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Applied Radiation and Isotopes
journal homepage: www.elsevier.com/locate/apradiso
Improved peak shape fitting in alpha spectra
S. Pommé a *, B. Caro Marroyo b
a European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium b Laboratorio de Metrología de Radiaciones Ionizantes, CIEMAT, Avenida Complutense 22, 28040 Madrid, Spain
CrossMark
HIGHLIGHTS
• A new algorithm for analysis of alpha-particle spectra is presented.
• The peak shape functions reproduce low- and high-energy tailing.
• Its proficiency is demonstrated by a fit to high-resolution spectra.
• It outperforms existing specialised fit functions.
ARTICLE INFO ABSTRACT
Peak overlap is a recurrent issue in alpha-particle spectrometry, not only in routine analyses but also in the high-resolution spectra from which reference values for alpha emission probabilities are derived. In this work, improved peak shape formulae are presented for the deconvolution of alpha-particle spectra. They have been implemented as fit functions in a spreadsheet application and optimum fit parameters were searched with built-in optimisation routines. Deconvolution results are shown for a few challenging spectra with high statistical precision. The algorithm outperforms the best available routines for high-resolution spectrometry, which may facilitate a more reliable determination of alpha emission probabilities in the future. It is also applicable to alpha spectra with inferior energy resolution.
© 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/).
Article history:
Received 1 September 2014
Received in revised form
17 November 2014
Accepted 27 November 2014
Available online 28 November 2014
Keywords:
Alpha-particle spectrometry Spectral interference Deconvolution Least-squares fitting Alpha-emission probabilities Maximum likelihood
1. Introduction
Spectral deconvolution has always been an issue in alpha-particle spectrometry (Garcia-Torano, 2006; Pommé, in press), as individual alpha peaks tend to partially overlap due to the limited attainable energy resolution, even in ideal conditions. The benefit of spectral analysis is the identification and separation of peaks, with the aim of determining activity ratios between different alpha emitters or calculating individual emission probabilities in the decay of a single radionuclide. In particular for the latter application, i.e. deriving reference decay data from high-resolution spectra, spectral deconvolution is performed by fitting complex mathematical functions to each peak. The normalised area of each fitted peak in the energy spectrum then represents its relative emission probability. The quality of the emission data depends on the realistic representation of the peak shape and the ability to closely reproduce the experimental energy spectrum.
* Corresponding author. Fax +32 14 571 864. E-mail address: stefaan.pomme@ec.europa.eu (S. Pommé).
Various peak models have been proposed in the past (Trivedi, 1969; Baba, 1978; Watzig and Westmeier, 1978; García-Toraño and Aceña, 1981; L'Hoir, 1984; Bortels and Collaers, 1987; Koskelo et al., 1996; García-Toraño, 1997, 2003; Lozano et al., 2000; Pommé and Sibbens, 2008; Semkow et al., 2010) and in recent work an attempt was made to include more physics in spectrum deconvolution (Siiskonen and Pollanen, 2011; Westmeier and Siemon, 2012; Pollanen and Siiskonen, 2014) or to use an artificial neural network (Baeza et al., 2011). One of the most successful analytical models to represent the shape of a mono-energetic alpha peak is the convolution of an exponential low-energy tail with a Gaussian distribution, as suggested by L'Hoir (1984) and introduced by Bortels and Collaers (1987)
f (u - f a, t) = Aexpl —
2T \ T
where A is the peak area, u - a is the distance to the peak position, s is the standard deviation of the Gaussian and t is the tailing parameter. They obtained a better fit after pre-treating the spectrum by subtracting a long tail distribution and then fitting a mix
http://dx.doi.org/10.1016/j.apradiso.2014.11.023
0969-8043/© 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
of two exponential functions with different characteristic lengths, t1 and t2, using normalised weighting factors n and 1 - n1. This was implemented in the software package ALFA (Babeliowsky and Bortels, 1993). Pommé and Sibbens (2008) abolished the pre-treatment of the spectrum and added the third exponential function into the fit function with a normalised weighting factor ^3 = 1 - n - ^2
F(u) = 2 nf (u
■Mi ;
This function, implemented as visual basic application ALPHA in Excel©, has been applied in spectral deconvolution of 235U (Garcia-Torano et al., 20 05), 240Pu (Sibbens and Pommé, 2004; Sibbens et al., 2010), 236238U (Marouli et al., 2014; Pommé et al., 2014) and the 230U and 225Ac decay series (Marouli et al., 2012; 2013). Whereas ALPHA generally performs very well, there are still noticeable residuals when fitting spectra that have millions of counts in each energy bin, since the extremely low random variations in the spectra reveal the slightest mismatch between fit and experiment. Similar imperfections show up with other analysis software, such as ALPACA (Garcia-Torano,1997) and ALFITEX (Caro Marroyo et al., 2013). Whereas ALFITEX applies the same peak model with the optional use of two or three exponentials (Eqs. (1 ) and (2)), ALPACA relies on a different peak shape model in which the low energy region is reproduced by a hyperbolic function.
A more adequate fit of the most challenging spectra requires more elaborate modelling. This adds to the complexity and computing time for reaching the optimum fit parameters via search routines. In this work, a logical extension to the analytical model in Eqs. (1) and (2) is proposed. The analytical model is implemented as a call function in an Excel spreadsheet and the built-in optimisation routine SOLVER is used to find the best match between fitted and measured spectrum. This spectral analysis tool will be referred to as 'BEST' and its performance will be tested on recently published 240Pu and 238U spectra and compared with existing models.
2. The 'BEST* algorithm
The typical line shape of a peak with triple tailing as defined in Eqs. (1) and (2) is shown in Fig. 1, with indications where the peak width a and the tailing parameters t1 < t2 < t3 determinate the
Line shape model with triple tailing
Energy
Fig. 1. Typical line shape applying a convolution of a Gaussian with three left-handed exponentials (Eqs. (1) and (2)) and indication where the peak width a and the tailing parameters t1 < t2 < t3 are the most influential.
peak shape. With increasing distance from the peak top, the tailing is dominated by exponentials with larger t values. The analytical peak shape is not completely smooth in transition regions where the dominance changes from one exponential tail to another.
One way of mitigating the abrupt changes in the slope of the tailing is adding more exponential tails with progressively increasing t values. On the other hand, the experimental spectra are also not perfectly smooth and may show mild curvatures that are not well reproduced by monotonic functions. Therefore the fit would not only benefit from a higher number of exponential tailing functions, but also from flexibility in the weighting factors, allowing some of them to be negative or larger than 1 on condition that the sum of all weighting factors remains one.
Another observation is that alpha peaks show slight distortions and even some tailing at the high-energy side. This aspect can be taken into account by applying also right-handed exponential tailing in addition to the left-handed tailing. The inversion of the tailing is easily established by changing the sign of the position in the function call, i.e. by replacing u - ¿u with ¿u - u. In the current implementation of the BEST algorithm, the peak shape can contain up to ten left-handed and four right-handed tailing functions
0 10 F(u) = 2 nf (Mi - u; t) + 2 nf (u - Mi;
i=-3 i=1
The sum of the weighting factors is normalised to one, but this is the only constraint on the individual values. In practice, this constraint is implemented by excluding ni from the list of free parameters in the fit and setting it equal to 1 minus the sum of the other factors
10 10 2 ni = 1 or n1 = 1 - 2 ni
In the decay of an alpha-emitting radionuclide, usually there are several alpha transitions involved with different energy and relative emission probabilities. Each of them corresponds to an individual peak in the alpha spectrum and the peak area Ak of peak k can be represented as the product of the total number of alpha decays Atot of this particular nuclide multiplied by the relative intensity Ik
Ak = Atot = 11
npeaks
2 Fk(u)
It is convenient to use Atot as a free fitting parameter to adjust all peaks of a radionuclide to the number of counts in a spectrum, without changing the spectral shape through the relative peak intensities. This feature can be used i.a. or e.g.. for determining the activity ratio of radionuclides in a mixed source. Some of the intensities may be released for more precise fitting, while others may be kept fixed, e.g. because literature values or information derived from gamma spectrometry may be more accurate. The normalisation of the intensities is enforced by constraining the intensity of the major peak to the value of 1 minus the sum of the intensities of the other peaks
Npeaks Npeaks
X Ik = 1or h = 1 - 2 4
Additional functional relationships between parameters, e.g. a constant ratio between emission probabilities, can be introduced as a constraint in the spreadsheet. Also the energy calibration, e.g. through a linear relationship between bin number and keV, can be part of the optimisation process, before individual peak positions are released in the fit for more precise positioning.
When fitting a spectrum within a selected energy region, all
peaks contributing to it should be included, including the tailing of possible high-energy peaks situated outside the considered energy region. This may include peaks from interfering nuclides, the spectrum of which can be introduced using the same energy calibration. In principle it is possible that there is a misalignment of the peak energies between two radionuclides, for example through a mismatch of the energy calibrations in the experiments on which the literature values are based. In that case, one can easily introduce a separate energy calibration for both nuclides to align both spectral shapes to the measured aggregate spectrum. Multiple energy calibration can also be useful in conversion electron spectrometry, involving mixed spectra of photons and particles (Perajarvi et al., 2014).
-3 8 101
(- 10J
3. Examples
3.1. High-resolution 240Pu spectrum
It is bad practice to demonstrate the adequacy of analytical functions by means of a fit to spectra with a low number of counts. Visually the fit may appear convincing, certainly in a log scale plot, and the residuals are bound to look stochastically distributed since the randomness of the Poisson process would be the dominant uncertainty component. Systematic functional misrepresentations of the spectral shape can only surface when they supersede statistical uncertainty. For this reason, a high-resolution 240Pu spectrum with extremely good statistical accuracy (3.6 x 108 events) was selected for a test of the algorithm.
In Fig. 2 a classical fit with triple tailing is shown, which was the best available fit for the determination of the alpha emission probabilities (Sibbens et al., 2010). The fit reproduces the spectral shape quite well, but slight discrepancies between model and reality do show up. The regions around the peak tops are well fitted, but the slope of the tailing is not rigorously reproduced by the model. The apparent subtle tailing at the high energy side was not included in the model and therefore the fit could not be extended to that region. The residuals varied between - 22 and + 27, owing to the extremely low statistical uncertainty in each bin.
For comparison, this challenging spectrum was fitted with BEST using the full capacity of 10 exponentials at the low-energy side and 4 at the high-energy side of the peak. The same shaping was applied to the three peaks and the result is shown in Fig. 3. It is clear that the increased flexibility of the model has made it
Energy (keV)
Fig. 2. Measured 240Pu alpha-particle spectrum 'IRMM HR01' from Sibbens et al. (2010) and fitted line shapes using the ALPHA software based on a convolution of a Gaussian and triple exponential low-energy tailing. The residuals spectrum (top) is shown in units of one standard deviation of the channel contents.
5000 5050 5100 5150 5200
Energy (keV)
Fig. 3. Fit to the 240Pu alpha-particle spectrum 'IRMM HR01' from Sibbens et al. (2010) with the BEST algorithm, using a convolution of a Gaussian with 10 exponential low-energy tails and 4 high-energy tails (Eqs. (1) and (3)). The residuals spectrum (top) is shown in units of one standard deviation of the channel contents.
possible to fit the tailing and the changes in its slope more adequately. Also the high-energy part of the spectrum could be reproduced more rigorously, including the high-energy tailing of the main peak and the interfering tailing from peaks around 5.5 MeV. The overall fit has been drastically improved and - the residuals being confined between - 4 and + 4 - approaches the desired level of perfection.
The fitted intensities of the three main peaks - 72.63%, 27.29% and 0.084% with a negligible statistical uncertainty - agree by one standard uncertainty with the data obtained by Sibbens et al. (2010), i.e. 72.70 (7)%, 27.21 (7)% and 0.085 (4)%. Nevertheless, the difference is significant compared to the uncertainty on model dependence estimated from a comparison among six participants with different software packages, which is the dominant uncertainty component (Sibbens et al., 2010). On the basis of the closer fit to the spectrum, it seems reasonable to assign a higher credibility to the result obtained with the BEST model than with the simpler models. However, this issue is open to future scrutiny. The provided data in this work are not intended to replace the previously published values, and therefore no extensive uncertainty budget is provided.
3.2. High-resolution 236U spectrum
The example of the 240Pu spectrum is extreme in the number of counted events and the complexity of the fit function used, i.e. with 14 exponentials. This level of complexity is not needed in more common spectra which are usually well reproduced with a double or triple tailing and lack the statistical accuracy to require or even justify the free fit of additional shaping parameters. The BEST algorithm is implemented in a way that the number of exponentials is freely chosen by resetting the weighting factors n of the superfluous exponentials to zero, or by simply erasing these numbers in their respective spreadsheet cells.
As an example of a spectrum with 'intermediate complexity', a 236U spectrum was fitted without using the full potential of the model. In Fig. 4 the classical fit with triple tailing is shown (Marouli et al., 2014) as well as the residuals, which vary between -15 and +15. Fig. 5 shows a fit result obtained with BEST, using 6 left-handed and 1 right-handed exponentials. The fit has improved considerably, the most extreme residuals being reduced to - 5 and + 5, but there are still issues with reproducing the finer details of the shape of the main peak. Also in this example, the fitted intensities of the main peaks - 74.25%, 25.64% and 0.119% - agree by
""w**i/vVA : ; ■
4150 4200 4250 4300 4350 Energy (keV)
Fig. 4. Measured 236U alpha-particle spectrum (with magnet) from Marouli et al. (2014) and fitted line shapes using the ALPHA software based on a convolution of a Gaussian and 3 exponential low-energy tails. The residuals spectrum (top) is shown in units of one standard deviation of the channel contents.
4150 4200 4250 4300 4350 4400 4450 4500 Energy (keV)
Fig. 5. Fit to the 236U alpha-particle spectrum (with magnet) from Marouli et al. (2014) with the BEST algorithm, using a convolution of a Gaussian with 6 exponential low-energy tails and 1 high-energy tail. The residuals spectrum (top) is shown in units of one standard deviation of the channel contents.
one standard uncertainty with the data obtained by Marouli et al. (2014), i.e. 74.20 (5)%, 25.68 (5)% and 0.123 (5)% and have the same uncertainty budget. The evaluated statistical uncertainty of the main emission probabilities was very small, 0.02%, but the uncertainties were increased to 0.05% to take into account spectral distortions (Marouli et al., 2014). Apparently, the introduction of a more sophisticated line shape demonstrates that an uncertainty increase of this magnitude was indeed needed.
3.3. Thick-source spectra of238U
Whereas the algorithm was primarily designed for high-resolution spectrometry, it can also be tested on spectra from thick sources, having a poor energy resolution. Two alpha spectra from the decay of 238U were selected from the work of Semkow et al. (2010), in which samples of various thicknesses were prepared and counted with a grid ionisation chamber suitable for fast alpha spectrometry in bulky matrices. Figs. 6 and 7 show spectra from a relatively thin (0.11 mg cm- 2) and thick (1.3 mg cm- 2) 238U source (with 230Th tracer), respectively. As the source mass increases, the peaks become broader and less intense due to energy loss and self-absorption in the sample. Nevertheless, there is always a fraction emitted from the top layer that can
Fig. 6. Fit of a peak shape with 5 left-handed and 4 right-handed exponentials to a thick-sample (0.11 mgcm-2) 238U alpha spectrum (Semkow et al., 2010).
Fig. 7. Fit of a peak shape with 5 left-handed and 4 right-handed exponentials to a thick-sample (1.3 mg cm-2) 238U alpha spectrum (Semkow et al., 2010).
be measured close to the emission energy.
The line shapes fitted to the spectra were based on 5 left-handed and 4 right-handed exponentials. The exponential models are well suited to reproduce the long tailing part (also below 3.5 MeV), but residuals remain between the maxima of the two main peaks. No substantial gain in quality was found by increasing the number of exponentials. For this type of spectra, the fit result is fairly good but not superior to the functions proposed by Sem-kow et al. (2010).
4. Uncertainty
The issue of uncertainty in alpha spectrometry has been discussed in a recent review paper (Pommé, in press). The statistical uncertainties on the fitted parameters, including the peak areas, can in principle be calculated from the covariance matrix. The corresponding equations have been published by Sibbens (1998), Ihantola et al. (2011) and Caro Marroyo et al. (2013). However, a minimum requirement is that the residuals are purely stochastic. This condition is not fulfilled in most of the spectra presented in this work.
Experience with complex spectra shows that fit results are badly defined if a spectrum can be fitted in several ways, e.g. increasing a tailing can have similar effects as adding a small additional peak, changing the peak width or energy calibration can significantly influence the areas of overlapping peaks, the area of
small peaks interfered by the tailing of a dominant peak are crucially dependent on the exact representation of the latter, not taking into account underlying peaks from impurities can change relative positions of freely fitted peaks and alter relative peak areas. Also scattering effects, electronic drift and coincidences between signals from alpha particles with beta particles, conversion electrons, x-rays or electronic noise give rise to a distortion of the energy spectrum which is not explicitly implemented in the analytical model (Pommé et al., 2014; Marouli et al., 2014).
For these reasons, the uncertainties obtained from the covar-iance matrix may perhaps suffice in routine measurements with low statistical accuracy, but should be interpreted with caution in reference measurements with high statistical precision in the measurement data. Comparison of fit results with different codes does reveal biases that are not under statistical control (Sibbens et al., 2010).
The preferred way of dealing with uncertainties in radionuclide metrology is to make for each peak a detailed uncertainty budget and perform proper uncertainty propagation towards the final result (see e.g. Sibbens et al., 2010; Pommé et al., 2014; Pollanen and Siiskonen, 2014). Uncertainty components include counting statistics, spectral interferences, impurities and background, residual deviations between fit and measurement, physical effects not included in the model and model dependence of the fit result. Normalisation and (anti-) correlation of uncertainty components are constraints that require specific propagation formulas. Equations and numerical examples can be found in Pommé (in press).
The alpha emission probability Pk is derived from the peak area Ak relative to the total area Atot of all peaks in the emission spectrum of the same radionuclide. The sum of all relative intensities EPk equals 100% by definition. Through normalisation, all intensities receive a degree of correlation which comes on top of the (anti-)correlation of the peak areas due to imperfect deconvolu-tion of spectral interferences. In the hypothetical case in which the alpha peaks are perfectly separated, i.e. their peak areas are un-correlated, the uncertainties on the areas propagate to Pk via
APk) = p2(i - PkY
Mk) - 2( z^A)
( Zu,AY
Statistical uncertainties are readily introduced into Eq. (7), and the same equation can be used to propagate the interference of an impurity that affects part of the spectrum (Pommé, in press). Also a mismatch between fit and measured spectrum can be included in the uncertainty budget. Explicit uncertainties can be assigned to fit model dependence, contrary to ignoring this component when relying on the covariance matrix.
Another propagation formula is required in the ubiquitous cases in which the uncertainties in the peak areas are anti-corre-lated
2(Pk) =
Eq. (8) is applicable in any situation in which adding an amount A(Ak) to peak k implies the subtraction of the same amount from the rest of the spectrum, so that the total area EA remains invariable and the corresponding emission probability changes to Pk =(Ak+A(Ak))/EA. This situation occurs in the fit of an unresolved doublet, subtraction of tailing from a higher-energy peak and correction for coincidence summing-in and summing-out effects (Pommé, in press).
To a lesser extent, positively correlated uncertainties may also appear for which the propagation factor is smaller (Pommé, in press). If the relative deviation is the same for all peaks, there is no change in the emission probabilities. Eq. (7) gives an upper
limit for the propagated uncertainty.
5. Conclusions
Whereas the convolution of a Gaussian with three left-handed exponentials is very successful in satisfactorily reproducing most high-resolution alpha particle spectra, a more elaborate modelling is needed to fit the most demanding spectra with extremely good counting statistics. A line shape model was proposed that expands the number of left-handed exponentials and also incorporates a number of righted-handed exponentials, which allows obtaining a smoother function, better reproducing changes of slope in the tailing and incorporating spectral broadening at the high-energy side.
A line model with up to 10 left-handed and 4 right-handed exponentials was implemented as a function in the spreadsheet application BEST. It uses the functionality of a spreadsheet to perform the search for optimum fit parameters, to select which parameters to keep fixed or to define a relationship between a set of parameters (e.g. normalisation, fixed reference data, maintain a ratio between parameters), to store spectral data and all specifics of the fit together in one file, to plot and export the results. Applications include the free fit of individual peaks to determinate alpha emission probabilities or of complete radionuclide emission spectra to determine activity ratios in a mixed sample. The algorithm outperformed existing software at fitting high-resolution 240Pu and 236U spectra with high count numbers. Its applicability extends to thick alpha sources of which the spectrum almost resembles a step function. Further extensions are possible in which different functional shapes are combined, e.g for application with mixed spectra of mono-energetic electrons and x-rays.
Acknowledgement
One of the authors gratefully benefitted from training possibilities in the frame of an Early Stage Researcher Mobility Grant (ESRMG) from the European Metrology Research Programme (EMRP) (1ND04-ESRMG2) within the EURAMET-EMRP Joint Research Project 1ND04-MetroMetal. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The authors thank Prof. Dr. T. Semkow (New York State Department of Health, Albany) for providing thick-sample alpha-spectra of 238U.
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