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Anal. Chem. 1986, 58,2300-2306
(4) Woodard, F. E.; Goodin, R. D.: Kinlen, P. J. Anal. Chem. 1984, 56, 1920. (5) Blagg, 8.;Carr, S. W.; Cooper, G. R.; Dobson, I. D.; Gill, J. B.; Goodall, D. C.; Shaw, B. L.; Taylor, N.; Boddington, T. J. Chem. Soc.. Dalton Trans. 1985, 1213. (6) Bard, A. J.; Faulkner, L. R. Nectrochsmicai Methods; Wiiey: New York, 1980; pp 236-242. (7) Abramowitz, M.; Stegun, 1. A. Handbook of Mathematical Functions; U.S. Government Printing Office: Washington, DC, 1964: section 7.2. (8) Pilla, A. J. Nectrochem. SOC. 1970, 717, 467. (9) Smith, D. E. Anal. Chem. 1976, 4 8 , 517A. (IO) Suprenant, H. L.;Ridgway, T. H.; Reilley, C. N. J. Electroanal. Chem. 1977, 75, 125.
(11) OMham, K. 8.J. Electroanal. Chem. 1982, 136, 175. (12) Anderson, J. E.; Myland, J. C.; Oldham, K. 8..submitted for publication in J . Nectroanal. Chem . (13) OMham, K. B.; Spanier, J. The Fractional Calculus; Academic Press: New York, 1974; pp 136-148.
RECEIVED for review March 5, 1985. Resubmitted April 24, 1986. Accepted April 24,1986. The generous financial support of the Natural Sciences and Engineering Research Council of Canada is acknowledged with gratitude.
Ion Intensity and Image Resolution in Secondary Ion Mass Spectrometry Margaret E. Kargacin and Bruce R. Kowalski*
Laboratory for Chemometrics, Department of Chemistry, BG-IO,University of Washington, Seattle, Washington 98195
wlth secondary Ion mas spectrometry (SIMS), mass spectra can be generated as a functkm of the rample surface spatlal coordlnetes. Often In a oample analyds, however, the nmbw d wrface components,thek characterktic mass spectra, and the extent of beam damage are unknown. Relylng on slngle peak Intensiilesto represent lndlvidual components can lead to error In the qualttatlve and quantltatlve interpretation of SIMS spectra or Ion images. The methods of cross-vaiklatlon and factor analyols are presented as a means for estimathrg the true number of components In a muttlcomponent sample. A multlvarlate curve resolutlon procedure Is used In the analysts d SIMS data from two and three component M u r e samples to estimate the pure component spectra and the relative intensity contrlkrtion of each component in the mlxture spectra. These methods are then applied to the resobtlon of lndlvklual colrponents over the surface of a sample uslng SIYS Ion Images and Image processing.
Secondary ion mass spectrometry (SIMS) is a widely accepted method for the characterization of surfaces and is often used in the analysis of thermally labile or nonvolatile organic compounds. The low detection limits obtainable with SIMS and the ability to analyze for all elements and isotopes have contributed to the importance of SIMS as an analytical method. Secondary ion mass spectra, containing monatomic and molecular ion peaks, yield information about the elemental composition of a surface and can give chemical information as well. Additionally, mass spectral information can be collected as a function of sample spatial dimensions. A depth profile analysis is an example with one spatial dimension. A two-dimensional description of a sample can be obtained by measuring series of ion images of different m / e values. Here an ion image consists of the secondary ion intensity for a particular m / e value peak at each lateral point (pixel) sampled (I, 2). In practice the complete analysis of SIMS spectra collected over a surface involves three steps. First, the number of surface components is determined, where a component is defined as an element, compound, or mixture that gives a distinct unchanging mass spectrum over the surface. Second, the components are identified using the spectra of the com-
ponents determined to be present. Third, the spatial distribution of the components over the sample surface is estimated. The SIMS spectra contain the information needed to determine the number of distinct surface components and their spectra. However, this information may not be easily extracted. A single component spectrum may contain monatomic and molecular ion peaks. Often more than one component is sampled at one time resulting in spectral interference between components ( 3 , 4 )making the determination of the number of components and their identification difficult. Components that have spectral peaks a t the same m / e values, when present in a multicomponent sample, will give extreme spectral overlap. An example of such overlap has been reported for the negative secondary ion mass spectra measured for the inorganic salts Li2S04,Na2S03,Na2S04,and NazS2O3(5).The same m / e value peaks appear in the negative ion spectra of each salt. None of the four salts, when analyzed alone, give spectra containing a unique signal that can serve to distinguish it. The different salts and their anion stoichiometry are distinguishable only by their relative peak intensities in the negative ion spectra. Other such examples include the SIMS analysis of poly(alky1methacrylates) (6) and the SIMS analysis of different oxides of Cr (7). Existing empirical and semitheoretical methods (1,8)for quantitation in SIMS can yield acceptable results but involve calculations in which an individual component is represented by a single mass spectral peak. Best results are achieved only if the peaks chosen are representative of the pure component and free from spectral interference. Proper peak selection then depends on prior knowledge of the pure component spectrum. Additionally, even if a unique mass is used for quantitation, the information contained in the other masses is wasted. Multivariate statistical methods, customized for the analysis of multicomponent spectral data, have been applied in analytical chemistry for interpretation of spectra from methods such as GC-MS, LC-UV, and fluorescence spectroscopy (9-12). These multivariate methods address some of the same problems often encountered in the analysis of SIMS spectra, namely, lack of knowledge of pure component spectra and/or presence of spectral interferences, and may therefore be useful in interpreting SIMS spectra. With one exception involving
0003-2700/86/0358-2300$01.50/00 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1988
the solution of simultaneous equations (131,multivariate methods have yet to be applied in SIMS spectral analysis. Since a series of SIMS ion images can be viewed as a spectral data set, with one mass spectrum per pixel, the multivariate methods of spectral analysis that hold promise for analysis of SIMS data sets with only one spatial dimension may also be applied in SIMS image analysis. A measured SIMS spectrum can be related to the surface component concentrations as described in the following formulation where a measured mass spectrum is represented by a column vector xi
xi = ( x l , x 2 ,
...,x j , ...,x p ) T
and x i is the intensity of the j t h m / e value peak and P is the total number of peaks monitored in the mass spectrum. A mixture mass spectrum, xi can be expressed as the sum of the spectra of the pure components according to NC
sample components is then estimated as the number of eigenvectors associated with systematic variation in the spectra. The cross-validation method involves the deletion of one or more of the experimental spectra followed by calculation of the eigenvectors using the reduced data set. The spectral intensities deleted are then predicted as a function of the number of eignevectors used in the following: D
x’(k)ij =
CUi,kUkj k=l
for i = 1, N’and j = 1, P
THEORY In factor analysis a data matrix of ion intensities, X N x p , consisting of N mass spectra with P different m / e values, can be represented as the product of two matrices U and V
x=uv
(3)
Where the matrix V, of “loadings“, is composed of P rows, each row being one eigenvector of the data scatter matrix, X%/N. The matrix U,of “scores”, consists of the projections of the N experimental mass spectra onto the P eigenvectors. The goal of cross-validation, as used here, is to determine the number of eigenvectors that are representative of systematic variation in the data set (Le., associated with changes in the relative amounts of the components, and not representative of random variation due to noise). The number of
(4)
The term ~ ’ ( kdenotes ) ~ ~ the prediction of the jth mass spectral intensity in the ith held out spectrum when calculated using the first k eigenvectors. U,,k is the score of the ith spectrum on the kth eigenvector, v k j is the j t h element of eigenvector k. N’ is the number of held out samples. Prediction error is then calculated as a function of the number of eigenvectors used in the calculation of ~ ‘ ( k ) This ~ ) quantity, the prediction residual error sum of squares, designated as PRESS(k), is given by N’ P
where i, is the primary ion current with units of ions per second, t is the count time in seconds, and q is the collection efficiency equal to the (number of ions detected)/(number of ions sputtered). sk, the sputtering yield, is equal to the number of sputtered particles per incident ion. I+k is the ionization yield or the degree of ionization for component k . x k is the j t h peak from the kth component spectrum, normalized to unit area, and cip is the mole fraction of component k in mixture spectrum i. NC is the total number of components. In gas-phase mass spectrometry the spectra of individual components in a multicomponent sample add linearly to give a mixture mass spectrum. In SIMS, matrix effects, or interactions between components, may lead to additional terms in the model (14).In sample systems where such interactions are absent, the mixture spectra can be considered to be linear superpositions of the component spectra and eq 2 is valid. It is necessary to test the validity of eq 2 for the sample under investigation because in SIMS eq 2 is often not true. This can be done by comparing the results of a cross validation with information obtained about the sample composition from sample preparation, visual examination, or complementary analytical methods. In this paper the results of the analysis of two SIMS data sets, varying in degree of spectral interference, in the number of components, and in the number of spatial dimensions are used to illustrate the use of a cross-validation method (15) involving factor analysis for estimating the number of components in the mixture spectra. A multivariate curve resolution procedure (9, IO),applicable when eq 2 is valid, is then tested as a method for finding pure component spectra and for estimating the relative spectral intensity of each component in mixture spectra.
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PRESS(k) =
C ( C ( x i j i = l j=1
-~’(k),j)~)l/’
(5)
The procedure is repeated with the deletion of a different portion of the full data set until each spectrum has been held out one time only. The PRESS(k) values are summed in each cycle. From a plot of PRESS&) for the entire data set vs. k , the number of components is estimated to be equal to the value of k at which the plotted PRESS(k) curve drops to a level of steady but slow decline. Relatively large decreases in the value of PRESS&) are seen as eigenvectors that account for compositional variance in the data are included in the calculation of PRESS(k). But, when eigenvectors representing noise contributions to the data are included, the PRESS&) value declines very slowly because the noise present in the group of spectra, for which the eigenvectors are found, are not useful for modeling the noise in the spectra held out. Recently Borgen and Kowalski (16)generalized multivariate curve resolution and developed algorithms for the three component case to allow determination of pure component spectra and estimation of the component signal contributions in mixture samples. In both the two and three component curve resolution procedures each mixture mass spectrum, xi, is expressed as a linear combination of the pure component spectra
where NC indicates the total number of components and x*k is a vector representing the kth pure component mass spectrum. ai,k is the relative contribution of the kth component in xi and NC
= 1.0 k=l
(7)
The measured spectra and pure componentspectra can each be expressed as a linear combination of the first NC eigenvectors of the data scatter matrix, XTX/N NC Xi
=
xUi,kVk k=l
i = 1, N
(8)
where Ui,k are scores or projections of the ith spectrum onto the kth eigenvector, vk. The asterisk indicates pure component spectra and scores. To estimate the unknown pure component spectra, the scores, U*i,k, must be found.
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
Table I. Operating Conditions" parameter beam current (gA) beam voltage (kV)
0.220 2.0
Y
-6.36
t-----3
2.55 -38.2 8.50
E; (V)
sample bias (V)
a30000 fis/pixel; 64 X 64 pixels; images collected at 46, 48, 49, 50, 55.
56, 57, 62, 63, 64, 65, and 66 amu.
For both the two and three component cases three restrictions enable estimates of these scores to be made. The first restriction arises from the fact that in mass spectra the peak intensity at each mle value must be nonnegative. The second restriction states that each measured spectrum be a linear combination of nonnegative amounts of the pure components; thus all uikare 10. The third restriction results from normalization of each spectrum to unit area. In the two component case the spectra can be represented in a two-dimensional plot by their score pairs ui,l and u ~ , ~ These score pairs are found by projection of the measured spectra onto the first and second eigenvectors. In this plot the three restrictions confine the location of the pure component score pairs to the normalization line segments. Possible pure component spectra can be calculated from score pairs located on the normalization line, one score pair from each of the solution bands (9, 10). The normalization line, with the pure component spectra located a t the two ends, constitutes a line analogous to a mole fraction line. The distance between a mixture point and the pure component points indicates the relative intensity associated with each component in the mixture spectra. In the three component case the spectra, represented by their scores on the three major eigenvectors (eq 8 and 91, will fall on a plane in a three-dimensional space. This plane can be viewed as a projection onto the plane defined by the second and third eigenvectors. By use of the three restrictions as before and algorithms developed by Borgen and Kowalski (16), allowed regions in this plane can be found for the score values in eq 9. Use of one score triplet from each of the allowed areas enables estimates of the pure component spectra to be calculated (eq 9). The three allowed regions form a triangle, analogous to a mole fraction triangle, from which the relative intensity of any of the three components in a mixture spectrum can be calculated.
EXPERIMENTAL SECTION In this paper sets of mass spectra are analyzed. The spectra in the first set were taken from the literature (13)and the pure component spectra of Cr02,Cr03, Cr203,CrO, and vacuum melted Cr were obtained from H. Werner, Phillips &search Laboratories.
Figure 1. Diagram of iron oxide brown and titanium white sample. Area 1 samples are titanium white paint alone, areas 2 through 5 are mixtures containing progressively more Fe brown paint. Area 6 is Fe brown alone.
A 2.5-h depth profile analysis was performed on the sample as described (13)using a Cameca IMS-300 spectrometer with 5.5-keV Ar primary ion beam. The metal and pure oxide spectra were obtained under the same conditions. Spectra for the second data set were collected from samples of a titanium white paint, an iron oxide brown acrylic paint (M. Grumbacher, Inc., New York), and four mixture samples which were painted onto a silver support in a checkered pattern as shown in Figure 1. Each area in the figure is numbered according to its composition. Ion images were collected with an 40Ar+beam . at 46, 48, 49, 50, 55, 56, 57, 62, 63, 64, 65, and 66 amu with a modified 3M (KFUTOS)535-BXsecondaryion and ion scattering spectrometer (minor peaks, for example, the Fe isotope peaks at 54 and 58 m u , were not imaged because they were of much lower intensity). Each image consisted of a grid of 64 X 64 pixels. The area of the sample analyzed is shown as shaded in Figure 1. The analyzed area measures 4.0 mm along the sample axis designated as the r axis and 6.5 mm along they axis. Operating conditions for the spectrometer are given in Table I.
RESULTS AND DISCUSSIONS Oxidized Chromium Metal, Depth Profile. The object of this experiment was to determine the number of oxides formed on the surface, to identify the oxide(s), and to calculate a concentration profile of the oxide with depth. The depth profile data present a difficult situation for spectral resolution because the possible pure components have very similar spectra and none have unique mass peaks. In the course of the depth profile, peak intensities for the positive ions Cr+, CrO+,Cr2+,Cr20+,Cr3+,Cr30+,and Cr302+were recorded (13). The ion intensities for the same mass ions for Cr metal and the four pure oxides were also obtained for comparison with the curve resolution estimates of the pure components. In the data analysis the intensity of the Cr30+peak is not included because its value was much higher in the depth profile than in any of the pure oxide spectra. In Table I1 normalized spectra are listed for the metal and oxides. Inspection of the spectra shows that differences exist between the individual samples. All oxides vary from the metal especially in ion intensity for the Cr+ ion and the cluster ions Crz+and Cr3+. The differences between the individual oxide spectra are not as great but they can be distinguished
Table 11. Spectra Normalized to Unit Area peak intensity Cr20+
Cr+
CrO+
Cr2+
0.587
0.012
0.176
0.035
0.188
0.015
Cr203
0.960 0.939 0.975
0.020 0.013
0.015 0.027
Crop CrOB
0.901
0.089
0.009 0.013 0.003 0.001
9.6 5.6 9.7 4.5
component one
0.939
0.018
0.013
0.031
0.000
0.002
0.551
0.004
0.178
0.084
0.171
0.016
sample
Cr metal CrO
0.010 0.009
Cr3+
x 10-5
x x 104 x 10-5
Cr302+ 0.003 3.6 x 10-4 7.8 X IO-" 1.6 x 10-4 2.5 x 10-4
Curve Resolution Results
component two
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
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Table 111. Root Mean Squared Error" for Comparison of Curve Resolution Estimator of Pure Component Spectra and Spectra of Pure Cr Oxides and Cr Metal
comparison spectra Cr CrO Cr203
CrOz CrOB Figure 2. PRESS(K) vs. K calculated for the depth profile of oxidized Cr metal. This plot indicates that two components are present.
IW
~
C
N
V
~
from one another. (In ref 13 it was noted that the presence of chromium-oxygencluster ions indicates that the chromium metal sample was slightly oxidized.) Eigenanalysis of the normalized depth profile spectra resulted in two major eigenvectors, the f i t accounting for 98.8% of the total data variance and the second for 1.17%. The plot of PRESS vs. the number of components, Figure 2, suggests that only two components are present. In the curve resolution two dimensional score plot for the experimentalspectra, Figure 3, the locations of the first and last spectra are labeled. The spectra fall in order on the normalization line and deviations from the line are small. This is further evidence that only two componentsare present and that rather good experimental techniques were employed. The solution band is narrow for the component most like the first measured spectrum and arbitrarily called the first component. This indicates that the first measured spectrum is close to the outer bound spectrum for component one. The solution band is wider for the second component. The solution spectra for both components calculated at the inner bounds of the solution bands are also listed in Table 11. On the basis of values calculated for the root mean square error between the solution spectra and those measured for pure oxides and Cr metal (Table 111), the two components can be identified. The f i s t component is seen to be most like that of Cr203while the second component spectrum is most like that of Cr metal. The relative intensities of the oxide and metal at each point in the depth profile are shown in Figure 4. The first six spectra in the depth profile are essentially due to the Cr203 with the relative contribution from the metal steadily increasing in the remaining spectra. Werner and co-workers (13)concluded that three components were present and that CrO was the third. The curve resolution result presented here is different but required fewer assumptions about the experimental spectra to be made. The
0.026 0.196 0.184 0.202 0.180
0.176 0.011 0.004 0.018 0.034
"RMS error = C,'ln ( i j - ~ , ) ' / ~ /where n , n = 6 , i , = curve resolution estimate of peak intensity at m / e = j . x, = peak intensity at m / e j measured from pure oxide or pure metal sample.
031
Figure 3. Twodimensional spectra representation of Cr oxide-Cr metal spectra as projections onto the first two eigenvectors of the covariance matrix. Spectra points are represented by the open clrcles. The narrow, upper solution band corresponds to component one, the oxide. The wider, lower band corresponds to Cr metal, component two.
root mean squared error component 1 component 2
1
A
s p o c I R u M ~
Flgure 4. Depth profile plot of Cr and Cr,OS. The relative intensity of each component, plotted vs. spectrum number, shows the oxide contribution decreasing with depth. Bands are used instead of lines in order to represent the ambiguity present in the resolution of the pure spectra (9- 7 7).
author, using only the peak intensities for Cr+, Cr2+,and Cr3+, assumed that the first spectrum was due solely to Cr203and the last was from Cr metal only. In the curve resolution no assumptions were made about the composition or identity of the first or last spectra. Also, CrO gives rise to a spectrum very similar to Cr203. Studies of the oxidation of metals and formation of oxide layers are common applications of SIMS. The identification of a metal oxide is possible only when its spectrum is resolved from the metal spectrum and those of other oxides. The use of higher mass spectral resolution would not improve the spectral resolution in such a case as presented here but instead would only lower the signal intensities. However, in this experiment curve resolution was used successfully to achieve spectral resolution, allowing calculation of component spatial profiles. Image Sample. In this experiment, images were collected for 12 different ions measured on a silver support painted with white and brown acrylic paints. Peaks due to Ti+and TiO+ with main isotopes at 48 and 64 amu, respectively, were expected from the titanium white paint. From analysis of samples of the brown paint, a peak at 56 amu from Fe and peaks at 55 and 57 amu were expected. The latter two peaks are presumably from organic compounds in the paint. At least two components, one for each paint, were expected to be found in the curve resolution analysis (the inorganic and organic compounds making up a single paint are expected, if homogeneously distributed, to comprise one component). The ion images (64 X 64 pixels) for 48,56, and 64 amu are shown in Figure 5. The expected checkerboard pattern is not observed in any of the ion images. The ion images do not reflect the sample composition for three possible reasons. First, low spectral resolution obscures the differences between compositionally different areas. Second, the ion collection efficiency is lower for ions sputtered away from the center of the imaged area. Third, the use of a beam with a diameter of 1000 Mm in imaging the 4.0 X 6.5 mm area causes consid-
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ANALYTICAL CHEMISTRY. VOL. 58. NO. 11. SEPTEMBER 1986
K F@N 6. FRESWK) v8. Kpbi cabdated for ton oxW bmm-titanlm whne sample. The presence of three componants ts Indicated.
C
r-m
9.M
491
0.0
0.02
QlarLllV2
F@N 7. Twc-dknenshmal spectra rwesentatbn for ttw Uve-xm. ponent case. The ton oxide-tnanlum W h b mixture specha are rep
resented by ttwir scores on ttw second and third e@nvectm. Stan mark the location of t b measured spectra and squares mark tha locations 01 points chosen as representative of the three pure corn ponent spscb-a. The s o l i line is the outer bound lina. all polnts Inside the outer bound saUsfy Uw frst m e resolution restridon. The dotted line defines an im bourd region swartding al me measued spema. Shaded areas are allowed regions for the three pure components.
64 amu5.(c). Flpure Raw Each ionimage imagesis collected 64 X 64 pixels for 46 and amuis(ai. Scaled 56 amu to1256 (b).gray and ~
levels. The pixel intensity is proportioned to tensdy. Brighter areas correspond to higher
~~
the recorded signal inmeasured signal.
erable blurring of the image in both spatial directions. Use of a smaller beam size would give sharper images but in this case lowered the signal level 80 much that peaks with intensity lower than ' T i + and "Fe+ could not be measured. Since each of the 12 ion images consisted of 4096 pixels, it is possible to construct 4096 separate spectra (only 12 mass peaks each) from this series of images. For curve resolution analysis a subset of SO representative spectra was chosen to determine the number of components and to find the eigenvectors spanning the mass spectral space. The spectral set was chosen so that each area of the sample would be represented by multiple spectra. Every sixth spectrum along the image x axis was chosen a t the lSth, 22nd, 3Oth, 37th, and 45th y axis positions. Eigenanalysis of the 50 spectra, after normalization to unit area, resulted in eigenvectors accounting for 97.87%. 1.7890, and 0.24% of the total data variance. Calculation of the
PRESS values indicates three components are present and that the three-component curve resolution procedure can be applied. Figure 6 shows the plot of PRESS vs. the number of components. In the threecomponent curve resolution analysis the spectra are represented by their scores on the second and third eigenvectors (Figure 7). The score on the second eigenvector is the x axis value. T h e y axis value is the score on the third eigenvector. In the figure, stars mark the location of data points (spectra). The outer bound resulting from the nonnegativity constraint is shown as the solid line. The dotted line is the inner bound line that surrounds the data points. The shaded areas are the areas where possible pure component spectra must lie. From the figure it can be seen that sample spectra were measured close to the allowed regions for the pure spectra of components one and two. Locations of the estimated spectral points of the pure components were arbitrarily chosen near inner bounds of the allowed regions. In Figure 7 these are marked with square points. By use of the scores correspondingto these points pure component spectra can be calculated (Table IV). Examination of the pure component spectra allows the first two components to he identified. Component one is seen to correspond to the Ti-white paint with highest intensity at 48 m u . The peak pattern for m/e ions 46,48,49, and 50 is close to the Ti isotope pattern and the peak pattern for the 62-66 mass range is close to that expected for Ti0 isotopes. The
ANALYTICAL CHEMISTRY, VOL. 58. NO. 11. SEPTEMBER 1986
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Table IV. Calculated Pure Component Spectra, Normalized to Unit Area m/e
46 48 49 50 55 56 57 62 63 64 65 66
peak intensity component one component two component three 0.084 0.719 0.046 0.021 0.005 0.022 0.002 0.013 0.013 0.058 0.088 0.002
0.031 0.334 0.086 0.085 0.139 0.249 0.041 0.021 0.018 0.010 0.019 0.013
0.307 0.192 0.187 0.129 0.015 0.007 0.027 0.020 0.027 0.046 0.011 0.007
intensities a t 5 5 , s . and 57 amu are much lower in comparison. The spectrum for component two is most noticeably different in having a much higher intensity for m / e value ions a t 55.56, and 57. This component is identified as the Fe brown paint. The third component has its largest peak a t 46 m u , with high intensities a t m/e 48, 49, and 50. Increased ratios for 46/48 amu, 49/48 m u , and 50/48amu are seen in the speetra measured from Ti-white paint after continued ion beam bombardment. This could poasihly he due to different sputtering rates between the inorganic and organic components in this paint. While appearance of components arising from beam damage to the sample might be erpected, this third component is not identified by us as a beam degradation product. Degradation of the sample would be expected to occulto a higher degree in the center of the imaged area where the ion beam is in better focus (with greater number of primary ionsfunit area). This component, however, is found in greater amount away from the center of the sample area; thus it cannot be positively identified as a degradation product, but it is spatially resolved from components 1 and 2. The relative signal contribution of the three components to all 4096 spectra can be calculated allowing a 64 X 61 pixel image to he created for each component of interest (Figure 8). In the first and second component images the expected checkerboard pattern is now apparent. Thus, the spatial resolution of the components has been improved by eliminating spectral interference from the unidentified component. In the Ti white image the intenaity in the center aren is higher than in the two areas composed of Ti white paint alone. This is due to a higher collection efficiency at the center and, also, more spectra from regions 2,4, and 6 were analyzed than from regions 1, 3, and 5. In the Fe brown component image the checkerboard pattern is more visible. Component three shows no checkered pattern hut has highest intenaity away from the center of the imaged area. The resolution of the third component from the first two allows the spatial distribution of the fmt and second components to be represented more accurately than any of the raw ion images. In these examples the crw-validation procedure gave good indications of the number of components. Often the intensities of some peaks in a SIMS spectrum d l be much greater than zero and may vary by orders of magnitude from other peaks in the spectrum. This is true for the chromium oxides and the metallic chromium. This, combined with high similarity between the component spectra, makes it difficult to determine the number of components based on the percent of total data variance amounted for by each eigenvector. For SIMS data the first eigenvector of the covariance matrix is similar to the mean vector of the data set in mass spectral space and often accounts for greater than 95% of the total
b
1
Flgwa 8. Estimated pure component images for component one. Ti white palnt (a);for component two. Fe brown paint (b); and for the unidentified third component (c). The third component shows highest intensky away from the center 01 the imaged area. Each image is scaled to 256 gray ieveis.
variance. With a noise level generally around 510% this may lead one to conclude incorrectly that no more than one component is present. Calculation of the PRESS value for different numbers of components was a better indicator of the actual numher. Multivariate curve resolution allows spectra for pure components to he estimated for two and three component cases when eq 2 is valid. As mentioned above, eq 2 may he found not to be true for many multicomponent samples. However, in cases where eq 2 is valid, the curve resolution analysis is a useful method for the determination of the component spectra and in these cases the curve resolution estimates of the component spatial distributions are more accurate than those found by relying on a single peak to represent a component. This was especially obvious for the image example. Our results for the first two components agree with what is
Anal. Chem. 1986, 58. 2306-2312
2306
known about the spatial distribution of the components from the sample preparation. Their component spectra also make chemical sense. Thus, it appears that, even when eq 2 is not valid, multivariate curve resolution can still yield useful results. In mixture samples where component spectra add linearly, the curve resolution estimate of a component’s relative signal intensity does not usually equal that component’s relative concentration because different components generally do not have equal sputter and ion yields. By comparison of eq 2 and 6 the curve resolution estimate, ai,k,of the relative spectral intensity of component k , in sample i, can be expressed as
where i,, n, and t are constants. The calculated ai,kvalues correspond to relative concentration only if the sputter and ionization yields are equal for all components. This is usually not the case; thus the concentration profiles and component images can only be considered qualitatively correct. Calibration samples are required in order to estimate actual component concentration.
ACKNOWLEDGMENT The authors thank H. W. Werner of Philips Research Laboratories, Eindhoven, The Netherlands, for providing spectra of Cr metal and Cr oxides.
LITERATURE CITED (1) Turner, N. H.; Dunlap, B. I.; Cokon, R. J. Anal. Chem. 1984, 56. 373R-416R. (2) Secondary Ion Mass Specfromefry SIMS I V : Benninghoven, A,, Okano, J., Shirnizu, R., Werner, H. W., Eds.; Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1964. (3) Slodzian, G. Surf. Sci. 1975, 4 8 , 161-186. (4) Colby, B. N.; Evans, C. A., Jr. Appl. Specfrosc. 1973, 2 7 , 274-279. (5) Ganjei, J. D.; Cokon, R. J.; Murday, J. S. Inf. J. Mass Specfrom. Ion Phys. 1981, 3 7 , 49-65. (6) Gardella, J. A.; Hercules, D. M. Anal. Chem. 1980, 52, 226-232. (7) Werner, H. W. Surf. Sci. 1975, 4 7 , 301-323. (8)Heinrich, K. F. J., Newbury, D. E., Eds. Secondary Ion Mass Spectromefry; National Bureau of Standards: Washington, DC, 1975; NBS Spec. Publ. No. 427, pp 79-127. (9) Sharaf, M. A.; Kowalski, B. R. Anal. Chem. 1981, 5 3 , 518-522. (10) Sharaf, M. A.; Kowalski, B. R. Anal. Chem. 1982, 5 4 , 1291-1296. (11) Osten, D.; Kowalski, B. R. Anal. Chem. 1984, 56, 991-995. (12) Warner, I.M.; Davidson, E. R.; Christian, G. D. Anal. Chem. 1977, 49,2155-2159. (13) Werner, H. W.; deGrefte, H. A. M.; Van den Berg, J. I n Advances in Mass Spectrometry; West, A. R., Ed.; Applied Science Publishers: Chichester, England, 1974; Vol. 6. (14) Blaise, G.; Lyon, 0.; Roques-Carmes, C. Surf. Sci. 1978, 7 1 , 630-656. (15) Eastmont, H. T.; Krzanowski, W. J. Technomefrics 1982, 24, 73-77. Kowalski, B. R. Anal. Chim. Acta 1985, 174, 1-26. (16) Borgen, 0.;
RECEIVED for review December 26, 1984. Resubmitted November 18, 1985. Accepted May 6, 1986. This work was supported in part by the Department of Energy under Contract DE-AT06-83ER60108.
Evaluation of Multipoint Kinetic Methods for Immunoassays: Kinetic Quantitation of Immunoglobulin G John W. Skoug and Harry L. Pardue* Department of Chemistry, Purdue University, West Lafayette, Indiana 47906
Thb paper desdbes reeults of a study of the kkretlc behavior of the reactions between the immunoglobulin, IgG, and antibodies to the protein. Stopped-flow mixing with nepheiometric measurements Is used to monitor the time course of the reaction and a variety of kkretlc parameters b evaluated for the quantltatton of IgG. Although several options can be used to quantify IgG, R is concluded that maxlmwn rates are most useful In ranges of excess antlbocty and excess antigen and that the pseudo-zero-order rate coefficient Is most useful for differentiating between these regions and for quantifying IgG at the maximum in the cailbratlon plot of rate vs. concentration. I t is shown that these options can be used to quantify IgG throughout the range of clinical intered from a single response curve for each concentration. For IgG concentratlons between 0 and 73 mg/dL, a least-squares fit of determined ( y ) vs. prepared ( x ) concentrations ylelded y = 1 . 0 4 ~- 0.54 mg/dL.
Because of the high selectivity associated with immunochemical reactions, immunoassay methods are becoming increasingly important. To date, most immunoassay procedures are based on measurements made after immunochemical reactions have approached equilibrium. Although these procedures are effective, it is possible that kinetic-based methods could offer complementary capabilities as has been the case with more conventional reactions. Examples of kinetic-based immunoassays reported to date include procedures for immunoglobulins (1-4) and drugs (5,
6 ) . Most of these procedures are based on single-point
measurements analogous of those used with more conventional reactions (7). In recent years we have developed and evaluated a variety of multipoint procedures for kinetic methods (8-10). An objective of this study was to evaluate the potential utility of these multipoint-kinetic methods for immunochemical reactions. The determination of the immunoglobulin, IgG, was chosen as a model system because this is an important but difficult assay, because it is representative of a group of similar assays, and because substantial amounts of kinetic data are already available for the reactions (11-14). Reactions of the immunoglobulins with antibodies produce precipitates that are monitored with light-scattering methods (15). All procedures reported to date produce calibration plots that pass through maxima such that sensitivities drop to zero a t the peaks and there are regions in which two different concentrations given the same measured response. Procedures reported to date usually require special features such as restricted concentration range or consecutive additions of antibody or antigen t~ overcome these problems. The objectives of this study were to determine if the kinetic characteristics of the antigenlantibody reaction could be used to differentiate unambiguously among the three regions of the so-called immunoprecipitin curve (22) and to quantify IgG over the full concentration range of clinical interest. These objectives have been achieved for measurements of purified IgG in synthetic samples. It was found that a kinetic parameter, an apparent zero-order rate coefficient,can be used not only to differentiate among the three regions in the calibration curve but also to
0003-2700/86/0358-2306$01,50/00 1986 American Chemical Society
