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Anal. Chem. 1991, 63,2735-2743
Sources of Uncertainty in the Experimental Determination of Sample Heterogeneity in Secondary Ion Mass Spectrometry Frank P. L. Michiels and Freddy C. V. Adams* Department of Chemistry, University of Antwerp (UIA), Universiteitsplein 1, B-2610 Wilrijk, Belgium
Although the sampllng constant is In principle a quantltatlve measure of sample heterogenelty, In practice, a number of llmltatlons arlse. Conventional errors and statistical errors Jeopardlzethe experimental determlnatlon of sample heta rogenelty. Computer sknulatlons were used In the determination of the sampling constant from the lateral variatlons of the Intensttles and showed that Important devlatkms are to be expected both wlth respect to the preclslon as well as wlth respect to the accuracy. Applled to real lncluston IntensHy dlstrlbuths, the shnulatbns yldded results whlch contlrmed seemingly discrepant experimentalobservations. Computer sknulatlon proves to be valuable to perform thls type of error analysis whlch Is hardly feasible by other means. Although theoretkally a quantltatlve measure of sample heterogenelty, the sampllng constants proves to be lhnlted to semlquantltatlve data In practlcal appllcatlons.
INTRODUCTION Due to the very small volume sampled in secondary ion mass spectrometry (SIMS), sample heterogeneity may contribute significantly to the overall uncertainty of the results. Estimation of the reliability of the analytical data is only possible if the influence of heterogeneity can be estimated. In the first article of this series (I), a new model to describe heterogeneity in SIMS was proposed. The model takes an intensity spread of inclusions into account, and it reveals the existence of a sampling constant which gives, in principle, a quantitative description of the heterogeneity of an element distribution. The model, however, does not explain a number of seemingly discrepant phenomena observed in experimental practice. Therefore, an investigation into the limitations of the sampling constant concept and possible error sources which may interfere in the experimental determination of a sampling constant was undertaken. Theory predicts that the sampling constant, as defined in the first article of this series, is a sample property, which means that it should be independent of the way in which it is determined and independent of e.g. the size of the area of analysis. Nonetheless, when sampling constants were determined in the conventional way, i.e. using the spot to spot variation of the total intensity, a number of discrepancies were observed. The error due to heterogeneity is expected to increase as the size of the area of analysis decreases. Although not typical, it was observed in a number of experiments that the heterogeneity as measured with a very small area of analysis is 1 order of magnitude lower than expected from the heterogeneity as measured with a very large area of analysis. Despite the care with which the experiments were performed, the obtained sampling constant values were not very reproducible. Optimization of the experimental conditions allowed the obtention of variations of isotope ratios across different areas of analysis of the order of 0.5%, indicating that the experimental errors are small. The obtained sampling 0003-2700/91/0363-2735$02.50/0
Table I. Compilation of Sampling Constants (mm) of a Number of Impurity Elements in a Steel Sample (NBS SRM 461), Measured during Six Different Experimental Sessions
element B Mg A1
Si Ti
400-pm area 0.19 0.43 0.13 0.11 0.16
0.077 0.25 0.12 0.10 0.16
0.27 0.12 0.10 0.12
170-pm area 0.044 0.270 0.065 0.057 0.13
0.068 0.15 0.074 0.051 0.16
0.061 0.083 0.088 0.055 0.11
Table 11. Experimental Conditions for the Determination of the Sampling Constants Listed in Table I
sample sample preparation
NBS SRM 461 low alloy steel embedding in Sn-Bi eutectic alloy grinding and polishing to 1pm diamond paste presputtering >7 min
analytical conditions 20-25 no. of analysis areas presputtering time, min >7 5 counting time, s analyzed masses, m l e 11, 24, 27, 28, 48, 51, 52, 54, 57, 98 primary bombarding species o*+ accelerating voltage, kV 12.5 1250 current, nA 500 raster, pM secondary + polarity accelerating voltage, V 4500 750 or 1800 field diaphragm, pm contrast diaphragm pm 60 170 or 400 analyzed area, pm energy window, V width 48 0' position For the measurement of the matrix species, 90-V sample voltage offset was amlied. (I
constant values, however, showed variations of the order of several tens of percent. This is shown in Table I, which summarizes experimental sampling constants measured on steel samples. The sampling constants were determined in six independent measurement sessions, three times with a 400 pm diameter area of analysis and three times with a 170 pm diameter analyzed area. The experimental conditions for the acquisition of these data are summarized in Table XI. The data clearly show a lack of reproducibility. The model does not address the problems related to the precision of experimental sampling constants. The model states that the sampling constant is the proportionality constant between the heterogeneity error and the square root of the area of analysis. Contrary to published data (2),it was observed that experimental sampling constants are not really constant but tend to decrease for a decreasing size of the area of analysis. Table I11 summarizes the average values of the sampling constants obtained with analyzed areas of 400 and 170 pm diameter. In all of the cases, the sampling 0 1991 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
Table 111. Average Sampling Constants (mm) in the NBS SRM 461 Steel Sample As Measured with a 400 and a 170 gm Diameter Field of View
n
5
28
1 '\
av samdinp const 400 pm 170 pm
element
B
0.134 0.317 0.123 0.102 0.146
Mg A1 Si Ti
0.0577 0.167 0.0755 0.0543 0.133
constant measured with a 170 pm field of analysis is smaller than that a t 400 pm. 4 -
EXPERIMENTAL SECTION It appears that the experimental determination of sampling constants k not straightforward. In what follows, we will describe the most important error sources. We discern mainly two types of error sourcea. The first type of errors is of physical nature and is related to nonperfect execution of the experiments and/or the basic limitations of the technique. The second type of errors is of statistical nature and is related to the uncertainty of sampling constants in case of perfectly performed experiments, in the absence of all other possible error sources. Instrumental Errors. In the determination of the sampling constant in the conventional way, the relative standard deviation due to heterogeneity must be known. Up to now, we have implicitly assumed that only heterogeneity effects had an influence on the total observed error. Generally speaking, though, it is and advisable to distinguish between instrumental errors (uimttr) errors due to heterogeneity (uh&). The total observed error (abL) is composed of both contributions: dbt2
=
0imu2
+ (rhe:
(1)
Sometimes, the signals of the impurity elements are normalized to the matrix reference signals to eliminate long-term drift of instrumental parameters. It was observed in a number of cases that normalization did not improve the instrumental error, and the normalization was therefore usually not performed. The instrumental error can be estimated by measuring the total error for an element which does not exhibit any heterogeneity. There are two main alternatives. First, the i n s d e n t a l error can be estimated from the variance of the ratio of the signals of two matrix isotopes. For our experiments, this yielded a typical population relative standard deviation of 0.4%, which shows that actual instrumental errors can be very small in optimized analytical conditions. Another way is to measure the variation of the normalized intensity of a homogeneously distributed element. The heterogeneity of the impurity element can be verified by visual observation of the ion images. Typical relative standard deviations of 1-4% were obtained, which are larger than the deviations for the isotope ratios. The second method gives probably a better estimate of the actual instrumental error since differences in experimental conditions may have a different influence on different elements. This is not taken into account when an isotope ratio of a matrix element is used for the estimation of the instrumental error. Estimation of a sampling constant value is only reasonable when the total observed error is significantly larger than the instrumental error. In the low alloy steel sample (NBSSRM 461), for instance, Cr, V, and Mo show variations between different areas of analysis of the order of 1-470, and this error is probably completely attributable to instrumental effects. The elements are homogeneously distributed in a steel matrix, and they were neglected in the sampling constant calculations. Throughout all the previous considerations, it is implicitly assumed that the contribution of the instrumental error is the same for all ionic species investigated. This hypothesis is reasonable, except in the case where counting statistics are an important or a predominant factor in the instrumental error. It is therefore better to write eq 1 as
0
0
a
A I
I
1
2
I
3
4
5
6
Log (total counts) Figure 1. Comparison of the experimental error (symbols) and the error predicted by counting statistics (full line).
Table IV. Total Intensity (c Different Analysis Areas 8 100
9 100
11OOO 12 900 9 400
9 200
14 500 14 300
8-l)
of Ti in Steel in 16 16 500 28 OOO 10400 6 200
7 900
13700 9 800 12 600
where uepis the contribution of counting statistics and uinstis the instrumental error, not including counting errors. Only if the experiments are performed in conditions where the counting statistics term in the above equation is negligible will the considerations above hold. To obtain this for our experiments, the counting times for all elements were 5 s and the count rate was high enough to ensure a small counting error. If the counting error cannot be minimized by increasing the secondary ion current and the counting time, the contribution of counting statistics to the overall error can be estimated from the total number of ions collected. When this is done, the electron multiplier efficiency has to be taken into account. It was experimentally verified that counting statistics equations indeed described the observed experimental errors, as appears from Figure 1. The solid line represents the theoretical error, calculated using an electron multiplier efficiency of 47%. The experimental data are indicated by the dots. The deviations from the solid line at low count rates ( 4 0 os-') can be explained in terms of the sampling distribution of the coefficient of variation of a Poisson distribution, as will be explained in considerable detail later in this article. Owing to its inherent characteristics, the failure to minimize the counting statistics effects can lead to the generation of virtual sampling constant values which are not related to heterogeneity effects. If, for instance, the sampled area is decreased by a factor of 4, the total number of counts collected decreases by a factor of 4, giving an increase (due to counting statistics) in the observed error of a factor of 2, which is also predicted by the introduction of a virtual sampling constant. Since both the counting and the heterogeneity error scale with the square root of the area of analysis, it is important to separate them well by choosing appropriate analytical conditions. If the heterogeneity error is small compared to counting statistics, a completely unrealistic sampling constant which reflects only characteristics of the experiment (transmission, sensitivity) and has no relation to the actual heterogeneity of the sample may be obtained. Detection of Rare Events. Table IV shows the Ti intensities recorded in 16 different areas of steel sample NBS SRM 461. In one spot an intensity of 28000 0s-l was measured. The next intensity is 16500 os-', which is only about half of the most intense spot. The question may be raised if the result at this very intense spot is really due to heterogeneity and not to an artifact, e.g. due to electromagnetical interferences of electrical equipment being turned on or off, which may induce spikes in the detection system.
ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
Elimination of discrepant results is always a questionable practice, in particular in the assessment of precision. Besides, the considerations in the previous article stress that the rare occurrences of very intense inclusions are essential in the determination of a sampling constant. To overcome this problem, all ionic species were measured three times at every analysis spot, which gives the operator a sufficiently high degree of certainty that the results obtained are indeed reliable. The problems with the detection of rare events also complicate sample preparation. Since large areas are investigated and since discrepant results, for instance caused by contamination due to embedding of an external particle in the sample surface, will distort the results appreciably, great care in the preparation of the samples is necessary. Shape of the Area of Analysis. Although this is not immediately obvious, the mathematical model presented in the first article of this series imposes restrictions on the shape of the area of analysis. This is implicitly built into the mathematical model when we assume that inclusions are either inside or outside the area of analysis (as discrete phenomena). As soon as a considerable fraction of the inclusions is only partially sampled, the model breaks down. The critical test is to consider the contribution of the inclusions which are at the edge of the area of analysis. If their contribution is important, the model will no longer be valid. In our analyses, the analyzed area was circular, giving a high surface to circumference ratio. An interesting application of this is the comparison of sampling constants determined with different analytical techniques. A sampling constant obtained with a volume sampling method, i.e. a method which essentially provides information about a certain contiguous volume element with approximately the same size in three dimensions, cannot be compared directly with a surface sampling method, i.e. a method which provides essentially information about a plane through a sample. A volume-sampling method provides, e.g., information about a cubic volume of 10 pm/side. If SIMS would probe this same volume, the measured heterogeneity would not be a unique value but it would depend on the size of the analyzed area. If, e.g., an analyzed area of 32 X 32 pm, with a sputtered depth of 1pm, was selected, this would show an appreciably larger deviation due to heterogeneity than in the case of a 100 x 100 pm area, with a sputtered depth of 0.1 pm. Nevertheless, in both cases, the actual sampled volumes are the same. The differernce is due to the fact that the size of the inclusions is usually much larger than the probing depth of SIMS. Only when the analytical conditions are such that the sputtered depth is appreciably larger than the size of the inclusions (so that boundary effects of the sampled area are negligible) are direct comparisons possible. Statistical Error Sources. Even in the ideal case of the absence of experimental errors and perfect agreement with all the conditions specified in the model, experimental sampling constants are subject to uncertainties due to the statistical nature of the measurement process. Since the experimental determination of a sampling constant in a certain matrix is a relatively tedious experiment, which usually requires a whole day, the precision of these sampling constants is very often not known, as it is very hard to determine it experimentally. Therefore, computer simulations were used to model the influence of these error sources. In the first part, we will confine ourselves to a very simple model where all the inclusions have the same intensity and a negligible size and are without background intensity. In the second part, we will expand our findings to the more complicated case where a spread in the inclusion intensities and a homogeneous contribution is present. If we confine ourselves to the simplest case of infinitely small inclusions with a constant intensity and without homogeneous background, the mathematical model can be simplified appreciably, as was shown in the first article. In this case the intensity in a random area of analysis is proportional to the number of inclusions in the area of analysis, and the variation in intensity is determined by the size of the area of analysis and the density of the inclusions. We can simplify the mathematical treatment if we assume that the number of inclusions in every analyzed area is small. In this case, the binomial distribution of the number of inclusions per area approaches a Poisson distribution (3). The
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experiment number analysis area
1
2
3
...
1000
...
00
1
2 0
1 1
0
0
0
2
... ... ... ... ...
2
0
... ... ... ... ...
2 3
1
...
... .
...
20
...
1
0
1
...
2 0
... 1
m
1
0 ... 3 ...
0
I
K,,
K,
Ksrm
Kr3
k..
Figure 2. Experimental layout to determine the precision of a sampling constant. The numbers xIJ refer to the total intensky in the ith analysis area during the /th experiment. For the simulation, they were replaced by Poisson distributed numbers.
Poisson distribution is a one-parameter distribution with following properties: u 2 = p = X
(3)
where X is the Poisson parameter, p is the population mean and u is the population standard deviation. The number X corresponds to the mean number of inclusions per area of analysis. The relative standard deviation can thus be written as U,,l
= x-112
(4)
which is identical to the expression encountered in counting statistics, where X corresponds to the total counts collected. Only if there are a limited number of inclusions per area of analysis is the relative standard deviation, and thus the error due to heterogeneity, significant, which supports the assumption of low X values. If we take a number of data from a Poisson distribution of a known A, we can calculate the coefficient of variation (cv) of this sample as cv = s/n (5) where is the sample mean and s is the sample standard deviation. If we consider the individual Poisson numbers as intensities measured at different areas of analysis, this coefficient of variation corresponds to a sampling constant. Theory predicts that this measured coefficient of variation is equal to X-'I2, if we take an infinite number of data points from our parent Poisson distribution. In reality, we are restricted to a very limited number of analysis points, and the measured coefficient of variation does not necessarily coincide exactly with the predicted value. Figure 2 shows a scheme explaining how the uncertainty of a sampling constant can be determined. Each column refers to one determination of the sampling constant. In such an experiment, a number of intensities at different analysis spots are measured. Each experiment gives one sampling constant value, which is equal to X-1/2 if an infinite number of analysis points are taken. In general, these K, values are not exactly the same. The ith experiment yields a sampling constant of KSi. The uncertainty on the KG value can be determined by repeating the experiment a number of times and calculating the standard deviation for the sampling constants obtained. Replacing the intensities in this scheme by Poisson data of a distribution with a known X value allows us to simulate the determination of a sampling constant for different values of the number of analysis spots. Moreover, if we do the simulation for a large number of experiments, we can even assess the sampling distribution of the sampling constant as a function of these parameters. This sampling distribution tells us how many times (frequency) a sampling constant in a certain interval (e.g. between 0.28 and 0.30 mm) has occurred. Physically speaking, it informs us on the likelihood of measuring a certain value of the sampling constant on a given sample in the given analytical conditions. They give us an insight into the accuracy and the precision of the determination of a sampling constant in the absence of any other error sources and represent a lower limit for the total experimental errors.
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 23, DECEMBER 1, 1991
enter
lambda number of spots seed number
I
n
Sh
Y reset poisson number
,
I
I
I
i
generate random number between 0 and n/lambda
I
I
m
t
U
1 number of data generated = n
poisson number = number of inclusions in area of analysis
KS (O/oI Flgure 4. Simulated sampling distribution of the coefficient of variation of a Poisson distribution for X = 10 and for 5, 10, 15, 20, 25, and 30
sampling areas (upper left to lower right). calculate K, value
*
e output data
Figure 3. Flowchart of the program to determine the sampling distribution of the coefficient of variation of a Poisson parent distribution.
In the simulations which follow, the variation of the intensity is expressed in a dimensionless relative way (in percent). For the sake of simplicity, we have assumed an analyzed area of unity area, and consequently, the sampling constant is also expressed in percent. Strictly speaking, the sampling constant has the dimension of distance. Computer Simulation Program. Figure 3 gives a flowchart of the basic program used for the simulations. Poisson numbers with a certain A were obtained by generating uniformly distributed numbers in a predefined interval and counting the number of times they were between 0 and 1. The Poisson number is the number of times a random number falls within the (0,l) interval. For each Poisson number, typically loo0 pseudorandom numbers in an interval between 0 and lOOO/X were generated using a FORTRAN random number generator. With X values between 0.1 and 10, this corresponded to a maximum probability of success of 0.01, confming the conditions of a Poisson distribution, stating that the probability of success must be low and that the total number of successes must be limited (3). Under these conditions, the binomial distribution generator behaves as a Poisson number generator. The generator was tested by generating a number of Poisson distributions of known A. They closely match the corresponding theoretical Poisson distributions. Generation of distributions with much higher probabilities of success yielded binomial distributions with the expected properties. Although this is a highly inefficient way to generate Poissonsince distributed data, it was preferred to other algorithms (4, it provides a very transparent and convenient generator.
Since the interest is mainly in inclusions with a low X value, values of 10,3,1,0.3, and 0.1 were selected. The rhulated number of analysis areas was 5, 10, 15, 20, 25, and 30, corresponding to experimentally accessible values. Typically loo0 K,values were generated in each simulation. The program was written in FORTRANon a VAX 11/780. A typical simulation required 5-10 min of CPU time. During the simulations, data were inspected to detect possible anomalies introduced by the pseudorandom number generator, but no indications of flaws were found. For every simulation, the random number generator was manually reseeded. In case all the Poisson numbers generated in one experiment are 0, it is mathematically impossible to calculate the coefficient of variation. In practice, however, an intensity of 0 would not be obtained, but rather the homogeneously distributed fraction would be measured. Therefore, these cases were treated by assigning them a K , value of 0.
RESULTS AND DISCUSSION Figures 4-6 show simulated sampling distributions of the sampling constant for values of X of 10, 1, and 0.1 and for 5, 10,15,20,25, and 30 analysis areas. A total of loo0 K,values was generated for each distribution. The values in the abscissa are the relative standard deviations in percent. They can be thought of as K, values if an analyzed surface of one area unit is assumed. Three main facts are apparent: (i) At high values of X and a high number of analysis areas, the sampling distributions show a bell-shaped distribution. For low values, however, they show an increasingly discrete character. In the case of a X value of 0.1 and 5 analysis areas, for instance, only sampling constant values of 0 and 224% are likely, together with a small probability of obtaining a sampling constant of 137%. This corresponds to respectively 0, 1, and 2 analyzed areas with each containing one inclusion. (ii) The apparent K, values depend on the number of analysis spots chosen. For a X value of 0.1, there is clearly a shift toward higher values as the number of analysis spots increases. Only for higher values of A and the number of analysis spots does the simulated values approach the theoretical value of X-lI2. Apparently, a too low number of analysis spots not only affects the precision of the determination but
ANALYTICAL
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VOL. 63,NO. 23, DECEMBER I , 1991
CHEMISTRY,
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t
160
-
120
-
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-
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-
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s
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240
/-
A=O. 1
--
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-
A=0.3
-
A= 1 4
A=3 t
-
0
4
w O
8
12
16
20
24
A=10 -e
28
32
Number of analysis areas Figure 7. Depencence of the simulated value of a sampling constant on the number of analysis areas, for five different values of A.
K, (YO) Flgurs 5. Simulated sampling dlstribution of the coefficient of variation of a Poisson distribution for A = 1 and for 5, I O , 15, 20, 25, and 30 sampling areas (upper left to lower right).
n
"I
In
-
n
K
120
2 L
0
c 100 .-0
ud
- \
h=O. 1
Y
'
P
c
.-0>
100
a
a
%
80
E0 U 6
60
iz
40
20 1
n
n
1
n
' 0
Fbure 6. Simulated sampling distribution of the coefficient of variation of a Poisson distribution for A = 0.1 and for 5, 10, 15, 20, 25, and 30 sampling areas (upper left to lower right).
also affects the accuracy. This is shown very clearly for the case of 0.1 inclusions per area of analysis and 5 analysis spots. In this case, the theoretical value of the K , is 316% (assuming unit surface area), but the maximum observable value is only 224%, and the most probable value is 0%. In this case, there is a systematic disagreement between the true value and the experimental value. (iii) The precision of the sampling constant determination deteriorates rapidly as the number of analysis spots and the average number of inclusions per area of analysis decreases.
4
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991 100
v)
.-0
b $1 y"
180 160
10
140
II
120 100 80 "'
& Underestimation of the sampling constant
1
0.001
10
1
100
Number of analysis spots Figure 9. Schematic representation of the errors occurring when sampling constants are determined in the conventional way.
the number of analysis areas is too low, systematic underestimations of the heterogeneity occur, in particular for low X values. Figure 8 shows the precision for different values of X and the number of analysis areas. It must be realized that these values are lower limits and that actual experimental determinations are also subject to a number of other error sources. Figure 9 shows X versus the number of analysis spots. The plane is divided into four different regions. Above X values of 50, the error due to heterogeneity is no longer significantly larger than the instrumental error, and this section is therefore unsuitable to measure K , values. The number of analysis spots is theoretically unlimited, but due to the extensive presputtering, an operational maximum of 25-30 analyses is assumed. The solid line shows the minimum X value which allows a reliable estimation of the sampling constant for a given number of analysis areas. Below this line, the experimental K,values will be inaccurate. This limits the accurate determination of sampling constants to the central section. The size of the analyzed area and the number of analysis areas must be chosen such that the mean number of inclusions and the number of analysis areas confirm these requirements. If these conditions are not met, incorrect K , values may result. To show that the data obtained above are independent of the Poisson number generator, Figure 10 shows the accuracy and precision data for a simulation with one inclusion per area of analysis. In the abscissa, the number of random numbers (n)used to generate one Poisson number is plotted. The mean of the sampling distributions is indicated by the dots, and the error flags indicate the width of the sampling distribution. It shows that the results are essentially independent of n for values above 25. In the simulations, n values of 1000 were used, ensuring that the binomial number generator behaved indeed as a Poisson number generator. Comparison with the Statistical Literature. Despite a major effort, no references were found in the statistical literature which deal in detail with the sampling distribution of the coefficient of variation of a Poisson parent distribution. However, the following equation can be derived (5) for the relative sampling error of the coefficient of variation of a Poisson parent distribution
/!+L
acv
=
v-
where n is the number of points used to sample the parent Poisson distribution, corresponding to the number of areas of analysis in the simulation. The values predicted by this
60 40
20
0
1
0
2
3
5
4
Log
(4
Figure 10. Plot showing the depencence of the simulated accuracy aiiu ~ I W L , I ~ I V I I VI
a
aaiiipiiiiy C I V I I ~ ~ ~ I I LIVI
A
- -.. ... .._..._-.-. I,
random numbers used in the generation of one Poissondistributed number (n). See also Figure 3.
formula should correspond to the values as predicted by the computer simulations, but the following limitations have to be taken into account. (i) The results of the computer simulation are subject to uncertainty, since they were not generated from an infinite number of simulations. Since a large number of simulations (1000) was performed for each data point, the uncertainty of the simulations is thought to be small. (ii) Equation 6 is an approximation where only the first two terms of a Taylor expansion were taken into account. (iii) The mathematical treatment leading to eq 6 assumes that the mean of the Poisson data is not equal to zero or that the probability that this occurs is negligible. When X values are low and, in particular, when the number of spots sampled is low, this is not negligible. In the simulations, these situations were assigned a K , value of 0. The mathematical treatment is therefore expected to deviate from the simulations for low values of X and low numbers of analysis areas. In this case, the simulations are believed to correspond better to experimental practice than the mathematical expression. (iv) The mathematical treatment gives only information about the width of the sampling distribution of the coefficient of variation. The simulations, on the other hand, give the actual sampling distribution of the coefficient of variation. In the statistical literature, only sampling distributions of the coefficient of variation for a normal parent distribution are described (6). (v) Theory predicts that the measured coefficient of variation is unbiased for the true (population) coefficient of variation. However, owing to the different treatment of the cases where all areas of analysis have 0 inclusions this is not the case. Also in this situation, the simulation is believed to be a better approximation of the reality. Taking all these limitations into account, Figure 11 compares the simulated values (dots) with the values as obtained from eq 6. We see that for high values of X and the number of spots, we have a good agreement. The deviations increase for lower values of X and the number of spots. The agreement between the values predicted by eq 6 and the simulated sampling distributions shows that the approach chosen to model the sampling distribution of the coefficient of variation of a Poisson parent distribution is justified. In what follows, we will extend our simulations to more complex
ANALYTICAL CHEMISTRY, VOL. 63,NO. 23, DECEMBER 1, 1991 I
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35
Number of analysis areas Flgure 11. Comparison of the precision of the coefficient of variation according to eq 6 (full lines) and the values obtalned from the simulations (symbols), for X values of 0.1, 0.3, 1, 3, and 10 (top to bottom). Table V. Simulation of the Determination of the Sampling Constant of Si in Brass Using Different Numbers of Analysis Areas no. of areas of analysis
K, simulation, mm
0
0.011
0.14 0.28 0.41 Sampling constant (mm) Flgure 12. Simulated sampling distributions of the sampling constant of Si in brass for 5 (full), 15 (shaded), and 30 (blank) spots of 100 pm diameter. The "true" value is 0.28 mm.
- 0.2 I -E 0'25
E
K:himul/K,thw std dev, %
4-
5 10 15 20 25 30 125
0.077 0.13 0.17 0.19 0.20 0.22 0.27
50
0.28 0.48 0.60 0.69 0.73 0.78 0.98
89 57 45 36 34 29 12
situations, taking the presence of an intensity spread and a homogeneously distributed fraction into account. Statistical Errors for the Extended Model. Previous simulations were all based on the assumption that all inclusions have the same intensity and that there is no homogeneous fraction. Simulations showed that determinations of sampling constants get difficult if low X values are involved. In reality, there is a spread in the intensities of the inclusions, and it appears that only the very high intensity inclusions contribute significantly to the sampling constant. Since these inclusions tend to be scarce, they have a low X value. Problems can therefore be anticipated. The program used to simulate the determination of sampling constants with inclusions of different intensities is based on the program described above. Intensities were calculated as the sum of the homogeneously distributed intensity and the intensities of inclusions in the different intensity intervals. The number of inclusions of each intensity interval was generated using the simple generator described above. The intensity intervals and the according X values were input by hand or by a command file. Simulations for Real Intensity Distributions. Similar to the examples shown above, simulationscan be used to assess the precision and the accuracy of sampling constants of samples with a known intensity distribution. Two examples with real intensity distributions will be presented here. In the first example, we simulate the determination of the sampling constant of Si in brass (NBS SRM 11021, using an analyed area of 100 bm diameter. The experimental intensity distribution of the previous article was used, and the contribution of the homogeneously distributed fraction was taken into account. The simulation data obtained are listed in Table V. The first column lists the number of analysis areas used in the simulation experiment. The second and the third column list
+
0.15 -
ln
s
u
-F .-
0.1
-
Q
5 * 0.05 0 1 0
5
10 15 20 25 30 Number of analysis areas Figure 13. Sampling constants of Mg in steel as determined with a 50 and a 100 pm dlameter analysis area (simuiatkn). The "true" vakre is 0.20 mm.
the simulated I(, values and their ratio to the theoretical value, calculated from the intensity distribution. The last column lists the simulated precision, determined from 1000 experiments. The precision of the simulation with 125 areas of analysis was determined from 25 experiments. As explained above, the precision obtained by these simulations is to be considered as a lower limit for actual experiments. We see that the experimental sampling constant depends on the number of analysis areas used. If we take 125 analysis areas, which is virtually impossible in experimental practice, the simulated value approaches the true theoretical value (0.28 mm). For lower numbers of analysis areas, however, there are appreciable deviations between the "true" and the simulated values. Similar to the simple model, there is a problem of accuracy. The precision is not very good either. For a determination with 15 areas of analysis, the expected lower limit for the precision is 45%. This tentatively explains the relative irreproducibility of the experimental K,values. Figure 12 shows the simulated sampling distributions of the sampling constant for 5, 15, and 30 spots of 100 pm diameter. As a second example, the determination of the sampling constant of Mg in steel, using analyzed areas of 50 and 100 pm diameter, is simulated. Figure 13 displays the simulated values of the sampling constant as a function of the number of analysis areas. The theoretical value of the sampling constant in this sample is 0.20 mm. In case of an analyzed
ANALYTICAL CHEMISTRY, VOL. 63,NO. 23, DECEMBER 1, 1991
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t
2 1 -
log (intensity)
Figure 14. Hypothetical inclusion intensity distribution explaining the lack of accuracy occurring in the determination of sampling constants in the conventional way. For details, see text. area of 50 pm diameter, large deviations from this value are observed, and to a lesser extent for a 100 pm diameter analyzed area. The results for the smaller area of analysis indicate consistently a smaller sampling constant, as was also experimentally observed. A simulation with 500 analyzed areas resulted in precise and accurate values, as expected.
DISCUSSION The examples shown here confirm the results of the simple simulation. Analytical conditions, mainly with respect to the number and the size of the analysis areas, may influence the observed value of experimental sampling constants. This seems to contradict that the sampling constant is independent of the size of the area of analysis. This calls for further explanation. Figure 14 shows a typical intensity distribution. We can distinguish three different regions in this distribution. Region 1,covering inclusions with a low density and a low intensity, has a negligible contribution to the overall intensity and heterogeneity. The inclusions in the high-density region 2 typically have a limited contribution to the overall intensity and an insignificant contribution to the heterogeneity. Only the high-intensity inclusions in region 3 contribute significantly to the heterogeneity. The difficulties in the experimental determination of a sampling constant originate from the small X values these high-intensity inclusions tend to have. Inclusions with a density lower than l/l,& are not likely to be present in a random area of analysis. This is indicated in Figure 14 with a horizontal line. Depending on the size of the area of analysis, the threshold is higher or lower. Inclusions with a density below this threshold, and therefore with a X value smaller than 0.1, are essentially “invisible” to the experiment. In the rare case that they happen to be in the area of analysis, their contribution to the heterogeneity is underestimated, as can be understood from Figures 6 and 9. If the inclusions below this threshold level contribute significantly to the heterogeneity, an underestimation of the sampling constant will occur. If the size of the area of analysis is increased, the effective threshold is decreased, the X value increases, and the estimation of the sampling constant improves. If the number of analysis areas increases, the low-density inclusions are more likely to be detected and the estimation of their contribution to the overall heterogeneity improves to some extent. Figure 9 shows, however, that not much gain is to be expected, since the number of analysis areas increases very rapidly if decreases. All this tentatively explains the observation that all sampling constants determined with an analyzed area of 170 pm
are smaller than when determined with a 400 pm analyzed area, as mentioned above. If a smaller area of analysis is selected, the threshold is increased, and the contribution of the inclusions with a low density is underestimated, resulting in a too small sampling constant. From the considerations above, it is clear that the experimental determination of sampling constants represents a lower limit of the actual heterogeneity. Depending on the contribution of the inclusions below the “visibility threshold” of the experiment, which can roughly be estimated as l/l&,, smaller or larger deviations occur. Only if a sufficiently large area is analyzed can reliable sampling constant values which are independent of the analytical conditions be generated. The experimental sampling constants as reported in this work are subject to the same sources of uncertainty. The visibility threshold in the experiments performed here is of the order of 0.5 inclusions/mm2. The contribution of inclusions with a density lower than this threshold is not properly accounted for. In our simulations, the analyzed areas were limited to a diameter of 50 and 100 pm, since the intensity distribution used in the simulation is cut off at densities below 0.5 inclusions/mm2. It ensures that the threshold level in the simulations is well below the threshold level of the experimental registration of the intensity distribution. The results of the computer simulations enable us to understand some of the seemingly discrepant observations, mentioned in the Experimental Section of this article. The simple model simulation (Figures 6 and 9) explains the observation that the heterogeneity as measured with a little field of view may be much smaller than the heterogeneity as measured with a large field of view. Decreasing the field of view may decrease the X value below the threshold of 0.1. In this case, the sampling constant will be underestimated. Checking the original data confirms that this is the reason for the discrepancy. The 400 pm areas of analysis contained typically between 1and 5 inclusions. None of the investigated 22 pm diameter areas of analysis contained any inclusions, giving a relative standard deviation equal to the instrumental error, since only the homogeneously distributed fraction contributed to the signal. Thii is a clear-cut example of Figure 6. The relative irreproducibility of experimental sampling constants is also readily explained. I t was shown that only the very high intensity inclusions contribute significantly to the second moment of the intensity distribution and hence to the heterogeneity. These high-intensity inclusions have low X values, and simulations show that, in this case, the precision deteriorates rapidly for a decreasing number of analysis areas. This semiquantitative conclusion is supported by the simulations using real intensity distributions. It appears from these simulations that a high precision is not to be expected, even in the ideal case of the absence of all other error sources. The observation that the sampling constant tends to decrease when the size of the field of view decreases is readily understood by comparison to the simulations for the extended model (Table V and Figure 13). I t is therefore concluded that, in contradiction with “common sense” and “intuition”, analyzing a sample and, e.g., 10 different spots and taking the standard deviation of the experimental data will not give a good idea of the contribution of heterogeneity to the data in the cases illustrated here. Sampling Constant Concept in SIMS. Sample heterogeneity in SIMS can, in principle, be quantitatively described by using a sampling constant, which is related to the size and intensity distributions of the inclusions in the solid matrix. This sampling constant is a sample characteristic,
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and it can, in principle, be used to predict the error due to heterogeneity when an analysis is performed. In experimental practice, however, the use of the concept is limited. Only a very small fraction of the total number of inclusions has an intensity which is high enough to contribute significantly to the overall heterogeneity. The low abundance of the inclusions impedes the experimental determination of a sampling constant and gives rise to imprecise and inaccurate sampling constant values. The obtained value is dependent on the analytical conditions and underestimates the actual uncertainty due to heterogeneity. In practice, therefore, the sampling constant concept is limited to semiquantitative information. Apart from the difficulties mentioned above, it is impossible to transfer absolute values of sampling constants from SIMS to a volume-sampling technique. The values as measured in SIMS are transferable to other two-dimensional analysis techniques only to the extent that the intensity maps as measured in SIMS correspond to the actual concentration distributions. Matrix effects may influence the width and the shape of the intensity distributions to some extent. The distinction between two-dimensional and three-dimensional sampling techniques is related to the size of the inclusions.
If the size of the inclusions is appreciably smaller than the smallest dimension of the sampled volume, a technique will behave as a volume-sampling technique.
ACKNOWLEDGMENT We are indebted to S. Leigh (NIST) for the derivation of eq 6 and helpful discussions.
LITERATURE CITED Michieis, F. P. L.; Adams, F. C. V.; Brlght. D.; Simons. S. Anal. Chem., previous article In this issue. Scilla, G. J., Morrison, G. H. Anal. Chem. 1977, 49, 1529. Piessens, B. I n WaarschJnlild?&kskenhg en Statisti& ; Stop-Scientia: Gent-Leuven, 1969; p 100. Yakowitz. S. J. I n CMpUtetlonal Probabllky and SlmulaMon, AddlsonWesley PublishingCo.: Readlng. MA, 1977; p 63. Michlels. F. Ph.D. Thesis, University of Antwerp, Department of Chem istry, 1990. Hendriks and Robey. Ann. Math. Stat. 1936, 7.
RECEIVED for review March 12, 1991. Revised manuscript received September 3, 1991. Accepted September 9, 1991. F.M. is a research associate of the National Science Foundation (NFWO/FNRS, Belgium). This research was sponsored by FKFO, Belgium, and DPWB, Belgium, in IUAP programs 11 and 12.
Pulse Voltammetry at Microcylinder Electrodes Mary M. Murphy, John J. O’Dea, and Janet Osteryoung* State University of New York at Buffalo, Department of Chemistry, 130 Acheson Hall, Buffalo, New York 14214
Mlcrocyllndrlcal electrodes are easler to construct and maintain than mlcrodlsk electrodes. I n the normal-pulse mode, ranges of time parameters and electrode slzes can be found wch that depletion of reactant Is unimportant and the response to the analysis pulse Is predicted by theory for planar condltlons. Similarly, ranges of parameters are found for reverse-pulse voltammetry such that the potentlaldependent response can be treated as a sequence of Individual doubie-pulse responses. Cylindrical diffusion and convection act to replenkh reactant quickly near the electrode and thus permit overal experiment times In the range of seconds. For square-wave voltammetry the shape and position of the net current response are Independent of the extent of cyllndrkal diffusion.
INTRODUCTION Microelectrodes have become a routinely used tool for electrochemical experimentation in recent years. Advantages of microelectrodes include the small size of the electrode assembly and improved performance with respect to electrodes of conventional size, especially in resistive media or for fast experiments. The most commonly used microelectrode is a circular conductor embedded in an insulating plane (a so-called disk electrode). Cylindrical microelectrodes, however, have special advantages with respect to ease of fabrication and theoretical properties which suggest they could be used to advantage more widely. In the case of a disk, the current density is in general nonuniform, with the highest current density at the edge (I). 0003-2700/9 1/0363-2743$02.50/0
Thus the uniformity of the circular geometry and quality of the seal between insulator and embedded conductor influence strongly the voltammetric response. In contrast, for cylindrical electrodes the configuration of the seal is unimportant to performance. Therefore fabrication of microcylindrical electrodes is technically undemanding. Microcylinders also have the practical advantage that area depends on length and radius, whereas diffusional properties depend only on radius (and time). Thus electrode area can be controlled independently over a fairly wide range of cylindrical radii. In harsh environments, for which there is no insulating material available, the electrode may be operated partly submerged without affecting the diffusional characteristics. Finally, the current density is uniform at a cylinder, which simplifies the interpretation of kinetic data (1). Voltammetry at cylindrical electrodes has been examined both theoretically and experimentally, primarily by Aoki and co-workers (2-8). Techniques which have been investigated include cyclic voltammetry (2-4),chronoamperometry (3,5, 6),chronopotentiometry (9, normal- and differential-pulse voltammetries (8), and staircase and square-wave voltammetries (9). Quasi-reversible (IO),irreversible (ll), and second-order catalytic (6) reactions have also been studied using cyclic voltammetry at microcylinder electrodes. Normal-pulse and square-wave voltammetry at a cylinder have been employed for chemical analysis (12),and normal- and reversepulse and staircase and square-wave voltammetry have been used for kinetic studies (13). In all cases interpretation of voltammograms has been complicated by the influence of cylindrical diffusion. The aim of the present study is to find procedures for using microcylinder electrodes which yield simplified results for a reversible system. The longer range 0 1991 American Chemical Society
