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of sulfur dioxide lost was about the same in all cases, which explains the straight calibration curve. The tailing increases, of course, with the volume of the column. We could eliminate the negative errors completely and obtain a higher precision of the sulfur results by placing a gas permeation tube ( 6 ) with sulfur dioxide into the sampler of the analyzer. T h e tube liberates slowly a small amount of sulfur dioxide, which diffuses into the carrier gas. The active sites in the apparatus, which can retain sulfur dioxide, are thus continuously occupied, and the correct amount of sulfur dioxide from the sample passes through the column on top of the constantly present sulfur dioxide and is measured accurately without losses. The added sulfur dioxide also reduces the tailing of the other peaks in the chromatogram. From the displacement of the recorder base line on addition of the gas permeation tube. the amount of sulfur dioxide liberated from the tube could be roughly estimated to be about 6 pg/min. Regeneration of the Reduction Zone. The regeneration of the copper(1) oxide requires a very accurate regulation of the furnace temperature. A too high temperature can cause sintering of the filling and, with a too low temperature, the decomposition of the copper(I1) oxide takes a very long time. With our old Carlo Erba instrument, the temperature cannot
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be just set, but it must be carefully followed and adjusted manually, and afterward it must be carefully readjusted again to the working state. We use, therefore, the copper filling, which can be brought to its regeneration temperature with a pushbutton, and which lasts longer. With an instrument with a more modern temperature regulation a copper(1) oxide filling would probably be more convenient than the copper filling. The analytical results are the same with both fi, ings.
LITERATURE CITED (1) (2) (3) (4) (5) (6) (7) (8) (9)
Dugan, G. U.S. Patent 3 838 969, 1974. Dugan, G. Anal. Lett. 1977, 10, 639-657. Pella, E.;Colombo, B. Mikrochim. Acta 1978 I. 271-286. Biandrate. P.;Colombo, B. Carlo Erba Elemental Analyzer, mod. 1106, manual. Ruer, R.; Nakamto, M. Recl. Tray. Chim. fays-Bas 1923, XLII, 675-682. O’Keeffe, A. E.: Ortman, G. C. Anal. Chem. 1966, 38, 760-763. De Souza. T. L. C.; Bhatia, S. P. Anal. Chem. 1975, 4 7 , 543-545. Pelia, E.;Colombo, B. Carlo Erba Strumentazione, Milano, Italy, private communication. Bethea, R. M.: Meador, M. C. J . Chromatogr. Sci. 1969, 7, 655-664.
RECEKEDfor revie January 18,1979. Accepted April 2 , 1979. The work was ma..e possible by grants from the Swedish Council for Forestry and Agricultural Research and from she Faculty of Agriculture of the Swedish University of Agricultural Sciences.
Evaluation of Several Semi-Theoretical Methods for Quantitative Secondary Ion Mass Spectrometric Analysis after Discrimination-Correction of Data M. A. Rudat’ and G. H. Morrison” Department of Chemistry, Cornell University, Ithaca, New York
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Four methods of quantitative analysis are evaluated using steel, copper, and aluminum standards. The mass spectra are corrected for instrumental discrimination effects in order that the models themselves rather than the instrument are tested. The effects of this correction and normalization of the calculated concentrationsto 100 % are described. The analytical results of the various approaches are presented in terms of error factors. None were found to be analytically quantitative.
Two basic approaches to the conversion of qualitative secondary ion mass spectrometric (SIMS) data into semiquantitative or “quantitative“ concentration values have been the use of sensitivity factors and the use of theoretical models combined with raw data for the determination of some “fundamental” parameter values. ‘The latter approach will be referred t o as “semi-theoretical“, since the values of the parameters are determined empirically. The former approach has been the most successful and accurate to date, particularly when some sophistication is introduced into the choice of the sensitivity factors ( I ) . However, for routine analyses the semi-theoretical approach is most desirable if it can be applied P r e s e n t address: C e n t r a l Research & Development D e p a r t m e n t . Experimental Station, E. I. d u P o n t K e m o u r s & Company, LVilmingon, Del. 19898. 0003-2700/79/0351-1179$01 O O i O
to all samples, and several methods have been claimed to be nearly universal in applicability (2-13). These semi-theoretical methods are the ones to be discussed here. A shortcoming of previous examinations of the accuracy of these computational methods has been the failure to include the effects of instrumental discrimination on the relative ion intensities. Furthermore, no direct comparison of these methods by using a single data set has previously been made, making comparisons of claims of accuracy difficult to perform. T h e present work is designed to serve several functions. Most importantly, the effects of instrumental discrimination (14-1 7 ) are virtually eliminated so that the unencumbered accuracy of the investigated methods can be clearly seen. If the overall accuracy is not improved by removing these effects, then the analytical usefulness of the methods studied here must be brought into question. Secondly, the results of the different semi-theoretical methods will be directly compared using the same data sets, so that no confusion regarding different analysis conditions in different laboratories can exist. These methods can also be compared to the simplistic approach of considering the relative intensities of the corrected signals to be directly related to the concentration. Finally, the universality of applicability of the methods to metal alloy matrices will be explored. In considering some preliminary results obtained using Simons et al.’s local thermal equilibrium (LTE) program ( 4 ) and Andersen and Hinthorne’s published accounts of the F 1979 American Chemical Society
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CARISMA program ( 2 , 3 ) ,it was noted that several features tended to improve the results: a careful choice of standard elements, normalization of the total sum of concentrations t o 100% while simultaneously minimizing the error in the standards’ concentrations, the elimination of certain elements from the calculations, and the use of “oxide corrections”, that is, adding to the atomic ion signal the signals of detected oxides of that element. In most analytical techniques, the validity of the use of such techniques is arguable a t best; thus, they have not been used here except for comparative purposes. As with most analytical techniques, the preferred method here is to use only the signals from the standards to optimize quantification. T h e absolute accuracy of the different computational techniques is the most important measure of usefulness for quantitative analysis; the accuracy should extend at least over the metal systems studied here, and preferably over semiconductors and insulators as well. The linearity should be good, so that, by using known concentrations of minor elements, the matrix and/or major element concentrations can be estimated with good accuracy and the (unaided) total of concentrations should be near 100%. The use of any set of elements as standards for analyses should be possible. Finally, no set of elements should be excluded from the analysis because of the calculations yielding unreasonably large or small concentration values. Utilizing this background and criteria, the semi-theoretical methods outlined below have been chosen for analysis. Simplified LTE Model. This is similar to the approach used by Simons et al. ( 4 ) ,in that simplified L T E equations and partition functions are used, except that no “oxide correction” has been included. In the first version of the program, only the positive ion calculations are included, whereas in the L T E 2 program, a correction is made for the production of negative ions simultaneously with the (detected) positive ions ( 2 , 3 ) . The partition functions, etc. are from standard references; see (2, 4). Werner’s Method. Werner et al. (8-10) have proposed using a single fitting parameter in the LTE model; calculations based on an equation equivalent to their expression have been made (14). Schroeer’s Model. Several variations of Schroeer et al.’s (11, 12) adiabatic ionization model are tested. The use of a correction term for the supposed interaction between the energy spectra of the sputtered particles (as used in one version) is not verified experimentally (14,18,19),so a similar expression without this term is used in the other versions:
where n+ is the number of positive ions, no is the number of neutrals (no>> n’), B is the atomic binding energy, I , is the ionization potential of the departing atomic, 4 m the work function of the surface, u the velocity of the particle, a the “interaction distance” (or a fitting parameter), and n is a fitting parameter. This model can he made to fit energy spectra data on the high-energy power dependence (19) by a suitable choice of n. Since a is an interaction distance for the particle with the surface, a reasonable choice for a is the van der Waals radius of the atom; this is also of the order of Schroeer et al.’s best-fit value for a from some published data ( a = 1.4 A ) . B has been taken as the average sublimation energy of the matrix and impurity atoms using the values from Honig (20). Gries and Rudenauer‘s Method. Gries and Rudenauer’s (13) theoretical improvement of Schroeer et al.’s model by using an expression from sputtering theory for L‘, and including the spectrometer energy window effect explicity, is tested for
the cases for n = 0, 1, and 2. Two expressions are required for each value of n: the first, for the energy range over which all ions are accepted (here E 5 0.25 eV), and the energy range over which the transmission is inversly energy-dependent. For the case n = 0, the expressions are:
where Eo = cutoff energy of immersion lens (0.25 eV for the present work), ol = -Eo/B, q2 = E o / B ,E , = lower boundary of window (= Eo),and E 2 = upper boundary of the window. For n = I, the first equation is as given by Gries and Rudenauer and the second has been modified for the transmission effect.
and
Y2(
( E + B/2)’/* E+B -+
where M , is the atomic mass of the matrix element and M I is the atomic mass of the impurity element. Finally for n = 2:
B2 4(E + B)’
+ In ( E + B ) ] : :
(6)
Since these equations explicitly include the energy discrimination and energy window effects, the only correction factors t h a t should be needed are those for detector discrimination (15, 16). By integrating over the energy range of zero to infinity, Equation 4 becomes the total expected ion current predicted by this combination of theories:
The energy discrimination/energy window correction factors (17)as well as the detector discrimination correction factors must be applied to the use of this equation. Jurela‘s Method. Jurela (7) has proposed equations to the LTE model, but based on nonequilibrium considerations. However, it has been shown that this model is conceputally
ANALYTICAL CHEMISTRY, VOL. 51, NO. 8 , JULY 1979
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Table I. Program Characteristics program
equation or ref.
LTE 1 LTE 2 LTE 3 LTE 4 Werner 1 Werner 2 Werner 3 Werner 4 Werner 5 Schro l a Schro 2b Schro 3c Ruden 1 Ruden 2 Ruden 3 Ruden 4 Ruden 5 Ruden 6d Normal 1 Normal 2
Ref. 4 Ref. 2, 3, 4 Ref. 4 Ref. 4 Ref. 1 0 Ref. 1 0 Ref. 1 0 Ref. 1 0 Ref. 1 0 Ref. 1 2 Eq. 1 Eq. 1 Eq. 4,5 Eq. 8 Eq. 8 Eq. 6, 7 Eq. 6, 7 Eq. 2, 3, 4, 5, 6, 7
bandpass corrected
detector corrected
sum = 100% no no Yes
no no Yes no Yes no no no no no no yes no Yes no ._no __yes Velocities calculated from most probable enera Velocity used as a fitting parameter; a = van der Waals radius of atom. n = 0 , n = 1,or n = Velocities calculated from average energy of ions from energy spectra. gy of energy distributions. 2: chosen for each element depending on the high energy dependence of the ion energy spectra. (21)and computationally ( 4 ) equivalent t o the LTE model, and will therefore not be tested here. Normalization. T h e simplest approach t o quantitative analysis is simply to assume t h a t the (corrected) ion signals are directly proportional to the concentrations of the elements. Thus the signals can be normalized so that the internal standards are best-fit, or the sum of the calculated concentrations can be set t o 100%. This simple approach is compared to the models above.
EXPERIMENTAL Apparatus a n d Conditions. A CAMECA IMS-300 Ion Microanalyzer (22) was used for this study. A 1-1.5 WA02+ion beam at 5.5 keV struck the sample a t an angle of -73” from the surface normal; secondary ions were extracted at 4.5 keV. Residual pressure for the determinations made in vacuum conditions was generally 6 - 9 X Torr, and for those runs in which an oxygen backfill was used, high purity oxygen gas was bled in through a Granville-Phillips leak valve to a pressure of 5 x lo4 Torr. The ion-to-electron converter section operated a t 6 X Torr. The energy window of the spectrometer was set to 0-15 eV, and the mass resolution was -400 at the base of the peaks. Samples. Several types of standard matrices were investigated for this study: steels, both low- and high-alloy; copper; and aluminum. The steels were NBS SRMs 661-664 and stainless steels M2 (270 Cr) and M25 (2570 Cr) manufactured by Sumimoto metals and Nippon Steel (see ( I ) ) . The copper standards were Johnson and Mathey (London) low-alloy standards for atomic absorption analysis (CBO, CB1, CB2) and contained small amounts of Si, Mn, Fe, Co, and Ni. The aluminum standards were also Johnson and Mathey atomic absorption standards (AA1, AA2, AA3, AA4) and contained small amounts of Ti, Mn, Fe, Ni, Cu, and Zn. Pure (99.9999%) polycrystalline samples of Cu and Al, and NBS-665 were used as interference references. All samples were cut, polished, cleaned, and mounted as described previously (15, 1 7 ) .
Procedure. Samples were first sputter-cleaned by a focused ion beam (1-2 mA/cm2) rastered over a 500 Fm X 500 Fm area for about 20 miri in residual vacuum. The raster was then stopped and the beam defocused to a 1-2 mm diameter ( - 100 pA/cm2). The detector accepted an area -75 fim in diameter from the center of this spot. A computer program (23) then collected the data. The program determined the pulse-count rate at each magnet setting by counting the pulses for 25 ms. For a single mass scan from m / e = 1 to m / e = 220, 26 000 points were used in order to yield the best possible resolution. The pulse counts were integrated over the entire peak, so that the final output was
an integrated intensity. The integrated intensities were corrected for the effects of the decreasing point density with mass. By utilizing the digital mass scans from the pure reference standards, approximate corrections for interfering molecular ions were deduced and applied. Calculation Method. The abundance-corrected integrated intensities were input into the various computer programs, which applied the instrumental correction factors from References 15 and 17 automatically as necessary. The ionization fraction was calculated for the standards using the appropriate expression(s) with the chosen values of the fitting parameters. For the LTE models, T was varied in steps of 25 K from 2500 K to 1 2 000 K and ne was varied in multiplicative steps of a factor of two, from 1 X lo8 cm-3 to 1 X lozo~ m - The ~ . refinement loop resulted in steps of 5 K and a factor of 1.5, respectively. For Werner et al’s method, T was varied as for the LTE models. For Schroeer et al.’s model, if n or a were allowed to vary, they were limited to the range 0.3-4.0 in steps of 0.1; the work function was set equal to that of the matrix. When u was used as a fitting parameter, it was allowed to vary between 1 X 10’ and 1 x 10’ cm/s in multiplicative steps of 2.0. For Gries and Rudenauer’s method, if a was allowed to vary, the same limits as above were imposed, and the work function was taken to be equal to that of the matrix element. (Work functions were taken primarily from Fomenko (24.) The general computational approach for all methods was similar to that outlined by other authors (4-6). For each sample, elements were chosen as internal standards based on microsampling ( I , 25-27) and concentration considerations. Every pairwise combination of these elements was used to calculate the “unknown” concentrations, so that a typical spectrum was calculated using three or more different pairs of standards. Only elements for which the concentrations are certified have been included in the error tables and histograms, although every detected ion was included in the calculations.
RESULTS AND DISCUSSION Table I is a list of the programs used for this study and the conditions and ionization model used with each program. The names of the programs are intended t o identify the general approaches which have been discussed above, with the version numbers referring to variations of the conditions imposed on the calculations. These program names will be used throughout the remainder of this paper t o unambiguously identify the method being discussed. Table I1 is a compilation of the internal standard elements used for each sample or sample type. The matrix elements
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Table 11. Elements Used as Internal Standards sample
internal standard elements
NBS-661 NBS-662 NBS-663 NBS-664 M 2 Steel M25 Steel A1 Standards Cu Standards NBS-610 Glass
Cr, Mn, N i V, Cr, Mn, Co, Ni, Cu V, Cr, Mn, Ni, Co V, Cr Cr, Mn, M o Cr, M o Ti, Mn, Fe, Ni, Cu Si, Mn, Fe, Co, Ni Mg, AI, K, Ca, V, Cr
Fe, Cu, and A1 were used only in a few cases out of the total of 290 internal-standard/spectra combinations calculated for this study, and have therefore not been listed. Table I11 lists the data sets, their composition, the number of spectra in each, and the total number of computer evaluations of the data by each program for each set. The latter value is obviously dependent upon the number of standards which could be used for the various spectra and how many combinations of those standards are possible. Altogether, over 2000 computer runs were made to produce the results presented here. From the computer runs, it was obvious that carbon was never assigned a correct concentration. Certainly, it is not homogeneously distributed since it segregates into carbide phases, not being very soluble in Fe or Cr (28). The problem arises from the very low secondary ion emission probability calculated from the models compared to the actual very high yield from the carbide phases. The result is that carbon is frequently overestimated by more than a factor of 100; if the condition that the sum of the calculated concentrations must equal 100% is imposed, then carbon frequently is calculated to be the matrix element even though it actually comprises less than 3% of any of the samples. This type of error is probably what led Simons to discard his results for carbon ( 4 ) . The results have been included here, since the inclusion of carbon in the analysis would be expected for a normal analytical procedure. Another problem with many of the results was the marked underestimation of the concentration of the matrix elements, which were not generally used for internal standards. Underestimation of the Fe, Cu, or A1 concentrations by factors of five or more was not uncommon for the programs not using the 100%-sum normalization condition. When the latter condition was added, the general result was to improve the estimation of the matrix element. Thus, the linearity of the methods was not, in general, satisfactory. Ruden 6 was an attempt to roughly fit the value of n in the equations to the observed power dependence of the high energy portion of the ion energy spectra, depending upon whether the observed dependence was close to E-2 ( n = 01, E-' ( n = l),or E-' ( n = 2). The result was so poor that the program will not be included in any further discussions; most
'
errors were of the order of factors of *50. Comparison of Methods. In this section, only those programs using discrimination-corrected data will be considered; a comparison of the results with and without the corrections will be presented below. Table I\' provides a direct comparison of the results of the program for the largest data set evaluated, the NBS steels. The 19 spectra included in this set were obtained in residual vacuum conditions. In Figure 1, a-d are histograms of the results for the L T E 1, Schro 1, Ruden 1, and Normal 1 programs; they have been adjusted so that the area within each bar accurately represents the population for that error factor range. Negative error factors are underestimations, positive factors are overestimations. The best results were obtained with LTE 3, which normalized the sum of the concentrations to 100%. In descending order, the quality-of-fit ranking from Table IV is: L T E 3, Werner 2 and Schro 1; L T E 2, L T E 1 and Schro 2; Schro 3; Werner 1 and Ruden 3; Ruden 1 and Ruden 5 ; Ruden 2; Normal I ; Ruden 4; and Normal 2. I t is surprising that Schro 1, Schro 2 , and Schro 3 yielded better results than the Ruden programs, and particularly surprising that Schro 1 should give such (relatively) good results. One would expect that the Ruden programs would rank higher, since the distribution of velocities is taken into account in the calculations, whereas in the Schro programs a single velocity has been used for each ion. The most successful of the Schro calculation methods was to allow u, to vary as a parameter (with a set to the individual van der Waals radii) and n = 1; this is essentially equivalent to using a constant value of urn and varying a. The other two methods are based on a calculation of u, from the average energies (Schro 2) and most probably energies (Schro 3) of the energy distributions, with a set to the van der Waals radii and n variable. The basis of the calculations are more physically meaningful for the Schro 2 and 3 programs, but there is a substantial reduction in the number of values falling within "analytical" range of the correct answers (factors of
