INTRODUCTION
A reliable computer program that gives an accurate determination of the peak
area of characteristic Xrays is of major importance for PIXE analysis. Knowing
the expected Xray 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 K_{α} and K_{β} lines
of neighboring elements; 
• 
Distortion of the lower side of the peak from the pure gaussian
shape due to incomplete charge collection; 
• 
Pileup peaks due to high counting rates; 
• 
Various physical sources contribute to the background; 
• 
Xrays 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.
DESCRIPTION OF THE MATHEMATICAL MODEL
Several approaches of PIXE spectra fitting and peak modeling are described in the literature^{[19]}. 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 NL_{I} are the number of elements in the sample and the number of lines of element l, respectively. P_{l} is a free parameter corresponding to the area of the element l. R_{k, l} is the relative intensity of K_{th} line of element I. Its’s correct Abs (E_{k, l}) is a function depending on xray energy which corrects xrays attenuation in the different absorbers. The absorption function at energy E_{I }corresponding to channel I is expressed as:
Ab (E_{I}) describes the xray 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]}. Ab_{s}
(E_{I}) represents the sample selfabsorption which is approximated
by:
where, b_{1}, b_{2 }are parameters.
G_{k, l} (I) is an ideal gaussian representing the K_{th} line of the element l. Tail_{k, l} (I) and Esc_{k, 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, I_{0} is a fixed reference channel and I_{e} is the
channel corresponding to the incident energy E_{e} transferred by incident
proton to free electron. E_{e} is approximated by:
Where, m_{e}, m_{p} are the mass of electron and proton, respectively and E_{p} 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 G_{k, l} (I) is expressed by:
Where,
depicts the Full Width at Half Maximum (FWHM). C_{1 }and C_{2 }are
the parameters of the linear energy calibration. C_{1 }, C_{2}
, C_{3} and C_{4 }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 Tail_{k, 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 SiK_{α }secondary Xrays 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 SiK_{α} energy.
The FWHM of the escape peak is chosen to be the same as for a characteristic Xray peak at the energy
NONLINEAR LEASTSQUARES FIT
The experimental data are fitted by using the leastsquares method. The parameters in the Eq. 1 are determined by minimizing the χ^{2} function. The χ^{2} function is defined as follows:
Where, Y (I) is the model function, Y_{I} is the yield at channel I and NC is the number of channels in the spectrum. f_{c }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 χ^{2} 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 K_{α} peak areas obtained with
AXIL code and our program for a prawn sample PIXE spectrum 

THE COMPUTER PROGRAM
The program is designed for a fast interactive operator control during offline 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 
APPLICATION
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. 2 (χ^{2} = 16) and 3 (χ^{2} = 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 (χ^{2} = 1.2). In Table 2, we compare the K_{α} 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.
CONCLUSIONS
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 pileup peaks stemming from high counting rates.