Electron-induced x-ray emission from solids. Simulation and measurements

Abstract

Theoretical methods to compute accurate x-ray spectra emitted from targets bombarded with kV electrons are required for quantification in Electron Probe Microanalysis (EPMA), especially for the analysis of non-homogeneous samples.<br/><br/>Monte Carlo simulation has proved to be the most suitable theoretical tool for the computation of x-ray spectra; it can incorporate realistic interaction cross sections and can be applied to complex geometries. Moreover, it allows us to keep track of the evolution of all secondary particles (and their descendants) generated by primary electrons.<br/><br/>A Monte Carlo program for the calculation of ionization depth distributions and x-ray spectra produced by kV electron irradiation has been developed. Inner-shell ionization by electron impact is described by means of total cross sections evaluated from an optical-data model. A double differential cross section is proposed for bremsstrahlung emission, which combines a modified Bethe-Heitler DCS with the Kirkpatrick WiedmannStatham angular distribution, and reproduces the radiative stopping powers derived from the partial wave calculations of Kissel, Quarles and Pratt [At. Data Nud. Data Tables 28, 381 (1983)]. These ionization and radiative cross sections have been introduced into the general-purpose subroutine package PENELOPE, which performs simulation of coupled electron and photon transport for arbitrary materials. The underlying electron scattering model combines elastic scattering cross sections calculated by the partial wave method with inelastic cross sections obtained from Liljequist's generalized oscillator strength model. The reliability of the electron trajectory generation algorithm has been verified through a comparison of simulation results with measured backscattering coefficients and spatial dose distributions. To improve the efficiency of the simulation, a variance reduction technique, interaction forcing, has been applied for both ionizing collisions and radiative events.<br/><br/>A systematic method for the measurement of ionization cross sections by electron impact, using the electron microprobe, has been proposed and applied. Measurements of ionization cross sections for Ni, er and Cu have been performed, from threshold up to 40 keV, combining the use of wavelength and energy dispersive spectrometers.<br/><br/>Our results provide the electron-energy dependence of the ionization cross section to an accuracy of about 3%. The transformation to absolute cross section values increases the global uncertainty to about 12%. These errors include the determination of the target thickness, detector efficiency, solid angle of collection, the number of incident electrons, the fluorescence yield and the line fraction. The comparison between experiment and various theoretical formulas confirms that the optical-data model yields a more reliable energy dependence of the ionization cross section in the energy range of interest in microanalysis. Further work to reduce errors in the determination of the target mass thickness is required to draw a definite conclusion about the accuracy of the theoretical model in absolute magnitude.<br/><br/>Simulated depth-distribution of ionizations and surface ionization, for different homogeneous targets and energies, have been shown to be in satisfactory agreement with experimental data taken from the literature. Comparison of simulated data, using various ionization cross sections, confirms again the validity of the optical-data model used.<br/><br/>In the case of non-homogeneous samples (e.g. thin layers on substrates or multilayered structures) or special geometries (e.g. oblique incidence) experimental measurements are very rare and there is a real need for experimental data to check the reliability of simulations and/or alternative approximate formulations. Thus, experimental measurements and Monte Carlo simulations of the surface ionization have been performed (for Ni KC(o) X-rays) on Cu films of various thickness (40.5, 67, 100 and 196 nm) deposited onto a variety of substrates and for accelerating voltages between 10 keY and 30 keY. Measurements have been performed using the wavelength-dispersive spectrometer. The main difficulties of the measurements have been i) to ensure that tracer films have the same thickness, ii) the large statistical uncertainties due to the smallness of the peak-to-background ratio of the tracer peak, iii) the contribution from the substrate and iv) the surface contamination. In spite of the uncertainties arising from sample preparation and from the smallness of the peak-to-background ratio, the results from the experiments and the Monte Carlo simulations are found to agree to within 5%. These measurements allow us to validate the developed code and to obtain information of interest for the analytical methods of microanalysis. In particular, we have derived a simple analytical formula, based in a new scaling rule, which gives the surface ionization in terms of the bulk values of the substrate and the overlayer. Finally, simulated ionization distributions for different layered targets have been presented, which allow us to study the influence of the substrate on the ionization of the film.<br/><br/>The reliability of the simulation code has been globally assessed by comparison of measured x-ray spectra with simulation results. X-ray spectra have been measured using the energy-dispersive spectrometer and converted to absolute units. Measurements in absolute units serve as the most stringent test of the physical parameters used in the simulation algorithm, although they may contain systematic uncertainties. Measurements have been performed for different targets and irradiation conditions, including multilayered targets and oblique incidence. The result of the various experimental sources of error leads to an overall uncertainty of 5-7%. The agreement between simulation and experiment has been shown to be satisfactory in the "meaningful" region of the spectra (say, between 3-15 keY), where the detector efficiency is essentially constant. Comparison of simulated and measured x-ray spectra obtained with the wavelength-dispersive spectrometer allows us to derive the absolute efficiency of the latter as a function of the x-ray incident energy. This information is essential for standardless x-ray microanalysis using wavelength-dispersive spectrometers.<br/><br/>In short, we have developed a realistic Monte Carlo code adequate for the simulation of x-ray generation by electron irradiation of samples with complex geometries. We have already demonstrated its usefulness for EPMA of layered specimens. Although further work is required to include L- and M-shell ionization, the code has an evident potentiality for quantitative EPMA of samples with complex geometries.<br/><br/>A Monte Carlo program for the calculation of ionization depth distributions and x-ray spectra produced by kV electron irradiation has been developed. The code is based on the PENELOPE subroutine package, which has been suitably modified to extend its range of application to lower energies. It incorporates a new double differential cross section for bremsstrahlung emission, which combines a modified Bethe-Heitler DCS with the Kirkpatrick-Wiedmann-Statham angular distribution. Ionization of inner-shells by electron impact is described by means of an optical-data model proposed by Mayol and Salvat. Interaction forcing is systematically applied to both bremsstrahlung and impact ionization to improve the efficiency of the simulation. The simulation program is applicable to samples with arbitrary geometries (multilayers, particulate samples, etc.).<br/><br/>The ionization cross section for Ni, Cr and Cu has been experimentally determined using the electron microprobe. Measurements confirm that the optical-data model yields a more reliable energy dependence of the ionization cross section.<br/><br/>Simulated depth-distribution of ionizations and surface ionization, for different homogeneous targets and energies, has been shown to be in satisfactory agreement with experimental data taken from the literature.<br/><br/>Systematic measurements of the surface ionization, for a Ni tracer, in Cu films of different thicknesses deposited on a wide variety of substrates have been performed. The results from the experiments and simulations have been found to agree to within 5%. A simple analytical formula is proposed, which gives the surface ionization in terms of the bulk values of the substrate and the overlayer. Simulated ionization distributions for layered targets have been presented.<br/><br/>Absolute x-ray spectra have been measured using the energy-dispersive spectrometer, for different targets and irradiation conditions. The agreement between simulation and experiment has been found to be satisfactory in the photon energy region of aprox. 3-15 keY, where the detector efficiency is constant. The absolute efficiency of a wavelength dispersive spectrometer has been obtained.