Electron contribution to photon transport in ... - Wiley Online Library

3 downloads 11387 Views 2MB Size Report
May 22, 2013 - Electron contribution to photon transport in coupled photon-electron problems: inner-shell impact ionization correction to XRF. Jorge E.
Research article Accepted: 21 December 2012

Published online in Wiley Online Library: 22 May 2013

(wileyonlinelibrary.com) DOI 10.1002/xrs.2473

Electron contribution to photon transport in coupled photon-electron problems: inner-shell impact ionization correction to XRF Jorge E. Fernandez,a* Viviana Scot,a Luca Verardib and Francesc Salvatc The Monte Carlo code PENELOPE (coupled electron-photon Monte Carlo) has been used to compute the effect of the secondary electrons on the X-ray fluorescence characteristic lines. The mechanism that produces this contribution is the inner-shell impact ionization. The ad hoc code KERNEL (which calls the PENELOPE library) has been used to simulate a forced first collision at the origin of coordinates. The electron correction (produced by the secondary electrons and their multiple scattering) has been studied in terms of angle, space and energy. The energy dependence has been quantified in the interval 1–150 keV, for all the emission lines (K, L and M) of the elements with atomic numbers Z = 11–92. For each characteristic line, the energy dependence is described by simple parametric expressions corresponding to the five energy regions delimited by the K, L1, L2 and L3 absorption edges. It has been introduced a new photon kernel comprising the correction due to inner-shell impact ionization. The new kernel is suitable to be adopted in photon transport codes (either deterministic or Monte Carlo) with a minimal effort. Finally, the new kernel has been studied for different elements and lines to trace a general behavior. Copyright © 2013 John Wiley & Sons, Ltd.

Introduction

X-Ray Spectrom. 2013, 42, 189–196

* Correspondence to: Jorge E. Fernandez, Laboratory of MontecuccolinoDepartment of Industrial Engineering (DIN), Alma Mater Studiorum University of Bologna, via dei Colli, 16, I-40136, Bologna, Italy. E-mail: jorge. [email protected] a Laboratory of Montecuccolino (DIN), Alma Mater Studiorum University of Bologna, via dei Colli, 16, 40136 Bologna, Italy b Department of Electrical, Electronic and Information Engineering “Guglielmo Marconi” (DEI), Alma Mater Studiorum University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy c Facultat de Física (ECM), Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain

Copyright © 2013 John Wiley & Sons, Ltd.

189

The most accurate description of the radiation field in X-ray spectrometry requires the modeling of coupled photon-electron transport, because Compton scattering and photoelectric effect give both photons and electrons as secondary particles. The solution for the coupled problem is time consuming because the electrons interact continuously with the medium and therefore, the number of electron collisions is always very high. For this reason, transport codes usually neglect the electron contributions shown in Fig. 1, and consequently, only photon transport is considered. Nevertheless, secondary electrons contribute to the photon field through electron-photon conversion mechanisms like bremsstrahlung (which produces a continuous photon spectrum) and inner-shell impact ionization (ISII) (which modifies the intensity of the characteristic lines) (Fig. 2). Other processes, not mentioned here, that contribute to the multiple scattering of electrons (Auger electrons, Coster-Kronig transitions, etc.) need to be also considered. In what follows, we will focus on ISII, which is the only effect that modifies the intensity of the characteristic lines, and therefore, it is of interest in X-ray fluorescence (XRF) analysis. Several authors have tried to describe the XRF due to secondary electrons by focusing separately on the contributions from Kphotoelectrons,[1] Auger electrons,[2] both effects,[3] and Compton electrons.[4] The whole problem is very complex because all of these mechanisms (plus other not mentioned) have to be considered together, and the fact that electrons interact continuously and locally (compared with the photons) makes it necessary to describe carefully the multiple scattering of the electrons. A practical approach to study the mentioned mechanisms in presence of multiple scattering is to recur to a coupled photon-electron Monte Carlo (MC) code. Some authors[5,6] have developed ad hoc MC codes on the basis of simplified models of electron

transport to evaluate the effect of secondary electrons in thin layers. To understand the extent of the electron contributions on multilayers,[5,7,8] some experiments have been performed using the polychromatic excitation of X-ray tubes, showing that the effect exists, but it is lower than expected. Kawahara et al.[8] described also an experiment with tunable monochromatic excitation at BESSY II. In this case, it appeared a very strong discontinuity of the Cr–L line intensity in correspondence with the Cr–K edge, which is due to the cascade effect. The apparent contradiction of these experiments needs to be explained by recourse to an MC code using a detailed description of the multiple scattering of electrons. In this work, it has been used the MC code PENELOPE[9] (coupled electron-photon MC) to compute the effect of the secondary electrons into the photon transport. In particular, PENELOPE has been used to compute a corrective term to the photon kernel, which fully describes the effect of ISII on the characteristic line emission. It is given a formal expression of a new photon kernel comprising the correction for ISII. The new kernel is suitable to be adopted in photon transport codes (either deterministic or MC) with a minimal increase in complexity.

J. E. Fernandez et al. Scattered photons

PRIMARY PHOTON

COHERENT SCATTERING

RAYLEIGH PHOTON

INCOHERENT SCATTERING

COMPTON PHOTON

PHOTOELECTRIC EFFECT

COMPTON ELECTRON

CHAR. X-RAYS

ELECTRON

Scattered electron

Figure 1. Main photon interactions in the X-rays energy range (1–100 keV). Photon transport codes usually neglect the electron contributions, and consequently, pure photon transport is considered.

SECONDARY ELECTRON

Scattered electrons

INNER SHELL IMPACT IONIZATION

BREMMSTRAHLUNG

PHOTON CONTINUOUS

CHAR. X-RAYS

ELECTRON

Scattered photon

Figure 2. Secondary electrons feedback new photons into the photon interactions cycle through conversion mechanisms like bremsstrahlung and inner-shell impact ionization.

To formulate the new kernel, the coupled photon-electron transport needs to be studied in terms of angle, space and energy. It is demonstrated that the angular distribution of the characteristic photons due to ISII can be safely assumed as isotropic and that the source of photons from electron interactions is well represented by a point source centered at the place of the primary collision. The energy dependence of the correction is quantified in the range 1–150 keV, for the main emission lines (K, L and M) of the elements with atomic numbers Z = 11–92. For each characteristic line, the energy dependence is represented using simple parametric expressions corresponding to the five energy intervals delimited by the K, L1, L2 and L3 edges, and requiring a total number of 20 parameters by line.

The code PENELOPE

190

PENELOPE[9] is a general-purpose MC subroutine package for the simulation of coupled electron-photon transport in arbitrary geometries for a wide energy range, from a few hundred of eV to about 1 GeV. It was developed at the Facultat de Fisica (ECM) of the Universitat de Barcelona. PENELOPE performs ‘analogue’ simulation of electron-photon showers in infinite (unbounded) or finite media of various compositions. Photon histories are generated by using a highly detailed simulation method. Secondary particles emitted with initial energy larger than an absorption energy defined by the user are stored and simulated after completion of each primary track. Secondary particles are produced in direct interactions

wileyonlinelibrary.com/journal/xrs

(hard inelastic collisions, hard bremsstrahlung emission, positron annihilation, photoelectric absorption, Compton scattering and pair production) and as fluorescent radiation (characteristic X-rays and Auger electrons). PENELOPE simulates fluorescent radiation, which results from vacancies produced in K, L and M shells by photoelectric absorption and Compton scattering of photons and by electron/positron impact. The relaxation of these vacancies is followed until the K, L and M shells are filled up, i.e. until the vacancies have migrated to N and outer shells. In previous versions of PENELOPE, ISII was simulated using cross-sections obtained from the atomic generalized oscillator strength model,[10] i.e. from an approximate formulation of the plane-wave (first) Born approximation (PWBA). However, the cross-section for ionization of a bound shell decreases rapidly with the shell ionization energy Ui (because energy transfers less than Ui, which would promote the target electron to occupied states, are forbidden). As a consequence, collisions occur preferentially with electrons in the conduction band and in outer bound shells. ISII by electron/positron impact is a relatively unlikely process, and the generalized oscillator strength model, used for electron inelastic interactions, is too crude to provide an accurate description of it. A more elaborate theoretical description of total ionization cross-sections is obtained from the relativistic distorted-wave Born approximation (DWBA), which consistently accounts for the effects of both distortion and exchange.[11] In the present version of PENELOPE (2010), an extensive numerical database of ionization cross-sections has been calculated for K, L and M shells of all the elements from hydrogen (Z = 1) to einsteinium (Z = 99), for projectiles with kinetic energies from threshold up to 1 GeV. The theoretical model adopted in these calculations combines the DWBA and the PWBA, as described by Bote and Salvat (2008).[11] The DWBA is used to calculate the ionization cross-section for projectiles with energies from ~Ui up to 16Ui. For higher energies, the cross-section is obtained by multiplying the PWBA cross-section by an empirical energy-dependent scaling factor, which tends to unity at high energies where the PWBA is expected to be reliable. This calculation scheme accounts for differences between the cross-sections for ionization by electrons and positrons.

The ad hoc code KERNEL PENELOPE is a subroutine package and cannot operate by itself. The user needs to build a main program for his/her particular problem, to read the parameters introduced through the input file, control the evolution of the generated tracks and keep score of relevant quantities. For our study, the ad hoc code KERNEL[12] was developed to simulate a forced first collision at the origin of coordinates. A point source of monochromatic photons is considered. The physics of the interactions was described using the PENELOPE subroutine library. All the secondary electrons were followed along their multiple scattering until their energy becomes lower of a predefined threshold value. All photons produced by the electrons at every stage were accumulated. Polarization was not considered at this time. The simulated process[12,13] is sketched by the flow diagram in Fig. 3 and may be summarized by the following points. 1. A monochromatic photon having energy E0, propagates along the z-axis and hits an atom in the origin of the reference frame. Photons are allowed to interact through three different

Copyright © 2013 John Wiley & Sons, Ltd.

X-Ray Spectrom. 2013, 42, 189–196

Inner-Shell Impact Ionization correction to XRF Primary photon N h = N h +1 E=E0, r=(X, Y, Z)=0, d=(U, V, W)=(0,0,1)

Firs t collision YES

N h < N max

NO

Particles in secondary stack?

YES

NO

Score of the distributions

END

YES

NO

Bremsstrahlung or relaxation?

NO

Simulation of slowing -down until E 60 are not meaningful.

Copyright © 2013 John Wiley & Sons, Ltd.

wileyonlinelibrary.com/journal/xrs

195

As an example, the top panel of Fig. 13 shows Q and Qcorrected for Na Ka1, as a function of the excitation energy. In both curves (Q and Qcorrected), the dominating term is the mass attenuation coefficient. The lower panel of Fig. 13 shows the absolute value of the correction ΔQ. It is apparent that the correction is significant only within a restricted energy interval. For Na Ka1, for example, the correction presents a maximum for an excitation energy of 5 keV, which represents a relative percent of ΔQ/Q  4.63%. Figure 14 shows the trend of ΔQ/Q in %, (corresponding to the maximum of ΔQ), for K-lines, for all elements from Z = 11–92. It is observed that the correction may be significant not only for low-Z elements (as considered by other authors[3,5]) but also for intermediate elements. In fact, ΔQ/Q is more than 1% up to Z = 26–27. Figure 15 shows for K-lines the energies of the maximum of ΔQ. This value increases almost linearly with the atomic number until 150 keV (the maximum energy considered in the simulations). Figure 16 shows Q, Qcorrected and ΔQ for Fe La1, as a function of the excitation energy. For the considered line, ΔQ has the maximum for the excitation energy of 5 keV. Here, the correction is very high (ΔQ/Q  31%) and narrow.

J. E. Fernandez et al. Most of the available measurements of this effect[5,7,8] use polychromatic excitation. Being the effect averaged on a wide energy interval, it suffers the mentioned reduction because of the contribution from less dominant energies. The only experiment using monochromatic excitation,[8] in our opinion, was not performed at the optimal energy conditions and therefore is not well representative of this effect. It should be necessary to design more adequate experiments for investigating better the electron contributions to XRF emission.

Conclusions

Figure 16. Top panel: Q and Qcorrected for Fe La1, as a function of the excitation energy. Lower panel: absolute value of the correction ΔQ = Qcorrected  Q. It is apparent that the correction is significant only within a restricted energy interval. For Fe La1, it has maximum at 5 keV (ΔQ/Q  31%). Note that the y-axis in the top panel is in logarithmic scale. The green line represents the K-edge, violet lines the L-edges.

Figure 17. Top panel: Q and Qcorrected for U Ma1, as a function of the excitation energy. Lower panel: absolute value of the correction ΔQ = Qcorrected  Q. For U Ma1, ΔQ has maximum at 17.17 keV (ΔQ/Q  3.6%). Note that the y-axis in the top panel is in logarithmic scale. The green line represents the K-edge, violet lines the L-edges.

196

Figure 17 shows Q, Qcorrected and ΔQ for U Ma1, as a function of the excitation energy. From the lower panel, it is apparent that ΔQ presents a peak near the L-edges, and then decrease smoothly with the energy. For narrowly shaped ΔQ, the contribution of secondary electrons appears to be relevant prevailingly for a monochromatic excitation lying within the energy interval where ΔQ is concentrated. For polychromatic excitation, it is computed a mean value of the correction over a wider energy interval. In this case, the lower contribution coming from outside the region, where ΔQ is relevant, reduces the overall effect to a less significant level. This is generally true, except when the correction spreads on a wide energy interval. An example of a wide correction can be seen in Fig. 17.

wileyonlinelibrary.com/journal/xrs

The MC code PENELOPE has been used to evaluate the contribution from secondary electrons to the characteristic lines in coupled photon-electron problems. The ISII correction has been studied in terms of space, angle and energy: The spatial distribution has been demonstrated to be very close to the interaction point and then can be safely considered as point wise; the angular distribution has been demonstrated to be isotropic; the energy dependence of the correction has been described using a parametric expression, which needs 20 parameters by line (distributed along five energy regions, four parameters each) for the main lines of the elements Z = 11–92 in the range of 1–150 keV. The analytical expression of the photon kernel for the photoelectric effect followed by atomic relaxation has been improved to include the correction due to ISII. In this way, the correction can be used either in deterministic or MC codes for photon transport, with minimum effort without the need to recur to the solution of the coupled photon-electron problem. The absolute correction of ISII on XRF lines has been studied. In some cases, it is concentrated in a limited energy interval, making the contribution of the electrons more significant for monochromatic excitation lying inside this interval than for a polychromatic one. The experiments found in literature are not definitive because most have been carried out using polychromatic excitation, which dilutes the effect. New experiments with monochromatic excitation need to be designed taking care that the excitation belongs to the energy interval where the effect is more relevant.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13]

H. Ebel. Z. Naturforsch. 1971, 26 A, 927–928. H. E. Bishop, J. C. Rivière. J. Appl. Phys. 1969, 40, 1740. G. V. Pavlinsky, A. Y. Dukhanin. X Ray Spectrom. 1994, 23, 221–228. K. N. Stoev. J. Phys. D: Appl. Phys. 1992, 25, 131–138. Y. Kataoka, N. Kawahara, T. Arai, M. Uda. Adv. x-ray anal. 1999, 41, 76–83. M. Mantler. Adv. x-ray anal. 1999, 41, 54–61. F. A. Weber, L. B. Da Silva, T. W. Barbee Jr., D. Ciarlo, M. Mantler. Adv. x-ray anal. 1997, 40, 301–309. N. Kawahara, T. Shoji, T. Yamada, Y. Kataoka, B. Beckhoff, G. Ulm, M. Mantler. Adv. x-ray anal. 2002, 45, 511–516. F. Salvat, J. M. Fernandez-Varea, J. Sempau, PENELOPE, a Code System for Monte Carlo Simulation of Electron and Photon Transport, Nuclear Energy Agency (NEA), Paris, 2008. R. Mayol, F. Salvat. J. Phys. B: At. Mol. Opt. Phys. 1990, 23, 2117–2130. D. Bote, F. Salvat. Phys. Rev. A 2008, 77, 042701. L. Verardi, Evaluation of electron kernels through Monte Carlo simulation in coupled electron-photon transport, Energy Engineering Master Thesis, Alma Mater Studiorum University of Bologna, Bologna, 2011. J. E. Fernández, V. Scot. Appl. Rad. Isot. 2012, 70, 550.

Copyright © 2013 John Wiley & Sons, Ltd.

X-Ray Spectrom. 2013, 42, 189–196