PIXE simulation in Geant4

Research Article Received: 13 January 2011

Accepted: 28 January 2011

Published online in Wiley Online Library:

(wileyonlinelibrary.com) DOI 10.1002/xrs.1301

PIXE simulation in Geant4 A. Mantero,a∗ H. Ben Abdelouahed,b C. Champion,c Z. El Bitar,d Z. Francis,e ` f S. Incerti,g V. Ivanchenkoh,i,j and M. Mairei,k P. Gueye, Geant4 is a general purpose and open source C++ Monte Carlo simulation toolkit, widely used in the scientific community. It is able to simulate physical interactions of particles through matter. According to the user’s needs, models for the simulation of electromagnetic (EM) interactions are provided in two Geant4 subpackages, the ‘standard’ EM subpackage, well suited for a wide range of applications and the ‘low-energy’ EM subpackage, able to reach the electronVolt regime. Particle-induced X-ray emission (PIXE) is a well known and a very useful technique for quantitative elemental analysis in environmental, archaeological, biological, medical and space applications. An atomic de-excitation module is part of the Geant4 ‘low-energy’ EM subpackage since 1999 and has been validated in recent years. PIXE simulation has been included in this subpackage in 2001 and new ionisation cross-sectional models following the ECPSSR theory have been added for the PIXE simulation in 2008. In 2010, these models have been further extended to higher energies. In this work, we present new results on the verification of these models and an overview of the new interface to PIXE modelling prepared for the recent public release of the Geant4 toolkit (December 2010) allowing a unified usage of the Geant4 de-excitation module by both ‘standard’ and ‘low-energy’ subpackages. Copyright c 2011 John Wiley & Sons, Ltd.


Introduction X-ray fluorescence (XRF) spectroscopy, and in particular particleinduced X-ray emission (PIXE), are two of the most used techniques for elemental composition analysis of simple and complex materials. Geant4 is an open source toolkit initially developed for the Monte Carlo simulation of particle–matter interactions in high-energy physics.[1,2] It can reproduce full experimental setups, transport particles in complex geometries, simulate physical interactions and model detector response. XRF and PIXE simulation capabilities have been included in the Geant4 toolkit since 1999. However, the usage of these Geant4 capabilities required expert knowledge of the toolkit to configure electromagnetic (EM) physics processes and was available only with the ‘low-energy’ subpackage of the Geant4 toolkit. Recently, significant efforts were introduced in order to develop a unified model approach of Geant4 EM physics software.[3,4] In this work, we report the most recent developments in the ‘standard’ and ‘low-energy’ EM physics subpackages allowing common usage of the de-excitation module including PIXE. This software is available with the 9.4 version of the Geant4 toolkit (December, 2010). We also present PIXE cross-sectional models for ionisation of K and Li shells as well as their implementation in the toolkit adopting this unified software approach for the first time in Geant4.

Status of PIXE Simulation in Geant4 EM Subpackages Geant4 EM physics The Geant4 toolkit includes several subpackages for different aspects of EM physics modelling. The standard EM subpackage[1 – 3] provides simulation of EM interactions of particles in the energy interval from 1 keV to 10 PeV. It is used for Monte Carlo production for the Large Hadron Collider experiments at CERN and for other high-energy physics experiments. This subpackage is also used in many other application domains like medical and space science.

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It includes models for ionisation, Bremsstrahlung, photoelectric effect, Compton scattering, gamma conversion and for many other physical processes. The Geant4 toolkit structure allows different approaches when for each physical process several models with different level of accuracy and CPU efficiency of simulation co-exist. The ‘lowenergy’ EM subpackage[2,5] includes an alternative set of models extending Geant4 applicability at lower energy (by default down to 100 eV). Separation of the two EM subpackages provided strong limitations for Geant4 users, which was resolved recently

∗

Correspondence to: A. Mantero, Istituto Nazionale di Fisica Nucleare, sez. di Genova, via Dodecaneso 33, 16146 Genova, Italy. E-mail: [email protected]

a Istituto Nazionale di Fisica Nucleare, sez. di Genova, via Dodecaneso 33, 16146 Genova, Italy b Centre National des Sciences et Technologies Nucl´eaires (CNSTN), Pˆole technologique, 2020 Sidi Thabet, Tunis, Tunisia c Laboratoire de Physique Mol´eculaire et des Collisions, Universit´e Paul VerlaineMetz, 1 Boulevard Arago, Technopˆole 2000, 57078 Metz, France d Institut Pluridisciplinaire Hubert Curien, CNRS/IN2P3, 67037 Strasbourg Cedex, France e Universit´e Saint Joseph, Science Faculty, Department of Physics, Beirut, Lebanon f Physics Department, Hampton University, 23668 Hampton, VA, USA g Universit´e Bordeaux 1, CNRS/IN2P3, Centre d’Etudes Nucl´eaires de Bordeaux Gradignan, CENBG, Chemin du Solarium, 33175 Gradignan, France h Ecoanalytica, Moscow State University, 119899 Moscow, Russia i Geant4 Associates International Ltd, Hebden Bridge, UK j LPMC, Institut de Physique de Metz, Technopole 2000, 57000 Metz, France k Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Annecy, France

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A. Mantero et al. (Geant4 version 9.3, December 2009) by migration to the common software interface.[4] The consolidation of all EM subpackages gives a boost for the development of new models and provides new opportunities for complex simulation of high-energy and low-energy effects. The most recent development of the deexcitation module is described in the Section on Geant4 Atomic De-excitation. Geant4 atomic de-excitation An atomic de-excitation module is available in Geant4 since 1999 and has been intensively validated.[6,7] The simulation of atomic de-excitation is a part of Geant4 particle transport and is initiated by Geant4 physical processes producing vacancies in the atomic electronic structures like the photoelectric effect, Compton scattering and ionisation. Physical processes must sample the element and the shell from which the electron is taken out, in accordance with the process cross-section. Once the shell where the vacancy occurs is known, this information is passed to the atomic de-excitation module which starts the simulation of the de-excitation cascade. In this process, the original vacancy is filled by an electron coming from an outer subshell, together with the emission of an X-ray (‘radiative’ transition) or of an electron from an outer shell (Auger effect) or from an outer subshell of the same shell (Coster–Kronig effect). The implementation of the Geant4 de-excitation module is data-driven: secondary particle energies and transition probabilities are based on the evaluated atomic data library (EADL).[8] Energies of relaxation products in EADL are calculated as the difference between the binding energies of the shells involved in the transition, derived from the theoretical calculation of Scofield.[9,10] Probabilities for radiative transition derive from Scofield Hartree–Slater calculations,[11,12] while nonradiative ones derive from Dirac–Hartree–Slater calculated from Chen et al.[13 – 17] with corrections from Hubbell.[18] PIXE models in Geant4 In order to simulate PIXE, Geant4 ionisation processes were required to calculate the ionised (sub) shells, for which values of ionisation cross-sections of atomic shells are needed. PIXE was implemented only inside the ionisation processes of the ‘lowenergy’ subpackage. First, attempts to determine ionised shells relied on Gryzinski cross-sectional semi-classical calculations[19,20] eventually substituted by another model described in Ref [21], developed to reproduce the compilation of experimental data from Paul and Sacher.[22] However, this model does not deliver the desired results, which affects the possibility to use it. To address the problem, in 2008 a new extension from the thesis work of Ben Abdelouahed[23,24] has been proposed in order to calculate proton and alpha K-shell ionisation cross-sections according to the ECPSSR theory[25,26] and Li -shells ionisation cross-sections according to an empirical model from Orlic et al.[27,28] The models proposed in Refs [23,24] were suited for the low-energy regime and their full availability for Geant4 user applications was still at a beta version as underlined in Ref [29].

Update of Geant4 Software The Geant4 software has been updated specifically for the modelling of atomic de-excitation and for the implementation of the upgraded ionisation cross-sectional models for PIXE

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simulation. It is available in the 9.4 (December 2010) release of the Geant4 toolkit. The main goal of this work was to deliver a userfriendly interface allowing the enabling of atomic de-excitation with a production threshold (cut in range[1] ) defined by the user for ionisation and Bremsstrahlung processes. Fluorescence, Auger electron production and PIXE may be enabled and disabled by the user using macrocommands for the entire simulated set-up geometry or for specific areas only (using G4Region objects[2]). The user has the possibility to select easily a model of PIXE ionisation cross-section as well. The set of PIXE cross-sectional ionisation models following this new interface in Geant4 version 9.4 is described in the Section on New Atomic De-excitation Software. For backward compatibility, all old models and interfaces are still available in Geant4 9.4 version.

New atomic de-excitation software The atomic de-excitation abstract process interface, named G4VAtomDeexcitation has been developed in order to be used both by the ‘low-energy’ and ‘standard’ Geant4 subpackages of EM processes, which was not possible in previous versions of the toolkit (e.g. see Refs [6,7]). Two new models of the ‘standard’ EM subpackage are released, which now include atomic de-excitation: the G4PEEffectFluoModel for the modelling of the photoelectric effect and the G4KleinNishinaModel for the modelling of Compton scattering. The concrete implementation of the interface has been included in the ‘low-energy’ EM subpackage with the name G4UAtomicDeexcitation. An important advantage brought by this design iteration is the possibility to add in a very straightforward way the atomic de-excitation module, leaving the possibility to users to select one of the available PIXE ionisation cross-sectional models and even substitute the whole de-excitation module by any alternative one following the same abstract interface. User can enable/disable the de-excitation module, Auger emission and PIXE for the whole simulated geometry or for specified detector regions (G4Region). The interface is provided in the form of User Interface macrocommands and also in the G4EmProcessOptions class.[4] By default, the new PIXE implementation is available with EM components of the reference Physics Lists ‘standard EM option 2’ and ‘standard EM option 3’.[4] It can be activated by the user for any Physics List.

New PIXE software In the new design of Geant4 PIXE processes, the sampling is performed in the base class G4VEnergyLossProcess for any kind of primary particle and model used for the simulation of ionisation. The de-excitation module is responsible for providing PIXE ionisation cross-sections and for sampling of de-excitation of an atom with a vacant shell. This sampling is performed in the same way as for the photoelectric effect and Compton scattering processes. In the current Geant4 version (9.4), the implementation of the interface to PIXE cross-sections is provided by the G4teoCrossSection and G4empCrossSection classes. The first class provides cross-section from the ECPSSR analytical and interpolated models while Paul et al.’s and Orlic et al.’s models are provided by the latter. These models were upgraded as described in the following section. Both empirical and theoretical models provide cross-sections for protons and alphas. Cross-sections for other heavy particles σh (E) are scaled from the proton cross-section

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PIXE simulation in Geant4 σp (E) according to the standard scaling relation: σh (E) = Q2 · σp (E · Mp /Mh )

(1)

where E is the kinetic energy of the incident particle, Mh its mass, Q2 is the square of the effective electric charge[30] in units of electron charge and Mp is the proton mass. For electron and positrons, shell ionisation cross-sections are already part of the ‘low-energy’ subpackage ionisation models, which sample shell vacancies themselves.

K- and Li -Shell Ionisation Cross-Sectional Models In the following, we present the two approaches here developed for calculating the K- and Li -shell ionisation cross-sections for proton and alpha particles. Both have been included into the Geant4 toolkit (December 2010). The first approach – so-called theoretical – is based on the ECPSSR theory,[25,26] whereas the second approach – so-called empirical – refers to the work of Paul et al.[22,31] for K shells and to that of Orlic et al.[27,28] for Li shells. ECPSSR-theory-based models As explained in Ref [24], the ECPSSR theory is nowadays the most acknowledged theory by the scientific community able to reproduce shell ionisation cross-section values. Consequently, ECPSSR has been considered to be the best choice for K- and Li -shell cross-sectional calculations in Geant4. It has been implemented in two categories of models, analytical models and interpolated models. The analytical model available for the calculation of K-shell ionisation cross-sections follows the ECPSSR calculations presented in Ref [26] for low-scaled impact velocities with the definition of the projectile velocity as proposed in Ref [32]. Rice et al.’s[33] calculations have been used to calculate the reduced universal cross-sectional function both for medium- and high-scaled impact velocities through data tables. ECPSSR can reproduce Li shell cross-sections better than other available theories,[34,35] even if it shows deviations below the MeV range.[36] The analytical model available for the calculation of Li -shell ionisation cross-sections follows calculations based on the work of Liu and Cipolla,[37] while Benka and Kropf’s[38] calculations have been used to calculate the reduced universal cross-sectional function. The alternative interpolated models use ECPSSR data tables generated with the ISICS2008 software in Ref [39] that we initially mentioned in Ref [24], in a similar way to what is suggested in Ref [29]. They are applicable to the energy range from 4 keV to 587 MeV, covering typical PIXE use cases. Out of these boundaries, the interpolation algorithm of Geant4 returns the cross-sectional value corresponding to the boundary energy value. Note that these interpolated models have been used exclusively for comparison of accuracy between models. Empirical models Given the success in the scientific community and the soundness of the works of Paul and Sacher[22] and Paul and Bolik,[31] we decided to provide them to Geant4 users as an alternative model for K-shell ionisation cross-sectional calculations, interpolating data tables given in Refs [22,31]. The Orlic et al.’s fitting formula

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described in Ref [27] from the compilation of experimental Li shell cross-sections for protons described in Ref [28] has also been included into Geant4. It should be noted here that during the preparation of this work the authors did not have the knowledge of the existence of an updated version of the database containing additional experimental measurements of Li -shells cross-sections. This database will undoubtedly be needed in the upcoming Geant4 releases in order to provide Geant4 users with more accurate empirical cross-sections. All the limitations and assumptions of the original works apply identically to their corresponding Geant4 implementation. In particular, the energy range in which the Orlic et al.’s model for protons is valid extends from 0.1 to 10 MeV and the atomic numbers for which the model provides all Li -shell values (L1 , L2 and L3 ) correspond to Z values ranging from 41 to 92.

Results ECPSSR models Models implementing the ECPSSR theory have been compared to calculations performed with the ISICS2008 software developed by Cipolla[39] which was taken as a reference. An example of direct comparison of K-shell cross-sectional values is shown in Fig. 1 for protons and alphas in copper indicating clearly a good agreement for this material over three orders of magnitude in energy. In Fig. 2, relative differences [σG4 (E)/σref (E) − 1] between K-shell Geant4 cross-sections σG4 (E) and reference values σref (E) are presented for copper, for both incident protons and alpha particles. The values in the energy regions of typical PIXE experiments (1–10 MeV) and above are in good agreement for all the presented models, except in some cases for the analytical model, for example in the case of gold K shell. The discrepancies at low energies mainly originate from the use of different atomic parameters between Geant4 and ISICS2008, such as shell-binding energies, derived from EADL[8] for Geant4 and derived from the study of Bearden and Burr[40] for ISICS2008. Besides, at low energies, analytically calculated cross-sections are very low and rapidly decrease towards lower energies, a scenario in which small differences in data handling by software can easily lead to big relative fluctuations. However, given the very low cross-sectional values, ionisation probabilities can be neglected in these energy ranges. In general, reasonable agreement for all elements is observed but for some elements (like gold) the Geant4 analytical implementation of ECPSSR in some energy ranges can be systematically lower (up to 20%) than the ISICS2008 results, which should be further evaluated. The same approach has been used for Li shells. A direct comparison is presented in Fig. 3, showing a qualitative good agreement of the Geant4 analytical model with reference data at low energies. Relative differences are plotted in Fig. 4 for copper. The analytical model for Li shells shows deviation lower than 10% from the reference data, on the whole energy range. In Fig. 4, the Y scale has been widened in order to show details on the medium- and high-energy regions. Generally, although differences are present, deviations from the reference values are below 5% for alphas and protons at typical energies used for PIXE experiments. Orlic and Paul models Empirical models have been compared to experimental reference data from which they derive and a very good agreement can be observed both for protons and alphas.

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A. Mantero et al.

Figure 1. Comparison of Geant4 (Geant4 analytical and Geant4 interpolated) and ISICS2008 implementations[39] of the ECPSSR theory for K-shell cross-sections for protons and alphas in copper.

Figure 2. Relative differences between the Geant4 (Geant4 analytical and Geant4 interpolated models) implementations[39] of the ECPSSR theory for K-shell cross-sections for protons and alpha particles in copper and the ISICS2008 values. The discrepancies at low energies, explained in the text, appear outside normal use cases.

As stated above, the Li -shells empirical model derives from a fit to experimental data from Orlic et al. As there were no published tabulation of Orlic et al. calculations, a subset of experimental data fitted by Orlic et al.,[28] has been used for comparison. Figure 5 shows a good agreement between Geant4 and experimental data, with the exception of the L3 shell for gold (other materials, not shown here, show good agreements). However, as Orlic et al.[27] fits are performed grouping data with similar Z (gold belongs to the group between Z = 71 and 80) and as experimental values are quite scattered (up to a factor 1 or 2 to the mean value), the present results can be considered to be in agreement with Orlic et al.’s work.

Conclusion For the first time in Geant4, a unified software approach for the simulation of atomic de-excitation has been implemented, which makes atomic de-excitation available for all particle types

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both for the Geant4 ‘standard’ and ‘low-energy’ EM physics subpackages. Following the first developments described in Ref [24], PIXE simulation has been fully included in this new software and is be available in reference Physics Lists with Geant4 version 9.4. Various models to calculate K- and Li -shell ionisation crosssections for all elements from carbon to uranium have been implemented. These shells are the main lines observed in experimental spectra for elemental composition analysis. The user can easily choose between theoretical models based on the ECPSSR theory and empirical models. Although the proposed models are not the most accurate and efficient models available for the modelling of ionisation of K and Li shells, these models have been verified and they return coherent results when compared to the used references (such as values calculated with the ISICS08 software) with reasonable agreement. The impact of accuracy of shell ionisation cross-sectional models on fluorescence spectra simulated with Geant4 needs to be evaluated carefully among

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PIXE simulation in Geant4

Figure 3. Comparison of Geant4 (Geant4 analytical and Geant4 interpolated) and ISICS2008 implementations of the ECPSSR theory for Li -shell crosssections for protons in copper. For clarity of the picture X values for L2 and L3 shells are scaled by a factor 0.5 and 0.1, respectively, while Y values are scaled by a factor 10 and 100, respectively.

Figure 4. Relative differences between the Geant4 (Geant4 analytical and Geant4 interpolated) implementations of the ECPSSR theory for Li -shell cross-sections for protons in copper and the ISICS2008 values. Note the large Y scale.

Figure 5. Comparison of Geant4 and Orlic experimental Li -shell cross-section for protons on gold. L2 and L3 data have been scaled by a factor 100 and 10 000, respectively.

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A. Mantero et al. other contributing sources of uncertainty in simulated set-ups such as photon attenuation, precision of tracking, etc. The ECPSSR theory remains the first choice for a wide range of energies. Corrections to ECPSSR, like Hartree-Slater (HS) and United Atom (UA),[41] are available and are known to work better in some energy and Z ranges. Other models based on experimental data, such as models for Li shells described in Ref [42], are available in the literature. The inclusion of such complementary and/or alternative models, as well as upper shells cross-sections, for PIXE modelling in both Geant4 EM subpackages using the unified software approach discussed in this work represents a clear continuation of the developments presented in this paper. Acknowledgements The authors thank Prof. Sam Cipolla for his valuable support with the ISICS2008 software, including customised versions kindly provided. The authors are also grateful to Dr. Claire Habchi and Dr. Philippe Barberet for their useful discussions about PIXE. This work has been partially funded by the CNRS grant PICS-4865, RFBR grant 09-02-91065 and by the ESA contracts 22712/09/NL/AT and 22839/10/NL/AT.

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