PII S0016-7037(00)00447-6

Calibration of a reflectance FTIR method for determination of dissolved CO2 concentrationin rhyolitic glasses

GORDON MOORE,1,* A NDREW CHIZMESHYA,2 and PAUL F. MCMILLAN1,2

1Department of Chemistry and Biochemistry2Center for Solid State Science, Arizona State University, Tempe, AZ 85287, USA

(Received August17, 1999;accepted in revised form May3, 2000)

Abstract—A technique based upon infrared reflectance spectroscopy is developed as an alternative to theFourier transform infrared (FTIR) transmission method for the quantitative measurement of dissolvedmolecular carbon dioxide in aluminosilicate glasses. The technique has the advantage that only a single samplesurface need be polished, and no thickness measurement is necessary. The reflectance spectrum is analyzedby Kramers–Kronig relations or classical oscillator analysis to yield the optical constants and the absorptioncoefficient at 2350 cm21, due to the asymmetric stretching (n3) vibration of molecular CO2. The valueobtained is in excellent agreement with values obtained by the transmission FTIR technique for a suite ofrhyolitic glasses. For practical application of the method to rhyolites, an empirical correlation is developedbetween the normalized change in reflectance at 2350 cm21 and the CO2 content, up to;0.40wt%. Copyright © 2000 Elsevier Science Ltd

1. INTRODUCTION

Given the significance of volatile components (H2O, CO2,S-bearing species, etc.) in magmatic systems, measurement oftheir concentrations in quenched glass (and particularly, inglassy inclusions in minerals) has become increasingly impor-tant for igneous petrology (Ihinger et al., 1994; Lange, 1994).Fourier transform infrared (FTIR) spectrometry has become atechnique of choice to determine concentrations of dissolvedH2O and CO2 in silicate melts and melt (glass) inclusions.Although samples must be polished for analysis, the techniqueis “nondestructive,” and the sample remains intact for furtherexperiments or archiving. Our understanding of the role andbehaviour of H2O and CO2 in volcanic systems has recentlybeen greatly increased through the application of FTIR trans-mission spectroscopy to the measurement of OH- and CO-bearing volatile species dissolved in synthetic and naturalglasses (Dixon et al., 1995; Wallace et al., 1995). The practicalapplication of the transmission IR method has several disad-vantages however. Sample preparation is tedious and exacting,because samples must be polished on both sides to give flatparallel faces of the desired thickness. This is particularlychallenging for glass inclusions in minerals, and it means thatgenerally only a single inclusion within a particular mineralsample can be examined. For valid statistical analysis of pre-eruption volatile contents, it would be desirable to measurevolatile concentrations in multiple inclusions in a single min-eral grain. The error introduced into the analysis from mechan-ical thickness measurement can also be significant, as well aserrors due to estimation of the density of the volatile-containingglass, both of which are needed in order to report the concen-tration (Ihinger et al., 1994).

In order to provide an alternative technique that does notpresent these difficulties, we have developed and calibrated a

technique based on infrared reflectance spectroscopy (Hadni,1967; Efimov, 1995) to measure volatile contents in natural andsynthetic aluminosilicate glasses (Grzechnik et al., 1996). Herewe develop and demonstrate the application of the technique todissolved molecular CO2 in rhyolitic glass samples. In reflec-tance FTIR spectroscopy, the infrared beam is reflected from asingle polished surface of a glass sample, rather than one thatis passed through the sample, and thus eliminates the doublypolished plate sample preparation necessary for transmissionmeasurements. The reflectance method is therefore ideal for useon samples that have been prepared for electron and ion mi-croprobe analysis (after removal of any conductive Au or Ccoating by light polishing), and can be more conveniently usedin conjunction with these other analytical techniques. The re-quirement of a single polished surface significantly reducessample preparation time and success rate, and the method betterlends itself to making many measurements in a single thinsection. It also offers the opportunity to measure volatile con-centrations in multiple melt inclusions hosted within a singlecrystal, by successive polishing steps to various levels withinthe mineral sample, to intersect inclusions occurring at differentdepths.

The sample reflectance throughout the wavelength range ismeasured relative to that of a gold standard, which has areflection coefficient of unity (100% reflectance) over the IRregion measured in the experiment, and so the absolute value ofthe sample reflectivity is known with no further calibration.There is no need for a sample thickness measurement becausethe optical constants giving rise to the reflectivity function arederived from reflection at a single surface, at the sample–airinterface. In spectroscopic measurements of volatile contents,the concentrations are initially obtained in molar quantities(through knowledge of an extinction coefficient for a charac-teristic spectra feature of the species of interest, in L/mol-cm).These are then transformed into the usual wt% via knowledgeof the density (r in g/cm3) of the volatile-containing glass. Intransmission FTIR studies,r must be estimated for this pur-

* Author to whom correspondence should be addressed (gordon.moore@asu.edu).

Pergamon

Geochimica et Cosmochimica Acta, Vol. 64, No. 20, pp. 3571–3579, 2000Copyright © 2000 Elsevier Science LtdPrinted in the USA. All rights reserved

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3571

pose, using previously determined relationships from volatile-containing glasses and melts (Newman et al., 1986). In the caseof FTIR reflectivity, the real part of the refractive index in themid-near IR is determined as one of the optical constantsderived from the reflectivity, and this quantity scales (generallylinearly) with the density. Finally, unlike the FTIR transmissionmethod, there is no upper limit to the volatile concentration thatcan be determined, which is a consideration for species withstrongly IR active vibrations (Ihinger et al., 1994). These ad-vantages make the FTIR reflectivity technique an attractive onefor the systematic determination of volatile content in silicateglasses, and particularly in glass inclusions. In a previous study,we described the application of the technique to CO3

22 deter-mination in basaltic (basanite) glasses (Grzechnik et al., 1996).Here, we develop the application of the method to molecularCO2 in rhyolitic glass compositions.

Carbon dioxide is an important volatile in most magmaticsystems, and it dissolves in silicate melts as either carbonate(CO3

22) or molecular CO2 species, depending on the composi-tion of the melt. In general, the molecular CO2 to carbonatespecies ratio increases with the silica content of the melt:molecular CO2 is undetectable in basaltic compositions, andcarbonate is not found in rhyolitic glasses (Blank and Brooker,1994; Brooker et al., 1999; Fine and Stolper, 1985; Fine andStolper, 1986; Stolper et al., 1987). Since most natural meltinclusions found in rocks of andesite or higher bulk silicaconcentration are dacitic to rhyolitic in nature, they containsignificant amounts of molecular CO2, and only minor carbon-ate species, if any. Therefore, we present here a study of theapplicability of the IR reflectivity technique to the determina-tion of molecular CO2 dissolved in rhyolitic glasses. We de-scribe a formal analysis of the reflectance data via the Kra-mers–Kronig technique to extract optical constants, and inparticular, the absorption coefficient associated with the char-acteristicn3 (asymmetric stretching) vibration of the dissolvedCO2 molecules, for direct comparison with transmission FTIRdata. We also develop an empirically calibrated method for thepractical application of the technique, based upon a relationshipbetween the measured (normalized) change in reflectivity at theposition of the CO2 (n3) oscillator.

2. EXPERIMENTAL METHOD

2.1. Sample Preparation, Synthesis, and Characterization

Six samples with molecular CO2 contents ranging up to;4000 ppm(0.4 wt%) were examined in this study. Three of the samples studiedwere CO2-bearing rhyolites previously analyzed by Fogel and Ruther-ford (1990). These samples were synthesized at pressures and temper-atures between 150 and 550 MPa and 950 and 1150°C, and werecharacterized using transmission IR spectroscopy on doubly polishedsections ranging between 170 and 136mm in thickness. In that study,the CO2 contents were determined using an extinction coefficient forthe (n3) C 5 O stretching vibration (2350 cm-1) of 945 L/mol-cm,determined by Fine and Stolper (1985) for sodium aluminosilicateglasses. In the present work, the CO2 concentrations of these sameglasses were recalculated using the more recent value of 1066L/mol-cm established by J. Blank (unpublished; personal communica-tion) for CO2-bearing rhyolites.

Two further samples (PC-2 and PC-3) were synthesized for thisstudy using a non–end loaded piston cylinder assembly. The startingmaterial used for the high-pressure syntheses was a rhyolite powderfrom the Mono Craters area (CAM-73) (Carmichael, 1967) that wasintimately mixed with a carbonate powder that had an equivalent

Na:K:Ca ratio to the whole rock. Sufficient carbonate powder wasadded to give approximately 0.4 wt % CO2 during the course of the run.This rock/carbonate mixture was then welded into a platinum capsuleand run in the piston cylinder at 1 GPa and 1200°C for 6 h. Aliquots ofthe glasses were mounted in epoxy and doubly polished to;100 mmthickness for optical examination and IR transmission spectroscopy, fordetermination of the CO2 content by this technique. The molecular CO2

content was determined from the IR absorbance at 2350 cm21 using theBouguer–Beer–Lambert law and an extinction coefficient of 1066L/mol-cm (as above). The samples were found to be homogeneous inCO2 content. The density used to calculate the concentration was 2348g/L, the same as that used by Fogel and Rutherford (1990). No featuresdue to CO3

22 (bands at;1400 and;1500 cm21) were detected in thetransmission spectra.

The molecular CO2 contents determined in this way for these suitesof samples by IR transmission spectroscopy were used to establish ourcalibration curve for the IR reflectivity method described below.

2.2. Reflectance IR Technique

Reflectance IR spectra were collected between 500 and 4000 cm21

on polished glass samples using a Bio-Rad FTS-40 spectrometer witha Ge-coated KBr beamsplitter, a broadband Mercury-Cadmium-Tellu-rium (MCT) detector on a microscope attachment, and a Globar infra-red source. In principle, only singly polished thick samples should beused for this technique, to avoid optical effects due to multiple reflec-tions from both sides of the sample. Some interference effects were infact observed in some samples that had been doubly polished andpreviously studied by the FTIR transmission technique.

Dry air was continuously pumped into both the interferometer benchas well as the microscope attachment. This is important to reduce anyinterference from atmospheric CO2 with the spectrum being analyzed,and also to avoid degradation of the sample surface by CO2 (and H2O),which can be adsorbed during the measurement (a weak but observablesignal due to surface CO3

22 was observed for nominally volatile-freebasanite glasses studied by Grzechnik et al. (1996), due to reaction ofatmospheric CO2 with the sample surface). It was usually found to bebeneficial to cover the microscope attachment with a glove bag in orderto control the atmosphere around the sample. The size of the micro-scope aperture was approximately 200mm, with 1200 scans beingtaken for each spectrum at a spectral resolution of 4 cm21 (fixed by thenumber of points sampled for the Fourier transform analysis), and withan interferometer speed of 20 kHz. Each analysis set of scans (and FTconversion) took approximately 12 min under these conditions. Back-grounds were taken on a silica glass slide thick-coated with gold (100%reflectivity over the frequency region scanned) before each sample wasmeasured.

3. RESULTS

3.1. The Observed Reflectance Spectra

A representative reflectance spectrum of a rhyolite glasscontaining;0.40 wt% dissolved CO2 is shown in Figure 1.The spectrum is dominated by the strong band due to IR-activeSi-O stretching vibrations in the 1000–1100 cm21 region, withmaximum reflectivityR ;43%. The resonance due to theasymmetric stretching vibration (n3) of dissolved molecularCO2 in the glass is observed as a weak feature at 2350 cm21,superimposed on the slowly varying reflectivity function of thesilicate glass above the frequency range of first-order glassphonons. The reflectivity is related to the refractive index (n) ofthe silicate glass matrix through the Fresnel relation:

R 5 S un 2 1uun 1 1uD

2

(1)

The frequency dependence of the refractive index [n(n)] in this“high-frequency” range is mainly determined by the interactionof light with the electron density in the sample, far from

3572 G. Moore, A. Chizmeshya, and P. F. McMillan

vibrational or electronic resonances. At a given frequency,n isgenerally well-correlated with the glass density.

The feature due to molecular CO2 is small, but is clearlydistinguishable from the “background” provided by thesmoothly varying glass reflectivity in this spectral region. Theobjective of our study is to establish a methodology to calibratethe extent of the CO2 reflectivity feature that will permit usefulanalyses of CO2-bearing rhyolitic glasses via infrared reflec-tance spectroscopy. In Figure 2, changes in the reflectancespectra in the 2200–2500 cm21 region as a function of dis-solved CO2 concentration are shown. There is an obviousvariation in the amplitude of the CO2 resonance in the reflec-tivity function and the CO2 content. Due to the mathematicalnature of the reflectivity function (see below), the featureappears either as a “negative” peak or develops a “sigmoid”shape, depending upon the CO2 content. To establish an easilyused empirical relation between the contribution to the reflec-tivity curve from the CO2 resonance and the CO2 content, wehave chosen to measure the maximum amplitude (DR2350%) ofthe signal at the 2350 cm21 resonance (Fig. 3).The amplitudeof this function is directly dependent upon the CO2 content fora given glass refractive index (correlated with its density), and

details of the instrument sensitivity, alignment, etc. Because ofvariation in absolute reflectivity that may occur for a singleinstrument on a day-to-day basis, it is necessary to normalizethis value by a reflectivity taken close to, but distinct from, theregion of resonance. We have chosenR2200 for convenience.The DR2350/R2200 ratio (used in Eqn. 2) measured in ourlaboratory remained constant within 10% for measurementstaken on a given sample within our laboratory, over a period ofseveral months, although the absolute reflectivities measuredshow a much greater variation owing to the reasons describedabove. The results for the samples studied here are given inTable 1.

In Figure 4, we plot the averaged values of the normalizedreflectivity (DR2350/R2200)% for the rhyolitic samples againstwt% CO2 (previously determined with infrared transmissionspectroscopy) to establish an empirical correlation. A best-fitline to the data (R2 5 0.985;r.m.s. error5 0.022) is shown inthe figure, and this has the equation:

CO2~wt%! 5 6.21~0.39! zDR2350

R22002 0.055~0.029! (2)

Fig. 1. A representative FTIR reflectance spectrum from 500 to 4000 cm21 for rhyolite glass (sample CM73CO2-3)containing 3952 ppm dissolved CO2, determined from FTIR transmission measurements. Note the distinctive feature thatappears at 2350 cm21 due to C5 O oscillators, in a featureless part of the reflectivity function (determined by the“high-frequency” [i.e., above the range of the first-order glass phonons] optical constants of the silicate glass matrix). Thereis also a very weak feature just visible at;3500 cm21, indicating that this sample also contained a small concentration ofO-H oscillators, corresponding to;1 wt% dissolved H2O (see Moore and McMillan, in preparation).

3573Reflectance FTIR determination of CO2 in rhyolitic glasses

The standard errors of the fitted intercept and slope are given inparentheses, and the regression error corresponds to approxi-mately6250 ppm CO2. We would expect the regression errorto become considerably smaller, once more samples are in-cluded in the data set, as it is likely that part of the correlationerror results from the fact that the results were derived fromtwo different sets of samples. For practical applications, how-ever, the empirical correlation must be robust enough to with-stand many minor variations in sample bulk chemistry, prepa-ration conditions, and likely presence of other volatiles, even inminor concentrations (a calibration for CO2 in the presence ofH2O is under way: Moore et al., in preparation). It is interestingto note that the best-fit line does not intersect the origin,indicating that a measurable “dissolved CO2” signal is detectedeven when none is indicated by FTIR transmission spectros-copy. There are two contributing factors to this observation.First, the IR reflectance technique is sensitive to atmospheric

CO2 molecules adsorbed on the sample surface (which weknow occurs for silicate glasses: see Pandya et al, 1992),whereas the transmission FTIR technique, which probes mainlythe bulk, is not (this is a potential advantage of the reflectancetechnique). Second, the “signal” “measured” for the nominal0% sample might be due to the peak-to-peak “noise” variationin the reflectivity, for our set of recording parameters. Thiswould indicate that the “0% CO2” point should be discarded,and then theDR2350/R2200 vs. wt% CO2 correlation becomesquadratic (or higher order) rather than linear. These points willbe explored in further experimentation, and as the technique istested on more suites of samples.

3.2. Modeling of the Reflectance Spectra Using a SimpleVibrational Model: Detection Limit

In Figure 5 we show a reflectance spectrum for a typicalrhyolite sample measured in this study, as well as a calculatedspectrum using a simple vibrational model (Burns, 1990) thatuses two damped oscillators to model the resonances associatedwith Si-O and C-O stretching vibrations (Table 2). It can beseen in Figure 5 that the model reproduces the observed spec-trum well, in that the weak C-O oscillator has the correct shape,superimposed upon the strong Si-O reflectance signal. Thisapproach can be used to calculate a theoretical detection limitfor molecular CO2 in rhyolite, given the signal-to-noise ratioobtainable within a set of instrumental parameters. We havedetermined that using 1200 scans (;12 min/measurement) on arhyolite blank with nominally no CO2 (sample CO2-1 of Fogeland Rutherford, 1990) gives a representative peak-to-peak vari-ation of 60.015% reflectance in the 2350 cm21 region, afterthe sample interferogram is divided by that of the reference (Austandard) signal. This is consistent with the peak-to-peak vari-ations in the “100% transmission line” recorded for this type ofinstrument, due to fluctuations in source intensity, mechanicalstability of the instrument, control of the atmosphere surround-ing the sample during the measurement, and round-off error inthe computing methods employed for digital calculation of theFourier transform and to perform the referencing to the stan-dard. For the suite of samples investigated here, the peak-to-peak variation in reflectivity due to background fluctuationscorresponds to the magnitude of theDR% associated with then3 resonance for samples containing approximately 350 ppmCO2, implying that the signal from dissolved CO2 would not bedistinguishable from the noise level. This observation fixes aneffective detection limit, for these instrumental conditions. Re-corded CO2 contents in rhyolitic to dacitic glass inclusionsrange from 10–1000 ppm, so that the effective detection limitfor our chosen set of instrumental parameters already fallswithin the lower half of this range (Mt. Pinatubo rhyolitecontains 278–416 ppm CO2; Johnson et al., 1994). Ways toreduce this detection limit would include (a) longer countingtimes (a fourfold increase in counting time, to;48 min perspectrum, would result in a twofold improvement in detectionlimit, to ,175 ppm CO2); (b) increasing the FT sampling for aneffective resolution of 2 cm21 or better (this would renderchanges in the reflectivity function due to the CO2 resonancemore clearly distinguishable from background, for a givencounting time); (c) evacuation of the sample chamber; and (d)improvement of the temporal stability of the IR source, the

Fig. 2. Reflectance spectra of rhyolite glasses containing between 0and 3953 ppm dissolved CO2 in the 2200–2600 cm21, showing thegrowth of the reflectance due to the C5 O oscillator (then3 asym-metric stretching vibration of molecular CO2) as the CO2 concentrationis increased. The large period oscillatory features that can just bedistinguished in the baseline of the 841 and 1586 ppm samples occurbecause these samples were doubly polished specimens prepared for aprevious study (Fogel and Rutherford, 1990) and the features corre-spond to optical interference due to reflection from upper and lowerfaces of the sample.

3574 G. Moore, A. Chizmeshya, and P. F. McMillan

detector, and the FTIR interferometer itself. Varying some orall of these parameters could result in lowering the detectionlimit by a factor of at least 3–5 and perhaps by as much as10-fold, to bring detection of molecular CO2 through the FTIRreflectivity technique into the lower end of the range reportedfor glass inclusions. Given the great potential of the techniquefor sampling multiple inclusions within a given mineral grainor suite of grains, this is a worthwhile goal.

3.3. Kramers–Kronig Analysis of the Reflectance Spectraand Comparison with Transmission Experiments

In order to compare the absorption coefficients observed inprevious transmission IR calibrations of dissolved CO2 to ourreflectance data, we have used the Kramers–Kronig transformprocedure (Burns, 1990; Efimov, 1995) to calculate the opticalconstants (real and imaginary indices of refraction, n1 and n2,

Fig. 3. Detail of rhyolite reflectance spectrum (3952 ppm CO2) showing the scheme devised to generate an empiricalcalibration to correlate CO2 concentration with features in the reflectivity spectrum, without requiring detailed mathematicaldata analysis, and that is independent of changes in glass density and instrument parameters.

Table 1. Rhyolite CO2 concentrations and measured change in reflectivity at 2350 cm21.

Sample No. CO2 (ppm) D R/Rp SD N

CO2-1 0.0a 0.009 0.003 5152 841a 0.019 0.004 3149 1586a 0.038 0.002 2147 2533a 0.054 — 1CM73CO2-3 3952b 0.070 0.006 4CM73CO2-2 3871b 0.069 0.005 4

a CO2 concentration calculated from absorbance data of Fogel and Rutherford (1990) and using an extinction coefficient of 1066 L/mol-cm (seetext).

b This study.

3575Reflectance FTIR determination of CO2 in rhyolitic glasses

as functions of the frequency,n) of our samples. The real andimaginary parts of the frequency-dependent refractive index arerelated to the measured reflectance by the equations:

n1~n! 51 2 R~n!

1 1 R~n! 2 2ÎR~n! cosQ~n!(3a)

n2~n! 52 2ÎR~n! sinQ~n!

1 1 R~n! 2 2ÎR~n! cosQ(n)(3b)

where R(n) is the measured reflectance function,n is thefrequency (given in wavenumber units), and the loss functionQ(n) (phase angle) is given by the integral relation:

Q~n! 5 21

2p E0

`

log F n 1 np

un 2 npuG d log R~n!p)

dnpdnp (4)

Note that, in this expression,n is the frequency value at eachpoint considered, andn* is the integration variable. Withn1

andn2 thus determined, we can then calculate the absorptioncoefficient (a) as a function of frequency using the relation:

a~n! 5 4pnn2 (5)

Because the form of the inversion algorithm is sensitive to thesmoothness of the data (Efimov, 1995), we developed a proce-dure involving the following steps: (i) the high-resolution re-flectivity spectrum is first smoothed using the Savitsky–Golaymethod with a 19- to 24-point window and a second-orderpolynomial; (ii) the reflectivity in the range below the lowestmeasured frequency is approximated by assuming a value atzero frequency that corresponds to a linear extrapolation fromthe last measured point (inaccuracies in this low-frequencyregion do not affect the results in the 2350 cm21 range); (iii)these smoothed and modified spectral data are spline-interpo-lated and then used in the subsequent integration using Simp-son’s rule (Press et al., 1989). A similar procedure was used byBortz and French (1989) who used a fast Fourier transform(FFT) based algorithm.

The numerical procedure employed to calculate the Kra-mers–Kronig inversion is based on numerical integration ofEqns. 3a,b and 4 using Simpson’s rule. Although this approachis robust, it is less efficient than the FFT-based approach of

Fig. 4. A plot ofDR2350/R2200 versus dissolved carbon dioxide in wt% (determined by the FTIR transmission technique.The errors shown for the reflectivity ratio represent61 SD of multiple measurements as indicated in Table 1. The best-fit(R2 5 0.985) linear relationship to the data as given by Eqn. 2 is shown also.

3576 G. Moore, A. Chizmeshya, and P. F. McMillan

Fig. 5. (A) The dark dotted line shows a representative reflectivity spectrum measured for a CO2-bearing rhyolite, from500–4000 cm21. The spectrum is dominated by the strongly IR active Si-O oscillators (primary atn0 ; 1050 cm21) thatdetermine the form of the reflectivity function at “high” frequency along with the slowly varyingn(n) function of the glassmatrix. The dashed line is generated from a model that includes only two damped (anharmonic) oscillators to model“effective” Si-O and C5 O stretching vibrations (Burns, 1990). The model parameters are given in Table 2 and reproducethe essential features of the Si-O resonance and the slowly varyingR(n) function due to the glass matrix in the 2000–3000cm21 region. (B) Detail of 2350 cm21 region for the same model spectrum, compared with the experimental spectrum.

3577Reflectance FTIR determination of CO2 in rhyolitic glasses

Bortz and French (1989). All computations were performed ona Pentium-based PC. In order to implement the inversion pro-cedure, the experimental reflectance data were first artificiallyextended to span the entire frequency range (i.e., from 0 to4000 cm21). This extension of the data is necessary for theevaluation of the phase integral function, which has theoreticalintegration limits of zero and infinity [see Eqn. (4)], whereasthe reflectance spectrum is only measured over a finite fre-quency range (e.g., 600 to 4000 cm21). For the low-frequencypart, a linear function is used to approximate the reflectancebelow the lowest measured frequency. A zero frequency valueis assumed for the reflectance (typically 1.0), and a linearfunction is fit between it and the lowest frequency reflectancedata. The influence of the assumed value of the zero frequencyreflectance was tested, and it was found that it has a negligibleeffect on the transformation less than 100 wave numbers abovethe beginning of the measured data. Therefore the assumedlow-frequency limiting value does not affect the results in thefrequency region of interest (in this case;2350 cm21). Thehigh-frequency limit is simply taken to be the last measuredfrequency, and this does not appear to affect the results in thehigh-frequency region in any way. The resulting extendedreflectance function is smoothed with a Savitsky–Golay proce-dure using a second-order polynomial and a 19- to 24-pointwindow. This latter step is necessary in order to eliminate theoccurrence of transients and numerical instabilities in the finaltransformed functions. Finally, the extended, smoothed reflec-tance function is interpolated onto an internal integration gridusing a cubic-spline approximation. Full numerical conver-gence (stability) with respect to the quantitiesn1(n) andn2(n)was achieved with a uniform grid of 2000 points.

Once calculated using the Kramers–Kronig relations, theabsorption coefficient obtained from Eqn. 5 can be directlycompared to the empirically determined transmission molarabsorptivity («) of the Beer–Lambert law (the transmissionextinction coefficient) using the expression:

a 51

2.303

r

M«C (6)

where r is the density of the glass, andM and C are themolecular weight and concentration of the absorbing speciesbeing measured. The random orientation of the dipoles in theglass sample must be taken into account (Paterson, 1982) andin so doing the extinction coefficient for the transmissionexperiment is multiplied by a factor of 3 to be compared withthe values normally reported in the literature (Ihinger et al.,1994). It must also be multiplied by a further factor of 2 to

account for the phase change that occurs upon reflection, com-pared with the transmission experiment (Grzechnik et al.,1996). Absorption coefficients calculated in this way for themolecular CO2 resonance for our sample spectra are given inTable 3, and are plotted in Figure 6 versus concentration ofcarbon dioxide. By taking the slope of the regression line anddividing by the constants in Eqn. 6, we obtain a transmissionextinction coefficient («) to be 10106 60 L/mol-cm. The errorgiven here is only the standard error of the fit, and does notinclude any errors in the estimates of the constants. This valueis in excellent agreement with the previously determined coef-ficient of J. Blank (personal communication) of 1066 L/mol-cm. There is some circularity in the argument, however, be-cause the wt% CO2 is derived from FTIR transmissionassuming this« value: it does show, however, the consistency

Table 2. Model oscillator parameters.

Oscillator «` n0 (cm21)Si

(3103)a gi

2.53 — — —Si-O 1050 750 90C-O 2350 1.35 10

a Note thatSi (oscillator strength) varies with concentration for agiven oscillator. The values ofSi given here are for the model curve inFigure 5.

Table 3. Absorption coefficients at;2350 cm21 calculated usingKramers–Kronig inversion of reflectance data.

Spectra No. a (KK)

CO2-3-1 581CO2-3-2 598CO2-3-3 606CO2-2-5 548CO2-2-7 525frCO2-152-2 150frCO2-152-1 95frCO2-147-1 400frCO2-149-1 228frCO2-149-2 261frCO2-149-3 309

Fig. 6. Absorption coefficients (a) for the CO2 resonance at 2350cm21 calculated from the reflectance spectra for CO2-bearing rhyolites,following Kramers–Kronig analysis of the reflectivity spectra, versusCO2 concentration in wt%. A linear best-fit line (R2 5 0.99) isshown,which has a slope of 14006 80. The resulting molar extinctioncoefficient («) is 10106 60 L/mol-cm, in close agreement with thevalue (1066 L/mol-cm) derived from transmission studies (J. Blank,personal communication).

3578 G. Moore, A. Chizmeshya, and P. F. McMillan

between FTIR transmission and reflectivity determination ofvolatile contents.

4. CONCLUSIONS

We have shown that the reflectance micro-FTIR techniquecan be used for quantitative measurements of the dissolvedmolecular carbon dioxide concentration in rhyolitic glasses,using the change in reflectivity at 2350 cm21 due to then3

resonance of dissolved CO2. The detection limit of the tech-nique for our present set of instrumental parameters is approx-imately 350 ppm. Kramers–Kronig inversion of the reflectancedata gives an absorbance coefficient that is in excellent agree-ment with that obtained by transmission FTIR measurements.Quantitative analysis does not require such extensive datatreatment. We have developed a simple empirical linear cali-bration of change in reflectance (DR2350%/R2000) at the res-onance frequency (Eqn. 2) that is determined to have a preci-sion of 6250 ppm (regression error), with our present limiteddata set. Scaling the reflectivity to a value measured away fromthe resonance (R2000) accounts for fluctuations in operatingparameters and changes in glass density, and should permitinterlaboratory comparisons of the technique. The reflectivitymethod is easy to implement (it requires only a single polishingstep), and multiple glass inclusions can be examined succes-sively. Unlike the FTIR transmission technique, there is noneed for sample thickness measurement, and errors introducedinto the volatile determination from the thickness measurementare avoided (Ihinger et al., 1994). For strongly absorbing spe-cies (e.g., the asymmetric stretching vibration of CO3

22), thevolatile species can be examined to arbitrarily high concentra-tions (Grzechnik et al., 1996). Therefore, the reflectance IRtechnique opens up the possibility of easily quantifying IR-active volatiles in multiple melt inclusions hosted in singlecrystals, as well as allowing volatile measurement on singlypolished samples prepared for electron and ion microprobeanalysis.

Acknowledgments—This work was supported by a NSF Earth SciencePostdoctoral Fellowship to Gordon Moore, as well as NSF grantEAR-9614229 to P. McMillan. We also thank Drs. Fogel and Ruther-ford for permitting use of their samples, and Dr. Hervig for makingthem accessible to us for this calibration.

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Bortz M. L. and French R. H. (1989) Quantitative, FFT-based, Kra-mers–Kronig analysis for reflectance data.Appl. Spectroscopy43,1498–1501.

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3579Reflectance FTIR determination of CO2 in rhyolitic glasses