Evaluation of the local thermal equilibrium model for quantitative

Aug 1, 1976 - Evaluation of the local thermal equilibrium model for quantitative secondary ion mass spectrometric analysis. David S. Simons, Judith E...
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Evaluation of the Local Thermal Equilibrium Model for Quantitative Secondary Ion Mass Spectrometric Analysis David S. Simons,' Judith E. Baker, and Charles A. Evans, Jr." Materials Research Laboratory, University of Illinois, Urbana, 111. 6 180 I

A simplified version of the local thermal equilibrlum model based on the Saha-Eggert ionization equation has been incorporated into a computer program for quantitative interpretation of secondary ion mass spectral data. The program was tested uslng standard reference metal alloys, composite glasses, and a natural mlneral. Over 80% of the determlnatlons of elemental concentratlons were within a factor of 2 of the known bulk values. The effects of indlvidual terms of the Saha-Eggert equation on the quality of fit were investigated, and the slmplified version of the model was found to be as accurate as the more elaborate version incorporated into the CARISMA computer program. Use of the Saha-Langmuir or Dobretsov equations with two adjustable parameters resulted In concentration values identical with those predicted by the Saha-Eggert equation.

Several methods have been proposed for the conversion of ion intensities to elemental concentrations in secondary ion mass spectrometry (SIMS). These include the use of empirical sensitivity factors ( I ) , the quantum mechanical model of adiabatic surface ionization advanced by Schroeer ( 2 ) ,and the local thermal equilibrium (LTE) approach of Andersen and Hinthorne ( 3 , 4 ) .The LTE model has been subjected to the most extensive testing, and it has achieved some success as a semiquantitative correction procedure, particularly when applied to mineral samples ( 4 ) .However, some questions remain about the general applicability of the LTE approach. Since it is available in the form of the CARISMA computer program (4) only t o users of one particular commercial instrument, the Applied Research Laboratories Ion Microprobe Mass Analyzer, one might wonder whether it would work as well on other types of SIMS devices which have different conditions of primary ion bombardment, secondary ion extraction, and energy acceptance of the mass spectrometer. Also, the CARISMA program incorporates as many as 80 constants per element in its data base, and one should question whether this amount of complexity is necessary to achieve an acceptable semiquantitative analysis. For these reasons, we have developed a much less complex computer program based on the LTE model, hereafter called the simplified LTE program, which is similar to the approach used by Shimizu and co-workers in Japan (5, 6). The basis of the model is the assumption of a plasma in local thermal equilibrium produced by ion bombardment. The Saha-Eggert ionization equation describes the dissociation equilibrium between neutral atoms and positive ions in a plasma of temperature T and electron density Ne. This equation has the form:

where Ni+ and Nio represent the concentrations of singly charged positive ions and neutral atoms of element i, Bi+ and Bio are the internal partition functions of the positive ion and Present address, Knolls Atomic Power Laboratory, General Electric Company, Schnectady, N.Y.

neutral atom, A is a numerical constant with a value of 4.83 X 1015 K-3/2, and Ii is the first ionization potential. AE = 2 X 10-s(N,/T)1/2is an ionization potential depression due to coulomb interactions in the plasma, which is calculated according to the Debye-Huckel model ( 4 ) .Once the temperature and electron density values of the plasma are furnished, the iodneutral concentration ratio of any element can be calculated using the Saha-Eggert equation. In addition to testing the accuracy of this equation as a quantitative correction procedure, a systematic study of the effects of the individual terms was undertaken to assess the validity of the underlying physical assumptions. Simplified LTE Model. Compared to the full CARISMA computer program described by Andersen and Hinthorne ( 4 ) , the simplifications used in the present work are as follows: 1)Corrections for the production of negative ions are not included. Such corrections may prove t o be important for elements of high electron affinity such as the halogens ( 4 ) ,but the standard materials on which the program was tested did not contain such elements. Some evidence suggests that the LTE approach does not work with the halogens even when the negative ion term is included (7). 2) An empirical oxide correction is used for each analyzed element which has an oxide ion intensity greater than 10%of the corresponding atomic ion intensity. In these cases, the oxide ion intensity of the element is added to the atomic ion intensity and the sum is used in the subsequent analysis. This procedure replaces the theoretical oxide correction incorporated into the CARISMA program which is based on the calculation of oxide dissociation constants ( 4 ) . 3) Gas-phase partition functions for the positive ions and neutral atoms are represented by fifth-order polynomial functions of temperature for 73 elements (8).Although this form is only stated to be accurate between 1500 K and 7000 K, we extrapolate the polynomial calculation to 10 000 K and use the 10 000 K value a t all higher temperatures. The dependence of the partition functions on electron density is ignored here. The partition functions for many elements used in the CARISMA program do incorporate electron density effects which greatly increases the number of numerical constants required in the data base (9). The procedure followed by the simplified LTE program in the calculation of atomic concentrations is the following: 1)The ion count rates, corrected for isotopic abundance and oxide ion intensity, are read into the program for each element detected in the sample. 2) A minimum of two elements are selected as internal standards, and their atomic concentrations are supplied to the program. 3) A grid of T and N e values is defined. Typically, the first pass of the grid search will cover values of T between 2000 K and 12 000 K a t 100 K intervals and values of log N e between 0 and 21 a t intervals of 0.1. The number of points on the grid where calculations are actually made is limited to a narrow band because of a correlation between T and N e discussed in a later section. 4) At each of the ( T , N e ) pairs on the grid, the following calculations are made: ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

1341

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SRM 661 40001- 'COW ALLOY S T E E L

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Figure 2. Contours of equal fit to internal standards in (T,Ne) space. Value of fit is indicated for each contour. X denotes relative minima. (a)low alloy steel, (b)labradorite, (c)monel

a) The iodneutral ratio Ni+/Niofor each element in the sample is calculated from the Saha-Eggert equation, b) The relative atomic concentration of each element is calculated from its ion count rate li+ according to the formula I

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Figure 1. Fits of simplified LTE model to standardized materials for three spots, denoted by triangle (Spot A), circle (Spot E), and square (Spot C). Underlined elements are internal standards. Dashed lines indicate factor of 2 error. Concentrations are given in atomic % , (a)low alloy steel, (b)stainless steel, (c)monel, (d~ labradorite, ( e )K series glass

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ANALYTiCAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

c) The relative atomic concentrations are usually normalized to 100%. If the matrix is an oxide, the constituent elements excluding oxygen, are normalized to the sum of the major cation concentrations, if known from other analytical techniques. If these concentrations are unknown, the cation sum is estimated to be 40% atomic. This procedure is necessary because a primary beam of oxygen ions is normally used for the SIMS analysis, preventing a direct determination of the native oxygen concentration in the sample by this technique. d) The error of fit a t this (T,N e )is calculated by summing the squares of the deviations between the calculated and known concentrations of the internal standards. Either absolute or relative (%) deviations can be used, but absolute deviations have the advantage that they weight the fit in favor of the elements at high concentration whose values are known most accurately. Minimization of relative deviations is the criterion used in the CARISMA program ( 4 ) .

Table I. Simplified LTE Results for Standard Reference Materials Material Spot Internal stds T(K) log N e Av. error internal stds, % SRM 661 A Fe, Ni 2 630 6.73 0 3 026 8.80 0 Low Alloy B Fe, Ni C Fe, Ni 2 488 5.85 0 Steel A Fe, Cr 4515 15.49 0 SRM 121d Stainless B Fe, Cr 5375 15.87 2 4555 14.00 0 Steel C Fe, Cr SRM 162a A Ni, Cu 5760 17.56 23 Monel B Ni, Cu 4775 16.02 17 Ni, Cu 4365 15.17 17 C Lake County A Si, A1 9895 19.85 6 Labradorite B Si, A1 9 180 18.23 0 C Si, A1 8725 18.21 0 K252 A Si, Ba 6005 15.65 0 Glass B Si, Ba 9250 19.77 9 C Si, Ba 6475 16.15 0 -Table 11. Effect of Partition Functions on Quality of Fit Material Case T(K) log Ne SRM 661 I 2 632 6.74 Low Alloy I1 5 710 19.25 Steel I11 9 540 19.80 SRM 121d I 4 881 15.45 Stainless I1 6 320 19.25 Steel I11 7 725 19.50 SRM 162a I 4 515 18.60 Monel I1 4 425 18.70 I11 5 500 19.07 Lake County I 9 180 18.23 Labradorite I1 8 355 17.69 I11 11285 20.03 K252 I 6 570 16.32 Glass I1 7 575 17.37 I11 9 105 19.73

Av. error internal stds, % 0 2 27

5 ) After steps a)-d) have been performed for each ( T ,N e ) pair on the grid, the ( T ,N e ) pair is selected which produced the minimum error for the internal standards, and the normalized concentrations of all elements are calculated and printed for this best-fit value of T and N e . 6) The calculation procedure in steps 3-5 can be repeated using a finer grid centered on the previously found best-fit values of T and N e . A total of three passes is usually made, with a final resolution of 5 K in T and 0.01 in log N e . T h e presence of two adjustable parameters in the SahaEggert equation suggests that the concentrations of two elements in any sample can be exactly fit by an appropriate choice of the parameters. However, this is not the case since the concentrations of other elements have an indirect effect on the calculated internal standard element concentrations through the normalization condition. In fact, when two elements in I I sample have a much higher concentration than any others, the two major elements can be fit essentially perfectly over ti wide range of N e values, with each N e determining a corresponding T value. This situation would occur, for example, in high purity binary alloys and binary semiconductors. In these cases, the predicted impurity concentrations can vary widely over the range of ( T ,N e )pairs which fit the major elements. Therefore, one of the impurity elements is required as a third internal standard to resolve the ambiguity of the fit. I n other types of samples, where the concentration of impurities in known to be 1% or higher, two internal standard elements are usually sufficient for a unique determination of the best values of the fitting parameters.

EXPERIMENTAL The simplified LTE correction procedure has been tested using three NBS standard reference alloys and two well characterized in-

2

Av. error remaining elements, % 73 72 88 36 32 31 160 124 121 43 14 26 10 61 8

Av. error remaining elements, % 79 60 93 40

16 31 18 19 33 0 0 22

158 284 14 16 98

1

22

3 16

34 51

54

94 121

sulators. Data from three different spots on each sample were taken in the ion counting mode on an AEI Scientific Apparatus Model IM-20 ion microprobe. For the metallic samples, a primary beam of positive oxygen ions was used with an impact energy of 15 keV. For the insulators, negative oxygen ions were used with an impact energy of 35 keV. The primary beam diameter was -20 pm, and the beam was rastered over an area 7 5 pm square or larger in an effort to average over small scale sample inhomogeneities. None of the materials studied have been certified for microhomogeneity. Independent tests for homogeneity were not generally made, but a comparison of results from the three spots per sample indicated that this was not a serious problem. During the analysis of each spot, the magnet current of the mass spectrometer was manually adjusted to detect each element of interest. Several isotopes of each element were measured if available. Over the time period of 1 hr typically required for the complete analysis of a single spot, the ion count rate from a major element varied by an average of 15%due primarily to instrumental drifts. This figure is taken as the limiting precision of a concentration determination.

RESULTS Only those elements which could be positively identified in the SIMS spectrum have been included in the analyses. Carbon has not been included because of possible contamination from hydrocarbons in the residual vacuum. In all cases, the two most abundant elements have been taken as internal standards. The resulting fits are shown in Figure 1,a-e and the fitting parameters are summarized in Table I. SRM 661 Low Alloy Steel (Figure la). Iron and nickel were used as internal standards. The concentrations of most other elements were corrected to within a factor of 2 of their bulk values by the L T E computer program. Exceptions were B, Al, and Mo. Significant oxide corrections were required for Nb and Mo. In the case of Cu, a correction for interference with T i 0 was required. These contributions a t mle 63 and 65 ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

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Flgure 3. Correlation of Tand Ne values determined from various LTE computer programs: Simons (this work), Shimizu (Refs. 5,S),Lovering-CARISMA (Ref. 1 4 , Newbury-CARISMA (Ref. 10, 11)

Solid line is function log Ne = 25 067 - 43196/T. Shaded region IS searched by simplified LTE computer program Dashed line is functionalrelationshipquoted by Andersen and Hinthorne (Ref 4)

were deduced from the count rate of *STil6Oa t rn/e 64. It has been reported that Ti and B are inhomogeneously distributed in this alloy (10, 11). Since the variation of these elements among the three spots in Figure l a is not excessive, the scale of the inhomogeneities must be much smaller than the rastered area used here. SRM 121d Stainless Steel (Figure lb). Iron and chromium were used as internal standards. The concentrations of all elements in this sample were corrected to within a factor of 2. An oxide correction was applied to Mo, and Cu was corrected for a T i 0 interference. SRM 162a Monei (Figure IC). Nickel and copper were used as internal standards. The fit here is not considered satisfactory for semiquantitative analysis. The calculated Ca concentration is so low that a segregation is suspected. The program assigned a value to the Cu internal standard which is 30%-45% too low, and the trace elements are assigned excessively high values to make up for the deficiency. The problem lies in the fact that the ionization potential and partition function values for Cu and Ni are very similar. Hence, the model predicts nearly equal ionization efficiencies for both elements, while the data indicate that Cu is less efficiently ionized than Ni by about 40%. This does not create a problem in semiquantitative analysis unless the sample has a high concentration of both Cu and Ni (note previous two examples). In such a case, the deficiency of Cu calculated by the program must be compensated by a general excess concentration of the trace elements. This is a result of the normalization technique which requires that all elemental concentrations must sum to 100%. Lake County Labradorite (Figure la). In this feldspar mineral, silicon and aluminum were taken as internal standards. The ion counting of all elements was done at high mass resolution ( M / A M c 3000) in order to separate the 56Fe peak from 4oCa160,and 2sSi2 (12). All elements were corrected by the LTE program to within a factor of 2 of their known values. K252 NBS Glass (Figure le). This synthetic glass contains 6 cations at concentrations ranging from 1to 22 at. %. Silicon and barium were used as internal standards. All elements were assigned concentrations by the computer program that are within a factor of 2 of the bulk values. 1344

ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

II

12

13 14 log Ne

15

16

17

Figure 4. Degree of ionization for each element in stainless steel sample as a function of Ne at T = 4555 K. Shaded area represents range of log Ne searched at this value of T, as discussed in text

DISCUSSION In order to investigate the simplified L T E model in more detail, one data set from each sample was selected for further analysis. The purpose here was to study the effects of several parameters on the fitting procedure, including partition function approximations, the role of the coulomb interaction term AE, and the method used to define the error of fit of the internal standards. Contour Maps. Maps with contours of equal fitting error for the two internal standard elements as a function of T and log Ne are shown for three of the analyzed reference materials in Figures 2, a-c. The fit value used here is defined as the sum of squares of the deviations between calculated and known bulk concentrations of the internal standards. In the low alloy steel and labradorite contour plots (Figures 2a and 2b), two minima are observed. They occur a t the same value of T , but at different values of log Ne, and the calculated concentrations are identical a t the two minima. This same general shape of the contours was also noted in the contour plots of the stainless steel (SRM 121d) and glass (K252) analyses, not shown here. The minimum which occurs a t 20 < log N e < 21 is the result of the hE(T,N e ) term in Equation 1.When that term is deleted, only a single minimum near the lower Ne value remains, with the calculated concentrations the same as before. In fact, an examination of Equation 1reveals that any results obtained with parameters Neoand T o and with the AE term included can be duplicated by excluding the AE term and changing Neoto Ne* = Neo exp [-AE(To, NeO)/KTO], assuming that the internal partition functions are independent of N e . We therefore conclude that the hE term has no effect on the quality of our analysis. We have included it here only to facilitate comparison with CARISMA results. Andersen and Hinthorne have stated that the inclusion of the AE term improves the accuracy of the CARISMA computer program (4).This apparent discrepancy probably arises from the Ne-dependent partition functions used in the CARISMA program which can break the degeneracy of the double minima seen in Figures 2a and 2b. Whether this results in a significant improvement in accuracy has not been assessed; however, as discussed in a later section, a direct comparison of the CARISMA program with the simplified L T E program using common data sets does not seem to favor the more elaborate CARISMA program.

Figure 2c illustrates the qualitatively different type of contour map associated with the monel data. There is a single minimum which is not well localized and has a high fit value indicating a poor fit to the internal standard elements. The fit of the remaining elements is also unsatisfactory (see Figure IC).I t is possible that such contour maps of fit to internal standards may be a diagnostic aid which would indicate a satisfactory analysis for contour shapes of the type shown in Figures 2a and 2b, and unsatisfactory results for the type illustrated in Figure 2c. Further testing will be required to assess the validity of this hypothesis. P a r t i t i o n Function Approximations. The contribution of the partition functions in Equation 1t o the quality of fit has been evaluated using various approximations. For Case I, the temperature dependence of the partition functions is expressed in fifth-order polynomial form which requires 12 constants per element (8).This may be reduced to six constants with no loss of accuracy by generating a single polynomial for the ratio Bi+(T)/Bio(T).For Case 11, the partition functions are approximated by the statistical weights of the lowest lying group of states with the same azimuthal quantum number L ( 1 3 ) .These are temperature-independent and require two constants per element (or one for the ratio). In Case 111, all partition functions are set equal to 1. The effects of these approximations on the quality of fit are summarized in Table 11. Only a slight loss in accuracy is realized by eliminating the temperature dependence of the partition functions in Case 11. This difference is not statistically significant when the 15% precision of the measurement is considered. There is a significant loss of accuracy when the partition functions are completely eliminated from the Saha-Eggert equation (Case 111),supporting the observation of Andersen and Hinthorne that the partition functions are required to achieve their reported degree of accuracy ( 4 ) . Since the partition functions and statistical weights incorporate information about the relative probability of occupation of the initial atomic and final ionic states of the sputtered particle, it should not be surprising that their inclusion improves the accuracy of the analysis. It appears from this study that the temperature dependence of the partition functions is not necessary to the quality of fit. More extensive testing of this approximation is required before a firm conclusion can be drawn. Definition of E r r o r of Fit. Two alternative definitions of error of fit to the internal standards were tested with the five sample data sets. Both absolute and relative deviations between the calculated and known concentrations were considered. No significant difference between the two cases could be discerned, either for the fit of the internal standards or of the remaining elements. The use of more than two elements as internal standards was also found to make little difference in the analysis. These results are attributable to the excellent fit of the internal standards which usually occurs (see Table I), allowing little freedom for variations of T and N e with definition. The additional constraint imposed by the normalization of concentrations also removes some freedom from the analysis. We do not preclude the possibility that in certain cases a third internal standard might be required to better define the optimum ( T ,N e ) pair. In these instances the contour maps such as Figures 2a-c would be extremely valuable diagnostic aids. Empirical Correlation between T a n d Ne. Andersen and Hinthorne have noted that a functional relationship exists between the T and N e values calculated from their analyses of many standard samples ( 4 ) . They expressed this relation15.67 log T. We have ship by the equation log N e = -44.15 plotted the results of our simplified LTE analyses along with (T,Ne) pairs obtained by others (5, 6, 10, 11, 1 4 ) and have confirmed that such a correlation does indeed exist. Figure 3 indicates that the data are better described by the function

+

log N e = 25.067

- 43 196/T

(3)

especially a t low values of T . This functional form corresponds to a roughly constant, high degree of ionization Ri for a given element i along the curve, where

and Ni+/Niois calculated from Equation 1. The simplified LTE computer program searches an area of ( T ,N e )space with a width of 2 units of log N e on each side of the curve described by Equation 3. This band, shown as the shaded area in Figure 3, encompasses nearly all of the plotted data points. Restricting the computer search to this area greatly reduces the computer time required for an analysis. The reason why the best-fit (T,N e )pair must lie in this region is best illustrated with a particular example. The data from spot C of the stainless steel sample SRM 121d were best fit with the parameters T = 4555 K and log N e = 14.00. In Figure 4, the degree of ionization Ri of each element in the sample, calculated from Equations 1and 4,is plotted a t this value of T as a function of log Ne. The figure illustrates two regions of asymptotic behavior: a t sufficiently low values of log N e ,Ri 1for all elements and a t sufficiently high values Ri Ne-l for all elements. Because of the method of normalizing the calculated concentrations, only the relatiue values of Ri are significant. Hence, the results will be independent ofithe value of N e in either the low or high asymptotic regions. The value of N e can only affect the analysis in the “crossover” region indicated by the shaded area in Figure 4 for this particular value of T . This “crossover” region, which also corresponds to the shaded band in Figure 3, is therefore the only part of (T, N e ) space which need be searched. The fit in the “crossover” region need not always be better than the fit in the asymptotic regions. In most cases, we do find that the optimum ( T , N e ) values do fall within the boundaries of the crossover band. However, in some analyses, e.g., the ,monel, we have found that the true optimum values of T and N e fall outside of the crossover band (see Figure 2c). In these cases, the restricted search produces optimum (T,N e )values on the edge of the band, with an insignificant loss in fitting accuracy when compared with the results of an unrestricted search. Direct Comparison between Simplified L T E a n d CARISMA Analyses. In order to test whether the complexity of the CARISMA computer program produces a more accurate analysis than the simplified LTE program, a comparison has been made using data produced by both the AEI IM20 and ARL IMMA ion microprobes (D. E. Newbury, National Bureau of Standards, Washington, D.C., provided unpublished data from the ARL IMMA and processed the data presented here through the CARISMA program.) The samples run on the AEI instrument are those previously discussed, and the results of the comparative analyses are shown in Table 111. In most cases, the simplified LTE analysis produces a slightly better fit, although only in the low alloy and stainless steel results are the differences considered statistically significant. An indication of the distribution of fit to the AEI data is given in Table IV, where the percentage of determinations falling in a given range of F ( F = calculated/bulk concentration) is shown for each of the computer programs. For the simplified LTE case, only 2% of the determinations have a discrepancy larger than a factor of 5, while the corresponding figure for the CARISMA analysis is 11%.The larger percentage of major errors in the latter case is primarily attributable to the theoretical oxide correction procedure used in the CARISMA approach ( 4 ) .When the theoretical oxide correction is bypassed and the empirical one substituted in the CARISMA program, the number of large errors is reduced substantially and the -+

ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

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Table 111. Comparison of Simplified LTE and CARISMA Analyses of AEI IM20 Data Av. error remaining elements,

Program Internal stds LTE Fe,Ni CAR Fe,Ni LTE Fe,Cr,Ni CAR Fe,Cr,Ni LTE Ni,Cu CAR Ni,Cu LTE Si,Al CAR Si,Al LTE Si,Ba CAR Si,Ba

Material SRM 661 Low Alloy Steel SRM 121d Stainless Steel SRM 162a Monel Lake County Labradorite K252 Glass

TW) 2 630 13 840 4 948 5 020 4 805 4 200 9 180 13 000 6 005 9 760

log Ne

6.73 20.70 16.33 15.38 18.88 14.27 18.23 20.63 15.65 19.46

Table IV. Distribution of Fit of Simplified LTE and CARISMA Analyses AEI IM20 ARL IMMA Range 0.5 5 F I 2 0.2 I F 5 5

LTE 82% 98%

CARISMA

68% 89%

LTE 83% 97%

CARISMA

81% 90%

results are closer to those of the simplified L T E analysis (10, 11). The comparison of the two computer programs using common data from the ARL IMMA ion microprobe is summarized in Table V. A primary beam of l 6 0 - ions was used a t an impact energy of 21.5 keV. The beam was rastered over an area 75 pm square. All samples are from the NBS K glass series, and each contains between 4 and 6 cations. These glasses are not Standard Reference Materials, and the composition is assumed to be that of the initial mix. Most data sets were analyzed with at least two different pairs of internal standard elements. The results in this table do not indicate a bias in favor of either program. However, the distribution of fit shown in Table IV for the ARL IMMA data again reveals a substantially larger percentage of major errors with the CARISMA analysis. This table also indicates that the simplified L T E program works equally well with data from either instrument. The analysis of the K326 sample in Table V is of special interest. Three different pairs of internal standard elements have been used, and in each case the accuracy of the analysis is excellent. However, the value of T varies by a factor of 2 or more depending on the internal standard pair chosen, an observation which has also been made by others (10, 2 1 ) . Such a result strongly argues against a physical interpretation of the T and N e parameters as a true temperature and electron density, since their values can vary widely depending on which elements are chosen as internal standards in a given data set. Similarity between LTE a n d O t h e r Thermodynamic Models. The Saha-Langmuir equation, originally used to describe surface ionization with a heated filament (15),has also been discussed in relation to sputtered ion formation (16). This equation is derived from the Saha-Eggert equation under the additional assumption that the electrons in the gas are in equilibrium with a surface of work function 4 and temperature T . It has the form:

where the symbols have the same meaning as in Equation 1. The applicability of Equation 5 to sputtered ion formation has been questioned because it predicts ion/neutral ratios which are many orders of magnitude lower than those observed when bulk sample temperatures and experimentally determined work functions are used for T and 4 (16). However, in the same spirit in which T and N e are determined in Equation 1, 1346

ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

Av. error internal stds, % 0

%

73 440 30 84 125 115 14 25

2 2

6 17 18 0 0 0

10

8

28

we can allow T and 4 to be fitting parameters in Equation 5 . The two equations then become formally identical when

+

4 = kT In (AT3I2/Ne) AE

(6)

Therefore, any of the fits obtained using the Saha-Eggert equation with parameters T and N e can be duplicated using the Saha-Langmuir equation with the same T value and a 4 value calculated from Equation 6. Values of 4 determined in this way are listed in Table V for the NBS glasses. These values are uncorrelated with the corresponding T parameter, and range between about 7 and 9 eV. The average value of 4 for all of the glasses is 8.1 eV ( u = 0.5 eV) from the simplified L T E parameters and 8.0 eV (u = 0.6 eV) from the CARISMA parameters. Andersen has used a Richardson plot to derive a work function value of 11.18 eV for a group of silicates (7). The physical significance, if any, of these work function parameter values is obscure because the state of the sample surface under ion bombardment is not well understood. Jurela has objected to the Saha-Langmuir equation on the basis that temperatures higher than the thermodynamic critical temperature of the material were required to explain the sputtered ionheutral atom ratios from pure metals and semiconductors (16). He suggested that the Dobretsov equation

was more appropriate for a nonequilibrium process such as sputtered ion emission. Here AI = e2/4x, is the lowering of the ionization potential due to the surface image force at the critical distance for charge exchange. Using Equation 7, Jurela was able to fit the measured ion/atom emission ratios from a number of pure elements with values of T below the critical temperature. He assumed that x, = 0.55a, where a is the largest lattice parameter, and used literature values for the work function 4. We note that the Dobretsov equation is also compatible with our two-parameter fits to multielement samples if the value of 4 determined using Equation 6 i s the sum of a work function and the image force term AI. For the glasses, the work function derived from the Dobretsov equation would then be several electron volts lower than the values given in Table V. The exact value would depend on the choice of x,. For example, if x, = 3 A, the work function value would be reduced by 2.18 eV. The three thermodynamic models represented by Equations l, 5, and 7 give equivalent results when two parameters are free to vary in each equation. T h a t a reasonable fit to the data can be achieved does not by itself validate the physical assumptions of a particular theory. Such a validation requires independent corroboration or justification of the values found for the parameters. The high temperatures required to fit sputtered ion data are usually justified on the basis of-a thermal spike model (7, 16). However, a true corroboration would require a n additional simultaneous determination of

-Table V. Comparison of Simplified LTE and CARISMA Analyses of ARL IMMA Data from NBS Glasses Av. error remaining elements, Material Internal stds Program T(K) log Ne 4(eV) Av. error internal stds, % % K249 Si, A1 LTE 4085 11.01 8.18 20 72 Si, A1 CAR 5 060 12.89 8.39 19 72 K326 Si, Mg LTE 6 740 15.21 8.33 0 14 Si, Mg CAR 7 020 15.34 8.53 0 17 0 11 K326 Si, B LTE 6 170 14.63 8.26 Si, B CAR 6 640 15.01 8.45 0 12 LTE 12035 18.71 7.81 10 11 K326 Mg, Ca 0 7 CAR 16120 20.26 7.69 Mg, Ca 77 K252 Si, Ba LTE 3 795 10.84 7.69 0 17.82 6.92 12 48 8 960 CAR Si, Ba 12.59 7.79 0 55 4 570 LTE K252 Si, Zn 10.89 7.67 22 79 3 800 CAR Si, Zn 17.16 9.06 40 42 10000 LTE K409 Si, Fe 18.61 8.38 27 24 12540 CAR Si, Fe 8.30 3 63 3350 8.49 LTE K409 Si, A1 10.61 8.98 4 62 4 300 CAR Si, A1 18.43 8.68 24 32 12445 K411 Si, Fe LTE 18.61 8.02 1 17 12060 CAR Si, Fe 8.56 5 48 11720 18.21 K411 Si, Mg LTE 19.95 8.39 0 47 16840 CAR Si, Mg K309 Si, Ca LTE 10250 18.29 7.21 0 14 Si, Ca CAR 13780 20.90 7.55 0 23 7.52 0 20 8 375 17.09 LTE K309 Si, A1 17.38 7.50 0 8 8 820 CAR Si, A1

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the temperature based on one or more of the following measurements of the sputtered species (17, 18): negative ion/ neutral atom ratios; doubly charged ionlsingly charged ion ratios; molecular ionlneutral molecule ratios; optical emission intensity ratios of excited atoms or ions; kinetic energy distributions of secondary ions or electrons. Andersen has recently discussed whether the data from each type of measurement above is internally consistent with thermal equilibrium (7). His results are inconclusive, since the temperatures of the various species mentioned above have generally been determined from different samples. The energy distribution of sputtered ions has been well established (7,19, 20). The high energy tails are non-Maxwellian and the most probable energies of several eV correspond to temperatures an order of magnitude higher than those deduced from the Saha-Eggert equation. This evidence suggests that a local thermal equilibrium does not exist at the sputtering site. The work function $C is another parameter which might be independently determined. This would require a measurement under conditions simulating ion bombardment-that is, an oxygen-implanted, damaged, amorphous surface. Blaise and Slodzian have developed a technique for measuring work function changes of a surface under ion bombardment in a secondary ion microanalyzer (21). Such a measurement could test whether the 4 parameter derived from a fit of sputtered ion data to the Saha-Langmuir equation is physically meaningful. CONCLUSION This study has verified that a correction procedure based on an equation of the Saha-Eggert form can be used to convert relative sputtered ion intensities to elemental concentrations, with semiquantitative accuracy in most cases. A comparison of this procedure with the CARISMA computer program, based on data from 11different samples and 2 different ion microprobes, suggests that the corrections for negative ions and for the lowering of the ionization potential in a plasma do not measurably improve the accuracy of the analysis. In addition, the comparison indicates that the theoretical correction for oxide ions is more likely to produce large errors than an empirical oxide correction procedure.

The fitting of sputtered ion data with the Saha-Eggert equation does not by itself prove that local thermal equilibrium exists a t the sputtering site. It is a necessary but not a sufficient condition for LTE. In order to establish the existence of thermal equilibrium, each type of species must be shown to obey the appropriate equilibrium statistical equation and the temperatures determined for each type must have the same value (17, 18).Until such evidence can be shown, the physical basis of the L T E model as applied to the sputtering process will remain in question. Despite these reservations, the Saha-Eggert and the other thermodynamic models have been undeniably successful in predicting the relative degrees of ionization of sputtered ions. This success can be rationalized using an empirical argument. The exponential dependence on ionization potential is a likely choice to account for the well established periodicity of the sputtered ion yield as well as the large magnitude of variation from element to element ( 3 ) .A fitting parameter here (5") sets the scale of this variation. The ratio of partition functions or statistical weights of the ion and atom accounts for density of state differences. A second degree of freedom is gained by adding another fitting parameter ( N e ) to allow for saturation of the degree of ionization. ACKNOWLEDGMENT The authors thank Dale Newbury of the National Bureau of Standards for providing unpublished data and for data processing assistance and Peter Williams for valuable discussions throughout this research. LITERATURE CITED (1) J. A. McHugh, in "Secondary ton Mass Spectrometry", K. F. J. Heinrich and D. E. Newbury, Ed., NBSSP-427, U.S.G.P.O.,Washington, D.C., 1975, pp 129-134. (2) J. M. Schroeer, T. N. Rhodin, and R. C. Bradley, Surf. Sci., 34,571 (1973). (3) C. A. Andersen and J. R. Hinthorne, Science, 175, 853 (1972). (4) C. A. Andersen and J. R. Hinthorne, Anal. Chem., 45, 1421 (1973). (5) R. Shimizu, T. Ishitani, and Y. Ueshima, Jpn. J. Appl. Phys., 13, 249 (1974). (6) R. Shirnizu, T. ishitani, T. Kondo, and H. Tamura, Anal. Chem., 47, 1020 (1975). (7) C. A. Andersen. in "Secondary ton Mass Spectrometry", K. F. J. Heinrich and D. E. Newbury, Ed., NBS SP-427, U.S.G.P.O.,Washington, D.C., 1975, pp 79-119. (8) L. de Galan, R. Smith, and J. D. Winefordner, Spectrochim. Acta, Part B, 23, 521 (1968). ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

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(9) H. W. Drawin and P. Felenbok, "Data for Plasmas in Local Thermodynamic Equilibrium", Gauthier-Villars, Paris, 1965. (IO) D. E. Newbury, K. G. J. Heinrich, and R. L. Mydlebust, Proceedings of ASTM Symposium on Surface Analysis Techniques for MetallurgicalApplications, Cleveland, Ohio, March, 1975, ASTM Special Technical Publication

596. (11) D. E. Newbury, National Bureau of Standards, Washington, D.C., personal communication, 1975. (12) P. Williams and C. A. Evans, Jr., in "Secondary Ion Mass Spectrometry", K. F. J. Heinrich and D. E. Newbury, Ed., NBS SP-427, U.S.G.P.O., Washington, D.C.. 1975,pp 63-68. (13)C. E. Moore, "Atomic Energy Levels", Vols. 1-111, NSRDS-NBS 35, U.S.G.P.O., Washington, D.C., 1971. (14)J. F. Lovering, in "Secondary Ion Mass Spectrometry", K. F. J. Heinrich andD. E. Newbury,Ed., NBSSP-427, U.S.G.P.O., Washington, D.C., 1975, pp 135-178. (15) M.J. Dresser, J. Appl. Phys., 39,338 (1968). (16)2. Jureia, Int. J. Mass Spectrom. /onPhys., 12, 33 (1973),and references therein.

(17)P. W. J. M. Boumans, "Theory of Spectrochemical Excitation", Plenum Press, New York, 1966,pp 78-80. (18)H. W. Drawin, in "Reactions under Plasma Conditions", M. Venugopalan, Ed., Vol. 1, Wiley, New York, 1971,pp 123-125. (19)G. Carter and J. S. Colligon, "Ion Bombardment of Solids", American Elsevier, New York, 1968,p 88. (20) R. F. K. Herzog, W. P. Poschenrieder, and F. G. Satkiewicz, Radiat. Eff., 18, 199 (1973). (21) G. Blaise and G. Slodzian. Surf. Scb, 40, 708 (1973).

RECEIVEDfor review January 12, 1976. Accepted April 26, 1976. This research was supported in part by the National Science Foundation under Grants DMR-72-03026 and MPS-74-05745.

Ana Iys is of Organic Mixtures Using Metastable Transition Spectra E. J. Gallegos Chevron Research Company, Richmond, Calif. 94802

Metastable transitions obtained by accelerating voltage scanning are used in the analysis for specific organic components in a mixture. This technique Is described and applled to the analysis of terpanes, steranes, and phthalates in complex organic mixtures. The results obtained using metastable transition spectra and normal mass spectra compare well with that obtalned by GC-MS.

This paper describes the use of metastable transition (MT) spectra in the analysis for specific organic components in a mixture. In many cases, this technique gives much of the same information obtained by single ion detection GC-MS, the advantage being that M T spectra are obtained in only a small fraction of the time required for GC-MS. M T analysis is based on the ability of a double-focusing mass spectrometer to detect specific ionic decomposition processes and identify products and related progenitors without interference from other reaction processes and their related ion participants. This, in effect, provides an isolation technique with a resulting capability similar to GC-MS. The experimental technique is described, and three examples of its application are given. These are terpane, sterane, and alkylphthalate analysis in complex organic mixtures. EXPERIMENTAL Metastable transitions in mass spectrometry were first reported by Hipple and Condon ( I ) in 1945. The theories of metastable transitions and the techniques of obtaining these kinds of data have been summarized recently (2,3 ) . This work was done on an MS-9 double focusing high resolution mass spectrometer. The instrument was modified so that the accelerating voltage can be scanned from 2-8 kV. The electrostatic sector voltage is held constant at 170 volts which, with an accelerating voltage of 2 kV, will produce normal mass spectra. The sample is introduced through an all-glass, hot inlet ( 4 ) or by using a direct insertion probe. The magnet is adjusted to a current which will bring into focus a daughter ion Mz+ of interest. The accelerating voltage is then scanned from 2-8 kV to produce the metastable transition spectra used in this

study. This arrangement will allow metastable transitions to be observed due to precursor ions MI+ with a mass up to four times that of the daughter ion Mz+. 1348

ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976

RESULTS M T analysis is possible only if a decomposition process can be found that is uniquely characteristic of a compound or compound type. There are many systems which obey the requirements for M T analysis. Three examples of this are given. Terpanes and Steranes. Terpanes and steranes are two systems that fall well inside these requirements. Many terpanes, either pentacyclic- or tricyclic-saturated hydrocarbons found in nature, fragment on electron impact t o give a base peak at m / e 191 fragment ion ( 5 , 6 ) .See Figure 1.

Similarly, steranes which are saturated tetracyclic hydrocarbons fragment to give a base peak at mle 217 fragment ion. See Figure 2. Figures 3 and 4 show the metastable spectra of authentic cholestane, a C27 sterane, and authentic gammacerane, a C30 terpane, respectively. Each shows only one major metastable peak. I n each case, it is due to decomposition of the molecular ion, M f , to give the characterizing fragment ion m/e 217 in the case of cholestane and m/e 191 in the case of gammacerane. The well-analyzed saturated fraction of the extractable portion of Green River shale (7) is used to demonstrate the use of the method. Figure 5 shows the metastable scan of m/e 191 of this fraction. The most important peaks are those due to C31, C30, and C29, followed by the C20 and C2l terpanes. Similarly, Figure 6 shows the metastable results for a scan of m / e 217. These results show that the largest metastable peak is due to a C29 sterane followed by the C28 and, finally, C27 steranes. The normal mass spectrum was used to calculate the total concentration of terpanes and steranes in this sample. This is done in the following manner. The mle 191 fragment ion measured 2.33% of the total ionization of the saturate fraction of Green River shale. This fragment ion represents one eighth of the total ionization in the mass spectrum of the average terpane in this sample so that the total terpane contribution to the total ionization of the sample is 18.6%. The m / e 217 fragment ion measured 1.25% of the total