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Research Article

Nonlinear Least-squares Fitting for PIXE Spectra

A. Tchantchane, M.A Benamar , A. Azbouche , N. Benouali and S. Tobbeche
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An interactive computer program for the analysis of PIXE ( Particle Induced X-ray Emission) spectra was described in this study. The fitting procedure consists of computing a function Y (I, a) which approximates the experimental data at each channel I. a is a set of fitting parameters (energy and resolution calibration, X-rays intensities, absorption and background). The parameters of fit were determined by using a nonlinear least-squares fitting based on the Marquardt`s algorithm. The program takes into account of low energy tail and escape peaks. The program was employed for the analysis of PIXE spectra of geological and biological samples. The peak areas determined by this program are compared to those obtained with AXIL code

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  How to cite this article:

A. Tchantchane, M.A Benamar , A. Azbouche , N. Benouali and S. Tobbeche , 2005. Nonlinear Least-squares Fitting for PIXE Spectra. Journal of Applied Sciences, 5: 1645-1649.

DOI: 10.3923/jas.2005.1645.1649



A reliable computer program that gives an accurate determination of the peak area of characteristic X-rays is of major importance for PIXE analysis. Knowing the expected X-ray yield of an element in a sample, the peak area is converted to elemental concentration. Therefore, a good estimation of the peak area is very crucial for the trace element analysis. PIXE spectra are complicated because of the existence of following effects:

Interference between L and K lines of heavy and light elements respectively and between Kα and Kβ lines of neighboring elements;
Distortion of the lower side of the peak from the pure gaussian shape due to incomplete charge collection;
Pile-up peaks due to high counting rates;
Various physical sources contribute to the background;
X-rays absorption into the sample and in filters and Si(Li) detector;
Radiative Auger transitions.

This study deals with the description of a computer program that was developed to perform PIXE spectra analysis. The modeling method and the performance of the program were also described.


Several approaches of PIXE spectra fitting and peak modeling are described in the literature[1-9]. In this program, the model function Y (I) at channel I is defined as the superposition of an analytical function B (I) that accounts for the background and a second term which accounts for all the element lines that may be present in the spectrum.


Where, NE and NLI are the number of elements in the sample and the number of lines of element l, respectively. Pl is a free parameter corresponding to the area of the element l. Rk, l is the relative intensity of Kth line of element I. Its’s correct Abs (Ek, l) is a function depending on x-ray energy which corrects x-rays attenuation in the different absorbers. The absorption function at energy EI corresponding to channel I is expressed as:


Ab (EI) describes the x-ray absorption into the Si(Li) detector and the filters. It is estimated by using the mass attenuation coefficients from the compilation of data of McMaster et al.[10]. Abs (EI) represents the sample self-absorption which is approximated by:


where, b1, b2 are parameters.

Gk, l (I) is an ideal gaussian representing the Kth line of the element l. Tailk, l (I) and Esck, l (I) describe the low energy asymmetry and the escape peaks, respectively.

Background model: The background model is expressed as the sum of three components. It is given by the following formula:





Where, I0 is a fixed reference channel and Ie is the channel corresponding to the incident energy Ee transferred by incident proton to free electron. Ee is approximated by:


Where, me, mp are the mass of electron and proton, respectively and Ep is the incident energy. NB1, NB2 and NB3 are constants.

The third term in the Eq. 4 accounts for the secondary electron bremsstrahlung background whereas the first two terms account for the other types of background, i.e. the Compton scattering and the proton bremsstrahlung background.

Gaussian shape: The ideal gaussian function Gk, l (I) is expressed by:


Where, depicts the Full Width at Half Maximum (FWHM). C1 and C2 are the parameters of the linear energy calibration. C1 , C2 , C3 and C4 are handled as free parameters but varied under some constraints. They are included in the nonlinear gaussian model in order to achieve the energy calibration and the energy dependence of FWHM.

Low energy tail: A low energy tail is added to the pure gaussian. It takes into account the asymmetry of the lower side of the peak. In some cases, this tail may be neglected since more than 95% of the detected pulses are contained in a gaussian peak. However, when a weak peak is superposed to the tail of an intense neighboring peak, the relative area can be over or under estimated. In the program, we have used the method described by Marageter et al.[11]. The tail function Tailk, l (I) is given by:



Escape peaks: The escape peaks are included in the program. They are treated as pure gaussians. The inclusion of the escape peaks tends to correct the Si-Kα secondary X-rays escaping from the sensitive volume of the crystal during the photon absorption. Their relative intensities are estimated by Clayton[4]:


The energy of the escape peak is related to the energy E k, l of its parent element by:


1.739 keV is the Si-Kα energy.

The FWHM of the escape peak is chosen to be the same as for a characteristic X-ray peak at the energy


The experimental data are fitted by using the least-squares method. The parameters in the Eq. 1 are determined by minimizing the χ2 function. The χ2 function is defined as follows:


Where, Y (I) is the model function, YI is the yield at channel I and NC is the number of channels in the spectrum. fc is the constraint function and a is a set of fitting parameters (Table 1). We have based the nonlinear fitting on Marquardt’s algorithm[12] which combines both the linearisation of χ2 function and the use of the gradient search in order to assure the convergence.

Table 1: List of fitting parameters (NP)

Table 2: Comparison of Kα peak areas obtained with AXIL code and our program for a prawn sample PIXE spectrum


The program is designed for a fast interactive operator control during off-line PIXE spectra analysis. The experimental data, the modeled spectrum and the modeled background may be visualized on a graphic terminal. The program is flexible and easy to use. All the input data are stored in one sequential file. The flow chart of the program is depicted on the Fig. 1. An additional option is included in the program which allows the determination of the background prior to the fitting. In this case, the background is modeled using a non polynomial approximation[13] and stripped off from the original spectrum before its fitting. We have found this option very useful for fitting large region, small peaks and complex spectrum. The peak areas which are evaluated in the fitting process are converted to element concentrations using a separate program.

Fig. 1: Flow chart of the computer program

Fig. 2: Fits to the soil PIXE spectrum. Dots: experimental data, solid line: modeled spectrum, dashed line: modeled background

Fig. 3: Fits to the prawn PIXE spectrum. Dots: experimental data, solid line: modeled spectrum, dashed line: modeled background


The computer program used to is analyze PIXE spectra of geological and biological samples. The spectra were measured using 2.5 MeV protons of 10 nA beam current delivered by 3.75 MV Van de Graaff accelerator at Algiers. Two examples of fitted spectra are shown in Fig. 22 = 16) and 3 (χ2 = 15)[14]. The experimental spectrum is well reproduced by the fit and a good estimation of the background is noted (Fig. 2). The same spectrum was also analyzed by using AXIL code[2]. The fit is obtained with (χ2 = 1.2). In Table 2, we compare the Kα peak areas of different elements evaluated with our program and AXIL code[2]. One can notice that the values of peak areas are very similar. The observed deviations are due mainly to the statistical errors.


The described computer program is widely used in the laboratory to perform PIXE spectra analysis. The inclusion of escape peaks and the low energy tail leads toa more accurate evaluation of peak areas. The peak areas determined by this program are similar to those obtained with AXIL code. The program will be improved by the treatment of pile-up peaks stemming from high counting rates.

1:  Kaufmann, H.C., K.R. Akselsson and W.J. Courtney, 1977. REX: A computer program for PIXE analysis. Nucl. Instr. Meth., 142: 251-257.

2:  Van Espen, P., H. Nullens and F. Adams, 1977. A method for the accurate description of the full-energy peaks in nonlinear least-squares analysis of X-ray spectra. Nucl. Instr. Meth., 145: 579-582.

3:  Johansson, G.I., 1982. Modification of the HEX program for fast automatic resolution of PIXE spectra. X-Ray Spectrom, 11: 194-2000.

4:  Clayton, E., 1983. BATTY83: A computer program for thick target PIXE analysis. Nucl. Instr. Meth. Phys. Res., 218: 221-224.

5:  Maxwell, J.A., R.G. Leigh, J.L. Campbell and H. Paul, 1984. Least-squares fitting of PIXE spectra with a digital filter treatment of the continuum. Nucl. Instr. Meth., 3: 301-304.

6:  Maenhaut, W. and J. Vandenhaute, 1986. Accurate analytic fitting of PIXE spectra. Bull. Soc. Chim. Belg., 95: 407-418.

7:  Bombelka, E., W. Koening, F.W. Richter and V.Watjen, 1987. Linear least-squares analysis of PIXE spectra. Nucl. Instr. Meth., 22: 21-28.

8:  Marageter, E., W. Wegscheider and K. Muller, 1984. A novel method for nonlinear least-squares analysis of energy-dispersive X-ray spectra. Nucl. Instr. Meth., 1: 137-145.

9:  Marquardt, D.W., 1963. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Indust. Applied Math., 11: 431-441.

10:  Kajfosz, J. and W.M. Kwiatek, 1987. Nonpolynomial approximation of background in X-ray spectra. Nucl. Instr. Meth., 22: 78-81.

11:  Benamar, M.A., I. Toumert, S. Tobbeche, A. Tchantchane and A. Chalabi, 1999. Assessment of the state of pollution by heavy metals in the surficial sediments of Algiers bay. Applied Rad. Isot., 50: 975-980.
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12:  McMaster, W.H., N.K. Del Grande, J.H. Mallet and J.H. Hubbell, 1969-1970. Compilation of X-ray cross sections. University of California, Lawrence Livermore Laboratory Report, UCRL-50174.

13:  Clayton, E., P. Duerden and D. Cohen, 1987. A discussion of PIXAN and PIXANPC: The AAEC PIXE analysis computer packages. Ibid, pp: 64-67.

14:  Duffy, C.P., S.Z. Rogers and T.M. Benjamin, 1987. The Los Alamos PIXE data reduction software. Ibid, pp: 91-95.

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