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Anal. Chem. 1982, 5 4 , 1778-1782
Empirical Evaluation of the Use of Moments in Describing Fluorescence Spectra Michael G. Mulkerrin and John E. Warnpier" Bioluminescence Laboratory, Department of Biochemistry, University of Georgia, Athens, Georgia 30602
Moments as a dlstrlbution-free method for the classlflcation of curves were developed in the late elghteen hundreds. I n our laboratory we have demonstrated the use of moments in describing spectral and klnetlc profiles. Other laboratorles are applying this technique to experlmental data as a method of unlquely definlng the underlying curve in analysis of chromatograms, In characterlratlon of saturation functions, and In computer retrlevai of spectra. With the ald of real-time, online data processing and utliizlng fluorescence spectrometry we have evaluated the spectral moments In a rigorous investlgatlon of their power and llmltatlons for identlflcation, characterlratlon, and measurement control. The results show that (1) these moments can serve as unlque fingerprlnt for a spectral curve, (2) the maxlmum standard deviation for repetitive measurement for any of the four moments under our Instrumental and experimental condltlons Is 1.2%, (3) the moments are stable to changes In spectrometer band-pass from 2 to 20 nm, (4) they are stable to variatlons In concentratlon providing reabsorptlon Is avoided, and (5) slgnificant varlatlons In the moments occur when spectra are altered experlmentaliy or numerically.
The use of moments as a distribution-free method for identifying and classifying functions was first developed by Karl Pearson in the late eighteen hundreds in his early papers on Mathematical Contributions to the Theory of Evolution ( I ) . Work in the use of moments was further developed by Elderton (2). Since then, the use of moments per se has received little attention due to the development of other, more efficient, but distribution-dependent statistical tests. However, when the distribution function is not known a priori or it is composed of multiple complex components, the description of the underlying function by the moments of the curve can still be very useful. In chromatography, such uses have been coupled to analysis of chromatographic models (3), optimization of separation ( 4 ) ,and detection of unresolved peaks (5,6). Similarly, Boyanhames and Contraine (7) have detailed the use of moments in analysis of data which follow the Langmuir isotherm. In spectrometry, brief studies have shown that the moments of a spectrum can be used for classification and identification (8) and for computer retrieval of IR data (9). We have also suggested their potential use in measurement validation in analysis (IO). To this date, though, none of these studies have rigorously analyzed the power of this method when applied to spectral data with experimental variations in noise and spectral resolution. This study is an investigation of the specificity, reproducibility, and sensitivity of the spectral moments as tools in identifying, validating, and describing fluorescence spectra. EXPERIMENTAL PROCEDURES Numerical Procedures. Calculation of the moments of a spectrum is a straightforward process, easily and quickly carried
out with a modest computer or programmable calculator. When the data consist of discrete measurement values, y , equally spaced on the measurement axis, x , the moments are calculated and described by the general equation
where the denominator term normalizes the spectral curve to unit area. In classification of distribution functions, the first four moments are sufficient to define the shape of the distribution. Similarly, for applications in spectrometry and chromatography, only four moments are generally needed. Moments generated by eq 1 are dependent on the absolute magnitude of the units of the x axis. Since one of the purposes of their use is to compare the shapes of spectra, the higher order moments are generally calculated in relation to the first moment, the mean position, in order to remove this effect of absolute x position. These reduced moments (also called central moments by some authors) are calculated according to eq 2, where a1 is the mean x position, E ,
also referred to as the center of moment or the central tendency. The second reduced moment is nothing more than the variance of a distribution function, the square of the standard deviation. U2is a measure of the dispersion of a spectrum or chromatographic peak. The common symbol for this moment is, of course, u2. The third and fourth reduced moments also describe the shape of the curve. U3reflects the degree and direction of asymmetry. U4 is a measure of peakedness or flatness. The reduced moments still have a magnitude dependent upon the absolute range of the measurement axis. Typically, the third and fourth moments, U3and U4,are further normalized relative to the dispersion, U2,in order to remove this dependence. This defines two additional parameters, the skewness, S, and the kurtosis, K . s = U3/(u2)'I2 (3) K = U4/(U2)2-3 (4) The difference in eq 4 is used to calculate the kurtosis relative to the normal distribution (where U 4 ( U 2 ) 2= 3) so that the value is less than zero when a distribution is sharper (more peaked) and greater than zero when it is flatter (less peaked) than the normal distribution. For time efficiency in calculation, rather than two "passes" through the data to calculate the moments and the shape parameters using eq 1through 4, one "pass" can be used where each of the following sums is calculated: CY C X Y C X 2 Y C X 3 Y C X 4 Y (5) Then each of the moments of interest is calculated by eq 6 through 9; the skewnessand kurtosis calculated by using eq 3 and 4.
x =CXY/CY
(6) (7)
u 3 =C-X-3 Y CY
3-cx2yx
CY
+
2x3
CX4Y CX3Y C X 2 Y X 2 - 3x4 u,= - -4 X+6CY CY CY
0003-2700/82/0354-1778$01.25/00 1982 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 54,NO. 11, SEPTEMBER 1982
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Table I. The Reproducibility of the Calculated Moments of Spectra dye (no.)“
mean
DPA ( 1 0 )
430.52
RHD (10)
580.56
i. 0.05 i. 0.06 i. 0.5
u,(~10-5) 0.709 i. 0.003 0.946 i. 0.002 1.38 i. 0.02
u,(x10-7)
u,
1.94 i 0.02 1.62 f 0.01 3.07 ~t 0.03
2.19 k 0.02 3.33 t 0.01 5.6 * 0.1
(xio-10)
S
1.033 f 0.005 0.556 t 0.002 0.60 f 0.01
K
1.36 * 0.02 0.714 k 0.005
-0.04 i. 0.01 APT (9) 499.4 “ DPA = diphenylanthracene, R H D = rhodamine B, APT = 3-aminophthalamide, all dyes in ethanol with absorbance below 0.1. Numbers in parentheses indicate the number of different spectra taken. For APT each of the nine spectra were taken using a different concentration covering a 5-fold range.
This procedure must be carefully done, since eq 6 through 9 involve small differences in large numbers. Therefore, the sums in eq 5 must be calculated with high precision. If this is not possible, then eq 1 through 4 should be used. Instrumentation. For this work spectra are taken using an on-line spectrofluorimeter designed in this labroatory. The hardware components of this instrument are described by Wampler (11). The instrument utilizes a SPEX Doublemate double-grating monochromator (SPEX Industries), with a 150-W xenon lamp as an excitation source, and employs a SPEX Minimate single-gratingmonochromator with a cooled photomultiplier (EM19558QB) for the anabysis of emission. A. computer-controlled mechanical shutter has been included in the excitation optical path in order to acquire data with a precise measurement of the base line. The spectral sensitivity of this instrument is corrected by numerical processing as previously described (12). This instrument is operated by software developed in this laboratory using SPECOS (Spectroscopy Operating System, Wampler, unpublished). The major features of this software have been previously describedl (IO). Data Collection. Instrumental parameters, digitization intervals, and numerical processing during data collection were performed with the procedlures and guidelines described by Wilson and Edwards (13). As previously described (14),each datum in each spectrum represents a very large number of individual readings from the A/D converter. During an individual scan, at each wavelength position, the photomultiplier signal is read 100 times. This 100 sample average is in turn repeated a selected number of times (10 times, except where noted), so that each point in a scan is the average of 1 OOO A/D readings. The whole process takes about 0.1 s per datum. All spectra reported here are also scan averaged 10 times except as noted in the figure legends. The time for acquiring a 10-scan averaged spectrum is 5.1 min under the measurement conditiions employed. Materials. The dyes wed were purchased from the following vendors: Sigma, J. T. Baker, Eastman Kotlak, K and K Laboratories, Allied Chemical and Dye. They were used without further purification and dissolvedl in spectrograde solvents. RESULTS AND DISCUSSION Figure 1 shows a series of dyes and their moments. This figure demonstrates that for a set of spectra the four moments uniquely define each dye. The mean, increasing from left to right coupled with the three reduced moments, the skewness, and the kurtosis, demonstrate the capacity of these parameters to act as a “fingerprint” for each of these dyes. Note, that with these 10 dyes even if the mean position is ignored, the shape defined by the reduced moments is still unique for each dye. Thus 10 dyes emil,i,ing in the same wavelength region would still be expected to be easily distinguished from each other by their moments. Table I demonstrates the repeatability and stability of the moments and shape parameters for three different dyes. Their insensitivity to noise is illustrated by the data of Figure 2. Since calculation of the moments involves integration of the data set, they are rather noise immune when compared to parameters normally used for describing spectra such as peak positions and half-widths. The higher order moments, Us and U,, are more senciitive to noise than are the mean and the variance, since each isuccessive power of the difference, (x - x),“amplifies” the effects on noise at the extremes of the spectrum. Even so, the (data in Table I and Figure 2 show that the third and fourth moments of these spectra are stable
T
n
m
n
Figure 1. The moments of each of 10 fluorescent dyes. Data were normalized so that the largest value in each case Is 100% and arranged in order of increasing means. The abbreviations are QUI = quinine sulfate, APZ = aminopyrazine, ANA = analine blue, LUM = luminol, APT = 3-aminophthtalmide,ACD = acridine yellow, COR = coriphosphene 0, EOS = eosine Y, RHD = rhodamine B, MEB = methylene blue. The parameters for each dye are plotted in order: mean, second, third, and fourth reduced moments, skewness, and kurtosis. Hashed bars Indicate negative values.
WAVELENOTW (nm)
510
Increase in the signal to noise ratio of a multimodal spectrum, the spectrum of diphenylanthracene in ethanol wRh signal to noise improvement via scan averaging: A = 1 scan, B = 4 scans, C = 9 scans, D = 16 scans, and E = 36 scans. The insert gives the moments of the spectra with U , X and U, X lO-’O. U, X Flgure 2.
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
Table 11. The Effect of Smoothing fwh h/SRa mean 2.1 1.0
580.5 580.5 580.6
0.5
580.0
u,( ~ 1 0 - 4 ) u,(xio-7) 9.51 9.50 9.50 9.81
u,
(xio-10)
1.59 1.60 1.64 1.56
3.37 3.35 3.34 3.43
S
K
0.542 0.548 0.560 0.503
0.720 0.714 0.704 0.564
a Ratio of the full width at half-height (fwhh) of the spectrum to the smoothing range (SR) where oversmoothing occurs with a ratio of atmroximatelv 1.4 (see Wilson and Edwards. ref 13).
within a few percent. Similar results were obtained when the experiment of Figure 2 was repeated with both multimodal and unimodal spectra. This empirical variation is slightly less than the variation predicted by the theoretical studies of Chelser and Cram (15) and Petitclerc and Guiochon (16) in their studies of the effects of noise on the moments of chromatographic peaks. There are two possible reasons for this difference. First, even the most asymmetric fluorescence spectrum is fairly symmetrical as compared to the type I1 chromatographic peak of Cheder and Cram (15). It was with these very asymmetric peaks that they observed the largest variation of the higher order moments with noise. Secondly, the synthetic noise applied to the curves in these studies was signal independent whereas measurement noise with our instrument system is primarily signal dependent. The result of both of these effects is that the third and fourth moments are, in fact, fairly noise immune. The shape parameters are more sensitive to noise since they combine the effects of noise on two of the moments. In the case of the kurtosis, the difference calculated by eq 4 “amplifies”the error further. In spite of this the actual kurtosis values of the ten dyes of Figure 1 remain uniquely representative of each dye. In the case of 3-aminophthalamide (Table I) the nine spectra which are used to calculate the variation of the momenta each involve a different solution containing a different concentration. Here, though a 5-fold difference in absolute signal is seen in the various spectra, there is little variation in the moments. Again, the moments clearly identify the spectrum providing there is no physical or chemical change due to dilution of the dye. Changing the slit width on the emission monochromator has little effect on the momenta of a spectrum even when the spectrum is multimodal (Figure 3). Even when the slit width is too large to resolve the structure, the moments still identify the fluorophore correctly. In Figure 3, though the appearance, signal strength, and noise levels of each spectrum are progressively changing, the moments remain quite constant. An analogous procedure to the experimental variation of slit width is the numerical process of smoothing via convolution (13). Smoothing involves convolution of a symmetrical function and, when applied to spectral data, can change the shape and appearance of a curve. Typically, a full width at half-height (fwhh) to smoothing range (SR) ratio of less than 1.4 is considered oversmoothing, and the spectral shape will be significantly altered (13). In Table I1 a curve smoothed using a (fwhh/SR) ratio of 2.1 shows no significant difference in the moments over unsmoothed data. When the ratio is decreased to 1.0 there is still no appreciable difference, although the spectrum is significantly oversmoothed. However, when the ratio is reduced to 0.5, then the momenta begin to show the effects of drastic oversmoothing. The spectra of Tables I and I1 and of Figures 1through 3 show the stability of the moments of individual fluorophores with respect to variations in intensity, signal-to-noise ratio, and resolution. However, to be useful the momenta must also be sensitive to real spectral changes or instrumental artifacts, and indeed they are as the following data shows. Figure 4 shows a spectrum of rhodamine B which has been numerically altered in various ways. When the alterations
M A 430 E 430 C 429 0 430 E 430
370
WAVELENOTH ( n n )
0’
UJ
U4
090 190 2 1 0 7 00 I95 2 13 7 0 2 1 95 2 10 707 193 210 7 4 7 189 2 4 0
S
K
I O 1 140 1 0 5 1 30 1 0 5 I38 103 133 008 131
)IO
Figure 3. The effect of increasing slit width on the moments of diphenylanthracene, the slit width increasing from top to bottom: A = 0.25 mm, B = 0.5 mm, C = 1.25 mm, D = 2.5 mm, and E = 5.0 mm. U3 X The insert gives the moments of the spectra with U , X IO-’, and U 4 X lO-‘O.
change the shape (curves B through D), even if the change is simply addition of a small offset, there is a pronounced difference in the moments with a relativly minor difference in the appearance of the curve. By inspection of the third and fourth moments and the shape parameters, the nature of the change in the original spectrum can be analyzed. On the other hand, in curve E where random noise has been added, there is again a negligible change in the moments. In Figure 5 the spectrum of coriphosphene 0 is shown along with the spectrum of coriphosphene 0 when contaminated with acridine yellow. Though these two spectra are quite similar,the contamination causes a slight blue shift and change in asymmetry. These changes are easily detected via the moments. The mean is lower, indicating the blue shift. The second moment is larger, indicating the addition of acridine yellow as adding to the breadth of the spectrum. The third moment is larger, though one might initially expect it to be less. However, since the third moment is sensitive to the tails of the spectrum, ( x - x), and there is very little short wavelength tail, the additions to the long wavelength tail make a
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
480
680
Flgure 4. The effect of numerical perturbations on the spectrum of rhodamine B: A = normal spectrum, B = spectrum of A with exponentially increasing base llne (maximum added value 3.6% of the peak of A) added, C = spectrum (of A with exponentially decreasing base llne added (maximum added ivalue 5 % of the peak of A), D = spectrum of A with constant offset added (1 % of peak), and E = spectrum of A with added noise. The inwrt gives the moments of each spectrum U, X lo-’, and U 4 X 1Oi0. with the U , X M 5410 5403
450
5 I U 1 Ug U, 1.30 3.01 5.11 M2 ,077 1.32 3.06 5.3’1 ,636 ,085
550 WAVELENGTH (nm)
650
Flgure 5. A comparison of the spectrum of coriphosphene 0 (curve A) with the same sample contaminated by acridine yellow (curve B). The insert gives the moments of the spectra with U, X lo-’, U 3 X lo-’, and U, X lo-‘’.
greater numerical contrilbution. The skewness parameter, which is normalized to the variance, shows that the shape becomes more symmetric,al. The fourth moment, again very sensitive to the tails, shows the spectrum to be flatter as does the kurtosis. When this procedure was reversed with coriphosphene 0 added to acridine yellow, the moments easily revealed the addition is t o the longer wavelength side of the spectrum.
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In the case of the purturbations of Figures 4 and 5, though there is very little difference in the curves upon visual inspection, there are statistically significant and quantitative differences in the moments which can be easily seen. This study indicates that the moments of fluorescence spectra are well suited for use in characterization of spectra either for the purpose of verification and validation or as library search parameters. They are much more immune to noise, to changes in absolute amplitude, to variations in resolution, or to smoothing than are the typically used parameters (peak and minima positions, peak ratios, half-widths, etc.). They can be calculated more quickly and with more modest computing power than can Fourier terms. I t is a calculation which can, in fact be carried out “on the fly” in real time. An additional advantage, particularly as regards library searching, is that regardless of whether a spectrum is structured or unimodal, four and only four parameters are needed to characterize it. Nevertheless they do have the required sensitivity to perturbations which cause real changes in the shape of the spectrum such as the presence of impurities, base line errors, or changes in the wavelength-dependent sensitivity profile of the instrument. The major disadvantage which could limit the use of the spectral moments in analysis of fluorescence spectra is the sensitivity of the higher order moments (the third and fourth) to small base line errors and subtle changes at the extremes of a spectrum. Thus if the scan range includes the tail of the excitation peak or second-order stray light, variations in these components will change the values of the moments. Similarly, base line drift can have a significant effect. However, as demonstrated by this study, good instrumentation and careful choices of scan range and exciting wavelengths can overcome these problems. The data acquisition algorithms of our fluorimeter system (11)give accurate and reproducible scan positioning and base line correction making possible the high reproducibility of the higher order moments reported here. A common problem in all fluorescence spectrometry is the correction of spectra for the wavelength sensitivity profile of the instrument. A variety of procedures have been discussed for determining the correction factors (17). Common procedure is to rely on secondary fluorescence standards as calibration checks and for interlaboratory comparisons. The moments of the corrected spectra of these fluorescent standards could be widely used both to verify that proper correction factors have been evaluated and as an easy way to communicate essential information about a spectral standard or the accurate correction capabilities of an instrument system. A simple statement of the moments of the corrected spectrum of the standard would relay a complete description of the calibration of the instrument and could be easily checked by any laboratory. Good agreement of these values with those from another laboratory would verify that the wavelength sensitivity correction procedures of the two labs are consistent over the entire spectral range of the standard fluorophore.
ACKNOWLEDGMENT The authors wish to thank Clark H. Groover for his technical assistance with this work and Edwin S. Rich, Jr., MEE, for his assistance and design input with the fluorimeter system. LITERATURE CITED (1) Pearson, K. “Karl Pearsons Early Statistical Papers”; Cambridge Unlversity Press: London, 1948. (2) Elderton, D. “Frequency Curves and Correlation”; Cambridse Universlty Press: Cambridge, England, 1938. (3) Mott, S.D.; Grushka, E. J. Chromafogr. 1978, 148, 305-320. (4) Carbonell. R. G.: McCov. B. J. Chem. €no. J. 1975. 9. 115-124. (5) Oberholtzer, J. E.; Rogek, L. 6 . Anal. Cheh. 1969, 4 1 , 1234-1240. (6) Grushka, E.; Myers, M. N.; Giddlngs, J. G. Anal. Cbem. 1970, 42, 21-26. (7) Boyanhames, J. M.; Contrane, F. R. L. Anal. Blochem. 1978, 91, 32-45.
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Anal. Chem. 1082, 5 4 , 1782-1786
(8) Wampler, J. E. Amerlcan Society for Photobiology, Program and Ab-
(9)
(IO) (11) (12)
(13)
stracts, 1977, p 75. Tamura, T.; Tenabe, K.; Hlrashl, J.; Saeki, S. Bunsekl Kagaku 1979, 28, 591-595. Wampler, J. E.; Mulkerrin, M. G.; Rlch, E. S. Clln. Chem. (WinstonSalem, N . C . ) 1979, 25, 1628-1834. Wampler, J. E. "Modern Fluorescent SPectroscopy"; . . Plenum: New York,. 1976; Chapter 1. Wampler, J. E. "Blolumlnescence in Action"; Academlc Press: London, 1978; Chapter 1. Wilson, P. E.; Edwards, T. H. Appl. Spectrosc. Rev. 1976, 72, 1-81.
(14) (15) (18) (17)
Wampler, J. E.; DeSa, R. J. Appl. Spectrosc. 1971, 25, 823-827. Chesler, S. N.; Cram, S. P. Anal. Chem. 1971, 43, 1922-1933. Petitclerc, T.; Guiochon, G. Chromatographla 1975, 8, 185-192. Demas, J. N.; Crosby, G. A. J. Phys. Chem. 1971, 75, 991-1024.
RECEIVED for review January 28,1982. Accepted June 4,1982. This work was supported in part by the National Science Foundation (Grant PCM 8012433).
Analysis of Steels by Energy Dispersive X-ray Fluorescence with Fundamental Parameters Kirk K. Nielson" Rogers & Associates Engineering Corporation, P.O. Box 330, Salt Lake City, Utah 841 10
Ronald W. Sanders and John C. Evans Pacific Northwest Laboratory, Richland, Washington 99352
A simple fundamental parameters method was developed for energy dlsperslve X-ray fluorescence analysls of nonradloactlve and radloactlve steels. The method utilizes a thinfilm multlelement Calibration of the spectrometer, wlth mathematlcal matrix correctlons for self-absorptlon and enhancement. The method allows direct analysls of steels of varylng physical conflguratlon and composltlon with hlgh accuracy and preclslon and does not requlre specialized standards or sample preparation. Preclslons are dominated by countlng statistics, and slgnlflcant relative errors due to sample form averaged 3.8%. Analyses of radloactlve steels were accomplished by subtractlng the radlonucllde spectrum from the energy dlsperslve X-ray fluorescence spectrum before spectral analysls.
A fundamental parameter method has been developed for energy dispersive X-ray fluorescence (EDXRF) analysis of steels without reference to standards of similar thickness, composition, or physical form. The method is based on a spectrometer calibration from thin-film standards, and on mathematical corrections for matrix self-absorption and enhancement effects. The excitation and detection conditions of the EDXRF spectrometer are suitable to detect nearly all significant constituents of the steels simultaneously, and thereby provide the basis for accurate matrix corrections. The method is validated by analyses of three National Bureau of Standards steels in various physical configurations to demonstrate the insensitivity of the method to sample configuration and to illustrate its precision and accuracy. One important application of the method is in the analysis of radioactive steel from the internal structures of a nuclear power reactor. The determination of niobium in the steel is of particular importance for the eventual decommissioning of the reactor. This is due to the neutron activation of niobium to "b (20000 year half-life),an important source of long-term radioactivity. Niobium is difficult to determine by many analytical techniques; however, EDXRF provides an inexpensive and reliable means for simultaneous determination of Nb along with the other constituents. The method reported
here is particularly applicable to the analysis of radioactive metals because it minimizes sample handling and preparation and permits subtraction of the radionuclide contributions prior to spectral analysis. The application of X-ray fluorescence for steel analysis is well established and has traditionally and competently relied on empirical matrix corrections such as those of Rasberry and Heinrich ( I ) , Lucas-Tooth and Pyne (2),Beattie and Brissey (3), LaChance and Trail1 (4), or Claisse and Quintin (5). Although these methods provide accurate results for steel analyses, the calibrations and requirements for standards for many of them are time-consuming and costly. Fundamental parameter methods avoid elaborate multielement empirical calibrations for steels and are suitable over virtually any range of concentrations of any of the observed elements. Equations for these methods have been developed over the past few decades (6-B), and comprehensive computer programs such as that of Criss and Birks (9) have been developed to implement them for both wavelength and energy dispersive X-ray fluorescence. Particularly simple applications of fundamental parameter methods have been reported (IO) for EDXRF spectrometers using monochromatic or secondary source excitation. The steel analysis method presented here is included as an option of a more general fundamental parameters program, SAP3 ( I I ) , which was designed for monochromatic and secondary source excitation. SAPB normally estimates light element concentrations (carbon, oxygen, etc.) from backscatter peaks (12)and also allows the user to enter known concentrations of analyte elements not observed in the spectrum. However, in the analysis of steels, the incident radiation provided by Ag Ka,P X-rays (22.104, 24.987 keV) provides simultaneous excitation and measurements of virtually all contributing constituents. Therefore, only observed elements were utilized for matrix corrections with the exception of the known silicon concentrations in two of the NBS steels. If the silicon were in greater concentration or of greater interest, it could be determined by using a lower atomic number fluorescer such as titanium, and could utilize matrix corrections from the Ag-excited elements with appropriate modifications of the present procedure. The general SAPB program also has
0003-2700/82/0354-1782$01.25/00 1982 American Chemlcal Society
