American Mineralogist, Volume 77, pages 399-406, 1992

Crystal chemistry and symmetry of a birefringent tetragonalpyralspiteTs-grandite2s garnet

D.qsa T. GnrrrnxDepartment of Geology, Brigham Young University, Provo, Utah 84602, U.S.A.

DonreN M. HarcrrDepartment of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, U.S.A.

Wrr,r,rau RBvBr-r, Pnrr,r,rpsDepartment of Geology, Brigham Young University, Provo, Utah 84602, U.S.A.

Spvrr Kur,arsrzMinerals Engineering Department, Hacettepe University, Ankara, Turkey

AssrnA,cr

The crystal chemistry of a natural, birefringent garnet of composition (Fe, ,rCa" ,rMgo ,o-Moo ,o)r=, n, (Al, ,uFeo o.Ti' 0r)::2 00Si3 0rO,, (approximately pyralspiterr-granditerr), producedby subduction-zone metamorphism, has been investigated by single-crystal X-ray diffrac-tion, electron microprobe analysis, Mdssbauer spectroscopy, and Fourier-transform infra-red spectroscopy. By FTIR the garnet is apparently indistinguishable from synthetic gar-nets of similar compositions and is anhydrous. The crystal structure has been refined inspace group 14,/acd, using 592 unique, observed reflections measured with an automatedfour-circle X-ray diffractometer. Dimensions of the tetragonal unit cell are a : 11.6207(7),c: 11.6230(8) A. fne final conventional residuals are R : 0.033 and R* :0.024.X-raysite refinement indicates a slight difference in electron density at the two unique X sites,and this is apparently the cause ofbirefringence (-0.001). It is proposed that the differencein electron density arises from slight ordering between the pyralspite X components (Fe2*,Mg, and Mn) and the grandite X component (Ca). The high pressure of the geologicenvironment evidently favored substantial solid solution between pyralspite and granditeend-members, whereas comparatively low temperatures prevented complete disorder.

A group theoretical analysis based on the Landau formalism and including the theoryofinduced representations shows that the observed space group could have arisen by phasetransformation from a parent cubic garnet (space group la3d), driven by a single orderparameter in each case. Such transitions may occur (l) by the Tr, irreducible representation(IR), leading to possible space groups R3c, Fddd, C2/c, or Il; (2) by the T,, IR, leading topossible space groups R3, 14,/a. C2/c, or 1T; or (3) by the E, IR, leading to possible spacegroups 14,/acd or lbca. The first two ordering paths (Tr, and T,r) yield octahedral ordering;if the sizes of the octahedral cations are similar (as with Al and Fe3t in grandites), orderingoccurs by Tr* and if they are very different (as in MgSiO. and MnSiO, garnets), orderingoccurs by T,r. Garnets that possess cations ordered in the eightfold-coordinated sites (forexample, some with both significant pyralspite and grossular components) order by Er.Thus far, the noncubic garnets with refined crystal structures possess space groups Fdddand 1l (Tr"), 14,/a (T,"), and 14,/acd (E").

The approach presented is based on the assumption that phase transitions may haveoccurred to produce garnets of noncubic symmetry. The mathematical treatment used,however, yields the same predictions about possible space groups if ordering is assumedto have arisen as a crystal-growth phenomenon rather than by phase transformation. Theexperimental and field evidence accumulated so far does not provide clear evidence infavor ofone or the other, and it is possible that each plays a role under different circum-stances.

INtnooucrroN

Among the major rock-forming silicates, garnets con-stitute one of the few groups that crystallize with cubic

0003-004x/92l03044399$02.00

symmetry. Garnets occur in a wide variety of rocks andshow great variation in chemical composition, but nearlyall have symmetry described by space group la3d. lnthe

399

400 GRIFFEN ET AL.: BIREFRINGENT GARNET

general formula, {X}rlYl,(Z)rO,, (notation of Geller,1967), {X} is most commonly Fe2*, Mg, Mn, or Ca ineightfold coordination on the Wyckotr 24c position, IYlis most commonly Fe3* or Al in octahedral coordinationat the Wyckoff 16a position, and (Z) is tetrahedrally co-ordinated Si at the Wyckoff24d position. Natural garnetsare often divided into two groups: the pyralspites, con-sisting of solid solutions of pyrope (MgrAlrSirO,r), al-mandine (FerAlrSirO,.), and spessartine (Mn.AlrSi.O'r);and the grandites, solid solutions of grossular(CarAlrSirO,r) and andradite (Ca3FerSi3O,r). Garnets in-termediate in composition between the pyralspites andthe grandites are much less common than those withinthe two groups, although this seems to reflect the fact thatthe conditions of pressure, temperature, and compositionrequired for their formation are rarely simultaneously at-tained rather than to imply a crystal-chemical limitationof the structure (Nemec, 1967).

Although nearly all garnets found in the crust of theEarth are isotropic, consistent with their cubic symmetry,some are not. Optically anisotropic members of the gros-sular-andradite solid-solution series exist, and crystal-structure refinements of five of them (Tak6uchi et al.,1982; Allen and Buseck, 1988; Kingma and Downs, 1989)have shown that they crystallize in space groups Fddd or1T. Other space groups are possible (Hatch and Griffen,1989) but have not yet been observed. In each case, theoptical (and thus structural) anisotropy has been causedby ordering of the [Il cations, Fe3t and Al. [The garnetof Allen and Buseck (1988) has nearly as much alman-dine component as andradite component, as well as asmall amount of OH; it shows ordering of Ca and Fe2*on the {X} sites and a noncubic distribution of OH groups,in addition to ordering at the [I] sites.l

In this paper we report the crystal chemistry of a nat-ural anisotropic garnet intermediate in composition be-tween the pyralspite and the grandite groups. Anisotropicgarnets in this composition range have not previouslybeen reported. There has been one study ofa pyrope sam-ple containing 10 molo/o grossular component (Dempsey,1980) that showed difraction maxima on precession pho-tographs that violated space group la3d, but the probablespace group was deduced as I2r3 and the garnet was notanisotropic. Additionally, the average, cubic structure ofa natural birefringent spessartine sample was refined bySm1"th et al. (1990), but the true structure was not.

The garnet reported here came from a rock sample des-ignated as T-KAY-K-l (Kulaksiz and Phillips, 1983) andwill be referred to in the present paper as garnet T-KAY-K-1. The regional setting is a paleosubduction zone ofmid-Cretaceous age (85.7 + 1.67 Ma), located in north-west Turkey between Sivrihisar and Balgikhisar. The gar-net occurs as highly fractured, euhedral, dodecahedronporphyroblasts in an eclogite fragment from a melangeconsisting mostly of low-grade marble. The rock consistslargely of omphacitic pyroxene, epidote, and white mica,with less abundant garnet, rutile, and sodic amphibole.

ExprnrlmNTAL METHoDS

Chemical and spectroscopic methods

A room-temperature Mtjssbauer spectrum was ob-tained with the spectrometer in the Chemical EngineeringDepartment of Brigham Young University. Velocitieswere varied between *4 and -4 mm/s, with metallic Feas standard.

Chemical analyses were carried out using the Camecaelectron microprobe at the Department of Geology andGeophysics of the University of Utah, at an operatingvoltage of 15 kV and beam current of 15 nA. Specimensanalyzed consisted of polished fragments from severalcrystals. Well-characterized silicates and oxides were cho-sen as standards. Data were obtained using wavelength-dispersive spectrometers, and an energy-dispersive spec-trometer was used simultaneously to monitor the analysisof each spot to ensure that all elements present aboveEDS detection limits were accounted for. Data reductionwas carried out on-line. Analyses were performed for Si,Ti, Al, Fe, Mg, Mn, Ca, K, Na, and F; the latter threewere not detected.

A Perkin Elmer 1600 series FTIR was used to obtainan infrared spectrum from a KBr pellet containing 0.25wto/o garnet. The spectrum is similar to those for syntheticalmandine-grossular and almandine-pyrope garnets(Geiger et al., 1989), with bands A through I prominent(Moore etal., l97l). The spectrum showed the mineralto be anhydrous.

The garnet is weakly birefringent Oiref. t 0.001) withmean index of refraction 1.797 + 0.002, measured bystandard oil-immersion techniques.

Crystal structure refinement

Long-exposure precession photographs were taken ofasmall euhedral crystal that displayed nonuniform extinc-tion, and no indications of twinning, noncubic lattice pa-rameters, or reflections violating space group IaSd wereobserved. A birefringent fragment with uniform extinc-tion, approximately cubic in shape and 0.08 mm on eachside, was then cut from a larger chip and mounted on aNicolet R3 automated four-circle, single-crystal X-ray dif-fractometer equipped with a graphite monochromator andMoKa radiation. Unit-cell parameters were deterrninedby least-squares refinement of25 reflections in the range44 < l20l < 59o automatically centered at both *20 ar;'d-20.The unit cell is pseudocubic, but the results of thestructure refinement, given below, show that the actualsymmetry is tetragonal. The cell parameter obtained byassuming cubic symmetry is a : I1.6215(4) A.

A total of 4988 reflections were measured in the 0'20mode from half of the Ewald sphere out to sin 0/^,: 0.70.Scan speeds were automatically varied from 3.0 to 29.3"20/min depending on diffracted intensity. Three stan-dards were monitored once in every 100 reflections, andthe observed intensities were corrected according to vari-ations in the intensities of the standards. Lorentz and

GRIFFEN ET AL.: BIREFRINGENT GARNET 401

polarization coffections were applied, as well as an em-pirical absorption correction (p = 59 cm ') based on ry'scans (North et al., 1968) and an isotropic extinction cor-rection.

As a first step in determining the crystal structure, thedata were refined in the average (cubic) space group 1a3d.Of the 4988 intensities measured, 807 were rejected asunobserved (zero or negative intensities), and the re-maining 4181 were reduced to a symmetry-merged set of201 unique reflections, of which 190 had I > 3o(I) andwere used in the refinement. Starting positional parame-ters for the O atom (the only atom not on a special po-sition) were approximated from the regression equationsof Novak and Gibbs (1971). Atomic scattering factorswere taken from Cromer and Mann (1968) and anoma-lous-dispersion factors from Cromer and Liberman(1970). Data processing and refinement were carried outwith the program package SHELXTL (Sheldrick, 1983).

By varying the positional parameters for O, isotropictemperature factors for all atoms, and site-occupancy fac-tors for the {X} and [Il cations, refinement convergedrapidly to R : 0.06. Adoption of anisotropic tempera-ture-factor coefficients for all atoms and of a weightingscheme lw : l/o2(F)l led to conventional residuals of R: 0.045 and R* : 0.028. The final refined O position inthis average space group has x : 0.0347(2), y: 0.0476(2),z : 0.6525(2); based on modeled occupancy by Ca andAl, the site occupancy factors of the {X} and [ts] sites arel. 168(4) and 0.99 l(8), respectively.

In an attempt to determine the true symmetry, thestructure was refined in space group 1T and a difference-Fourier map was calculated with Ca in the six symmet-rically unique X sites, Al in the eight I sites, and Si inthe six Z siIes. The only signiflcant peaks on the differ-ence map were at the Xpositions and ranged from 5.5 to5.9 e/ A. in heigh! they were not different enough to pro-vide a clear indication of the true symmetry. Metrically,however, the triclinic unit cell, in the same orientation asthe cubic cell used above, had dimensions a : I 1.622(1),b: 11.625(l), and c:11.622(l) with interaxial anglesall statistically 90'; this suggested the possibility of a te-tragonal unit cell with basis vectors o, E c", b, = &", c, =b", where subscripts t and c refer to tetragonal and cubic,respectively. Possible tetragonal space groups that aresubgroups of la3d, and thus would allow the garnet struc-ture to be maintained, are 14,/acd and 14,/a. Refinementin space group 14,/acd was attempted.

The refined unit-cell dimensions, assuming tetragonalsymmetry and using the same 25 reflections obtained forthe cubic-cell refinement, are a : 11.6207(7) and c :11.6230(8). Merging of the original 4988 reflections inLaue group 4/mmm resulted in 592 symmetry-indepen-dent reflections, ofwhich the 428 that had 1 > 3o(I) wereused in the refrnement. Starting values for variable po-sitional parameters were determined by adding small,random amounts to the corresponding values obtained inone of the intermediate stages of the cubic refinement.

The initial four cycles ofunweighted least-squares refine-ment, in which positional and isotropic thermal param-

eters but not site occupancy factors were varied, con-verged to R : 0.06. The introduction of anisotropictemperature-factor coefficients, variable site-occupancyfactors for the two unique X sites and the single uniquey site in this space group, and a weighting scheme [w :

l/o'(F)l resulted in final residuals of R : 0.033 and R* :

0.024. Observed and calculated structure amplitudes forthe tetragonal refinement are given in Table 1.'

Rnsur,rs AND DIScussIoN

Chemical relations

Results of the microprobe analysis are presented in Ta-ble 2, with end-member concentrations calculated by themethod of Rickwood (1968). FeO obtained by micro-probe was converted to FeO * FerO, by assuming thatthe deficiency of Al + Ti available for the I site is madeup by Fe3*. Minor chemical heterogeneity was noted inthe analyses from spot to spot and from grain to grain.At the spots atalyzed, Mn varies from 0.89 to 2.17 wto/o,corresponding to 0.07 to 0. I 5 atoms per l2 O atoms, andMg varies from 0.05 to 0.18 wt0/0, corresponding to 0.21to 0.28 atoms per 12 O atoms. Moreover, Mn and Mgvary antipathetically, whereas concentrations of all otherelements remain virtually constant, indicating mutualsubstitution of spessartine and pyrope components. Giv-en the low concentrations of both components, the com-positional variations do not significantly atrect the resultsof the crystal-structure refinement.

The Mdssbauer spectmm (Fig. l) was fitted with fourLorentzian peaks constrained to be equal in width andintensity as doublets. The result yielded one Fe2* doubletwith an isomer shift of 1.31 mm/s and quadrupole split-ting of 3.56 mm/s and one Fe3* doublet with an isomershift of 0.37 mm/s and quadrupole splitting of 0.44 mm/s. The 12 value for the fit is 1.095. Peak areas indicateIhatg4o/o of Fe,., is Fe2* and 60/o is Fe3*, but the spectrumexhibits substantial scatter in the Fe3* region and consid-erable error in these percentages is suspected (see, forexample, Dyar, 1984, concerning errors in peak areas).Further, the X-ray site refinement discussed below doesnot support this much Fe3*. We have taken the micro-probe results reported above to be a more accurate ap-proximation of Fe3* concentration, but the Mossbauerspectrum does confirm the presence of minor trivalentFe.

The site-occupancy factors for the two unique X sitesrefined to l.l82 and l.l7l, corresponding on (weighted)average to an atom with 17.80/0 more scattering power

than Ca (the model atom used in the refinement), in good

' Copies ofTables l, 4, 10, and I I may be ordered as Docu-ment AM-92-493 from the Business Ofrce, Mineralogical So-ciety of America, 1130 Seventeenth Street NW, Suite 330,Washington, DC 20036, U.S.A. Please remit $5.00 in advancefor the microfiche.

402 GRIFFEN ET AL.: BIREFRINGENT GARNET

TABLE 2. Chemical composition of garnet T-KAY-K-1

Ox. wt% Cations*

3.010 0 11.96

0.51 0.0328.28 1.88

0.100.240.7s7.98

(Fe, BsCao 75Mgo 21Mno jo)>=2 e7(Al1 s6Feo @Tio 01),=2 mSi3 01O12(alm6rTprpsosps34Xgrs45adrl 3xschor)t r pyralspiteTs-grandite25

. Normalized to 12 O atoms.-'See text for discussion ot t**rt""t.t See Kretz (1 983) for abbreviations of components; sch : schorlamite.

agreement with the chemical formula {X} : Fe, *rCso rr-Mgo roMno ,o (16.60lo more scattering power than Ca). Theoccupancy factor for the I site refined to a value corre-sponding to occupancy by Al within a standard deviation,about l.8o/o less than implied by the formula [I/j : Al, rr-Feo o.Tio 0,.

Crystal structure

Because of the somewhat unusual composition of thisgarnet (approximately pyralspiteo rr-granditeo rr) and thefact that it deviates so little from cubic symmetry, thebond lengths and angles for the average structure are giv-en in Table 3 and the anisotropic temperature-factor co-efficients are given in Table 4' to facilitate comparisonwith other garnet structures refined in space group la3d.

Owing to the pseudosymmetry of garnet T-KAY-K-I,each ofthe several possibilities for the correct space groupmay yield apparently acceptable refinement results. An-isotropic garnets of quite different compositions from thisone have been refined in space groups 11 (Tak6uchi etal., 1982; Allen and Buseck, 1988; Kingma and Downs,1989), Fddd (Tak6uchi et al., 1982), and I4'/a (Prewitt

TABLE 3. Selected interatomic distances (A) and angles (") fromrefin€ment of garnet T-KAY-K-1 in the averge spacegroup, /a3d

- 3 .0 - 20

Velocity (mm/s, relative o iron)

Fig. 1. The Mdssbauer spectrum of garnet T-KAY-K-1.

and Sleight, 1969; Fujino et al., 1986; Angel et al., 1989),and other space groups are possible (see Hatch and Grif-fen, 1989). Although R*, values obtained on averagingequivalent reflections sometimes clearly indicate the cor-rect Laue group, the values obtainedfor m3m,4/mmm,4/m, and mmm were too close to be conclusive-0.024,0.023,0.021, and 0.021, respectively. During the courseof this study, structure refinements were attempted inspace groups 14,/acd, 14,/a, Fddd, 1T, and lbca (thelastis a subgroup of 14r/acd), and the generalized weighted Rfactor, Rr, (Hamilton, 1965) was used to test each refine-ment against the cubic model. Table 5 shows the resultsof these tests. Note that tests to determine if a given mod-el is significantly better than the cubic model are shownonly for 14,/acd and lbca because the other three refine-ments yielded higher Ru values than did the cubic refine-ment and thus cannot be preferred. The clear choice forthe correct space group is l4r/acd. Final positional pa-rameters, site-occupancy factors, equivalent isotropictemperature factors, and anisotropic temperature-factorcoemcients for refinement in space group I4'/acd are dis-played in Table 6, and selected interatomic distances andangles are in Table 7.

Why is garnet T-KAY-K-I noncubic? The anisotropicgrandites thus far submitted to crystal-structure refine-ment (Tak6uchi et al., 1982; Kingma and Downs, 1989)have shown ordering ofAl and Fe3* at the octahedral sites(Wyckoffsymbol l6a in the average cubic space group).A chemically more complicated garnet of compositionGronoAnduAlmo (Allen and Buseck, 1988) with a signifi-cant HrO content shows ordering of Al and Fe3* at theoctahedral sites, ordering of Ca and Fe2* at the dodeca-hedral sites (cubic 24c positions), and a noncubic distri-bution of OH ions.

Table 8 shows a comparison of the crystallographic po-sitions occupied in cubic garnets and the analogous po-sitions in space group l4r/acd, and Figure 2 shows a slabof the garnet stmcture with the unique cation sites indi-cated. Given that there is no evidence of anionic substi-tution in garnet T-KAY-K-I, that the tetrahedral sitesare completely filled with Si, and that there is only onesymmetrically unique kind of octahedral site, the X sites

sio,Tio,Alr03FerO..*FeO-.MnOMgoCaO

28.741 5 32028.84

100 03

37.860 . 1 1

20.93 c u )

-3 ro

1 5

3.01 0

si-o201-024 01-O3Y-O6 01-O46 01-O54X1-O4De.-O4

201-024 01-044 04-06204-074 01-07208-07

1.640(2)2.521(412.7s2(3\1.902(2)2.673(3)2.706(3)2.244(2)2.407(2\2.3262.519(3)2.674(3)2.818(4)2.826(4)3.367(4)3.948(4)

2 01-Si-O24 01-Si-O3

6 01-v-o46 01-v-o5

201-X2-O24 01-X2-O44c4-)8-06204-X2-O74 01-X2-O7208-X2-O8

100 5 (1 )1 1 4 . 1 ( 1 )

89.3(1)s0.7(1)

68.3(1 )70 .1 (1 )74.5(1)71 .9(1)92.7(1)

1 1 0.2(1)

A/otei Numbers in oarentheses are estimated standard deviations andrefer to last decimal place. This convention applies to subsequent tables.

r###** #

GRIFFEN ET AL.: BIREFRINGENT GARNET 403

Tleue 5, Significance tests on refinements

Ho: The /a3d model (4 : 0.0205) is the better of the two in question1. l4, la

& : 0.0210 Rs(la3q < Rs(t4/a),.'. Ho is accepted2 Fddd

Rs:0.0220 R,(lfid) < Rs(Fdctd), '. Ho is accepted3. /T

& : 0.0314 Rs(tf,d) < 4(/T), .' 4 is accepted4. lbca

Fs : 0.0190 F/ : 0 0205/0.0190: 1.079 ) i * , . .oooou: 1.094tt > R', .'. Ho is accepted at the 0.005 significance level

5. l4,lacd& = 0.0185 R : 0.0205/0.0185: 1.108 Ss,375.s66, : 1.076tt < R', . Ho is reiected at the 0.005 significance level, /4,/acd is thebetter model

turn out to be the only possible locations on which or-dering could occur. Table 6 shows that the electron den-sity at the Xl sites is slightly greater than that at X2,whereas Table 7 shows that Xl is the slightly smaller ofthe two sites. From Table 2 and the ionic radii of Shan-non (1976), it can be determined (1) that the eightfold-coordinated ionic radius of the mean pyralspite X com-ponent is 0.92 A and that the radius of I8lCa2+, the onlygrandite X cation, is 1.12 A and (2) that the averageweighted atomic number of the pyralspite X componentts 24.4, whereas that of Ca is 20. Although both the dif-ferences between the mean bond lengths of the two Xsitesand the difference between the electron densities at thesesites are very small (< I esd for the bond lengths and -2combined esd for occupancy factors) and might be con-sidered trivial considered alone, they are consistent withslight ordering of (Fe,Mg,Mn) into Xl and Ca into X2 ofthe tetragonal garnet.

Sylrllmrny ANALysrs

Hatch and Griffen (1989) have considered noncubicgarnets in general, and birefringent grandites in particu-lar, within the framework of the Landau theory of phasetransitions (Landau, 1937a, 1937b, with extensions byBirman, 1966; Goldrich and Birman,1968;Jaric and Bir-man,1977; Hatch l98la, l98lb; Hatch and Stokes, 1985,1986) and the theory ofinduced representations (Hatchet al., 1987). On the assumption that noncubic garnetsmay arise by phase transformations from an initially cu-bic phase ofspace group la3d and that the transitions are

Trele 7. Selected interatomic distances (A) and angles ('), poly-hedral volumes (43), quadratic elongations, and anglevariances* for gamet T-KAY-K-1 refined in space groupl4'lacd

2 Si1-O1 1.638(2) 01-O1 2.517(412 Si1-O3 1.642(2) O3-O3 2.52s(sl

1-6-;10- 4 01-O3 2.75q3)2.676

v : 2 . 2 2 2 { I ) : 1 . 0 1 24Si2-O2 1.U0t2) 2O2-O2 2.527(41

4 O2-O2 2.7sq4\2.676

v : 2.224 (^) : 1.01 12 Y-O2 1 .898(2) 2 O1-O2 2.671(3)2 Y-O3 1.899(2) 2O1-O2 2.70343)2 Y-O1 1.902(2) 2 O1-O3 2.67343)

Igo0- 2 O1-O3 2.701(3)2O2-O3 2.672(3)2O2-O3 2.698(3)

2.686v : 9 . 1 3 6 ( X ) : 1 . 0 0 0

2 x1-O1 2.244(2) O1-O1 2.51714)2 x1-O3 2.244(2) O3-O3 2.525(5)2 x1-O2 2.408(2) 2O1-O2 2.671(3)2 x1-O1 2.410(2) 2 01-O3 2.673(3)

2.327 2 O1-O1 2.821(5)2O2-O3 2.825(3)2O1-O2 2.827(3\2 01-O1 3.366{4)2O2-O3 3.367(3)2O1-O2 3.95q3)

3.003v : 2 1 . 6 7 ( r ) : 1 . 0 0 1

4 X2-O2 2.249(2) 2O2-O2 2.s27(4\4 x2-O3 2.407(2) 4 O2-O3 2.67213],

2.328 2 O3-O3 2.828(5)4 O2-O3 2.825(314 O2-O3 3.370(3)2 O3-O3 3.948{5)

3.004v : 2 1 . 7 2 ( r ) : 1 . 0 0 1

o1-si1-o1 100.4(2)o3-si1-o3 100.5(2)

4 01-Si1-O3 114.2(1)109.6

o 2 : 5 O 0 22O2-S12-O2 100.8(2)4O2-Si2-O2 114.0(1)

109.6d: 46.282O1-Y-O2 89.3(1)2O1-Y-O2 90.7(1)2O1-Y-O3 89.4(1)2 01-v-O3 90.6(1)2O2-Y-O3 89.5(1)2O2-Y-O3 90.s(1)

90.0o 2 : 0 . 4 1

o1-x1-o1 68.20)o3-x1-o3 68.5(1)

201-X1-O2 70.0(1)2 01-x1-O3 70.0(1)2 O1-X1-O1 74.5(112 O2-X1-O3 74.7(112 O1-x1-O2 71 .9(1)2 O1-x1-O1 92.6(1)2O2-X1-O3 92.7(1)2O1-X1-O2 110.3(1)

80.6o'z : 199.412O2-x2-O2 68.4(1)4O2-X2-O3 70.0(1)2O3-x2-O3 72.0(114 O2-X2-O3 74.6(1)4 O2-X2-O3 92.7(1)2O3-X2-O3 110.20)

80.6r : 198.65

- For method of calculation of quadratic elongation ((I)) and angle vari-ance (o,), see Robinson et al. (1 971 ). For the X sites, the mean values ofthe bond lengths and bond angles were used as the ideal values.

driven by a single order parameter (not necessarily thesame one in all cases), they have determined that thereare three ordering routes possible: one involving an orderparameter that transforms according to the T2s irreduci-ble representation (IR), one involving an order parameterthat transforms according to the T,, IR, and one involv-ing an order parameter that transforms according to the

TABLE 6. Final fractional coordinates, site-occupancy factors, anisotropic temperature-factor coefficients (x1@), and equivalentisotropic temperature factors ( x 1 04) for garnet T-KAY-K-1 refined in space group l4rlacd

z k' Uu" U"" Us U"" Ur U." U*

1 1 3(2)1 08(3)s8(4)96(4)94(4)

123(71't20(7)122(7)

00

-6(6)00

-20(10)2(1 0)7(111

x1x2Ysi1si3o l02o3

0.1249(1) 00 V 40 00.3749(1) 0u ,/4

0.2848(2) 0.0973(2)0.0977(2) 0.2024(2)o.2o24(2) 0.2847(2)

v4 1.182(41 76(4)vo 1.171(5) 128(4)0 1.001(5) 81(7)va 106(6)3h 86(6)0.2025(2) 126(13)0.28s112) 114(13)0.0976(2) 136(12)

134(5) 130(4)(4,) 68(s)84(7) 83(5)e1(7) 8e(6)(u , , ) 111 (9 )

126 (13 ) 116 (11 )114(12) 133(11)11802 ) 113 (11 )

12(4\0

-5(6)-1(s)

01 ( 1 1 )

- 9(1 0)1(10)

01 6(5)1(6)00e(e)

1 0(s)-5(s)- Site occupancy. Scattering factor for Ca was used in the X site and {or Al in the y site.

* Utare coefflcients in the expression expl-2n2(U1i2a*2 + U22Kt2 + U$12c"2 + 2uehka.b' + 2ulshla'C + 2uBklb-c-ll.

404 GRIFFEN ET AL.: BIREFRINGENT GARNET

Fig.2. A slab of the gamet structure (after Novak and Gibbs,1971), with Wyckoffletters for the unique cations sites in spacegrotp 14 I acd indicated.

E, IR. The T2g route leads to possible space groups R3c,Fddd, I2/c, and 1T; the T,, route leads to possible spacegroups R3, I4r/a, I2/c, and 11; and the E, route leads topossible space groups l4r/acd and lbca.

The transformation from la3d to l4r/acd is a properferroelastic transition that must be associated with the fpoint of the Brillouin zone. The space group 14,/acd isnecessarily related to the parent space group by an orderparameter that transforms according to the E, IR (If inthe notation of Stokes and Hatch, 1988).

Table 9 gives the two space groups, and their relation-ships to la3d, thatmay result from phase transformationsthat are driven by a single order parameter using the E,IR. The 4, shown in the last column are components ofthe order parameter, 4, in the equation p(r) : po(r) +2n,Q,Q). Here p(r) is the density function that specifies theprobability distribution of atomic positions in the crystal,with po(r) being the density function in the parent (high-symmetry) space group. The d,(r) are the basis functionsof a physically irreducible representation of the parentspace group (Ia3d in this case), so that the order param-eter components, 4,, give the contribution of each basisfunction to the density function. The ordering, in thiscase, is between the (Fe2*,Mg,Mn) that constitute the pyr-alspite Xcomponents and Ca, the grandite Xcomponent;

TraLe 8. Occupied sites in /a3d and l4rlacd garnet structures

14,lacd

x ,0 , Vq0 ,Vq , l a0 , 0 , 0x, 0,1/a0, Va,3/ax , y , zx , y , zx , y , z

TABLE L subgroups of ,agd allowed by the En lR

SubgroupDirection of

Origin Basis vectors vector 4

(0,0,0)(0,0,0)

0,0,0),(0,1,0),(0,0,1) ( ' r ' ,0)(1,0,0),(0,1,0),(0,0,1) (n,,n")

it takes place on tl;re 24 {X} sites of the cubic unit cell,thus lowering the symmetry ftom laid to l4r/acd.

In Table l0' the splitting of the equivalent points ofthe Wyckoff 24c position of la3d into the 8b and l6epositions of 14,/acd are listed. In Table I I' the two basicfunctions that transform according to the E, representa-tion (Ir+) are shown. The relative degree of ordering onthe 24c points of la3d represents the ordering of the Xcomponents; their positions in Iaid are also shown. Forthe order parameter (4,, 0) shown in Table 9 (that is, forspace group l4r/acfi, only the first basis function of TableI I contributes. Note that any individual site on an 8bWyckoffpoint shows twice the ordering of an individualsite on a l6e point, as required by the relative numbersof points constituting the Wyckoffpositions.

From Stokes and Hatch (1988), the lf representationdoes not satisfy the Landau condition; thus the transitionmust be discontinuous. The free energy for If is

AF : a(n1 + n',) + b(q1 - 3n,ql) + c(q? + n)'.

The free energy form will yield many physical propertiesthat undergo a change of symmetry at the transition as aresult of a l;f order parameter becoming nonzero. Forexample, heat capacity and enthalpy as a function of tem-perature can be obtained. The spontaneous strain com-ponents (e ,, * e22 - 2r.., \/Tr" - V-3eJ also transformas the two components of f3+. Thus these strain compo-nents, denoted (Q,,Qr), will couple bilinearly to (4',4r) inthe form 1rQ, * nzQrand, will also become nonzero atthe transition. Because only the first component is non-zero at the transition to I4r/ acd, only the first componentof the spontaneous strain Q, (= .,, * er, - 2ett) ,rllllbecome nonzero.

CoNcr-usroN

On the basis of the discussion above, we interpret theanisotropy of garnet T-KAY-K- I as due to slight orderingof the pyralspite (Fe'?*,Mg,Mn) and grandite (Ca) Xcom-ponents on the 24c positions of the average la3d struc-ture. There are, of course, small shifts in the O and Sipositions relative to those positions in the average cubicstructure, but we take these to be secondary consequencesof X-site ordering, inasmuch as there seems to be no rea-son for such displacements independent ofthe ordering.

It is not possible to determine unambiguously whetherthe three pyralspite components participate equally in thesite exchange with Ca. One could speculate that crystal-field effects might favor Fe2* as the key component, giventhat Mn2*, Mg, and Ca do not experience crystal-fieldstabilization effects; using the mean O-X-O angles fromTable 7 as ideal angles and applying a measure of distor-

Ajlacdlbca

la3d

\x\lvl@

o

24c

16a

24d

96h

n,o,n l l0 , 0 , 0 - Y.z', o, z ljl-r,, zi

16e8b

16c16e8a

329329329

GRIFFEN ET AL.: BIREFRINGENT GARNET 405

tion analogous to octahedral angle variance (Robinson etal., l97l), the Xl site does turn out to be very slightlymore distorted than the X2 site, which would presumablyenhance the crystal-field stabilization energy of Fe2t inXl relative to X2 (see, for example, Burns, 1970, con-cerning the effects of polyhedral distortion on crystal-fieldstabilization energy). One could also argue that Ca + Mgcannot be the primary substitution because that wouldbe inconsistent with the observed difference in electrondensities of the two X sites.

In emphasizing the Landau approach to symmetry re-lationships, we have treated the ordering observed in thenoncubic garnets discussed as if it arose by phase tran-sition from a preexistent cubic phase with space groupIa3d. Al<rzul,t (1984) has argued that birefringence ingrandites is a crystal-growth phenomenon that arises asordering of Al and Fe3* on different (110) growth stepsoccurs, making the steps crystallographically inequiva-lent, with each resulting growth sector having space group11. A treatment from a symmetry viewpoint based on thesame group theoretical formalisms we have used here canalso be applied to crystal growth, yielding the higher sym-metry space groups that can result from superposition oftriclinic growth sectors (Gali, 1983).

Hatch and Griffen (1989) argue that definitive evidencefor discriminating between crystal growth and phase tran-sition is lacking for the grandites. MgSiO. (Angel et al.,1988) and MnSiO. (Fujino et al., 1986) were both syn-thesized in high-pressure-high-temperature devices inexperiments of short duration. Although the times re-quired to bring the synthesized crystals to room condi-tions were not specified in either paper, it seems likelythat they were quite short-perhaps too short to allowthe nearly complete ordering of octahedral cations froman initially disordered state.

It would seem that the combination of high pressure(perhaps 0.8-1.2 GPa) and relatively low temperature(probably 200-400 "C) in the geologic environment inwhich garnet T-KAY-K-I crystallized is the likely causeofthe ordering. It is well known that increased pressurefavors increased miscibility between pyralspites andgrandites (e.g., Nemec, 1967), but it has been shown thatheating of anisotropic grandites causes them to becomeoptically isotropic (Hariya and Kimura, 1978; Allen andBuseck, 1988), that is, to become disordered. Garnets ofcompositions similar to ours but from geologic environ-ments quite different from a paleosubduction zone havebeen described but have not been identified as birefrin-gent (e.g., Sobolev et al., I 968), again suggesting that spe-cific P, ?'conditions are important in determining wheth-er a particular garnet chemically capable of orderinganisotropy will, in fact, order. Investigation of many moreanisotropic garnets of various compositions and from dif-fering geologic environments is required before moremeaningful discussion on the influence on ordering ofpressure, temperature, and composition can be undertak-en.

Given an ordered garnet, the ordering route followed

1 .2

1 . 1

Grandites

T-KAY-K-I

0.4 0.5 0.6 0.7 0.8<rry1> (A)

Fig. 3. Plot of mean cation sizes in the X and I sites ofnoncubic garnets with structures refined to date. Gray bandsindicate size ranges suggested by solid-solution seriesCa.(A1,Fe3*)rSirO,, (Tr.), (Ca,Fe'z*,Mg,Mn)rA1rSi.O,, (E"), and(Mn,Fe'zt,Mg)SiO3 (T,,). AB indicates the garnet of Allen andBuseck (1988). FeSiO, garnet has not been synthesized, but itfalls on the T,u band and is likely a component of any tetragonalgarnet that exists in the upper mantle.

(i.e., Tr' T,r, or E' regardless of whether ordering is con-sidered to arise from a phase transformation or crystalgrowth) seems to depend primarily on the crystallograph-ic sites potentially available for ordering and secondarilyon the relative sizos of the cations in those sites. MgSiO.,MnSiOr, and the ordered grandites of Tak6uchi et al.( I 9 82) each contain a single type of cation in the X sitesand two in the I sites; this restricts potential ordering tothe I sites, with the Tr. route being followed if the fcations are of similar size (e.g., Fe3* and Al) and the T,,route being followed ifthe Ications are ofvery differentsizes (e.g., Mn and Si). The grandite of Kingma and Downs(1989) evidently contains too little Mn in the X site forordering to be detectable, if it occurs, so it too followsthe Tr, route. Garnets that have two or more kinds of Xcations in significant concentrations but only one signif-icantly abundant kind of I cation follow the E, route;garnet T-KAY-K-I is of this type.

These generalizations appear to apply when ordering isdriven by a single order parameter, regardless of whethera phase transition or a primary growth phenomenon isassumed. They leave several important issues unad-dressed, however. For example, only four of the eightspace groups related to IaSd by these IRs have been ob-served, and it is not at all clear what factors control theparticular ordering pattern within a given ordering route;thus the specific factors that discriminate between theformation of an l4r/acd ordering pattern and an lbca or-

x l oLr

0.9

0.8

406 GRIFFEN ET AL.: BIREFRINGENT GARNET

dering pattern are unknown. We have also not consideredgarnets that have the potential to order on more than onesite; the near-grandite of Allen and Buseck (1988) is ofthis type, exhibiting ordering at both the X and Y sites.Figure 3 shows the mean radius of the I cations plottedagainst that of the Xcations for the noncubic garnets withstructures refined thus far. The gray bands indicate sizeranges predicted for the solid solution seriesCar(Al,Fe3*)rSi3Or, (TJ, (Fe'z*,Mg,Mn)rAl2Si3O,2 @), and(Ca,Mn,Fe2*,Mg)SiO. (T,r). Investigation of the first is-sue-factors that determine the specific ordering patternwithin an ordering route-will require examination ofmore noncubic garnets on the gray bands from as large avariety of geologic environments as possible. Investiga-tion of the second issue-the nature of ordering drivenby more than a single order parameter-will require ex-amination of noncubic garnets that do not occur on thegray bands. The relative scarcity ofsuch garnets does notminimize the fact that understanding them may shed lighton processes rangrng from the shallow cnrst to the uppermantle.

RrrnnnNcrs crrED

Akizukr, M. (1984) Origin of optical variations in grossular-andraditegamet. American Mineralogist, 69, 328-338.

Allen, F.M., and Buseck, P.R. (1988) XRD, FrIR, and TEM studies ofoptically anisotropic gossular gamets. American Mineralogist, 7 3, 56r-.584.

Angel, R.J., Finger, L.W., Hazen, R.M., Kanzaki, M., Weidner, D.J.,Liebermann, R.C., and Veblen, D-R. (1989) Structure and twinning ofsingle-crystal MgSiO, garnet synthesized at l7 GPa and 1800 .C. Amer-ican Mineralogist, 7 4, 509-512.

Birman, J.L. (1966) Simplified theory of symmetry change in second-order phase transitions: Application to VrSi. Physical Review I-etters,l7, 1216-1219.

Burns, R.G. (1970) Mineralogical applications of crystal field theory.Cambridge University Press, Cambridge, England.

Cromer, D.T., and Liberman, D. (1970) Relativistic calculation of anom-alous scattering factors for X-rays. Journal of Chemical Physics, 53,1 89 1- I E98.

Cromer, D.T., and Mann, J.B. (1968) X-ray scattering factors computedfrom numerical Hartree-Fock wave functions. Acta Crystallographica,A^24, t21-324.

Dempsey, M.J. (1980) Evidence for structural changes in garnet causedby calcium substitution. Contributions to Mineralog,y and Petrology,7r ,281-282.

Dyar, M.D. (1984) Precision and interlaboratory reproducibility of mea-surements of the Mtissbauer effect in minerals. American Mineralogist,69,1127-1144.

Fujino, K., Momoi, H., Sawamoto, H., andKumazawa, M. (1986) Crystalstructure and chemistry of MnSiO, tetragonal garnet American Min-eralogist, 7 l, 781-785.

Gali, S. (1983) Grandite garnet structures in connection with the $owthmechanism. Zeitschrift fiir Kristallographie, 163, 43-52.

Geiger, C.A., Winkler, 8., and Langer, K. (1989) Infrared spectra of syn-thetic almandine-grossular and almandine-pyrope garnet solid solu-tions: Evidence for equivalent site behaviour. Mineralogical Magazine,53 ,23 r -237 .

Geller, S. (1967) Crystal chemistry ofthe garnets. Zeitschrift fiir Kristal-lographie, 125, l-47.

Goldrich, F.E., and Birman, J.L. (1968) Theory of symmetry change insecond-order phase transitions in perovskite structure. Physical Re-view. 167. 528-532.

Hahn, T., Ed. (19E3) International tables for crystallogaphy, vol. A. Rei-del, Boston.

Hamilton, W.C. (1965) Significance tests on the crystallographic R factor.Acta Crystallographica, 18, 502-5 10.

Hariya, Y, and Kimura, M. (1978) Optical anomaly garnet and its sta-bility field at high pressures and tempetatures. Journal ofthe Faculty

ofScience, Hokkaido University, series IV, 18,611-624.Hatch, D.M- (l98la) Structural phase transitions in crystals of R3c sym-

metry. Physical Review, B23, 2346-2349.-(1981b) Order parameter symmetry for the F-z transition in

Pb(VO4),. Physica Status Solidi, 106, 4'13-479Hatch, D.M., and Griffen, D.T. (1989) Phase transitions in the grandite

garnets. American Mineralogist, 74, 151-159.Hatch, D.M, and Stokes, H.T. (1985) Phase transitions in solids of dt-

periodic symmetry. Physical Review, B3l, 4350-4354.- (I986) Systematic classification ofsymmetry changes in solid-solid

phase transitions. In L.H. Bennett, Ed., Computer modeling of phase

diagrams, p. 145-162. Metalluryical Society of AIME, Warrendale,Pennsfvania.

Hatch, D.M., Stokes, H.T., and Putnam, R.M. (1987) Symmetry analysisof the microstructure and phase transitions of a crystallographic spacegroup: Applications. Physical Review, 835, 4935-4942.

Jaric, M.V., and Birman, J.L. (1977) Group theory of phase transitionsin A-15 (Oi-Pz3z) structure. Physical Rewiew, Bl6, 2564-2568.

Kingma, K.J, and Downs, J.W. (1989) Crystal-structwe analysis of abirefringent andradite. American Mineralogist, 7 4, 1307 -1316'

Kretz, R (1983) Symbols for rock-forming minerals. American Miner-

alogist, 68, 277-279.Kulaksiz, S., and Phillips, W.R. (1983) Mineralogy of a paleosubduction

zone (Sivrigisar-Balgikhisar.l. Hacettepe Universitesi YerbilimleriUygulama ve Arastirma Merkezi Biilteni, 10, 95-104 (in Turkish).

Landau, L.D. (1937a) On the theory ofphase transitions. I. PhysikalischeZeitschrift der Sowjetunion, 11,27-47 (in German).

-(1937b) On the theory ofphase transitions II. Physikalische Zeit-schrift der Sowjetunion, 1 1, 545-555 (in German).

Moore, R.K., White, W.B., and Long, T.V. (1971) Vibrational spectra ofthe common silicates: I The garnets. American Mineralogist, 56, 54-7 1 .

Nemec, D. (1967) The miscibility of the ppalspite and grandite mole-cules. Mineralogical Magazine, 37, 3E9-4O2.

North, A.C.T., Phillips, D.C., and Mathews, F.S (1968) A semi-empiricalmethod of absorption correction Acta Crystallogaphica, 424, 351-359.

Novak, G.A., and Gibbs, G.V. (1971) The crystal chemistry of the silicategamets. American Mineralogist, 56, 79 1-825.

Prewitt, C.T., and Sleight, A.W. (1969) Garnetlike structures of high-pressure cadmium germanate and calcium germanate. Science, 163,386-387.

Rickwood, P.C. (1968) On recasting analyses ofgarnet into end-membermolecules. Contributions to Mineralogy and Petrology, 18, 175-198.

Robinson, K., Gibbs, G.V., and Ribbe, P.H. (1971) Quadratic elongation:A quantitative measure of distortion in coordination polyhedra. Sci-ence. 172. 567 -57O.

Shannon, R.D. (1976) Revised effective ionic radii and systematic studiesofinteratomic distances in halides and chalcogenides. Acta Crystallo-graphica, 432, 7 5|-7 67 .

Sheldrick, G.M. (1983) SHELXTL, An integrated system for solving, re-fining, and displaying crystal structures from diffraction data (4th re-vision). University of G6ttingen, Gdttingen, Germany.

Smyth, J.R., Madel, R.E., McCormick, T.C., Munoz, J.L., and Rossman,G.R. ( 1990) Crystal-structure refinement of a F-bearing spessartine gar-

net. American Mineralogist, 75, 314-318.Sobolev, N.V., Jr., Kuznetsova, LK., and Zyuzin, N.I. (1968) The pe-

trology ofg;rospydite xenoliths from the Zagodochnaya kimberlite pipe

in Yakutia. Journal ofPetrology, 9, 353-380.Stokes, H.T., and Hatch, D.M. (1988) Isottopy subgtoups of the 230

crystallographic space groups. World Scientific, Singapore.Tak6uchi, Y.,Haga, N., Umizu, S., and Sato, G. (1982) The derivative

structure of silicate garnets in grandite. Zeitschrift fiir Kristallographie,l 58, 53-99.

MAr.ruscRrpr REcETvED FBsrumy 5, 1991MnNuscnrpr AccEprED NowMBER l. 1991