Anal. Chem. 2009, 81, 6851–6859
Determining Elemental Composition Using Prompt γ Activation Analysis Zsolt Re´vay† Institute of Isotopes, H-1525 Budapest, Hungary Prompt γ-ray spectra sometimes contain hundreds of characteristic peaks, and the masses of the sample components can be determined from dozens of γ-ray peak areas, some of which are affected by spectral interferences. Reliable qualitative and quantitative analyses can only be performed by using a precisely calibrated detector system, an accurate spectroscopic data library, highquality spectroscopy software, and a sophisticated method to convert raw data into chemical composition. This article describes the steps of the chemical analysis with prompt γ activation analysis (PGAA). A complete data reduction method has been constructed that handles the large number of spectral peaks using statistical procedures, identifies the chemical components based on statistical criteria, and determines the sample composition with a least-squares fit of the masses calculated from the individual peaks. The calculation of the uncertainties as well as the detection and analytical limits are discussed in detail. The validation of the method is also presented. The method can also be used in the evaluation of the results of other spectroscopic techniques. While many nuclear and radioanalytical techniques have lost their earlier importance, prompt γ activation analysis (PGAA) has gained more and more attention in the past decade. This rapidly developing chemical analytical method is based on radiative neutron capture (or the so-called (n,γ) reaction), in which a nucleus absorbs a neutron followed by the immediate release of characteristic γ radiation.1 Qualitative analysis is based on the identification of energies of the detected γ rays, while a quantitative analysis can be performed using their intensities. The highestperformance PGAA systems are located at neutron beam facilities of research reactors and are equipped with high-purity germanium (HPGe) detectors for the collection of the γ spectra.2 Specific industrial and in-field applications allow the uses of lowerresolution scintillator detectors and other neutron sources, which may require somewhat different analytical algorithms than presented here.3 † To whom correspondence should be addressed. Phone: +361-392-2539. Fax: +361-392-2584. E-mail:
[email protected]. (1) Re´vay, Zs.; Belgya, T. In Prompt Gamma Activation Analysis with Neutron Beams; Molna´r, G. L., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp 1-31. (2) Lindstrom, R. M.; Re´vay, Zs. In Prompt Gamma Activation Analysis with Neutron Beams; Molna´r, G. L., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp 31-58. (3) Alfassi, Z.; Chung, Ch. Prompt Gamma Neutron Activation Analysis; CRC Press: Boca Raton, FL, 1995.
10.1021/ac9011705 CCC: $40.75 2009 American Chemical Society Published on Web 07/10/2009
The in-beam PGAA facilities located at research reactors can be used for panoramic analyses of a large variety of samples.4 With its nondestructive nature, PGAA can be used for the investigation of precious objects like archeological artifacts.5 Neutron and γ radiations deeply penetrate into matter. Hence this technique can be used for the identification of materials inside sealed containers6 and even for the in situ investigation of processes inside chemical reactors.7,8 Prompt-γ spectra are much more complex than those acquired in traditional neutron activation analysis (NAA). The latter contain only the decay γ-ray peaks from radioactive nuclides, typically up to a few megaelectronvolts, while the characteristic prompt-γ peaks are much more numerous and appear over a much broader energy range from a few tens of kiloelectronvolts up to 11.5 MeV. Light elements have relatively simple spectra, containing at maximum a few dozen peaks fairly evenly distributed in this energy range. Many heavy elements however have several hundred distinguishable peaks, sometimes superimposed on a continuum originating from the unresolved overlaps of an uncountable number of peaks.1,9 One of the reasons for the late development of PGAA as a useful analytical tool was the computational difficulty in evaluating prompt-γ spectra. A large variety of laboratory applications can be carried out by PGAA only if a reliable qualitative and quantitative analysis method is available. This requires the following conditions to be fulfilled: (1) a detector accurately calibrated for both energy and counting efficiency, (2) a spectroscopic database containing the γ-ray energies and the cross-section data for the chemical elements, (3) a high-quality spectroscopy software for the evaluation of the spectra, which provides reproducible positions and net areas for the γ-ray peaks, and finally (4) a set of algorithms, i.e., the data reduction procedure that identifies the detected elements and calculates their masses, giving the composition of the sample. (4) Anderson, D. L., Kasztovszky, Zs. In Prompt Gamma Activation Analysis with Neutron Beams; Molna´r, G. L., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp 137-172. (5) Kasztovszky, Zs.; Biro, K. T.; Marko, A.; Dobosi, V. Archaeometry 2008, 50, 12–29. (6) Re´vay, Zs. J. Radioanal. Nucl. Chem. 2008, 276, 825–830. (7) Teschner, D.; Borsodi, J.; Wootsch, A.; Re´vay, Zs.; Ha¨vecker, M.; KnopGericke, A.; Jackson, S. D.; Schlo ¨gl, R. Science 2008, 320, 86–89. (8) Re´vay, Zs.; Belgya, T.; Szentmiklo´si, L.; Kis, Z.; Wootsch, A.; Teschner, D.; Swoboda, M.; Schlo ¨gl, R.; Borsodi, J.; Zepernick, R. Anal. Chem. 2008, 80, 6066. (9) Re´vay, Zs.; Firestone, R. B.; Belgya, T.; Molna´r, G. L. In Prompt Gamma Activation Analysis with Neutron Beams; Molna´r, G. L., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp 173-364.
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Because of their complexity, PGAA spectra have to be carefully calibrated for γ-ray energy before the peak evaluation can be done. The identification of γ peaks requires an energy precision better than 0.1 keV (about 0.001% relative to the whole energy range). In our laboratory, the following two-step procedure is followed to achieve this. First a pair of peaks with known energies is identified and fitted at the high- and the low-energy ends of the spectrum. With the use of this two-point energy calibration, the mentioned precision cannot be reached because of the nonlinearity of the electronic modules (more than 1 keV in our case); hence, a correction function is also applied. A set of energy standards (152Eu, 37Cl(n,γ)38Cl reaction, etc.) are measured, and then an eighth-order ortho-normal polynomial representing the energy difference between the linear ideal and the measured values as a function of the channel number is used to correct for this discrepancy. The nonlinearity function obtained in this way is characteristic of the instrument and the setup, and it does not change significantly with time.10 The counting efficiency of the detector is determined using a similar set of radioactive calibration sources (152Eu, 60Co, etc.) and (n,γ) reactions (such as 14N(n,γ)15N and 37Cl(n,γ)38Cl). Their activities are fitted together, using an eighth-order orthonormal polynomial on a log-log scale. The procedure also calculates the uncertainty of the efficiency function. Since we use more than a hundred γ peaks to determine the efficiency, its uncertainty is less than 1% in the middle energy range where most of the characteristic peaks appear.11 The efficiency function together with the nonlinearity is determined twice a year in the course of the regular calibration procedures at the beginning of each reactor campaign. A spectroscopic database containing the energy and partial γ-ray production cross-section data has been developed in our laboratory during the past 12 years. Pure chemical elements were measured in the thermal and cold neutron beams at the Budapest PGAA facility. The spectra were energy calibrated using 35 Cl(n,γ)36Cl as the energy standard.9 The partial cross sections were determined using the so-called standardization method, where stoichiometric compounds or homogeneous mixtures made of the elements of interest and the comparators were measured.12 The spectroscopic data of chemical elements obtained in this way have been compiled into data libraries.9,13 Prompt-γ spectra have been evaluated using the HypermetPC code,10 which has proved to be an appropriate tool in PGAA, especially for handling the asymmetric shapes typical for highenergy peaks.15,14 The nonlinearity and the efficiency calibrations ¨ sto (10) Fazekas, B.; O ¨r, J.; Kis, Z.; Molna´r, G. L.; Simonits, A. In Proceedings of the 9th International Symposium on Capture Gamma-Ray Spectroscopy, and Related Topics, Budapest, Hungary, October 8-12, 1996, Molna´r, G.; Belgya, T.; Re´vay, Zs., Ed.; Springer Verlag: Budapest, Hungary, 1997; pp 774778. (11) Molna´r, G. L.; Re´vay, Zs.; Belgya, T. Nucl. Instrum. Methods 2002, A 489, 140–159. (12) Re´vay, Zs.; Molna´r, G. L. Radiochim. Acta 2003, 91, 361–369. (13) Choi, H. D.; Firestone, R. B.; Lindstrom, R. M.; Molna´r, G. L.; Mughabghab, S. F.; Paviotti-Corcuera, R.; Re´vay, Zs.; Trkov, A.; Zerkin, V.; Chunmei, Z. Database of Prompt Gamma Rays from Slow Neutron Capture for Elemental Analysis (STI/PUB/1263), International Atomic Energy Agency: Vienna, Austria, 2007; http://www-pub.iaea.org/mtcd/publications/PubDetails. asp?pubId)7030. (14) Re´vay, Zs.; Belgya, T.; Ember, P. P.; Molna´r, G. L. J. Radioanal. Nucl. Chem. 2001, 248, 401–405.
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are carried out using built-in routines of the Hypermet-PC program. All the measurements presented here were performed at the PGAA facility of the Budapest Research Reactor between 1997 and 2008. This 10 MW nuclear reactor was equipped with a liquidhydrogen-containing cold-neutron source in 2000. The original natural-nickel-coated neutron guides have been replaced with supermirrors during the past years. The PGAA system is operated by the Institute of Isotopes and is located at the end of the 35 m long guide in a well-shielded low-radiation environment. The flux of the thermal beam at the sample position was 2.5 × 106 cm-2 s-1 and is now 1.2 × 108 cm-2 s-1 (thermal equivalent flux) with the cold beam. The beam used for the measurements has a cross-section of 2 × 2 cm2. γ Radiation is detected with a HPGe detector surrounded by a Compton-suppressor and passive shielding. In this way a uniquely low spectral background could be achieved.16,17 A new method to convert the γ-ray yields to chemical composition has been developed in the Institute of Isotopes at Budapest in parallel with the establishment of the spectroscopic database. The procedures and methods programmed into this software are presented in this paper. The validation of the methods is also shown. THEORY Peak Detection Limit. Currie’s well-known criterion is often used for the characterization of detection limits of peaks in spectra of nuclear radiation, though his original considerations cannot always be implemented for spectra collected with multichannel analyzers (MCAs). His basic idea18 was to compare two values: a previously measured background and the signal, while in MCA spectra the background is the baseline below the peaks. Therefore, to calculate the detection limit, one has to apply the filtering criterion of the actual peak-search algorithm built into the spectrum-evaluation software. The most common peak-search algorithm looks for those channels whose counts exceed the average height of the baseline by 3 times the standard deviation in the baseline channel counts. Assuming Poisson statistics, the following expression is obtained: Ci - B > 3σB ) 3√B
(1)
where Ci is the total count number in the given channel, B is the average value of the baseline as calculated from the neighboring channels, and σB is its standard deviation. Thus, the net signal Ci - B has to be 3 times greater than the statistical fluctuation of a single background channel. This algorithm does not account for Currie’s error of the second kind (i.e., when no signal is found, even though the sample contains a significant amount of the analyte), which would (15) Re´vay, Zs.; Belgya, T.; Molna´r, G. L. J. Radioanal. Nucl. Chem. 2005, 265, 261–265. (16) Belgya, T.; Re´vay, Zs.; Fazekas, B.; He´jja, I.; Dabolczi, L.; Molna´r, G. L.; ¨ sto Kis, Z.; O ¨r, J.; Kasza´s, Gy. In Proceedings of the 9th International Symposium on Capture Gamma-Ray Spectroscopy, and Related Topics, Budapest, Hungary, October 8-12, 1996, Molna´r, G.; Belgya, T.; Re´vay, Zs. , Ed.; Springer Verlag: Budapest, Hungary, 1997; pp 826-837. (17) Re´vay, Zs.; Belgya, T.; Kasztovszky, Zs.; Weil, J. L.; Molna´r, G. L. Nucl. Instrum. Methods 2004, B 213, 385–388. (18) Currie, L. Anal. Chem. 1968, 40, 586–593.
imply a constant term on the right-hand side. When actually programmed, a smoothing algorithm calculates the moving average over 3-7 channels and whenever an outlier is found, a peak is registered. However, this ignores the natural width of the peaks. In Hypermet, a more sophisticated filter is used to find peaks, taking into account the natural peak width. It is a symmetric, zeroarea transformation that calculates the smoothed second difference of the data.19 The number of channels over which the moving averages are calculated equals the expected full width at the half of maximum (fwhm) of the peaks as determined from the width calibration. The filter consists of three equal, fwhm-long regions: in the first and the third ones, the channel contents are multiplied by -1, in the middle by +2, and then summed. At the two joining points, a numerical differentiation takes place. It can be shown that the expectation value for this filter, when used on a constant background fluctuating around b, equals zero with a variance of 6bw (where w is the fwhm). If the filter output exceeds 3 times the standard deviation (the square root of the variance), a peak is registered. This filter is highly sensitive for peak shapes of the proper width and is insensitive for broader peaks together with other slow variations in the baseline or for sudden fluctuations.19 Calculation of Masses. The measured areas of prompt-γ peaks are directly proportional to the masses of the emitting nuclides: m A ) NAσγΦ εt M
(2)
where A is the net peak area, ε is the counting efficiency, t is the measurement time (live time), m is the mass of the emitting nuclide, M is its atomic weight, NA is the Avogadro number, σγ is the partial γ-ray production cross section, and Φ is the thermal equivalent neutron flux.1 Partial cross-section values are listed, e.g., in the mentioned databases,9,13 and are expressed as thermal cross sections. (The partial γ-ray production cross section is a derived nuclear constant appearing in the activation equations directly: σγ ) σθPγ, where σ is the thermal neutron capture cross section, θ is the isotopic abundance, and Pγ is the emission probability for the given transition.)9 It has been shown that the convention of using thermal cross sections together with the thermal equivalent flux is valid for all regular nuclides (whose cross sections follow the so-called 1/v law) in any beam that avoids excitation of neutron resonances.1 The direct proportionality between prompt-γ peak areas and masses is valid for ideally thin samples, i. e., when the sample is fully transparent to neutrons and γ-rays. This can be approximated in real experiments quite well. When this criterion is not fulfilled and neither neutron self-shielding nor γ-ray selfabsorption are negligible, they have to be corrected for. When irradiating a homogeneous plate with thickness d, one gets the product of the two correction factors in the following analytic form:
where R is the angle of the plate surface relative to the beam direction (the direction of the γ-ray detection is assumed to be perpendicular to the beam), µγ is the linear absorption coefficient for γ-rays as a function of the energy, while µn is the same for neutrons: µn ) FΝΑσ/Μ (where F is the mass density and σ is the neutron capture cross section of the material at the given neutron energy). Systematic error sources such as an inhomogeneous neutronflux profile and inaccurate dead-time correction can be avoided by using the relative approach, where cross-section ratios or mass ratios are calculated from the ratios of the peak areas: σγ,1 Aσ,1 ε(E2) m2 M1 ) σγ,2 Aσ,2 ε(E1) m1 M2
(4a)
m1 Am,1 ε(E2) σγ,2 M1 ) m2 Am,2 ε(E1) σγ,1 M2
(4b)
The variables here are the same quantities as in eq 2. Aσ and Am represent the peak areas obtained in the cross-section measurement and in the elemental analysis, respectively, while numbers in indices refer to two different chemical elements. For a homogeneous sample, the peak areas are biased by the same systematic errors, which cancel when calculating their ratios. Equation 4a is the basic equation for the standardization, i.e., for the determination of partial cross sections of the elements relative to a comparator, while eq 4b is used for mass analysis, i.e., for the determination of the mass ratios and the elemental composition. The partial cross-section data listed in the mentioned databases9,13 have been determined from peak ratios measured on samples with known compositions using our accurately calibrated detector. The ultimate comparator was the 2223 keV peak of H, whose cross section is 0.3326(7) barn.12 In the case of γ-rays from a radioactive decay, measured peak areas must be corrected for saturation by dividing Ameas with the correction factor fdecay to get Acorr:
fdecay ) 1 -
1 - e-λtmeas λtmeas
(5)
(3)
where λ is the decay constant, tmeas is the measuring time (true time), assuming that the sample is activated and counted simultaneously. If the irradiation is much longer than the halflife of the nuclide, the correction factor can be approximated as fdecay f 1 - 1/(λ tmeas).14 (The in-beam decay correction factor for the nuclides of the more complicated decay class IV/A has also been determined and can be found elsewhere.20) Calculation of Uncertainties. One of the major issues in quantitative analysis is the determination of uncertainties. Throughout these calculations, the recommendations of ISO guide “GUM” are followed,21 though with our own notation introduced because of the many different uncertainties used. According to eq 4a, the relative uncertainty of the partial cross-section ratio equals
(19) Phillips, G. W.; Marlow, K. W. Nucl. Instrum. Methods 1976, 137, 525– 536.
(20) Szentmiklosi, L.; Revay, Z.; Belgya, T. Nucl. Instrum. Methods 2007, B263, 90–94. (21) International Organization for Standardization (ISO). Guide to the Expression of Uncertainty in Measurement; ISO: Geneva, Switzerland, 1993.
I ) I0
∫e ( d
0
-x
µγ
+
µn
cos R sin R
)
(
µγ
µn
1 - e-d cos R + sin R dx ) µn µγ + cos R sin R
)
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6853
δ
σ1 ) σ2
(
(δA1)2 + (δA2)2 + δ
ε(E1) ε(E2)
)
2
(6)
where δ means the relative uncertainty of the quantity standing after it. (For standardization measurements, atomic weights and the mass ratio should be known accurately.) The uncertainty of the efficiency and of the efficiency ratio is determined as a part of the detector calibration. The relative uncertainty for the ratio of two independent quantities equals the quadratic sum of the individual relative uncertainties, while for the efficiency ratio at two close-lying energies it is often negligible due to the correlation.11 The uncertainties of the ratio of the efficiency-corrected cross sections based on eq 4a and of the masses based on eq 4b are the following: δ
δ
m1 ) m2
ε(E2) σ2 ) √(δAσ,1)2 + (δAσ,2)2 ε(E1) σ1
(
(δAm,1)2 + (δAm,2)2 + δ
[) √(δA
2 m,1)
ε(E2) σ2 ε(E1) σ1
(7a)
)
2
]
+ (δAm,2)2 + (δAσ,1)2 + (δAσ,2)2
(7b)
The relative uncertainty of the mass ratio in eq 7b consists of two parts: the terms of statistical uncertainties from the peak areas (Am,1 and Am,2) and the systematic uncertainty from the efficiency-corrected cross-section originating from the standardization. As can be seen from eq 7a, this latter term equals the quadratic sum of the uncertainties of the peak areas observed in the standardization (δAσ,1, and δAσ,2; see the expression in square brackets). This replacement is usable only when the two efficiency ratios are the same, i.e., if the analysis takes place on the same system as the standardization. Then the relative uncertainty of the mass ratio becomes the quadratic sum of the uncertainties from the two components, and both types of uncertainties simplify to those obtained in peak area determinations. When using cross-section data measured on another instrument, the uncertainties from the efficiencies and from the cross sections have to be separately included in the systematic uncertainty. The above derivation is sufficient for a two-element system, but a similar one applies to the general case. In our laboratory, the elemental cross sections were all determined relative to a single comparator: the cross-section of hydrogen. In the case of measuring mass ratios in samples containing no hydrogen, the relative statistical uncertainties of the H comparator peak areas from standardization measurements have to be added to the expression in square brackets.11 These results underline the importance of using both systematic and statistical uncertainties. It is also important to note that in the second step (i.e., the determination of mass ratios) the statistical uncertainty of the first step (i.e., the standardization) has to be regarded as systematic since it cannot be further lowered in the actual experiment. METHODS The algorithms that calculate the composition are programmed in Excel macros. The result is given in the form of an Excel 6854
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spreadsheet, which gives the possibility of interactive modification of the tables. The program loads the peak list generated by the spectroscopy software and then compares this list with the spectroscopic library. On the basis of different statistical criteria, it makes the qualitative and quantitative analysis. Peak Matching. For each peak, all possible matching records are selected from the spectroscopic database. The criterion for a match is that the energies agree within 3σ (where σ is the total uncertainty of the measured and literature energy values) or that the energy difference is smaller than 0.1 keV. (The actual match criteria can be adjusted; in our case these proved to result in the needed agreement.) Besides the characteristic peaks, the so-called escape peaks, i.e., the ones with energies lowered by 511 and 1022 keV are also identified. With the use of these criteria, five to ten library records typically match for each γ-ray at low energies (below 1 MeV) and only at the highest energies (above 6 MeV) is one match per γ-ray found. These latter cases enable the unambiguous identification of certain elements, while those with peaks detectable only at lower energies need another approach for the identification, introduced below. The matching records are copied into Excel spreadsheets, giving an energy list combined from the peak list and the spectroscopic library. Qualitative Analysis. Because of the large number of library matches to peaks at lower energies, finding one low-energy characteristic peak cannot be regarded as a proof of containing the given element in the sample. An element can only be reliably identified barring the identification of a unique high-energy singlet if all or most of its strongest peaks are found in the spectrum. A qualifier number was introduced for this purpose: the sum of squared relative peak areas (εσγ) for the detected peaks divided by that for all the peaks of the element. This number is insensitive to the omission of weak peaks but becomes significantly lower than 1 if strong peaks are excluded. The program automatically selects elements whose qualifier number is greater than 0.4 (an empirically found limit) and those with unambiguously identified singlets. This automatic selection can be overridden by the analyst. The relative intensities of the selected peaks also have to follow the same pattern as those in the spectra of the pure elements. Though the distribution of γ-ray energies is not random, due to the large number of possible matches a statistical approach can effectively be used in judging the validity of a selection. For each measured γ-ray peak, the masses are calculated using eq 2 and for all possible energy matches. These energy-mass records are then sorted according to element, and the weighted average22 of the mass is calculated for each element as follows: n
mi
∑ (d m )
2
m ¯ )
i)1 n
∑ i)1
i
(8)
1 (d mi)2
where mi is the mass value as calculated from the area of the ith peak of the element and the matching partial cross-section, d mi, is the absolute statistical uncertainty, i.e., d mi ) miδmi (22) Bevington, Ph. G. Data Reduction, and Error Analysis for the Physical Sciences; McGraw-Hill, New York, 1969.
(and δmi equals the relative statistical uncertainty from the peak fit, δAi). Z scores for masses and for their associated energy values are calculated for each energy-mass record as follows:
Zm,i )
¯ mi - m d mi
(9a) Dm ¯ ) √(max(σm, ξm))2 + s2
Ei - Elib,i
ZE,i )
(9b)
√(d Ei)2 + (d Elib,i)2
where Zm,i and ZE,i are the Z scores for the mass and energy values, respectively, m ¯ is the average mass as defined above, Ei is the energy of the measured peak, Elib,i is the energy value of the matching peak in the data library, and d Ei and d Elib,i are their respective uncertainties in kiloelectronvolts. If the most important characteristic peaks of the given element appear in the spectrum, their measured energies should match the library values and at the same time the calculated masses should not differ significantly. If the relative intensities show the known pattern of the elemental peaks, the Z scores in eq 9a will follow the standard normal distribution. If the identification is correct, the same is true for eq 9b. Z scores with absolute values greater than 3 are either the result of a systematic error, e.g., an uncorrected spectral interference or a wrong identification. If the Z scores after the removal of the outliers still do not follow the standard normal distribution, it means that the peak areas do not show the elemental pattern. Then the agreements of the peak energies are just random coincidences. An element like that is not selected as a component of the sample. Also χ2 values are calculated for each element to help the decision:
2 χm,i )
1 n
n
∑ i)1
(
mi - m ¯ d mi
)
2
1 n
2 χE,i )
(Ei - Elib,i)2
n
∑ (d E )
2
i)1
i
+ (d Elib,i)2 (10)
where n is the number of selected peaks for an element, the other quantities are the same as in eqs 9a and 9b. For data having a standard normal distribution, χ2 has the expected value of 1 with the variance of 2/n. Other quantities calculated to support the decision making are the internal and external uncertainties22 for the mass values: σm )
1 m ¯
1
∑ n
i)1
ξm )
1 n
(11a)
1 (d mi)2
n
(m ¯ - mi)2
i)1
i
∑ (dm ) m 2
n
∑ i)1
uncertainty, which is the weighted average of the discrepancies of the calculated masses from the average. When the masses follow the normal distribution, the internal and the external uncertainties agree well. The total relative uncertainty of the mass value is then calculated as22
2 i
where s is the relative systematic uncertainty from the standardization: the weighted average of the relative systematic uncertainties of the peaks for the given element. In summary, an element is selected as a component of the sample (1) if the element has unambiguously identified highenergy singlet γ-rays or its qualifier is large enough, (2) if the χ2 values in eq 10 are close to 1, and (3) if the internal and external uncertainties for the mass (eqs 11a and 11b) do not differ significantly. Components with one or more strong peaks always meet these criteria with a high confidence. In some cases, the selection procedure needs the decision of the analyst, which is supported by a set of statistical quantities. Determination of the Composition. PGAA is mainly used for the analysis of major and minor components and in some cases of trace elements (B, Cd, etc.). For samples where all the major and minor components appear in the prompt-γ spectra, we can assume that the composition can be determined from the mass ratios of the components, without knowing the sample masses. This applies to several types of material: minerals, glasses, ceramics, many metal alloys, organic compounds, etc. Trace elements do not modify the concentration values nor the uncertainties of the major and minor components. (Components whose peaks are close to the detection limits may modify the composition, and their handling also needs analytical expertise.) For spectra in which all major and minor components can be expected to appear, the following simple process can be followed: (1) Spectroscopic data are taken from a library that contains the energies and the partial γ-ray production cross sections with their statistical and systematic uncertainties. (2) The mass values are determined from eq 2 using an approximate flux and measurement time. (3) The masses are then summed to get a total mass for the sample. (4) The concentrations are calculated from the elemental masses divided by the total mass (systematic errors again cancel). (5) The uncertainties of the concentrations cannot be directly derived from the individual uncertainties, as the uncertainty of the total mass is not independent of the masses of the components. Instead of the simple quadratic summation, the following expression has to be applied, as was shown elsewhere:11
δck ) (11b)
1 (d mi)2
where σm is the relative internal uncertainty, calculated as the weighted average of the relative uncertainties of the individual masses, using δm ) δA and ξm is the relative external
(12)
(1 - 2ck)(Dmk)2 +
n
∑ c (Dm ) 2 i
2
i
(13)
i)1
where δck is the relative uncertainty for the concentration ck of the component “k”, Dmi is that for the mass of the component “i” from eq 12. (If one neglects the above-mentioned correlation of the concentrations, the first term in eq 13 cancels, resulting in the usual quadratic sum. This simplification can be used as a rough approximation of uncertainties.) Use of eq 13 lowers the uncertainty of the dominant component, while those of the minor Analytical Chemistry, Vol. 81, No. 16, August 15, 2009
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constituents usually do not change much. However, in the case of a dominant component determined from peaks with poor statistics, the uncertainties of the other major and minor components can increase significantly. Corrections. After selection of the constituents, individual peaks are corrected for decay where appropriate and for selfabsorption together with self-shielding according to eqs 3 and 5. The spectral interferences are also taken into account. In many cases single peaks are fitted for multiple γ-ray lines appearing at the same energy within uncertainty limits. A peak area like this is decomposed into the areas of the interfering peaks, which are calculated from the masses according to eq 2. For the interference correction with escape peaks, a straight line has been fitted to the escape-to-full-energy peak-ratio as a function of energy during the detector calibration. Boron is a very special case. It has a characteristic Dopplerbroadened peak at 478 keV with the width of approximately 16 keV, i.e., an order of magnitude broader than the natural peak width at this energy. Its shape cannot be fitted with the peak shapes built into Hypermet-PC. Therefore its area is determined by summing the counts in the energy range of 470-486 keV after a linear background subtraction. More than a hundred peaks from other elements occur in this energy range. Those elements which have interfering peaks here are identified by their other peaks, and the peak areas calculated from the masses of the interfering elements are subtracted from the boron area. (The present routine will be replaced with the algorithm of La´szlo´ Szentmiklo´si,23 which fits the actual shape of the boron peak together with the interfering characteristic peaks.) The masses are updated automatically, and then the composition is finalized. Analysis of Oxides. Oxygen has a neutron capture cross section of 0.19 mb, which is about 2 orders of magnitude smaller than other light elements. Oxygen peaks can be identified in the matrix of light elements as a major component but with a large statistical uncertainty, typically 10-20%. In analytical chemistry a better precision is usually required for a major component. In the case of materials that contain oxygen as the dominant component, its imprecise determination would worsen the precision of the other constituents. For many materials (like rocks, minerals, glass, or cement), the oxygen content can also be calculated from stoichiometry: all the elements (with the exception of halogens) are supposed to appear in the oxide form with the maximum oxidation number. The oxide composition can simply be calculated, in eq 2 the elemental mass and the atomic weight are replaced by the mass and the molecular weight of the oxide (assuming that the stoichiometric coefficient of the element equals 1). If the stoichiometric coefficients are supposed to be accurate, this modification does not change the calculation of the uncertainties, though the actual uncertainty values for the oxides may be different than those for the elements. In the case of elements like iron, additional information is necessary on the oxidation state. Analytical Limits. The program calculates the detection limit (DL) and the determination limit of peaks in mass units and concentration units. The detection limit in our terminology is the mass calculated from the minimum peak area that can be found
by the peak-search algorithm at the characteristic energy (peak detection limit transformed to a mass unit). The determination limit (DL(X%)) is a mass value with a given relative uncertainty. The mass values close to the detection limit typically have relative uncertainties of about 50% using our settings. In our case these two quantities can be used interchangeably (DL and DL(50%), respectively). The uncertainty of the peak area depends partly on the complexity of the fitted region. If the background below the peak is zero, then according to Poisson statistics the standard uncertainty is the square root of the counts in the peak. When the peak stands on a background, the uncertainty is always higher than this ideal value. Thus from the fitted peak area and its uncertainty, the background can be estimated: d B ) √B
(14a)
d(A + B) ) √A + B
(14b)
d Afit ) d(A + B - B) ) √A + 2B
(14c)
where B is the background value and A is the peak area. B is the number of counts (or the integral) of the background under the peak and is determined over a given range around the peak. That range depends on the peak fitting algorithm. In the case of a leastsquares fit of a Gaussian peak on a constant background, this range is approximately 3 times the peak width (fwhm). The uncertainty of B can be approximated with its square root (eq 14a). If a peak with the net area of A also appears in this range, the uncertainty of the total counts equals (A + B)1/2 (see eq 14b). If the background B is subtracted from the total count number (A + B - B), according to the laws of error propagation its uncertainty will be (A + 2B)1/2 (see eq 14c). Thus the relative uncertainty in A and the background value can be formulated as
δΑ )
B)
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A2(δΑ)2 - A 2
(15a)
(15b)
In this calculation, no assumption is made about the background. If a large number is obtained for B, it can mean either a high constant background or a complex spectrum region. B thus characterizes the difficulty of the fit. From this effective background, a hypothetical peak area (a) for which the relative uncertainty is a predefined value (c) can now be determined from eqs 15a and 15b). This value equals
c)
√a + A2(δA)2 - A √a + 2B ) a a
(16)
The solution of the equation is
DL(c) ) a ) (23) Szentmiklosi, L.; Gmeling, K.; Revay, Z. J. Radioanal. Nucl. Chem. 2007, 271, 447–453.
d Afit √A + 2B ) A A
1 + √1 + 4Ac2(A(δA)2 - 1) √A2(δA)2 - A ≈ c 2c2 (17)
Table 1. Standardization of the 1884 keV Peak of Nitrogen Using Different Nitrogen Comparatorsa sample
peak ratio × nH/nN
relative uncertainty (%)
pyridine (NH4)2SO4 NH4NO3 NH4NO3 NH4NO3 NH4Cl melamine
0.05033 (0.05106) 0.05029 0.05009 0.04998 0.04763 0.05000
0.5 (0.9) 0.6 1.1 0.7 2.4 0.3
a
weighted average
Z score
efficiency ratio εH/εN
σγ (barn)
statistical uncertainty (%)
systematic uncertainty (%)
total uncertainty(%)
0.05018
1.5 (2.4) 1.1 0.23 0.06 -1.9
1.1456
0.01457
0.32
0.21
0.38
1.1456
0.01452
0.3
0.21
0.37
0.05000
Weighted average and Z scores are calculated according to eqs 8 and 9a, respectively. Values in parentheses were omitted from the calculation.
RESULTS AND DISCUSSION The validation of the method is shown in two steps: first for the calculation of the uncertainties and then for the determination of the chemical composition. Validation of Uncertainty Calculation. The validation of the uncertainty calculation is shown through the standardization of nitrogen, one of the most important comparators. The average of six different measurements is compared to the seventh, the most precise one. The seven measurements were made using five, highpurity stoichiometric compounds: e.g., inorganic salts or organic solvent. They were chemically different enough to have different contaminants and absorbed water or other sources of systematic
errors. The solid samples were sealed in Teflon bags, and pyridine was poured into a Teflon capsule. The samples were analyzed in the thermal beam at the Budapest Research Reactor. The 1884 keV peak of nitrogen was standardized to the 2223 keV peak of hydrogen (partial cross section, 0.3326 barn with a systematic uncertainty of 0.21%). As discussed earlier, the two energies are so close to each other that the uncertainty of the efficiency ratio could be neglected. Table 1 shows the measured data, the weighted average, and other quantities. One of the measurements was rejected because of its high Z score value, as is usually done with such a low number of measurements. The average cross section is then compared to the one determined in the most precise measurement. The discrepancy between the two values is 0.36%, which is within the mutual uncertainty limit of 0.44% (i.e., the quadratic sum of the two statistical uncertainties). As expected, by averaging the five measurements, the statistical uncertainty could be reduced to a low value. The good agreement of this low value with the statistical uncertainty of the highest-precision measurement validates the present method including the assumption that many sources of uncertainty can be neglected. Validation of the Analytical Method. The validation of the analytical method is demonstrated with the analysis of a standard. As there is no PGAA standard available, a cement standard was used for this purpose. The material is used for the qualification of chemical laboratories of cement factories; the major components are certified with a high degree of accuracy. The 2003 test material was available from one of the laboratories (Cemkut Ltd., Hungary) and was analyzed in round robin tests of 174 laboratories in 35, mainly European, countries using classical analytical chemistry, atomic absorption spectroscopy (AAS), X-ray fluorescence spectroscopy (XRFS), gravimetry, etc. Table 2 contains the averages of all the analytical results, sometimes from more than 200 measurements.25 The ignition loss (due to release of humidity and carbon dioxide) was taken out of the composition. The sum of the reference concentrations after this correction was 99.86 ± 0.06%, where the uncertainty is obtained as the weighted quadratic sum of the external errors of the reference concentrations. (The rest is in the form of trace elements or their oxides.) Similarly, the hydrogen (water) content was left out from the composition measured with PGAA. The two compositions are compared in Table 2 based on the concentration values. The external uncertainties from the interlaboratory test25 and the relative uncertainties of the PGAA results are also shown. The first value shows the accuracy of the reference concentration values and the second
(24) Olivieri, A. C.; Faber, N. M.; Ferre´, J.; Boque´, R.; Kalivas, J. H.; Mark, H. Pure Appl. Chem. 2006, 78, 633–661.
(25) Bonnet, M. Interlaboratory Testing Programme 2003-Final Report; ATILH: Paris, France, 2003.
This a value is by definition the determination limit as a function of c. In many analytical applications, c ) 0.2 is chosen. In our approach, c ) 0.5 is used, i.e., peak areas with relative uncertainties of 50% are still used in the quantitative analysis. The determination limit DL(c) can be well approximated with an expression which is inversely proportional to the relative uncertainty c. Thus it can be easily transformed to other values to conform to any other choice of c (right-hand side of eq 17). As mentioned earlier, the DL(0.5) value can replace the detection limit in our analytical method. As discussed above, in many cases the detection of several peaks is necessary for the identification of an element. Similar to the Currie approach, the recommended multivariate calibration approach could not be adapted for PGAA either.24 A two-level definition of analytical limit has been introduced for the elements based on the detection limits (or 50% determination limits) of their strongest peaks and has proved to be useful in our analytical method. Detecting the three strongest peaks of an element has proved to be a sufficient criterion for the identification of those elements which do not have high-energy singlets. The minimum of the DL values for the three masses is the rejection limit: if the mass of an element is below that limit, it cannot be identified in PGAA (a negative analytical limit for the element). The maximum of these DL values is the acceptance limit: if the mass of the element is above this value, it can be identified in the spectrum and the uncertainty of its mass will be e50% (a positive analytical limit for the given element). The two limits can sometimes differ by an order of magnitude in cases when the third peak is much weaker than the first. Between the two limits, the decision can only be made based on other information on the material. Examples are shown in the next section.
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Table 2. Analysis of the Cement Standard
a
oxide
reference concn (%)a
external uncertainty (%)a
PGAA measurement concn (%)
PGAA relative uncertainty (%)
Z score
Na2O MgO Al2O3 SiO2 P2O5 SO3 Cl (chloride) K2O CaO TiO2 Fe2O3 SrO
0.411 0.984 6.975 1.31 0.703 3.082 0.065 0.93 61.40 0.422 3.385 0.184
1.42 0.97 0.34 0.1 1.3 0.22 1.4 0.50 0.07 0.9 0.33 2.5
0.42 0.9 7.2 21.4 1.0 3.1 0.067 0.98 61.0 0.43 3.3 0.18
5.9 8.5 2.4 2.4 31 3.1 2.7 4.2 1.2 3.2 2.8 25
0.5 -1.5 1.1 0.2 1.2 0.2 1.1 1.2 -0.6 0.5 -1.2 -0.1
Taken from ref 23.
Table 3. Trace Element Content of the Cement Standard As Determined by PGAA oxide
concn (ppm)
relative uncertainty (%)
B2O3 Cr2O3 Mn2O3 CdO Sm2O3 Gd2O3
213 270 830 1.0 7 9
1.7 16 5 6 2.4 10
the precision of the PGAA measurement. In the case of any systematic errors (e.g., false selection of components or wrong library data), the PGAA results would be significantly different from the reference values. The results show a good agreement. The Z scores (calculated as in eq 9a), which ought to follow the standard normal distribution with the expectation value of 0 and with the variance of 1, here have the average of 0.18 with the estimated uncertainty of 0.27 and the standard deviation is 0.92. The result of PGAA measurement agrees well with the reference values. Since the concentration values are correlated, any systematic errors would shift the concentration values, and then the Z scores would not follow the normal distribution, as shown by the average and the variance. Using PGAA it was possible to determine a few trace elements, which did not appear in the certification of the standard. They are listed in Table 3. Their presence really does not modify the concentrations of the major and minor elements. The sum of the masses of the trace elements from the PGAA analysis proved to be 0.13%, while it is expected to be 0.14 ± 0.06% from the sum of the average concentrations. Though this quantity was not certified, the agreement again is good. Other analyses also confirm both the analytical method and the relevant spectroscopic database. Geological standards were analyzed with PGAA and compared with X-ray fluorescence (XRF) and inductively coupled plasma mass spectrometry (ICPMS) results.26 Several other measurements during the past 10-12 years also used the earlier versions of the analytical method presented here for the analysis of archeological samples,5,27 metal (26) Marschall, H. R.; Kasztovszky, Zs.; Gme´ling, K.; Altherr, R. J. Radioanal. Nucl. Chem. 2005, 265, 339. (27) Kasztovszky, Zs.; Biro, K. T.; Marko, A.; Dobosi, V. J. Radioanal. Nucl. Chem. 2008, 278, 293–298.
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Table 4. Analytical (Rejection and Acceptance) Limits for a Few Elements in the Cement Standard in Weight Percents element
RL (%)
AL (%)
Li Be C N Mg P Cu Zn
0.06 1.6 4.4 0.3 0.13 0.2 0.008 0.07
0.7 2.4 5.5 1.2 4.4 0.25 0.09 0.3
alloys,28 geologic materials,29 in material science,30,31 soil samples from meteor-impact sites,32 etc. with reliable results. Earlier versions of the program have been installed in other PGAA laboratories, too: at the National Institute of Standards and Technology, at the University of Texas at Austin, and the recent one at the Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II) in Garching, Germany. The analytical method was verified in a series of applications.33 Examples for the Analytical Limits. The same cement standard was used to calculate the rejection and the acceptance limits for the elements. A few of them are shown in Table 4. Some light elements, like Li, Be, C, and N, have low cross sections (45, 9, 4.5, and 80 mb, respectively). On the basis of the spectrum, their presence cannot be excluded, but it is well-known that heattreated cement does not contain these elements as minor components. Magnesium has a low cross section, too (67 mb). In the given spectrum, its first peak could be evaluated easily, the second was obscured by an interference, and the third was (28) Manescu, A.; Fiori, F.; Giuliani, A.; Kardjilov, N.; Kasztovszky, Z.; Rustichelli, F.; Straumal, B. J. Phys.: Condens. Matter 2008, 20, 104250. (29) Gmeling, K.; Harangi, Sz.; Kasztovszky, Zs.; Pecskay, Z.; Simonits, A. Geochim. Cosmochim. Acta 2007, 71, A331–A331. (30) Perry, D. L.; English, G. A.; Firestone, R. B.; Leung, K. N.; Garabedian, G.; Molnar, G. L.; Revay, Zs. J. Radioanal. Nucl. Chem. 2008, 276, 273–277. (31) Kovacs-Mezei, R.; Krist, T.; Revay, Zs. Nucl. Instrum. Methods 2008, A586, 51–54. (32) Firestone, R. B.; West, A.; Kennett, J. P.; Becker, L.; Bunch, T. E.; Revay, Zs.; Schultz, P. H.; Belgya, T.; Kennett, D. J.; Erlandson, J. M.; Dickenson, O. J.; Goodyear, A. C.; Harris, R. S.; Howard, G. A.; Kloosterman, J. B.; Lechler, P.; Mayewski, P. A.; Montgomery, J.; Poreda, R.; Darrah, T.; Hee, S. S. Q.; Smitha, A. R.; Stich, A.; Topping, W.; Wittke, J. H.; Wolbach, W. S. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 16016–16021. (33) Harrison, R. K.; Landsberger, S. J. Radioanal. Nucl. Chem. 2009, 267, 513– 518.
below the detection limit. The expected concentration of magnesium in cements is below the acceptance limit but above the rejection limit. As its presence in the sample was obvious in this case, its concentration was determined based on its strongest peak only, which has agreed well with the reference value (see Table 2), though with a relatively high uncertainty. The other elements are just illustrations of the analytical limits. CONCLUSION A new method has been developed for reliable qualitative and quantitative PGAA analysis. The algorithms handle the large number of γ-ray peaks, using statistical criteria to identify the elements and fiting the masses to the individual values calculated for each of the observed peaks. A careful analysis of the error propagation and the calculation of uncertainties are also part of the procedure. A two-level analytical limit was introduced to characterize the identification of elements based on the detection
of one or several peaks. The described method has proved to be a proper tool in the quantitative analysis of a large variety of samples. ACKNOWLEDGMENT The author thanks the supporters of the GVOP project (Contract Number GVOP-3.2.1-2004-04-0268/3.0.), the NAP VENEUS 2008 project (Contract Number OMFB 00184/2006), and the EU FP6 NMI3 project (Contract Number RII3-CT-2003505925). The help of Jesse L. Weil is appreciated. The suggestions of the staff members in the Department of Nuclear Research, especially of La´szlo´ Szentmiklo´si, are also acknowledged. The author also thanks the Cemkut Ltd. for providing the cement standard. Received for review May 29, 2009. Accepted June 26, 2009. AC9011705
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