characterization and thermodynamic properties of …...american mineralogist, volume 76, pages...

American Mineralogist, Volume 76, pages 1249-1260, 1991

Characterization and thermodynamic properties of andradite, CarFersi3or2

Mrcnnr, M,cooND6partement des G6omat6riaux, Institut de Physique du Globe de Paris, 4 place Jussieu,75252 Paris cedex 05, France

Joss I. Grr,InlncucHr. JuLTAN VreDepartamento de Mineralogia y Petrologia, Universidad del Pais Vasco-EHU, Aptdo. 644, 48080 Bilbao, Spain

J.lceuns Grn{non,c,uLaboratoire de Pdtrophysique, IPGP-Universit6 Paris VII, 2 place Jussieu, 75251 Paris cedex 05, France

Garnet with composition of almorr r::ffi:]tr" (cu,r",si,o,,) occurs in fissures cut-ting the foliation of pyroxenites at the Herbeira ultramafic massif (Cabo Ortegal, northwestSpain). The infrared and Raman spectra ofthe natural andradite have been recorded atroom pressure and room temperature in order to model the thermodynamic properties.The IR spectrum exhibits three modes (112,704, and 1070 cm-') not previously describedin the literature. Using these new data in the lattice vibrational model of Kietrer (1979a,1979b, 1979c), the thermodynamic properties of andradite have been calculated. Agree-ment between measured and calculated specific heat and relative enthalpy is excellent (ca.l0lo difference between 300 and I100 K). The specific heat (G,zqs x 346.5 J/mol.K) andentropy (Srn, = 313.6 J/mol'K) at room temperature agree with reported experimentalvalues. The model of density of state proposed for andradite is used to calculate C"(T),H(7"), and S(Z) of Ca.Fe2Si3O,, between 0 and 1500 K. Equations expressing the temper-ature dependence ofheat capacity and relative heat content ofandradite are thus proposedthat allow its thermodynamic properties G(7"), H(T), and S(7") to be calculated at hightemperatures (f > 298 K).

INrnooucrrou

A green gem variety ofandradite garnet, often referredto as demantoid, is generally associated with serpentinitesor variably retrogressed peridotites (Peters, 1963; Web-ster, 1970; Amthauer et al., 197 4; Schmetzer et al., 197 5;Fediukov6 eIal.,1976). During the petrological study ofthe ultramafic Herbeira massif at the Cabo Ortegal com-plex (northwestern Spain), garnet with composition of al-most pure andradite (CarFerSirO,r) was found associatedwith pyroxenites. Because the garnet family is one forwhich many heat capacity measurements have been per-formed (Berman, 1988, 1990) but for which the pub-lished experimental and theoretical values of thermody-namic parameters (C" and S) substantially disagree(Kieffer, 1980), we have recorded infrared and Ramanspectra of andradite from this area to see if another mod-eling scheme could be proposed and the disagreementbetween experimental and calculated values be ex-plained. The results of this study along with data on thechemical composition and unit-cell parameters of garnetsanalyzed are presented herein.

Gror,ocrc,q.L SETTTNG

The Cabo Ortegal complex is one of a series of alloch-thonous complexes exposed in the northwestern part of

the Iberian massif as klippen of a large scale, sheetlikestructure produced during the Hercynian collision (Are-nas et al., 1986). In the Cabo Ortegal area, the uppermostthrust unit consists mainly of basic granulites, eclogites,and ultramafic rocks (Fig. l). The main metamorphicevent in this unit was of high-pressure, high-temperaturetype (garnet-clinopyroxene granulites and eclogites, Vo-gel,1967; Gil Ibarguchi et al., 1990) and has been datedas Early Ordovician (Peucat et a1., 1990). The ultramaficrocks comprising mainly harzburgites and pyroxenitesoccur in three main rnassifs: Limo, Herbeira, and Uzal(Girardeau et al., 1989).

\rope-rich garnet is often see growing at the expenseof oriented spinel in pyroxenites of the Herbeira massifand seems to be related to the episode of deformationand recrystallization under high-pressure, high-tempera-ture conditions (Girardeau et al., 1989). Andradite gar-net, instead, occurs associated with chlorite and serpen-tine in fissures cutting across the foliation ofthe pyroxeniteand was probably developed at the end of a retrogradegreenschist episode. The actual P-Z conditions underwhich andradite was formed cannot be estimated fromthe available data. They should not be higher than thoseconditions necessary for the development ofgreenschistassemblages in the pyroxenites (<450 "C).

0003-004x/9 1 / 07 08-t249s02.00 r249

1250

Fig. l. Schematic geologic map of the northern part of theCabo Ortegal complex (after Girardeau et al., 1989) with loca-tion of the Herbeira massif and andradite samples, here labeledby the varietal name demantoid. Also shown are a cross sectionthrough the pyroxenite sheet (A-B line and inset) and the 400m contour line.

Slrtrpr,n DESCRTPTIoN

Andradite has been found on a loose block ofapprox-imately 40 cm in diameter at the base of a cliff 500 m inheight at the southeastern extremity of the Herbeira mas-sif (Fig. 1). The block is composed essentially of slightlyserpentinized, spinel-bearing clinopyroxenite. The garnetis exposed on the surface of the block, forming a crust inwhat must represent part of the filling of a fissure thatcuts the foliation ofthe pyroxenite. Garnets at the surfaceare pale green, euhedral, millimeter-size crystals exhib-iting a brilliant luster and distinct dodecahedral and tra-pezohedral forms and also combinations of these twoforms. Toward the host pyroxenite they become finergrained, and the euhedral shape becomes less evident.The total thickness ofthe garnet layer is usually less thanI cm.

Several types of'garnets may be distinguished under themicroscope. They correspond to different petrographiczones. From the pyroxenite toward the surface these areas follows: (l) A first zone of about 2 mm in thickness,close to the pyroxenites, with sparse garnet forming thinrims (ca. 0.05-0.1 mm) around spinel and, less often,around pyroxene crystals (Fig. 2A). Garnet around spineltends to develop a euhedral shape and quite often exhib-its a fibrous or radial aspect at the inner and middle partsand a clear, massive outer rim (Fig. 2A); it could not beascertained whether the fibrous aspect was caused by thepresence of minute radial (horse-tail) inclusions of asbes-tos as has been sometimes reported for this kind of garnet(e.g., Webster, 1970). Garnet also appears in this zone,rimming cavities filled by serpentine and chlorite (Fig.

MADON ET AL.: THERMODYNAMIC PROPERTIES OF ANDRADITE

a

t \ * !

e

Fig.2. (A) Zones 1,2, and 3 ofthe garnet vein, designatedas i, ii, and lii.Znne 1 is in contact with the pyroxenite host rockand contains clinopyrgxene (Cpx), small garnet (Gt) around spi-nel, and magnetite (Mt) in a chlorite + serpentine matrix. Zone2 contains small subhedral garnet (Gt) within chlorite + serpen-tine. Zone 3 exhibits a rim ofradial chlorite (Chl) and progres-sively larger and more euhedral garnet (Gt) with increasing dis-tance from the pyroxenite. (B) Small garnet (Gt) of zone I withan inner part rich in inclusions and a core of spinel (Sp). (C)Garnet (Gt) of zone I around spinel cores and rimming a cavityfilled by serpentine and chlorite; small dark spheres within thecavity are garnet with radial aspect.

MADON ET AL.: THERMODYNAMIC PROPERTIES OFANDRADITE

Tlau 1. Representative chemical analyses of andradite, spinel, and clinopyroxene

1r 1c 21 2c 3c 3r 4Gt.cavSample Gt.3 Gt.3 Gt.3 Gt.1 Gt.1

t251

cpxSp

sio,Tio,Alro3CrrO"Fe"O"FeOMnONioMgoCaONarol(rO

TotalFeO.,SiTiAICr

Fe*MgNiMnCaNaK

35.420.040.410 . 1 1

30.09bdbdbd

0.0434.13

bd0.01

100.2527.082.9730.0030.0410.0071.901

bd0.005

bdbd

3.070bd

0.001

0.4bd

30.82bdbdbd

0.0733.580.020.03

100.927.733.004

bd0.039

bd1.937

bd0.009

bdbd

3.0050.0030.003

34.64bd

2.39bd

27.88bd

0.1bd

0 . 1 134.01

bdbd

99.1325.092.913

bd0.237

bd1.765

bd0.014

bd0.0073.065

bdbd

35.840.030.460.04

30.66bd

0.02bd

0.0633.790.01

bd100.9127.592.9920.0020.0450.0031.926

bd0.008

bd0.0013.0220.002

bd

35.76bd

3.92.04

23.20bdM

0.041.54

32.89bd

0.0399.4020.882.948

bd0.3790.1331.440

bd0.1890.0027

bd2.905

bd0.003

35.170.041.490.04

28.97bdbd

0.030.2

33.97bd

0.0299.9326.072.9450.0030.1470.0031.826

bd0.0250.002

bd3.048

bd0.002

35.090.014.58

bd25.10

bdbd

0.010.78

33.7bd

0.0199.2822.592.9040.0010.447

bd1.563

bd0.0960.001

bd2.988

bd0.001

35.98bd

bd 54.340.31 0.07

26.72 1.0430.81 0.2111.4 bd22.32 2.050.41 0.070.13 bd8.94 17.320.02 24.46

bd 0.070.07 0.01

101.13 96.6432.58 2.05

M 1.9810.007 0.0450.972 0.0020.752 0.0060.265 bd0.576 0.0620.411 0.9410.003 bd0.011 0.0020.001 0.955

bd 0.0050.003 bd

/VoteiAbbreviat ions:1to4:garnet ; Gt.3:euhedral ,>1 mmandradi te(zone3); Gt.1 :smal ler , radia l garnetaroundspinel (zone1); c:core,r: rim; Gt.cav : small garnet in cavity walls (zone 1). Sp : spinel; Cpx : clinopyroxene; bd : below detection limits of 0.01 Mo/o. Fe3* determined bycharge balance calculations: eight cations and 12 O atoms for garnet, three cations and four O atoms for spinel, four cations and six O atoms forpyroxene.

2B). (2) An intermediate zone, ca. I mm thick, rich inchlorite and serpentine with some minute grains of clear,massive garnets (Fig. 2C). (3) The third zone has a vari-able thickness between 2 and 7 mm and usually startswith a layer of chlorite perpendicular to the vein walls; itfollows several layers of clear, massive garnet, progres-sively larger toward the outer parts of this zone (Fig. 2C).The outermost garnet may be >2 mm in diameter; thegarnet crystals are euhedral and sometimes exhibit alighter green rim. All the garnet displays anomalous bi-refringence. A twinning pattern consisting of six or morewedge-shaped sectors is best seen in the well-developedmillimeter-size garlaLet of zone 3. Only the millimeter-sizegarnet of zone 3 has been used in the X-ray, infrared,and Raman measurements.

MrNnrul cHEMrsrRy

Minerals were analyzed with a Camebax Micro auto-mated microprobe equipped with three spectrometers atthe University of Clermont-Ferrand. Operating parame-ters included a l0 s integration time, a ca. lO-nA beamcurrent, and a 1 5-kV acceleration voltage. Standards wereoxides and silicates provided by the French GeologicalSurvey (BRGM), and the ZAF conection procedure wasused. Table I presents a selection of minerals analyzed;the complete set of data used in this report is availablefrom J.I.G.I. on request. The procedures used to estimateFe3+ contents in the minerals analyzed are indicated inthe table. Thermo-gravimetric measurements of garnet

were caried out in a DSC-TGA Perkin Elmer System-7thermobalance at the University of Bilbao. A sample ofapproximately 25 mg of millirneter-size garnet from zone3 was heated in N, atm at a rate of l0 "C min-'. Theinitial temperature was 40 'C and the final temperaturewas 900'C.

Garnet

The garnet samples arnlyzed are rich in andradite end-member (Table l, Fig. 3). The main substitution in thegarnet samples with diameter >2 mm from zone 3 is Alfor Fe3*, some samples being almost pure andradite (Figs.3 and 4A). These garnets are practically unzoned andtheir Cr content is, in most cases, below the detectionlimits of the microprobe, which suggests that the pale-green color must be due to their content of Fe3*. Thethermo-gravimetric measurements did not show presenceof HrO in the structure of these garnets. The small garnetsfrom zone I that grow around spinel are distinctly zoned.Their inner parts with a fibrous or radial aspect are richin Cr, Al, or both (generally richer in these elements thanthe larger euhedral garnets from zone 3) and have clear,massive rims rich in Fe3* (Table l, Fig. 3). These garnetsexhibit a poor correlation between I6lAl and Fe3* (Fig. 48)and a good negative correlation between Fe3* and t61Al +Cr (Fig. 4C), which rnay reflect Cr : Al substitution inthe original spinel. These observations suggest a contin-uous trend. with increase in Fe3* and decrease in Cr andAl, during the development of andradite-rich garnets.

r252

Andr - I00Vo

(1)

80

Uvar Gr+Alm+Sp+Py

Andr - l00%o

Uvar Gr+Alm+Sp+Py

Fig. 3. Composition of garnet. (A) Garnet of zone 3; opencircles : euhedral millimeter-size garnets; (1) : smaller garnetsat the beginning ofzone 3. (B) Garnet ofzone 1; solid circles :cores, open squares : nms.

Pyroxene and spinel

Pyroxene has been found only in zone I ofthe andra-dite vein (Fig. 2C), where it may exhibit a rim of andra-dite (Table l). Pyroxenes analyzed are almost pure diop-side with relatively small amounts of Al, Cr, and Na.Spinel occurs generally as minute grains (<0.05 mm) inzones 1 and 2 of the andradite vein, being systematicallysurrounded by a rim of garnet often rich in uvarovite(Table l, Fig. 3A). The spinel is chromite with importantproportions of magnetite and spinel solutions (Table l).

X-nav na-r.c.

X-ray diffraction data for andradite garnet were takenon dried material, ground to pass through a 60 pm sieve.


2.0

r .9

+ l R

,9.'

t . t

t . 6

1 . 8

1 . 5

1 . 6

+1 1

c ) r . ,tJ{

1 . 9

1 . 8

+ "1 7()

l f

0.0 o.2Alt6l

0.40.30 . 1

1 . 5

Eh

a \o ' .

a

t . d ' .\ t ri ' o .

0.0

A l t6 l+ Cr

Fig. 4. (A) ret41 vs. Fe,* for garnets ofzone 3; least-squaresregression l ine is y :1.9717 - 1.0301x (r:0.96). (B) Iur4l .rrr.

Fe3* for garnets ofzone l; solid circles : cores, open squares :rims. (C) (turAl + Cr) vs. Fe3* for garnets of zone 1; solid circles: cores, open squares : nmsl least-squares reg.ression hne is y: 1.9512 - 1.1055x ( r : 0 .99) .

The powder patterns were taken at room temperaturewith a Philips PW l7l0 diffractometer (graphite mono-chromatized, CuKa radiation). Patterns were obtainedfrom 3 to 60 20 at goniometer speeds ranging from 0.2to 2 20 min-t with appropriate time constants, ratemeter: J x lQz cps at 40 kV and 20 mA. Final unit-cell pa-rameters for the cubic mineral were obtained by least-

1 . 6

0.50.40.30.20 . 1

' . o

B

aa o

1 O

a

N

o

N 6

rN

1 200 800 600 400

W A V E N U M B E R S c m - 1

i l 3

x$lhu r

12oo looo ',1,'ou=*.-itlr*, #'

2oo o

Fig. 5. Infrared (a) and Raman (b) spectra ofnatural andra-dite-rich garnet.

square refinements of the indexed powder diffraction,patterns using the LSUCRE program (Appleman andEvans, 1973). The values obtained for these parametersare o : I1.980(9) A and V : l7l9(4) A,. This result isslightly different from the values reported by the JCPDS(1983) for pure synthetic andradite (a: 12.059 A, JCppSfield l0-288) and by Hazen and Finger (1989) for andra-dite from Val Malenco la: 12.031(1) Al and probablyreflects the solid solution of Al for Fe3*.

TrrnnrvronrxAMrc pRopERTIES oF ANDRADTTE FRoMSPECTROSCOPIC MEASUREMENTS

Spectroscopic measurements (IR and Raman) are use-ful for predicting, by vibrational modeling (Kieffer, 1979a,1979b, 1979c), the thermodynamic properties of a min-eral. The garnet family is one for which many heat ca-pacity measurements have been performed (Berman,1988, 1990) but for which the published experimentaland theoretical values of thermodynarnic parameters (C"and ,S) substantially disagree (Kieffer, 1980). For in-stance, the modeling scheme of thermodynamic proper-ties proposed by Kiefler (1980) for andradite CarFerSirO,,

t253

leads to a value of C" and ,S at 298 K of 339.4 and,292.6J/mol.K, respectively. These values differ widely fromthose experimentally measured, i.e., CB.rr, : 351.9 andS9rs : 316.4 J/mol.K (Robie et al., 1987) or Co,.,ss: 350.6(Kiseleva et al .,1972). To see if another modeling schemecould be proposed to explain this disagreement betweenexperimental and calculated values, we have recorded in-frared and Raman spectra of natural, nearly pureCarFerSirO,, garnet (millimeter-size garnets of zone 3).The infrared spectrum exhibits three modes (112,704,and 1070 cm ') not previously described in the literature.Using these new data in the lattice vibrational model ofKieffer, a value of 343.5 J/mol.K for specific heat of an-dradite at 298 K is obtainedl if anharmonic effects areconsidered, a value of 346.5 J/mol'K is obtained, whichis in good agreement (within about 1.50/o) with the exper-imental values of Robie et al. (1987) and Kiseleva et al.(1972). Taking into account the magnetic contribution tothe total entropy, we obtain a value of 3 I 3.6 J/mol.K forentropy at 298 K, which is very close (within less than10/o) to the experimental value of Robie et al. (1987). Themodel also leads to a value of the enthalpy of andradite(H\r, - IQge : 301 .4 kJ/mol) in excellent agreement withthe value recently measured (301.7 kJlmol) by Kiselevaet al. (1989). We therefore propose a model of density ofstates for andradite, allowing its thermodynamic prop-erties [C"(7), H(T), and S(I)] to be calculated between0 and 1500 K.

Experimental procedure

Mid- and far-infrared powder transmission spectra wereobtained in CsI (mid-IR) or parafrn (farJR) discs, usinga Bruker IFS ll3 FI-IR spectrometer. Spectra were re-corded between 500 and 1500 cm-' with a MCT detectoror between 50 and 700 cm-' with a liquid He-cooledbolometer detector. The amount of CarFerSi.O, in eachdisc was approximately l0lo (in mass). Spectra were mea-sured at room temperature, and each spectrum resultsfrom the accumulation of 500 scans at a resolution of 2cm r.

Raman spectra were recorded with a multichannel Ra-man microprobe (Dilor Microdil 28). A conventional lightmicroscope was used to focus the laser beam (ionized ArI : 5145.3 A; into a l-mm spot on the sample. TheRaman light scattered by the sample was collected in thebackscattering direction through the same objective (kitzUTK 50). A multichannel detector (512 diodes) was usedto detect and store the sigral. Peak positions are locatedwithin + I cm-r. Spectra measured at room temperatureresult from the accumulation of 12 spectra, the integra-tion time being equal to 16 s.

Infrared and Raman spectra

The room-temperature and room-pressure infrared andRaman spectra of andradite are presented in Figure 5.The peak positions are listed in Table 2. The infraredspectrum recorded is in good agreement with those ofandradite previously described in literature (Moore and


t!

z

co

a(I]

ztI|Fz

1 0 0 0

t254

White, l97l; Farmer, 19741, Kieffer, 1979b), except fortwo reproducible weak lines appearing at 704 and 1075cm '. The Raman spectrum agrees moderately well withthat of Moore and White (197l\. The six Raman modesappearing between 580 and 816 cm ' in the spectrum ofMoore and White are unobserved in our spectrum (Fig.5b).

The garnet structure of andradite is cubic with 8Ca.FerSirO,, units per unit cell. The smallest Bravais cellis primitive with 4 formula units per unit cell. In thisstructure, the Ca atoms are eightfold coordinated, the Featoms sixfold coordinated, and the Si atoms fourfold co-ordinated. The vibrational spectra of andradite clearlypossess two different groups of vibrational modes locatedat low and high frequency (Fig. 5). The high-frequencyregion extends from 816 to 996 cm-' in the Raman spec-trum and possesses three strong and one weak bands; itextends from 704 to 1070 cm-' in the IR spectrum andpossesses two strong and two weak bands. In the garnetstructure, these high-frequency modes are commonly as-signed to the Si-O stretching motions within rotSiO+ groups(Moore and White, l97l; Omori, l97l). The low-fre-quency region extends from 175 to 576 cm ' in the Ra-man spectrum (12 modes) and from I 12 to 589 cm-' inthe IR spectrum (12 modes). The upper limit of this re-gion (589 cm-r) has been assigned as an Fe3*Ou octahedralmode (Omori, 197l). Indeed, the position of the top ofthe low-frequency region (=620-630 cm-') does notchange significantly for different garnets having Al in theoctahedral site (grossularite, pyrope, spessartine, and al-mandine) but is substantially lower for andradite (589cm ') that has Fe in the octahedral site. The lowest modesare not as easy to identifu, but they are usually assignedas CaO, modes, possibly coupled with Fe3*O6 octahedralmodes.

A critical parameter in the lattice vibrational model ofKieffer is the position of the lowest mode of the vibra-tional spectra. The lowest mode of andradite observed inboth the IR spectrum of Moore and White (1971) andthe Kiefer (1979b) IR spectrum is at 132 cm-'. A weakline at 113 cm-'was also observed in one sample byMoore and White (1971). In our infrared spectra (Fig.5a), we observe the strong mode at 132 cm-'but also areproducible shoulder at about ll2 cm', which agreeswith the observation of Moore and White (1971).

Vibrational modeling

Quasi-harmonic model. The lattice vibrational modelof Kieffer provides a method for calculating the ther-modynamic functions of a solid that is independent ofany calorimetric data; the only data required are elasticwave velocities, crystallographic parameters, and spectralfrequencies. To calculate the thermodynamic functionsfrom vibrational spectra, the solid is assumed to consistof a collection of harmonic oscillators of frequency n,containing the total energy, U, of the solid. If the fre-quency distribution, i.e., the density of states g(z), of thesolid is known, the specific heat at constant volume, Cr,


TABLE 2. Measured Raman and infrared absorption bands ofCa.FerSi.O,,

996.87484381 6*

1 070'

880--840

704-589."c / o

5545 1 8495455372354326314267238175

514480440388352

304

212185151132112*

' lsolated Einstein oscillators..'Cutoff of optical continuum at high and low frequency.

is known ICv : @U/dT)ul. This yields (Lewis and Ran-dall, 1961)

c"(T):3,,R J'#; g@) dv (r)

where x : hv/kT (h being the Planck's constant, k theBoltzmann's constant, R the ideal gas constant) and n isthe number of atoms in the formula unit. The 3nR factortherefore represents the harmonic limit of the specific heatat constant volume at high temperature, i.e., the limit ofDulong and Petit.

The knowledge of C" then leads to the specific heat atconstant pressure and, consequently, to the knowledge ofthe enthalpy and the entropy ofthe solid:

C,(7): C"(n + ToPl/Kr Q)

H(T) _

s(D: i" 'qQ p e)

where Z is the temperature, a the thermal expansion co-efficient, Z the molar volume of the solid, and K" itsisothermal bulk modulus.

The lattice vibrational model of Kiefer gives an esti-mation of the density of states, g(v), of a solid from itsspectroscopic properties. The vibrational unit of the crys-tal is taken as the primitive unit cell with 3n total degreesof freedom. Of the 3n degrees of freedom, three areacoustic modes and the remaining 3n - 3 modes areoptic modes. The three acoustic modes are characterized

H(rJ: [ ' , r ,1r1n, (3)


C(v) TABLE3.

t255

Volume, thermal expansion, bulk modulus, and acous-tic data for Ca"FerSi"O,,

Vl"a:129.42do :2 .0 10 -5a1 : 0 ' 8 ' 10 IK ' : 159V":82O0Vs: 482O

cm3/molK 1K 2GPam/sm/s

(1 )(21(2)(3)(4)(4)

589 112

frequency 1cm-\Fig. 6. Density ofstates g(z) ofandradite vs. frequency (cm r).

by acoustic velocities, assuming a simple sine wave dis-persion of these modes toward the edge of the Brillouinzone. The 3n - 3 optic modes span a broad range offrequencies. An estimate of the range is obtained frominfrared and Raman spectra. The optic modes are dis-tributed uniformly between a lower cutoff frequency z,and an upper cutoff /r given by spectral data. At highfrequency, any isolated modes are represented as separateEinstein oscillators or by a second optic continuum. Thus,the assumed frequency distribution g(z) is a sum of theacoustic branches, optic continuum, and Einstein oscil-lators (all these functions are explicitly described in Kief-fer ,1979c) .

Such an approach has been used to estimate the ther-modynamic properties of the andradite garnet analyzed.The primitive unit cell contains four formula units, andhence there are 80 atoms and 240 degrees of freedom.The molar volume, thermal expansion, bulk modulus,and acoustic velocities used in the model are given inTable 3.

Acoustic velocities of andradite have been studied byBabuska et al. (1978). Andradite has longitudinal andshear wave velocities of 8200 and 4820 m/s, respectively.The corresponding directionally averaged acoustic veloc-ities (Kieffer, 1979a) are ur: 4627 m/s, ur: 5050 m/s,and u3: 8200 m/s. These velocities characterize acousticbranches that reach 64,70, and I l4 cm-r at the Brillouinzone boundary Gig. 6). The acoustic modes constitute| .25o/o (3/240) of the total number of degrees of freedom.

The optic modes constitute 98.75o/o (237/240) of thetotal number of degrees of freedom. From vibrationalspectra (Fig. 5), these modes can be distributed amongthree isolated Einstein oscillators (at704,996, and 1070cm-') and two optic continua, with limits given by IRand Raman spectra. The continuum at low frequency ex-tends from 112 to 589 cm-'and that at high frequencyfrom 816 to 880 cm-'. The high-frequency modes cor-respond to the Si-O stretching modes, which represent20o/o of the total number of the optic modes (Kieffer,1980). Eight modes (four IR and four Raman modes) arelocated at high frequency, and five modes (two IR andthree Raman modes) are located in the frequency range816-880 cm-'. We thus assume that the continuum athigh frequency contains 12.5o/o(5/e) of the total number ofthe high frequency optic modes uniformly distributed be-

A/ofe; (1) this work; (2) Skinner (1966); (3) Hazen and Finger (1989);(4) Babuska et al. (1978).

tween 816 and 880 cm-1, and the Einstein oscillators at704,996, and 1070 cm-r each contain 2.5o/o (t$ of thehigh frequency optic modes. At low frequency, the re-maining optic modes (800/o) are distributed uniformly inthe second continuum extending from l12 to 589 cm-'.The resulting model of frequency distribution, g(z), isshown in Figure 6.

Using the density of states of Figure 6 and the param-eters ofTable 3, we are able to calculate the specific heatat constant volume with temperature from Equation Iand, consequently, the thermodynamic functions C"(7),H(7), and S(7).

To test the accuracy of Kieffer's model, it is useful tohave some calorimetric data to compare the calculatedvalues with the experimental values. For andradite, wehave used one of the two thermodynamic data sets exist-ing on this compound, i.e., the recent heat capacity mea-surements obtained by Robie et al. (1987) in a large tem-perature range (10-1000 K). Below 365 K, heat capacitymeasurements were made by an intermittent heatingmethod under quasi-adiabatic conditions using a cryostattogether with a calorimeter. The accuracy of measure-ments was estimated to be +0.150/o above 40 K (Robieet al., 1987). The heat capacity of andradite exhibits asharp maximum (\ peak) at about 12 K. This heat max-imum arises from the antiferromagnetic ordering of themagnetic moments (spins) of the Fe3* ions. Above 350K, the heat capacity of andradite was measured by dif-ferential scanning calorimetry. Accuracy was estimatedto be around +lol0. The experimental values of C" ofRobie et al. (1987) and those obtained by modeling aregiven in Table 4 and Figure 7.lt can be seen that themodel reproduces very well the calorimetric data at lowtemperature (T < 200 K), except the ), peak which cannotbe observed by vibrational spectroscopy. At higher tem-perature (T > 200 K), Kieffer's model systematically un-derestimates heat capacities. The deviation is of about2.50/o between 200 and 1000 K (Fie. 8). The departure athigh temperature of the calculated specific heat from theexperimental data has been observed previously in somecompounds such as MgrSiOo (Gillet et al., 1990a), SiO2,and GeO, (Gillet et al., 1990b; Madon et al., 1991); itwas attributed to anharmonic effects not taken into ac-count in the quasi-harmonic model of Kieffer describedpreviously. The calculated vibrational entropy S" strong-ly differs from calorimetric data (Fig. 9). The differenceresults principally from the magnetic contribution to the

A

r256

400

r50 200

T K

250 300

300 400 500 600 700 800 900 1000

T K

Fig. 7. Comparison of calculated (heavy line) and experi-mental (squares) values of heat capacity of andradite. Experi-mental values are from Robie et al. (1987). (A) Low-temperatureadiabatic calorimetric data. (B) High-temperature (I > 300 K)differential scanning calorimetric data.

total entropy, which is unestimated from vibrationalspectra. For andradite, the ideal magnetic entropy is as-cribed to the transition from the fully antiferromagnet-ically ordered spin state at 0 K to the completely randomspin arrangement at high temperature. There will be anincrease in entropy of2R ln(2S + l) for a first transitiongroup ion. For Fe3*, S : %, and the ideal magnetic spinentropy for andradite is 29.8 J/mol'K. Robie et al. (1987)have shown that significant short range order must beretained by the spins at high temperature and that only580/o of the ideal magnetic entropy, i.e., 17.3 J/mol'K,contribute to the total entropy. Including this magneticcontribution S- to the calculated vibrational entropy S",Kieffer's model thus reproduces the calorimetric data verywell (Fig. 9). A value of 312.2 J/mol'K for entropy at298 K is obtained, which is close (1.30/o) to the experi-mental value (316.4 J/mol.K) of Robie et al. (1987).

The second thermodynamic data set existing onCarFerSirO,, is that of Kiseleva et al. (1972). Measure-


z )

s€ o oO

I 300

€ 2oo

v loo

r guasi-harmonic

a

t

a l l r r l l l l l l t l

l r

1000

T K

Fig. 8. Difference (in percent) between experimental valuesofC" ofRobie et al. (1987) and calculated values from the quasi-harmonic model of Kieffer.

ments of the change of enthalpy with temperature were

made on a Tian-Calvet type microcalorimeter by heating

the sample from room temperature to the temperature of

the calorimeter. The accuracY of measurements was es-

TABLE 4. Values of heat capacity, vibrational entropy, and rel-ative enthalpy of Ca.FerSi.O',

35010050

5 0

Itr 450:?

O

3s0T

(K)C" CP

(J/mol.K) (J/mol.K)s, H, - H"""

(J/mol.K) (kJ/mol.K)

0 050 26.6

100 107.2150 187.9200 253.3250 303.5298 340.4300 341.73s0 370.9400 393.3450 410.7500 424.4550 435.3600 444.1650 451.2700 457.17s0 462.0800 466.1850 469.5900 472.5950 475.0

1000 477.21050 479.11 100 480.81150 482.31200 483.61250 484.71300 485.81350 486.71400 487.51450 488.31500 488.9

0 027.0 9.0

108.1 51 .6189.3 111.2255.2 175.0306.0 237.7343.5 294.9344.8 297.'l374.6 352.6397.7 404.1415.9 452.1430.4 496.7442.'t 538.3451.7 577 .1459.8 613.6466.6 648.0472.5 680.3477.7 71 1 .0482.3 740.1486.4 767.8490.2 794.2493.7 819.4496.9 843.6500.0 866.8502.9 889.1505.7 910.6508.5 931.3511.2 951 .2513.8 970.6516.4 989.3519.0 1007.5521.6 1025.1

00.7

18.738.058.479.5

101.4123.7146.5169.7193.2216.9240.9265.1289.6314.1338.9363.8388.9414.1439.5465.0490.6516.4542.2568.3

Note: G!: heat capacity at constant volume, C" : heat capacity atconstant pressure, S" : vibrational entropy, (H, - Hrss) : relative enthalpy.Values calculated from quasi-harmonic model of Kieffer based on densitiesof state.


500000

r257

1100 1300

E zuuF)

a

A

Svib + Sm

Svib

150

T K

Bquasi-harmonic

200

T K

Fig. 9. (A) Comparison between calculated (heavy lines) andexperimental (squares) values ofentropy ofRobie et al. (1987).Kiefer's model allows the determination of vibrational entropy(S",0); the magnetic contribution (,S*) to the total entropy isabout 17.3 J/mol'K. @) Diference (in percent) between exper-imental values of ,S of Robie et al. (1987) and calculated values(^S" + S-) from the quasi-harmonic model of Kieffer.

timated to be 1.5-2.50/o (Kiseleva et al., 1972). The ex-perimental values of AIl of l(iseleva et al. (1972) andthose obtained by modeling are shown in Figure 10. Itcan be seen that the model reproduces very well the ca-lorimetric data of Kiseleva et al. (1972). The differencebetween experimental and calculated values is less thanl.5olo between 300 and ll00 K and therefore in goodagreement with the experimental error (Fig. l0B). A val-ue of 301 .4 kJlmol for the relative enthalpy of andraditeat 973 K (H3,, - f/!rr) is obtained, which is in excellentagreement with the value recently measured (301.7 kI/mol) by Kiseleva et al. (1989).

-2.5

700

T K

Fig. 10. (A) Comparison of calculated fteavy line) and ex-perimental (squares) values of relative enthalpy of andradite.LH(T-) - H(298) is in J/moll. Experimental values are from Kis-eleva et al. (1972). (B) Difference (in percent) between experi-mental values of AIl of Kiseleva et al. (197 2) and values calcu-lated from the quasi-harmonic model of Kieffer.

Another thermodynamic data set existing onCarFerSirO,, is that of Devyatkova and Tikhonov (1967).They have measured the specific heat of andradite byadiabatic calorimetry between 90 and 384 K. Unfortu-nately, no details are given on the experimental proce-dure, and it is not easy to estimate the accuracy of theirC" measurements at low temperature. Moreover, theirvalues are not very reliable because specimens were foundto contain appreciable amounts of oxides that had notreacted in the synthesis (Kiseleva et al., 1989).

The proposed density of state thus satisfactorily repro-duces the diferent calorimetric data sets for andradite butunderestimates the specific heat at hlgh temperature,which could be due to anharmonic effects not taken intoaccount in the quasi-harmonic model of Kieffer.

Anharmonic model. Gillet et al. (1990a) have intro-duced a correction in the lattice vibrational model of Kief-fer to consider the effects of anharmonicity. Their modelaccounts more readily for the observed departure at hightemperature of the specific heat from the harmonic Du-

coct

3

400000

200000

100000

7005000300 900

T K

100

st r 00

a

2.5

0.0

s

6o\ct

F<

500300

5.0

A

B

l 258

long and Petit limit (3nR). Anharmonicity basically re-sults from the interactions between the normal modes ofvibrations of a mineral. It can be estimated from thepressure and temperature shifts of the optical vibrationalmodes. A mode anharmonic parameter la,

-- @ In v,/67")"1,

expressing a change with temperature (at constant vol-ume) in the frequency of a vibrational mode v,, carr berelated to the pressure and temperature shifts of themode by

ai : a ' (^Yir - ^Yi)

:"(n+.a-1ry). (5)r d P a 6 T l '

The anharmonic specific heat at constant volume C$(and consequently the anharmonic specific heat at con-stant pressure Cf and the anharmonic entropy S*) canthus be related to the harmonic specific heat (C") by (Gil-let et al., 1990a):

.*: ? [c" ft - @tr") - \qru] (6)

where (a) is the mean value of the anharmonic param-eters a, of the modes 2,, uniformly distributed in a givenoptic continuum (in the case of isolated modes repre-sented by an Einstein oscillator, (d) is taken as a,), andwhere U is the internal energy in excess of that at 0 K,1.e.,


TABLE 5. Values of quasi-harmonic heat capacity, anharmonicheat capacity, and anharmonic vibrational entropy ofCa.FerSi.O,, calculated from anharmonic model

T C v q(K) (J/mol.K) (J/mol.K)

q a F r q(J/mol K) (kJ/mol K) (J/mol K)

050

100150200250298300350400450500550600650700750800850900950

1 0001 0501 1001 1501 200125013001 3501 4001 4501 500

026.6

107.2187.92s3.3303.5340.4341.7370.9393.3410.7424.4435.3444.1451.2457.1462.0466.1469.5472.5475.0477.2479.1480.8482.3483.6484.7485.8486.7487.5488.3488.9

026.6

107.5188.6254 7305.7343.4344.7374.8398.1416.5431.1443.0452.7460.8467.7473.6478.6483.1487.0490.5493.7496.6499.3501.7504.0506.1508.2510.1511 .9513.6515.3

027.0

108.4190.0256.6308.2346.5347.8378.5402.6421.6437.1449.8460.4469.4477.2484.1490.2495.8500.9505.7510.1514.4518.4522.4526.2529.9533.5537.2540.8544.4547.9

0 00.4 9.03.7 51.7

11.2 1 1 1 .422.4 175.636.6 238.752.4 296.353.0 298.57't.2 354.590.8 406.7

111.4 455.3132.9 500.5155.0 542.8177.8 582.4201 .1 619.6224.7 654.7248.8 687.8273.1 719.3297.8 749.2322.7 777.7347.9 804.9373.3 830.9398.9 855.9424.7 879.9450.7 903.1476.9 925.4503.3 946.9529.9 967.8556.7 988.0583.6 1007.6610.8 1 026.6638.1 1045.1

This anharmonic correction in the lattice vibrationalmodel of Kieffer leads to calculated thermodynamicproperties of a solid in excellent agreement with experi-mental data.

Anharmonicity of garnets has been recently studied(Fiquet and Gillet, in preparation); a mean value of-2.10-5 K-' for the anharmonic parameters c, can beconsidered for the garnet family (Gillet, personal com-munication). This datum can thus be used to express theanharmonic thermodynamic properties G(T), II*(T), andS*(7) of andradite from Equations 6, 7, l, 2, 3, and 4.The variation of these parameters between 0 and 1500 Kis listed in Table 5. The anharmonic model greatly re-duces diflerences between calculated and experimentalcalorimetric data at high temperature (Figs. I I and l2).Calculated heat capacities agree with experimental valuesbetween 300 and 1000 K within lol0, which is in goodagreement with the experimental error of Robie et al.(1987) (Fig. llB). A value of 346.5 J/mol.K for specificheat ofandradite at 298 K is obtained, which is in goodagreement (about 1.590) with the experimental values ofRobie et al. (1987) and Kiseleva et al. (1972) (Table 5,Fig. I lA). Taking into account the magnetic contributionto the total entropy (,S*: 17.3 J/mol.K), we obtain avalue of 3 I 3.6 J/mol.K for entropy at 298 K (Fig. 12A),which is very close (less than l0lo) to the experimentalvalue of Robie et al. (1987) (Fig. l2B).

Nofe: The Lff : H(n - H(0), G, : quasi-harmonic heat capacity atconstant volume, q : anharmonic heat capacity at constant volume'G : anharmonic heat capacity at constant pressure, S : anharmonicvibrational entropy.

We thus propose, for 7 > 298 K, equations expressingthe temperature dependence of specific heat (in J/mol'K)and relative enthalpy (in J/mol) of andradite that are dif-ferent from those of Robie et al. (1987) and Kiseleva etal. (1972):

C,(T) :839.65 - 13.21 x 10-27- 7 .691 x l 03 f 05- l l . 3 l 9 x l 05T -2 + 4 .702 x l 0 572

H, - H,nr : 839.657 - 6.61 x 10-272- 15 '38 x l o3zo5+ l l ' 3 1 9 x l o 5 Z - '+ 1.567 x l0 57"3 + 16972.

CoNcr,usroNs

The infrared and Raman spectra of CarFerSirO', havebeen recorded at room pressure and room temperature.The IR spectrum exhibits three modes (112,704, and1070 cm-') not previously used in the modeling schemeof the thermodynamic properties of andradite. Using thesenew data in the lattice vibrational model of Kieffer, thethermodynamic properties of andradite have been cal-culated. The agreement between measured and calculated

u(n:rrnrJ F5 g@)dv. (7)

A

600

I quasi-harmonicO 1

I i . l l r r l l r l l l r l r r

t r t r oO E E

t r E o t r

anharmonic o o

DD

t259

A

Svib + Sm

Svib

150 200

T K

Bquasi-harmonic

II t ro

l to t r

anharmonic

-2.5

-5.0

T KFig. 12. (A) Comparison of experimental (squares) values of

entropy ofRobie et al. (1987) and calculated values (heavy lines)from anharmonic model. (B) Diference (in percent) among theexperimental values ofS ofRobie et al. (1987) and values cal-culated (S" + S-) from the quasi-harmonic model of Kiefferand the anharmonic mode^.

RnrBnnNcrs crrEDAmthauer, G., Kurtz, W., Rost, F., and Schloemer, H. (1974) Chemismus

und Genese des Andradits aus dem Serpentinit des Val Malenco (Ber-nina-Gebiet, Oberitalien). Schweizerische Mineralogische und Petro-graphische Mitteilungen, 54, 691-706.

Appleman, D.E., and Evans, H.T. (1973) Indexing and least squares re-finement ofpowder diftaction data. NTIS Document no. PB-216188,U.S. Geological Survey, Washington, DC.

Arenas, R., Gil Ibarguchi, J.1., Gonziiez Lodeiro, F., Klein, E., MartinezCatalin, J.R., Ortega Giron6s, E., Pablo Maci6, J.G. de, and Peinado,M. (1986) Tectonostratigraphic units in the complexes with mafic andrelated rocks of the NW of the Iberian massif. Hercynica, 2, 87 -l lO.

Babuska, V., Fiala, J., Kumazawa, M., and Ohno, I. (1978) Elastic prop-erties ofgarnet solid-solution series. Physics ofthe Earth and PlanetaryInteriors. 16. 157 -17 6.

Berman, R.G. (1988) Internally-consistent thermodynamic data for min-erals in the system NarO-KrO-CaO-MgO-FeO-FerOr-AlrOr-SiOr-TiOr-HrO-COr. Journal of Petrology, 29, 445-522.

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

Derryatkova, E.D., and Tikhonov, V.V. (1967) Teploprovodnoct'i teplo-emkost' ittrii-kal'tsievykh granatov (in Russian). Fizika Tvergodo Tela,9 ,772 -777 .


400

500300

E 2oo

U)100

I 4oo

- 300

3o. 200

Q

5.0,

2.5

5.0

z )

s\- n nE v . v

v)

100

35030025010050600200 400

T K

800 1000

N

r )

0.0

-2.5

-5.0

200 400 600

T K

800 1000

Fig. I 1 . (A) Comparison of experimental (squares) values ofheat capacity ofRobie et al. (1987) and calculated values (heavyline) from the anharmonic model. (B) Difference (in percent)among the experimental values of C" of Robie et al. (1987) andvalues calculated from the quasi-harmonic model of Kieffer andthe anharmonic model.

specific heat and relative enthalpy is excellent (about l0lodifference between 300 and ll00 K). The specific heat(Cr,rn, * 346.5 J/mol.K) and entropy (Srn, = 313.6 J/mol.K) at room temperature are in good agreement withthe experimental values reported by Robie et d. (1987)and Kiseleva et al. (1972). The model of density of stateproposed for andradite is used to calculate C"(7), H(T),and ̂ S(D of CarFerSirO,, between 0 and 1500 K. Equa-tions expressing the temperature dependence ofheat ca-pacity and relative heat content ofandradite are thus pro-posed that allow its thermodynamic properties C"(7),H(T), arrd S(Z) to be calculated at high temperature (Z> 298 K).

AcxNowr,nlcMENTs

Financial support by UPV and DGICYT grants (139.89, E040/90,89.4 I l) to J.I.G.I., by the French-Spanish Cooperation Project (HF-077)to J.I.G.I. and J.G., and by INSU (DBT 732Ap88) ro J.c. is acknowl-eged. The present study was done as a part of EEC contract Sd.0046. qH).IPG contribution no. 1146, CNRS/INSU/DBT contribution no.2l2.

400300200100

t260

Farmer, V.C. (1974) Orthosilicates, pyrosilicates and other finite chainsilicates. In V.C. Farmer, Ed., The infrared spectra of minerals, mono-graph 4, p. 285-303. Mineralogical Society, London

Fediukovd, E., Hovorka, D., and Gregus, J. (1976) Compositional zoningof andradite from serpentinite at Dobsina (West Carpathians). VestnikUstredniho Ustavu Geologickeho, 51, 339-345.

Gil Ibarguchi, J.L, Mendia, M., Girardeau, J., and Peucat, J.J. (1990)Petrology of eclogites and clinopyroxene-garnet metabasites from theCabo Ortegal complex (northwestern Spain). Lithos, 25,133-162

Gillet, Ph., Guyot, F., and Malezieux, J.M. (1990a) High pressure andhigh temperature Raman spectroscopy of CarGeOo: Some insights ofanharmonicity. Physics ofthe Earth and Planetary Interiors, 58, 541-554.

Gillet, Ph., k Cl6ac'h, A., and Madon, M. (1990b) High temperatureRaman spectroscopy of SiO, and GeO, polymorphs: Some insights onanharmonicity and thermodynamic properties of the SiO, polymorphsat high temperatures. Journal of Geophysical Research, 95, 21635-21655.

Girardeau, J., Gil Ibarguchi, J.I., and Ben Jamaa, N. (1989) Evidence fora heterogeneous upper mantle in the Cabo Ortegal Complex, Spain.Science. 245. 123 l-1233.

Hazen, R.M., and Finger, L.W. (1989) High-pressure crystal chemistry ofandradite and pyrope: Revised procedures for high-pressure diffractionexperiments. American Mineralogist, 7 4, 3 52-3 59.

Joint Committee for Powder Diftaction Standards (JCPDS) (1983) Groupdata book. International Centre for Diffraction Data, Swarthmore,Pennsylvania.

Kieffer, S.W. (1979a) Thermodynamics and lattice vibrations of minerals.1-Mineral heat capacities and their relationships to simple lattice vi-brational models. Review ofGeophysics and Space Physics, 17, l-19.

-(1979b) Thermodynamics and lattice vibrations of minerals. 2-Vibrational characteristics of silicates. Review of Geophysics and SpacePhysics, 17, 20-34.

-(1979c) Thermodynamics and lattice vibrations of minerals. 3-Lattice dynamics and an approximation for minerals with applicationto simple substances and framework silicates. Review of Geophysicsand Space Physics, 17, 35-38.

-(1980) Thermodynamics and lattice vibrations of minerals. 4-Application to chain and sheet silicates and orthosilicates. Review ofGeophysics and Space Physics, 18, 862-886.


Kiseleva, I.A., Topor, N.D, and Mel'chakova, L.Y (1972)Experimentaldetermination ofheat content and heat capacity ofgrossularite, andra-dite and pyrope. Geokhimiya,11,1372-1379 (in Russian).

Kiseleva, I.A., Ogorodova, L.P., and Sokolova, Y.L (1989) The enthalpyof formation of andradite. Geokhimiya, 1, 1 25-1 3 l.

lrwis, B.N., and Randall, M. (1961) Thermodynamics,T23 p. McGraw-Hill, New York.

Madon, M., Gillet, Ph., Julien, Ch., and Price, G.D. (1991) A vibrationalstudy of phase transitions among the GeO, polymorphs. Physics andChemistry of Minerals, in press.

Moore, R.K., and White, W.B. (1971) Vibrational spectra of the commonsilicates: I. The garnets. American Mineralogist, 56,54-71.

Omori, K. (1971) Analysis of the infrared absorption spectrum of alman-dine pyrope garnet from Nisojan, Osaka prefecture, Japan. AmericanMineralogist, 56, 84 l-849

Peters, T. (1963) Mineralogie und Petrographie des Totalpserpentins beiDavos Schweizerische Mineralogische und Petrographische Mitteil-ungen,43, 529-685.

Peucat, J.J, Bernard-Griffiths, J., Gil Ibarguchi, J I, Dallmeyer, R.D.,Menot, P., Cornichet, J., and Iglesias Ponce de Leon, M. (1990) Geo-chemical and geochronological cross-section ofthe deep Hercynian crust:The Cabo Ortegal high-pressure nappe (NW Spain). Tectonophysics,r77, 263-292.

Robie, R.A., Bin, 2., Hemingway, B.S., and Barton, M.D. (1987) Heatcapacity and thermodynamic properties of andradite garnet,CarFerSi,O,r, between l0 and 1000 K and revised values forAGS(298.15 K) ofhedenbergite and wollastonite. Geochimica et Cos-mochimica Acta, 51, 2219-2224.

Schmetzer, K., Traub, I., and Medenbach, O. (1975) Demantoid aus Ko-rea. Zeitschrift der Deutsche Gemmologischen Gesellschaft, 24, l-3.

Skinner, B.J. (1966) Thermal expansion. Geological Society ofAmericaMemoir, 97, 7 5-96.

Vogel, D.E. (1967) Petrology ofan eclogite and plrigarnite bearing poly-metamorphic rock complex at Cabo Ortegal, NW Spain. kidse Geo-logische Mededelingen, I-eiden, 40, l2l-213.

Webster, R. (1970) Gems, their sources, descriptions and identification,836 p. Butterworth, London.

MeNuscnrrr RECETvED Jeruenv 2, 1990Mnr.ruscnrrr AccEprED MencH 15. l99l

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