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American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method D.q.vro L. Brsn Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A. JBppnry E. Posr Department of Mineral Sciences, Smithsonian Institution, Washington, DC 20560, U.S.A. Ansrnlcr Quantitative phaseanalysisof a number of multicomponent standard and natural min- eral mixtures has been done using an adaptation of the Rietveld method. Binary mixtures (most 50:50 by weight) of corundum with quartz, hematite, ilmenite, magnetite, biotite, analcime,mordenite, or clinoptilolite were analyzed,using digital powder X-ray diffraction (XRD) data. In addition, a suite of standardmixtures of hematite and corundum, a natural feldspar mixture, the G-l standard granite, two natural bauxite samples, and a mixture of biogenic carbonate minerals were also analyzed. Quantitative information was extracted from refined individual scalefactors and unit-cell voiumes (derived from refined unit-cell parameters), obtained with a Rietveld refinement program modified to analyze up to ten phases. The quantitative results for standard mixtures were within 2.50/o (absolute)of the true values,with the exception of the hematite, ilmenite, and magnetite mixtures. Results for the latter mixtures using CuKa data were severely afected by microabsorption, but analysis of the hematite mixtures using FeKa radiation gave results with absolute errors <2010. Results for the G-l granite and the natural feldspar agreed well with optically determined modes, and the method facilitated separationof the significant overlaps in the pattern of the carbonate minerals. Quantitative mineralogical analysis by the Rietveld method has several significant ad- vantages over conventional methods of quantitative analysis. The method uses all intensity data in a pattern rather than a few of the most intense reflections,partially compensating for preferred orientation and extinction. In addition, standard dala arecalculatedfor each phase during analysis, overcoming the troublesome requirement of obtaining standards representative of the materials in an unknown. It is also possible to gain a wealth of information from each sample in addition to amounts of phases.Because some of the most troublesome systematicerrors, including sample displacementand zero-point shift, can be refined, the method yields unit-cell parameters of an accuracy comparable with that obtained when using an internal d-value standard. The method should find a wide application in geology,including in modal analysisand compositional determinations of individual mineral components using unit-cell parameter systematics. INrnonucrroN addressedin order to obtain accurate results; these in- clude particle statistics, primary and secondary extinc- X-ray powder diffraction has been a popular method tion, microabsorption, preferred orientation, separation of quantitative mineralogical analysisfor over 40 yr. The of overlapping and broad reflections, variation in stan- power and convenience of the method lie in its simplicity dard data with composition, availability of pure stan- and speedand in the fact that the method can be applied dards, and detectionofamorphous and tracephases (Klug to a wide range of materials.It is especially useful for and Alexander, 1974;Bish and Chipera, 1988).In addi- fine-grained sedimentary or volcanic rocks that are not tion, traditional methods require that standard data be amenable to quantitative study by optical methods. The acquired for every phase present in the mixtures to be importance of the powder diffraction method to quanti- analyzed. tative analysisis illustrated by the large number of papers The Rietveld method of quantitative analysis mini- onmanydifferentclassesofmaterials(e.g.,Brindley, 1980; mizes or eliminates many of these problems (Bish and Snyder and Bish, 1989). However, in spite of the large Howard, 1986,1988;HillandHoward, 1987; Snyderand amount of routine work in the area of quantitative X-ray Bish, 1989). Rietveld refinement was originally devel- powder diffraction analysis, the method still largely re- oped as a method of refining crystal structures using pow- mains semiquantitative except in special cases. Numer- der neutron diffraction data (Rietveld, 1969),and the ap- ous instrumental and sample-relatedproblems must be plication of the method to mineralogy was summarized 0003-004x/93l09 10-0932$02.00 932

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Page 1: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

American Mineralogist, Volume 78, pages 932-940, 1993

Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

D.q.vro L. BrsnEarth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A.

JBppnry E. PosrDepartment of Mineral Sciences, Smithsonian Institution, Washington, DC 20560, U.S.A.

Ansrnlcr

Quantitative phase analysis of a number of multicomponent standard and natural min-eral mixtures has been done using an adaptation of the Rietveld method. Binary mixtures(most 50:50 by weight) of corundum with quartz, hematite, ilmenite, magnetite, biotite,analcime, mordenite, or clinoptilolite were analyzed,using digital powder X-ray diffraction(XRD) data. In addition, a suite of standard mixtures of hematite and corundum, a naturalfeldspar mixture, the G-l standard granite, two natural bauxite samples, and a mixture ofbiogenic carbonate minerals were also analyzed. Quantitative information was extractedfrom refined individual scale factors and unit-cell voiumes (derived from refined unit-cellparameters), obtained with a Rietveld refinement program modified to analyze up to tenphases. The quantitative results for standard mixtures were within 2.50/o (absolute) of thetrue values, with the exception of the hematite, ilmenite, and magnetite mixtures. Resultsfor the latter mixtures using CuKa data were severely afected by microabsorption, butanalysis of the hematite mixtures using FeKa radiation gave results with absolute errors<2010. Results for the G-l granite and the natural feldspar agreed well with opticallydetermined modes, and the method facilitated separation of the significant overlaps in thepattern of the carbonate minerals.

Quantitative mineralogical analysis by the Rietveld method has several significant ad-vantages over conventional methods of quantitative analysis. The method uses all intensitydata in a pattern rather than a few of the most intense reflections, partially compensatingfor preferred orientation and extinction. In addition, standard dala are calculated for eachphase during analysis, overcoming the troublesome requirement of obtaining standardsrepresentative of the materials in an unknown. It is also possible to gain a wealth ofinformation from each sample in addition to amounts of phases. Because some of themost troublesome systematic errors, including sample displacement and zero-point shift,can be refined, the method yields unit-cell parameters of an accuracy comparable withthat obtained when using an internal d-value standard. The method should find a wideapplication in geology, including in modal analysis and compositional determinations ofindividual mineral components using unit-cell parameter systematics.

INrnonucrroN addressed in order to obtain accurate results; these in-clude particle statistics, primary and secondary extinc-

X-ray powder diffraction has been a popular method tion, microabsorption, preferred orientation, separationof quantitative mineralogical analysis for over 40 yr. The of overlapping and broad reflections, variation in stan-power and convenience of the method lie in its simplicity dard data with composition, availability of pure stan-and speed and in the fact that the method can be applied dards, and detection ofamorphous and trace phases (Klugto a wide range of materials. It is especially useful for and Alexander, 1974; Bish and Chipera, 1988). In addi-fine-grained sedimentary or volcanic rocks that are not tion, traditional methods require that standard data beamenable to quantitative study by optical methods. The acquired for every phase present in the mixtures to beimportance of the powder diffraction method to quanti- analyzed.tative analysis is illustrated by the large number of papers The Rietveld method of quantitative analysis mini-onmanydifferentclassesofmaterials(e.g.,Brindley, 1980; mizes or eliminates many of these problems (Bish andSnyder and Bish, 1989). However, in spite of the large Howard, 1986, 1988;HillandHoward, 1987; Snyderandamount of routine work in the area of quantitative X-ray Bish, 1989). Rietveld refinement was originally devel-powder diffraction analysis, the method still largely re- oped as a method of refining crystal structures using pow-mains semiquantitative except in special cases. Numer- der neutron diffraction data (Rietveld, 1969), and the ap-ous instrumental and sample-related problems must be plication of the method to mineralogy was summarized

0003-004x/93l09 10-0932$02.00 932

Page 2: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

by Post and Bish (1989). Refinement is done by mini-mizing the sum of the weighted, squared differences be-tween observed and calculated intensities at every 20 stepin a digital powder diffraction pattern. Thus, refinementis done on a point-by-point basis rather than on a reflec-tion basis. The calculated intensity at a given step (y") isdetermined by summing the contributions from back-ground and all neighboring Bragg reflections (k) for allphases (p) as

, , " : 4 So ) puLo lFk l2G(L?&)P t * y ,o (c ) ( l )

where S is the scale factor, Lo is the Lorentz and polar-ization factors for the kth Bragg reflection, Fu is the struc-ture factor, p* is the multiplicity factor, P* is the preferredorientation function, d* is the Bragg angle for the kthreflection, G(L?,) is the reflection profile function, andy^(c) is the refined background (Wiles and Young, I 98 I ).The parameter most often used to assess the fit is R*o,the weighted profile residual, and

R*o:12w,$t,"- y,.)2/2wyl"los (2)

where y- and y,, are the observed and calculated inten-sities, respectively, at point i in a diffraction pattern andw, is the weight assigned to each intensity. For a perfectrefinement, the final R*o would equal R"*o, and

R""": KN - P)/2wyl"lo ' (3)

where N is the number of data points in the pattern, andP is the number of parameters refined. Typical values ofR*o range from a few percent for very good neutron re-finements to 20-30o/o for X-ray refinements, dependingin part on the count times used, the degree of preferredorientation, and the number of variable parameters.

Quantitative analysis by the Rietveld method requiresa knowledge of the approximate crystal structure of allphases ofinterest (not necessarily all phases present) in amixture. The input data to a refinement include spacegroup symmetry, atomic positions, site occupancies, andunit-cell parameters. In a typical Rietveld quantitativeanalysis, individual scale factors (related to the amountsofeach phase) and peak-shape parameters for each phaseare varied along with background and unit-cell parame-ters. Atomic positions and site occupancies can also oftenbe successfully varied for major phases if such detailedinformation is of interest, although in some cases (e.g.,zeolites), site occupancies of some sites must be varied toobtain accurate scale factors. Information on the weightfractions (l4f) of phases present in a mixture is calculatedfrom the scale factors for each phase obtained from therefinement

/W,: S,p,Vl

f 2 S,o,Vi @)

where 5',, p,, and V, are thescale factor, density, and unit-cell volume, respectively, of phase i, and the summationis over all phases present.

933

Few applications of Rietveld quantitative analyses havebeen published, particularly for geological samples, al-though the strength and versatility of the method suggestthat it should be a powerful tool for analyzing geologicsamples. Among others, Hill and Howard (1987), Bishand Chipera (1988), Bish and Howard (1988), Bish andPost (1988), and Reid et al. (1992) presented examplesof quantitative analysis of a variety of mixtures, and wehave successfully applied the method to mineral mixturescontaining up to ten components.

The Rietveld method of analysis provides numerousadvantages over conventional quantitative analysismethods. As the method uses a whole pattern-fitting al-gorithm, all lines for each phase are explicitly considered,and even severely overlapped lines are usually not a prob-lem. Thus it is not necessary to decompose patterns intoseparate Bragg peaks, as is often the case for traditionalmethods. The use of all reflections in a pattern rather thanjust the strongest ones minimizes both the uncertainty inthe derived weight fractions and the effects of preferredorientation, primary extinction, and nonlinear detectionsystems. Also, failure to consider a phase in the analysiswill yield obvious differences between the observed andcalculated diffraction pattems and reveal unsuspectedminor phases. Along with a quantitative analysis, it isalso possible to refine structural parameters, includingatom positions, site occupancies, and unit-cell parame-ters for each phase in the mixture. One attractive featureof this method is that it is possible to calculate tailor-made standards to match the phases in a given mixture,e.g., the composition, unit-cell parameters, or extent oforder-disorder in solid-solution phases can be refined tomatch the unknown as closely as possible. The methodis not standardless, however, as it uses calculated stan-dard data.

Preferred orientation of crystallites is one of the mostserious problems inherent in conventional quantitativeanalyses, and numerous sample preparation methods havebeen proposed to minimize or to eliminate the problem.However, because the Rietveld method uses all classes ofreflections, the effects of preferred orientation on quan-titative analyses (scale factors) are greatly minimized.Orientation effects on unit-cell parameters are also min-imal, but they can significantly affect site occupancies andatomic positions. The Rietveld method presents the op-portunity to correct for preferred orientation using theMarch function (Dollase, 1986) or symmetrized harmon-ics (Jarvinen, I 992). Although older preferred orientationcorrection methods do not work well with quantitativeanalysis, the March function works better as it is nor-malized to the unit integral, and a change in its value willnot affect the scale factors (e.g., Hill and Howard, 1987).

In this paper, we describe the application of the Riet-veld method of quantitative analysis to a variety of geo-logically significant systems. These applications show thepower and versatility of this method and show the typesof information, in addition to amounts of phases, thatcan be obtained in a Rietveld quantitative analysis.

BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

Page 3: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

934 BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

ExpBntuBNTAL METHoDS

Samples

Numerous standard mixtures were prepared to assessthe precision and accuracy of the Rietveld quantitativeanalysis method. Binary mixtures (50:50 by weight) wereprepared by mixing 1.0-pm metallurgical-grade corun-dum with the given mineral (Table l). In addition to thelaboratory-prepared mixtures of minerals, several naturalsamples were examined, including a feldspar separate (p:2.51-2.60 {cmr) from a rhyolite lava near the TimberMountain caldera complex adjacent to the Nevada TestSite, Nevada, and a mixture of coralline algae and echi-noid spines containing aragonite and two composition-ally distinct calcites. As a test of the method on complexgeologic mixtures, the U.S. Geological Survey standardgranite G-1 (Chayes, l95l) was examined. Finally, sev-eral natural bauxite samples were analyzed to assess theability of the method to accommodate samples contain-ing poorly crystalline minerals and minerals with a widerange of peak widths.

Data measurement

X-ray diffraction data for this study were obtained onSiemens D-500 and Scintag PAD V automated diffrac-tometers, both with incident- and diffracted-beam Sollerslits. The Siemens instrument used either a diffracted-beam graphite monochromator with a scintillation detec-tor or a Kevex Psi solid-state Si (Li) detector, and theScintag diffractometer used a solid-state intrinsic-Ge de-tector. Conventional data measurement parameters wereused in these analyses, usually scanning from 2 to 70 20and counting for 2.0 s every 0.02 or 0.03" 20. Most anal-yses used CuKe radiation, but FeKa radiation was usedwith some of the hematite-corundum mixtures, and aHawaiian sample of bauxite was measured with CoKaradiation. Prior to analysis, all samples were milled underacetone to a mean particle size of <3 pm using a Brink-mann Micro Rapid mill. Particle size distributions wereverified using a Horiba centrifugal particle size analyzercalibrated using Duke Scientif ic glass microanalysisspheres. Powders were mounted in a variety of ways, in-cluding front- and back-packed cavities, smear mounts,and glass-fiber mounts, to allow assessment of the effectsof sample preparation on quantitative results.

Rietveld refinement

All Rietveld quantitative analyses reported here useda significantly modified version of the program DBW3.2(Wiles and Young, 198 l; Bish and Howard, 1988). TheDBW program has been adapted to perform both inter-nal- and external-standard analyses and can thus be usedto determine the amount ofnoncrystalline materials pres-ent in a mixture. These two modifications are analogousto the matrix-flushing and adiabatic methods, respective-ly, of Chung (1974a, 1974b\. The internal-standard meth-od involves the addition to a sample of a known amountof a crystalline substance, usually corundum in our anal-

TABLE 1. Rietveld quantitative analysis results for various binarymineral mixtures*

Known RefinedSource wlo/" wt"h

OuartzCorundumAnalcimeCorundumClinoptiloliteCorundumClinoptiloliteCorundumMordeniteCorundumBiotiteCorundumllmeniteCorundumMagnetiteCorundum

Hot Springs, Arkansas

Wikieup, Arizona

Nevada Test Site (NTS)

Castle Creek, ldaho

Union Pass, Arizona

Bancroft, Ontario

Koidu, Sierra Leone

Espanola, Ontario

50 49.9(3)"50 50.1(5)50 48.2(5)50 51.8(7)50 50.1(7)s0 49.9(7)50 52.7(9',,50 47.q5)50 48.8{1.7)s0 51.2(4)s0 47.5(3.0)50 s2.s(7)s0 25.7(31s0 74.3(1.2)50 38.1(2)50 61.9(4)

'CuKd radiation.". Values in parentheses represent estimated standard deviations in the

last quoted place.

yses. The external-standard method imposes the con-straint that the sum of all phases determined in the mix-ture total to l00o/0. Thus the presence of amorphous phasesor crystalline materials not explicitly included in the anal-yses will give incorrect absolute but correct relative anal-yses. These methods of analysis differ significantly fromthat used by O'Connor and Raven (1988), in which theydetermined the constant parameters in Equation I frommeasurements of a single sample. In light of their resultson a 50:50 quartz to corundum mixture, in which theyconcluded that their quartz contained 180/o amorphouscomponent, it appears that some pitfalls may exist withthis approach. A further difference in our implementationof the Rietveld quantitative analysis method involves thetreatment of the unit-cell volume. Because the volumecan be precisely determined during Rietveld refinementfrom the unit-cell parameters, we have included the cellvolume with the variable, phase-specific parameters inthe refinement. Other authors have retained the cell vol-ume with other constant and phase-specific parametersin the scale factor (Hill and Howard, 1987; O'Connor andRaven ,1988 ) .

Analyses were typically conducted on either an IBM4381 or a DEC MicroVAX III computer, although theprogram has been modified to run on a personal com-puter. In most cases, refinement involved the sequentialvariation of scale factor for each phase, background co-efficients, unit-cell parameters for each phase, specimen-displacement correction, peak-width parameters for eachphase, and Pearson VII or pseudo-Voigt profile functioncoemcients for each phase in the mixture. It is importantto note that the use of a single, simple profile functionfor each phase in a mixture will not allow accurate mod-eling ofobserved profiles resulting from highly anisotrop-ic crystallite size and strain effects. Apart from applica-tions to very fine-grained or disordered materials such assoils, this shortcoming should not present a significant

Page 4: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

l8 2l U 27 30 33 35 39 42 rts 48Two-lhetq (degrees)

Fig. l. Observed (pluses) and calculated (line) diffraction pat-terns for a 50:50 quartz-corundum mixture (R*, : 18.4010, R".,: 6.70/0). The relatively straight curve near the bottom of ob-served and calculated data is the refined background. The lowercurve shows the difference between observed and calculated pat-terns. Vertical marks at the bottom indicate the positions ofallowed Ko, and Ka, reflections.

problem. During the course of refinement, atomlc coor-dinates, occupancies, and temperature factors for eachphase were usually fixed. However, in some cases, atomiccoordinates and site occupancies were refined to deter-mine additional structural information on the individualphases in mixtures. Starting atomic structural parameterswere taken from the literature. The original Rietveld pre-ferred-orientation correction in our version of DBW wasnot refined, as preliminary tests with the correctionshowed that quantitative analytical results were invari-ably better when preferred orientation was ignored.

935

Rnsur-rs AND DlscussroN

Quantitative analysis results for the binary mixturesare summarized in Table l, and a discussion of each mix-ture follows. Estimated standard deviations for each val-ue in Table I were derived from the estimated standarddeviations on individual scale factors for the respectivephases, and other error contributions were not included.The 50:50 quartz-corundum mixture gave a mineralogiccomposition statistically identical to the known compo-sition, as well as unit-cell parameters of high precisionand accuracy (Table 2). Because the Rietveld method at-tempts to fit every data point in a pattern rather thandealing with individual peak positions, and because com-plex and overlapping patterns are explicitly fitted withthe method, unit-cell parameters obtained by Rietveldrefinement are generally superior to those obtained bymore conventional powder or single-crystal methods.Thus, the Rietveld method usually yields high-precisionunit-cell parameters, and, if sample displacement andtransparency corrections are refined, the values are alsovery accurate (Post and Bish, 1989). The observed andcalculated patterns for the quartz-corundum mixture areshown in Figure l Most of the detail in the differencecurve in Figure I is due to the inability of the profilefunction in the Rietveld program to fit the observed pro-files accurately. Quantitative analysis results for this andmost mixtures were relatively insensitive to variations inprofile parameters and overall temperature factors, andthe quantitative results usually changed little after refine-ment of unit-cell parameters, when the starting values forunit-cell parameters were nearly correct. Refinement ofsite occupancies is required only with minerals of com-plex composition or extensive solid solution. For exam-ple, analysis of a mixture containing an intermediate ol-ivine sample of unknown composition would benefit fromrefinement of the olivine divalent cation site occupanciesto determine both the approximate composition of the

BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

B 6r.sx

> 41.0

eE 20.s

TABLE 2. Unit-cell parameters obtained from the refinement of binary mixtures

i tr,* *^ ,tr ,I

I I

I

t t t l i l i l i l l l l i l l

a (A) b (A) c (A) Bf )

Quartzwiil er al. (1988)CorundumThompson et al. (1987)AnalcimeClinoptilolite (ldaho)Clinoptilolite (NTS)MordeniteBiotiteHematiteNatural. FeKdKastalsky and Westcott (1968)llmeniteWechsler and Prewitt (1984)MagnetiteHill et al. (1979)

4.91 10(1)4.91 239(4)4.7565(1 )4.7586(1 )

1 3.6732(4)17.682(2)1 7.6375(1 )1 8.1 08(s)5.34s(2)

5.03481 (2)5.0340(7)5.06463(5)5.0884(1)8.39293(8)8.394

17.97412117.9622(sl20.451(5)9.248(3)

5.4021(4)5.40385(7)

12.984(1)12.9897(1)13.696(1)7.4158(8)7.4000(1 )7.514(2)

1 0.208(1 )

1 3.7493(3)13.752(3)13.9228(7114.08s5(4)

1 1 6.1 9s(8)1 1 6.220(1 )

100.25(3)

A/ote.' literature values of unit-cell parameters are @mpared with Rietveld values to demonstrate the apparent accuracy of the Rieweld results. lt isnot expected, however, that the two results should be identical, as the samples (and methods) differ for most materials. In particular, routine single-crystal X-ray diffraction measurements do not necessarily yield highly accurate unit-cell parameters.

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936

1 9 z 2 9 3 4 3 9 4 p 4 9 s 4 5 9 ( / - 5 9Two+helo (degrees)

Fig.2. Observed and calculated patterns for the 50:50 mix-ture of clinoptilolite (NTS) and corundum (conventions as inFig. 1; R"" : 14.7o/o, R.,, : 7.5olo).

olivine and the extent of any intracrystalline cation or-dering, and as a result the calculated olivine standardpattern would be a near perfect match to the observedpattern.

Analysis of the 50:50 analcime-corundum mixture alsogave accurate quantitative results (Table l), and the re-finement resulted in significantly noncubic unit-cell pa-rameters for analcime (Table 2). Although analcime hastraditionally been considered to be cubic,Mazzi and Gal-li (1978) showed that most analcime deviates slightly fromcubic symmetry because of Al-Si ordering. It is notewor-thy that analcime could be stably refined assuming te-tragonal symmetry, and the final a and c unit-cell param-eters differ by more than 20o. Comparison of the refinedunit-cell parameters for this analcime with data given byMazzi and Galli ( I 978) suggests that this sample is silicarich, typical of sedimentary analcime.

The 50:50 clinoptilolite-corundum mixture illustratesone of the most significant advantages of the Rietveldmethod in analyzing complex mixtures of minerals, theability to explicitly accommodate severely overlappingreflections. The observed and calculated diffraction pat-terns for this mixture (Fig. 2) show the very large numberof reflections (>300) used in the analysis. Figure 3 is anexpansion of the region between 21.0 and 25.0'2d, illus-trating the significant peak overlap in this region. Themost intense doublet in this region is composed of sixindependent reflections, a situation that would make in-dexing this pattern difficult even when applying tech-niques such as profile refinement, in which overlappingreflections are decomposed without relation to a struc-tural model (Howard and Preston, 1989). Such a severeoverlap would make unit-cell refinement by conventionalmethods very difficult, but the Rietveld method routinelygave very precise parameters (Table 2) in addition to ex-cellent quantitative results. It is difficult to assess unit-

BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

150

I rosx

i 7 0coE 3 s

22.6 23.O 23.4 23.O U2 U.6 25.OIwo-lhelo (degroos)

Fig. 3. Expansion ofthe 21.0-25.0" region ofFig. 2.

cell parameter accuracy for materials such as clinoptilo-lite (and analcime, mordenite, and biotite) because ofcompositional variabilities from sample to sample, al-though results for well-characterized materials suggest thatthe refined parameters for these phases are also accurate.

Mordenite is one of the more difficult minerals to dealwith in quantitative analysis of zeolitic tuffs, primarilybecause of its variable exchangeable-cation composition,fibrous morphology with consequent high degree of pre-ferred orientation, and peak overlap with coexisting phasessuch as clinoptilolite. In contrast to conventional meth-ods, which yield results of variable quality (e.g., Bish andChipera, 1989), the Rietveld method consistently gavegood results on the mordenite standard mixtures studied(Table l). Figure 4 shows observed and calculated pat-terns for the corundum-Union Pass mordenite mixture,and the refined mordenite unit-cell parameters obtainedin this refinement are given in Table 2. The data in Figure4 illustrate the ability of the method to accentuate phasesthat are unaccounted for in the model; note the discrep-ancy between observed and calculated patterns in the vi-cinity of 22.0 20 due to the presence of a small amountof opal-CT in the mordenite standard.

The 50:50 biotite-corundum mixture serves as an ex-ample to assess the effects of preferred orientation onRietveld quantitative analyses. The observed and calcu-lated data (Fig. 5) show a poor fit because of preferredorientation; the preferred-orientation correction was notvaried in these refinements. In spite of the poor fit andthe large amount of preferred orientation, quantitativeresults were reasonably good (Table l). Cell parametersobtained for biotite are given in Table 2. This analysishighlights one of the strengths of the Rietveld method ascompared with traditional methods that rely upon ratiosof only a few reflections. Apparently, the effects of pre-ferred orientation are somewhat moderated by the use ofall reflections in a given pattern. Some observed reflec-tions will be too intense and others will be too weak, butthe overall least-squares refinement yields good quanti-

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BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSN

I50

937

o / Jx

F 5 0coE 2 s

6 | mEi 8 0coE r c

t25

22 20 34 40 46 52 58 & 70 76 02Two+heio (degreesl

Fig. 6. Observed and calculated patterns for a 50:50 hema-tite-corundum mixture (conventions as in Fig. l; R*,: 21.4o/0,R.-o : 18.40lo).

for the highly absorbing material, hematite @/p : 219cm I compared with 31 cm-' for corundum). This prob-lem can be minimized by grinding samples to very fineparticle sizes or choosing a different radiation. However,extreme grinding should be avoided, as it can eventuallycause degradation of the diffraction pattern. Refinementswith data obtained on <400 mesh material (<38 pm in-stead of <3 pm) yielded even worse results (20:80 he-matite to corundum for a 50:50 mixture), consistent withthe effects of microabsorption. Analysis of the mixturesusing FeKa radiation data virtually eliminated the mi-croabsorption problem, yielding 500/o hematite and 500/ocorundum for the 50:50 mixtures (hematite p/p : 56compared with 62 cm-' for corundum). Table 3 tabulatesthe results for a series of hematite-corundum mixturesanalyzed with both CuKa and FeKa radiation. Thesemixtures are a good example of a case in which conven-tional methods will implicitly correct for the problemduring standardization, therefore potentially yielding re-sults superior to those obtained with the Rietveld meth-

TABLE 3, Rietveld quantitative analysis results for variousmixtures of hematite. and corundum using CuKa andFeKa radiation

RadiationHematite tocorunoum

iI I

il ,ll,l,r*",I ' ' r r I r ' - l

i l r r i l | | i l i l i l i l t{ l t i l l t i l l l l l l l i l l l l l l lt 9 z 2 9 3 4 3 9 / 4 4 9 5 4 5 9 6 / 6 9

Two-fheto (degrees)

Fig. 4. Observed and calculated patterns for a 50:50 mor-denite-corundum mixture (conventions as in Fig. 1; R* : 16.3o7o,R",o : 7.10/o).

tative results. However, attempting to refine structuralparameters could be disastrous; only scale and unit-cellparameters apparently remain largely unaffectec.

The mixtures of hematite and corundum illustrate oneof the most serious potential problems with the Rietveldmethod when analyzing mixtures with one or more phaseshaving a linear absorption coefficient significantly greaterthan the average for the sample. In spite of the goodagreement between observed and calculated patterns forthe 50:50 mixture of synthetic hematite and corundummeasured with CuKa radiation (Fig. 6), the quantitativeresults, 440lo hematite and 560/o corundum (Table 3), wererelatively poor. As emphasized by Bish and Howard(1988), the poor results for this sample are probably dueto microabsorption, yielding low relative weight percents

t5

Fig. 5.corundum:8 .5o lo ) .

m 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5Two+helo (degrees)

Observed and calculated patterns for a 50:50 biotite-mixture (conventions as in Fig. l; R*o : 32.2o/o, R.-o

Note.'results are in weight percent.. Synthetic hematite unless otherwise specified.

" Cleater Moor, England.f The abbreviation n.a. : not analyzed.

a 135?

F e 0co= 4 s

20:8040:6050:5050:50-.60:4080:2090:1 0

16:8434:6644:5620:8053:4777:23n a.f

19 :8139:6150:5050:5058:4278:2288:1 2

'*!"*{.L J -l-rdA[itr.]'iA./ktg "".+'-*ql.*

*l*l*'l;,-,]tr1"il';ilf 1/

I I I U i l t lY

t[flil ]l

Page 7: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

938 BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

20 25 30 35 /Ul 45 50 55 60 65 70Two-lhelo (degrees)

Fig. 7. Observed and calculated patterns for volcanic feld-spars (conventions as in Fig. l; R*o : 26.60/0, R.*,: 6.20/o).

od, if microabsorption effects are not considered in theanalysis. In spite of the microabsorption problem, preciseand accurate cell parameters were still obtained with CuKaradiation (Table 2). The microabsorption problem is mostsevere when the high linear absorption coefficient mate-rial is a minor component of the mixture, i.e., when trrlpfor phase a is very different from the average p/p of themixture. However, for typical rock mixtures containinga small amount, e.g., a few percent, of an Fe-bearing phase,the errors due to microabsorption will probably not bethe limiting factor in the accuracy of the analysis. Forsmall amounts of a phase, the error in the refined scalefactor will typically be comparable with that due to mi-croabsorption, and quantitative results will not have largeabsolute errors. As pointed out by Bish and Howard(1988), the Rietveld method provides an opportunity tocalculate the linear absorption coemcient for every phasein a mixture, thereby facilitating a correction for mi-croabsorption problems if an average crystallite shape andsize are assumed (see Klug and Alexander, 1974,p.541-542). Taylor and Matulis (1991) recently presented re-sults of a simple correction for microabsorption duringRietveld quantitative analysis that appears to work rea-sonably well for a mixture of LiF (p/p : 20 cm ') andPb(NO3), fu/p : 231cm-', CoKa radiation). The goodresults obtained with finely ground hematite-corundummixtures using FeKa radiation illustrate that the Rietveldmethod of quantitative analysis can be applied even tomixtures containing large amounts of Fe-bearing phasesthrough judicious selection of radiation and proper sam-ple grinding.

Results for the ilmenite-corundum and magnetite-co-rundum mixtures were similarly affected by microab-sorption. Both 50:50 mixtures yielded poor quantitativeresults with CuKa radiation (Table l), but precise andapparently accurate unit-cell parameters were obtained(Table 2). The differences between the two sets of ilmen-

I

t .llll,{i xtL.i. ff.f ,. -L

r r

m 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0Two-lhelo (degrees)

Fig. 8. Observed and calculated patterns for the G-l stan-dard granite (conventions as in Fig. l).

ite unit-cell parameters in Table 2 are probably due tovariations in composition. These results are comparablewith the results for the natural hematite and corundummixtures with CuKa radiation. The ilmenite-corundumand magnetite-corundum mixtures were not analyzed withFeKa radiation.

Natural geologic samples are more difrcult to use todemonstrate the accuracy of the Rietveld method ofquantitative analysis because of the lack of well-charac-terized materials for test standards. Figure 7 shows theobserved and calculated patterns for the mixture of twovolcanic feldspars separated from a tufr the fit is good,considering the complexity of the pattern, the large num-ber ofreflections, and the presence ofsignificant preferredorientation. Rietveld quantitative analysis yielded 930/osanidine and 7o/o high albite, agreeing well with the in-dependent point count results (100 grains, D. Broxton) of930/o sanidine andTo/o albite. Cell parameters for sanidinewere e :8 .4499 (4 ) , b : 13 .008 ( l ) , c :7 .1755 (4 ) A , andP : 116.069(5)"; parameters for high albite were 4 :

8.21 1(3), b : 12.926(6), c : 7.r28(D A, o : 93.35(5), 0: 116.37(3), and "y : 90.15(4)". The cell parameters de-termined from this analysis can be used with the familiarD-c plots (e.g., Stewart and Wright, 1974) to determinecomposition and the state of ordering of the feldspars.Use of the b-c plot with these cell parameters shows thatboth feldspars have near maximum disorder, as expectedfrom their paragenesis; the high sanidine plots near theK end-member, and the albite plots between high albiteand high sanidine in composition.

The analysis of the G- I standard granite provided amore rigorous test of the method and involved analysisfor four phases. The results for quartz, albite, microcline,and biotite (Fig. 8) agreed with the modes (recalculatedto weight percent) of Chayes (1951) within I sd (Table 4)and also provided precise unit-cell parameters for all ofthe major phases in the rock. Data obtained using three

r20

EO

9 o ox

BooEIE m

400

I 300x

F zoocI.E 100

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BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS 939

TABLE 4. Rietveld quantitative analysis results for the G-1 stan-dard granite using three sample preparation methods'

Chayes (1951) Rietveld method

Filter Packed

125

m 2 5 3 0 3 5 f i 4 5 s 0 5 5 6 0 6 5 7 0Two-lheto (degrees)

Fig. 9. Observed and calculated patterns for the 50:50 mix-ture of coralline algae and echinoid calcite (conventions as inFig. l; R*o : 21.9o/o, R"^o : 14.50/o).

(R*" : 16.3, R"^o: 9.7o/o), although the results (Table 5)may be affected by microabsorption; however, the veryfine crystallite size of the iron oxides in this sample willminimize microabsorption effects. Results for the Ha-waiian sample (R*o : 12.7, R.*e: ll.9o/o) are in goodagreement with results obtained by R. Jones using profilerefinement and conventional methods (Jones and Bish,I 99 1) and should not be affected by microabsorption dueto the use of CoKa radiation. It is noteworthy that thesmall amounts of rutile and ilmenite in these sampleswere reproducibly determined, and it appears that thismethod is well suited to analyzing for minor components.

AcxNowr-BncMENTS

We are grateful to D. Broxton of Los Alamos for the volcanic feldsparseparate, to R. Jones ofthe University ofHawaii for bauxite X-ray data,and to P Ildefonse of the University of Paris for bauxite samples. Themanuscript benefited from reviews by G. Guthrie, F. Hawthorne, and R.Snyder. This research was largely supported by the U.S. Department ofEnergy through contract no. W-7405-ENG-36 to Los Alamos NationalLaboratory.

RnrunnNcns crrEDBish, D.L., and Chipera, S.J. (1988) Problems and solutions in quanti-

tative analysis of complex mixtures by X-ray powder diffraction. InAdvances in X-ray analysis, p. 295-308. Plenum, New York.

-(1989) Revised mineralogic summary of Yucca Mountain, Neva-da, 68 p. Los Alamos National l-aboratory, Report LA-l1497-MS.

Bish, D.L., and Howard, S.A. (1986) Quantitative analysis via the Riet-veld method. Workshop on Quantitative X-ray Diffraction Analysis,National Bureau of Standards. June 23-24. 1986.

-(1988) Quantitative phase analysis using the Rietveld method.Journal of Applied Crystallography, 21, 86-91.

Bish, D.L., and Post, J.E. (1988) Quantitative analysis ofgeological ma-terials using X-ray powder diffraction data and the Rietveld refinementmethod. Geological Society of America Abstracts with Programs, 20,4223.

Brindley, G.W. (1980) Quantitative X-ray mineral analysis of clays. InG W. Brindley and G. Brown, Eds, Crystal structures of clay minerals

I t ' -- i

, I t I l

, r [^ . i l ̂ [ ,4 [ - L a J

t l tl

-1-- -rilt ilil[ilrl iltillt illt llllilllllilllilill|,flllllll

t50

QuartzAlbiteMicroclineBiotiteOther

27.6(1.4) 28.7(2.6)31.2(1.4) 33.5(2.0)34.4(1.4) 31 .7(2.813.6(0.6) 3.6-'3.3 2.4(0.31

28.2 27.3 29 131.9 32.6 32.633.0 35.1 32sn.d.t 1.7 2 1n.d. n.d. n.d.

9zsx

F s ocoE 2 5Note.'results are in weight percent.

. Mode and normative values of Chayes (1951) recalculated to weightoercent. Rietveld results have been normalized to account for unanalvzedDnases.

'. Not determined in norm, set equal to mode value.t The abbreviation n.d. : not determined.

sample mounting methods gave similar quantitative re-sults.

The prepared mixture of 500/o coralline algae and 500/oechinoid calcite was a test of the method's ability to sep-arate closely overlapping contributions from similarphases. The coralline algae contained both magnesiumcalcite and aragonite, and the echinoid contained mag-nesium calcite of crystallite size and composition signif-icantly different from the magnesium calcite in the cor-alline algae. Results of the Rietveld analysis (Fig. 9) gave54(2)o/o echinoid calcite, 35(2)oh coralline calcite, and10.8(5)o/o aragonite. Use of the determinative curve ofGoldsmith et al. (1961) with the cell parameters obtainedfor the two calcites gave a Mg content of 20 molo/o for thecoralline calcite and l0 molo/o for the echinoid calcite.Because of peak-width variations between the two dis-tinct calcites and severe peak overlap, compositional andquantitative phase analysis of this sample by convention-al methods would be extremely difficult.

The bauxite samples were the most complex mixturesstudied, with seven phases present in each sample. Thefirst sample, from France, contained detectable goethite,hematite, gibbsite, boehmite, rutile, anatase, and kaolin-ite; the Hawaiian sample (CoKo data) contained hema-tite, goethite, magnetite, ilmenite, gibbsite, anatase, andrutile. The refinement for the first sample gave a good fit

TaBLE 5. Rietveld quantitative analysis results for two bauxitesampres

French Hawaiian

HematiteGoethiteMagnetitellmeniteGibbsiteBoehmiteRutileAnataseKaolinite

11.9(2re,s(2)

n.d. '*n.o.62.7(5)12.6(210.44(5)1 .51(6)1.4(21

5.s(3)29.9(1.7)

7.7(310.20(6)

51.4(8)n.d.0.1 7(8)5.2(21n.d.

Nofe: results are in weight percent.. Numbers in parentheses represent the estimated standard deviation

in the last quoted place... The abbreviation n.d. : not detected.

Page 9: Quantitative mineralogical analysis using the …American Mineralogist, Volume 78, pages 932-940, 1993 Quantitative mineralogical analysis using the Rietvetd full-pattern fitting method

940 BISH AND POST: QUANTITATIVE MINERALOGICAL ANALYSIS

and their X-ray identifrcation, p. 4ll-438. Mineralogical Society,bndon.

Chayes, F. (1951) Modal analyses ofthe granite and diabase test rocks.U.S. Geological Survey Bulletin, 980, 59-68.

Chung, F.H. (l 974a) Quantitative interpretation of X-ray diffraction pat-tems of mixtures I. Matrix-flushing method of quantitative multicom-ponent analysis. Journal of Applied Crystallography, 7, 5 19-525.

-(1974b) Quantitative interpretation of X-ray diffraction pattemsof mixtures. II. Adiabatic principle of X-ray diffraction analysis ofmixtures. Journal of Applied Crystallography,7 , 526-531.

Dollase, W A. (1986) Conection of intensities for preferred orientation inpowder diffractometry: Application of the March model. Journal ofApplied Crystallography, 19, 267 -27 2.

Goldsmith, J.R., Gral D.L., and Heard, H.C. (1961) Lattice constants ofthe calcium-magnesium carbonates. American Mineralogist, 46, 453-457 .

Hill, R.J., and Howard, C.J (1987) Quantitative phase analysis fromneutron powder diffraction data using the Rietveld method. Journal ofApplied Crystallography, 20, 467 -47 4.

Hill, R.J., Craig, J R., and Gibbs, G.V. (1979) Systematics of the spinelstructure type. Physics and Chemistry of Minerals, 4, 317-339.

Howard, S.A., and Preston, K.D. (1989) Profile fitting of powder diffrac-tion patterns. In Mineralogical Society of America Reviews in Miner-alogy,2O,2l7-275.

Jarvinen, M. (1992) About analytical models for texture correction. Pro-ceedings ofAccuracy in Powder Diffraction, 2, 34.

Jones, R.C., and Bish, D.L. (1991) Quantitative X-ray diffraction analysisofsoils: Rietveld full-pattern mineral concentrations and pattern curvefitting vs P sorption of bauxite soils Proceedings of the Annual ClayMinerals Society Meeting, 28, 84.

Kastalsky, V., and Westcott, M.F. (1968) Accurate unit cell dimensionsof hematite (c-Fe.O). Australian Joumal of Chemistry, 21, 106l-1062.

Klug, H.P., and Alexander, L.E. (1974) X-ray diffraction procedures forpolycrystalline and amorphous materials, 966 p. Wiley, New York.

Mazzi,F., and Galli, E. (1978) Is each analcime different? American Min-eralogist, 61, 448-460.

O'Connor, B H., and Raven, M.D. (1988) Application of the Rietveldrefinement procedure in assaying powdered mixtures. Powder Diffrac-tion. 3. 2-6.

Post, J.E , and Bish, D.L. (1989) Rietveld refinement of crystal structuresusing powder X-ray diffraction data. In Mineralogical Society ofAmer-ica Reviews in Mineralogy, 20,277-308.

Reid, R.P., Macintyre, LG., Post, J.E. (1992) Micritized skeletal grains innorthern Belize lagoon: A major source ofMg-calcite mud. Journal ofSedimentary Petrology, 62, 145- | 56.

Rietveld, H.M. (1969) A profile refrnement method for nuclear and mag-netic structures. Journal of Applied Crystallogaphy, 2, 65-7 |.

Snyder, R.L., and Bish, D.L. (1989) Quantitative analysis. In Mineral-ogicaf Society of America Reviews in Mineralogy, 20, l0l-144.

Stewart, D.B., and Wright,'t.L. (1974) Al/Si order and symmetry of nat-ural alkali feldspan, and the relationship of strained cell parameters tobulk composition. Bulletin de la Soci6t6 frangaise de Min6ralogie et deCristallographie, 97, 356-377 .

Taylor, J.C., and Matulis, C E (l 99 l) Absorption contrast effects in thequantitative XRD analysis of powders by full multiphase profile re-finement. Journal of Applied Crystallography, 24, 14-17.

Thompson, P., Cox, D.E., and Hastings, J.B. (1987) Rietveld refinementof Debye-Scherrer synchrotron X-ray data from A!Or. Journal of Ap-plied Crystallography, 20, 79-83.

Wechsler, B.A., and Prewitt, C.T. (1984) Crystal stnrcture of ilmenite(FeTiO,) at high temperature and at high pressure. American Miner-alogist ,69, 176-185.

Wiles, D.B., and Young, R A (1981) A new computer program for Riet-veld analysis of X-ray powder diffraction patterns. Journal of AppliedCrystallography, | 4, | 49- | 5 l.

Will, G., Bellotto, M , Parrish, W., and Hart, M. (1988) Crystal structuresofquartz and magnesium germanate by profile analysis ofsynchrotron-radiation high-resolution powder data Journal ofApplied Crystallog-raphy, 2 l, 182-191.

Mrxtlscnrm RECETVED Ocrosen 30, 1992Mexuscnrp'r AccEPTED Mw 27, 1993