Shipley, T.H., Ogawa, Y., Blum, P., et al., 1995Proceedings of the Ocean Drilling Program, Initial Reports, Vol. 156

3. CALIBRATION OF AN X-RAY DIFFRACTION METHOD TO DETERMINE RELATIVE MINERALABUNDANCES IN BULK POWDERS USING MATRIX SINGULAR VALUE DECOMPOSITION:

A TEST FROM THE BARBADOS ACCRETIONARY COMPLEX1

Andrew T. Fisher2 and Michael B. Underwood3

ABSTRACT

X-ray diffraction methods are used routinely to identify detrital and authigenic minerals in bulk powders of marine sediment.Semiquantitative bulk XRD analysis, however, is difficult. We employed a mathematical technique using matrix singular valuedecomposition to solve for reliable normalization factors, thereby allowing accurate conversion of XRD data to relative mineralabundances. Calibration was achieved through measurements of laboratory mixtures with known abundances of mineral stan-dards. Error analysis demonstrates that this method is superior to other common shipboard procedures used by the Ocean DrillingProgram; the errors from known standards fall within the range of analytical reproducibility and are better than 3%.

INTRODUCTION

X-ray diffraction (XRD) is an essential tool for determining thepresence or absence of detrital and authigenic minerals within bulksamples of marine sediment. The method is fast, inexpensive, and aroutine part of Ocean Drilling Program (ODP) shipboard operations.Because the required sample volumes are relatively small, XRDanalysis can be completed on sample residues from other analyses,such as tests of index properties or carbon-carbonate content.

Identification of specific minerals on diffractograms is accom-plished routinely through the recognition of characteristic peak posi-tions, either by eye or through a computerized match with standardXRD responses. It is much more difficult to calculate relative mineralabundances. Often, qualitative results are shown in a table with sym-bols to indicate categories such as "dominant," "abundant," "minor,"and "trace," but some scientists convert these categories to numericalvalues and plots. In addition, there is no consistency between assess-ments of different workers. Peak intensities and peak areas can beused as indicators of mineral abundance, but relations among peaksfor multicomponent mixtures can be very complicated. For a singlemineral, each individual peak will display a different geometry. In amineral mixture, the intensity of any given peak will be influenced byits own abundance, the absolute abundance, crystallinity, and orien-tation of all other minerals in the specimen, and the amount of amor-phous solids such as volcanic glass and opal. With bulk powders,additional problems are encountered if the analyses include bothplatyminerals, such as clays, and nonplaty minerals such as quartz, feld-spar, and carbonates. Platy minerals produce the best XRD resultswhen analyzed as oriented aggregates, whereas nonplaty mineralsshould be analyzed as random mounts.

Many techniques have been proposed to improve quantificationof bulk mineralogy and clay mineralogy (Pierce and Siegel, 1969;Brindley, 1980). One approach is to derive peak-intensity weightingfactors from 50:50 mixtures of two mineral standards (e.g., Cook etal., 1975). The typical mineral for comparison is quartz. This approachis flawed, however, because the error attached to the single weightingfactor for each mineral pair increases as the mixture deviates from theideal 50:50 blend. A second method involves spiking samples with a

1 Shipley, T.H., Ogawa, Y., Blum, P., et al., 1995. Proc. ODP, Init. Repts., 156: CollegeStation, TX (Ocean Drilling Program).

2 Department of Geological Sciences and Indiana Geological Survey, Indiana Univer-sity, Bloomington, IN 47405, U.S.A.

3 Department of Geological Sciences, University of Missouri, Columbia, MO 65211,U.S.A.

known weight percentage of a foreign mineral such as talc or corun-dum; weighting factors then can be calculated from the talc-normal-ized peak areas (Heath and Pisias, 1979). These weighting factors,however, will change with the relative abundance of the spike min-eral. Single best-fit weighting factors can be determined for eachindividual mineral in a known multicomponent mix using laboratoryblends of mineral standards (e.g., Underwood et al., 1993). Linearinteraction coefficients for every mineral pair in a multicomponentsystem also can be determined empirically (Moore, 1968); accuracyimproves if each mineral pair is measured over its entire range ofpossible mixtures. Mineral intensity factors likewise can be calcu-lated from theoretical mineral reference intensities using computerprograms such as NEWMOD (Moore and Reynolds, 1989). Many ofthe software packages that support modern digital XRD systems alsocan be calibrated using results from known mixtures of standardminerals (e.g., Mascle et al., 1988); unfortunately, the inner workingsof these programs are generally protected by proprietary status, sothey are impossible to evaluate. A final method involves the use ofsimultaneous linear equations, and the input parameters can includeboth XRD and chemical data (Johnson et al., 1985). With this ap-proach, the minimum number of properties measured must equal thenumber of components in the samples being analyzed, and the mini-mum number of samples analyzed must equal the number of proper-ties measured (e.g., peak intensity for three minerals in three stan-dards). This constraint may not allow the full range of natural mineralconcentrations to be represented in a limited set of standards. Themethod described in this paper is a modification of this linear ap-proach. Our new method was used to calculate relative abundances oftotal clay minerals (smectite, illite, andkaolinite), quartz, plagioclase,and calcite, as reported in the respective site chapters in this volume.

Mathematical Determination of OptimalNormalization Factors

Like Johnson et al. (1985), we used linear algebra to determinefactors for converting XRD data to relative mineral abundances.Absolute abundances are virtually impossible to quantify withoutidentifying every mineral phase in the specimen, along with theweight percentages of each amorphous constituent; this is an unreal-istic goal for ODP shipboard investigations. Our goal was to repro-duce bulk abundances of known standards to better than the analyticalaccuracy of the bulk measurements.

To calculate relative proportions, we assumed first that there is aconsistent and quantifiable relationship between one chosen XRDindicator (peak intensity or peak area) and the actual relative abun-

29

A.T. FISHER, M.B. UNDERWOOD

dance of each mineral of interest. Next, we assumed that the strengthof the signal for one mineral is influenced by the strength of thesignals for all other minerals in the same sample. The nature of thisinfluence can be either positive or negative in terms of the signalindicator. Finally, we assumed that the factors that allow conversionfrom XRD signals to actual mineral abundances are constants. Thislast assumption is perhaps the weakest; nevertheless, the use of con-stant factors allowed us to predict mineral abundances for a set ofreference mineral mixtures, spanning a wide range of compositions,with a degree of accuracy better than the experimental reproducibilityof the instrument and the supporting software used to reduce digitaloutput. Later we discuss how one might relax this last assumption,through iterative analysis, to calculate XRD factors that vary withmineral abundances in a relationship that follows a linear, polyno-mial, exponential, or practically any other mathematical form.

A specific example should make these assumptions and relation-ships clear. Assume that Sample 1 contains unknown proportions ofquartz, plagioclase, and calcite, but no other minerals. One specificpeak must be chosen for each mineral as the indicator of its relativeabundance. The relationship among the three signals and the abun-dance of quartz in Sample 1 is:

SQ1FQQ + SC1FCQ — A0/' (1)

where SXi is the signal from each mineral in Sample 1, FXQ is the factorfor each mineral as an indicator of quartz, X is the mineral responsiblefor the signal in question (Q = quartz, P = plagioclase, and C=calcite),and AQ1 is the true abundance of quartz in Sample 1. Thus, if we knowthe values of the various factors, FXQ, we can calculate the abundanceof quartz from the diffractogram of the sample. Similar equations areused to determine the abundances of the other two minerals in Sample1. The XRD signals from the sample remain the same, but the factorsare different: Fcc = the factor for calcite as an indicator of calcite, FQC

= the factor for quartz as an indicator for calcite, etc. The problem,then, is to determine values for these various factors that will allowaccurate conversion from XRD signals to actual abundances over therange of standard mixtures.

As a first step, one can determine the values of the various factorsindependently for each target mineral, beginning with quartz. If a setof three or more standards is mixed, each with different proportionsof the three minerals, then we will have three or more equations withthree unknowns. With three standards, we have an exactly determinedsystem. With four or more standards we have an over determinedsystem, and the solution will optimize the values of the factors inorder to minimize the difference between actual and predicted quartzabundances in the set of standards. With a set of four standards wehave for quartz four equations and three unknowns:

SQIFQQ < + SCIFCQ — Δ•QI,

+ SC2FCQ = AQ2,

+ ‰^cρ = Δ•QS,

(2)

+ Sp + ^C

Similar sets of equations can be generated for the other two minerals.For the Leg 156 samples, we blended six standard minerals

(quartz, plagioclase, calcite, smectite, illite, and kaolinite) into ninestandard mixtures (Table 1). The selection and compositions of thesemineral standards are based on earlier XRD studies near the Leg 156sites. The combination of six standard minerals and nine mixturesgives nine equations and six unknowns. For quartz abundance deter-mination, the matrix representation of these equations is:

S F ß = Aß, (3)

where S is a rectangular (m × n) matrix of signals from m standardmixtures (rows) and n standard minerals (columns), F ß is a vector of

n factors, one for each mineral as an indicator of quartz abundance,and Aß is a vector of m quartz abundances, one for each standardmixture. This matrix equation can be inverted numerically to deter-mine the values of the solution vector, Fß , using singular valuedecomposition (SVD) (Press et al., 1986), such that:

F ρ = V W • R r • A£ (4)

ss s

ss

sssssss

~Fρ~

Fp

\

F/

U

_

~AQ~

A c

A;

A K .

With this construction, V W • R7" is equivalent to S ', V is acolumn-orthogonal matrix with the same dimensions as S, W is an n× n diagonal matrix containing inverse singular values (l/wn), and R r

is the transpose of an n × n orthogonal matrix. Similar constructionscan be made for the vectors containing factors for the other minerals:FP, F c , Fs, F;, and FK. In this way, the calculated abundance of a givenmineral is not only influenced by the intensity or area of its diagnosticpeak, it is also influenced by the signals generated by other mineralcomponents in the system.

One shortcoming to this approach is that the indicator factors foreach mineral are calculated independently of those for the otherminerals. To compensate, we added an additional constraint to thesystem; the total of all mineral abundances should add up to 100% foreach standard mixture. We then solved for all factors simultaneously.This was accomplished by creating and solving combined sets ofequations, shown in matrix form as:

(5)

where the combined signal matrix is sparse, single-bordered, and blocktriangular using the original signal matrix, S. The abundance vectorincludes abundances for all minerals in all standard mixtures plus theunity constraint, U ( a l × m matrix, filled with the value 1, if decimalfractions are used to indicate abundance, or 100 if percentages areused). A singular factor vector (the desired solution) includes factorsfor all indicator and target minerals. For the nine standard mixturesanalyzed during Leg 156, the combined signal matrix has 63 rows(m × [n + 1]) and 36 (n × n) columns; the abundance vector has 63components and the factor vector has 36 components. With this con-struction, the unity constraint is not absolute, but instead is given asmuch influence on the solution as each of the standards. More or lessweight could be applied to the unity relation (or to data from any ofthe standards mixtures or minerals), but as we will demonstrate, theequal weighting applied to our standard mixtures resulted in an excel-lent match between measured and calculated mineral abundances.

Shipboard Equipment and Mineral Standards

The X-ray laboratory aboard JOIDES Resolution is equipped witha Philips PW-1729 X-ray generator, a Philips PW-1710/00 diffractioncontrol unit with a PW-1775 35 port automatic sample changer, anda Philips PM-8151 digital plotter. Machine settings for all standardswere as follows: generator = 40 kV and 35 mA; tube anode = Cu;wavelength = 1.54056 Å (CuKαl) and 1.54439 Å (CuKα2); intensityratio = 0.5; focus = fine; irradiated length =12 mm; divergence slit =automatic; receiving slit = 0.2 mm; step size = 0.01 °2θ; count timeper step = 1 s; scanning rate = 2°2θ/min; ratemeter time constant = 0.2s; spinner = off; monochrometer = on; scan = continuous; scanningrange = 2°2θ - 35°2θ.

Digital data were processed using a Philips peak-fitting programthat subtracts background intensities and fits ideal curve shapes toindividual peaks or ranges of peaks, as specified by the operator.

30

CALIBRATION OF AN XRD METHOD

Table 1. Measured weight percentages for X-ray diffraction mineral standards.

StandardID

1234567

9

Smectite(wt%)

1.924.911.910.69.4

43.130.718.161.5

Illite(wt%)

1.317.78.57.66.74.8

10.14.80.0

Kaolinite(wt%)

2.026.712.811.310.15.25.2

15.10.0

Clay(wt%)

5.269.333.229.526.253.146.038.061.5

Quartz(wt%)

15.724.940.015.218.537.139.842.038.5

Plagioclase(wt%)

10.15.8

13.137.0

9.05.24.65.60.0

Calcite(wt%)

69.00.0

13.718.346.3

4.69.6

14.40.0

Typically, this program was used over the following scanning angles:3.5-10.5°2θ (smectite and illite), 10.5-13.5°2θ (overlapping kao-linite + chlorite) and 25.5-30.5°2θ (quartz, plagioclase, and calcite).Curve-fitting is most effective if there are fewer than 750 steps perscanning interval, and iterations continue automatically until a pre-scribed x2 test is satisfied. Output of a processed digital data includesthe angular position of each peak (°2θ), d-spacing (Å), peak width( °2θ), intensity or height (counts per second above background),and peak area (total counts above background). Precision of thepeak-fitting program deteriorates as peak intensities approach thebackground noise.

The six minerals selected for our standard mixtures are based on theresults of previous bulk-powder and clay-fraction XRD analyses ofDeep Sea Drilling Project (DSDP) and ODP specimens from thenorthern Barbados Ridge (Pudsey, 1984; Capet et al., 1990; Tribble,1990). The total abundance of each standard mineral in the standardmixtures is shown in Table 1. The quartz standard is National Bureauof Standards (NBS) SiO2 from Hot Springs, AR. Calcite is WardsIceland spar from Chihuahua, Mexico. Plagioclase is Wards oligoclasefrom Mitchell County, NC. Illite is Clay Mineral Society (CMS) stan-dard IMt-1. Kaolinite is a mixture of approximately 50% each poorlycrystalline (CMS KGa-1) and well crystalline (CMS KGa-2) kaolinite.The smectite mixture comprises three components in varying propor-tions: Wards generic montmorillonite from Cameron, AZ (75%); CMSSWy-2 Na-montmorillonite (13%-25%); and CMS SAz-1 Ca-mont-morillonite (0%-12%). All minerals were available either prepow-dered or were oven-dried and ground in a ball mill. After weighing andstirring, each standard mixture was blended further in the ball mill forseveral minutes to produce a powder with uniform texture. Subsamplesof each standard mixture were packed into rectangular sample holders,as randomly oriented mounts, and run sequentially. The randomly ori-ented powders were not pretreated with any chemicals. The powder ofeach standard mixture was blended a final time before subsampleswere extracted; in spite of these precautions, variations in the diffracto-grams for subsequent slides of the same mixture show that preferentialsettling of mineral particles did occur, particularly with well crystallinematerial such as calcite. This phenomenon is extremely difficult toavoid on the JOIDES Resolution because of the constant shipboardvibration. In partial compensation, the average peak areas and averageintensities from individual runs were used as input values for the SVDprogram, with the exception of Standard 9 (montmorillonite and quartzmixture), for which only one subsample was run (Table 2).

Typical diffractograms for each of the nine standards are shown inFigure 1, along with labels indicating the peaks used for specificmineral identification. Standard 9 was a mixture of three montmoril-lonite mineral "standards" available aboard the ship. Careful examina-tion of diffractograms from each individual component of this smectitemixture revealed that the Wards generic montmorillonite and CMSNa-montmorillonite both contain significant proportions of quartz. Wedetermined the amount of quartz contamination by conducting a seriesof tests independent of those described below, in which subsamplesof generic montmorillonite and Na-montmorillonite powders werespiked with known quantities of quartz to generate mixing curves (Fig.2). The samples were spiked only twice, but all of the mixing curves

for peak intensity and peak area are clearly nonlinear. These curveshelp illustrate the problem associated with using single mixtures of agiven mineral pair for calibration of weighting factors; errors increasewith increasing deviation from the ideal 50:50 mix. Diffractogramsfrom the spiked subsamples were evaluated along with those from thetwo unspiked subsamples, using the same SVD technique described inthe preceding section. As a result, we concluded that the genericmontmorillonite "standard" contains 45% quartz, whereas the Na-montmorillonite "standard" contains 39% quartz. These montmoril-lonite/quartz ratios were used to recalculate the compositions of allstandard mixtures. For example, when combined with the measuredweight of relatively pure Ca-montmorillonite, we determined that theStandard 9 mixture contains 38.5% quartz (Table 1).

Analyses of Standard Mixtures

Once the nine standard mixtures were prepared, and the respectivediffractograms were generated, the next task was to determine whichtype of signal provides the most consistent indicator of mineral abun-dance. Obvious candidates include peak intensity and peak area, butother decisions were less clear-cut. For example, should individualclay-mineral peaks be used (i.e., at approximately 6°, 8.5°, and 12°2θ),or should one rely on a single composite clay-mineral peak producedby overlapping reflections at approximately 19.8°2θ? This peak appar-ently represents a 020 reflection that is common to several clay miner-als. Should mineral signals be normalized or transformed in any wayprior to calculation of weighting factors? We elected to take an empiri-cal approach, by examining several alternative strategies; our goal wasto find out which method yielded the closest match between measuredand calculated mineral abundances. As a target for error reduction, theaccuracy of calculated mineral abundances should be no greater thanthe analytical reproducibility, as dictated by imperfections in samplepreparation, machine drift, and peak fitting using Philips softwarepackages. Tests completed during Leg 156 showed that the reproduci-bility averages ±2.4% for total clay, ±2.0% for quartz, ±2.8% forplagioclase, and ±1.7% for calcite. These are not the maximum errorsassociated with using the proposed method for the analysis of un-known mineral samples, but reflect instead our ability to replicate themineral abundances of known standards. These relatively low errors inreproducibility indicate that the proposed method is statistically stablefor the nine standards, and that errors associated with the inversion forstandard composition are lower than experimental variations for typi-cal bulk XRD analyses (Table 2).

Results of several approaches are tabulated in Table 3. In thattable, and in the following discussion, the calculated individual abun-dances of smectite, illite, and kaolinite are combined into a total claycomponent. This was done for two important reasons: to facilitatecomparisons with the XRD methodologies of Cook et al. (1975) andMascle et al. (1988) and because there are unresolved questions ofreliability for individual clay-mineral percentages when analyzed asrandom bulk powders, especially without ethylene glycol solvation.In particular, we were unable to eliminate the effects of overlapbetween the smectite (001) and illite (001) reflections. The peak inter-ference problem was exacerbated in the most of the Leg 156 clay stone

31

A.T. FISHER, M.B. UNDERWOOD

4000

Standard 1:Total clay minerals = 5.2°/Quartz = 15.7%Plagioclase = 10.1%Calcite = 69.0%

I • I ' I 'Standard 3:Total clay minerals = 33.2°/cQuartz = 40.0%Plagioclase = 13.1%

- Calcite = 13.7

1 I ' I > IStandard 7:Total clay minerals = 45.9%Quartz = 39.8%Plagioclase = 4.6%Calcite = 9.6%

200 1600 I ' I 'Standard 2:Total clay mineralsQuartz = 24.9%Plagioclase = 5.8%

- Calcite = 0.0%

- 100 800

300

- 150 32

1000

1 ' 1 ' 1Standard 4:Total clay mineralsQuartz = 15.2%

1 i

= 29.5%

Plagioclase = 37.0%- Calcite = 18.3%

CD

olin

ii

- i 2ε

Ilite

- iljillküfl

lase

30

lag

IL i i J i ihiiiiL I i ii

WlüiiliIf Ffippi

1 i •

B

mpo rtz)

8 |— To

1ULI nJJUJ ™

P i

i >

•

c

ise)

ocl;

agi

1

a>

s —3 O

lag

LL

I

J u

1

αiIIOI

CO

—

cfl

iocl

lag

YΛ

300

-1150

- 2 0 0

I • I •Standard 6:Total clay minerals = 53.1°ΛQuartz = 37.1%Plagioclase = 5.3%

- Calcite = 4.6%

400

1 ' I • I ^Standard 8:Total clay minerals = 38.0%

- Quartz = 42.0%Plagioclase = 5.6%Calcite = 14.4%

2000

1000

12 16 20

Degrees 2θ

12 16 20Degrees 2θ

32

ü

700

525

350

175

Figure 1. Annotated X-ray diffractograms from nine mixtures of laboratory standards showing characteristic peaks used to estimate mineral abundances. Measuredweight percentages are shown for each mineral group in each mixture. For each standard, a portion of the diffractogram with key clay mineral peaks is shown atan expanded scale of peak intensity (upper diffractogram). Peaks labeled in parentheses correspond to reflections that are diagnostic of mineral presence but werenot used in the calculations described in this paper. The quartz peak at 3.34 Å was selected as a primary indicator so that direct comparisons could be made withCook et al. (1975) and because the cluster of quartz, calcite, and plagioclase peaks was convenient for fitting.

CALIBRATION OF AN XRD METHOD

1000 I • I ' I • IStandard 9:Total clay minerals = 61.5%Quartz = 38.5%Plagioclase = 0.0%Calcite = 0.0%

O

400

-300

-200

- 100 CL

12 16 20Degrees 2θ

Figure 1 (continued).

specimens, which were obtained from physical properties residues.Smectite interlayer water is partially expelled during oven-drying;even though this is accepted part of the shipboard procedure formeasurements of water content and dry density, it creates an XRDartifact by distorting the d-spacing of the smectite (001) reflection.Thus, whereas the values of [smectite + illite] may be reasonablyaccurate, the smectite-to-illite ratios definitely are not. Proper post-cruise investigations of clay-sized separates will yield much betterestimates of the clay-mineral abundances, based on oriented aggre-gates and appropriate solvation and heating techniques.

Use of a composite clay-mineral peak (e.g., the peak at approxi-mately 19.8°2θ) is also problematic because its intensity varies withboth the abundance of each individual clay mineral and the specificchemical and crystallinity characteristics of that mineral. Ideally, thecomposite peak changes as a function of total clay-mineral abun-dances, but this change will also be consistently proportional tovariations in the sum of the individual clay-mineral peaks. For thisideal to be tested successfully, similar responses must be recorded forclays in the calibration mixtures and clays in the natural sedimentmixtures. We discovered that the match between composite peakresponse and the sum of individual peak responses was erratic whennatural Leg 156 claystones were compared to the artificial laboratorymixtures (Fig. 3). In other words, there are significant differences inclay chemistry, crystallinity, or both, suggesting that use of the com-posite clay peak will give erroneous results in most cases. Accord-ingly, unless clay mineral standards are extracted from the same Leg156 claystones that are to be analyzed for detailed clay mineralogy,we have no way of accurately calibrating the composite peak at19.8°2θ. This problem provides an additional explanation for theimperfect match between data derived from Leg 110 and Leg 156shipboard methods, as documented below.

The Cook et al. (1975) method uses peak intensities relative to thatof quartz, such that:

A, = (6)

π=l

where A, is the abundance of mineral i, It is the peak intensity, C, is aconstant used to transform intensity into abundance, and m is thenumber of minerals. Cook et al. (1975) provided the following valuesof Cj for the minerals of interest: smectite (3.00), illite (6.00), kaolinite(4.95), quartz (1.00), plagioclase (2.80), and calcite (1.65). In the caseof the Leg 156 standard mixtures, this formulation leads to largesystematic underestimates in total clay content, systematic overesti-mates of calcite abundance, and inconsistent errors with respect toquartz and feldspar (Table 3).

Mascle et al. (1988) calibrated proprietary Philips software that wasdesigned for quantitative analysis with measurements of six mixtures

of unspecified mineral standards. The corresponding constants for netpeak intensities (after background correction) are: clay composite =0.6012 × 10"2; quartz = 0.3002 × 10"3; plagioclase = 0.1365 × 10"2;calcite = 0.3087 × 10~3 (J. Tribble, pers. comm., 1994). Application ofthese values toward the data from Leg 156 standard mixtures yields abetter fit for calcite, at least with respect to the Ct values of Cook et al.(1975), but the fits for quartz, plagioclase, and total clay remain rela-tively poor (Table 3). We attribute part of the problem to the use of thecomposite clay-mineral peak by Mascle et al. (1988).

We achieved better results using simple linear regressions of eitherpeak area or peak intensity vs. standard mineral abundance, but errorsfor some minerals are erratic and still exceed 10% (Table 3; Fig. 4).These errors are greater than experimental uncertainty and are stillconsidered unacceptable. The best match between actual and calcu-lated mineral abundances was obtained through matrix SVD (Table 3).We evaluated both peak area and peak intensity as indicator signals foreach mineral; both types of values were multiplied by the appropriateset of target mineral factors, as in equation 5, to calculate the raw"abundance" of each mineral. Raw abundances less than zero were setequal to zero (but still tabulated as present in trace amounts), andremaining abundances were normalized (proportionately adjusted) sothat they total 100%. Errors turned out to be 4% or less for all mineralsin the nine standard mixtures, and in most cases the errors are less than2% (Table 3). These errors are well within the range of experimentalreproducibility. In no case did we encounter a negative raw abundanceless than -2%, despite the wide range in sample compositions. Similarresults were obtained using peak areas and peak intensities, althoughwe believe that peak areas are more robust overall, particularly withrespect to analysis of broad peaks generated by poorly crystalline clayminerals such as smectite. The peak-area normalization factors for theLeg 156 standards are listed in Table 4.

Figure 5 illustrates a direct comparison of calculated mineralabundances for Leg 156 claystones, as determined by the Cook et al.(1975), Mascle et al. (1988), and matrix SVD methods. These data arebased on analyses of trimmings from interstitial water samples fromSite 948, and the compositional range is representative of the overallstratigraphy recovered during Leg 156. Differences among the calcu-lated mineral abundances are as large as 40% for calcite, plagioclase,and total clay. In addition to these tests, bulk mineral abundances werecalculated for all of the shipboard physical properties residues andinterstitial water trimmings using the matrix SVD method; thoseresults are presented in the "Lithostratigraphy and Sedimentology"sections in the "Site 948" and "Site 949" chapters (this volume).

Discussion

What is the actual physical meaning of the individual normaliza-tion factors that we calculated using matrix SVD? Strictly speaking,these numbers are no more than a mathematical convenience. It isworth noting, however, that the factor with the largest value is alwaysthat of the target mineral. In other words, the factor for quartz exertsthe most influence on the calculated weight percentage of quartz, andthe factor for calcite is the most important for calculating the weightpercentage of calcite. For some minerals, the difference in factormagnitude is several orders magnitude (e.g., illite, kaolinite), whereasfor others the differences are smaller (Table 4).

Some factors for nontarget indicators (e.g., quartz as an indicatorfor illite) are negative. Peak overlap should result in the calculationof positive nontarget factors. Unlike earlier workers (e.g., Heath andPisias, 1979), however, we believe that it may be experimentallyreasonable to have negative nontarget factors, particularly becausewe did not normalize signals prior to processing. Clay minerals havea preferred orientation due to the nature of their crystallinity. Thepresence of varying proportions of nonclay minerals could contribute(in a nonlinear fashion) to a lesser degree of clay mineral orientationin bulk-powder, random mounts. There may be other negative contri-butions related to the density of different mineral grains and the

33

A.T. FISHER, M.B. UNDERWOOD

σ

120 160 200 240Na-montmorillinite peak area

(counts)

120 160 200Montmorillinite peak area

(counts)

3500 5000

Figure 2. Mixing curves for two montmorillonite stan-dards and quartz, expressed in terms of peak intensity andintegrated peak area. Specimens of generic Wards mont-morillonite and CMS Na-montmorillonite were spikedwith measured weight percentages of quartz to solve forthe amount of quartz contamination in each standard.These curves were not used quantitatively for the methoddescribed in this paper, but they do illustrate the nonlinearinterference one mineral can exert on another.

70 105 140Na-montmorillinite peak intensity

(counts/s)Montmorillinite peak intensity

(counts/s)

Table 2. X-ray diffraction data for mixtures of mineral standards.

Mineral

standard

1AIBICAve.#l2 A2B2CAve. #23 A3B3CAve. #34A4BAve. #45 A5BAve. #56A

6B6CAve. #67A7BICAve. #78A8BAve. #89

Smectite

6.310.09.98.7

38.735.742.939.119.613.121.318.018.117.717.918.415.817.152.438.059.750.039.942.838.340.333.1*2.833.072.6

lllite

3.14.53.53.7

26.023.729.126.312.810.813.712.41 1.512.812.1

11.08.19.55.7

3.69.66.3

15.520.115.016.97.68.88.2

0.0

Peak intensity above background (counts/s)

Kaolinite

15.817.617.316.9

222.9223.6270.6239.0

78.691.7

100.890.482.882.882.871.463.967.73 1.327.739.732.938.940.534.538.090.8

111.0100.9

0.0

Clay

7.16.06.56.5

75.072.877.074.934.435.135.134.830.826.828.82.3.020.921.981.176.382.880.164.459.761.862.049.643.446.5

117.2

Quartz

635.5803.4691.5710.1589.9700.7899.9730.2

1513.33731.12951.72732.01144.3681.9913.1585.9564.6575.2832.2

1286.91449.51189.51850.71386.92019.01752.22341.71933.62137.7

639.0

Plagioclase

158.2259.5167.9195.2192.6227.2214.7211.5215.2469.7571.2418.7

1204.1871.6

1037.9181.9262.9222.4

39.847.2

161.782.9

208.272.190.8

123.787.699.393.5

0.0

Calcite

3708.24215.84729.74217.9

0.00.00.00.0

574.6985.8996.7852.4

1120.91872.61496.82346.82596.82471.8

170.8768.3463.7467.6752.6892.5735.5793.5

1378.41221.51300.0

0.0

Smectite

21.735.745.034.1

124.1106.4149.6126.761.763.766.764.056.565.260.867.463.265.3

185.2136.1202.6174.6144.6143.8122.4136.9115.5116.6116.1231.3

Integrated area after peak fitting (total counts)

lllite

0.43.62.22.1

14.413.913.714.05.75.87.36.37.77.47.55.94.25.02.00.61.61.49.48.67.48.55.15.75.40.0

Kaolinite

2.94.03.43.4

66.061.372.466.62 1 . 4

22.326.523.424.422.823.617.116.316.79.27.4

10.69.19.1

12.09.3

10.131.532.532.0

0.0

Clay

2.33.20.82.1

54.255.133.547.620.819.929.52.3.415.119.917.518.915.817.353.651.353.652.843.250.839.444.536.033.734.972.2

Quartz

69.3111.890.890.699.2

100.3140.9113.5274.9428.8348.8350.8144.296.6

120.490.682.186.4

139.2172.7182.3164.7282.9218.1235.8245.6349.4315.3332.4

98.8

Plagioclase

24.341.536.934.228.026.528.727.763.667.980.070.5

188.6170.5179.641.042.741.852.138.644.144.937.835.331.334.833.329.131.2

0.0

Calcite

511.5677.4657.3615.4

0.00.00.00.0

96.0151.4136.8128.1161.5251.6206.5387.1418.2402.7

32.673.360.755.5

107.0144.2127.8126.3222.5191.4207.0

0.0

Notes: The quantitative composition of mineral standards are described in the text and listed in Table 1. Most standard mixtures were run several times; resultsof individual runs and averages (when there was more than one run) are listed above. XRD values for "Clay" are based on the composite peak, while those forindividual clays are based on separate peaks. See discussion in text.

degree to which various phases settle prior to and during mountpreparation. Negative factors certainly should be smaller in magni-tude than those for the target minerals. So long as a wide enough rangein standard compositions is selected, the appearance of negative ap-parent raw abundances will be kept to a minimum. If large negativefactors appear, then it is likely that one or more of the primaryassumptions was violated. Given this circumstance, additional stan-dards should be mixed to be more representative of the natural min-eralogy, or a nonlinear construction should be considered. If the goal

is to develop an accurate experimental method for XRD analysis, thenpreventing the determination of negative factors should be avoided,as such prevention can mask the violation of a primary assumption.

Negative normalization factors (and perhaps all factors for miner-als other than the target) can be considered as corrections, necessarybecause of interference on the diffractograms. The use of only "pri-mary" normalization factors (quartz for quartz, etc.) determined withSVD results in errors that are greater than the errors based on inde-pendent linear regressions for each mineral. This is perhaps not surpπs-

CALIBRATION OF AN XRD METHOD

P 1 0 I I I I I I I I l I I I I I I I I I I I I I I l I I I I IE3 A-ratio (muds)

A-ratio (standards)

12

I l l l • l I I I I6 8 10 12 14Area ratio

(sum of individual peaks/composite peak)

i i I I I I I I I I I I i i il-ratio (muds)l-ratio (standards)

2 3 4Intensity ratio

Figure 3. Histograms showing the relations between inten-sity and peak area for the composite clay mineral peak (at19.8°2θ) and the sum of individual intensity and peak-areavalues for three individual clay peaks [smectite (001), illite(001), and kaolinite (001)]. There is one standard valuewith an area ratio of 62, off the right of the plot. Ideally, formixtures involving uniform clay chemistry and crystallin-ity, these values should be consistently proportional. Thepoor match between natural Barbados clays and laboratoryclay-mineral standards indicates that the composite peak

(sum of individual peaks/composite peak) c a n n o t b e c aiib r ated accurately using shipboard standards.

250

i 125

Wt% smectite = 0.79 x peak intensity, r = 0.952Wt% smectite = 0.24 x peak area, r = 0.923

250 3000

125

20 40Smectite (wt%)

60

Wt% illite = 0.66 x peak intensity, r = 0.991. α wt% illite = 1.25 x peak area, r = 0.957

I ,

14

10Illite (wt%)

20

Wt% kaolinite = 0.13 x peak intensity, r = 0.981Wt% kaolinite = 0.45 x peak area, r = 0.981

I ,

80

20Kaolinite (wt%)

40

55 1500

Φ

θ Wt% quartz

Q Wt% quartz• 0.024 x peak intensity, r = 0.677 Q= 0.17 x peak area, r = 0.720 D

400

20Quartz (wt%)

1200

"55 600

Wt% plagioclase = 0.036 x peak intensity, r = 0.986Wt% plagioclase = 0.20 x peak area, r • 0.981

200

20Plagioclase (wt%)

5000,

40 e £ 2500-2

Wt% calcite = 0.016 x peak intensity, r =0.982Wt% calcite = 0.11 X peak area, r = 0.985

700

350

20 40Calcite (wt%)

60

Figure 4. Peak areas and peak intensities vs. mineral abundances for the six mineral standards used in this study. See text for identification of specific mineralsources. Linear least-squares best-fitting lines are shown along with regression coefficients. The correlations are significant to the 99% confidence level, exceptfor quartz, where the correlation is significant to the 97% level. The errors associated with using these linear relationships are greater than experimental uncertainty,as shown in Table 3.

ing, as a simple linear regression more effectively optimizes for thebest-fitting solution when only a single dependent variable is allowed.

We also experimented mathematically with normalizing peakareas and peak intensities prior to solving for weighting factors bySVD. While this did allow calculation of raw abundances that comecloser to totaling 100%, the factors generated through these analysesseemed less physically reasonable. In several cases, the largest factorswere those of minerals other than the target, and in one case, the factorwith the greatest sealer value was negative. We believe, therefore, thatit is better to solve for normalization factors without overly constrain-ing their values through prior normalization of signals.

Recommendations for Improvement

The normalization factors calculated during this shipboard inves-tigation are valid only for the minerals and the ranges of abundancesused in our standard mixtures. The presence of either additionalphases or similar phases having different chemical compositions orcrystallinities could create significant mismatches. However, the con-sistency of results with natural samples, and the lack of predictednegative raw abundances with magnitudes greater than 2%, indicatethat the method is effective for semiquantitative comparisons. It prob-ably is not possible to derive a fully quantitative method involving

35

A.T. FISHER, M.B. UNDERWOOD

420

Total clay minerals(wt%)

50 100 0

Quartz(wt%)

35 70 0

Plagioclase(wt%)

30 60 0

Calcite(wt%)

30 60

.§.500

Figure 5. Comparison of relative mineral percentages cal-culated from a subset of Leg 156 XRD data using the SVDmethod described in this paper, and with the methodsdescribed in Cook et al. (1975) and data from Leg 110(Mascle et al., 1988). Leg 156 specimens were obtainedfrom trimmings of interstitial water samples at Site 948.

580

random mounts of bulk powders, as variations in crystallinity andpreferred orientation of individual minerals will limit reliability, reso-lution, and the degree of dependance on preparation techniques.

While it is clear that the SVD method works well with standardsof known composition, this study was made easier because previousinvestigations of the Barbados accretionary prism had identified thedominant mineral phases. In cases where the mineralogy is com-pletely unknown, it might be best to run several pilot samples and thengenerate a set of preliminary standard mixtures for a first attempt atcalibration. As more diffractograms are generated and samples areanalyzed using the first set of normalization factors, additional stan-dard mixtures can be created to improve the match. Two or threeiterations might be necessary to optimize the selection of minerals andranges of abundance in the standard mixtures, ideally with many morestandard mixtures than minerals.

If, in a particular geologic setting, the use of constant normalizationfactors proves incapable of providing sufficient accuracy, or if exces-sively negative raw abundances are generated, then one might use asimilar iteration scheme to develop more comprehensive relationships.Such a relationship could include factors that vary with abundance(i.e., FQQ = FQQ2 × AQ, etc.). One could find an initial solution usingconstant factors, then iterate for offset and slope factors, or offset andexponential factors, or factors with whatever mathematical form isrequired to provide a sufficiently accurate fit for the standard mixtures.The primary limitation of this approach is that investigators must createa large enough number of standard mixtures to assure that there aremore equations than unknowns in the matrix being solved. One mustalso be careful that the varying mineral proportions in these mixturesare sufficiently different so as to avoid linear redundancy (i.e., wherethe ratios of several mineral abundances are the same in several mix-tures). The SVD technique also lends itself to solution of undeterminedsets of equations (Press et al., 1986), but this might be particularly riskyin the case of bulk XRD analysis, as there are already errors of un-known magnitude introduced by the presence of unanticipated mineralphases and amorphous solids.

ACKNOWLEDGMENTS

We thank the crew and staff aboard the JOIDES Resolution fortheir assistance during sample acquisition. Mary Ann Cusimano wasparticularly helpful during preparation and analysis of the standardmineral mixtures. The manuscript benefited from reviews by BillBusch and Jane Tribble.

REFERENCES*

Brindley, G.W., 1980. Quantitative X-ray mineral analysis of clays. In Brin-dley, G.W., and Brown, G. (Eds.), Crystal Structures of Clay Minerals andtheir X-ray Identification. London Mineral. Soc. Monogr., 5:411-438.

Leg110

Capet, X., Chamley, H., Beck, C , and Holtzapffel, T., 1990. Clay mineralogyof Sites 671 and 672, Barbados Ridge accretionary complex and Atlanticabyssal plain: paleoenvironmental and diagenetic implications. In Moore,J.C., Mascle, A., et al., Proc. ODP, Sci. Results, 110: College Station(Ocean Drilling Program), 85-96.

Cook, H.E., Johnson, P.D., Matti, J.C., and Zemmels, I., 1975. Methods ofsample preparation and X-ray diffraction data analysis, X-ray MineralogyLaboratory, Deep Sea Drilling Project, University of California, Riverside.In Hayes, D.E., Frakes, L.A., et al., Init. Repts. DSDP, 28: Washington(U.S. Govt. Printing Office), 999-1007.

Heath, G.R., and Pisias, N.G., 1979. A method for the quantitative estimationof clay minerals in North Pacific deep-sea sediments. Clays Clay Miner.,27:175-184.

Johnson, L.J., Chu, C.H., and Hussey, G.A., 1985. Quantitative clay mineralanalysis using simultaneous linear equations. Clays Clay Min., 33:107-117.

Mascle, A., Moore, J.C., Taylor, E., and Shipboard Scientific Party, 1988. ODPLeg 110 at the northern Barbados Ridge: introduction and explanatorynotes. In Mascle, A., Moore, J.C., et al., Proc. ODP, Init. Repts., 110:College Station (Ocean Drilling Program), 5-25.

Moore, C.A., 1968. Quantitative analysis of naturally occurring multicompo-nent mineral systems by X-ray diffraction. Clays Clay Min., 16:325-336.

Moore, D.M., and Reynolds, R.C., Jr., 1989. Quantitative analysis. In Moore,D.M., and Reynolds, R.C., Jr. (Eds.), X-ray Diffraction and the Identifica-tion and Analysis of Clay Minerals: New York (Oxford University Press),272-309.

Pierce, J.W., and Siegel, F.R., 1969. Quantification in clay mineral studies ofsediments and sedimentary rocks. J. Sediment. Petrol, 39:187-193.

Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., 1986.Numerical Recipes: The Art of Scientific Computing: New York (Cam-bridge Univ. Press).

Pudsey, C.J., 1984. X-ray mineralogy of Miocene and older sediments fromDeep Sea Drilling Project Leg 78A. In Biju-Duval, B., Moore, J.C., et al.,Init. Repts. DSDP, 78A: Washington (U.S. Govt. Printing Office), 325-342.

Tribble, J.S., 1990. Clay diagenesis in the Barbados accretionary complex:Potential impact on hydrology and subduction dynamics. In Moore, J.C.,Mascle, A., et al., Proc. ODP, Sci. Results, 110: College Station (OceanDrilling Program), 97-110.

Underwood, M.B., Orr, R., Pickering, K., andTaira, A., 1993. Provenance anddispersal patterns of sediments in the turbidite wedge of Nankai Trough.In Hill, I.A., Taira, A., Firth, J.V., et al., Proc. ODP, Sci. Results, 131:College Station (Ocean Drilling Program), 15-34.

Abbreviations for names of organizations and publication titles in ODP reference listsfollow the style given in Chemical Abstracts Service Source Index (published byAmerican Chemical Society).

Ms 156IR-103

36

CALIBRATION OF AN XRD METHOD

Table 3. Calculated mineral abundances and errors of standard mixtures using

various methods.

Standardnumber

Cook et al. (1234561

9

Mascle et al12345o7

9

Clay(wt%)

Quartz(wt%)

1975) weighting1.6

52.49.87.97.8

13.810.712.525.4

8.526.346.413.410.146.845.941.274.6

Plagioclase(wt%)

factors6.5

21.319.942.610.99.19.15.00.0

. (1988) weightinε factors2.2

47.011.27.49.6

43.928.419.378.6

11.722.944.011.812.632.640.144.321.4

14.730.130.760.922.210.312.9

8.80.0

Simple linear regression of abundance vs.12345

789

11.276.026.628.431.855.144.737.178.9

Simple linear regress12345

7

9

12.075.827.830.530.452.643.436.476.8

16.516.851.018.915.332.839.342.721.1

6.87.3

11.732.2

8.93.44.22.80.0

ion of abundance vs.15.118.849.118.015.230.837.742.023.2

6.75.4

11.63 1.6

8.69.96.34.60.0

: Calcite(wt%)

83.40.0

23.936.271.330.334.341.3

0.0

71.50.0

14.119.855.613.118.627.6

0.0

ΔClay(wt%)

-3.6-16.9-23.4-21.6-18.4-39.3-35.3-25.5-36.1

-3.0-22.3-22.0-22.1-16.6

-9.2-17.6-18.7

17.1

peak intensity65.5

0.010.620.644.0

8.611.917.3

0.0

peak area66.2

().()11.620.045.7

6.712.616.9

0.0

Singular value decomposition based on peak area1234567;:

9

5.968.832.829.825.85 1.946.138.660.8

16.325.242.314.918.438.338.639.537.9

9.96.0

15.335.610.43.94.34.11.2

67.90.09.7

19.845.4

5.911.017.8

0.0

Singular value decomposition based on peak intensit>1234561v9

5.569.432.630.324.651.545.538.561.6

15.824.943.215.418.436.938.839.538.4

9.55.7

14.135. λ

10.47.84.84.90.0

69.1().()

10.219.146.6

3.810.917.00.0

6.7-6.6-1.1

5.62.0

-1.3-0.917.4

6.86.5

-5.41.04.2

-0.5-2.6-1.615.3

0.7-0.5-0.4

0.3-0.4-1.2

0.10.6

-0.7

0.30.1

-0.60.8

-1.6-1.6-0.5

0.50.1

ΔQuartz(wt%)

-7.21.46.4

-1.8-8.4

9.76.1

-0.836.1

^ . 0-2.0

4.0-3.4-5.9-4.5

0.32.3

-17.1

0.8-8.111.03.7

-3.2-4.3-0.5

0.7-17.4

-0.6-6.1

9.12.8

-3.3-6.3-2.1

0.0-15.3

0.60.32.3

-0.3-0.1

1.2-1.2-2.5-0.6

0.10.03.20.2

-0.1-0.2-1.0-2.5-0.1

ΔPlagioclase(wt%)

-3.615.56.85.61.93.94.5

-0.60.0

4.624.317.623.913.25.18.33.20.0

-3.31.5

-1.4-4.8-0.1-1.8-0.4-2.8

0.0

-3.4-0.4-1.5-5.4-0.4

4.71.7

-1.00.0

-0.20.22.2

-1.41.4

-1.3-0.3-1.5

1.2

-0.6-0.1

1.0

-1.71.42.60.2

-0.70.0

ΔCalcite

(wt%)

14.40.0

10.217.925.025.724.726.9

0.0

2.50.00.41.59.38.59.0

13.20.0

-3.50.0

-3.12.3

-2.34.02.32.90.0

-2.80.0

-2.11.7

-0.62.13.02.50.0

-1.10.0

-4.01.5

-0.91.31.43.40.0

0.10.0

-3.50.80.3

-0.81.32.60.0

Note: Δ values are determined as (calculated abundance - actual abundance).

Table 4. Normalization factors for Leg 156 bulk XRD samples, based on peak areas, determined with matrix singular value decomposition using nine

standards.

Indicator

mineral

SmectiteIlliteKaoliniteQuartzFeldsparCalcite

Smectite

2.6680 × 10"'7.3737 × 10"2

-1.4424 × 10"'-1.1867 × 10"2

9.2418 × 10~3

-1.1726 × 10~2

Illite

5.5772 ×1.1768 ×

7.5415 ×-6.5690 ×

2.1527 ×-1.6797 x

io-3

10"l O " 1

IO-4

io- 'io-3

Target mineral

Kaolinite

-2.4129 × IO-3

-7.7366 × IO-2

4.1043 × 10-'7.3326 × IO-3

8.0372 × IO-3

1.7838 × 10"3

Quartz

1.2937x10-'2.0973 × 10"'

-6.1388× 10"2

8.7705 × 10"2

-2.1873 × IO-2

2.3147 × IO"3

Feldspar

-4.3829 × 10"3

-8.1253 × IO"2

3.8812 × 10"2

-7.4016 × IO-3

2.0165 × 10"'5.5948 × IO-3

Calcite

2.5689 ×9.0305 ×

-4.2443 ×-1.4640 x-1.5821 ×

1.1432 ×

io-3

ur2i o -

io--10"'

37