Canodian MineralogistVol.28, pp. 93-109 (1990)

EVALUATION OF THE RIETVELD METHOD FOR THE CHARACTERIZATIONOF FINE-GRAINED PRODUCTS OF MINERAL SYNTHESIS:

THE DIOPSIDE_HEDENBERGITE JOIN

MATI RAUDSEPP, FRANK C. HAWTHORNE ANDALLAN C. TURNOCKDeportment of Geological Sciences, University of Monitoba,

Winnipeg, Manitoba R3T 2N2

ABSTRAcT

We have evaluated the Rietveld method for characterizingfine-grained products of mineral synthesis by refirting thecrystal structures of a series of synthetic clinopyroxenesalong the diopside - hedenbergitejoin. The Rietveld methoduses the whole powder-diffraction pattern to characterizetlte structure of a material; the structural parameters (atornicpositions, site occupancies, displacement parameters),together with various instrumental parameters, are refinedby least-squares procedures to minimize the differencebetween tlte complete observed and calculated diffraction-patterns. With this technique, more than one phase can berefined simultaneously. Step-scan X-ray powder diffrac-tion data were collected over the range l7-130o2d usingCul(a X radiation. The structures of the syntheticclinopyroxenes were refined to Rs indices of 1.8-3.590 andR, indices of 8.0-11.490. For diopside, comparison withsinlle-crystal results on natural material shows good agree-ment between the struclural parameters, but half-normalprobability analysis shows the Rietveld standard deviationsto be underestimated by a factor of - 1.6; this was expected,as the Durbin-Watson d-statistic (1.39) indicates thepresence of significant serial correlation. For the Fe-bearingpyroxenes, the observed stereochemistry agrees with thatexpected for pyroxenes intermediate between the knownstructures of diopside and hedenbergite. Unconstrained site-occupancy refinement (witi isotropic displacement factorsfixed at appropriate single-crystal values) indicatesMQ)=Ca; the refined M(l) occupancies are within 2 stan-dard deviations (o = 0.01 a.p.f.u.) of the nominal com-positions, but are systematically lower, suggesting that thesynthesized pyroxenes may be slightly off-composition. Forthe two-phase products, the (fixed) structure of ferrobusta-mite was incorporated into the refinement procedure, andthe site occupancies, cell dimensions and scale factor (ameasure of modal amount) were refined concurrently s/iththe accompanfng pyroxene. Model calculations show thatthe minimum in the site-occupancy refinement is well-defined, provided that the displacement factors are fixed(at appropriate values). Site occupancies are dependent onthe values of the displacement factors used, but model cAl-culations show that a fairly large range in B (a 0.25i\2)spans only I standard deviation (0.01 a.p.f.u.) of a Mg-Fe site occupancy. Thus site-occupancy refinement is notimpractically dependent on the displacement factors used.Unconstrained site-occupancy refinement gives us not onlythe site occupanciesn but also the bulk compositions of thepyroxene; the close agreement between the nominal andrefined compositions indicates that this may be a viabletech-nique for compositional determination of suitable fine-grained minerals. In addition, multiphase mixtures may be

93

analyzed, with the possibility of determining cationoccupancies, bulk compositions and modal proportions foreach phase. These factors indicate that Rietveld structurerefinement will be of major importance in characterizationof products of synthesis in the future.

Keywords: Rietveld method, diopside, hedenbergite, struc-ture refinement, fine-grained materials, multiphaseassemblages, products of syntheses.

SOMMAIRE

Nous avons 6valud la m6thode de Rietveld pour carac-tdriser les produits de synth0se i grains fins par I'affine-ment de la structure cristalline de clinopyroxbnes de la sdriediopside - hddenbergite, Cette m6thode utilise le clich6 dediffraction au complet pour calculer Ia structure du mat6-riau. Les parambtres structurarD( (positions atomiques,rdpartition parmi les sites, paramBtres de d€placunent), ainsique les divers parambtres instrumentaux, ont 6td affinds parmoindres carr€s afin de minimiser la diffdrence entre leclichd de diffraction observ6 et calcul6. Il est m€me pos-sible de caractdriser plus d'une phase d la fois. La lecturedes intensitds diffractom6triques s'est faite selon le mode"step-scan" entre 17 et l30o 20 (rayonnement Cu/(a). Lesstructures ont 6te affinees jusqu'i un r6sidu R6 entre 1.8et 3.590, et une valeur de R' entre 8.0 et ll.49o. PourIe diopside, les rdsultats obtenirs sur cristal unique naturelmontrent une bonne concordance des parambtres stnrctu-raux, mais une analyse de probabilit€ demi-normaleddmontre que les dcarts-types pour la m6thode de Rietveldseraient sous-estim€s par un facteur d'envhon 1.6; c'est uned6viation anticipee, puisque la statistique d de Durbin-Watson (1.39) indique la pr€sence d'une corr6lation sdrielleimportante. Pour les compositions riches en fer, la st6r6o-chimie observde concorde avec les rdsultats anticipds pourles compositions interm€diaires entre les p6les diopside eth6denbergite. Les affinements d'occupation de sites sanscontraintes (avec des facteurs de ddplacement isotrope fixesselon les rdsultats obtenus sur cristaux uniques correspon-dants) indiquent que MQ) contient le Ca; M(l) contient,d deux ecarts-type prbs (o : 0.01 atomes par unit6 formu-laire), les populations prdvues selon les compositions prd-par6es, mais les occupations sont systemadquement incom-plOtes, ce qui fait penser que la composition des produitssynthdtiques est legerement non stoechiom€trique. Pour lesproduits d deux phases, la structure (fixde) de la ferrobus-tamite a €t€ incorpor6e dans la proc€dure d'affinement, etles occupations des sites, les parambtres rdticulaires, et lefacteur d'ajustement (mesure de la proportion des volumes)ont dte affinds simultan€ment avec ceu( du pyroxdne pr€-

94 THE CANADIAN MINERALOGIST

sent. Le minimum dans la fonction d'occupation des sitesserait bien ddfini, d'aprbs nos calculs, pourvu que les fac-teurs de d€placement soient fix6s (i des valeurs appropri6es).Les occupations des sites d6pendent des valeurs choisiespour ces facteurs, mais des calculs modbles mongent qu'unintervalle assez grand de valeurs de B (+ 0.25A2) corres-pond i seulement un 6cart-type (0.01 atomes pax unit6 for-mulaire) dans I'occupation du site Mg,Fe). C'est donc direque I'affinement de l'occupation d'un site ne d€pend pasde faqon non pratique des facteurs de d6placement choisis.L'affinement sans contraintes de l'occupation des sites mbneen plus d la composition globale d'un pyroxdne; la con-cordance 6troite entre la composition pr6dite et affin6emontre que cette technique pourrait bien servir pour d6ter-miner la composition de mat€riaux i grain fin approprids.De plus, la possibilit€ existe d'analyser des assemblagesmulti-phasds et de d6terminer la r€partition des cations, Iacomposition elobale et la proportion modale de chaquephase. Pour ces raisons, il paralt dvident que l'affinementdes structures par la mdthode de Rietveld aura une grandeimportance dans la caract6risation des produits de synthdse.

(Traduit par la R6daction)

Mots-clds: m6thode de Rietveld, diopside, hfienbergite,affinement de la structure, matdriaux i grains fins,assemblages multiphas€s, produits de synth&se.

TABLE 1. NOI'INAL COMPOSXNONS, CONDITIONS AND PRODUCTS

BuNo. CoDp@itid r cc)

INTRoDUcTIoN

A significant problem in studies of mineral syn-thesis and stability is the characterization of the runproducts. These are commonly quite fine-grained (ofthe order of a few micrometers), and consequentlyare difficult to characterize adequately. The tech-niques traditionally useC are optical microscopy andX-ray powder diffraction; however, these can rarelyprovide quantitative information on phase compo-sition and intracrystalline order. Although small par-ticles can be analyzed by electron microprobe (e.g.,White 1964, Spear 1981), this approach provides noinformation on the degree of order of the constituentphase(s). Spectroscopic techniques are now beingused more frequently for this purpose (Hawthorne1988), but tend to be too problem-specific to con-stitute a general-purpose method. We have advocatedthe use of the Rieweld method for better run-productcharacterization (Raudsepp et ol.1982, 1984, Haw-thome et al. 1984). The method (Rietveld 1967, 1969)uses the whole powder-diffraction pattern to charac-terize the structure of a material; the structureparameters of the mineral latomic coordinates, siteoccupancies and displacement (thermal) parametersl,togethernith various experimental parametersaffecting the pattern, are refined by least-squaresprocedures to minimize the difference'between thecomplete observed and calculated diffraction-patterns. Although the Rietveld method was usedoriginally for the refinement of fairly simple struc-tures with X-ray-diffraction data (Young et al. 1977,Khattak & Cox 1977, Young 1980), it has sinceproved quite powerful for complicated structures(Baerlocher 1984, Baerlocher & Schicker 1987, Ercitet al.1985, Raudsepp et al.1987a, b). In addition,it is also possible to refine the structures of more thanone phase simultaneously, and thus the method isapplicable to ossemblages of synthesized minerals;indeed, the modal amounts of each phase also canbe derived by this method. Additional advantagesare: (i) the equipment used, an X-ray powder diffrac-tometer, is widely available; (ii) there is extensivereadily available software for structure refinement(Hill & Howard 1986, Larson & von Dreele 1988,Wiles & Young 1981, Baerlocher 1982).

We have long been interested in the synthesis,characterization and phase relations of Ca-Mg-Fepyroxenes (Turnock 1962, 1970, Turnock et al.1973). In these previous studies, the synthesizedpyroxenes were assumed to be of nominal composi-tion, and the degree of order over the M(l) and MQ)sites was not characterized. For single-phasematerial, examination by Rietveld structure refine-ment can provide information on degree of order,and information on stereochemistry (although thiscannot be as good in quality as analogous single-crystal data). For multiphase material, Rietveld struc-

D0 CaMgSi2Oo

D2 CallgorFqrSi2O!

D$ CaMto.rFqrSi:Oo

D6 CaM&rFqr$:Or

D7 CaMg:Fo,rSir Oo

1:tito

12EE

|m,1160

1120

"pt + (*)

cpr + gle

cpx + (Ibs)

cF -r' (lb)

c p : + I b s + ( @ ) + ( o l )

oo tzoo

cpr cliaop5uoc; lba: fecoburtaoik; o: eiatobalite; ol oliri*.

t400

r 300

1 1 0 0

't000

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

1O0 Fe/(Fe+Mg)

Frc. l. Temperature-composition section along thediopside-hedenbergite join at I atm and low oxygenfugacity, showing T, X and, phases of refined producrs(after Huebner & Turnock 1980). Symbols: A augite;B pyroxenoid, ferrobustamite; L liquid.

Woa60

L

RIETYELD METHOD: DIOPSIDE_HEDENBERGITE JOIN 95

ture refinement can provide compositions, informa-tion on tlte degree of order of all phases, and theirmodal proportions. This being the case, it will greatlyincrease the information derived from synthesis andphase-equilibrium studies, and provide much morecomplete information for thermodynamic modeling.

In the past 30 years, many new techniques ofcharacterization have been introduced into miner-alogy. These have usually fbllowed the same cycleof development. They have been introduced tomineralogy with strong (usually exaggerated) claimsof accuracy, but with no adequate study of eitherprecision or accuracy. This has eventually led to the(undeserved) discreditation of the method, and it hastaken a further 10 years or so to re-establish it asa standard credible method of characterization. Theprincipal thrust of the current work is to establishthe utility of the method and to demonstrate the pre-cision and accuracy that can be expected for geolog-ically useful minerals; hopefully we can bypass thediscreditation phase of the cycle outlined above. Inorder to provide a realistic evaluation of accuracy,we examine here a simple but geologically relevantsy$tem: single- and multi-phase products of synthesisalong the diopside - hedenbergite join, from diop-side (Wo5Bnr6Fs) to hedenbergite (Wo5sEn15Fs35).Nominal compositions are given in Table l, togetherwith conditions of synthesis and nature of theproducts; phase relations are shown in Figure l. Thecrystal structures of diopside (Clark et ol. 1969,Levien & Prewitt 1981) and hedenbergite (Cameronet ol. L973) are well known, and provide good con-straints on the possible stereochemical variations tobe expected in our intermediate binary compositions.In addition, the bulk-composition constraintsimposed by the simple nominal chemistry providea good check on the accuracy of the refined bulk-compositions. If the refined structures arereasonable, we can expect good results for the chem-ically more complex ternary pyroxenes.

ExpBnrMeNtaL METHoDS

Synthesis

Clinopyroxenes were synthesized from mixtures ofdry reagent-grade CaCOr, FqO3, SiO2 and MgO byrepeated heating and grinding cycles (Turnock et al.1973). Sample pellets were suspended from Pt wirehangers and heated at temperafures l0 to 60'C belowthe solidus in the range ll20 to 1330'C at I atm pres-sure. Oxygen fugacity was maintained near the iron-wiistite buffer by passing a gas mixture(COICO2:3) through the furnace. After eachexperiment, the product was ground for 15 minutesunder xylol in an alundum or agate mortar. Afterl0 cycles, the Mg-rich run products were found tocontain mainly prismatic clinopyroxene crystals up

to 30 pm in length, with rare (< 0.590) olivine andcristobalite; Fe-rich products also contain up toabout 8Yo ferrobustamite. Nominal compositions,conditions of synthesis, and products are listed inTable I and are shown graphically in Figure l.

Data collection

Samples were mounted in standard aluminumsample holders with 20 x 15 x 1.6 mm cavities. Thuswith a l' divergence slit, the irradiated area was con-fined to the sample at20 angles greater than l9o.Two different methods of mounting were used. Forcell-dimension determination, powders were denselypacked from the back of the mount against a frostedglass slide; this procedure gave a flat surface levelwith the top of the holder, a surface that was con-sistent from sample to sample; thus specimen-displacement and transparency errors are minimized(Wilson 1963). Intensity data collected in this fashionare unsuitable for structure refinement because theresultant tight packing against a flat surface greatlyexaggerates preferred orientation. Mounts forintensity-profile data were made by loading thepowder from the front of the sample holder, levelingthe surface flush with the mounl with a straight-edge,and finely serrating the surface of the sample witharazor blade several times; each pass with the razorblade was at right angles to the previous one. Thistechnique tends to randomize the orientation ofanisotropic crystals that are aligned during filling'but maintains a generally flat surface.

2t @ m8c (o)

Siep inkeal ('2r)

Iat€gBdoa tine/dep (s)

MqiEm tupirbity (c@b)

No. of pha6 refaed

No, ot uiqui rddi@ (cFr)

No. of mique rcddio (lbr)

No ddnd@pM(cpr)

No. of dmdNps@tu (lbs)

No. otcpsi@talp@tu(ar)

No of*pedmtalpm(fba)

Selcfacar(cp:) xld

SBle Ctur (lbd) xld

N - P

na

ig (q)

Aa (rba)

Dubia-Wtud.didic

A (cF:)

Y ( q )

r (cpr)

I (cP:)

? ("pr)

r?-r3o r7-1m 17-130 1?-?00.:2 0.t2 0.12 0.12

3185 45S8 igln An

v-mo.t2

2

70

s0310

10

1

0t46

0.01t2

89

19

10

1

0.613

0.012

1 t 1

394 3S3 3S6

19 l0 l9

t, t, t"

o.tso 0.688 0.701

911 011 9t2 390 4lb

72 7i E.4 6.9 6.1

0.7 10.4 11.4 8.0 8.8

427 3.15 3$2 l.7E 2.4Jt

0.0690.040

0.104 0.030 0.0:10 o.rts6 0.1t8003!5 0372 0&2 0.664 0.,1@0.008 0.001 0.r!t6 0.tn5 0.qI2

139 1.83

0.0{9 0.041-{.(rrr} 0.043

3.13 3.87L.n Ldl

0.41)0 0rs1-.011r -0.134

ai clhopt@qfb6: fembtuik

iV - P: rc, olobmiim (d@) - ao. otld.ss'I3G @tu.

96 THE CANADIAN MINERALOGIST

Step-scan powder-diffraction data were collectedwith a Philips automated diffractometer systemPWl710, using a PWl050 Bragg-Brentano goniom-eter equipped with incident- and diffracted-beamSoller slits, 1o divergence and anti-scatter slits, a0.2-mm receiving slit and a curved graphitediffracted-beam monochromator. The normal-focusCu X-ray tube was operated at 40 kV and 40 mA,using a take-off angle of 6'. The profiles were takenusirig a step interval of 0.12o N, with a step countingtime of 2 s. As discussed by Hill & Madsen (1986),these are approximately the optimum parameters forreducing serial correlation without adverselyaffecting the accuracy of these results. Informationpeftinent to data collection is given in Table 2.

Rietveld structure refinement

Structures were refined with the Rietveld programLHPMI (Hill & Howard 1986). The peaks weredefined as pseudo-Voigts with percentage Lorentziancharacter varied according to the function

'l : ^tr * 1220, (l)

where .y' and y2 are refinable parameters. The var-iation of the peak full-width at half-maximum(FWHM) was defined by the function of Caglioti eral. (1958)

Ho : (Utan2| + Vtan| + W)0.5, A)

where U, V and W are refinable parameters. Back-grounds were fitted with a simple polynomial func-tion. The profile-step intensity was calculated overthe interval of four FWHM on either side of eachpeak centroid; peak asymmetry was corrected as afunction of 20. Initial structural parameters weretaken from the single-crystal study of diopside(Levien & Prewitt 1981); isotropic displacementfactors were fixed at the single-crystal values. Infor-mation pertinent to the structure refinements is givenin Table 2.

TSt

P (') r (') Y (t!)

PPrc

Di1 9.?456(?) 8.9r9E(E) 5.2i1q5)

D0 9.?4?0(?) 8.035(4) 6.2,5%(4\

D2 e.7634(8) E.e4E8(7) 5260/:(4)

D3 9.m30(7) 8.s523(6) s.*U(S)

D6 0.796(1) 8.97S(1) 62u5(1)

D? 0.E14(1) 8.9960(9) 6r!84(6)

F€frbtuib

7.891 7.rt2 13.785?.?33(7) 7.14s(6) 13.?S0(e)7.1t2(8) 1.11s(2) 13.7s(4)

ldioplide (Itri@ & Ptritt 19t1)'feebutuib,Il9qFa5o (RaFI,ct & Bunhu 19?3)

After estimating as closely as possible the initialstructural and experimental parameters both fromthe single-crystal structure and by inspection of thepattern, refinements were done in the followingsequence. First, the scale factor, zero-point correc-tion and background parameters were refined withall other parameters fixed, followed by the celldimensions. Next, the half-width parameters wereadded in the order W, U, V; these are the mostdifficult paxameters to refine, and occasionally somemanual adjustment was necessary in order to achieveconvergence. The remaining parameters were addedto the refinement in the order: peak shape (.y1), peakasymmetry, atomic positions, site occupancies, peakshape (7) and correction for preferred orientation.In the refinements of two-phase products, only thescale factor, cell dimensions and octahedral siteoccupancies of ferrobustamite were refined; peakparameters were fixed to be those of pyroxene, andstructural parameters were fixed at those fromthe single-crystal refinement of Ca6.5Fe6.rSiO3(Rapoport & Burnham 1973). These constraints werenecessary because the low symmetry of ferrobusta-mite (large number of reflections) results in a largenumber of variable parameters, coupled with adecrease in the information content of the patterndue to increased peak-overlap with the majorpyroxene phase. Final convergence of the refine-ments was assumed when the parameter shifts in thefinal rycle were less than 3090 of their respective stan-dard deviations.

EXPERIMENTAL RESULTS

Single-phose structure refinement

The Rietveld method uses the whole powder-diffraction pattern (point by point) to characlerizethe structure of the material examined: the differ-ence between the calculated and observed patternsis minimized by least-squares procedures. Such step-scan data are prone to serial correlation of the least-squares residuals, and incorrect estimates ofparameter variances (precision) are inevitable if thestep widths and counting times at each step are notoptimized. Thorough studies by Hill & Flack (1987)and Hill & Madsen (1984, 1986) have shown thatserial correlation is significantly influenced by thechoice of step widths and counting times at each step;however, the accuracy of the structural parametersis zor sigi,rificanlly affected (within sensible limits).They also showed that the Durbin-Watson d-statistic(Durbin & Watson 1950, 1951, 1971) is a sensitivemeasure of the presence of such serial correlation,and it should routinely be used as a means ofassessing the reliability of the derived standard devi-ation of a parameter. We obtained d-statisticsbetween 1.39 and 1.77 for the pyroxene refinements,

d (')"

(t)b (a)"(A)

105.8n(1)105.939(4)105.72q4)106.676(4)10!.6rx)(8)1o53in(8)

s627 103.97 7272

06.2(1) 103.72(8) ?30.69

95.16(4) 103.S(2) n0.E2

439.13{0)

4l9rE

4L.8

442,44

448.34

4J7,€

D7

$37q.2Q)

s0.{6(6)

RIETVELD METHOD: DIOPSIDE_HEDENBERGITE JOIN

TABLE 4. ATOMIC POSITIONS FOR CLINOPYBOXENE

97

Dil

M(1) e

g

,B (A')

M(2) ag

B (L,)

T t

g

z

s (A')

O(1) .g

B (L")

o(2) cg

I (A')

o(3) .g

. B (4,)

0 0 0

0.e08r4(5) 0.e071(8) 0.e0e0(6)

0.25 0.25 0.25

0.3?(1)

0 0 0

0.30144(3) 0.29e5(5) 0.2e99(5)

0.25 0.25 0.25

0.635(8)

0.28627(3) 0.2857(5) 0.2875(4)

0.09330(3) 0.094i1(5) 0.0e28(5)

0.22e36(5) 0.2312(8) 0.2323(8)

0.34e(8)

0.11550(7) 0.1144(9) 0.1162(8)

0.08728(7) 0.0e00(11) 0.0896(10)

0.1422(1) 0.1420(16) 0.1403(16)

0.51(1)

0.36136(7) 036re(10) 0.3610(e)

0.2501E8) 0.2516(e) 0.2488(8)

0.318:r(1) 0.3176(u) 0.3184(16)

0.65(r)

0.35083(7) 0.34ee(11) 0.3520(10) 0.3505(1r) 0.3514(16) 0.3525(17)

0.0175e(8) 0.01E5(e) 0.01?5(e) 0.01?8(10) 0.0160(15) 0.0188(16)

0.9e53(1) 0.9974(2r) 0.ee68(20) 0.9980("2) 0.ee70(36) 0.9916(38)

0.56(1)

0 0 0

0.e080(7) 0.e078(8) 0.e056(8)

0.25 0.25 0.25

0 0 0

0.3000(6) 0.2e73(e) 0.2e72(10)

0,25 0.25 0.25

0.2873(5) 0.28e4(8) 0.287e(8)

0.0e32(6) 0.0922(e) 0.0920(10)

0.2323(e) 0.x2r(15) 0.2310(16)

0.1r6e(e) 0.1182(13) 0.11e2(r4)

0.0e12(11) -

0.08e7(17) 0.0e0e(20)

0.14$(17) 0.1608(28) 0.1458(28)

0.3623(9) 0.36,44(12) 0.3607(14)

0.2488(e) 0.2480(14) 0.24e8(16)

0.3227(18) 0.323:|(28) 0.3210(30)

rdiopaide, singlecrystal etruciure (Invien & Prewitt 1981)

consistent with moderate positive serial correlation.At the outset of this study, we tried refinements

with step widths of 0.08 to 0.24'20 and countingtimes of 2 to 5 s at each step. Our results confirmedthose of the previous studies. However, step widthswide enough to avoid serial correlation (- 8090 ofthe minimum FWHM) in pyroxene refinements didnot give structural parameters that were asreasonable for the A/c pyroxene structure as didstep scans using narrower intervals (- 5090 of theminimum FWHM). These step-dependent differ-ences in the refinements presumably result from thecomplexity of the pyroxene stnrcture (as comparedto the simpler compounds used in the above studies),with severe peak-overlap at high diffraction-angleslikely being responsible for ill-conditioned refine-ments of the peak-width parameters, especially U,V and W. Indeed, the derivation of these parameterswas quite difficult for some refinements. The max-imum step-width that still gave realistic atomicparameters (based on comparison with single-crystalp,'roxene structures) vtas0.l2oW, with step countingtimes of 2 s. Step counting times of 2 s gave max-imum step intensities of 2300-4600 counts (Iable 2),

within a rnnge where counting statistics dominateother sources of uncertainty; values of the standarddeviations are correctly estimated in this case.However, as the step width was less than optimum,values of the standard deviations in these refinementsare probably slightly underestimated owing to effectsof minor serial correlation; this will be shown to bethe case for diopside. However, the accuracy ofthestructural parameters is good.

Details of data collection and structure refinementare given in Table 2, cell dimensions are given inTable 3, and final atomic parameters are presentedin Table 4, where they are compared with cor-responding data from the single-crystal structuralstudy of diopside (Levien & Prewitt 1981); refinedsite-occupancies are given in Table 5. The intensity

TABIE 5. R.EFfI{ED SITBOCCUPA

DO D2 D3 D6 D7

1.oo 0.82(1) 0.74(1) 0.542)o.1E(1) 016(1) o.4E(2)

0.Es(2) 1.00 1.00 1.oo0.r(2)

Mg

Fc

Mt

034(2)o,06(2)

1.00

98 THE CANADIAN MINERALOGIST.

data are available from the Depository of Unpub-lished Data, CISTI, National Research Council ofCanada, Ottawa, Ontario KlA 0S2. In general, thevarious indices of agreement show that the fitsbetween the observed and calculated patterns arequite good, with the whole-pattern index (R)varying between 9.7 and 11.40/o. The conventionilBragg indices (R6) evaluate the model fit of theindividual peaks, and vary between 3.2 and 3.5t/0.In these single-phase refinements, there were otherminor phases present in amounts too small to be ade-quately refined as additional phases. Their presencegenerally contributes far more to the backgroundthan to the Bragg peaks of the major phases, andconsequently the RM is slightly higher than onewould expect for R, values of 3-3.590 on purephases.

The key pieces of information derived from eachrefinement are the site occupancies. For the Fe-bearing pyroxenes, the occupancy of the M(2) posi-tion was refined first as Ca + Fe; this resulted in insig-nificantly small negative occupancies of MQ)byFe.As expected, further refinement of M(2) occupancyas Ca + Mg produced insignificantly small positiveMQ) occupancies of Mg, and consequently the MQ)occupancy was considered as fixed at 1.0 Ca. Thiswas not the case for diopside, which refined to sig-nificant occupancy of MQ) by Mg. Several sets ofdata were collected on this diopside sample, and therefinements consistently showed - l09o substitutionof Mg at rhe MQ) site.

For the M(l) site, the occupancy was refined asMg + Fe, except for diopside, in which it was fixedat 1.0 Mg, making the crystal-chemically reasonableassumption (borne out by the observed <M(l)-O >distance) that Ca will not substitute for Mg at M(l).

Unconstrained site-occupancy refinement convergedto values that do not deviate by more than 2.7 stan-dard deviations from values suggested by the nominalcompositions of the run products.

Structure refinement o! two phoses

A significant feature of the Rietveld method is itsintrinsic capability to refine more than one phasesimultaneously. This attribute is very important tothe characterization of synthesis products, becausgthey commonly are multiphase. In this study, threeof the products (D0, D2 and D3, Table l) comprisemore than 9890 pyroxene, and each was refined asa single phase; the others (D5, D7, Table l) containup to - 890 ferrobustamite, and were refined as twophases. Ferrobustamite has a complex structure; itis triclinic, PI, with 16 independent atoms in theasymmetric unit. Because ferrobustamite is presentin only small amounts, it was not possible to totallyrefine its structure; however, this constraint is nota problem, as the ferrobustamite structure is wellknown from previous single-crystal refinements(Rapoport & Burnham 1973), and we are interestedin producing accurate refinements of the hostpyroxene by allowing for the presence of ferrobusta-mite. By fixing the ferrobustamite positions andrefining only the scale factor and site occupancies,we could greatly improve the corresponding pyroxenerefinements; thus for the two-phase samples D5 andD7, the one-phase l?r, and clinopyroxene R6 indiceswere 12.1, 18.070 and 4.32, 6.48a/0, respectively,compared with the corresponding two-phase R,,and R3 indices of 8.0, 8.890 and 1.78, 2.49t/o',respectively. The ferrobustamite site-occupanciesalso give us the ferrobustamite composition, and thescale factors give us a relative measure of the modalamounts of each phase. R, indices (3.13, 3.5790)for the second-phase refinements of ferrobustamiteare surprisingly low, in spite of refinement of onlyone experimental parameter (scale factor) and tenstructure parameters (cell dimensions, M-siteoccupancies) out of a possible fifty-eight variables.

The clinopyroxene site-occupancy results for thetwo-phase refinements parallel those of the single-phase refinements. There is insignificant Fe or Mgat MQ), a\dlhe MQ) occupancies were set at 1.0 Cafor the final cycles of refinement. The unconstrainedM(l) site occupancies agree fairly well with thoseexpected for the nominal compositions. The resultsfor all the clinopyroxenes refined here are shown inFigure 2. There is a slight but systematic deviationbetween the refined and nominal compositions,which will be discussed in detail in a later section.

For the two-phase refinements, it is probable thatthe clinopyroxene does deviate from the nominalcomposition, otherwise there seems no reason for theexistence of the second (fenobustamite) phase. Is this

0 .

o.4

-ql

E o .g

EL

octxo o .

olr

o.2

o . 2 0 . 4 0 . 6 0 . 8Fe (nominal)

Frc. 2. Comparison of refined synthetic clinopyroxeneM(l) site-occupancies with nominal values for l00Vosynthesis of clinopyroxene.

RIETVELD METHOD: DIOPSIDE-HEDENBERGITE JOIN

TABLE O. INTERATOMIC DISTANCES (A) AND ANGLES (') FOR CLINOPYROXENE

99

Di1

r-o(1)r-o(2)T-o(3)ar-o(3)D

(r-o)

(T-O)"u,(T-o)t'

M(1)-o(1)c x2M(1)-o(1)d x2M(1)-o(2)e x2

(M(1)-o)

M(2)-o(1) x2M(2)-o(2)/ x2M(z)-o(3)e xzM(2)-o(3)a x2

(M(2)-o)

1.602(3) 1.606(8)

1.58e(1) 1.5e4(8)

1.66e(3) 1.667(10)

1.68?(3) 1.6e7(10)

r.ffi7(1) 1.641(4)

r.610(7) 1.604(8)

1.57e(7) 1.586(8)

1.674(10) 1.662(11)

1.681(10) r.688(11)

1.636(4) 1.635(5)

1.616(11) 1.5e6(12)1.5e3(r1) 1.6M(12)1.662(r7) 1.685(re)1.673(13) r.676(15)

1.636(7) 1.640(7)

1.5e6(2) 1.600(6) 1.5e4(5)

1.678(2) 1.682(7) 1.678(7)

2.1Ls(2) 2.13e(11) 2.r4r(e)

2.060(7) 2.055(8) 2.056(8)

2.051(2) 2.031(10) 2.071(8)

1.5e5(6) r.605(8) 1.600(e)

1.665(8) 1.668(11) 1.681(12)

2.153(10) 2.146(15) 2.L87(t7)

2.082(8) 2.115(13) 2.oee(14)

2.065(10) 2.060(14) 2.059(15)

2.076(2) 2.075(4) 2.08e(4) 2.100(4) 2.107(6) 2.115(5)

2.363(2)

2.346(8)

2.561(4)

2.721(4)

2.4e8(3)

x25(r0)2.345(8)2.580(e)2.741(10)

2.348(e)

2.345(8)

2.570(8)

2.730(10)

2.333(10)2.325(8)2.577(s)2.742(t2)

2326(16) 2.335(1e)

2.316(13) 2.!a$5)

2.590(13) 2.6n$4)

2.761(20) 2.7X(2r)

2.4e8(3) 2.4e8(3) 2.4s4(4) 2.4e8(6) 2.506(6)

ri-o(3)-Ti r35.?e(5) 136.1(6)o(3)Fo(3)-o(3)i 166.3?(6) 165.?(7)

135.6(6) 136.5(?) 137(r) 135(1)

166.4(?) 1662(E) 167(10) 165(r)

ai r, g, -7 + zi b: z, -y, -f, + z; c: e, | + g, 4 d.: 4 L - g, ! + z; e: ! - a, * * y, I - r;

t t i - " , I - v , l - z i g z I - a , I + s , l - " ; h : l - a , I - v , l - " ; i : x , g , l * 2 1 i : c , - g , ! + 2 .

ldiopside, einglecryatal stnctre (Ievim & Prewitt f981)

deviation significant? The ferrobustamite is more Fe-'rich than the nominal composition of theclinopyroxene. Using the site occupancies and rela-tive scale-factor for ferrobustamite in sample D7, thecomposition of the coexisting pyroxene can be cal-culated from the original bulk-composition of themix. An M(1) occupancy of 0.68 Fe results, com-pared with the refined occupancy of 0.66(2) Fe andthe nominal occupancy of 0.70 Fe. Thus the devia-tion of the refined pyroxene composition from thenominal composition is in line with the occurrenceof (and site occupancies obtained for) the coexisting(more Fe-rich) ferrobustamite.

PyRoxENE SrgrsocHEN{IsrRy: CoMpARISoNWITH SINGLE.CNYSTA,I RESULTS

The Rietveld-derived interatomic distances andangles (Table 6) and site occupancies (Table 5) ofpyroxene were used to evaluate the reliability of themethod by comparison with single-crystal structuredata for diopside (Levien & Prewitt 1981), heden-bergite (Cameron et al, 1973) and CaCoSi2O6(Ghose et ul.1987).

Diopside

It is worthwhile to examine the diopside separatelyas we have both Rietveld and single-crystal resultson what is ostensibly material of the same composi-tion. Figure 3 shows a half-normal probability anarysis of the Rietveld and single-crystal results for theatomic positions. Data sets containing randomnormal distributions of errors should give a linearplot of unit slope with zero intercept. The plot is notquite linear, suggesting the presence of a slight sys-tematic error in one or both sets of data. In addi-tion, the slope of - 1.6 (as compared to the idealvalue of 1 .0) indicates that the pooled standard devi-ations are underestimated by a factor of 1.6. Thestandard deviations for the Rietveld data are approx-imately an order of magnitude larger than those forthe single-crystal data, and thus the Rietveld valuestotally dominate the values of the pooled standarddeviations. As Figure 3 shows that the lattsr areunderestimated, either the Rietveld values are slightlyunderestimated or the single-crystal values areunderestimated by about an order of magnitude.Comparison of different single-crystal refinements

100 THE CANADIAN MINERALOGIST

o .4 0 .8 1 .2

G'

co.Eoo.xut

2 .4

2 . O

1 .6

1 .2

0 .8

o.4

ExpectedFrc. 3. Half-normal probability analysis of Rietveld and single-crystal positional

parameters for diopside.

1 . 6 2 .O

in diopside are markedly wider than the Braggdiffraction peaks in the Fe-bearing pyroxenes.

There is obviously a trade-off here between get-ting accurate atomic parameters and getting correctstandard deviations. If we increase the step intervaltoo much (we examined values up to 0.25o20), therefined parameters begin to degrade, particularlywhere there is significant overlap of peaks in the pat-tern. Conversely, too small an interval gives badlyunderestimated standard deviations. The step-scanvalues we used (based on the work of Hill & Madsen1984) seem a reasonable compromise, but a closerexamination of this point on a series of known struc-tures is desirable (and is currently under way).

This kind of analysis gives us a good statisticalmeasure of agreement between the two techniques,but leaves us with no intuitive feel for the agreementbetween crystal-chemical parameters. Such a com-parison is made for selected interatomic distances in

of diopside (Clark et al. I 969, Levien & prewitt l9g l)shows that the single-crystal standard deviations areapproximately correct. Consequently the underesti-mation of the pooled standard deviations must vir-tually entirely be a result of the underestimation ofthe Rietveld standard deviations. Of course, this iswhat we expect, as this is the result of serial correla-tion, which leads to a systematic under€stimation ofthe assigned standard deviations in Rietveld refine-ment. As discussed previously, the Durbin-Watsonstatistic is a measure of this effect (Hill & Flack1987). The value of 1.39 for the diopside refinemenr(Table 2) is considerably less than the ideal value of2.0t:0.3, indicating thepresence of significant serialcorrelation that will adversely affect the standarddeviations. The Durbin-Watson statistic for diop-side is also less than the values obtained for the othirpyroxene refinements (Table 2). This is a result ofthe fact that the individual Bragg diffraction peaks

q) o

d,

a

ooq

d,o

'o

oa

. /

t

o

oa

RIETVELD METHOD: DIOPSIDE_HEDENBERGITE JOIN 101

Table 6. The mean polyhedral distances agreeextremely well, within one pooled mean standarddeviation in each case. If this agreement proves tobe generally true, it should be possible to use suchmean bondJengths with pre-existing mean bond-length - site-occupancy curves to derive degree oforder and bulk chemical information for those casesin which there is not sufficient difference in scatteringpower to do so directly from the intensity data. ThusAl,/Si order nay possibly be accessible from (Z-O)bond lengths for example.

As expected, deviation of individual bondJengthsfrom the single-crystal values is a little greater thanthat for the mean polyhedraldistances, with max-imum lifferences of 0.038 A for MQ)-O(I) and0.020 A for M(l)-O(l)c and M(l)-OQ)e. However,

1 . 6 8 0

1 . 6 6 0

1 . 6 4 0

1 . 6 2 0

1 . 6 0 0

in terms of the assigned standard deviations, thereis only one significant discrepancy: M(2)-O(l)' witha drfference of 3.8 pooled standard deviations' Eventhe chain angles li-o(3)-Ti afi O(3)b-o(3)-O(3)ishow good agreement. Thus the agreement for diop-side stereochemistry is excellent. Of course, we cannever achieve results as good as for single-crystaldata, because we are collapsing our data into twodimensions, with the consequent loss of informationdue to overlap. Nevertheless, the method seemscapable of accurate results with useful precision'

Int ermediat e co mP osit ions

For the pyroxenes of intermediate composition,we have no analogous crystal-structure results for

o(

I T- I (r-o)nrr_iJ__l_"

L11Il(T-o)n6,

o

D O D 2 D 3 D 5 D 7

o . 7 2 o . 7 4 0 . 7 6 o . 7 8

(rrr.,)) (A)Ftc.4. Variation in <I-O>, (?-O)r. and <7-O>n6, with mean ionic radiusof the constituent cations at the M(l) site (solid symbols). Open symbols are single-crystal data for diopside (Levien & Prewitt l98l), CaCoSi2O6 (Ghose el al. 1987)and hedenbergite (Cameron et al.1973).

r02

1 . 6 8 0

1 . 6 6 0

1 . 6 1 0

1 . 5 9 0

1 . 6 1 0

1 . 5 9 0

1 . 5 7 0

o .72 0 .74 0 .76 0 .7a

(trt , t) (i)

FIo. 5. Variation in individual tetrahedral distances withmean ionic radius of the constituent cations at the M(l)site. Symbols are as in Figure 4.

THE CANADIAN MINERALOGIST

o . 7 4

comparison. However, we can assume that the vari-ations in (at least most) metric properties are linearbetween those of diopside and hedenbergite.

Tetrahedral chain. There is one unique, tetra-hedrally coordinated site in diopside - hedenbergitepyroxenes; this is occupied solely by Si, and themean I-O distances should be similar across thejoin. Figure 4 shows the variation in <I-O>,(Z-O)u, and <Z-O)nu. distances with meanionic radius of the constituent cations at the M(l)site; in addition, single-crystal-derived bond lengthsof diopside, CaCoSi2Ou and hedenbergite areshown. All values are consistent with the single-crystal trends. Variation of individual ?"-O distanceswithM(l) occupancy is less regular (Fig. 5), and thevariation from structure lo structure is larger thanfor the average distances, but all bond lengths arewithin one or two standard deviations of the corrE-lations for the single-crystal data. Thus within theprecision of these data, there is no systematic varia-tion of the I-O bond lengths as a function of M(l)site occupancy, the scatter being a result ofthe inher-ently lower resolution of powder technique of struc-ture refinement.

Octohedral strip. Ca-rich pyroxenes on the diop-side - hedenbergite join have two unique sites in thisstructural unit, the [6]-coordinated M(l) site withoctahedral coordination, and the larger [8]-coordinated MQ) site.In ordered C2lc pyroxenes,

0 . 7 6 0 . 7 8

ril

o (

z ,

I

2. " t20

2 . 1 0 0

2 . 0 8 0

D3 .l/"

D2 I;

V':,/'

o . 7 2

{ t. I ? o_r_T_-oI-

t l r_o(3)b

T -Ii_-+"1 L_-"

r I r_o(3)a

- r Ib-?-l-'-1 ,-*""T T {i----t_-

l 1r T_O(2 )

0 .76

rit

(rrc,r) tAtFtc. 6. Variation in <M(l)-O > with mean ionic radius of the constituent cations

at the M(l) site, Slrnbols are as in Figure 4.

RIETVELD METHOD: DIOPSIDE-HEDENBERGITE JOIN

D 2 D 3 D 6 D 7

103

tine MQ) site is occupied by Ca, with Fe and Mgordered at the M(l) site. Many groups of iso-morphous structures show a nearly linear relation-ship between the mean bondJengths of cation poly-hedra and the mean ionic radius of the constituentcations at those sites. Figure 6 shows the variationin <M(l)-O > as a function of the mean ionic radiusfor the constituerLr M(l) cations. Samples D3 and D5are displaced (one to two standard deviations) toslightly larger <M(l)-O> distances from the trendline for the single-crystal structures and the othersamples from this study. The small displacementsshown in Figure 6 are likely due to discrepancies inone or more of the individual M(l)-O bond lengths,rather than incorrect site-occupancies; on this plot,any reasonable changes in site occupancies have littleeffect on the magnitude of the mean ionic radius.Furthermore, our work on amphiboles (Raudseppet al, 1987a, b) has shown that even though quiteinaccurate bondJengths may sometimes be obtainedfrom some Rietveld refinements (owing to pseu-dosymmetry), the site occupancies are generallyaccurate and not as sensitive to minor discrepanciesin the refinements (provided they are unrelated tothe pseudosymmetry aspects of the structure). Figure7 shows the variation in the individual M(l)-O bondlenglh$ with the mean ionic radius of the constituentM(l) cations. All distances are rvithin two standarddeviations of the single-crystal trends, but theredo seem to be some systematic differences. ForM(l)-O(l)d and M(l)-OQ)e, the powder values seemto scatter randomly about the single-crystal correla-tions. However, for the M(2)-O(1)c bond, althoughthe differences between the powder and single-crystaldata do not exceed two standard deviations, theRietveld-derived distances are systematically longer.

In ordered C2,/c calcic pyroxenes, the M(2) site issolely occupied by Ca. However, in spite of iden-tical M(2) occupancy, <M(2)-O> increases withincreasing size of the M(l) cation (Cameron & Papikel98l). Figure 8 shows the variation in <M(Z)-O>apd individual MQ)-O distances with the mean ionicradius of the constituent M(l) cations. The Rietveld-derived <MQ)-O> distances follow the single-crystal trend fairly closely, and show an increase asexpected. The individual MQ)-O distances vary moreirregularly.ForMQ)-OQV,MQ)-O(3)qandM(2)-'O(3)&, the Rietveld values are consistent with thesingle-crystal trends, but forM(2)-O(l), although thedeviations from single-crystal values are all withinabout two standard deviations, the Rietveld-deriveddistances are systematically shorter than those fromthe single<rystal refinements. Note that there seemsto be a correlation between the maximum systematicdeviations from the single-crystal trends for theM(l)-O and M(2)-O bond lengths: M(l)-O(l)c issystematically longer, and M(2)-O(1) is systemati-cally shorter. Examination of the local

M ( 1 ) - o ( l ) c

T I O

/-'--T -o l o - l - l)L a-./- | |

---./l ? ?o l I

{ , t , , -o , r r "

19?)M ( 1 ) - o ( 1 ) d

o<

2 .1 80

2 .140

2 . 1 0 0

2 .060

2 . 6 6 0

2.44O

2 .7

o .72

Frc. 7. Variation in individual M(l)-O distances withmean ionic radius of thq constituent cations at the M(l)site, Symbols are as in Figure 4.

o .74 ^0 .76 0 .74(rr1.'1) (A)

IL I I T

"-? - : -?-o

M ( 2 ) - o ( 3 ) h

c-..'...--.- o-"-.-r;ffio

-i

- i I 1

M ( 2 ) - O ( 1 )

. - r I o- O - rt

I M(2) -o (2) t

o(

2 . 3 6 0

2.920

o .7 2 o .74 o .76 0.78

(r",,1) (A)

Frc. 8. Variation in <MQ)-O> and individual MQ)-Odistances with mean ionic radius of the constituentcations at the M(l) site. Symbols are as in Figure 4.

t04 THE CANADIAN MINERALOGIST

+

D7

3D5

a

D3D2

450

446(V)

o<

442

438

0 .38 0 .40 o.42 o .44

tA'l0 .46 0 .48

Ftc. 9. Variation in cell volume with mean ionic radius cubed of the constituenr carions at the M(l) site (solid circles).Open circles represent data from Turnock et al. (1973); crosses are data from the single-crystal studies analogousto Figures 4-8.

(trt ' , ,) '

FIa. 10. Variation in cell parameters a, b, cand 0 with mean ionic radius of the constituent cationsat the M(L) site, Symbols are as in Figure 9.

stereochemistry about O(l) suggests that theobserved systematic variations result from underes-timation of the y coordinate of O(1). Comparisonof this parameter with a linear interpolation of y O(l)for diopside and hedenbergite indicates that this isindeed the case. This deviation is reproducible, andseveral different refinements showedy O(l) to be sys-tematically larger than the corresponding interpo-lated values; we have no explanation for this, butnote that it does not lead to major disparities in thestructural parameters of interest.

Cell dimensions. Cell dimensions are sensitive indi.cators of compositional variation in an isostructuralseries. Prewitt & Shannon (1969) have shown thatthe cell volume is a linear function of the cube ofthe octahedral radius of the variable cation; although'Hawthorne

(1978) has shown that this relationshipis intrinsically nonlinear, the variation in cationradius in this pyroxene series is sufficiently small thata linear model is adequate. As two octahedral cationsare involved here, the plot involves the cubedweighted average of the Mg and Fe radii calculatedfrom site occupancies (Fig. 9). Also shown are cellvolumes compiled by Turnock et ql. (1973) for thesame samples and from other pyroxene slmtheses;the correspondence between the former and the latteris excellent. Both these powder-data sets are slightlydisplaced to larger volumes with respect to the single-

1 05 .60

tir

.840

8.800

9 . 7 6 0

C - c c , c i

*

7(r",. , ,) tA)

RIETYELD METHOD: DIOPSIDE-HEDENBERGITE JOIN 105

crystal results, but the trend is clo$e to being linear.Figure l0 shows the variation in the a, b, c and

B cell dimensions with the mean radius of the con-stituent M(1) cations. Again the trends are close tobeing linear, and the correspondence between theRietveld-derived parameters and those both meas-ured and compiled independently by Turnock e/ a/.(1973) is excellent. However, there are some dis-crepancies between our data and the single-crystalvalues. Parameters a, c and B show linear trends closeto those of the single-crystal structures, but D syste-matically diverges with increasing Fe content, beingabout 0.2590 larger than the extrapolated single-crystal value at Fe/(Fe+MB)=0.70.

Cation order. Characterization of order amongoctahedral cations in the diopside - hedenbergiteseries is facilitated by the large difference in scatteringpower between Mg and F&+ . M$) site occupancieswere refined without compositional constraints. M(1)site occupancies (Table 5) show that the pyroxenesfrom run products D2, D5 and D7 are essentiallynominal in composition (within one to two standarddeviations), but D3 is somewhat more Mg-rich. Italso deviates most on a plot of <M(l)-O> versusmean M(l) cation radius (Frg. 6); however, its posi-tion here would not be improved significantly byassuming the nominal composition.

Initially, theMQ) occupancywas fixed at 1.00 Ca,and the M(1) site occupancies were refined. Insubsequent refinements, the occupancy of the M(2)site also was allowed to vary. The M(2) siteoccupancies of all the Fe-bearing pyroxenes refinedto within one standard deviation of 1.00 Ca; theM(2) occupatcy of diopside, however, refined to0.89(2) Ca+0.11(2) Mg. In the light of previouswork on the synthesis and characterizarion of tremo-lite, this is a significant result. Jenkins (1987) showedthat tremolite synthesized from many startingmaterials (oxide mixes, gels, crystalline phases) isenriched by - 1090 in Mg (Ca depleted). If this isthe case for diopside, there should be significantadditional Ca-rich phases presenq the only observedextraneous phase is alrace of cristobalite. We areforced to conclude that this occupancy of MQ) byMg is an artifact of the refinement.

Bun CouposITIoN FRoM RIETvELD REFINEMENT

Diffraction is a spatially resolved electron-countingtechnique, and with sufficient accuracy and preci-sion, is therefore a technique of chemical analysis.As is apparent from the previous discussion, we aredetermining site-occupancies with quite reasonableprecision (even allowing for the problem of itsunderestimation). We can deal with the site occupan-cies in two ways: (i) we can constrain them such thattheir sum must equal the bulk value for the material;(ii) we can refine them unconstrained. If we choose

4 .40

3 .60

3 .20

o . 7 2 0 .80 0 .88

Mg Occupancy M(1)

Frc. ll. Variation in Rs index as a function of M(I)occupancy for clinopyroxene D2; note the sharp asym-metrical minimum.

4 .00

a!t

(E

to follow method (i), we have mode an assumptionconcerning the synthesis process, an assumption thatprevious work (Hawthorne 1983b and referencesiherein) has shown to be not necessarily true. It isobviously preferable to follow method (ii) and deter-mine tlre mineral composition, provided thot this canbe done accurately. Although we examine the detailsof unconstrained site-occupancy refinement else-where (Hawthorte et a/., unpublished ms.), a briefdiscussion along these lines is warranted for the cur-rent results.

How well-defined is the minimum with regard tothe occupancy parameters in an unconstrained least-squares refinement? In principle, this is indicatedbythe assigned standard deviation(s); however, thiscriterion ignores the presence of neighboring falseminima. We can examine this point by mapping outthe value of the residual as a function of theparameter(s) of interest. This we have done for theM(1) occupancy of sample D2. The structure ofclinopyroxene D2 was rdfined with the occupancyof M(l) fixed at a series of values spanning thenominal composition +0.12 Mg p.f.u. The resultsare shown in Figure I l. The minimum seems verywelldefined; it is sliehtly displaced from the nominalcomposition of 0.80 Mg, and it is apparent that theR, index is quite sensitive to the occupancyparameter. Similar results for the MQ) occupancy

a

III

a

a

o

\ at'

106 THE CANADIAN MINERALOGIST

1 . 0 4oc(tlCL3ooo

IRlD

TE

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

B (42)Frc. 12. (a) Variation in M(l) (solid symbols) znd Me)

(hollow symbols) occupancy [M(l) =16149 1 11-";a".MQ)--yca+(l-l)Fel as a function of isorrooic dis-placement factor B for clinopyroxene D2; (b) variationin Rp index as a function of M(l) (solid symbols) andM(2) (hollow symbols) isotropic displacement facror forclinopyroxene D2..

show that we can expect quite precrse results fromsuch occupancy refinements.

What about accuacy? We have discussed thisaspect somewhat with regard to diopside, but needto focus a little more specifically on site occupan-cies. In least-squares refinement, it is well known thatthere is significant interaction nmong occupancyparameters, displacement parameters, and the scalefactor (Hawthorne 1983a). If there is sin0,4r-dependent systematic error in the intensity data, thesehighly correlated parameters can interact with thesystematic error to produce inaccurate results. Withimprovement in techniques of absorption correction,these problems have been largely overcome for

single-crystal intensity data. However, the effects ofabsorption are a possible source of inaccuracy inRietveld structure refinement, particularly as thistechnique has inherently less resolution. Conse-quently, we have taken a more conservative approachto the refinement procedure. We set tlte isotropic dis-placement factors for all atoms equal to the valuesderived from the single-crystal refinement of thestructure. Obviously if these values are notappropriate, the refined site-occupancies (and hencethe bulk composition) will not be accurate.

We examined the possible effect of using inap-propriate displacernent factors by doing a series ofrefinements with the M(l) and M(2) displacement-factors fixed at arbitary values, but refining all otherparameters except the displacement factors for siliconand oxygen, which were still fixed at the cor-responding single-crystal values. The results forsample D2 are shown in Figure l2a. As expected,there is an inverse correlation between isotropicdisplacement factor and site occupancy [expressedas M(l):.rcMg + (l-x)Fe; MQ): yCa+ (l-y)Fel. Thusit is obviously important to choose appropriatevalues for the isotropic displacement factors. Com-paring the slopes of these curves to the values of thestandard deviations on the site occupancies gives anindication of the sensitivity of this relationship. Ineach case, t I standard devialion in the siteoccupancy corresponds to +0.25 A2 in the isotropicdisplacement factors; this is quite a significant differ-ence in the values of the isotropic displacementfactors, suggesting that significant differences in siteoccupancies will only be caused by use of (whatshould seem to be) unrealistic displacement factors.

We also examined the behavior of the R-indexduring this procedure; it is shown in Figure l2b asa function of the isotropic displacement parameters.For small values of .B (including the values used fromthe single-crystal refinement of the diopside struc-ture), the Ru index is fairly insensitive to theisotropic displacement factor; as B becomes larger,Rs climbs relatively rapidly. Comparison of Figuresl?-a and l2b indicates that the situation is fairly wellconstraine{. Any significant changes (of the orderof +9.25 A2) to the chosen values of the isotropicdisplacement factors will lead to less satisfactoryresults. Increase in the B values by this amount willsignificantly raise the R6 index. Decrease by thisamount will not significantly raise or lower R6, butwill give us a less realistic model (i.e., displacementparameters that we know aretoo small). Thus in thepresent circumstances at least, the method chosenfor the site-occupancy refinements seemssatisfactory.

Of course, the optimum method would be to refineboth the occupancies and the isotropic displacementfactors. When this was done for the D2 sample, theresults were as follows: Ra:3.0390, M(1)=9.3511;

1 . 0 8

0 . 9 6

0 . 8 4

o . 8 2

4 . 2

3 .8

s.4

3 .0

(a)

-{-t-

l....-l..-- i.-i.-l____.-...-p

(b)

RIETVELD METHOD: DIOPSIDE-HEDENBERGITE JOIN t0'7

Me+0 .15 ( l ) Fe , B : -0 .Q4 A2 ; M(2 )= l . 0 l ( 3 )Ca+-0.01(3) Fe, B:0.75 F. The values for M(2)seem reasonable, with the occupancy essentially 1.0Ca and the isotropic displacement factor not signifi-cantly different from the single-crystal value.However, B for M(l) refined io a physicallyunreasonable value, shifting the M(l) site occupancyby approximately three standard deviations. Themost important factor in the assessment of the resultsof any least-squares refinement is the physicalvalidity of the results. We can discount the resultsof the simultaneous refinement of occupancy anddisplacement factor on this basis, and prefer ouroriginal refinement. This does indicate the presenceof some systematic error in the intensity data, anerror that is slightly biasing the refined struciure.However, the results of this work do suggest that thiserror is in fact quite small, and that the Rietveldmethod can be used to derive site occupancies ondbulk compositions of minerals when the samplecomposition and constitution are suitable for thistechnique.

CoNcr-usroNs

The Rietveld method has been used to refine thecrystal structures of synthetic fine-grainedclinopyroxenes along the diopside - hedenbergitejoin. The following points are of particular interestwith regard to the characterization of fine-grainedsynthetic minerals:(D For compositions in which there is significant

difference in scattering power among differentcomponents, site occupancies can be determinedwith fairly high accuracy and precision(although problems may arise when the struc-ture has pseudosymmetry); thus structlual statemay be determined.

(ii) Unconstrained site-occupancy refinement cancommonly lead to the determination of bulkmineral compositions, subject to the provisosgiven in (i); this may be of particular interestwhen synthesis products rue very fine grained.

(iii) More than one phase can be refined simultane-ously, and modal amounts of the phases can berecovered from the refinement, together with thedegrees of cation order and bulk compositions;how well this can be done depends on the com-plexity and number of the phases involved,together with the degree of overlap in thepowder-diffraction pattern.

(iv) As well as compositional information and veryprecise cell parameters, we also recover crystal-structure information. Of course, these data arenot as precise as corresponding single-crystalinformation, but for the first time we have beenable to retrieve such data from fine-grainedpowders; lower precision is not an issue whenthis is the only method of deriving such data.

The Rietveld method is a very powerful way toexamine fine-grained single- and multi-phaseproducts of mineral syntheses; it is capable ofproviding information on structural state (cationorder), bulk composition, modal composition andcrystal structure. Data collection is rapid, there isgood public-domain software for the refinementprocess, and the necessary equipment (a powderdiffractometer) is widely available. It should becomea standard technique of product characterization inexperimental mineralogy and petrology in the future.

ACKNOWLEDGEMENTS

We thank J.J. Papike, an anonymous reviewerand R.F. Martin for constructive criticism that sub-stantially improved the manuscript. Financialassistance was provided by the Natural Sciences andEngineering Research Council of Canada in the formof operating grants to F.C. Hawthorne and A.C.Turnock, and an infrastructure grant to F.C.Hawthorne.

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Received September 14, 1989, revised manuscriptaccepted December 4, 1989.