f t

o \ - o\

o \

\ s

Inxo Gornet-Biot i te

B Pecos

\Po^\g^ors

8d o_s^

Boldy

&

v a

a \ aa \ a

4 \

A \\

AA

Cerro Colorodo

A

A

\ ^

In6o Gornet-Bio i i te


cient (Devore, 1982). Coefficients with / ratios less than2 arc not distinguishable from zero at the 950/o confidencelevel and were not considered significant.

Simplification of regression model

Several terms can be eliminated from Equation 3 be-cause they are of only minor importance in samples fromNew Mexico. W**.(Xr, - Xr*)", can be eliminated be-cause ffi,." is suspected to be near zero (see above).AWM^(XM")B'can be eliminated because X|ln is near zero.Both of these terms are strongly correlated with X,o. and(X^r^ - Xo.). Readers should note that, by eliminatingthese terms, any nonideal behavior actually carried bythem will be assigned to other coefficients.

Because X,^ is small (<0.10) and A,Wfi has been con-strained at approximately 12550 J/mol in other studies(Ganguly and Saxena, I 984; Ganguly and Kennedy, 197 4),the term AWc^(Xs*) has been incorporated into the leftside of the equation for most of the regressions that fol-low. With these simplifications, Equation 3 can be re-wntten:

-RZln KD - 12550(Xs"): -Ri| h K* * lW*rr"(X"^ - X*) + AI,f/M"(X"p")lc.

- lLWrt(X) - LWN(X) - LWr.t,(Xr"t*)18,. (5)

Equation 5 may be simplified further. ,l}i and 45- areboth very small and vary only slightly. -X!!, although sig-nificant in magnitude, is also nearly constant. If thesecomponents are eliminated, Equation 5 becomes

-RIln KD - 12550(Xo.): -RZ ln K"o

+ lwMsF.(xd^ - X,*) + azM"(x""Jlc". (6)

These final simplifications are not completely justifiedbecause none of the eliminated variables are rigorouslyconstant. Therefore, regression analyses have been car-ried out on both Equations 5 and 6. Correlation matriceshave been constructed for the three data sets incorporat-ing the left side of Equations 5 and 6. However, corre-lation coefficients are not significantly different from thosein Table 8, and thus they are not reproduced here. Read-ers should note that the two remaining nonideal mixingterms in Equation 6 are associated with the two moststrongly correlated compositional parameters, (X^r^ - X,*)and X"p., and thus even with all of the simplifications,regtession analyses of Equation 6 must be afected bymulticolinearity.

Biased multiple regression (ridge regression)

Ridge regression is based on the conceptthat"a biasedestimator with a small variance may be preferred over anunbiased estimator with a large variance" (Devore, 1982,p. 513). The technique, described by Devore (1982, p.513-5 16), is briefly summarized here. Ridge regressionuses a biasing constant, c, which is multiplied by the co-variance of the individual compositional variables. As cincreases, the amount of bias in the estimated coefficientsincreases. Theoretically for multicolinear data, as c in-

creases, the total mean-squared error of the ridge esti-mators decreases to a minimum and then increases. Nearthis optimum (minimum) value, the ridge estimators mayrepresent an improvement on the least-squares esti-mators. However, the optimum value of c is unknownbecause it depends on the unknown regression parame-ters. Its value must be selected based on the sample data.Typically, a plot of c vs. the ridge-estimated coefficients(ridge trace) is constructed and c is chosen as the smallestvalue for which the estimators have stabilized. The ridgetrace for a ridge regression ofdata from Pecos Baldy pre-sents an example (Fig. l0). Typical of all New Mexicodata, ridge coefficients change dramatically at first, butthen quickly stabilize at a very small value of c. Thus,with a rather small amount of bias, significant improve-ment can apparently be made to the estimated coeffi-crents.

As noted by Devore (1982), statisticians disagree aboutwhich value of the biasing constant (c) to select. The choiceof the stabilization point on the ridge trace is somewhatarbitrary. Even if ridge coefficients are not accepted, how-ever, ridge regression analysis can be useful for selectingcoefficients in a multicovariant data set that are least im-portant. Coefficients that quickly converge on zero withincreasing c are candidates for elimination from theregression model. The remaining coefficients can be re-regressed using either least-squares or additional ridgereg.resslons.

Regression results

Results ofregression analyses for each ofthe three studyareas in New Mexico are summarized in Table 9.

Ca mixing in garnet. Preliminary stepwise regressionanalyses carried out using Equation 3 (Table 9, Model l)yield values of AW"^ of approximately 20000 (o: 5000)J/mol for least-squares or 14000 (o : 5000) J/mol forridge regressions. The latter estimate is indistinguishablefrom the value of Affif; : 12550 (o : 2090 J/mol), sug-gested by Ganguly and Saxena (1984). It provides an ex-ample of the success of the ridge regression technique.The larger value of the least-squares estimate probablyreflects the positive correlation between X.o" and X^ ob-served in the data (Table 8).

Ti,AloFe3* mixing in biotite. Stepwise regressions usingEquation 5 suggest that octahedral Fe3* and Al in biotitedo not significantly contribute to the regression model.This does not require that these components mix ideally,but only that their nonideal effects do not account forvariation in ln K" within the range of compositions ex-amined.

The term Alfii is significant in most least-squaresregressions, and it is significant in several models usingridge regression. However, the derived coefficients exceed-50000 J/mol, which seems unreasonable. Using regres-sion analyses of natural data, Indares and Martignole(1985) suggested that AW+l is approximately -30000

J/mol. We suspect that the large values obtained here areexaggerated because of multicolinearity. This is suggested

WILLIAMS AND GRAMBLING: EQUILIBRIUM BETWEEN GARNET AND BIOTITE 90r

1 . 4

1 . 2

-P

c.Q)

tso

q)c',3E.

0.6

oon

>< o.4

0.8

0 . 6

0 . 4

o.2

-o.2

-o.4

-0 .6

0.0

In6o Gornet-Biot i te

Fig. 9. Mole-fraction spessartine in garnet vs. ln Kg"-", 1o. uttsamples from Rio Mora. The ln Ko values have been correctedfor the effects of nonideal Ca mixing (Wg : 12500 J/mol). FeB,includes only Fe,*; Fe3* estimates are from Figure 2. Sampleswith X,o" greater than 0.5 were excluded from all regression anal-yses (see text).

by the tendency for -l}1 to be eliminated from most ridgeregression models. It is possible that slight alteration orreequilibration ofbiotite is reflected in a decrease in )Gland a slight reduction in Kg"'n,. This effect could contrib-ute to the large values for AW\\in the regression analyses.Because of the strong possibility that other factors arecontributing to the derived value of AW\i, we have cho-sen to eliminate it from our subsequent regressions. Al-though not rigorouslyjustified, this simplification will notsignificantly influence our conclusions. With a modestvalue of Atr41, such as -30000 J/mol (Indares and Mar-tignole, 1985), the small magnitude of Xli in our samplesminimizes the importance of this term.

Mn and Fe-Mg mixing in garnet. Regression analysesusing Equation 6 were carried out by excludingWr,, Wr4*,atd Wo, in biotite from Model 2 in Table 9. Assumingthat Fe-Mg mixing in garnet follows symmetric simplemixture behavior (i.e., Wffir" is constant), these analysesconsistently yield small values for Al,yg* (5000 J/mot)and near zero values for Wf;;!o.. This indicates that theminimum sums of both the squared residuals (SSE) andmean-squared error (MSE) occur when most variationdue to nonideal mixing is incorporated into the A,Wffiterm. Because of the near perfect correlation between X"o"and (X",- - Xo*), a range of paired values of AZfii andZg1o. results in only slightly larger model errors. Error(SSE and MSE) and the regression variance both increasemonotonically, but rather slowly, with increasing AWffi.The data allow but do not require ideal Mg-Fe mixing ingarnet.

Fig. 10. Ridge trace for ridge regression ofln K" against X"o",X",", (X^,^ - Xo-), and -Xli from Pecos Baldy (Table 9, Model l).Regression data (mole fractions) were normalized to the meanand standard deviation ofthe particular variable (Devore, I 982).Increasing c (biasing constant) produces increasingly biased es-timates. The regtession coefrcients are chosen at the minimumvalue of c where relative stability occurs. Coefrcients that ap-proach zero are relatively unimportant to the regression model.

Several attempts were made to incorporate into theregression analyses the hypothesis that Mg-Fe mixing be-havior in garnet is asymmetric (Ganguly and Saxena,1984; Hackler and Wood, 1989). The first attempt uti-lized regressions of Equation 6 with Wfi;r.fixedaccordingto Equation 4 and the Fe-Mg mixing parameters of eitherGanguly and Saxena (Table 9, Model 3a) or Hackler andWood (Table 9, Model 3b). The paramet€rs of Gangulyand Saxena (1984) imply a significant amount of Mg-Fenonideality, approximately 8000 J/mol for Fe-rich gar-nets. With this model, AWffiis 19780, 19630, and,17570J/mol for Pecos Baldy, Rio Mora, and Cerro Colorado,respectively (Table 9, Model 3a), values considerablylarger than those suggested by Ganguly and Saxena (1984).The parameters of Hackler and Wood imply only a mod-erate amount of Mg-Fe nonideality, less than 2000 J/molfor Fe-rich garnets. With this model, A Zgi is 7 120,7 630,and 9720 J/mol for Pecos Baldy, Rio Mora, and CerroColorado, respectively (Table 9, Model 3b). The coeffi-cient of multiple determination (r,) for these models areimpressive, approximately 0.90 for Model 3a and 0.80for Model 3b, and considerably higher than the values of0.60 to 0.70 derived from Models I and 2 in Table 9.However, the SSE and MSE of the regressions are largestfor Model 3a, intermediate for Model 3b, and smallestfor models assuming ideal Mg-Fe mixing.

It is suspected that neither 12 nor regression error (SSEor MSE) can be used to evaluate the success of the various


TABLE 9. Summary of least squares and ridge regression analyses

Model 1: Regression using Equation 3

Data Reg.source type AW"" AWc" Wrn . Aw-, LWF€. LW^l

PB L-Sq

PB Ridgeo

PB L-Sqo

L-Sqq

Ridge6

L.Sqo

Ridgeo

10458 20605 6080(27041 (s448) (223s15425 14080

(1921) (5058)3745 14176(71 1 ) (4185)

8498 22340 1926(21s1) (s264) (1926)4347 14561(805) (4566)

17921 - 11615(4444) (3772)11935 ' 6744(3877) (3291)

83074 0.85 288

95 370 0.82 309

108 323 0.78 329

245270 0.78

296253 0.72

966 880 0.54 983

1 098970 0.49 1048

-77 714(27 414)

-47 571(20 042)

1 993 781

2288887

2 816 409

5 886 473

7 702576

13 536 325

1 5 385 584

33

28 0.03

46

495

544

RM

RM

21.5

33 0.03

8.2

6 .4 0 .12


Datasource

Reg.rype

Garnet Biotite

LW.. LW"^ W.n . ^w, LWF+' LW^l

PB

PB

PB

PB

RM

cc

L-Sq 8197o (2276)L-Sq 5011o (2160)L-Sq 3930o (563)Ridge 4073o (1516 )

L-Sq 4581o (729)L-Sq 20423o (4818)L-Sq 7724o (3419)Ridge 12966o (4173)

3928 -75 019(1747) (27 s71)

, N . t .

N . t . N . t .

1 3008(408s)N. t .

6970(3542)

2 170 062

2794 454

2823 423

2 461 191

7 752977

15 913 332

27 4't4't09

18772777

0.73 295 22

0.65 328 24

0.65 3236 49

0.69 314 19

0.60 536 39

0.57 1066 9

o.25 1351 5

0.50 1 158 7

86 802

107 479

104 571

98 448

287 147

1 136666

1 827 607

1 340 912

0.04

0.42

0 .14

Model 3a: Regression using Equation 6, yyPg,q fixed according to Equation 4, Fe-Mg mixing model of Ganguly and Saxena (1984)

Data Reg.source type LW*^ LW.. Wrn . L W awFd* aw^l MSE R 2

PB L-Sqq

PB L-Sq

PB RidgeRM L-Sq

o

RM L-Sq

RM RidgeCC L-Sq

o

CC L-Sqq

CC Ridge

19779 N.t.(1 061 )

No improvement on least-squares model

16518(e0s)

14420(1 s85)19634

-165 827(28 432\

-99 698(23 025)

N. t .

560 737(53 303)

N. t .

4341 024

10 020 521

1 4 898 403

25 640 970

14 991 416

16 176 427

1 070 815 0.77

1 078 428 0.74

166 962 0.97 409

371 130 0.93 609

573 015 0.94 757

949 665 0.89 974

403

348

191

2't9

23

44

(1 326)No improvement on least-squares model

1 8655(2815)1 7566

1 034

1 038(26271

No improvement on least-squares model

regression analyses with respect to Mg-Fe mixing in gar-net. Variations in SSE, MSE, and 12 result, in part, froma model-dependent magnifying effect: constant errors inmicroprobe analyses produce increasing SSE, MSE, and12 as Wf;;|r. and AWffi increase. SSE and MSE increasebecause microprobe data are multiplied by Zfl!.. in theregression model. Thus, when Wf;;|r. is fixed at a large

value, scatter in the data leads to increased scatter in theregression solution. The multiple correlation coefficient(r'z) is defined as I - (SSE/SST) (Devore, 1982) whereSST is the sum ofthe squared total error ofthe dependentvariable. As the paired values of IZfilF" and AZS\ in-crease, both SSE and SST increase because of the mag-nifying effect described above. However, SST increases


TaEte 9.-Continued

903

Model 3b: Regression using Equation 6, Wpg,n fixed according to Equation 4, Fe-Mg mixing model of Hackler and Wood (1989)

Datasource

Reg.rype AW., AW". Wn. awFe' aw^l SSE MSE

102

154

63

108

PB

PB

PBRM

RM

RM

L-So 5907 -58 749o (646) (202971L-So 7122 N. l .o (573)Ridge No improvement on least-squares modelL-So 5983 -31 471o (1067) (1s 507)L-So 7629 N.l.o (7321Ridge No improvement on least-squares modelL-Sq 12067 120587o (3093) (58 s74)L-Sq 9724 N.l.o (31711Ridge No improvement on least-squares model

2212171 85 083 0.89 292

2925041 108 334 0.85 329

6757 682 259 910 0.83 509

7 828 160 2 899 315 0.80 538

18 103 033

23 583 475

r 293 073 0.53 1137

1 572231 0.39 1254

8

1 0


Data Reg.source rype AW." Wrn

"Wr"*n ssE

PB

RM

cc

L-sqo

L-sqq

L-sq6

4853(21 s0)61 17

(2O71121472(1210 )

632(1 41 8)1 067

(1 339)1 2556(1 028)

N. t .

N . l .

N . t .

2767 536

7 513 286

17 667 785

608 010

288 969

1 261 980

0.65 326

0.60 540

0.57 1121

24

20

9.2

Notej L-Sq: stepwise least-squares regression; Ridge: stepwise ridge regression; PB: Pecos Baldy study area; RM: Rio Mora study area; CC: CerroColorado study area; N.l.: variable not included in regression analysis; t: variable eliminated during stepwise regression (see text); SSE: sum of squarederror; MSE: mean-squared error|. R2i coefficientof multipledetermination; s: regressionstandarddeviation; F: Fstatistic(Devore, 1982); and C ridgeconstant (biasing constant for ridge regressions).

more rapidly than SSE, which causes 12 to increase. Thus,the increase in 12 and regression error may simply be anartifact ofgreater sensitivity ofthe dependent variable toroutine analytical errors, instead of a better regtessionmodel. Consequently, one cannot determine uniquelywhich statistical model is more appropriate.

As a second attempt to evaluate the asymmetric mixinghypothesis, Equation 6 can be rewritten to explicitly in-corporate the symmetric approximation (Equation 4) toasymmetric Mg-Fe mixing in garnet, maintaining Zr.-..and WF.-M' as variables:

-R?"ln KD - 12550(X*,"): -R7"ln K"o + lf(X,,,)W,"*"

+ f(x"t^)wMe_F. + azM"(x.p.)lc" (7)

wrth f(X) : lxt/(Xery + X",-)l(X,,- - X,*).Preliminaryleast-squares stepwise regressions ofln K" against the threecompositional variables in Equation 7 yielded large neg-ative values for W*"-o, and WF"-M,. These values, whichcontradict the estimates of Ganguly and Saxena (1984),Geiger et al. (1987), and Hackler and Wood (1989), seemunreasonable. This is not surprising, considering the highlycorrelated nature of the X",*- and Xo*-dependent vari-ables.

Ridge regression suggests a third way to treat the data.As the ridge regression biasing constant (c) increases, theJ(Xo*)(Wr.*") term quickly converges on zero, suggestingthat it may not be important to the regression model.This is consistent with all existing mixing models; all pro-pose that Wl!*"is less than 1000 J/mol. Elimination ofItrf|r, from the model yields small values for the tworemaining interaction parameters, Wffi.r" and A,Wf;;] (T a-ble 9, Model 4). The results are similar to those obtainedusing Equations 5 or 6 with no significant improvementin the regression statistics.

Toward independent estimates of AW$il and ffi!".Least-squares and ridge regression analyses su ggest that

Equation 6 characteizes nonideal mixing in garnet andbiotite from New Mexico. Nonideality in the mixing ofTi, Al, and Fe3t into Fe2*-Mg biotite has little or no efectwithin the compositional range of biotite samples treatedhere, but the Ca, Mn, and Fe-Mg interaction parametersfor garnet are all nonzero. The magnitude of Ca nonideal-ity (LW39 seems to fall near 12550 J/mol, as suggestedby Ganguly and Saxena (1984).

As noted above, garnet mixing behavior within eachindividual study area can be described with a range ofpaired, AWffiand Wffir" values. The relationship between


!4&'n*,l/Fl"nwFn-..at4fi'"

Vlfi'.'"lI&'.rn

All models:

TABLE 10. Suggested interaction parameters (J/mol, one-sitebasis)

Model 1 Model 2 Model 3

obtain unique and realistic estimates of Z$!o. andLWff;.One of us (Williams) has begun such an analysis,which requires knowing the complete chemistry of gametand biotite, including Fe3*/total Fe, for a suite of sampleswith a limited range in T and P, data not commonlyavailable in the literature.

Suggested mixing parameters

Data from this study permit three sets of mixing pa-rameters for garnet (and biotite) in the Mn-Mg-Fe system(Table l0). Model I assumes ideal Mg-Fe mixing in gar-net and biotite. The assumption leads to the minimumpossible value for AWg,\,5500 (+759; J/mol (one-sitebasis). Model 2 uses the Mg-Fe mixing model of Gangulyand Saxena (1984). This model leads to alarge estimateof AWffi (19000 + 800 J/mol), and almost certainly rep-resents a maximum estimate for this parameter. Model3 uses the Mg-Fe mixing model of Hackler and Wood(1989) and leads to an intermediate estimate of AWffi(8200 + 700 J/mol). Because this model seems to char-acteize Mg-Fe mixing at high X",- better than that ofGanguly and Saxena (1984) (see discussion in Hacklerand Wood, 1989), we prefer this model.

O'Neill et al. (1989) concluded that ll'ffir. is approxi-mately 1800 J/mol based on the partitioning of Mn be-tween gamet and ilmenite. Based on this value, Wffi."isapproximately 3700, 11200, and 6400 J/mol for Modelsl, 2, and 3, respectively (Table l0). Again, we prefer thethird estimate. Consistent with predictions, Mn-Mg mix-ing is significantly less ideal than Fe-Mn mixing.

Impr,rcllloNs FoR GARNET-BIorrrEGEOTHERMOMETRY

Geothermometry in the Mn,Ca-free system

Many studies have derived useful temperature esti-mates by applying the Ferry and Spear (1978) experi-mental calibration to rocks low in Mn and Ca, even thoughthe calibration does not treat nonideal mixing in garnet.We have applied their calibration directly to specimensfrom New Mexico in which garnets have low Mn and Ca,yielding temperatures consistent with those of coexistingaluminum silicate assemblages (Grambling and Wil-liams, 1985; Grambling, 1986). If Mg and Fe do not mixideally in garnet and if the Fe3*/Fe2* ratio of biotite isvariable, then one must question why the Ferry-Spearcalibration of the geothermometer works so well.

Garnet and biotite in the low-Mn, low-Ca rocks fromNew Mexico have compositions close to those in the Fer-ry and Spear experiments with (X,,- - X".") near 0.80and Fe3*/(Fer+ + Fe3+) in biotite near 0.10. At this com-position, the calibration and the field data are affectedequally by nonideal Mg-Fe mixing in garnet and by Fe3*in biotite. From the relationships in Figures 3 and 5, itseems likely that most relatively reduced, amphibolite-facies garnets that are low in X"o" and X*^ have similarvalues of (X.,^ - Xo*).The compositional similarity ap-pears to explain the success ofthe Ferry and Spear cali-bration.

0

5485- 745

1 800+368s.

Asymmetric835-'

10460- 209218975. 8181 800+

17'175'

Asymmetric6951 42O

2115t 2958231- 680

1 800+6431'

o

vtr:.t/VF,'n*WrlAWFiu'

1 2550"0

0 or -312000 or -6600$

2090

5000$(?)

'This work.'Ganguly and Saxena (1984); Geiger et al. (1987).t Hackler and Wood (1989).+ o'Neiil et ar. (1989).$ hdares and Martignole (1985).

the two interaction parameters is linear for each area:

AWgi:: l.45WW. + 4580 (Rio Mora) (8)

LWF[;: l.46WW. + 3930 (Pecos Baldy) (9)

AW$fr: 0.80t29t . + 11164 (Cerro Colorado). (10)

These relationships incorporate all of the multicolinear-ities discussed above, and their differences probably re-flect variations in temperature, pressure, bulk composi-tion, or degree ofretrogression. As noted above, the datafrom Cerro Colorado, in particular, may have been af-fected to a small extent by retrogression or alteration.Assuming that A,Wffiand, Wffir. are independent of tem-perature, the slopes and intercepts from the three equa-tions can be regressed against each other to suggest theunique values of AIZgf and Wf;6.. Such a regression yields19400 and 10400 J/mol for AWffiand Wf;fto., respective-ly. The value of WW" 00400 J/mol) is remarkably con-sistent with the asymmetric mixing model of Gangulyand Saxena (1984) and Geiger et al. (1987), whose modelsyield values on the order of 10000 J/mol for the garnetcompositions presented here. The value of AIIzS{ (19400J/mol) is slightly larger than that suggested by Gangulyand Saxena (1984) and may, in part, reflect the fact thatthe biotite samples were generally not corrected for Fe3*in the data sets that they regressed.

We realize that these mixing parameters are not wellconstrained; they are defined primarily by differences be-tween Cerro Colorado and the other two map areas. Weinterpret the results to suggest that a nonideal Mg-Femixing model may be preferable to an ideal one, but wesuspect that the large value of W*"o. may have, in part,resulted from scatter in the Cerro Colorado data. We sug-gest that this methodology could be applied to garnet-biotite pairs from numerous other study areas in order to


Fe3" in biotite

All available data (this study; Goldman and Albee,1977; Wones and Eugster, 1965; M. Darby Dyar, person-al communication) show that biotite can contain signifi-cant Fe3* and that the Fe3*/Fe2* ratio of biotite is quitevariable. Consequently, accurate garnet-biotite geother-mometry using microprobe data must employ biotitecompositions corrected for Fe3*. Corrections for Fe3* maybe made directly using Mdssbauer or wet-chemical anal-yses. Alternatively, the relations among biotite and com-positions of Fe-Ti oxide (Fig. 2) can be used to estimateFe3* in biotite.

For accurate geothermometry, it is also necessary toestimate the amount of Fe3* in the biotite used by Ferryand Spear (1978). They did not measure Fe3* in theirexperimental biotite but conducted their experiments inthe presence of graphite and methane. If graphite equili-brated with methane at the P and T of the experiments,/o, should have been below those of the quartz-fayalite-magnetite (QFM) buffer (Eugster and Skippen, 1967;Chou, 1987). AII biotite samples analyzed in this studyappear to have crystallizedatfo,above QFM (Grambling,1986). Thus, the experimental biotite should have hadFe3+/(Fe3+ + Fe2*) below 0. 12, the most-reduced biotitefrom New Mexico. Wones and Eugster (1965) and Dyar(personal communication, 1989) suggested that some Fe3*is present in virtually all biotite. We estimate that thebiotite used by Ferry and Spear had Fe3+/(Fe3+ * Fe2*)near 0.10.

An empirical garnet-biotite thermorneter forMn-Fe-Mg garnets

Ferry and Spear's (1978) calibration ofthe garnet-bio-tite geothermometer was derived for the Fe-Mg systemassuming that Fe and Mg mix ideally and that all Fe wasFe2*. Their calibration can be modified to incorporate Fe-Mg nonideality (following Ganguly and Saxena, 1984) andto exclude Fe3t:

RZInK*: -17042 - 79.5(P)+ RZ[0.782 - ln (Fe2+/Ferd)h*]+ [(X"* - X^^)(W*"r.)o^*

- (Xo. - Xv,)(WMEF)8,*]/R (l l)

where P is in kbar, Z is in K, R in J/mol K, and thestarred terms refer to the composition used in the cali-bration experiments. Garnets used by Ferry and Spear

Fig. ll. Mole-fraction Mn in garnet vs. garnet-biotite tem-perature for samples from Rio Mora, New Mexico, calculatedusing Equation (12). (A) Temperatures calculated using inter-action parameters from Table 10, Model 1, one Mn-dependentparameter. (B) Temperatures calculated using interaction pa-rameters from Table 10, Model 2, nonideal Fe-Mg mixing ingarnet, based on Ganguly and Saxena (1984). (C) Temperaturescalculated using interaction parameters from Table 10, Model3, nonideal Fe-Mg mixing in garnet, based on Hackler and Wood(1989) .

0.3

oo6

>< o.2

0.5

0.0 -F400

0.3

0.0 -l-400 5oo

500 600

Temperoture ( 'C)

o.4

I

700

700

6oo

>< o.2

0 . 1

600

Temperoture ('c)

0.3

oq6

x o.2

^ R io MoroA

o

tr

tr

% { t r

trtrn

m t r n-otro]otr

Rio MoroE

tr

tr

f t tr tr

tr

cft

C R io Moro

tr

tr

t r g

% E D

tr

Temperoture ("C)


Tlau 11. Geothermometry (€)

Rio Mora (P: 4.5 kbar) Pecos Baldy (P: 3.8 kbar)

Sample In Ko Sample In KD

81-23a80-'t4280468046bc81-17480-3080-4380-1 8780-4280-2080-51c81-18180-1 78a81-22580-1 8a80-192-281-234sCc81-148c81 -1 05c?81-21381-13ac81-31080-1 7881 -1 4581 -70a80-21 I82-62c81-24482-12

77-85876-42077-12577-1577-81E77-10376-400877417644677-3977-677-4276-44578-22477-7377-2383-4577-46D84-163-284-169184-1 69-27G'43784-169476-40078-2576-37977-4584-162-284-162-1

- 1 .930- 1 .788-1.902-1.874- 1 .961-1.847-1.782- 2.010-1.780-1.790-2.001- 1.891-2.046- 1 .976-2.113-2.O94- 1 .894- 1 .898-2.042-2.129-2.257-2.028-2.033-2.O77-2.100-2.084-2 .168-2.326-2.215

Meang

539577541551543563583522580581531coo5 1 45545 1 1525585584564543538573577594555579586529558

53758054555454856358552158358753055851352550852156358430c

540534570b /c573554575578526551

-1.993-2.034-2.154-2.O20-1.973-2.058-2.188-2.177-2.154- 2.1 39-2.001-2.091-2.103-2.063-2.134-2.187-2.187-2.181-2.209-2.195-2.269-2.167-2.223-2.219-2.208-2.302-2.240-2.403-2.373

Mean

5185024915185274994854824884885275084885044934844875215145 1 6505523520521527503525509510

528592559569569JOO

5935155936165255255 1 1404499507473584574526519JCC

570492549561549514526

54043

540 522507 503484 490547 523541 530534 505475 483488 483494 489501 490545 530523 511542 498567 516514 497553 497531 496508 518501 512502 513493 502581 535511 518514 520523 526499 502523 524495 506495 507

518 50825 14

s061 5

55324

5 5 /

24

had (Xo- - Xr-) : 0.8 and, based on the discussion above,(Fe2+/Fsrot)at* may have been approximately 0.90.

Equation I I can be combined with Equation 5 to yielda general expression for the nonideal geothermometer:

IK) : I- 17042 - 79.s(P) + 0.8 l4lgl;- w*"r"(X^_ - Xr*) - awg*(Xs")c"- awffi(x",")"" + AI'?l(Xr,)',- aw;16(x^rc)B,f

+ R[n KD - 0.782 * ln (Fe'z*BtlFerot'Bt)*] (12)

where Ko includes only Fe2* and, as above, starred termsrefer to the composition of the minerals in the calibrationexperiments. This form of the geothennometer assumesthat Mg-Fe and Mn mixing are ideal in biotite but allowsfor nonideal mixing of octahedral Ti, Al, and Fe3+.

Substitution of the three sets of interaction parametersfor Mn-Fe-Mg mixing (Table 10) into Equation 12leadsto three expressions of the geothermometer. All threeforms treat only Fe2* in biotite. The first, using mixingModel I (Table 10), is relatively simple; all mixing termsare excluded except those for Mn and Ca in garnet. Es-sentially, this relies on the correlations between X*- (X^^- Xo*), and (X." - Xnnr)u', in order to place all interre-lated nonideality artificially into a single X"*-dependentparameter. Because of this limitation, it can be appliedonly to suites of rocks with linear relationships amongthe compositional parameters similar to those seen here.

Fortunately, to a first approximation, these relationshipsare common. The small magnitude of the X"o"-dependentparameter (5500 J/mol) is consistent with the observa-tion that at relatively low X"o", the assumption of idealMn and Mg-Fe mixing does not produce dramaticallyerroneous results (Hodges and Spear, 1982; Williams andGrambling, 1984).

The two remaining expressions of the thermometer ex-plicitly incorporate nonideal Mg-Fe mixing in garnet (Ta-ble 10, Models 2 and 3). Wf!""is calculated using Equa-tion 4. ttl$ti" is 9500 J/mol for Model 2 and 2000 J/molfor Model 3. Terms characterizing nonideal Ti and Almixing in biotite can be inserted into both equations ifW\t,and Wfiare known. As noted above, mixing Model3 (Table l0) is our preferred model at present, and thus,our preferred geothermometer for Mn-Fe-Mg garnets is

r(K): l-r7042 - 79.s(P) + rs79- w**"(x^^ - X,*) - 12550(X*)- 8230(X""")l

+ Run KD - 0.782 + ln (0.90)1. ( l 3)

We have applied the three forms of the garnet-biotitegeothermometer to specimens from New Mexico, withbiotite corrected for Fe3* using the correlation betweenFe3* in biotite and Xr"ro, in hematite-ilmenite solid so-lution (Table I l). Mean temperatures are within error forthe three thermometers, and all are compatible with in-

WILLIAMS AND GRAMBLING: EQUIUBRIUM BETWEEN GARNET AND BIOTITE

TABLE 11.-Continued

907

Cerro Colorado (P: 4.5 kbar) Rio Mora high-Mn samples, not used in regression analyses

Model

Sample In Ko Sample In Ko

w83-302w84-3sw84-86w84-94bW84-117aW84-130aw83-300w84-145bw85-195bw84-58w84-153bw84-87w85-253w84-7sw83-242w84-133w85-254

- 1 .689-2.290-2.154-2.278-1.834-2.176- 1 . 8 1 6-2.25'l-2.231-2.094-2.164-2.184- 1 .962-2.334-2 .172-2.391-2.283

Mean

5294954355136 1 1527614526526561521544569490549493547

6134725 1 1494638524652507520560536537604493545485542

-2.166-2.123-2.064-2.O75-1.302-1.182

632639723716

1 0701 155

6 1 0631782797

1 2081307

626637734730

10921 179

634467530489645523662503519s605395356't2494544484541

84-147e84-147d81-22181 -1 75a81 -1 3981 -1 39a

546 532 54357 42 52

dependent estimates of the peak metamorphic tempera-ture. The data from Rio Mora are plotted in Figure 1l asan illustration. The standard deviations are generally largerfor Model 2 than for Model 3. For example, data fromRio Mora yield 560 'C + 24 (o), 543 "C + 43 (o), and557 'C ! 24 (o) for Models l, 2, and 3, respectively. Asdiscussed above, the increased standard deviation fortemperatures calculated using Model 2 probably reflectsthe increased sensitivity of this model to minor compo-sitional variation. Although within the limits of error, theCerro Colorado mean temperature is somewhat less thanpredicted, and the standard deviation is considerablylarger than expected, presumably reflecting retrograde re-sorption or reequilibration.

We suggest limiting these thermometers to garnets withX"o" less than 0.50, Xr^ less than 0.10, and biotite withcompositions similar to those reported here. Significantvariations in Ti and AL either within a data set or be-tween the data set and the biotite used for these calibra-tions, may affect calculated temperatures. Variations inTi and Al in biotite seem to be most extreme at temper-atures above 650 "C, which may place an upper limit onthe calibrated thermometer presented herein.

Theoretical uncertainties for the three forms of thegeothermometer were calculated using the methodologyof Powell (1985) and Powell and Holland (1988). Al-though strong correlations exist among the compositionalparameters, errors associated with the compositional datawere assumed to be essentially uncorrelated. As expected,the calculated uncertainties vary significantly as a func-tion of composition and as a function of the chosen mix-ing model. For Model I, uncertainties range from 25 " forIow-Mn samples to 35 'for high-Mn samples (X."" : ap-proximately 0.40). For Model 2, uncertainties range from35 " for low-Mn samples to 65 " for high-Mn samples.For Model 3, uncertainties range from 25 o for low-Mn

samples to 45 " for high-Mn samples. These estimateduncertainties are generally consistent with the calculatedstandard deviations for the three New Mexico data sets(Table I l).

AcxNowr,nocMENTS

This research was supported by NSF grant EAR-8309503 to J.A. Gram-bling at the University of New Mexico, and funding for the electron mi-croprobe at the University ofNew Mexico was provided under NSF grant

EAR-8201211. We gratefully acknowledge R. Burns and M. Darby Dyarfor providing Mdssbauer analyses and M. Darby Dyar for helpful discus-sions. Helpful comments were provided by T. Hoisch, J. Lieberrnan, andS. Seaman. The manuscript was greatly improved by reviews by F. Spear,J. Ganguly, M. Kohn, and an anonymous reviewer.

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M.a.Nuscnrrr REcETvED Jur.rs 3, l9EEMelltncnrrr AccEmD Mav 25, 1990

Manganese, ferric iron, and the equilibrium between garnet ...888 WILLIAMS AND GRAMBLING:...

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