On the Determination of Carbon, Phosphorus, Sulfur and Silicon in Grey Cast Irons Using Glow Discharge Optical Emission Spectrometry MICHAEL R. WINCHESTER*a AND CHARLES MAULb aAnalytical Chemistry Division, Chemical Science and T echnology L aboratory, National Institute of Standards and T echnology, Gaithersburg, MD 20899, USA bL ECO Corporation, 3000 L akeview Avenue, St. Joseph,MI 49085, USA Glow discharge optical emission spectrometry is used to with the spark. When graphitic C is present, P and Si are usually aVected as well.This is why the metalloid, Si, was determine C, P, S and Si in grey cast irons. White cast irons included in the set of analytes. are used as calibrants, and data are acquired while sputtering In a recent paper,Weiss discussed the grey cast iron problem is non-stoichiometric. Type standardization is employed to at length and elegantly demonstrated the potential of GD-OES correct the resulting matrix eVect. For comparison, analytical to alleviate fully these matrix eVects.4 The success of GD-OES mass fractions of P, S and Si are also calculated from the in this regard depends on the use of a judiciously chosen data while omitting the matrix eVect correction.A statistical presputtering routine that assures that stoichiometric sputter- evaluation of the data shows that while all except one of the ing5 is established prior to data acquisition. Using such a experimental mass fractions, computed with or without type routine, Weiss was able to cause both grey (graphitic) and standardization, are in agreement with corresponding certified white (non-graphitic) cast irons to fall along the same cali- values at the P=0.05 level of significance, the use of the type bration line for each element investigated.It was shown that standard is indispensible for the unbiased determination of C if data are acquired before sputtering becomes stoichiometric, and S and may be helpful in the case of P.In contrast, the white and grey cast irons may tend to fall along diVerent data indicate that type standardization may in some cases calibration lines. This is especially true for C, owing to the introduce analytical bias into the determination of Si. relatively low sputtering rate of graphite. No analytical deter- Keywords: Analytical bias; cast iron; glow discharge optical minations were performed in that particular work. emission spectrometry; metalloids; non-metallic elements The temporal behavior of emission intensities during the approach to stoichiometric sputtering is highly predictable within a given matrix.By making use of this fact, it is possible Quantitative determination of non-metallic elements in solids to analyse samples of one matrix with calibration equations is an important analytical problem in a wide range of technol- developed using calibrants of a diVerent matrix while ogies. As an example of its importance, manufacturers of jet employing a presputtering routine that does not allow stoichioengine turbine blades have learned that sub-mg kg-1 mass metric sputtering to be attained prior to the acquisition of fractions of S in the alloys used to make the blades can lead data.For example, grey cast irons may be analysed using to engine failure. Similar findings are likely to be forthcoming calibration equations based on white cast irons only. In this for P as well.1 The successful control of S and P necessitates approach, one or more grey cast iron reference materials are the use of reliable, quantitative analytical methods. Given the used to determine correction factors that can be applied to the widespread need to quantify non-metals in solids, it is surpris- analytical mass fractions for the unknown samples.This ing that this analytical problem has historically been somewhat method of correcting the matrix eVect after data acquisition is ignored by the analytical community.Although methods exist known as ‘type standardization’. for the determination of these elements, there is a need for The studies discussed in the present paper build upon the better methods to be developed. work reported by Weiss. They were conducted at the LECO At NIST, GD-OES is being investigated for its potential for World Headquarters in St. Joseph, MI, USA, as part of a quantitative determination of non-metals. Although other ana- collaboration between NIST and LECO.Prior to undertaking lytical figures of merit such as sensitivity and detection limit the studies, it was anticipated that a presputtering routine that are important, the emphasis is on accuracy. This is because it would eliminate the matrix eVect associated with nonis hoped that GD-OES may eventually prove useful for the stoichiometric sputtering would be used. In this way, the plan certification of non-metallic elements in reference materials. Of was to analyse grey cast irons using only white cast iron calibrants, without the need for any sort of matrix eVect the figures of merit, accuracy has historically been one of the correction. Although this routine would have required only an least discussed among GD practitioners.extra 2 min per burn, a shorter presputtering routine was used The present paper reports the initial studies in this project. to save analysis time, and the resulting matrix eVect was The work discussed involves the use of GD-OES for the corrected with a type standard.For comparison, analytical determination of several non-metallic elements (C, P and S) mass fractions for P, S and Si were also calculated from the and a metalloid (Si) in grey cast irons. In the past, GD data without matrix eVect correction. A statistical evaluation techniques have occasionally been used for this sort of analysis, of the data is presented. though very little has been published. The existing publications, for example, refs. 2 and 3, do not focus on accuracy, and no statistical evaluation has been reported. EXPERIMENTAL The analysis of grey cast irons is particularly problematic Instrument for spark-based spectrometries, owing to severe matrix eVects associated with the presence of graphitic C. It is very diYcult, These studies were performed on a GDS-750A glow discharge optical emission spectrometer (LECO, St. Joseph, MI, if not impossible, to determine C accurately in grey cast irons Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12 (1297–1305) 1297Table 1 Wavelengths, PMT models and PMT voltages employed for are strictly matrix matched, not only in terms of elemental the elements of interest composition, but also in terms of metallurgical structure. As seen in Table 2, the four grey cast irons are very similar in Element Wavelength/nm PMT Model* PMT voltage/V terms of elemental composition. Considering metallurgical C 165.701 R306 -850 structure, the important characteristics in this case are the P 177.499 R306 -900 proportion of total C that exists as graphite and the form of S 180.731 R306 -900 that graphite (i.e., nodules, flakes, etc.).The four grey cast irons Si 288.158 R300 -900 should be very similar in this regard as well, because they were Fe† 249.318 R300 -900 all manufactured from white cast irons through annealing. * All PMTs manufactured by Hamamatsu Photonics. Finally, type standardization should be most eVective when † Used as an internal standard.the analyte mass fraction does not vary appreciably from sample to sample and from sample to type standard. Table 2 shows that this criterion is at least approximately met by the USA).† Briefly, the instrument incorporates a Grimm-type GD set of grey cast irons. For these reasons, the grey cast irons lamp that can be operated in either dc or rf mode. Using all used in these studies present an ideal case for the use of type reflective optics, optical emission is focused onto the entrance standardization. slit of a 750 mm focal length, f/10, Paschen–Runge polychromator.The polychromator provides a nominal spectral reso- Data Reduction lution of <25 pm over the full spectral range from 120 nm to 800 nm. The spectrometer chamber is pumped with a turbo- Even though the software accompanying the instrument promolecular pump for operation at wavelengths below 200 nm. vides data reduction capabilities, the raw data were transferred Emission intensities are measured and quantified by means of to a separate computer at NIST for processing.This is because PMTs and associated electronics. The entire instrument is the available time at LECO was insuYcient to allow a complete monitored and controlled with a software package running on evaluation of the data. All data reduction and analysis were a PC. Full data reduction routines are also provided within performed in the Excel (Microsoft, Redmond, WA, USA) and the software.SigmaPlot (Jandel, San Rafael, CA, USA) software packages. For the present studies, the Grimm lamp was operated in the dc mode, and the anode diameter was 4 mm. The power RESULTS AND DISCUSSION supply was operated in voltage regulation, while the discharge EVect of Uncertainty in the Independent Variable on Calibration current was maintained by adjusting the source pressure in Equations real time. In this way, the lamp was actually operated in constant discharge power. The argon support gas was obtained Before calibration equations for the four analytes could be from the house supply.For all samples, each burn consisted calculated from the raw data with confidence, it was necessary of a 5 s evacuation, a 55 s presputtering period at 1000 V and to determine whether least squares regression would be an 35 mA, and a 10 s integration under the same conditions. acceptable method. As noted earlier, internal standardization Elemental analytes were C, P, S and Si.Emission from the was used in these studies. This means that the calibration major matrix element, Fe, was also monitored and used as an graphs consist of mean intensity ratios plotted against ratios internal standard. The wavelengths employed for the elements, of mass fractions, with the latter being the independent variable. along with the PMT models and voltages, are listed in Table 1. The issue at hand is the uncertainties associated with the mass Though the instrument is capable of background correction, fraction ratios for the calibration points.As seen in Fig. 1, it was not used for these studies. these uncertainties are generally comparable in magnitude to the uncertainties in the intensity ratios. Least squares regression assumes that all uncertainty is in the dependent variable. Samples Because of this assumption, it will tend to underestimate the Samples used for the preparation of calibration lines consisted slope and overestimate the intercept if significant uncertainty of nine white cast iron reference materials (CRM Nos. 241–249, is associated with the independent variable. CKD Research Institute, Prague, Czech Republic). Prior to There are variations on least squares regression that are analysis, these calibrants were dry sanded using 120 grit ZrO2 designed to accommodate uncertainties in the independent paper. The grey cast irons consisted of four reference materials variable.6 These methods are fairly eVective, so long as the (CRM Nos. 21G, 22G, 23G and 24G, Brammer Standard, magnitudes of these uncertainties are the same at each cali- Houston, TX, USA). These samples were wet sanded using bration point, or can at least be modelled. For these experi- 600 grit paper prior to analysis. Known mass fractions for the ments, the uncertainties in the mass fraction ratios meet neither four analytes and Fe in both the white and grey irons are of these criteria.Under such circumstances, the eVectiveness of presented in Table 2. these methods for reliably estimating regression coeYcients Samples 21G, 22G and 23G served as unknowns, while 24G tends to be compromised. Moreover, it is diYcult to estimate served as the type standard from which correction factors were uncertainties associated with unknown values predicted with derived. The choice of which of the four grey irons to use as equations computed with these methods when such circumthe type standard was made randomly.The randomness of the stances are present. Uncertainties in predicted values are choice is important, because, with certified values for the four essential for the statistical analyses reported herein. For these analytes in hand, there is a natural tendency to choose the one reasons, the use of these variant methods was not considered that will provide the most eVective corrections. further. Type standardization should be most eVective for matrix The validity of using least squares regression in these experie Vect correction when the type standard and unknown samples ments was tested by simulating the actual calibration data in Excel using the Visual Basic programming language.The † In order to describe experimental procedures adequately, it is simulation was done four times, once for each analyte. This occasionally necessary to identify commercial products by manufac- was necessary, because the noise associated with the calibration turer’s name or label.In no instance does such identification imply points is diVerent for each analyte. endorsement by the National Institute of Standards and Technology, The simulation for a particular analyte consisted of nine nor does it imply that the particular products or equipment are necessarily the best available for that purpose. ‘dummy’ calibration points with means positioned exactly 1298 Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12Table 2 Elemental mass fractions and uncertainties, expressed as 95% confidence intervals, for the four analytes and the matrix element, Fe, in the white and grey cast irons Certified mass fractions (%) Sample C P S Si Fe* White cast irons: 241 1.71±0.007 0.006±0.0016 0.14±0.004 3.20±0.016 93.23±0.52 242 2.21±0.022 0.040±0.0015 0.033±0.0014 2.88±0.020 92.55±0.52 243 2.23±0.026 0.162±0.0054 0.085±0.0016 2.44±0.020 93.03±0.52 244 2.60±0.017 0.024±0.0015 0.019±0.0015 2.11±0.024 92.95±0.52 245 2.78±0.028 0.40±0.008 0.049±0.0018 1.60±0.017 92.89±0.52 246 2.82±0.028 0.60±0.006 0.022±0.0024 0.62±0.016 92.92±0.52 247 3.12±0.028 0.095±0.0048 0.005±0.0015 1.16±0.010 92.82±0.52 248 3.41±0.018 0.050±0.0014 0.006±0.0009 1.81±0.028 92.96±0.52 249 3.75±0.025 0.25±0.009 0.008±0.0011 0.34±0.016 92.94±0.52 Grey cast irons: 21G 3.98±0.030 0.057±0.0020 0.028±0.0024 1.61±0.038 92.60±0.29 22G 3.70±0.013 0.102±0.0025 0.032±0.0012 1.96±0.037 92.61±0.28 23G 3.17±0.019 0.047±0.0017 0.024±0.0013 2.61±0.032 92.49±0.28 24G 2.42±0.019 0.022±0.0019 0.019±0.0019 2.93±0.019 92.65±0.28 * Uncertified, with uncertainties estimated from statistical information provided on the Certificates of Analysis and/or general knowledge of grey cast irons.Fig. 1 Internally standardized calibration lines obtained from triplicate burns on each of the nine white cast irons ($) for (a) C, (b) P, (c) S and (d) Si. Points for the 24G grey cast iron type standard (#), also based on triplicate burns, are plotted as well.All error bars are 95% confidence intervals. Weighted least squares regression was employed for the calibration lines. along a straight line of known slope and intercept, but with the mean slope and intercept was estimated with simulated experimental data. noise in both the x and y coordinates that mimicked as closely as possible the observed noise associated with the experimental As expected, the slopes were underestimated and the intercepts were overestimated for all four analytes.For all analytes calibration points. With the use of random number generation, 10 000 sets of dummy calibration data were computed, each except C, their mean values diVered from the corresponding design values at the P=0.05 level of significance. Neither the set diVering from every other set in terms of the (x, y) coordinates of the points. Linear least squares regression was per- slope nor the intercept diVered significantly in the case of C.Fortunately, in terms of predicting an unknown x-value from formed on each set, thus producing 10 000 slopes and 10 000 intercepts. The t-test was then used to determine if the mean a known y-value, the biases in the slope and intercept being in opposite directions has a partially compensating eVect. As slope and mean intercept diVered significantly from their corresponding design values. Finally, the magnitude of bias a result, the biases expected to be introduced into the final elemental determinations in these experiments were found to expected to be introduced into unknown values predicted with Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12 1299be negligible for all four analytes. Therefore, it was decided to on which the data in the figure are based, S=s/2. Corresponding 95% confidence intervals are extremely broad use least squares regression to compute the calibration equations.(0.52s<s<6.28s).8 In the figure, the standard deviations of the means of the observations, rather than the SDs of the observations, are plotted. As a result, the uncertainties associated Heteroscedasticity in the Dependent Variable with the points may be quantitatively somewhat diVerent from the example. Nevertheless, it should be obvious that the Next, it was necessary to determine whether weighted or standard deviations of the means are very uncertain.This fact unweighted least squares regression should be employed. accounts for the considerable scatter in the points. Owing to Weighted regression is usually more appropriate when there is these large uncertainties, the data for all four analytes are significant heteroscedasticity associated with the dependent better treated as an ensemble, rather than separately. This is variable (i.e., when the magnitudes of the uncertainties associwhy the data for the four analytes have been plotted on the ated with the ordinate values of the calibration points are not same graph. equivalent).Therefore, the mean intensity ratios for the cali- The heteroscedasticity of the mean intensity ratios is clearly bration points were evaluated for heteroscedasticity. revealed in Fig. 2(a). From this, it was concluded that weighted The first means of evaluation was a close visual examination least squares regression should be employed to calculate the of the error bars associated with the mean intensity ratios in calibration equations for the various analytes.Weighted the various plots of Fig. 1. Though somewhat inconclusive, an regression is typically performed using the reciprocals of the examination of this sort seems to reveal that the error bars variances in the ordinate values of the calibration points as generally increase in magnitude with increasing mean intensity weighting parameters for those points. In this work, the ratio.This is most clearly evident in the P and Si data in experimentally observed standard deviations of the mean inten- Fig. 1(b) and (d), respectively. In order to assess this apparent sity ratios were not used to generate these weighting param- heteroscedasticity more eVectively, the standard deviations of eters, because of the large uncertainties associated with them. the mean intensity ratios were plotted against the mean Instead, estimates of these standard deviations derived by intensity ratios themselves [see Fig. 2(a)]. plugging the experimental mean intensity ratios into the equa- In looking at these data, it is important to remember that tion for the regression line in Fig. 2(a) were used. This line, standard deviations usually have very large uncertainties. For which approximates the noise behavior of the mean intensity example, the standard deviation, S, of the SD, s, of n obserratios, was calculated using all 36 points and was forced vations is: through the origin, since, in the ideal case, if there is no signal, there should also be no noise.Weighting parameters derived S= s Ó2(n-1) (1) in this way should be more meaningful. Finally, it should be noted that assuming a linear dependence assuming that the observations are drawn from a normally between the mean intensity ratios and their standard deviations distributed population.7 For n=3, the number of observations is synonymous with assuming that the data are flicker noise limited.One way to provide some justification for this assumption is to plot the mean intensity ratios against their relative standard deviations (RSDs) [see Fig. 2(b)]. If flicker noise is dominant, there should be no dependence of the RSD on the mean. As shown in the figure, this appears to be at least approximately true. Given the assumption of flicker noise dominance, the slope of the regression line in Fig. 2(a) is essentially an estimate of the flicker factor.Calibration and Determinations Employing Type Standardization Weighted least squares regression lines were calculated for each of the four analytes using the data obtained from the nine white cast iron calibrants (see Fig. 1). The resulting calibration equations and correlation coeYcients accompany the graphs. All four calibrations are acceptably linear, though the linearity of the S data is somewhat undesirable. The linearities of the P and Si data are exceptionally good. All four intercepts are reasonably close to the origin.Before proceeding to the analytical determinations using the calibration equations, it is helpful to discuss type standardization in a little more detail. As noted before, type standardization is a method by which the matrix eVect associated with non-stoichiometric sputtering can be corrected after data acquisition. The method assumes that the bias introduced by the matrix eVect is additive and consistent from sample to sample within a given matrix.Type standardization is performed as follows. First, the mass fraction of the analyte in a known sample that is matrix matched to the samples to be analysed (i.e., the type standard) Fig. 2 Standard deviations (a) and relative standard deviations (b) of is determined experimentally from the calibration equation. the mean intensity ratios plotted against the mean intensity ratios Since the calibration equation is based on calibrants of a themselves for the C ($), P (#), S (&) and Si (%) calibration data.diVerent matrix, the mass fraction determined in this way may The regression line in Fig. 2(a) was computed using all 36 points as an ensemble and was forced through the origin. disagree considerably with the known value. The diVerence 1300 Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12between the experimental and known mass fractions for the corresponding certified values. Statistical tests are performed in the next section to evaluate whether the remaining determi- type standard is a measure of the bias introduced by the matrix eVect.This diVerence serves as the correction factor for the nations agree with the corresponding certified values on a statistical basis. analysis of the unknowns. Next, the mass fractions of the analyte in the unknown samples are determined experimentally The relatively large errors associated with the S determinations for samples 21G and 22G may be attributable in part from the calibration equation.These experimental mass fractions are then adjusted using the correction factor. to digitization noise. The same is true for the determination of P in sample 23G. The digitization noise results from the fact Points for the 24G grey cast iron type standard used in this work are plotted on the graphs in Fig. 1. Considering where that only two significant figures are reported for these mass fractions.Digitization noise should play a lesser role in the they each fall in relation to their respective calibration line, it seems that type standardization is only necessary for the remaining determinations, since all of these are reported with three significant figures. determination of C. The grey iron points for P, S and Si are essentially indistinguishable from the white iron calibration The relatively large errors associated with the C determinations are probably caused by the fact that C required the points.However, on a theoretical basis, if type standardization is necessary for even one element, it should be necessary for largest matrix eVect correction. The relative magnitudes of the corrections for the various analytes are indicated by the every element, since stoichiometric sputtering was not attained prior to data acquisition. Given this fact, type standardization proximities of the points for the 24G type standard to their respective calibration lines in the graphs in Fig. 1. was initially used for all four analytes. As a matter of interest, P, S and Si mass fractions computed without matrix eVect correction are reported in a subsequent section. Statistical Evaluation of the Results Analytical determinations for the three grey cast iron unknowns were based on triplicate burns. Mathematically, As alluded to above, 9 of the 12 analytical mass fractions in analytical mass fractions were calculated using the following Table 3 do not agree numerically with the corresponding equation: certified values.For each of these determinations, the t-test was performed to determine if the observed diVerence between the experimental and certified values can be attributed to [A]unk,exp=GA[A] [Fe]Bunk,exp [Fe]unk,certH random noise. Since the precisions associated with the certified mass fractions and the mass fractions determined in this work +G[A]type,cert- A[A] [Fe]Btype,exp [Fe]type,certH cannot be assumed to be equivalent, it was necessary to employ a version of the t-test that does not pool the uncertainties from the two sets of data.With this constraint in mind, the exper- (2) imental t-statistic is defined as: In this equation, square brackets symbolize mass fraction, and A refers to the analyte. The subscripts unk and type refer to texp= |xE-xC| ÓuE2+uC2 (3) the unknown sample and type standard, respectively, while the subscripts exp and cert refer to experimentally determined and where xE and xC are the means for the experimental and certified values, respectively.In this regard, it is important to certified data sets, respectively, and uE2 and uC2 are the note that the mass fraction ratios are the experimentally variances associated with those means. The eVective degrees determined values calculated from the mean intensity ratios of freedom for this t-test are given by: and the calibration equations. The remaining mass fractions on the right side of eqn. (2) are the known values from Table 2. 1 veff = 1 vE A uE2 uE2+uC2B2 + 1 vC A uC2 uE2+uC2B2 (4) Examination of the equation reveals that the experimentally determined mass fraction for the unknown sample prior to matrix eVect correction is given by the first bracketed term, where vE and vC are the degrees of freedom associated with the data sets.9 while type standardization is performed with the addition of the second bracketed term. In order to perform these t-tests, it is obviously necessary to know the means, variances and degrees of freedom associ- The determinations of the various analytes in the unknowns are presented in Table 3.As seen in the table, only 3 of the 12 ated with both the experimental and certified data. All of the necessary statistics for the certified mass fractions were avail- analytical mass fractions are numerically identical to the Table 3 C, P, S and Si determinations in three grey cast irons based on triplicate burns.Brammer CRM 24G was used as the type standard Sample Certified Mass fraction determined from Analyte No. mass fraction (%) this work (%) Relative error (%) C 21G 3.98 3.88 2.51 22G 3.70 3.62 2.16 23G 3.17 3.20 0.95 Average relative error: 1.87 P 21G 0.057 0.057 0 22G 0.102 0.102 0 23G 0.047 0.048 2.1 Average relative error: 0.7 S 21G 0.028 0.027 3.6 22G 0.032 0.031 3.1 23G 0.024 0.024 0 Average relative error: 2.2 Si 21G 1.61 1.59 1.24 22G 1.96 1.94 1.02 23G 2.61 2.60 0.38 Average relative error: 0.88 Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12 1301able from the Certificates of Analysis, whereas the statistics for experimental t-statistic is: the experimentally determined mass fractions had to be calculated. The variances and degrees of freedom associated with tpaired= xdÓn Sd (5) the experimental values were computed by applying the law of the propogation of error and the Welch–Satterthwaite where n=3 (i.e., the number of grey cast iron samples), xd is formula, respectively, to eqn.(2). These methods are well the average of the signed diVerences between the experimental known in the statistical literature, for example, reference 10, and corresponding certified mass fractions, and Sd is the SD and so they will not be reproduced here. associated with these diVerences. For the Si data, the value of After these preliminary computations were completed, the tpaired calculated using eqn.(5) is 4.18. (In calculating tpaired, t-tests using eqns. (3) and (4) were performed (see Table 4). As all available significant figures for the various mass fractions seen in the table, at a level of significance of P=0.05, the null were employed, rather than the rounded-oV values in Tables 3 hypothesis, H0, is rejected in only one of the nine cases, the and 4.) Given that the degrees of freedom for the paired t-test determination of C in sample 21G. Note that for this determi- are n-1, or 2 in this case, the corresponding critical value of nation, texp is just barely larger than the corresponding critical t at the P=0.05 level of significance is 4.30.Therefore, at the value of t. This means that the observed diVerence between 95% confidence level, H0 is retained. However, since P=0.053 the experimental and certified C mass fractions for this sample for tpaired=4.18 and 2 degrees of freedom, H0 would be rejected is significant at the 95% confidence level, but just barely so.at a confidence of 94%. Therefore, it was concluded that there In fact, given that P=0.043 for 18 degrees of freedom and is statistical evidence for systematic bias in these Si determitexp= 2.18, H0 would have been retained at the 96% confidence nations, but this evidence is somewhat weak. The cause of this level. In view of these observations, it was concluded that, on apparent systematic bias will be explained in a later section. the basis of these t-tests, there is evidence of analytical bias associated with this one determination, but that the evidence P, S and Si Determinations Without Type Standardization is weak.As discussed earlier, the points for the 24G type standard are Another way to assess the presence or absence of analytical essentially indistinguishable from the calibration points in the bias is to employ the paired t-test.11 In contrast to the t-tests cases of P, S and Si. Nevertheless, on the basis of theoretical just performed, the paired t-test does not compare an individual considerations, type standardization was used for all four experimental mass fraction with its corresponding certified analytes. It is informative at this point to recalculate the value.Instead, it compares the full set of experimental mass analytical mass fractions for these three analytes without fractions for a particular analyte with the corresponding set of making use of the matrix eVect correction.certified mass fractions. While the paired t-test provides no Omitting type standardization, eqn. (2) simplifies to: information on any single determination, it can be especially helpful for discerning whether the set of experimental values [A]unk,exp= GA[A] [Fe]Bunk,exp [Fe]unk,certH (6) for a particular analyte as a whole is systematically biased relative to the set of certified values. Such systematic bias may sometimes remain hidden with the use of the non-paired t-test.where all symbols and subscripts are used exactly as before. The determinations using eqn. (6) are presented in Table 5. The validity of the paired t-test rests on the assumption that the uncertainties associated with the mass fractions for the As seen in the table, only one of the nine determinations agrees numerically with the corresponding certified mass frac- various samples under study are independent of mass fraction. Certainly, this is not strictly true for any of the four analytes tion.Non-paired t-tests were performed for the remaining determinations to ascertain whether the observed diVerences in this study. However, as shown in Table 4, the variances in general do not vary greatly with mass fraction, and so this between the experimental and certified mass fractions can be attributed to random noise. Eqns. (3) and (4) were again assumption was considered to be eVectively satisfied. Obviously, the paired t-test may not need to be performed employed for these tests.The necessary statistics for the data sets were computed exactly as before. for all four analytes, but only for those whose sets of analytical mass fractions appear to be biased. Referring to the data in The results of the non-paired t-tests (see Table 6) indicate that all of the analytical mass fractions are in agreement with Table 3, the only one of the four analytes for which the set of analytical determinations as a whole appears to be biased the corresponding certified values at the 95% confidence level.This is surprising for S, because the S determinations have relative to the corresponding set of certified mass fractions is Si. The analytical Si mass fractions for all three grey cast iron large relative errors. The agreement in this case can be attributed to the fairly large uncertainties in the experimentally samples are underestimated. Furthermore, they are each underestimated by approximately the same amount.Therefore, the determined values. These large uncertainties are probably a result of the relatively low mass fractions of S in the samples. paired t-test was used to evaluate the Si data only. For the application of the paired t-test to the data, the Next, in order to assess the presence or absence of systematic Table 4 Results of t-tests for the determinations in Table 3 that did not agree exactly with certified mass fractions Statistics from Certificates of Statistics from this work Analysis Results of t-test Sample Analyte No.x� E (%) uE2 (%2) vE x� C (%) uC2 (%2) vC veff texp tcrit,double-sided,P=0.05 H0 rejected? C 21G 3.876 2.349×10-3 16 3.985 1.402×10-4 5 18 2.18 2.10 Yes 22G 3.624 2.064×10-3 17 3.700 2.400×10-5 5 17 1.66 2.11 No 23G 3.204 1.694×10-3 18 3.170 4.500×10-5 4 19 0.81 2.09 No P 23G 0.0483 4.13×10-6 16 0.0467 4.27×10-7 5 18 0.75 2.10 No S 21G 0.0271 1.25×10-5 18 0.0281 9.66×10-7 6 21 0.28 2.08 No 22G 0.0314 1.47×10-5 17 0.0319 2.41×10-7 6 17 0.14 2.11 No Si 21G 1.588 1.604×10-3 15 1.609 2.401×10-4 6 18 0.49 2.10 No 22G 1.939 1.777×10-3 17 1.959 2.042×10-4 5 20 0.45 2.09 No 23G 2.599 2.204×10-3 19 2.608 1.352×10-4 4 21 0.18 2.08 No 1302 Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12Table 5 P, S and Si determinations in the three grey cast irons without the use of type standardization Sample Certified Mass fraction determined from Analyte No. mass fraction (%) this work (%) Relative error (%) P 21G 0.057 0.058 1.8 22G 0.102 0.103 1.0 23G 0.047 0.049 4.3 Average relative error: 2.4 S 21G 0.028 0.024 14 22G 0.032 0.028 13 23G 0.024 0.021 13 Average relative error: 13 Si 21G 1.61 1.59 1.24 22G 1.96 1.94 1.02 23G 2.61 2.61 0 Average relative error: 0.75 Table 6 Results of t-tests for the determinations in Table 5 that did not agree exactly with certified mass fractions Statistics from Certificates of Statistics from this work Analysis Results of t-test Sample Analyte No.x� E (%) uE2 (%2) vE x� C (%) uC2 (%2) vC veff texp tcrit,double-sided,P=0.05 H0 rejected? P 21G 0.0577 4.14×10-6 9 0.0567 6.91×10-7 6 12 0.46 2.18 No 22G 0.1029 1.294×10-5 9 0.1020 1.041×10-6 6 10 0.24 2.23 No 23G 0.0488 2.98×10-6 9 0.0467 4.27×10-7 5 11 1.12 2.20 No S 21G 0.0241 7.58×10-6 9 0.0281 9.66×10-7 6 11 1.37 2.20 No 22G 0.0284 9.75×10-6 9 0.0319 2.41×10-7 6 9 1.11 2.26 No 23G 0.0211 6.25×10-6 9 0.0245 2.40×10-7 5 10 1.34 2.23 No Si 21G 1.594 3.417×10-4 9 1.609 2.401×10-4 6 15 0.63 2.13 No 22G 1.945 5.150×10-4 9 1.959 2.042×10-4 5 14 0.54 2.14 No analytical bias more completely, the paired t-test was per- Final Consideration of Type Standardization formed for those analytes in Table 5 whose sets of analytical Before drawing this discussion to a close, it is helpful to plot mass fractions appear to be systematically biased relative to the points for all four grey cast irons, using known mass the corresponding certified values.An examination of the data fractions, on each of the calibration graphs. Doing so provides in the table indicates that such systematic bias almost certainly a clearer understanding of type standardization and the results exists for S and may exist for P. However, no such systematic reported in this paper. Plots of this sort, focusing on the bias is implied for Si. Therefore, the paired t-test was applied domains of the grey iron points, are presented in Fig. 3. to the P and S data only.The values of tpaired for the P and S As seen in the plots in Fig. 3(a) and (c), the points for the data were calculated using eqn. (5) and all available significant greys and the points for the whites clearly fall along diVerent figures for the various mass fractions. The critical value of t calibration lines for C and S. This behavior, which is a result that tpaired must exceed for H0 to be rejected at the P=0.05 of the matrix eVect associated with non-stoichiometric sputter- level of significance is 4.30.ing, is known from previous work.4 The wide separation For the P data, tpaired=3.58, indicating that H0 should be between the S calibration lines for the whites and greys in retained at the 95% confidence level. However, since P=0.070 Fig. 3(c) explains the systematic analytical bias in the determi- for tpaired=3.58 and two degrees of freedom, H0 would be nations performed without type standardization demonstrated rejected at the 92% confidence level.Therefore, it was conin the preceeding section. cluded that there is statistical evidence for systematic bias in Referring now to the P data in Fig. 3(b), all four grey cast the P determinations performed without matrix eVect correciron points are indistinguishable from the white cast iron tion, but that the evidence is somewhat weak. Comparing the points. The error bars associated with each grey iron point non-corrected P determinations in Table 5 with the P mass fully overlap the white iron calibration line.This is why fractions determined with type standardization in Table 3, it reasonably acceptable determinations of P could be done was found that the use of matrix eVect correction improved without type standardization. However, a closer examination the relative errors by about a factor of three on average. The of the data indicates that all four grey iron points lie slightly apparent systematic bias in the non-corrected P mass fractions above the white cast iron calibration line.This means that the accounts for this improvement. greys may in fact lie along a calibration line of their own. It For the S data, tpaired=19.5, indicating sound rejection of also accounts for the systematic bias in the non-corrected H0 at the 95% confidence level. In fact, given such a large determinations demonstrated in the preceeding section. value of tpaired, H0 would be rejected even at the 99% confidence Referring to the Si data in Fig. 3(d), it is apparent that, as level. Therefore, there is strong statistical evidence for systemfor the P data, all four grey cast iron points are indistinguish- atic bias in the S determinations performed without type able from the white cast iron points. Again, the error bars standardization. Comparing the non-corrected S determicharacterizing each grey iron point fully overlap the calibration nations with those performed with matrix eVect correction in line developed with the white cast irons.This accounts for the Table 3, it was found that the use of the type standard improved low relative errors associated with the Si determinations the relative errors dramatically. The systematic bias in the non-corrected mass fractions accounts fimprovement. obtained without type standardization. Unlike P, however, all Journal of Analytical Atomic Spectrometry, November 1997, Vol. 12 1303Fig. 3 Points for the four grey cast irons (#), using known mass fractions, plotted with the calibration data from Fig. 1 ($), for (a) C, (b) P, (c) S and (d) Si.For clarity, the graphs focus on the domains of the grey iron points for P, S and Si. All error bars are 95% confidence intervals. As for the white cast iron calibration lines, weighted least squares regression was used to generate best-fit lines through the grey cast iron points. four grey iron points do not lie on one side of the white cast helpful in the case of P.However, in some cases it may introduce analytical bias into the determination of Si. iron calibration line. In fact, the grey irons seem to lie along a calibration line of their own that intersects the white iron For all four analytes, the relative errors observed for the determinations performed with matrix eVect correction are calibration line. This is why the Si determinations performed without type standardization appeared to be free of system- acceptably small for many applications.As discussed earlier, the set of grey cast irons employed in these studies constitute atic bias. A more careful examination of the figure reveals that while an ideal case for the application of type standardization. Poorer results would be expected in less ideal cases. the 24G type standard point lies slightly above the white cast iron calibration line, the three remaining grey cast iron points Among the determinations for which matrix eVect correction was performed, the largest relative errors were associated with lie either directly on the line or slightly below it. This is why matrix eVect correction using the 24G sample as the type the C determinations, with one of the three analytical C mass fractions disagreeing with the corresponding certified mass standard seemed to introduce systematic bias into the Si determinations.If one of the other grey cast irons had been fraction at the P=0.05 level of significance.The relatively large errors associated with C were attributed to the fact that this chosen for the type standard, or if a diVerent set of grey irons altogether had been used, the results may have been diVerent. element required the largest matrix eVect correction. Certainly, better results for C would be expected if data were acquired Finally, as discussed earlier, type standardization assumes that the error introduced by the matrix eVect associated with after sputtering becomes stoichiometric.This may constitute a future study. non-stoichiometric sputtering is additive and consistent from sample to sample within a given matrix. If this assumption is Regarding the determinations performed without type standardization, it is important to note that the relative errors justified for the work reported herein, the calibration lines for the whites and greys should be parallel. The plots in Fig. 3 associated with the analytical P and Si mass fractions were also small enough for many applications.This can be attributed show that this is at least approximately true. It is important to note that, theoretically, they should not be exactly parallel, to the fact that these two elements required little or no matrix eVect correction. The fact that P and Si can be determined in since this would mean that one or the other would have an intercept far removed from the origin, possibly even negative, this way while sputtering is still non-stoichiometric demonstrates the relative lack of matrix eVects, compared with most in the case of an element such as C.This inherent aparallelism is why type standardization should be most eVective over small other analytical methodologies, that characterizes GD-OES. ranges of analyte mass fraction. The authors thank David Valensi (Product Manager— CONCLUSIONS Material Characterization) and Joel Mitchell (Manager— Atomic Spectrometry) of LECO for their kind hospitality and Three non-metals (C, P and S) and a metalloid (Si ) have been determined in grey cast irons using GD-OES. The results the use of the LECO facilities. Robert Watters (Analytical Chemistry Division, Chemical Science and Technology reported in this paper demonstrate that, under the circumstances of these experiments, type standardization is indispens- Laboratory, NIST) is acknowledged for many helpful discussions. ible for the unbiased determination of C and S and may be 1304 Journal of Analytical Atomic Spectrometry, November 1997, Vol. 127 Statistical Methods in Research and Production, ed. Davies, O. L., REFERENCES and Goldsmith, P. L., Hafner, New York, 4th edn., 1972, pp. 51–52. 1 Measurement of Sulfur in Superalloys Workshop, held at the 8 Statistical Methods in Research and Production, ed. Davies, O. L., National Institute of Standards and Technology, Gaithersburg, and Goldsmith, P. L., Hafner, New York, 4th edn., 1972, p. 66. MD, USA, March 22, 1996. 9 Statistical Methods in Research and Production, ed. Davies, O. L., 2 Radmacher, H. W., and De Swardt, M. C., Spectrochim. Acta, and Goldsmith, P. 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