American Mineralogist, Volume 78, pages 583-593, 1993

Syntheses, volume, and structural changes of garnets in the pyrope-grossularjoin: Implications for stability and mixing properties

Jrn,l.lltrru, GlNGur,v, Wrr.rr CrmNcDepartment of Geosciences, University of Arizona, Tucson, Arizona 85721, U.S.A.

HucH Sr.C. O'Nnrr.r,Bayerisches Geoinstitut, Universitiit Bayreuth, D-8580 Bayreuth, Germany

Ansrucr

We have synthesized 1l garnet compositions in the pyrope-grossular join from glassstarting materials by a combination of hydrothermal synthesis and recycling of most ofthe products at -40 kbar, 1400 "C, in graphite capsules. Synlheses ofhomogeneous crystalswere also successful within the compositional range Grro-Grro, which represented a con-spicuous gap in earlier studies on synthetic garnets.

The lattice parameters of the synthetic garnets have been determined with a precisionof 0.0002-0.0004 A. The results show a slightly positive excess volume of mixing, whichis asymmetric toward the grossular end, and can be described by a subregular or Margulesformulation with W(^-":0.36 + 0.23 arrd W{r".^-- 1.73 + 0.3 cm3/mol (12 O atombasis). Powder X-ray diffraction (XRD) structural refinements show linear dependence ofthe O positional parameters and cation to O distance in dodecahedral, octahedral, andtetrahedral sites on the Ca content ofgarnet.

The partial molar volume of grossular in pyrope-grossular and almandine-grossularjoins, determined from our data and those of Geiger et al. (1989), varies smoothly andsimilarly between the terminal compositions and exceeds the molar volume by <lo/o,which has an effect of no more than 1.50/o on the calculation of equilibrium pressure ofthe assemblage grossular + aluminosilicate * plagioclase + quartz. The partial molarvolumes of pyrope and almandine change very similarly as a function of the Ca contentof garnet. The V-X relation of the pyrope-grossular join determined in this work has beenused to calculate the effect of lattice strain on the enthalpy of mixing. We have alsoextended the formulation of Ferreira et al. to derive a simple but general expression toaccount for the effect of multiatom interaction on lattice strain.

INrnonucuoN l98l) and a somewhat larger positive deviation towardthe grossular end. (A very similar volumetric behavior

Most mineralogical equilibria that are commonly used was reported by Cressey et al., 1978, for the almandine-as geothermometers and geobarometers involve garnet as grossular binary.) Subsequently Wood (1988) made sev-one of the phases. Calibration of these equilibria as P-T eral additional measurements on synthetic pyrope-gros-sensors requires a thorough understanding of the ther- sular solid solution and regressed the data along with themodynamic mixing property of garnet solid solution at others to suggest a small asymmetric positive Ceviationleast within the range of compositions commonly en- from ideality toward the grossular end, but without a neg-countered in the natural assemblages. The present paper ative deviation near the pyrope end. Berman (1990) re-is part of a continuing project in the development of a viewed the available data on garnet solid solution andcomprehensive solution model of aluminosilicate garnet concluded that the "excess volumes in the grossular-py-solid solution in the Ca-Mg-Fe-Mn quaternary system, rope join are poorly constrained by the data." He arguedand it deals primarily with the syntheses of garnets and that the scatter of the data is too large to justify the useexperimental determination of volume change in the py- of anything but the simplest mixing model and thus fittedrope-grossular join, along with its thermochemical im- the available volume data by a symmetrical nonidealplications. model. The various excess volume models, along with

Haselton and Newton (1980) and Newton and Wood the available experimental data, are illustrated in Figure(1980) concluded that the molar volume in the pyrope- l. The experimental data are taken directly from thegrossular binary has an S-shaped form, with a small neg- compilation of Berman (1990). It is evident that the dataative deviation near the pyrope end (also see Delaney, are too scattered to lead to a reliable model. Further,

0003-o04x/93l0506-0583$02.00 583

A ocbncy (1981)O wood (1988)tr t{.uton .t ol. (1977)

584 GANGULY ET AL.: GARNETS IN THE PYROPE.GROSSULAR JOIN

c v.zo

> U.U

X."

Fig. 1. Excess molar volume data in the pyrope-grossularjoin, as compiled by Berman (1990), along with the variousmodels fitted to part or all of the data.

there is a conspicuous gap ofdata between -50 and 80molo/o grossular composition. This gap, which is alsopresent in the calorimetric enthalpy data in the pyrope-grossular join (Newton et al., 19'17), is a consequence ofthe failure to synthesize homogeneous garnets that havecomposition within the above range.

We report in this work syntheses of homogeneous gar-net crystals in the pyrope-grossular join, including the gapbetween 50 and 80 mol0/o grossular content, and carefulindependent measurements of cell volumes of these syn-thetic crystals, which would remove the ambiguity of theexcess volume model in this join. We have also carriedout structural investigation by the Rietveld method onseveral of the synthetic samples and calculated the con-tribution ofelastic strain caused by solid solution on theenthalpy of mixing.

ExpnnrunNTAL PRoCEDURE

Garnets of all compositions in the pyrope-grossular joinwere synthesized from glasses in a piston cylinder appa-ratus. First end-member glasses were prepared by meltingstoichiometric oxide mixtures. Glasses of intermediatecompositions were prepared by remelting finely groundmixtures of the end-member slasses.

Preparation of glasses

The following oxides were used to prepare end-mem-ber glasses: MgO [Johnson and Mathey (JM) puratronic],CaCO, (Baker, >99.98o/o pure),.y-AlrO. (JM puratronic),SiO, (Alfa Co.,99.9o/o pure). All starting oxides were driedat ll0 oC prior to weighing. Five grams of both end-member oxide compositions were mixed manually underalcohol for about I h and then further homogenized in aSPEX vibrating mixture for 20 min. The grossular mix-ture, in which CaCO. was used as the source for Ca, wasfired at 800 "C for 4 h after preparation ofthe stoichio-metric oxide mixture to remove CO, before melting. Theoxide mixtures were melted in a Deltech MoSi, furnacefor about 45 min to t h and quenched. Microprobe anal-yses of the end-member and a few selected intermediateglasses showed them to be homogeneous to within +30lo

(!2o) of the average composition. The temperatures atwhich the oxide mixtures were melted are as follows: py-rope: 1620'C; grossular: 1400 "C; Py"Gr,.": 1450-1580'C, depending on the composition. The end-membermixtures were melted in Pt crucibles, whereas the inter-mediate ones were melted in graphite crucibles under Aratmosphere (to prevent the graphite crucible from burn-ing off). The graphite flakes were burned off the glassesby heating at 500'C for 2 h in air (there was no specialreason for choosing graphite crucible for the intermediatemixtures excepl convenience).

Synthesis of garnets

The synthesis conditions and results are summarizedin Table l. Pyrope and grossular were synthesized hydro-thermally in a piston cylinder apparatus using a pressurevessel fitted with a carbide core3/q in. in diameter. Glassesof the respective compositions were sealed (without anyseed crystal) with approximately 20-30 wto/o distilled HrOin Au capsules 3 mm in diameter and were held at 25kbar, 1000 "C, for 20 h in a pressure cell with CsCl outerbushing. The products were l00o/o pure garnets (withinthe limits of detection by optical and X-ray methods)with crystal sizes ranging from -200 to 400 pm. Drysynthesis of pyrope at 25 kbar, 1200 "C, for 20 h with 7wto/o pyrope seed was not completely successful; the prod-uct had -50lo extraneous phases.

Syntheses of garnets of intermediate compositions havebeen complicated by the fact that in the system CaSiOr-MgSiO,-Al ,03 (CMA), the c l inopyroxene jo in Ca-Al,SiOu-CaMgSirOu (CTs-Diop) can pierce the grossular-pyrope join, depending on the P-T condition, thusrendering intermediate garnet unstable relative to pyrox-ene (Boyd, 1970). This explains the lack of any mixingproperty data for garnets in the compositional range Grro-Grro. Higher pressure or lower temperature are expectedto stabilize garnet relative to clinopyroxene. Prior to mi-croprobe analysis, all experimental products were checkedfor the homogeneity of garnet composition and the pres-ence ofextraneous phases by XRD scan using Cu radia-tion. The extent of splitting of the Ka, and Ka, doubletabove 50' 20 value was found to be a good indicator ofthe homogeneity of the garnet composition. Microprobeanalysis of garnets, for which the 642 reflection was clear-ly separated into the a, and a, doublet, always showedhomogeneity of composition within +0.50/0.

We found that the best way to synthesize homogeneousgarnet with - l00o/o yield was first to perform a hydro-thermal synthesis at -25 kbar, 1000 'C, for 25 h, thengrind (grain size 5-10 prm) and remix the product in anagate mortar and recycle itat 40-42 kbar, 1350-1400 "C,for -48 h under dry conditions if there was any extra-neous phase or the garnet was heterogeneous in compo-sition, as indicated by the poor resolution of the a,-a,doublet. An example is shown in Figure 2.Here the prod-uct JW7 (Grro) showed broad reflections with only par-tial separation of the doublets, indicating heterogeneity

r.00.80 .60.4a a

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN

TABLE 1. Summary of selected experimental data for syntheses of pyrope-grossular garnet solid solutions

585

Conditions

Expt.- Starting material P (kba0 rcc) r (h) Results..

JWl6JWl25JWl4JWSOJWl3JWI't7JWl28JW18JWt29JW19JW126

JW31

JWOTJWl21JWl20JWl22JW/03JWlsJW23

grassproduct of JWl6grassproduct of JW/14glassglassproduct of JW/17glassproduct ot JW18glassproduct of JW19

product of JW19+3% of JW29as seeo

grassproduct of JW07glassproduct of JW209rassgrassglass (GrroPyso)

25402540252540'25

404240

40

25404240302540

1 00014001 00014001 0001 00014001 0001 4001 0001 400

1 400

1 0001 4001 0001 4001 0001 0001 350

2448204820244824482448

48

Gt, 40-50 pm, heterogeneousGt, -60 pm, &: 0.073(3)Gt, 60-100 pm, heterogeneousGt, x"" : 0.134(3)Gt, -50 pm, Xc": 0.196(8)Gt, -5% CPx, -1% othersGt, -30 pm, Xc": 0.258{4)Gt, -5% CPxct, xc" : 0.384(7)Gt, -15 pm, -5% CPxGt, -15 pm, <3% others,

xc " : 0 .491 (1 )Gt, -3% others, Xc" :0.473(7)

Gt, -20 pm, heterogeneousGt, -20 pm, Xc" : 0.536(3)Gt, 20-40 pm, heterogeneousGt, X"" : 0.702(3)Gt, -35 pm, Xc.: 0.797(21Gt, X"":0.849(3): 22 analysesGt, 5-8% others

254824481 2

48

A/ofe; All experiments at 1000'C were in Au capsules sealed with H,O; others were in graphite capsules (dry).. The experiments are arranged in the order of increasing Ca content (Xc").

-. Stated garnet (Gt) compositions represent average of 12-14 analyses, except for experiment no JW15.

of composition. The latter was, therefore, recycled (JW/21) as described above, which resulted in sharp X-rayreflections with clear separation of the a,-a, doublet. Hy-drothermal experiments at 25 kbar, 1000 'C, yielded ho-mogeneous garnet when the grossular content exceeded80 mol0/0. Attempts to synthesize homogeneous garnetswith lower grossular content directly through dry exper-iment at 40 kbar were not successful (e.g., experimentJW/23).

We have illustrated in Figure 3 the compositions ofsynthetic garnets on the CMA phase diagram of Boyd(1970) at 30 kbar. Boyd's original diagram is in weightpercent and has been recast in terms of mole percent ofthe components. All experiments that are indicated bycircle, excepting the one with garnet composition Gr70(experiment JW/22), yielded single phase garnet underhydrothermal condition aI 25-30 kbar, 1000 'C, 25 h. Ofthese, the products of the experiments JW3 (Groro) andJW/ I 5 (Gro ,r) were very homogeneous and therefore didnot require any recycling. The experiment JW22 wasunder hydrothermal condition at 40 kbar, 1000 "C, andalso yielded single phase garnet (Gr?.). We believe, afterreviewing the results of all synthesis experiments, thatgarnet with Grro composition can be synthesized hydro-thermally at 25 kbar, 1000'C (also see Gasparik, 1984).The products of the experiments indicated by a trianglein Figure 3 (experiment JW / 17 and JW /29) showed smallbut significant amounts of clinopyroxene, along with oth-er extraneous phases, after hydrothermal syntheses at 25kbar, 1000 oC, but these phases disappeared after theproducts were recycled at the higher P-?" condition, as

described above. However, we failed to synthesize singlephase well-crystallized garnet of composition Grro (ex-periment JW I 9 and JW /26), which is suitable for precisecell dimension measurement.

Microprobe analysis

AII synthetic garnet solid solutions were analyzed in aCameca SX50 electron microprobe at the University ofArizona, using an accelerating voltage of 15 kV and abeam current of 30 nA. X-rays were collected for 30 s oneach spot. Synthetic end-member pyrope and grossulargarnets were used as standards. The composition of eachsample reported in Table I represents an average of I 1-l4 spot analyses, except for the product of experimentJW/ 15. It represents an average of 22 spot analyses madein two independent sessions because of the anomalouscell volume of the garnet, as discussed later.

X-n-c.v DrFFRAcrroN sruDIES

Lattice parameter and excess volumemeasurements

Lattice parameters were measured using CoKa, radia-tion (\ : 1.78897 A; on a STOE STADIP focusing dif-fractometer with a curved Ge monochromator. The dif-fractometer operates with a flat, rotating sample in thetransmission geometry mode and is equipped with a po-sition-sensitive detector, with an operating width of about8" 2d. All samples were measured with an internal Si stan-dard from the U.S. National Bureau of Standards (NBS).The diffraction pattern was collected from 6l to l4l" 20,

586 GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN

Al2o3

59 5820(Cu,Q

Fig.2. X-ray diffractometer scans (Cu radiation) of syntheticgarnets, illustrating the distinctive effect of compositional het-erogeneity on the resolution of Ka,-a, doublet around 60' 2d.The product of experiment JW/'l (20 kbar, 1000 'C, 20 h, hy-drothermal) showed poor resolution caused by compositionalheterogeneity, but the resolution improved greatly on recyclingthe product to 40 kbar, 1400 qC, for 48 h.

using y2" increments of the position-sensitive detector.Proprietary software from STOE was used to determinepeak positions by fitting entire peak profiles. The six Sipeaks in the 20 range covered were used to derive a two-term (linear) correction to the observed values of20. Thelattice parameter was then determined by a weightednonlinear least-squares refinement of the positions of thel2-15 most intense reflections (except for samples witha bimodal population of compositions, for which onlythose highest angle reflections in which the different pop-ulations were resolved were used-see below). The peakpositions were weighted according to the estimated stan-dard deviation from the peak profile fitting. The resultsare given in Table 2. The estimated uncertainties includenot only those found from the least-squares procedure,but also those from fitting the internal standard, as wellas an estimated contribution from the effects of thermalexpansion and variation in laboratory temperature. It isto be noted that the relative error in the lattice parametermeasurements is <0. lolo of the variation across the solidsolution and is therefore proportionately less than theuncertainty in the composition.

For most samples the relatively narrow peak half-widths(approaching those of the Si internal standard) confirmthe high degree of compositional homogeneity found byelectron microprobe analysis. Three samples (JW13,JW27, and JW30) were found to consist of two discretegarnet compositions (Table 2). In each case, only thedominant composition was encountered during the mi-croprobe analysis, suggesting that the subordinate com-position might be finer grained. Although the bimodalnature of these samples was readily apparent from ex-amination of the highest angle reflections, it could, in our

I

CaSiO, MgSiO.

Fig. 3. Compositions ofgarnets in the pyrope-grossularjoin,synthesized in this work, compared with the phase diagram ofBoyd (1970) in the system CaSiO.-MgSiO3-Al,O3 (CMA) at 30kbar, I 200 "C (Boyd's original diagram is in weight percent scale,which has been converted here to mole percent). Co*: corundumsolid solution; Di..: diopside solid solution; En*: enstatite solidsolution. The unlabeled fields represent two phase fields con-sisting of garnet solid solution and one of the above phases.Circles indicate successful syntheses of single-phase garnet (butnot necessarily homogeneous) under hydrothermal conditions at25 kbar, 1000 'C, except the one with Xc": 0.70 (experimentJW/20), which was ar 42 kbar, 1000 "C, under hydrothermalcondition. The triangles indicate that the products ofhydrother-mal syntheses at 25 kbar, I 000 "C, contained pyroxene and otherphases, which disappeared on recycling at 40 kbar, 1400'C,under dry condition, yielding single-phase homogeneous garnets.

opinion, easily have been overlooked ifonly lower angle(20 < 90o/o) reflections had been measured, or if unmono-chromated a, I a, radiation had been employed. In suchcases the lattice parameter would be an average of thetwo populations, but the microprobe analysis might bethat of the dominant population only.

Excess molar volumes were obtained by weighted least-squares fitting ofthe data in Table 2, first using a second-order polynomial (equivalent to a regular solution mod-el), and then using a third-order polynomial (subregularor Margules model). The reduced x2 fx2,: x2/(N - n -

1); Bevington, 1969, p.2021for the regular solution mod-el was 8. l, which is unacceptably high; for the subregularmodel, x? decreased to 4.7, a significant improvement,but still high. Higher order expressions produced insig-nificant decreases in x7. The high y2" was entirely due toone apparently anomalous datum, that for JWl5 (Xc" :

0.849), which appeared to have too large a molar volumefor its measured composition. Elimination of this datumfrom the regression gave y2,: 2.5 for the regular solutionmodel and X2r: 0.7 5 for the subregular model. Thus, thesubregular model provides a significantly better represen-

JW21 (Gr54)

40 kb, 14oooc,4g h,ory

o * /

9 X8 l t

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULARJOIN 587

TreLe 2. Compositions, lattice parameters, and molar volumesof pyrope-grossular garnet solid solutions

126

125

124

123

LatticeComposition parameter

Sample X.. (A)

Molarvolume Com-(cm') ments

JW12JW25JW3OJW13JW28JW27JW21JW22JW3JW15JW11

0.000(0)0.073(3)0.134(3)0.1 96(8)0.258(4)0.384(7)0.536(3)0.702(3)0.797(210.849(2)1.000(0)

11.4566(2)1 1.4883(2)1 1 .s125(4)11.5420(4)1 1.5670(2)1 1 .6156(4)1 1.6800(2)'t1.744s(311 1 .7817(3)1 1.8063(3)1 1 .851s(2)

1 1 3.1 9s(6)1 1 4.1 37(6)1 1 4.860(1 2)115.745(1211 16.499(6)117.974(12)1 19.947(6)1 21 .945(9)1 23.1 07(9)1 23.880(9)12s.308(6)

122

" 8 1 2 1oI tzoE

€ 1 l e

b

o

Note: a: Two garnets, other ao : 11.492. Ratio of intensities is about1:0.5. b: Two garnets, other ao: 11.481. Rat io of intensi t ies is about1:0.2. c: Two garnets, other ao : 11 568 Ratio of intensities is about 1:0.6. d : High-angle peaks show some asymmetry to high 2d, indicatingsome zoning toward more Mg-rich compositions.

tation of the V- X relationship in the pyrope-grossular j oin,and is. therefore. recommended in this work. The third-order regression, excluding JWl5, led to the followingexpression of the molar volume of garnet on a 12 O atombasis:

v(cm3): l13.l94 + 0.006 + 12.477(+0.235)Xc^

+ 1.002(+0.604)X?" - 1.36(-19.330)xa.. (l)

Reanalysis of JW15 by electron microprobe confirmedthe init ial analysis (0.848 + 0.002 vs. 0.850 + 0.002;mean 0.849). The lattice parameter measurement was alsorepeated, first using the original mount and both Co andCuKa, radiations, and then with a second mount and Coradiation. The data from these experiments were also re-fined using the Rietveld method (see below). All mea-surements and both determination methods produced alattice constant essentially identical to that determinedinitially. We are therefore unable to explain this anomalyat this stage, but further work is in progress to resolvethis problem.

The V-X relation in pyrope-grossular join is illustratedin Figure 44, whereas the excess molar volumes (Alrys),derived from Equation I and end-member volumes givenin Table 2, are plotted vs. composition in Figure 48. OnlyWood's model (Wood, 1988; see Fig. I of this paper) isin qualitative agreement with our results. The error barsrepresent + 1o and include the errors in the measurementof cell volumes of intermediate and end-member com-posltrons.

Powder X-ray diffraction structuralrefinements

Several samples for which sufficient material was avail-able to make a good mount were selected for furtherstructural investigation by powder XRD. Most of thesespecimens already contained the Si internal standard, andin addition most contained detectable amounts of quartzand some the noble metal (Au or Pt) from the capsule.

F 1 1 8o= 1 1 7

1 1 6

1 1 5

1 1 4

Xc"

Xc"

Fig. 4. (A) Molar volume and (B) excess molar volume Ys.composition in the pyrope-grossular join, as determined in thiswork. The anomalous datum JWl5 has been finally rejected.The error bars represent + lo. The molar volume data are preciseto within the range +0.006 to +0.012 cm3 (see Table 2).

Some samples also contained small amounts of anotherunidentified phase (either indigenous or from the pres-sure assembly used in the synthesis). Because this phasecould not be satisfactorily accounted for in the refinementprocedure, these samples produced inferior results andwill not be considered further.

XRD data were collected using the same diffractometeras for the lattice parameter measurements, but withMoKa, radiation (^ : 0.709300 A;. five or six duplicate

1 13 .0

oEoo 0.3E5o

6

ooooUJ

1 . 0u . 6

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN

oo( ! t A 4

.9,!t.9Eo(!

Q r . ^ ,E r o +

i;=o

0 038

XooES oose(uo-

.9ooo- 0.034

oCDxo

0 032

0 0

Xc"

u.ocu

Xc"

Fig. 5. O positional parameters x (A), y (B), and z (C) as afunction of composition. All estimated standard deviations are0.0002.

scans from 5 to 88" 2d [sin(d/I):0.98] were collected foreach sample and were checked for systematic drift in in-tensity. If that was negligible, the scans were summed toobtain the final powder diffraction pattern. Structures wererefined by the Rietveld method, using the program DBW(version 3.2, locally modified; Wiles and Young, l98l).

1.63

Xc"

1 . 8 6

0 0 0 . 2 0 4 0 . 6 0 . 8 1 0

Xc"

Fig. 6. Metal-O distance in the (a) dodecahedral, (b) octa-hedral, and (c) tetrahedral sites as a function of composition inthe pyrope-grossularjoin. All uncertainties are 0.002 A.

The structural model assumed space group la3d withcomplete occupancy of the 24 c sites by Ca and Mg, the16 a sites by Al, the 24 dsites by Si, and 96 ft sites byO. The ratio of Ca to Mg in the dodecahedral site wastaken from the EMP analyses (Table 2). Refined param-eters were a scale factor, the O atom positional parame-

1 9 40.050

ooE(u(U 0.048TL(6co

o

f; o.o+ecq)EDxo

0.044

N

fi o.ssscoo(!

(U

E o6s2ahoTL

o ^ ^ - .o D u o c rxo

ooE 1 a 2o - -

.2E

.9

O r o n.!

o

i;t 1 a R

u . 6

Xc"Xc"

6ootr(!

.!2!,.9 2.35

o(go.=O 225

(oo-E t=

2 . 1 50.20.0

o

b

c

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULARJOIN

TABLE 3. Powder XRD structural refinements of garnets synthesized on the ioin Mg.AlrSi.O,r-Ca3Al2Si3O12

Composi-tion

Sample X.,

O parameters' lsotropic temperature factors (4'?)

589

8., R","nn RFB*,B*4*JW12JW13JW28JW22JW3JW15JW11

00.1 960.258o.7020.7970.8491

0.03290.033s0.03400.03590.03670.03680.0376

0.05010.04900.04810.04630.04600.04540.0454

0.65280.65270.65250.65180.65140.65150.651 3

0.707(45)0.54s(44)0.579(29)0.486(29)0.423(35)0.345(29)0.323(21)

0.3s7(33)0.1 35(33)0.337(23)o.297(271o.276(37)o.222(33)0.233(28)

0.293(31)0.310(24)0.395(26)0.323(33)0.383(46)0.344(40)0.265(31)

0.290(36)0.215(43)0.399(26)0.216(30)0.245(44)0.191(39)0.189(33)

6.44 8.085.12 5.484.92 5.985.87 6.646.14 7.685.36 6.314.73 5.62

'A l l +0.0002 1 esd.

ters x, y, and z, and the temperature factors. In order tokeep the ratio ofobservations to structural variables rea-sonably high, a simple isotropic thermal motion modelwas employed, with separate isotropic temperature fac-tors for each site (i.e., Bo"o, Bou, Bn, and 8".). Previoussingle-crystal refinements of pyrope and grossular (e.g.,Novak and Gibbs, 1971) have shown that the thermalvibrations of the dodecahedral site atoms and the O at-oms are in fact markedly anisotropic, but that using theapproximation of the isotropic model produces no dis-cernible error in the determination of the positional pa-rameters. Profile functions were selected from experiencewith fitting spinel powder patterns (e.g., O'Neill et al.,l99l). The quartz and noble metal impurity phases andthe Si internal standard were treated similarly, using asingle effective temperature factor for quartz and posi-tional parameters constrained to literature values. Theabsorption for each sample was measured, and the ab-sorption correction (which is virtually negligible) wasmade accordingly.

The results of structural refinement are summarized inTables 3 and 4 and illustrated in Figures 5 and 6. Thedata for the end-member pyrope and grossular comparewell with published data from single-crystal XRD stud-ies. Within the limits of the uncertainty of the data, Opositional parameters (Fig. 5) and cation to O distancesin all three polyhedra (Fig. 6) show linear dependence on

composition. The results of structural refinement are notsufficiently precise to explain the nonlinear change of cellvolume vs. X.". The trends of compositional dependenceof the structural parameters exhibited by the data in Fig-ures 5 and 6 are in agreement with the observations ofNovak and Gibbs (1971), who noticed these trends pri-marily from the results of structural refinements of end-member aluminosilicate garnets. They have cast their datain terms of mean radius of dodecahedral cations r{X}: itis, however, easy to see that linear dependence agalnstr{X} implies linear dependence against X.

The isotropic temperature factors for the octahedral,tetrahedral, and O sites do not show any significant vari-ation with composition; mean values arc B",t: 0.27 +

0.08 Ar, -8,., : 0.33 + 0.05 A', and B.-: 0.25 + 0.08A'. However, .Bo* decreases from 0.7 Lt for pyrope to0.3 At in grossular. These temperature factors agree rea-sonably well with those reported by Meagher (1975) andHazen and Finger (1978).

Inrpr,rcluoNs FoR THERMoDYNAMICMIXING PROPERTIES

Excess and partial molar volumes:Geothermobarometry

The excess molar volume data illustrated in Figure 48can be expressed in terms of Margules formulation as

Taau 4. O positional parameters and isotropic temperature factors of end-member pyrope and grossular

O positional parameters lsotropic temperature factors (4'z)

Reference 4,a8*,Bo*

Gibbs and Smith (1965)Novak and Gibbs (1971)Meagher (1975)Hazen and Finger (1978)This study

Prandl (1966)Prandl (1 966) (neutron diffraction)Novak and Gibbs (1971)Meagher (1975)Hazen and Finger (1978)This study

Pyrope (Mg!Al,Si30l,)'0.0s08(4) 0.6s31(4)0.0502(1) 0.6534(1)0.0503(4) 0.6s34(s)0.0502(2) 0.6534(2)0.0501(2) 0.6528(2)

Grossular (Ca3Al,SiaO1,)*'0.0458(1) 0.6s12(1)0.0457(1) 0.6511(1)0.0449(1) 0.6514(1)0.0447(5) 0.6512(4)0.04s0(1) 0.6518(1)o.o454(2) 0.6513(2)

0.13(4) 0.38(s)0.1s(2) 0.50(3)0.29(1 1) 0.47111)0.32(3) 0.3s(3)0.2s(3) 0.2s(4)

0.24(2) 0.22(210.18(5) 0.33(2)0.56(2) 0.76(2)0.30(1 1) 0.37(9)0.29(4) 0.35(2)0.27(3) 0.19(3)

0.0329(4)0.0329(1 )0.0328(4)0.0328(2)0.0329(2)

0.0381(1)0.0384(1 )0.0381(1)0.0380(5)0.0380(1 )0.0376(2)

0.67(8)0.7e(3)0.93(20)0.70(5)0.71(5)

0.26(2)0.56(5)0 .61(1)0.39(10)0.33(2)0.32(2)

0.1 8(5)0.40(2)0.40(6)0.31(3)0.36(3)

0.20(3)0.31(7)0.66(2)0.40(6)0.25(1)0.23(3)

' Pyrope data from synthetic samples" Grossular data from natural specimens, except this study

590

Fig. 7. Partial molar volumes of end-member components(12 O atom basis) in the pyrope-grossular (solid lines) and al-mandine-grossular (dashed lines) joins. (a) This study. (b) Cres-sey et al. (1978).

follows:

AW" : X-^XM,(W[^M"X*" * W("".^X.") (2)

where W(^*": 0.36 + 0.23 and WkEc^: 1.73 + 0.3 cm3.The partial molar volumes of pyrope and grossular com-ponents, expressed as a function ofcomposition accord-ing to standard thermodynamic principles (e.g., Gangulyand Saxena, 1987, p. 8), is as follows:

2".(Gr-Py) : A + lW(^*" + z(Whsc^ - W(^.")

.X."lX'-" (3a)

2",(Gr-py) : A + lw{o"." + 2(w(^ME - w(,".^)

.X-"|X"^ (3b)

where I : X."VZ,, * X-"Qr.Geiger et al. ( I 989) have determined the molar volume

in the almandine-grossular join, which disagrees with theresults of Cressey et al. (1978), but is very similar to thevolumetric behavior of the pyrope-grossular join deter-mined in this work. The partial molar volumes of gros-sular and almandine components derived from the dataof Geiger et al. are as follows:

7".(Gr-Alm) : B + lW("o. + 2(W(..^ - W[",.)

x.^lx?.

7",-(Gr-Alm) : B + lW{".^ * 2(W(^r. - W(".")

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN

Calculation of the equilibrium pressure of the assem-blage garnet + aluminosilicate * plagioclase + quartz(GASP), which is one of the most extensively used geo-barometers, requires data on the partial molar volume ofthe grossular component in multicomponent garnet. Nat-ural garnets in the GASP assemblage have 2- 18 mol0/ogrossular (Newton and Haselton, l98l) and thus haveZo. values that are often significantly different from themolar volume of grossular, according to the data of Cres-sey et al. (1978) and Newton et al. (1977), but they arenot so according to our data and those of Geiger et al.( 1989). Since Zo, changes very little with the Fe/Mg ratio(Fig. 7), one would make a negligible error by setting Zo,in ternary Ca-Fe-Mg garnet equal to the weighted averageof Zo. in the two limiting binary joins, as suggested byNewton and Haselton (1981). It can be easily shown, us-ing the barometric expression of the GASP assemblage(e.g., Ganguly and Saxena, 1987, their F,q.4.27), that evenif the slight volumetric nonideality given by our data iscompletely neglected, one would underestimate the equi-librium pressure of the GASP assemblage by no morethan l.5ol0.

The partial molar volumes of almandine and pyropecomponents are essentially independent ofgrossular con-centration up to -50 molo/o Ca, which covers most geo-logically important situations. For extremely Ca-richcompositions, such as in grosspydites (Sobolev et al.,1968), the partial molar volumes of almandine and py-rope depart significantly from their respective molar vol-umes. However, no correction for volumetric nonidealityof garnet is needed for the calculation of the pressureeffect on Fe-Mg exchange geothermometers, since the Cahas a very similar effect on the partial molar volumes ofboth pyrope and almandine components.

Elastic enthalpy of mixing

The process of formation of a solid solution lxAQ + Q- x)BO - @P,)61can be formally thought to consistof the following steps (Ferreira et al., 1988): (1) changeof the molar volumes of the end-member components tothe molar volume Z of the solid solution, i.e., xVo(A"6)- xV ard(l - x)w(B d) - (l - x)V, and (2) mixing ofthese components to form a chemically homogeneousmixture (which coffesponds to spin-flips in classic Isingmodels). Steps (l) and (2) are often referred to as theelastic and chemical components of the enthalpy of mix-ing, respectively. Ferreira et al. (1988) have developedexpressions to calculate the elastic effect and have shownthat ignoring the effect of multiatom interaction leads tosubstantial overestimation of the elastic enthalpy of mix-ing. In the zeroth approximation, i.e., ignoring the effectof multiatom interaction. one has

where -B : Xr^Vo", * Xr"V\,^, W("r. : 0, and W(".^: 3.0cm3. The partial molar volumes of garnet components inthe binary joins pyrope-grossular and almandine-gros-sular, as derived above, are illustrated in Figure 7 andcompared with the results of Cressey et al. (1978), whoderived these values from their own volume data on thealmandine-grossular join and from the data of Newton etal. (1977) on the pyrope-grossular join.

r ro"Gr

vol'

rro'Pv

X,.lXL^

@a)

(4b)

AHo,"ru*: (l - 4 ["" xz14 ax

--,=-=f-!!:;::::---a="'-1:-,

** l ,e-nz(ndx (5a)

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN 591

where

t:t(f, (5b)

and,B is the bulk modulus. The subscript zero indicateszeroth approximation.

With a subregular expression for AV* (Eq. 2),

{ * : v l - vg )+w ,z+A ,+A2(6)

whereA, : 2(Wr, - 2Wn)X and Ar: 3(W,, - Wr,)Y and X: X, .

Using Equations 5 and 6, and the volume data deter-mined in this work (Eq. 2), we have calculated Af10,",u., inthe pyrope-grossular join. For simplicity, we have as-sumed that the ratio B/V changes linearly between theterminal compositions (a similar assumption was madeby Ferreira et al., 1988). This assumption implies thatthe slope of V-P curve at ary P-T condition changes lin-early with composition. The bulk moduli for pyrope andgrossular were taken from Leitner et al. (1980) and Bass(1989), respectively. The calculated elastic enthalpy canbe represented by a subregular form, A,F1 : W(x)x(l -

x), where W(x): WIl,x I WI;!.(L - x).Following Ferreira et al. (1988), the effect of multiatom

interaction can be incorporated as follows. Ifonly nearestneighbor interactions are significant, and there are m(Mg* Ca) atoms in the nearest neighbor cell with subregularmixing behavior, then the net elastic enthalpy of mixingis given by

l - rA11.,u",: W(x)lx(t - x) - ) r14X,1t - X,)l (7)

where n : mx,,x. being rn" .rrr,...o-porition]so that0 - n - m), X,: n/m, and P"(x) is the random proba-bility for m atom clusters at a composition x of the solidsolution. The second term in Equation 7 embodies theelastic effect of a medium-cluster interaction.

It can be shown (Appendix l) that

2 P"@)x.(rn = O

so that

- x , ) : ry -J rg - 11 (8 )

. - . . . f . m- t 1Afl.,u.,: W(x\lx(t - x) - .i-*<l - x)l

W(x): --'--x(l - x)

:^!#- (e)In other words, the net elastic enthalpy of mixing is lowerby a factor of m (i.e., by the number of atoms in thenearest neighbor cell participating in solid solution) thanthat calculated in the zeroth approximation.

Fig. 8. Unit cell of pyrope showing only Mg atoms. The near-est neighbor Mg atoms, which are 3.509 A apart, form triangularclusters.

In garnet, the divalent (dodecahedral) cations consti-tute triangular nearest neighbor cells. This is illustratedin Figure 8 by removing all but the Mg atoms from theunit cell of pyrope and rotating the unit cell arbitrarilyfor the sake of clarity. The drawing was done by usingthe program Atoms (Dowty, l99l: a computer programfor displaying atomic structures), which is available com-mercially (copyright: Eric Dowty). The nearest neighborMg atoms, which are 3.509 A apart, were identified byusing the program Distance, written by Antonio dellaGuista of the University of Padova, Italy, and then joinedtogether to form the nearest neighbor clusters (the nextnearest neighbors are 5.36 A apart).

Setting m : 3, we have calculated the elastic enthalpyof mixing in garnet according to Equations 5 and 9. Theresults are illustrated in Figure 9A,; also shown for com-parison is the A.FI.,"", if the V-X relation in the pyrope-grossular join were ideal. Figure 98 shows a comparisonof A.FI",,., with the enthalpy of mixing calculated from theheat of solution measurement by Newton et al. (1977) ofend-member and solid solutions in the pyrope-grossularjoin. The calorimetric enthalpy of mixing was calculatedaccording to the procedure and weighting scheme sug-gested by Haselton and Newton (1980). These data, whichform the basis of existing mixing models in the pyrope-grossular join (e.9., Ganguly and Saxena,1987; Berman,I 990), lie close to the enthalpy of mixing calculated fromthe volume of mixing (elastic model), but have an asym-metry toward the opposite (i.e., pyrope) end. The reasonfor this difference in the nature of the asymmelry is notclear. The calorimetric enthalpy of mixing incorporatesboth chemical and elastic effects. Thus, the difference be-tween the calorimetric and elastic enthalpy of mixing,includine that in the nature of the asvmmetrv. mav be

592

Pyrope-Grossulor

f't=o-=.._ - -__

Pyrope-Grossrlol. Xco 1 -Cot ion

TElost ic model

f : Color imetr ic doto

0.0 0-2 0.4 0.6V̂Co

Fig. 9. (A) Effect of elastic strain energy on the enthalpy ofmixing in the pyrope-grossular join. (B) Comparison of the elas-tic and calorimetric enthalpies of mixing. The calorimetric dataare from Newlon el al. (1977). Vertical bars: + lo.

due to chemical factors. We note. however. that all otherbinary joins in the garnet solid solution investigated sofar (Mg-Fe: Hackler and Wood, 1989; Fe-Ca: Geiger etal., 1987) show maximum (net) positive deviation fromideality toward the larger cation, which is in contrast tothe form of the calorimetic AH data in the pyrope-gros-sular join.

Reciprocal solution effect and stability

The dependence ofthe octahedral bond distance on thesize of the dodecahedral cation, as illustrated in Figure68, suggests that the mixing property of trivalent cationsin the octahedral site ofsilicate garnet should, in princi-ple, depend on the composition of dodecahedral site, andvice versa. As discussed by Ganguly and Saxena (1987),in the treatment of multisite or reciprocal solid solutions,this interdependence of the site mixing properties is usu-ally ignored in the formulation of activity expression ofmacroscopic end-member components. Whether this ef-fect is significant in rnultisite garnet solid solutions com-pared with the other effects, especially that arising fromthe standard free energy change of the reciprocal ex-change reaction, remains to be seen.

In garnets formed under crustal conditions, the con-

GANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN

1-Co t i on

F 2c

l rox nr -

- 2t-

l r

r "

- 1

centration of Fe3* and Cr3+ in the octahedral site canalways be expressed in terms of the andradrite(CarFerSirO,r) and uvarovite (Ca.CrrSirO,r) compo-nents, respectively. This is because Fe3* and Cr3* are toolarge for the octahedral site when the dodecahedral siteis filled with the divalent cations Fe, Mg, and Mn. Therequired expansion of the octahedral site is achievedthrough the substitution of Ca in the dodecahedral site.It is only in mantle-derived garnets that we find t6lcr3+partially decoupled from Ca. Here the high pressures fa-cilitate substitution or squeezing of the relatively largertrivalent cations into the octahedral site. Ringwood (1977)has synthesized MgrAlrSirO,r-Mg.Cr2Si3O,' (pyrope-knorringite) solid solutions at 60-80 kbar, I 400- I 500 "C.

AcxNowr,nocMENTS

This research was supported by a U.S. National Science Foundationgrant no. EAR 8903995 and9ll7927 to J.G We are grateful to Antoniodella Guista for his help in drawing Figure 8. Thanks are due to BenjaminBurton and an anonymous reviewer for their constructive comments, andto Luciano Ungaretti for comments and insightful discussions about thestructural properties of garnets.

RBronrNcBS crrED

Bass, J.D. (1989) Elasticity of spessartine garnets by Brillouin spectros-copy. Journal of Geophysical Research, 94, 7 621-7 628.

Berman, R. (1990) Mixing properties of Ca-Mg-Fe-Mn garnets. AmericanMineralogist, 7 5, 328-344.

Bevington, P.R.C. (1969) Data reduction and error analysis for the phys-ical sciences, 336 p. McGraw-Hill, New York.

Boyd, FR. (1970) Garnet peridotites and the system CaSiO,-MgSiO'-Al,O.. Mineralogical Society of America Special Paper, 3,63-75

Cressey, G., Schmid, R., and Wood, B.J. (1978) Thermodynamic prop-erties of almandine-grossular gamet solid solutions. Contributions toMineralogy and Petrology, 67, 397-404.

Delaney, J.M. (1981) A spectral and thermodynamic investigation of syn-thetic pyrope-grossular garnets, 201 p. Ph.D. thesis, University ofCal-ifornia, Los Angeles, Californra.

Dowty, E. (1991) ATOMS: A computer program for displaying atomicstructures. Copyright E. Dowty, 521 Hidden Valley Road, Kingsport,Tennessee 37663, USA.

Ferreira, L.G., Mbaye, A.A., and Zunger, A. (1988) Chemical and elasticeffects on isostructural phase diagrams: The e-G approach. PhysicalReview B. 37. 10547-10570.

Ganguly, J., and Saxena, S.K. (1984) Mixing properties of aluminosilicategarnets: Constraints from natural and experimental data, and applica-tions to geothermo-barometry. American Mineralogist, 69, 88-97.

-(1987) Mixtures and mineral reactions, 291 p. Spnnger-Verlag,Berlin.

Gasparik, T. ( I 984) Experimentally determined stability ofclinopyroxene+ garnet + corundum in the system CaO-MgO-AlrOr-SiOr. AmericanMineralogist, 69, 1025-1035.

Geiger, C.A., Newton, R.C., and Kleppa, O.J. (1987) Enthalpy of rnixingof synthetic almandine-grossular and almandine-pyrope garnets fromhigh temperature solution calorimetry. Geochimica et CosmochimicaActa.5l . 1755-1763.

Geiger, C.A., Winkler, B , and langer, K. (1989) Infrared spectra of syn-thetic almandine-grossular and almandine-pyrope solid solutions: Ev-idence for equivalent site behavior. Mineralogical Magazine, 53,231-23'1.

Gibbs, G.V., and Smith, J.V. (1965) Refinement of the crystal structureof synthetic pyrope. American Mineralogist, 50, 2023-2039.

Hackler, R.T., and Wood, B.J (1989) Experimental determination of Feand Mg exchange between garnet and olivine and estimation of Fe-Mggarnet mixing properties. American Mineralogist, 7 4, 994-999.

Haselton, H.T., and Newton, R.C. (1980) Thermodynamics of pyrope-

0.4

CANGULY ET AL.: GARNETS IN THE PYROPE-GROSSULAR JOIN 593

grossular garnets and their stabilities at high temperatures and pres-sures. Journal of Geophysical Research (G.C. Kennedy volume), 85,6973-6982

Hazen, R.M., and Finger, L.W. (1978) Crystal structures and compressibilities of pyrope and grossular to 60 kbar. American Mineralogist, 63,297-303.

I-eitner, B.J.D., Weidner, J., and Liebermann, R.C. (1980) Elasticity ofsingle crystal pyrope and implications for garnet solid solution series.Physics ofthe Earth Planetary Interiors, 22, lll-121.

Meagher, E.A. (1975) The crystal structure of pyrope and grossularite atelevated temp€ratures. American Mineralogist, 60, 218-228.

Newton, R.C, and Haselton, H.T. (1981)Thermodynamicsof thegarnet-plagioclase-Al,SiO,-quartz geobarometer. Advances in Physical Geo-chemistry, l , l3 l -147.

Ne*on, R.C., and Wood, B.J. (1980) Volume behavior of silicate solidsolutions. American Mineralogist, 65, 7 33-7 45.

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Novak, G.A., and Gibbs, G.V. (1971) The crysral chemistry of the silicategarnets. American Mineralogist, 56, 7 9 l-825.

O'Neill, H.SI.C., Dollase, WA, and Ross II, C.R. (1991) Temperaturedependence of the cation distribution in nickel aluminate (NiAlrO.)spinel: A powder XRD study Physics and Chemistry of Minerals, 18,302-319.

Prandl, W. (1966) Verfeinerung der Kristallstruktur des Grossulars mitNeutronen- und Rdntgenstrahlbeugung. Zeitschrift fiir Kristallogra-ph ie , 123 , 8 l - l 16 .

Ringwood, A.E. (1977) Synthesis of pyrope-knorringite solid solutions.Earth and Planetary Science l-etters, 36,443-448.

Sobolev, VN, Jr, Kuznetsova, I.K., and Zyuzin, N.I. (1968) The pe-trology ofgrosspydite xenoliths from zagadochnaya kimberlite pipe inYakutia. Journal of Petrology, 9, 253-280.

Wiles, D.8., and Young, R.A. (1981) A new computer prog.ram for Riet-veld analysis ofX-ray powder diffraction pattems Journal ofAppliedCrystallography, 14, 149-1 5 l.

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MrNuscnrrr RECETvED Aucusr l l, 1992Merusctrsr AccEprED JeNu,,,nv 8. 1993

AppnNrrx l: DpnryarroN oF THE EFFECT oFMEDIUM-CLUSTER INTERACTION ON THE

ENTHALPY OF MrxrNG (Snn rrxr, Eq. 8)

Let us consider a solid solution (A,B | ,)C in which thenearest neighbor cell consists of m(A * ,B) atoms. Let.{. be the fractional concentration of A in an m-atomcluster. The composition of such a cluster can also beexpressed as (A,8,_,")C,,, where n : mxc (so that 0 < n- m). The random probability for m-aIom clusters ofcomposition x. is given by (Ferreira et al., 1988):

P,(x): (Cr)x '( t - x)- , (A l )

Consequently, the second term within the square bracketof Equation 7, which accounts for the effect of medium-cluster interaction, can be expressed as follows:

:2#6,'.(, ,^ .*(t;)----)l)J------ yn ( l- l ) l ( m - n - l ) t - ' \ '

-.r-'C) (A2)- s- Z J

,=o

(m - l ) l(n

)ZJ,=0

)n:o

The first and last terms in the above summation equalzero. Thus,

2 p,@)x,(r - x,)'

: ; . , \T- r ) t , . ,x , ( t _" )__, (1)? , t n - l ) | . (m - , ? - l ) ! - ' ' -

\ ^ /

( m - t \ l I(m - 2y." ' ' " / m

( m - t ) l I*

I t l . - t x ' ( l - x)" ' ' '

m

* = ( , ! - l \ , 1 , x 3 ( l - x ) - + ' ' ''

2 ! (m - 4 ) l ^ ' '

( m - 1 ' 1 r ' ' ' I*

1^ - +1ZY'n- t ( l - 1)3 ' -

(m - l ' l t .- I+ - Y " t - ! l l - Y l z . -'

(m - 3) l t l " -

( m - l ) l I+ 1^ - 2y*-

' | t - x) ' i

( m - l ) l: (m - 'z)tnx(l

- x)

I t m - 2 \ t' l t t - x ; ' ' ' + 1 1 ; ; x ( l - ; 6 ) - - rL l i \m - J) , .

(m - 2\r.+ --l-----------::- v2( | - X),,, r +

2 l ( n - 4 ) ! - - '

* (m - 2 ) t y_a( l _ x )2' (m - 4)t2t^

(m - 2\r. -l

+ . . - , . r , , , ( l _ x ) + x__z l

(m - 3) l . t l ' - Im - 1

: * t t - x) [ ( l - x) - z + Cl" 'x( l - x) - t

+ C l "_ rx2 ( l - x ) ' " - a + ' . .

+ Cn-tx- +(l - a)z

+ C',1,1x--3(l - x) + n- 'z]. (A3)

Noting that the terms within the square brackets ofEquation A3 represent a binomial expansion of [(1 - x)+ xl--', which is unity, we have

P,(x)X^(t - X,):--J*1 - ,1. (A4)P"(x)X,(t - X,)