X-ray spectroscopy techniques are widely employed in elemental analysis. Energy Dis- persive X-ray Fluorescence technique (EDXRF) is one of the most commonly used for elemental analysis, due to its several benefits, such as being non-destructive, relatively easy, fast, and low-cost when comparing with other analytical techniques. Hence, it has been employed in a broad range of areas such as: biology, geology, forensic sciences, cul- tural heritage, etc. The setups employed for EDXRF varies from bench-top, to hand-held, diversifying the employment of the technique. Furthermore, setups can be employed in a triaxial geometry using a secondary target, allowing a considerate improvement on the detection limits, which is very convenient for the analysis of trace elements. In Fig. 1.1 it is presented a spectrometer in a triaxial geometry being employed in cultural heritage studies.
Nowadays, the analysis of the EDXRF spectrum, and consequent elemental quantifi- cation, is achieved using codes or software dedicated for such purposes, which employ one of three different methods, namely, Comparison with Standards method, Influence Coefficient method, or Fundamental Parameters method. In Comparison with Standards, quantification is achieved resorting to a linear calibration using standards of similar com- position to the unknown sample. This method considers that the matrix effects are the same among the different standards and the unknown sample, which is an approximation that limits to some extent the accuracy of the quantification [2]. Furthermore, the accu- racy of the unknown sample quantification is dependent on the accuracy of the standards analysis. Other drawbacks that must be considered are the unavailability of standards for certain type of materials, and the monetary costs associated with obtaining the standards.
Figure 1.1: Setup CFAUL-eclipse II, in situ, at Museu Nacional de Arte Antiga, Portugal. Figure adopted from Pessanha work [1].
In both Fundamental Parameters method and the Influence Coefficient method, quan- tification is achieved by employing mathematical methods to solve the equations relating peak intensity I and concentration c. These equations account for the absorption and enhancement of all elements in the sample, the matrix effects, and include physics pa- rameters related to the several physics processes occurring in the technique, the Funda- mental Parameters (FP). While in the Influence Coefficient method approximations are performed to lower the complexity of the equations, in the FP method no considerations are employed, making it more accurate and, nowadays, the most employed method. The accuracy of the quantification is affected by the contribution included in the equations to be solved, and the accuracy of the Fundamental Parameters present in these equations. For more accurate results, other contribution are sometimes included in these equations, such as: tertiary- and higher-order fluorescence, enhancement by electrons, non-parallel beam geometries, sample inhomogeneity, among others. The inclusion of such contri- bution improves the accuracy of the FP method. Codes and software usually resort to built-in tabulations of Fundamental Parameter values, which are in almost all cases, if not all cases, inaccessible to the user. As such, the inaccuracy of Fundamental Parameters can not be addressed by simply changing the parameters values.
The FP are crucial not only in methods such as the FP method and Influence Coeffi- cient method, but also in many other scientific areas. As such, the calculation of such parameters, using atomic structure calculations, has been the subject of many scientific
works, using Dirac-Hartree-Slater (DS) method, or the Dirac-Hartree-Fock (DF) method. While these approaches solve the multi-electronic atomic systems accounting for the electron-electron interaction, the interaction is not fully taken into account, for which several methods have emerged that better describe that interaction, allowing for more accurate results, such as the multiconfiguration Dirac-Fock method.
Although EDXRF analysis is performed in most cases by evaluation of the charac- teristic peaks in the X-ray spectrum, further analytical information can be obtained by exploiting the X-ray scattering features. A calibration curve of Compton-to-Rayleigh intensity ratios RC/Rin function of average atomic number Zavgcan be obtained by evalu-
ation the scattering features of several standards with the same material type, and under the same geometrical conditions [3, 4]. The Zavg of an unknown similar sample can be
obtained by direct correlating its measured RC/Rratio with Zavgthrough the calibration
curve. This method is quite sensitive for low-Z materials due to elastic and inelastic
scattering different dependence on atomic number. It can be used in the identification and characterization of low-Z materials, as shown in Pessanhaet al. [4] work. By com-
paring the measured RC/R for a sample with the calibration curve, this method allows for a calculation of light elements which are "invisible"in XRF, or even calculation of heavier impurities in light materials which may not be detected by XRF [5]. Using this method, materials can easily be distinguished from each other, by measuring the RC/R ratio. Such is useful for choosing samples to be used in experimental measurements. In light materials, even differences of Zavg as low as 0.1 can be distinguished due to the
method particular sensitivity in low-Zavg materials. Furthermore, the method can be
used to correct standards.
Nowadays, there are quite a number of codes and software for quantitative and quali- tative EDXRF analysis using the FP method, such as NRLXRF [6, 7] (from Naval Research Laboratory), NBSGSC [7, 8] (from National Bureau of Standards, actual National Institute for Standards and Technology), XRFAES [9] (from Chalmers University of Technology), WinAxil (from Canberra, and originally included in the software package QXAS, spon- sored by the International Atomic Energy Agency) [10], PyMCA [11, 12] (developed by the European Synchrotron Radiation Facility (ESRF) software group), among others. These codes and software mostly differ from one-another in the completeness of the physics models they employ. As such, some of them can be more suited for specific conditions such as the type of sample matrices to analyse, and the experimental setup conditions. Monte Carlo (MC) methods have also been employed in experiments involving X-ray fluorescence, providing assistance for the construction of X-ray spectrometers and anal- ysis of the data acquired. Some of the simulations where performed from dedicated in-house codes, such as 3D µ X-ray [13], X-ray optics [14], and X-ray tomography [15, 16], while others were performed using general-purpose codes, such as MCNP [mcnpRep, 17], PENELOPE [18, 19], and GEometry ANd Tracking 4 (Geant4) [20–22]. Monte Carlo methods present several advantages for the simulation of X-ray fluorescence experiments. They can account for contributions that are usually not included in the FP equations,
such as electron contribution to the fluorescence emission, tertiary- and higher-order of enhancement, and non-parallel beam geometry. Furthermore, the method present advan- tages in describing inhomogeneous or irregular samples, and further variations of the experimental geometry. The disadvantages of this method are the increased computa- tion time and required computational capacity. Regarding the EDXRF technique, Monte Carlo simulations have been applied for a quantitative analysis from the comparison of the simulated X-ray spectrum with the experimental spectrum, such as XMI-MSIM [23] and XRMC [24, 25]. MC simulations have also been applied in studies for in vivo measurements [26, 27] and in cultural heritage studies [28, 29].
MC simulations have also been applied to triaxial geometry spectrometers, such as in Lewiset al. [30, 31], Hugtenburg et al. [32] and Al-Ghorabie Fayez [33] works.