1433
Anal. Chem. 1985, 57, 1433-1436 Am-241 50 keV BACKSCATTER
in these results preclude a judgement at this time as to whether the SOAP or the XRF results are correct. Additional studies in this area are necessary. Registry No. Ar, 7440-37-1;Ne, 7440-01-9;Xe, 7440-63-3;Fe, 7439-89-6; Cu, 7440-50-8; Ti, 7440-32-6; Ag, 7440-22-4; 244Cm, 13981-15-2;55Fe,14681-59-5;241Am,14596-10-2;ImCd,14109-32-1; isobutane, 75-28-5.
LITERATURE CITED
X-RAY ENERGIES (keV)
Figure 12. Americium-241 lubricant backscattering spectrum.
analyses were done on samples carefully removed from the containers after standing undisturbed for 15 h at 75 OF (see Table VI). These results indicate that particle settling was not a problem, but the scatter and inconsistencies reflected
(1) Packer, L. L.; Miner, J. R. AFAPL-TP-75-6, AFAPL Contract No. F33815-744-2024, Jan 1975. (2) Internal UTRC Memo, L. Packer to H. Zickwolf (P&WA/CPD), 10 Jan 1980. (3) Sipila, H. Nucl. Instrum. Methods 1976, 733, 251-252. (4) Sipila, H.;Kiuru, E. Adv. X-ray Anal. 1978, 27, 187-192. (5) Jarvinen, M.-L.; Sipila, H. Nucl. Instrum. Methods 1982, 193, 53-56. (8) Jarvinen M.-L.; Sipila, H. I€€€ Trans. Nucl. S d . 1984, NS-31 356. (7) Sipiia, H.; Jarvinen, M.-L. Finnish Patent Applications 83-3547 and 83-353. (8) Kittirlgs, D. C.; Ellis, J. ASD-TR-68-2, 1984. (9) Brown, J. R.; et al. Anal. Chem. 1980, 52,2385-2370. (10) Schrand, J. B., et al. AFAPL-TR-75-77, Feb 1976.
RECEIVED for review November 28,1984. Accepted February 4, 1985. This work was supported in part by U.S. Air Force Wright AeronauticalLaboratory Contract F33615-81-C-2065.
Optimization of Multielement lnstrumental Neutron Activation Analysis Donald D. Burgess
Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4K1
A method for obtaining optimum conditions in muitielement Instrumental neutron actlvatlon analysis Is described. The technique of simplex optlmlratlon is employed through the prediction, by calculatlon, of y-ray spectra for samples of typical composttion. Response functions sultable for optlmization of muitieiement anaiysls are discussed and the performance of the proposed method Is evaluated.
Neutron activation analysis (NAA) has often been used for the simultaneous determination of several elements in a single sample. This capability is especially useful where multivariate data are required. The determination of the sources of environmental pollution and geological studies are examples. The procedures adopted for multielement analysis must be carefully designed if comprehensive and accurate analytical results suited to the task in hand are to be obtained. The design of procedures in NAA requires that the analyst take into account not only the widely varying properties of the analyte elements but also those of matrix elements. Effects such as instrumental dead time that occur during measurements must also be considered. Although other authors (1-4) have proposed computational methods that assist in this task, method design in multielement NAA has traditionally been carried out on the basis of experience and informed intuition. This paper proposes a new approach to the determination of 0003-2700/85/0357-1433$01.50/0
optimum conditions and procedures in multielement NAA.
THEORY The approach adopted in this work employs the method of simplex optimization by advance prediction of y-ray spectra reported earlier ( 5 ) for the optimization of single-element NAA. The process of optimizationis separated into a number of interdependent modules: task definition, spectrum prediction, spectrum evaluation, and simplex optimization. Figure 1 illustrates the organization of these modules and the following paragraphs describe their operation. The optimization process is most easily understood if the adjustable analytical parameters such as irradiation, decay, and counting times are considered to define the several dimensions of a geometrical space. Each point of this space can be associated with a value of a quantity (response) that represents the performance of a procedure that uses the corresponding parameter values. Optimization is then the task of discovering the point in parameter space that possesses the most favorable response. The first module defines the optimizationtask. The typical composition of samples of the kind under consideration is entered by the analyst and necessary nuclear data are retrieved from a data base. Escape peaks are then added and the saturation activity for unit neutron flux and unit sample size is computed for each y-ray peak obtained from the data base. The analytes and their respective analytical y-ray lines are 0 1985 American Chemical Society
1434
5
ANALYTICAL CHEMISTRY, VOL. 57, NO. 7, JUNE 1985 constraints
initial simplex
I
I
samp 1 e composition
TASK DEFINITION
nuclear data
I
rial parameter values
SPECTRUM PREDICTION
determinations, detection limits and measurement precision are attractive choices and can be readily computed. For the multielement case, however, single-element responses must usually be combined into a single function that yields satisfactory compromise conditions upon optimization. In this study, the single-element responses used were the detection limit F1' calculated according to Currie's definition (9) and the relative standard deviation F2' of the net peak area obtained by subtracting the spectral baseline
F1' =
(2.7 +4.65B09C W A
(1)
spectrum final parameter values
response
SPECTRUM EVALUATION
Figure 1. Organization of optimization calculations.
selected by the analyst from tables displayed a t a video terminal and constraints are entered that limit searches of parameter space to regions that are accessible to practical laboratory procedures. These constraints concern the values of the adjustable parameters and any other quantities such as total count rates that must be controlled. Finally, an initial simplex is described to begin the optimization process. This module is used only once for each optimization. The second module is used to model the y-ray spectrum that is to be expected for the given sample composition and a particular set of parameter values. The computations employ the usual equations for activation as described previously (5), a rectangular approximation to the Compton continuum, and a correction for counting dead time that will be described in a separate publication. Peak overlap is treated by considering the contribution from an overlapping peak to be a contribution to the base line of the peak of interest. Peaks found to be subject to overlap are flagged in the final output. The result is one set of data that describes the peaks and and another that describes the base lines underlying the peaks. A thorough characterization of the irradiation and counting facilities under consideration must be carried out to supply information used in this step. Guinn et al. (2-4) have shown that calculations of this kind yield results that are sufficiently accurate for use in method design. Once a spectrum has been predicted, it is subjected to an analysis in the third module that yields a value for the response of interest. Any continuously varying response that can be derived from a spectrum and the parameter values associated with it can be used provided that it expresses the quantity to be optimized. The final module consists of the actual simplex optimization process. A geometrical figure (simplex) is moved by reflection, expansion, and contraction in parameter space until a convergence criterion is satisfied, indicating that an optimum has been located. This module communicates with the spectrum prediction and evaluation modules to obtain a response for each new point of parameter space examined. The modified simplex method of Nelder and Mead (6) was used in this work with several modifications (ref 5 and references therein). When convergence has been obtained, the parameter values for the point of convergence are taken for use in the design of the desired laboratory procedure for the analysis of the samples of interest. The parameters studied here were irradiation, decay and counting time, and sample size. The formulation of response functions for multicomponent determinations presents certain difficulties, as has been pointed out by other researchers (7, 8). In single-element
where A is the net peak area, B is the base line area beneath the peak, C is the concentration of the element in the sample, and W is the sample size. Both of these responses become more favorable as their magnitudes diminish. That is, optimization corresponds to minimization of the response. A simple approach to multielement optimization is to use the sum of the individual responses as the combined response n
F1 = CF1,'
(3)
i=l
n
F2 = CF2,'
(4)
i=l
where n is the number of analyte elements and F1 and F2 are multielement responses. An alternative is to weight the individual responses in forming the combination. The problem here lies in determining what weighting factors to use. It is clearly desirable to select these on an objective basis and a reasonable goal is to give each element a roughly equal chance of being determined satisfactorily. If this is not done, difficult elements may dominate the optimization too strongly. The weighting factor chosen for this study was the reciprocal of the response, Fli" or F2i0, obtained by single-element optimization for each analyte. These quantities depend on the inherent difficulty of determing each element. The weighting was implemented in the form of a weighted average
F2, =
where F1, and F2, are the weighted multielement responses.
EXPERIMENTAL SECTION Spectrometer characterization was carried out as described previously (5) and included the determination of the variation with energy of total efficiency, photofraction, full peak width at half maximum, and dead time per registered count. The computer program SOAP was written in the C programming language and was developed and used on an eight-bit microcomputer. Although the advance prediction portion of this program naturally has much in common with the APCP program developed by Guinn et al. (2-4) and kindly provided for inspection during this work, the data structures and the details of many of the calculations are implemented differently in the two programs.
ANALYTICAL CHEMISTRY, VOL. 57, NO. 7,JUNE 1985
1435
Table I. Airborne Particulates, Effects of Weighting and Half-Life on Optimization of Detection Limitsa
unweighted LD,wg
element
LD, pg
LD/RI0
Mg
48.4 0.0664 0.00825 0.893 24.7 3.71 52.7 0.5* 77.8
1.13 1.00 1.93 5.82 6.85 12.6
I Mn
cu Zn As ti td
R1
LDIR~"
13.9 1.02 1.00 1.31
0.0596 0.157 3.60 0.388 3321 527 4.21
weighted LD? pg
LD, /%
LD/R1"
195 0.147 0.00431 0.290 9.44 1.25 349 10.1
4.54 2.21 1.01 1.88 2.62 4.24
0.00870 0.176 5.15 0.701 872 147
2.03 1.14 1.43 2.37
0.0286
0.153
R1w
LD/RI0
element
tl/2
RI", pg
ti
td
Mg
9.45 25 154.8 750 828 1578
42.9 0.0664 0.00428 0.154 3.60 0.296
25.5 49.0 291.6 2198 3235 6000'
0.5* 0.5* 10.1 379 520 2414
I Mn
cu
Zn As
OTimes are in minutes (* minimum, 33 min, sample size = 141 m3.
+ maximum),LD and R1" = detection limit, ti = irradiation time, t d = decay time, counting time =
Additional bank-switched random-access memory was used to eliminate recalculation of data and a floating-point mathematics unit was used to further reduce execution times.
F1
I
h
RESULTS AND DISCUSSION Optimization runs carried out for several data sets yielded reasonable results. For example, a marine sediment standard reference material, BCSS-1, prepared by the National Research Council of Canada was considered for the determination of Ti, Dy, Al, Mg, Na, Ca, I, V, Mn, and C1. A threeparameter optimization was done using F2 as the response function and irradiation time, decay time, and sample size as adjustable parameters. Counting time was fixed at 10 min. The optimum analytical conditions predicted were 0.24 rnin irradiation time, 18 min decay time, and 0.58 g sample size. These values are in good agreement with the values selected by an experienced analyst (IO)on the basis of experience alone: 0.17 min irradiation time, 15 min decay time, 10 min counting time, and 0.5 g sample size. Tables I and I1 give results for two-parameter optimizations done for a hypothetical airborne particulate sample. Four-parameter optimizations were also carried out with sample size and irradiation, decay, and counting times as parameters. The counting times generated, however, were often too long in comparison to the analyte half-lives for accurate dead-time corrections or converged at the upper boundary set by the counting time constraints. If counting time is used as an adjustable parameter, great care must be used in setting constraints. Response surfaces were plotted for several of the data sets examined in order to verify proper location of the optimum. These response surfaces were invariably smooth and, except as noted below, showed no more than one optimum. In most cases, the program correctly located the region of the optimum. The exceptions occurred when the location of the starting simplex allowed the simplex to encounter a local optimum created by a count rate boundary. This problem can be detected by comparing the results of several optimizations started from different locations. The value selected for the convergence criterion often determined the difference between the calculated and actual locations of the optimum and must be selected carefully. As the plots shown in Figures 2 and 3 demonstrate, however, response surfaces are often relatively flat near the optimum and unnecessarily long computation times can be avoided bv not imDosing extremelv strict con..~~ ._ .. .. .
Y
~
~ 1 1
25
'd
.5
Figure 2. Unweighted detection limit response surface for an airborne particulate sample.
Weighted detection limit response surface for an airborne particulate sample.
Flgure 3.
vergence criteria. For the same reason, minor deviations from the conditions specified by the optimization process are un-
1436
ANALYTICAL CHEMISTRY, VOL. 57, NO. 7, JUNE 1985
Table 11. Airborne Particulates, Effects of Weighting and Half-Life on Optimization of Precisiona unweighted element
61,
%
20.7 4.73 0.0759 0.762 9.04 35.2 257 3.70 70.5
Mg
I Mn
cu Zn
As ti td
R2 R2w
utf R2"
2.58 1.65 1.02 2.07 2.98 5.02
QI,
weighted
%
(u,/R2")
0.373 0.389 3.11 7.93 5988' 649 11.81
u,/R2"
Qr,
23.5 5.06 0.0750 0.722 8.57 33.2 284 4.98
5.04 1.06 1.03 1.13
2.91 1.78 1.01 1.96 2.82 4.74
0.888
Q,,
ur/R2"
%
0.130 0.377 3.40 12.8 1437 275
1.76 1.02 1.12 1.83
0.344
element
tlj2
R2", %
ti
td
Mg
9.45 25 154.8 750 828 1578
8.02 2.86 0.0741 0.368 3.03 7.00
25.5 48.2 354.5 2178 3180 6000*
0.5* 0.5* 10.9 404 520 2379
I
Mn cu Zn
As
+
'Times are in minutes (* minimum, maximum), u, and R2" decay time, counting time = 33 min, sample size = 141 m3.
= relative
likely to lead to significantly degraded performance. The weighting scheme tested in this study tended to equalize the ratios of the responses of the individual elements to their respective single-element-optimizedvalues. This can be seen in Tables I and I1 where the ratios for the weighted trials are more similar than those for the unweighted trials. The inclusion of weighting should therefore give each element a more nearly equal chance of satisfactory determination than would be obtained if weighting were omitted. The magnitude of this effect is not always very great, however, and does not invariably justify the increased computational effort. This weighted multielement response assumes that all analytes are of equal interest. Other weightings could be devised for specific tasks. For example, it has been speculated (11)that it might be possible to tailor the weighting to provide optimum discrimination in the application of pattern recognition techniques to the results of multielement analyses. Tables I and I1 also demonstrate the importance of half-lives in the choice of elements that are to be determined in a single activity measurement. The values given in these tables concern only a single irradiation and a single activity measurement. If a wide range of half-lives is present among the activities to be determined, the effect of half-life may dominate the optimization. Columns one and two of Table I and Figures 2 and 3 clearly show this trend. More useful results would be obtained if the selection of activities for determination were made to vary from point to point in parameter space. Such a procedure would probably produce multiple optima but could be used to design schedules in which counting is done more than once. The y-ray energy used for the determination of each element must be selected before optimization in the calculations described above. This requires a prior knowledge of any interferences that may be present. This requirement could be relaxed by selecting analytical peaks automatically on the basis of interferences revealed by calculation. Multiple optima are likely to occur in this instance also. The dynamic selection of activities and of peaks is now under study. The accuracy of the optimizations obtained using the method described here is naturally limited by any discrepancies between the assumed and the actual sample compositions. Any other method, however, would be subject to the same
standard deviation
of n e t peak area, ti = irradiation time,
td
=
limitation. If a close match is desired, a preliminary analysis of the sample of interest can be employed to provide the sample composition used in optimization. Care in choosing values for constraints is important in the optimization of both single- and multielement determinations. True optima are occasionally absent or lie outside accessible regions of parameter space. Consequently, the simplex process often converges at a boundary corresponding to a constraint. The modular organization described here has greatly aided the implementation of this technique. Changes to one module require minimal adjustment of others. Consequently, new response functions are easily tested and the spectrum prediction process can be readily modified. The heuristic approach used allows a full treatment of activation and counting without the necessity of separate treatment of special cases in the calculation of responses. Source code for the computer program used to implement this optimizationtechnique will be released in the near future. Interested persons are invited to communicate with the author for information concerning distribution of the program.
LITERATURE CITED (1) Davydov, M. G.; Naumov, A. P. Radlochem. Radioanal. Lett. 1978, 35,77-84. (2) Guinn, V. P. I n "Nuclear Methods of Environmental and Energy Research"; Universlty of Missouri: Columbia, 1980; CONF-800433, pp 2-14. (3) Guinn, V. P.; Leslie, J.; Nakazawa, L. J. J . Radioanal. Chem. 1982, 70, 513-525. (4) Guinn, V. P.; Dahlgren, L. N.; Leslie, J. C. J . Radioanal. Chem. 1984, 8 4 , 103-108. (5) Burgess, D. D.; Hayumbu, P. Anal. Chem. 1984, 56,1440-1443. (6) Nelder, J. A,; Mead, R. Compuf J . 1964, 7, 308-313. (7) Leary, J. J.; Brookes, A. E.;Dorrzapf, A. F., Jr.; Golightly, D. W. Appl. Spectrosc. 1982, 36,37-40. (8) Weyland, J. W.; Bruins, C. H. P.; Debets, H. J. G.: Bajema, 8. L.; Doornbos, D. A. Anal. Chlm. Acta 1983, 153, 93-101. (9) Currle, L. A. Anal. Chem. 1988, 4 0 , 586-592. (IO) Landsberger, S., McMaster Nuclear Reactor, unpublished work, Oct 1984. (1 1) Fleming, R., U S . National Bureau of Standards, personal communication, Oct 1984.
RECEIVED for review November 16, 1984. Accepted January 23,1985. This work was carried out with the financial support of the National Science and Engineering Council of Canada and the Ontario Ministry of the Environment, Air Resources Branch.
