PIXE Simulation With Geant4

3614 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 PIXE Simulation With Geant4 Maria Grazia Pia, Georg Weidenspointner, Mauro Augelli, Lina Quintieri, Paolo Saracco, Manju Sudhakar, and Andreas Zoglauer Abstract—Particle induced X-ray emission (PIXE) is an important physical effect that is not yet adequately modelled in Geant4. This paper provides a critical analysis of the problem domain associated with PIXE simulation; it evaluates the conceptual approach, design and implementations of PIXE modelling so far available in Geant4, and describes a set of software developments to improve PIXE simulation with Geant4. The capabilities of the developed software prototype are illustrated and applied to a study of the passive shielding of the X-ray detectors of the German eROSITA telescope on the upcoming Russian Spectrum-X-Gamma space mission. Index Terms—Geant4, ionization, Monte Carlo, PIXE. I. INTRODUCTION HE use of charged particle beams like protons, particles and other heavy ions as a means of characteristic X-ray production has received considerable attention in recent years. The origin of the emission lies in the ionization of target material atoms by incident energetic charged particles: some atoms are ionized by removing an electron from an inner electronic shell; this inner shell vacancy is subsequently filled by an electron from an outer shell. Such an electron transition may give rise to the emission of characteristic X-rays at energies corresponding to the difference in the binding energies of the involved atomic shells. The application of particle induced X-ray emission (PIXE) to non-destructive trace element analysis of materials has first been proposed by Johansson and co-workers in 1970 [1]. Today, this experimental technique is widely exploited in diverse fields, from industrial applications to biological, environmental, geological and forensic sciences, as well as in archeometry and studies of the cultural heritage. An overview of PIXE experimental methods and scope of application is documented in [2]. T Manuscript received April 03, 2009; revised September 06, 2009. Current version published December 09, 2009. M. G. Pia and P. Saracco are with INFN Sezione di Genova, 16146 Genova, Italy (e-mail: [email protected]; [email protected]). G. Weidenspointner is with the Max-Planck-Institut für extraterrestrische Physik, 85740 Garching, Germany, and also with MPI Halbleiterlabor, 81739 München, Germany (e-mail: [email protected]). M. Augelli is with the Centre d’Etudes Spatiales (CNES), 31401 Toulouse, France (e-mail: [email protected]). L. Quintieri is with INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy (e-mail: [email protected]). M. Sudhakar is with INFN Sezione di Genova, 16146 Genova, Italy and also with the Department of Physics, University of Calicut, India. She is on leave from ISRO, Bangalore, India (e-mail: [email protected]). A. Zoglauer is with the Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA 94720 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNS.2009.2033993 The physical process of PIXE may also give rise to unwanted instrumental background X-ray lines, as is the case for space missions and for some laboratory environments. It also affects the spatial distribution of the energy deposit associated with the passage of charged particles in matter: in this respect, its effects may become significant in the domain of microdosimetry. The wide application of this experimental technique has motivated the development of several dedicated software systems; nevertheless, despite its large experimental interest, limited functionality for PIXE simulation is available in general-purpose Monte Carlo codes. This paper discusses the problem of simulating PIXE in the context of a general-purpose Monte Carlo system; it analyzes the current status of PIXE simulation with Geant4 [3], [4] and describes a set of developments to improve it. Finally, it illustrates an application of the developed PIXE simulation prototype to a concrete experimental problem: the study of the passive shield of the X-ray detectors of the German eROSITA [5] (extended Roentgen Survey with an Imaging Telescope Array) telescope on the upcoming Russian Spectrum-X-Gamma [6] space mission. II. SOFTWARE FOR PIXE: AN OVERVIEW Software tools are available in support of PIXE experimental applications as specialized codes or included in general-purpose simulation systems. Their main characteristics are briefly summarized below with emphasis on modelling the physics interactions underlying PIXE. A. Specialized PIXE Codes Dedicated PIXE codes are focussed on the application of this technique to elemental analysis. They are concerned with the calculation of experimentally relevant X-ray yields resulting from the irradiation of a material sample by an ion beam: primarily transitions concerning the K shell, and in second instance transitions originating from vacancies in the L shell. For this purpose various analysis programs have been developed, which are able to solve the inverse problem of determining the composition of the sample from an iterative fitting of a PIXE spectrum; some among them are GeoPIXE [7], GUPIX [8]–[10], PIXAN [11], PixeKLM [12], Sapix [13], WinAxil [14] and Wits-HEX [15]. A few codes concern PIXE simulation [16]–[18] specifically. Physics modelling issues are considered in specialized codes insofar as they can affect the measurable X-ray spectrum; other physics effects are often subject to simplification or neglected. These codes adopt similar strategies to address the problem domain: they share basic physics modelling options, like the 0018-9499/$26.00 © 2009 IEEE PIA et al.: PIXE SIMULATION WITH GEANT4 adoption of the ECPSSR (Energy-Loss Coulomb-Repulsion Perturbed Stationary State Relativistic) [19] model for the calculation of ionization cross sections; they handle simple experimental geometries, such as target materials consisting of layers, and impose limitations on the type of samples that can be analyzed. Approximated algorithms are adopted to calculate the X-ray yield, like artificially dividing the material sample into thin slices and summing-up yields derived from each one. The capabilities of various dedicated software systems for PIXE spectra analysis have been compared in controlled conditions [20]; the results have shown similar performance, as well as room for improvement, for all the examined codes. B. PIXE Simulation in General-Purpose Monte Carlo Systems Dedicated PIXE software systems have a limited application scope, as they lack the capability of dealing with complex experimental configurations. Comprehensive modelling capabilities are usually associated with general-purpose Monte Carlo systems. However, the simulation of PIXE is not widely covered by large scale Monte Carlo codes treating hadron interactions, while the conceptually similar simulation of electron impact ionisation is implemented in the EGS [21]–[23] and Penelope [24] electron-photon Monte Carlo systems. The Geant4 simulation toolkit addresses X-ray emission induced both by electrons and heavy particles like protons and particles. The physics that needs to be considered for the simulation of PIXE includes the energy loss and scattering of the incident charged particle, atomic shell ionization cross sections, and atomic transition probabilities and energies. On top of these physics features, consistency should be ensured, when modelling PIXE, with the particle transport schemes governing the Monte Carlo simulation. Intrinsically, PIXE is a discrete process: X-ray emission occurs as the result of producing a vacancy in atomic shell occupancy, in competition with Auger electron emission and Coster-Kronig transitions. Nevertheless, this discrete process is intertwined with the ionization process, which determines the production of the vacancy; this process, for reasons which are elucidated below, is treated in general-purpose Monte Carlo codes with mixed condensed and discrete transport schemes. The transport scheme to which ionization is subject affects the simulation of PIXE. The simulation of the energy loss of a charged particle due to ionization is affected by the infrared divergence of the cross section for producing secondary electrons, leading to a large production of low energy electrons. In the context of general purpose Monte Carlo systems, a discrete treatment of each individual ionization process, and subsequent atomic relaxation, poses a problem: explicitely considering the production of soft secondary particles may soon become inhibitive, since, due to their large number, the required computation time becomes excessive. Infrared divergence is usually handled in general-purpose Monte Carlo codes by adopting a condensed-random-walk scheme [25] for particle transport. In such a scheme, the particle’s energy loss and deflection are treated as averaged net 3615 effect of many discrete interactions along the step, thereby substituting in the simulation a single continuous process for the many discrete processes that actually occur. In a mixed scheme, like the one adopted by Geant4 [3], two different régimes of particle transport are introduced, which are distinguished through a secondary production threshold setting, i.e. a threshold for the kinetic energy of the electron that is kicked out of an atom as a result of ionization (the so-called -ray): all ionizations that would generate -rays below the threshold are treated as a continuous process along the step, while the ionizations that produce -rays above the threshold are treated as a discrete process. While this combined condensed-random-walk and discrete particle transport scheme is conceptually appealing and appropriate to many simulation applications, it suffers from drawbacks with respect to the generation of PIXE. One drawback is that atomic relaxation, therefore the emission of characteristic X-rays or Auger electrons, occurs only in connection with the discrete part of the transport scheme, where the event of producing a -ray can be associated with the creation of a vacancy in the shell occupancy. For the same reason, the fluorescence yield depends on the threshold value set for the kinetic energy of the electron resulting from ionization. This problem cannot be avoided by simply setting the -ray production threshold to zero, which effectively would eliminate the continuous part of the energy loss, because, apart from the prohibitive computational time demands, the theoretical assumptions on which the cross section calculations rely are only valid for -ray kinetic energies that are much greater than the mean excitation energy of the material. Another drawback of the current transport scheme is that the cross section for discrete hadron ionization, i.e. for production of a -ray, is calculated in Geant4, like in other general-purpose Monte Carlo codes, from a model for energy loss that is independent of the shell where the ionization occurs. While theoretical calculations are available to determine the spectrum of the emitted electron for each sub-shell in the case of electron impact ionization [26] for any element, to the authors’ knowledge no such facilities are currently available for the ionization induced by protons and ions. Experimental data are not adequate either to complement the lack of theoretical calculations. It is worthwhile stressing that a cooperative theoretical and experimental effort into this direction would be desirable. Attempts to deal with fluorescence emission outside the scheme of ionization outlined above have not found a satisfactory solution in other Monte Carlo systems either. For instance, the Penelope code treats fluorescence emission resulting from electron impact ionization as an independent discrete process along with a conventional mixed transport scheme; however, this strategy has been reported to lead to negative dose computations in spatial regions of the experimental setup that are subject to large statistical fluctuations [27]. The simulation of PIXE in the context of Geant4 must deal with the constraints imposed by the condensed-random-walk schemes adopted for charged particles transport, as well as with the limitations of the theoretical calculations which are currently available in this problem domain. 3616 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 III. PIXE IN GEANT4: EVALUATION OF PREVIOUS MODELS Software models for the simulation of PIXE induced by protons and particles are currently available in the Geant4 low energy electromagnetic package [28], [29]. Their main features are briefly reviewed here, and their shortcomings are discussed. A. First PIXE Development Cycle The first development cycle [30] to provide functionality for PIXE simulation in Geant4 involved a fairly general software design to handle ionization cross sections, which was originally developed by one of the authors of the present paper, and a set of implementations. It concerned the simulation of PIXE associated with the discrete part of the continuous-discrete hadron ionization process in Geant4 low energy electromagnetic package. The software design allows the ionization process to use interchangeable models of cross sections, detailed at the shell level, through a Strategy design pattern [31]: a common abstract base class G4VhShellCrossSection is provided, while a variety of physics models can be implemented in derived concrete classes. A user application can be configured by instantiating the desired concrete cross section model; the hadron ionization process uses shell cross section models polymorphically through their abstract base class in its algorithm for the simulation of PIXE. Based on the cross section calculations, the process determines the creation of a vacancy in the shell occupancy, and uses Geant4 Atomic relaxation [32] package for producing fluorescence photons and Auger electrons as a consequence of it. Three implementations of cross section models for the determination of the vacancy in the shell occupancy are currently available in Geant4, based on the above described design. The G4hShellCrossSection class, derived from G4VhShellCrossSection, implements ionization cross sections based on Gryzinski’s theoretical model [33], [34] for all elements from hydrogen through uranium and all atomic shells. This implementation appears to be affected by a severe problem: this is illustrated in Fig. 1, which compares the cross section for K shell ionization of gold by protons calculated by this class to the cross section compilation by Paul and Sacher [35] based on experimental data. It is not clear whether the observed large discrepancy is due to an error in the software or the deficiency of the theory or both, but clearly this implementation is inadequate to describe PIXE correctly. Apart from the gross discrepancy with respect to reference values associated with experimental data, Fig. 1 shows another undesirable feature of the G4VhShellCrossSection cross section implementation: the value of the K shell ionization cross section as a function of the proton energy appears to depend on the threshold set by the Geant4 user for producing -rays (respectively 1 keV and 1 MeV in the test case illustrated in the figure). K shell ionization cross sections based on fits to tabulated data are implemented [36]–[38] in two other classes derived from G4VhShellCrossSection: G4hShellCrossSectionExp and G4hShellCrossSectionDoupleExp; the latter involves more refined fitting functions to describe the data. These models have been documented by their implementers [36], [37] as cross sections for K shell ionization by protons based on Paul and Sacher empirical compilation [35]. Nevertheless, these classes Fig. 1. A comparison of the K shell ionization cross section for Au, implemented in G4hShellCrossSection (open symbols), with the experimental reference [35] (black dots); the two sets of cross sections calculated by G4hShellCrossSection correspond to 1 MeV (triangles) and 1 keV (squares) settings of the threshold for producing secondary electrons from ionization. Fig. 2. A comparison of K shell ionization cross sections, implemented in G4hShellCrossSectionDouble [36], [37] (black circles), with Paul and Bolik [39] empirical reference for particles (open symbols): the curves are respectively for Si (squares), Cu (triangles), Cd (crosses) and Au (white circles). The line represents ECPSSR K shell cross sections for particles resulting from the software developments described in Section IV. appear to describe the cross sections for K shell ionization by particles compiled by Paul and Bolik [39]: this is evident in Fig. 2, where cross sections calculated by G4hShellCrossSectionDoupleExp are compared to data in [39] for various atomic numbers (Z); G4hShellCrossSectionExp shows similar behaviour consistent with Paul and Bolik tabulations. The software verification plots in [36], [37] also show the fitting functions to be consistent with the cross sections for particles listed in [39]. Therefore, one can conclude that these implementations calculate cross sections correctly; rather, they have been incorrectly documented by their implementers regarding the type of particle they are pertinent to. PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 3. An illustration of the effect of the -ray production threshold on the discrete part of the ionization cross section, concerning silicon as target atom. The filled symbols represent ionization cross sections for individual shells (summed over all sub-shells) from an ECPSSR calculation: K shell (black circles) and L shell (black squares). The total cross section for the discrete part of ionizations as computed in Geant4 is depicted with open symbols for different -ray production thresholds: 1 keV (white circles), 10 keV (white squares) and 100 keV (white triangles). For some -ray production thresholds it may occur that the total cross section is smaller than the cross section for ionising any shell. Nevertheless, even if K shell ionization cross sections appear to be correctly calculated, the simulation of PIXE, currently based in Geant4 by default on G4hShellCrossSectionDouble, is affected by other issues concerning the production of a vacancy in the K shell. Both these classes determine the probability of creating a vacancy in the K shell according to the ratio of its ionization cross section with respect to a total ionization cross section. The software design of their base class assumes that the value of the total ionization cross section is supplied from an external source. In the current Geant4 PIXE implementation the total ionization cross section value is provided by the low energy hadron ionization process, and coincides with the cross section at a given proton energy for producing a -ray above the secondary electron production threshold [40]. The secondary production threshold is set in a Geant4 user application in terms of the electron range; for convenience of computation, it is converted by Geant4 kernel into an energy cut associated with each material and region of the experimental set-up. Therefore, like in the case illustrated above with the class implementing the Gryzinski cross section model, this algorithm introduces a dependency of PIXE simulation on the user-defined electron energy cut. This feature is prone to generate inconsistent behavior: for instance, as it is illustrated in Fig. 3, for some -ray production threshold settings the cross section for ionizing the K shell (which is smaller than the cross section for ionizing outer shells) may be larger than the total cross section for producing a -ray supplied by the hadron ionization process. The logic of the implemented algorithm fails in association with 3617 the occurrence of these conditions: it prevents the generation of PIXE, despite the occurrence of a discrete ionization. Fig. 3 also shows that, even when the K shell cross section is smaller than the value of the total cross section supplied to the vacancy creation algorithm, other issues may arise, since the determination of the probability of creating a vacancy in the K shell neglects outer shells, whose cross section may in turn become larger than the supplied total cross section. Anomalous behavior of the software occurs in Geant4 user applications related to these features of the algorithm. At least in principle, this drawback could be overcome by setting a lower value for the -ray production threshold (e.g. well below 1 keV in a case like the one depicted in Fig. 3): however, this would not represent a satisfactory solution, both considering the required large computational time and the assumptions under which the applied theoretical models are well grounded. The key issue is that the validity itself of a mixed continuous-discrete simulation scheme relies on the assumption that the relevant processes (i.e. those producing particles that the user wants to be tracked) happen at a much higher energy scale than the applied cuts. A further evolution of the software belonging to the first development cycle implemented the capability of generating PIXE associated with the continuous part of the hadron ionization process. An algorithm implementing this functionality is currently available in Geant4 G4hLowEnergyIonisation hadron ionization process and can be optionally activated by the user; however, its underlying physical model does not appear to be documented. To the authors’ knowledge, no usage of this algorithm has been reported in experimental applications, nor its validation. B. Software Released in Geant4 Version 9.2 A set of ionization cross section implementations [41] for PIXE simulation were first released in Geant4 9.2-beta version, and subsequently in 9.2 version in identical form. These developments originate from the same project described in this paper; therefore, the selection of cross section models in [41], out of those documented in literature, partly overlaps with those described in this paper, while the modelling strategy, software design, implementation and test process are different in the two cases. The implementations in Geant4 9.2 include two cross section models for K shell ionization by protons and particles, and one model for L shell ionization by protons. The cross sections for K shell ionization in Geant4 9.2 are based on interpolations of empirical data compilations (Paul and Sacher [35] for protons, Paul and Bolik [39] for particles) and on the ECPSSR theory [19]; they are respectively implemented in the Geant4 G4PaulKCrossSection and G4ecpssrCrossSection classes. The cross sections for L shell ionization by protons are based on the semi-empirical model of [42], which consists of polynomials derived from fits to experimental data complemented by data calculated through the ECPSSR theory; they are implemented in the G4OrlicLCrossSection class. Comparison plots are reported in [41] to demonstrate the capabilities of the implementations released in Geant4 9.2 with 3618 respect to the previous Geant4 cross section models and experimental data. Nevertheless, they compare cross section implementations for K shell ionization by protons released in Geant4 9.2, and proton ionization experimental data, with the cross section models for particles described in Section III-A: this comparison is not pertinent. The software for the calculation of ionization cross sections described in [41] and released in Geant4 9.2 is affected by several drawbacks. G4PaulKCrossSection calculates cross sections for particles consistent with those provided by the previously released G4hShellCrossSectionExp and G4hShellCrossSectionDoupleExp classes, and with the Paul and Bolik data on which all the three implementations are based: in this respect, it does not contribute new functionality to Geant4. However, it is much slower in execution than the previously existing classes described in Section III-A implementing the same functionality; as documented in Section IV-F, it is slower than G4hShellCrossSectionDoupleExp (the more complex of the two previous models) by a factor 40 to 60 approximately depending on the target atomic number. The computational performance of G4PaulKCrossSection for proton cross section calculation is two orders of magnitude slower than the one measured with the developments described in this paper. The slow performance of G4PaulKCrossSection resides in its design: this class is deprived of attributes, therefore objects instantiated from it cannot be characterized by a state, which may vary along with the simulation execution. As a result of this design feature, the calculation of a cross section for a given incident particle energy and type, and target particle type is charged with the heavy burden of instantiating and constructing an object responsible of data management, and of loading the entire data tabulations associated with the target particle, every time a client invokes the corresponding function. The G4PaulKCrossSection [41] class released in Geant4 9.2 attempts to open an external file, which should contain the data corresponding to K shell ionization cross section values tabulated as a function of energy derived from [35] and [39] respectively for protons and particles, and reformatted for use by this class; nevertheless, the required associated data files are not released in Geant4 9.2. This omission prevents the execution of the code by any Geant4 user. The evaluation reported here has been possible by supplying the necessary data files derived by the authors of the present paper from the original references [35] and [39]. In addition, this class exhibits a memory leak that causes an abnormal end of execution of simulations using a sufficiently high number of histories. The G4ecpssrCrossSection class computes K shell ionization cross section for protons and particles nominally based on the ECPSSR formulation [19]. However, it does not appear to reproduce ECPSSR cross sections correctly: for instance, Fig. 1 to 3 in [41] exhibit an evident divergence at higher energies of these implementations with respect to the empirical data by Paul and Sacher. Reference ECPSSR cross sections for the corresponding elements are tabulated in the same Paul and Sacher publication [35]: they are documented to reproduce the ECPSSR theory of [19] with accuracy better than 1.3%, and are close to the empirical values. The ECPSSR cross sections calculated with the IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 4. Cross sections for copper K shell ionization by protons: reference ECPSSR tabulations reported in [35] (stars), ECPSSR as calculated by the G4ecpssrCrossSection class released in Geant4 9.2 [41] (white crosses), ECPSSR model as calculated by the software described in Section IV based on ISICS tabulations (solid line), and empirical model of Paul and Sacher [35] (black circles). G4ecpssrCrossSection does not appear to implement the ECPSSR theoretical model correctly. models described in this paper, based on the interpolation of values produced by ISICS [43], [44], are also consistent with the reference ECPSSR tabulations and the empirical values of Paul and Sacher [35]. As an example, Fig. 4 compares software implementations and reference values for K shell ionization of copper by protons: cross sections calculated by the G4ecpssrCrossSection class of Geant4 9.2, ECPSSR cross sections from the model described in Section IV, ECPSSR reference cross sections tabulated in [35] and the empirical cross sections of [35]. Similarly, one can observe in Fig. 2 that the ECPSSR cross sections for particles described in Section IV are in agreement with the empirical cross sections of Paul and Bolik [39], while the ECPSSR implementation of G4ecpssrCrossSection is shown in [41] to produce significantly different results at higher energies with respect to the same empirical model. Therefore, based on these verifications, the discrepancy shown in [41] between the ECPSSR cross sections calculated by Geant4 9.2 models and the empirical models of [35] and [39] hints to a flawed software implementation of the ECPSSR theoretical model in G4ecpssrCrossSection rather than a deficiency of the ECPSSR theory. G4ecpssrCrossSection implements the analytical formulae required for the ECPSSR cross section calculation; apart from not reproducing the expected theoretical values correctly, this software strategy determines significantly slower code with respect to interpolating pre-calculated look-up tables, which is a common practice in Monte Carlo codes when intensive computation of physical quantities is required. The computational performance of the cross section models released in Geant4 9.2 is documented in detail in Section IV-F in comparison to the previous models and those described in PIA et al.: PIXE SIMULATION WITH GEANT4 3619 Fig. 5. The unified modelling language [51] class diagram of the developments for PIXE simulation described in this paper, illustrating the main features of the software. this paper. The significant performance penalty charged by the models released in Geant4 9.2 with respect to the already available ones and the models presented in this paper makes them inefficient for practical experimental applications. The semi-empirical model by Orlic et al. [42] implemented in G4OrlicLCrossSection for the calculation of L shell ionization cross sections has limited capabilities. It computes cross sub-shells only for elements with sections for individual atomic number greater than 40; for lighter elements, it provides total L shell ionization cross sections. This limitation of the model prevents the simulation of fluorescence for such lighter elements, since the sub-shell where a vacancy in the shell occupancy is produced by ionization cannot be univocally determined to initiate the process of atomic relaxation. As it is emphasized in [42], this semi-empirical model is valid only in a restricted interval of proton energies, up to approximately 10 MeV, corresponding to the range of the data on which it is based; this limits its usefulness for experimental applications concerned by higher energy ranges, like astrophysics, precision dosimetry or high energy PIXE techniques. This model is also affected by physics limitations: in fact, as demonstrated in Section V-D, it is less accurate than other available models, when compared to experimental data. The cross section models released in Geant4 9.2 exhibit severe software design limitations, that prevent client code from exploiting polymorphism, i.e. one of the key features of the object oriented technology, in the simulation of PIXE. The related classes do not derive from a common base class, which would allow the client to treat different models through dynamic polymorphism according to basic object oriented programming techniques. They do not share common syntactic constructs either, which would make them suitable for static polymorphic usage consistently with generic programming techniques. The public interface is different for each one of the cross section classes released in Geant4 9.2: the basic operation of providing the client a cross section, which is conceptually identical for any atomic shell and any physics modelling option, is associated with a CalculateCrossSection member function in G4ecpssrCrossSection, a CalculateKCrossSection member function in G4PaulKCrossSection and CalculateL1CrossSection, CalculateL2CrossSection and CalculateL3CrossSection member functions in G4OrlicLCrossSection. Moreover, the member functions for K shell cross section calculation have a different argument list than those for L shell cross sections. G4OrlicLCrossSection contains a semantic flaw: the CalculateL1CrossSection public member function returns the total cross section for L shell ionization for target elements with atomic number between 14 and 40 instead of the cross section for L1 sub-shell ionization, as its name suggests and as it indeed implements for targets with atomic number greater than 40. This design would oblige the client to implement ad hoc code to use each physics model, both for dealing with either model for K shell ionization (ECPSSR or empirical) and for handling the ionization of different shells or sub-shells. The low energy hadron ionization process released in Geant4 version 9.2 still relies on the design described in Section III-A 3620 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 for the generation of PIXE. The cross section models of [41] released in Geant4 version 9.2 are not compatible with it, nor any client process using them has been released in Geant4 9.2. Due to their design features, these models are not practically usable for PIXE simulation applications based on Geant4; their implementation features—correctness, computational performance and physical accuracy—motivate concerns about their use in experimental studies. charged particle transport schemes in a general-purpose Monte Carlo environment. The developments described in this paper adopt a software process model based on the Unified Process [46] framework. The Unified Process is use case driven and architecture-centric; the software developments are motivated by concrete experimental applications, one of which is discussed in Section VI, and significant effort is invested in the software design, as described in Section IV-B. The adopted software process framework involves an iterative and incremental life-cycle: the final product is the result of incremental extensions of functionality and iterative refinements. This software process model is well suited to the complex problem domain associated with PIXE: it allows the developers to address the identified physics and software issues through a series of coherent developments, but at the same time producing functional versions of the software at intermediate stages of the development process. The concrete produced deliverables—which include both code and other work products of the software process, like validation results—can be used to address experimental requirements and to get feedback for further improvements. The development cycle described in this paper addresses a significant subset of the simulation requirements associated with PIXE, while it is still based on the current conventional Geant4 particle transport scheme. The adopted component-based design provides the ground for reusing the components for PIXE simulation developed at this stage in future evolutions of Geant4 kernel; in this respect, it mitigates the risks associated with ongoing investigations about particle transport schemes concerning impact ionization [47]. The test process included the verification of the software, the validation of its components, and the evaluation of computational performance. As part of the validation process, consistently with [48], the developed software prototype was applied to a concrete use case to demonstrate whether it satisfies its intended use and user needs. Its application to a real-life experimental study is illustrated in Section VI. The software described in the following sections is meant to be available for public use following the publication of this paper. Until the subject software is approved for release through the Geant4 collaboration, it is available from the corresponding author. Users obtaining the software prior to release through Geant4 will be required to acknowledge that they are using prototype software. IV. DEVELOPMENTS FOR PIXE SIMULATION WITH GEANT4 At the present time, the Geant4 toolkit does not provide adequate capabilities for the simulation of PIXE in realistic experimental use cases. The first development cycle had a limited scope: the implementations concerned only PIXE induced by particles and involving K shell ionization—apart from the implementation based on Gryzinski’s theoretical model, which produces physically incorrect results. Even with the models which calculate K shell ionization cross sections correctly, inconsistencies arise in the simulation of PIXE related to the algorithm implemented for determining the production of a K shell vacancy. The deficiencies exhibited by the software released in Geant4 9.2 did not contribute to improve PIXE simulation with respect to the previous version. A set of developments for PIXE simulation is described in the following sections, which followed the first development cycle described in Section III-A. A preliminary overview of their progress was reported in [45]; this paper documents them in detail, and shows that they can be concretely used to study a realistic experimental problem. It also discusses the remaining open issues and proposes a path for further improvements to PIXE simulation with Geant4 over a longer time scale. The vision underlying these developments derives from the problem domain analysis discussed in Section II-B. The complexity of PIXE simulation in a general-purpose Monte Carlo code and the issues which are still open in this respect motivate the adoption of sound software methods and technology: their purpose is to ensure the development of reusable solutions, which would stand the challenge of evolutions in transport schemes. The adopted software process model, described in Section IV-A, supports this vision; the domain decomposition and software design, described in IV-B, enable it. The physics aspects associated with PIXE involve the creation of a vacancy in the shell occupancy due to ionization, and the emission of X-rays from the following atomic deexcitation. The former requires the knowledge of ionization cross sections detailed at the level of individual shells: for this purpose several models have been implemented and validated against experimental data. The latter exploits the existing functionality of Geant4 Atomic Relaxation package. The physics features involved with PIXE simulation are described in Sections IV-C through IV-E, while the computational performance and accuracy of the cross section models are documented in Sections IV-F and V. A. Software Process A satisfactory treatment of PIXE in Geant4 requires addressing its specific requirements along with the complexity of B. Analysis and Design This discipline plays a critical role in PIXE software development, considering the complexity of the problem domain described in Section II-B. It is a key issue to provide operational software for concrete experimental applications within the current constraints, while setting the ground for further smooth evolution and reuse, should new theoretical physics developments or novel transport schemes become available. The domain decomposition at the basis of PIXE simulation with Geant4 identified three main entities with associated responsibilities: the hadron ionization process, the creation of a PIA et al.: PIXE SIMULATION WITH GEANT4 vacancy in the shell occupancy resulting from ionisation, the deexcitation of the ionised atom with the associated generation of X-rays. The simulation of PIXE is the result of the collaboration of these entities. The ionisation process is a client of both the vacancy creation and the atomic relaxation sub-domains; it interacts with the Geant4 tracking domain regarding the evaluation of the mean free path of a particle and the effects of its occurrence, like delivering secondary products for further tracking or depositing energy in the volume that the particle traverses. The interaction with Geant4 particle transport (Geant4 tracking domain) is limited to the ionisation process; the vacancy creation and the atomic relaxation sub-domains interact with the ionisation process, but do not with Geant4 kernel. The software design addresses the issues identified by the problem domain analysis with the support of appropriate object oriented techniques. It adopts a component-based scheme, where components are associated with the rigorous underlying domain decomposition and are responsible for the required functionality. On top of the specific analysis issues to be considered, PIXE simulation should take into account performance issues as well. For this purpose a data-driven modelling strategy has been adopted for the calculation of the physical quantities involved with creating a vacancy in the shell occupancy. Ionisation cross sections as a function of energy are pre-calculated from analytical models or experimental compilations, and provided for use in the simulation as ASCII text files; cross sections at a given energy are calculated through interpolation over the tabulated values. This approach minimizes the necessary computational investment; its previous application in the library-based processes of the Geant4 low energy electromagnetic package has demonstrated that satisfactory accuracy can be achieved [49], [50]. The requirement of dealing with physical computations through a data-driven approach determines another component in the software design, specifically responsible for handling physics data. A class diagram in the Unified Modelling Language (UML) [51] illustrates the main features of the software design in Fig. 5. The data management component deals with the low level management of ionisation cross section data. Its interface, the G4VDataSet abstract base class, defines the operations associated with the component; a set of derived concrete classes implement the functionality specific to the data management associated with a single entity, like an atomic shell, or a composed system, like an atom with an underlying shell structure, or a material consisting of atoms. The software design exploits a Composite design pattern [31] to handle all such entities transparently through a common interface. Interpolation over tabulated values can be performed according to different algorithms, which are handled transparently through a Strategy pattern [31]. The data management component also provides functionality for randomly selecting a sub-shell based on associated cross section data. The design of the data management package is more general than the functionality required for PIXE simulation: it can manage generic data, which may correspond to individual or 3621 composed structures. It extends and improves the preliminary functionality for this purpose developed by one of the authors of this paper for use in Geant4 low energy electromagnetic package. Thanks to its generality, it can find application in other physics simulation domains. The client deals with the data management only through the interface class. Therefore a variety of data sets can be supplied for use by the client software, which can either be interchangeable alternatives for the same functionality, for instance tabulations deriving from theoretical or empirical models, or correspond to complementary functionality, e.g. data for different particle types or (sub-)shells. The originally developed G4PixeCrossSectionHandler class is responsible for the physical management of cross sections and the creation of the initial shell vacancy by hadron impact. It provides functionality for the calculation of various types of cross sections: the cross section for emitting a ray above a given energy threshold (implemented as in [40]), the ionization cross sections corresponding to a single (sub-)shell, and the ionisation cross section calculated from the summation over individual (sub-)shells. It exploits the lower level functionality of the data management package for obtaining cross section values at a specified energy according to the model selection indicated by the user. The vacancy produced by ionisation is determined by delegating the lower level data management package the responsibility of randomly selecting a shell according to the relative weight of shell ionization cross sections. The deexcitation of the atom, once a vacancy has been created, is delegated to Geant4 Atomic Relaxation package [32], which is also responsible for the generation of secondary particles: X-rays and Auger electrons. Since Geant4 Atomic Relaxation package provides functionality for elements with atomic number larger than 5, this limitation also applies to the simulation of PIXE. The G4RDHadronIonisation class is an improved version of the hadron ionization process implemented in the Geant4 low energy electromagnetic package [52]. Its responsibilities regarding the simulation of PIXE have been minimized: they are entirely delegated to G4PixeCrossSectionHandler and Atomic Relaxation, respectively for what concerns the creation of a shell vacancy and the following creation of secondary products resulting from atomic deexcitation. For what concerns PIXE, its role is limited to triggering the operations of G4PixeCrossSectionHandler and of the Atomic Relaxation interface (G4AtomicDeexcitation) in its PostStepDoIt member function, which is responsible for the actions pertinent to the discrete behaviour of ionisation. The ionisation process is responsible for handing the products of atomic relaxation to Geant4 tracking together with the -ray electrons produced by ionisation. Thanks to the component-based design, the functionality provided by the components specific to PIXE simulation (G4PixeCrossSectionHandler and Atomic Relaxation) is not limited to being used in G4RDHadronIonisation according to the above outlined scheme. Other client processes can be envisaged to manipulate the functionality they provide—creation of a vacancy in the shell occupancy and deexcitation of the atom where a vacancy has been created, without requiring any modification of the PIXE specific software. For instance, these com- 3622 ponents could be used by a dedicated client process to handle PIXE resulting from impact ionization as an independent discrete process, similarly to the approach adopted in Penelope; they could also be reused in future evolutions of Geant4 kernel. This versatility of the design represents an improvement over the design of the first development cycle. The attribution of most of the responsibilities for PIXE simulation to specific components, along with the limitations of the responsibilities of the process class in this respect, is propedeutic to further evolutions in this physics sub-domain: thanks to this design, PIXE functionality can improve independently without requiring modifications to the ionisation process, while the process would benefit of such evolutions transparently to provide an overall improved description of ionisation and its associated physics effects. Even if presently applied only to protons and particles, this design is suitable to handle PIXE production by any particles. IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 TABLE I IONIZATION CROSS SECTION1 MODELS AVAILABLE FOR PIXE SIMULATION WITH GEANT4 C. Ionisation Cross Section Models The simulation of PIXE concerns a variety of experimental applications, that require the capability of calculating ionisation cross sections over an extended energy range: from a few MeV typical of material analysis applications to hundreds MeV or GeV range of astrophysical applications. Various theoretical and empirical models are available in literature to describe ionisation cross sections for different interacting particles, as well as compilations of experimental data. However, there is limited documentation in literature of systematic, quantitative assessments of the accuracy of the various models. The current software prototype has adopted the strategy of providing an extensive collection of ionisation cross section models as a function of element, atomic (sub-)shell, and incident particle kinetic energy. The availability of a wide choice of modelling options allows the study of the characteristics of the various models in a systematic and comprehensive way, also in relation to experimental data, and the evaluation of the different observable effects they may produce in concrete simulation use cases. According to the chosen strategy, the provision of a cross section model is reduced to the construction of tabulations of its values at preset energies. The cross sections associated with the models described in this paper have been pre-calculated either using existing software documented in literature, or developing ad hoc code. The data are stored in files, which make up a data library for PIXE simulation with Geant4—but could be used also by other codes. The cross section data sets selected by the user are loaded into memory at runtime; cross section values at a given energy are calculated by interpolation over the tabulated values whenever required. The adopted data-driven approach for the provision of ionization cross sections presents various advantages. It optimizes performance speed, since the calculation of the interpolation is faster than the calculation from complex algorithms implementing theoretical models. The performance of this approach is superior to that of the models currently released in Geant4. as is demonstrated in Section IV-F. This approach also offers flexibility: chosing a cross section model simply amounts to reading the corresponding set of data files; adding a new set of cross sections simply amounts to providing the corresponding set of data files, which are handled transparently. Cross section tabulations may be based on experimental data or derive from pre-calculating analytical formulae, that represent either theoretical or empirical models. Different modelling approaches, like exploiting cross sections based on experimental data or theoretical calculations for different shells, can be combined for use in a simulation as desired by the user. Finally, the cross section data are transparent to the user: the files are accessible to the user and human readable. Various sets of cross sections for K, L, and M shell ionization by protons and particles have been tabulated based on theoretical and empirical models. The available models are summarized in Table I. The lower limit of atomic number 6 for the provision of individual shell ionization cross sections is determined by Geant4 capability of handling atomic relaxation [32]. The documentation PIA et al.: PIXE SIMULATION WITH GEANT4 3623 Fig. 6. Comparison of implemented models of K shell ionization cross sections for incident protons: ECPSSR calculation (thick line), ECPSSR with HartreeSlater correction (dashed line), ECPSSR with United Atom correction (dotted line), ECPSSR with high energy correction as in [56] (thin line), PWBA calculation (triangles), empirical values from Paul &Sacher [35] (filled circles) and from Kahoul et al [58] (filled squares); overlapping curves may not be visible in the picture. The horizontally shaded band represents one standard deviation around Kahoul et al empirical values; the vertically shaded band represents one standard deviation around Paul and Sacher empirical values; ECPSSR and PWBA cross sections are interpolated from ISICS calculations. The plots span a wide range of ; (b) Cu ; (c) Cd ; (d) Au . target element atomic numbers. (a) Si of theoretical calculations of ionization cross sections for outer shells is very limited in literature [53], and experimental measurements are scarce as well; therefore they could not be provided at the present time. Tabulated data sets for outer shells can be added at a later stage, if they become available, and would be treated transparently by the software. The tabulations corresponding to theoretical calculations span the energy range between 10 keV and 10 GeV; empirical models are tabulated consistently with the energy range of validity documented by their authors, that corresponds to the range of the data used in the empirical fits and varies along with the atomic number and sub-shell. Theoretical cross section models include Plane Wave Born Approximation (PWBA) and variants of the ECPSSR model: the original ECPSSR formulation [19], ECPSSR with United Atom correction (ECPSSR-UA) [54], ECPSSR with corrections for the Dirac-Hartree-Slater nature of the K shell [55] (ECPSSR-HS), as well as calculations based on recent improvements to K shell cross section specific to high energy [56] (ECPSSR-HE). Tabulated data sets based on theoretical calculations are provided for K, L and M shells; some of the ECPSSR variants are specific to the K shell. ECPSSR tabulations have been produced using the ISICS software [43], [44], 2006 version; an extended version, kindly provided by ISICS author S. Cipolla [57], has been exploited to produce tabulations associated with recent high energy mod- elling developments [56]. Tabulations of ECPSSR calculations as reported in [35] are also provided: they differ slightly from those produced by ISICS, presumably due to different values of atomic parameters used in the calculations. Empirical cross section models based on experimental data for K shell ionization include the tabulations for protons documented in [35] and a more recent one [58], which also accounts for experimental measurements performed after the publication of [35]. An empirical cross section model for K shell ionization by particles is based on the tabulations published in [39]. The software implementation to produce tabulations from the analytical curves documented in [58] provided slightly different results with respect to the reference values at selected energies reported in [58]. This behaviour can be attributed to different values of K shell binding energies used by Geant4 and by Kahoul et al. in the calculation of the empirical formula describing the fit to the experimental data. The accuracy of the binding energies used by Geant4 is documented in [59]. Anomalous cross section values resulting from the analytical formula reported in [58] for elements with atomic number between 61 and 70 suggest that one of the polynomial coefficients listed in [58] may be affected by a typographic error; cross section values in line with expectations can be calculated changing its sign. The various K shell ionization cross sections for protons developed for use in Geant4-based simulations are plotted in Fig. 6 for a few representative target elements spanning a wide range 3624 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 of atomic numbers. The figures show the Paul and Sacher and Kahoul et al. empirical models, the ECPSSR model in its original formulation and variants (High Energy, Hartree-Slater and United Atom corrections) and the Plane Wave Born Approximation model. The empirical models are plotted with shaded bands corresponding to the uncertainties documented in their respective reference publications. The variants of the ECPSSR model show relatively small differences; in some cases the cross section values deriving from different models are undistinguishable in the plots, as they overlap. Empirical models for L shell ionization by protons, based on experimental data compilations [60], [61], have been developed by Miyagawa et al. [62], Sow et al. [63] and Orlic et al. [42]. The last model is actually semi-empirical, since it incorporates calculations from the ECPSSR theory to extend its applicability beyond the limited energy range covered by experimental data. Cross section tabulations are provided for all such models. Due to limitations of the available experimental data, these empirical models provide cross section values for L sub-shells only for elements with atomic number larger than 40 or 42. All such empirical models involve polynomial fits to experimental data; they differ for the pools of experimental measurements, the criteria to identify the data subsets to be fitted, and the degrees of the fitting polynomials. The empirical models for L sub-shell cross sections exhibit visible differences, as illustrated in Figs. 7 and 8 for two elements; the different functional behaviours resulting from the polynomial fits to the data and their large uncertainties derive from the spread of the available measurements. The Sow et al. model is characterized by much smaller uncertainties than the two other empirical models; it is also valid in a more limited energy range. Software implementing the empirical models of Kahoul et al. [58], Miyagawa et al. [62], Orlic et al. [42] and Sow et al. [63] has been developed to produce the tabulations; however, once the cross section data sets have been produced, this code is not further needed nor requires maintenance over all Geant4 supported platforms. The ISICS software allows the calculation of cross sections for heavier ions as well based on PWBA and ECPSSR theoretical models. Methods to calculate cross sections for ions derived from those for protons have been proposed [64] as well. Therefore, the current provision of PIXE simulation capabilities can be easily extended to heavier ions in future development cycles based on the already developed models. The creation of a data library, encompassing the cross section tabulations described in this paper, is in progress. This data library is intended for public distribution; it could be of interest to Monte Carlo and analysis codes dealing with PIXE. Fig. 7. A comparison of different sets of cesium L sub-shell ionization cross sections for incident protons: ECPSSR/ISICS calculation (thick line), empirical model by Miyagawa et al [62] (vertically shaded band), empirical model by Sow et al [63] (dark shaded band), semiempirical model by Orlic et al [42] (horizontally shaded band); the filled bands represent one standard deviation around the nominal values of the empirical models. (a) ; (b) ; (c) . D. Generation of a Vacancy The determination of which atomic (sub-)shell is ionised is related to its ionisation cross section with respect to the total cross section for ionising the target atom. However, as discussed in Section II-B, the condensedrandom-walk scheme raises an issue as to estimating the total ionisation cross section at a given energy of the incident particle. The secondary electron production threshold, that is introduced in Monte Carlo codes adopting a mixed transport scheme to distinguish the occurrence of a discrete ionization event from the continuous energy loss integrating the soft electron production, is prone to generate inconsistencies in the determination of the shell vacancy, as discussed in Section III-A. The solution implemented in the first development cycle, that related the determination of the vacancy to the cross section for PIA et al.: PIXE SIMULATION WITH GEANT4 3625 This algorithm provides a correct distribution of the produced vacancies as long as ionisation cross sections can be calculated for all the atomic shells involved in the atomic structure of the target element. Since cross section models are currently available for K, L and M shells only, at the present status of the software this algorithm overestimates PIXE for elements whose atomic structure involves outer shells, because of the implicit underestimation of the total ionization cross section. This approach, however, preserves large flexibility towards further improvements: adding the treatment of outer shells is straightforward. It is adequate for studies interested in higher energy X-ray emission deriving from inner shell vacancies, which are experimentally measurable, and provides better control on the simulation results than the algorithm implemented in the first development cycle: the outcome of the PIXE simulation component does not depend on the user’s setting of the secondary electron production threshold. A solution to the open issues would derive from the availability of hadron ionisation cross sections for outer shells, similar to calculations collected in the Evaluated Electron Data Library (EEDL) [26]. The exploitation of calculations methods for N shells as outlined in [53] is being investigated, as well as the possibility of extending the calculations to outer shells; nevertheless, such a theoretical effort would require a substantial investment of resources and an extended time scale. Once available, cross section tabulations for outer shells would be transparently incorporated into the existing software design. Further opportunities for improvement over the current approximation could derive from an ongoing effort [47] to explore particle transport in Geant4 kernel, that addresses in particular the issue of co-working condensed-random-walk and discrete transport schemes. E. Generation of Secondary Particles Fig. 8. A comparison of different sets of mercury L sub-shell ionization cross sections for incident protons: ECPSSR/ISICS calculation (thick line), empirical model by Miyagawa et al [62] (vertically shaded band), empirical model by Sow et al [63] (dark shaded band), semiempirical model by Orlic et al [42] (horizontally shaded band); the filled bands represent one standard deviation around the nominal values of the empirical models. (a) ; (b) ; (c) . producing a ray above the user selected threshold, demonstrated the weakness of this algorithm, as illustrated in Fig. 3. A different algorithm has been adopted in the developments described in this paper: the vacancy in the shell occupancy is determined based on the total cross section calculated by summing all the individual shell ionisation cross sections. The production of secondary particles by the atomic relaxation of an ionized atom is driven by the PostStepDoIt member function of the G4RDHadronIonisation class shown in the UML class diagram of Fig. 5, and is delegated to the Atomic Relaxation component. The PostStepDoIt function interacts with the vacancy creation and the atomic relaxation components through their respective interface classes; it retrieves the ionised shell from the former and hands it to the latter to initiate the atomic deexcitation process, which in turn generates fluorescence X-rays and Auger electrons. It applies a further filter to the candidate secondary products resulting from the atomic relaxation, based on the thresholds in range set by the user. X-rays that would be below the threshold for producing secondary photons are not considered for further tracking, and the corresponding energy is converted into a local deposit. An example of an X-ray spectrum produced by the PIXE model described in this paper is depicted in Fig. 9: it shows the energy distribution of characteristic photons originating from vacancies produced by proton ionization of Cu in K, L, and M shells, respectively. The relative importance of the different lines appearing in the spectrum is the result of the ionization cross sections for the various shells, combined with the respective probabilities of allowed atomic transitions. 3626 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 TABLE III PERFORMANCE EVALUATION OF DIFFERENT IMPLEMENTATIONS OF K SHELL IONIZATION CROSS SECTIONS, PARTICLES Fig. 9. X-ray spectrum produced by proton ionization of Cu respectively in K, L and M shells; the energy of primary protons is uniformly distributed between 1 keV and 10 MeV. TABLE II PERFORMANCE EVALUATION OF DIFFERENT IMPLEMENTATIONS OF K SHELL IONIZATION CROSS SECTIONS, PROTONS Fig. 10. Relative difference, expressed in percent, between the ECPSSR cross sections for K shell ionization by protons, calculated as described in Section IV, and reference ECPSSR cross sections tabulated in [35]. TABLE IV TEST COMPARING K SHELL IONIZATION P-VALUES FROM THE CROSS SECTIONS FOR PROTONS AGAINST IONIZATION CROSS SECTION MEASUREMENTS F. Performance Tests The computational performance of the cross section models described in this paper has been measured and compared to previously available Geant4 implementations with similar physics functionality. The performance has been evaluated on an Intel Core2 Duo Processor E6420, 2.13 Gz, 4 GB RAM, by instantiating the calls to the cross section cross section classes and making calculation functions. The timing results in seconds are reported in Tables II and III, respectively for protons and particles; those concerning the models described in this paper are identified as “this work”. The models described in this paper are faster than all the previous implementations. The various cross section models (based on theory or empirical) are characterized by similar computational performance; their differences in performance are related to the size of the tabulated data sets, since this parameter affects the time spent to search across the tabulation the appropriate values to be interpolated. The data-driven approach determines a gain of a factor 6 to 8 approximately with respect to performing the t calculations on the fly during the simulation execution and of about a factor 2 with respect to using analytical fits to the data; the better design of the K shell empirical models is responsible of a gain factor that is in general larger than 100 with respect to the equivalent cross section models released in Geant4 9.2. PIA et al.: PIXE SIMULATION WITH GEANT4 P-VALUES FROM THE 3627 TABLE V TEST COMPARING K SHELL IONIZATION CROSS SECTIONS FOR PROTONS AGAINST DATA DERIVED FROM X-RAY PRODUCTION CROSS SECTION MEASUREMENTS V. SOFTWARE VERIFICATION AND VALIDATION The component-based design adopted for the simulation of PIXE facilitates the software verification and validation: the components can be tested independently, while the correctness of their collaboration in a simulation application is ensured by Geant4 kernel design. The validation of simulation software is a key issue to estimate the accuracy achievable in experimental applications; for a toolkit like Geant4, the comparative evaluation of the available models also provides experimentalists objective guidance for selecting an appropriate model for a use case. As described in the previous sections, the production of X-rays, once a vacancy has been created in the shell occupancy due to the ionisation of the target atom, is delegated to Geant4 Atomic Relaxation component. The validation of the energies of fluorescence photons and Auger electrons generated by this component is documented in [59]. The accuracy of the fluorescence transition probabilities tabulated in the Evaluated Atomic Data Library [65], on which Geant4 fluorescence generation is based, is documented in [66]; consistent qualitative results specific to the Geant4 use of EADL have been reported [67]. Cross section calculations were subject to verification tests to ensure that the software reproduced the underlying theoretical or empirical models correctly. Special emphasis was invested into evaluating the differences betweeen theoretical and empirical models, and among the variants of the ECPSSR model. The availability of a wide variety of cross section models for the first time in the same computational environment allowed a detailed comparative assessment of their features. The various cross sections were compared to experimental data to estimate their validity, and among them to evaluate their similarities and differences. The use of statistical methods for the comparisons provides objective data to appraise the accuracy of the simulation quantitatively and to guide the choice of the models in an experimental application. The analysis focussed on proton cross sections, due to the larger availability of cross section data for the validation of simulation models and the greater variety of models. Detailed evaluations concerning cross section models for K and L shell ionization are reported below; the scarce availability of experimental data prevented a similar study for the M shell. A. Verification of Ionization Cross Section Models The verification of the software developed for the calculation of ionisation cross sections was performed by means of unit tests, that ensured the consistency of the software with the specifications of the physics models it implements. The most critical part of this process consisted of the verification of the correctness of the cross sections reproducing the ECPSSR theory. For this purpose ECPSSR cross sections calculated as described in Section IV were compared to reference compilations of ECPSSR cross sections for protons and particles available in literature. An example is illustrated in Fig. 10, which shows the distribution of the percent difference with respect to tabulations of ECPSSR cross sections for K shell ionization by protons published in [35] over all the elements with 3628 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 TABLE VI PERCENTAGE OF TEST CASE WITH COMPATIBILITY AT A DEFINED CONFIDENCE LEVEL BETWEEN SIMULATION MODELS AND EXPERIMENTAL DATA OF K SHELL IONIZATION BY PROTONS TABLE VII CONTINGENCY TABLE RELATED TO SIMULATION MODELS COMPATIBILITY WITH K SHELL IONIZATION CROSS SECTION MEASUREMENTS: COMPARISON OF ECPSSR AND ECPSSR WITH HARTREE-SLATER CORRECTION atomic number between 6 and 92. The small differences observed between the ECPSSR cross sections calculated by the simulation software and reference tabulations can be ascribed to different atomic parameters used in the computation of the cross section and, to a smaller extent, to the interpolation procedure. The observed differences are compatible with the precision of ECPSSR calculations reported in [35]. A significant discrepancy with respect to the reference tabulations was observed only for europium (atomic number 63); investigations are in progress to ascertain its origin. Similar verifications against published ECPSSR cross section references have been performed for the produced L sub-shell data sets and for particles, and to a smaller extent for M shell cross sections due to the scarce literature on this topic. TABLE VIII CONTINGENCY TABLE RELATED TO SIMULATION MODELS COMPATIBILITY WITH K SHELL IONIZATION CROSS SECTION MEASUREMENTS: COMPARISON OF PAUL AND SACHER AND ECPSSR WITH HARTREE-SLATER CORRECTION TABLE IX CONTINGENCY TABLE RELATED TO SIMULATION MODELS COMPATIBILITY WITH K SHELL IONIZATION CROSS SECTION EXPERIMENTAL DATA: COMPARISON OF KAHOUL ET AL. AND ECPSSR WITH HARTREE-SLATER CORRECTION TABLE X CONTINGENCY TABLE RELATED TO SIMULATION MODELS COMPATIBILITY WITH K SHELL IONIZATION CROSS SECTION EXPERIMENTAL DATA: COMPARISON OF ECPSSR AND ECPSSR WITH HARTREE-SLATER CORRECTION IN THE LOWER ENERGY RANGE B. Methods for the Comparison of Cross Sections Cross sections were compared by means of statistical methods: goodness-of-fit testing complemented by categorical analysis based on contingency tables. The comparison of cross sections as a function of energy was test [68]. This performed for each element by means of the test was selected since, among non-parametric goodness-of-fit tests, it takes into account experimental uncertainties explicitly. The null hypothesis in the goodness-of-fit testing process was defined as the compatibility that the two cross section distributions subject to comparison derive from the same parent distribution. Unless differently specified, a 95% confidence level was set to define the critical region of rejection of the null hypothesis. test was performed on pairs of cross section distriThe butions, that were calculated or measured at the same energies for each element. The data samples corresponded either to a simulation model and experimental data for the validation of cross sections, or to two simulation models for the comparison of their respective features. The whole range of atomic num- bers from 6 to 92 was subject to testing for the inter-comparison of models; only the elements for which experimental data were available were considered in the validation tests to estimate the model accuracy. Some experimental data identified as outliers were excluded from the tests; they concerned measurements exhibiting large discrepancies ( 3 standard deviations) with respect to other data at the same energy, or anomalous data series from the same experiment suggesting the presence of systematic effects. Guidance for this analysis derived also from [35] and [69]. Contingency tables were built on the basis of the outcome test to determine the equivalent, or different behavior of the over model categories: for instance, the compatibility of dif- PIA et al.: PIXE SIMULATION WITH GEANT4 P-VALUES FROM THE 3629 TABLE XI TEST COMPARING L SHELL IONIZATION CROSS SECTIONS FOR PROTONS CALCULATED BY THEORETICAL MODELS AGAINST EXPERIMENTAL DATA ferent simulation models, or the agreement of a simulation model with experimental data, over the whole range of target elements. Their analysis estimates whether the number of elements for which a model is compatible with experimental data is independent of the model, in other words whether both models under test describe the data equally well. The input test: to contingency tables derived from the results of the they were classified respectively as “pass” or “fail” according to whether the corresponding p-value was consistent with the defined confidence level. The null hypothesis consisted in assuming, for each contingency table, the equivalence of the two models subject to comparison in it. The contingency tables were analyzed with Fisher exact test test applying Yates continuity correction [70] and with the [71]; the latter ensured reliable results even with small number test [72] was also performed of entries in the table. Pearson on contingency tables, when their content was consistent with its applicability. The use of different tests in the analysis evaluates the robustness of the results: it contributes to the identification of possible systematic effects, which could be due to the peculiar mathematical features of a test. A signigficance level of 0.05 was set to determine the rejection of the null hypothesis, corresponding to a 95% confidence level, unless specified differently. The results summarized in the following sections show that all the tests produce equivalent results in terms of rejection, or non rejection, of the null hypothesis at the defined confidence level—with one exception, which is discussed in detail in Section V-D. Fig. 11. L shell proton ionization cross sections of silicon (black symbols) and chlorine (white symbols): ECPSSR model (stars) and experimental data from [61]. C. K Shell Ionization Cross Sections The cross sections for K shell ionization by protons based on the various theoretical and empirical models were compared against experimental measurements to evaluate their accuracy. The reference experimental data were extracted from the compilation in [35]. This collection includes various types of 3630 P-VALUES FROM THE IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 TABLE XII TEST COMPARING L SHELL IONIZATION CROSS SECTIONS FOR PROTONS CALCULATED BY EMPIRICAL MODELS AGAINST EXPERIMENTAL DATA TABLE XIII PERCENTAGE OF TEST CASE WITH COMPATIBILITY AT A DEFINED CONFIDENCE LEVEL BETWEEN SIMULATION MODELS AND EXPERIMENTAL DATA OF L SHELL IONIZATION BY PROTONS TABLE XIV CONTINGENCY TABLES RELATED TO SIMULATION MODELS COMPATIBILITY WITH L SHELL IONIZATION CROSS SECTION EXPERIMENTAL DATA: COMPARISON OF ECPSSR WITH UNITED ATOM CORRECTION AND OTHER MODELS cross section measurements associated with the K shell: X-ray production cross sections, Auger electron production cross sections and ionization ones. K shell ionization cross sections can be calculated from X-ray and Auger production ones according to the formulae: (1) and (2) where is the ionization cross section, and are respectively the X-ray production and Auger electron production cross PIA et al.: PIXE SIMULATION WITH GEANT4 P-VALUES FROM THE 3631 TABLE XV TEST CONCERNING HIGH ENERGY EXPERIMENTAL DATA sections, and is the fluorescence yield for the K shell. The fluorescence yields compiled in [73] were used to calculate ionization cross section from production ones, consistently with the data handling reported in [35]; their uncertainties were propagated into the rescaled reference ionization cross sections. Two analyses were performed to evaluate possible systematic effects due to deriving ionization cross sections from production ones: the first concerned only the small data sample of ionization cross section measurements, while the second was performed over the whole experimental data sample, also including rescaled production cross sections. It should be noted that, apart from the different size of the data samples, ionization cross section measurements concern a low energy range, while the full sample including production cross sections spans a wider energy range. A few production cross section measurements at 160 MeV were not considered in this test; a dedicated test specific to high energy PIXE is reported in Section V-E. The experimental ionization cross sections and the ones calculated by the simulation models are plotted in Figs. 13 to 24. test are listed in Tables IV The p-values resulting from the and V, respectively concerning the validation against ionization cross section data and the whole set of measurement, including rescaled production cross sections too. The p-values concerning the comparison with the PWBA model are not listed, since they are less than 0.001 in almost all the cases. test fails to reject The fraction of test cases for which the the null hypothesis at the 95% and 99% confidence level are listed in Table VI. The ECPSSR model with Hartree-Slater correction stands out in the test concerning ionization cross section measurements as the model exhibiting the highest agreement at 95% confidence level with respect to the experimental data. Contingency tables are reported in Tables VII and VIII concerning the comparison of the ECPSSR-HS model and the Paul and Sacher one against experimental data; the contingency tables associated with the ECPSSR with United Atom and High Energy corrections, and the Kahoul et al. have identical entries as Table VII, therefore the results obtained for Table VII hold for the comparison of ECPSSR-HS with those models too. From the analysis of the contingency tables one observes that the ECPSSR model with Hartree-Slater correction appears more accurate than the original ECPSSR model, the ECPSSR with United Atom and High Energy corrections, and the Kahoul et al. empirical model at the 90% confidence level; its difference of accuracy with respect to the Paul and Sacher model is not statistically significant. test summarized in According to the results of the Table VI, all the cross section models implemented in the simulation exhibit equivalent behaviour regarding the compatibility with the full set of experimental cross section data, including the rescaled X-ray and Auger production cross sections, with the exception of the Kahoul et al. model. The contingency table comparing the accuracy of the Kahoul et al. and ECPSSR-HS models is documented in Table IX: it confirms that the two models show a statistically significant difference regarding their accuracy. The contingency tables associated with the other models show that they are statistically equivalent regarding their accuracy over the whole experimental data sample. However, when only the lower energy range (below 5–7 MeV, depending on the atomic number) is considered, a statistically significant difference at the 95% confidence level is observed between the ECPSSR model and the ECPSSR-HS one; the latter is more accurate with respect to experimental data. The related contingency table is reported in Table X. This result is consistent with the analysis concerning ionization cross section measurements only. From this analysis one can conclude that the K shell ionization cross section models implemented for PIXE simulation exhibit a satisfactory accuracy with respect to experimental measurements; in this context, one should take into account the discrepancies exhibited by experimental data. The results of the goodness-of-fit test show that in more than 90% of the test cases at least one simulation model is compatible at 95% confidence level with ionization cross section measurements, while all the simulation models, with the exception of Kahoul et al., are compatible at 99% confidence level with the extended experimental data sample in more than 80% of the test cases. The ECPSSR model with Hartree-Slater corrections shows the best accuracy in the lower energy range, while all the other models appear equivalent regarding their accuracy, apart from Kahoul et al. A few inter-comparisons were performed among the implemented cross section models to characterize their similarities and differences. The method of comparison was the same as for test over inthe comparison against experimental data: the dividual elements, complemented by categorical analysis over the whole set of test results. The tests were performed on all elements with atomic number between 6 and 92, with the exception of 63, since, as reported above, inconsistencies were observed for this element between the produced ECPSSR cross sections and those reported in [35]. The empirical models were subject to comparison in the energy range of their applicability. The two empirical models are equivalent within their respectest they are compatible at 95% tive uncertainties: from the confidence level over 85% of the considered elements, and in 91% of the test cases they are compatible at 99% confidence level. The Kahoul et al. model is compatible at 95% confidence level with the theoretical cross section models in the majority of the test cases (from 71% regarding ECPSSR with United Atom 3632 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 13. K shell proton ionization cross sections, C: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 14. K shell proton ionization cross sections, N: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 12. A comparison of the fluorescence background due to ionization by cosmic-ray protons in an L2 orbit for three different graded Z shield designs for the eROSITA X-ray detectors. (a) Cu shield; (b) Cu-Al shield; (c) shield. correction to 79% regarding ECPSSR with Hartree-Slater correction). test, the Paul and Sacher According to the results of the model appears to be compatible at 95% confidence level with ECPSSR-like cross section models in a small fraction of test cases, approximately 20%. In most cases the null hypothesis test when the sample under test covers is rejected by the the whole energy range of the compilation in [35], but it is not rejected over a large subset of the sample. In the same test cases, the unbinned Kolmogorov-Smirnov test over the Fig. 15. K shell proton ionization cross sections, O: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. whole sample does not reject the null hypothesis; unlike the test, this goodness-of-fit test does not account for uncertainties explicitly. These results suggest that the uncertainties of the Paul and Sacher model may be underestimated in some energy ranges. PIA et al.: PIXE SIMULATION WITH GEANT4 3633 Fig. 16. K shell proton ionization cross sections, F: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 19. K shell proton ionization cross sections, Sm: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 17. K shell proton ionization cross sections, K: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 20. K shell proton ionization cross sections, Tb: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 18. K shell proton ionization cross sections, Nd: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. The inter-comparison of cross section models confirmed the difference of the ECPSSR model with Hartree-Slater correction with respect to the original ECPSSR formulation; the differences are more substantial at higher energies and for heavier elements, as it is also visible qualitatively in Fig. 6. Fig. 21. K shell proton ionization cross sections, Ho: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. As a result of this analysis, one can recommend the ECPSSR model with Hartree-Slater correction for simulations concerning the lower energy range (below 10 MeV approximately), the ECPSSR model in its plain formulation or the empirical Paul and Sacher one (within its range of applicability) otherwise. 3634 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 22. K shell proton ionization cross sections, Tm: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 23. K shell proton ionization cross sections, Ta: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. Fig. 25. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 24. K shell proton ionization cross sections, Th: ECPSSR model (thick line), with Hartree-Slater (dashed), United Atom (dotted) and high energy (thin line) corrections, PWBA (triangles); Paul & Sacher [35] (filled circles) and Kahoul et al [58] (squares) empirical models, experimental data from [35] (empty circles). Shaded bands represent Kahoul et al (horizontal) and Paul and Sacher (vertical) uncertainties. Overlapping curves may not be visible. D. L Shell Ionization Cross Sections The cross sections for L sub-shell ionization cross sections were compared to the experimental data collected in two com- plementary compilations [60], [61]. The same method was applied as described for the validation of K shell cross sections. The experimental compilations report only total L shell cross section data for a few lighter elements; sub-shell data are listed only for elements with atomic number larger than 44. The experimental data often exhibit large discrepancies; systematic effects are clearly visible in some data sets for different elements deriving from the same experimental source. Fig. 11 shows the total L shell ionization cross section for and chlorine calculated by the silicon ECPSSR model compared to experimental data: while the theoretical model appears compatible with the the chlorine data, it PIA et al.: PIXE SIMULATION WITH GEANT4 3635 Fig. 26. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 27. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . shows a large disagrement for silicon. Considering the scarcity of L shell ionization cross section measurements for light elements and the large systematic effects documented in the L shell experimental compilations, it is hard to say whether the observed discrepancy for silicon is due to a deficiency of the theory, or to systematic effects in the measurements. test against experimental data are The p-values from the listed in Tables XI and XII, respectively for the theoretical and empirical cross section models. The fraction of test cases in which the various models are compatible with the experimental data at 95% and 99% confidence level is summarized in Table XIII. The ECPSSR model appears to provide a satisfactory representation of L shell ionisation cross sections with respect to experimental data, especially with its United Atom variant. The ECPSSR-UA model exhibits the best overall accuracy among the various models; it is compatible with experimental measurements at 95% confidence level in 89% of the test cases sub-shell, 79% for the sub-shell and 64% for the for the sub-shell. The degradation of the compatibility with experimental data observed for the sub-shell could be due either to the lower accuracy of the theory or to poorer quality of the experimental data. The Orlic et al. model exhibits the worst accuracy with respect to experimental data among the available theoretical and 3636 Fig. 28. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . empirical models. This semi-empirical model is the only option implemented in Geant4 9.2 for the calculation of L shell ionization cross sections. The accuracy of the various cross section models was studied by means of contingency tables to evaluate their differences quantitatively. The categorical analysis was performed between the ECPSSR model with United Atom correction, i.e. the model showing the best accuracy according to the results of the test, and the other cross section models. The contingency tables were built based on the results of the test at the 95% confidence level, summing the “pass” and “fail” outcome over the three sub-shells; they are reported in Table XIV. IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 29. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . The Orlic et al. semi-empirical model is found to be significantly less accurate than the ECPSSR-UA model: the hypothesis of equivalence of their accuracy with respect to experimental data is rejected at 99% confidence level. The p-values concerning the comparison of the Miyagawa et al. empirical model are close to the critical region for 95% confidence, and slightly different for the three tests performed on the related contingency table. The Sow et al. empirical model and the ECPSSR model in its original formulation appears statistically equivalent in accuracy to the ECPSSR model with United Atom correction. As a result of this analysis, the ECPSSR model with United Atom approximation can be recommended for usage in Geant4based simulation applications as the most accurate option for L PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 30. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . shell ionization cross sections. The ECPSSR model in its original formulation can be considered a satisfactory alternative; the Sow et al. empirical model has satisfactory accuracy, but limited applicability regarding the target elements and proton energies it can handle. E. Cross Section Models for High Energy PIXE PIXE as a technique for elemental analysis is usually performed with proton beams of a few MeV. In the recent years, higher energy proton beams of a few tens MeV have been effectively exploited for PIXE applications; they present various advantages, due to the larger penetration depth of higher energy 3637 Fig. 31. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . protons, and the possibility of exciting K-lines of heavy elements. High energy protons are a source of PIXE in the space radiation environment, which motivates simulation studies to estimate their effects. The interest in high energy PIXE has motivated recent theoretical investigations [56] concerning cross section calculations at higher energies. Despite the emerging interest of high energy PIXE, only a limited set of experimental data is available above the energy range of conventional PIXE techniques. The accuracy of the implemented K shell cross section models was evaluated against two sets of measurements at higher energy [74], [75], respectively at 66 and 68 MeV. The 3638 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 32. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 33. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . experimental measurement with uranium was not included in the comparison, since it appears affected by some experimental systematics. test was performed first separately on either experiThe mental data set to evaluate the possible presence of any systematics in the two test cases, then on the combined data set. The test against these experimental data are p-values from the listed in Table XV. The original ECPSSR formulation is compatible with both data sets at 90% confidence level; the ECPSSR with United Atom correction produces identical results. The ECPSSR model with Hartree-Slater correction is compatible at 95% confidence level only with the data in [74]; regarding the data of [75], its compatibility at 95% confidence level is limited to target materials with atomic number up to 83. The ECPSSR model with the correction in [56], specifically meant for cross section calculations at high energy, is compatible with each one of the experimental data sets at 95% confidence level; nevertheless, it is compatible with the combined data set only at 99% confidence level, while the ECPSSR model in its original formulation is compatible with the whole set of data at 95% confidence level. Over the limited data sample considered in this test, this model does not appear to provide better accuracy than the original ECPSSR formulation; nevertheless more high energy PIA et al.: PIXE SIMULATION WITH GEANT4 3639 Fig. 34. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 35. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . experimental data would be required to reach a firm conclusion. Also, this analysis should be verified over tabulation deriving from a published version of the ISICS code, when it becomes available. test over the experimental data at 160 MeV collected The in [35] results in p-values less than 0.001 for all the ECPSSR model variants. The rejection of the null hypothesis could be ascribed either to the deficiency of the theory or to systematic effects affecting the measurements; further data would be required for a sound assessment. A further test was performed on the experimental data in [76], that concern two target materials (aluminium and copper) rele- vant to the shielding study demonstrated in Section VI and cover energies up to 40 MeV. The p-values from the test varied between 0.999 and 1 for all the ECPSSR modelling variants. While this test did not provide any discriminant power among the various modelling options, it confirmed the validity of the physics models used in the simulation application described in the next section. VI. APPLICATION OF THE PIXE PROTOTYPE SOFTWARE The prototype components for PIXE simulation described in the previous sections were applied to a study of the passive, graded Z shielding of the X-ray detectors of the eROSITA tele- 3640 Fig. 36. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . scope [5] on the upcoming Russian Spectrum-X-Gamma space mission. The purpose of the passive shielding is two-fold. Firstly, the passive shielding prevents abundant low-energy cosmic-ray particles from reaching the detectors and thus from causing radiation damage. Secondly, the passive graded Z shielding serves to reduce instrumental background noise to a minimum [77]. This background noise consists of both fluorescence lines and continuum background due to bremsstrahlung photons and rays from cosmic-ray interactions. The event timing accuracy of current imaging Si detectors for X-ray astronomy (photon energy range 0.1–15 keV) is limited by the signal integration time of these devices. For IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 37. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . such detectors, an active anti-coincidence system cannot be used for background reduction because discarding complete readout frames correlated with an anti-coincidence signal would result in unacceptable dead time. Detector triggers due to primary cosmic-ray particles can be discriminated in imaging detectors due to their high energy deposit and their pixel image pattern. However, interactions of primary cosmic-ray particles in the detector and surrounding passive material give rise to secondary X-rays and charged particles. These may in turn lead to detector triggers within the accepted energy range. Such triggers contribute to the instrumental background noise because they cannot be distinguished from valid events due PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 38. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . to cosmic X-ray photons that were focused by the telescope mirror system onto the detector. The production of secondary photons and particles by the ubiquitous cosmic radiation is inevitable, but graded Z shielding permits the shifting of the energy of secondaries from atomic deexcitation to low values. The probability that an atomic shell vacancy is filled by radiative transitions (emission of fluoresence X-ray photons) decreases with decreasing atomic charge number Z; by contrast, the probability for non-radiative transitions (emission of Auger and Coster-Kronig electrons) increases. The average energy of secondaries from atomic deexcitation decreases with decreasing charge number Z. Therefore, 3641 Fig. 39. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . in a graded Z shield cosmic-ray induced fluorescence X rays produced in an outer, higher Z layer of the shield are absorbed in an inner, lower Z layer. Subsequent atomic deexcitation, for instance following photoionization, in this inner layer gives rise to fluorescence photons and Auger electrons with energies that are lower than the energies of the deexcitation particles from the outer layer; in addition, there will be relatively more deexcitation electrons than X-rays. If the effective charge number Z of the innermost shield layer is sufficiently low, ionization results in the generation of mainly Auger electrons with energies below 1 keV, which can easily be stopped in a thin passivation layer on top of the detectors. Ionization can also create rare fluorescence X rays of similarly low energy. 3642 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 40. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 41. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . A first set of graded Z shield designs was studied by Monte Carlo simulation, using the PIXE prototype software together with Geant4 versions 9.1-patch 01. The detector chip was modthick square slab of pure Si with dimensions elled as a 450 5.6 cm 3.7 cm. The sensitive detector, which is integrated into the chip, covers an area of 2.88 cm 2.88 cm or 384 384 square pixels of size . This detector model was placed inside a box-shaped shield; figures of the actual design can be found in [78]. In its simplest form, the passive shield consisted only of a single 3 cm thick layer of pure Cu; the outer . dimensions of the shield were A second version of the graded Z shield had a 1 mm thick Al layer inside the Cu layer, and a third version in addition a 1 mm layer inside the Al layer. The physics configuration in thick the simulation application involved the library-based processes of the Geant4 low energy electromagnetic package for electrons and photons, along with the improved version of the hadron ionisation process and the specific PIXE software described in Section IV. Among the ionization cross section models documented in Table I, the ECPSSR ones were selected for K, L and M shells. Spectrum-X-Gamma is expected be launched in 2012 into an L2 orbit. The background spectra due to cosmic-ray protons were simulated for the three different eROSITA graded Z shield designs. A model for the detector response, taking into account PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 42. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fano statistics (and hence the energy resolution) and detector noise, was then applied to obtain a simulated data sets. These simulated data were further processed with algorithms that have been developed to eliminate charged particle tracks or invalid pixel image patterns and to select and reconstruct events that are most likely due to single photons [79]; these procedures are identical to those applied to actual laboratory or satellite data. A comparison of the results is depicted in Fig. 12. The spectra represent the average background in a detector pixel. Qualitatively, the PIXE prototype implementation is working properly (see also Fig. 9): protons produce the expected fluorescence lines with correct relative strengths. In case of a pure Cu shield, 3643 Fig. 43. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . and fluorescence lines shown in Fig. 12(a), strong Cu at about 8.0 and 8.9 keV are present in the background spectrum. In case of a combined Cu-Al shield, depicted in Fig. 12(b), the Cu fluorescence lines are absorbed in Al, but ionization in fluorescence line at about 1.5 Al gives rise to a clear Al keV. In case of the full graded Z shield configuration, shown in layer absorbs the Al line, but at the same is Fig. 12(c), the not a source of significant fluorescence lines, which is expected due to the low fluorescence yield of these light elements. Except for an excess below 0.3 keV for the case of the graded Z shield, which appears because the 3644 Fig. 44. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . simulated detector model does not yet include a thin passivation layer, there is no significant difference in the continuum background for the three different designs. The inclusion of a thin layer for the treatment of the low energy background will be the object of a further detector design optimization. This application demonstrates that the developed software is capable of supporting concrete experimental studies. Nevertheless, the concerns outlined in Sections II-B and IV-D should be kept in mind: while the present PIXE simulation component can provide valuable information in terms of relative fluorescence yields from inner shells, the intrinsic limitations of the mixed transport scheme in which ionization is modelled and the lack IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 45. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . of cross section calculations for outer shells prevent an analysis of the simulation outcome in absolute terms. Improvements may come from the ongoing efforts mentioned in Section IV-D. VII. CONCLUSION AND OUTLOOK This paper presents a comprehensive overview of the status, open issues and recent developments of PIXE simulation with the Geant4 toolkit. The analysis of the related models currently available in Geant4 showed their limitations under various aspects: physics functionality, correctness of implementation, accuracy, consistent behavior, software design, performance and usability. PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 46. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . The developments described in this paper represent a significant step forward regarding PIXE simulation with Geant4. They extend the simulation capabilities of the toolkit by enabling the generation of PIXE associated with K, L and M shells for protons and particles; for this purpose a variety of cross section models are provided. Methods to extend the simulation to heavier ion projectiles have been devised and can be easily incorporated in the new design. The adopted data-driven strategy and related software design contribute also a significant improvement in computational performance with respect to previous models available in Geant4. 3645 Fig. 47. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . The validation of the implemented models has been quantitatively estimated with respect to extensive collections of experimental data. The availability of several modelling options for the first time in the same computational environment has allowed their comparative evaluation through rigorous statistical methods. The results documented in this paper provide objective guidance for the optimal selection of simulation models in experimental applications. Experimentalists can further compare their effects in real-life use cases and full-scale Geant4-based simulations. An extensive ionisation cross section data library has been created as a by-product of the development process: it can 3646 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 48. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . Fig. 49. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . be of interest to the experimental community for a variety of applications, not necessarily limited to PIXE simulation with Geant4. Some issues identified in the course of the development process are still open: they concern the consistency of PIXE simulation in a mixed condensed-discrete particle transport scheme, in particular for heavier target materials. A possible approach to address them, while preserving the current continuous-discrete design scheme of hadron ionisation in Geant4, would involve the calculation of ionization cross sections for outer shells than M, similarly to what has been done for electron ionisation. This calculation is feasible, exploiting known theoretical methods, yet it would require a significant investment of resources. Further improvements would also derive from more refined models of hadron ionization, capable of correlating the spectrum of emitted -rays with the determination of the ionized shell. In parallel, a project is in progress to address in a more general way design issues concerning co-working condensed and discrete transport methods in a general purpose simulation system. In this respect, the sound component-based design described in this paper would allow the reuse of the developed PIXE software in connection with other modelling PIA et al.: PIXE SIMULATION WITH GEANT4 Fig. 50. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . 3647 Fig. 51. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . ACKNOWLEDGMENT approaches of hadron ionization or evolutions of particle transport schemes. Despite the known limitations related to mixed transport schemes, the software developments described in this paper provide sufficient functionality for realistic experimental investigations. An application to a concrete use case of physical interest, concerning the optimization of the shielding of the German eROSITA telescope on the upcoming Russian Spectrum-X-Gamma space mission, has demonstrated the capabilities of the new simulation software in response to a real-life experimental problem. The authors express their gratitude to A. Zucchiatti for valuable discussions on PIXE experimental techniques and to S. Cipolla for providing a prototype version of ISICS (ISICS 2008) including updates for high energy K shell cross sections. Also, the authors thank the INFN Genova Computing Service (A. Brunengo, M. Corosu, P. Lantero, and F. Saffioti) for technical support throughout the project, and S. Bertolucci and U. Bratzler for helpful comments and advice regarding this paper. Finally, the authors are grateful to the RSICC staff at ORNL, in particular B. L. Kirk and J. B. Manneschmidt, for the support to assemble a ionization cross section data library for 3648 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 6, DECEMBER 2009 Fig. 52. L sub-shell proton ionization cross sections for : ECPSSR model (solid line) and ECPSSR with United Atom correction (dashed line); empirical models by Miyagawa et al (vertical shaded band), Orlic et al (horizontal shaded band) and Sow et al (solid shaded green band); experimental data from [60] (squares) and [61] (circles). Shaded bands represent model uncertainties. Overlapping curves may not be visible. (a) ; (b) ; (c) . public distribution resulting from the developments described in this paper. REFERENCES [1] T. B. Johansson, R. Akselsson, and S. A. E. Johansson, “X-ray analysis: level,” Nucl. Instrum. Methods, Elemental trace analysis at the vol. 84, no. 1, pp. 141–143, 1970. [2] S. A. E. Johansson, J. L. Campbell, and K. G. Malmqvist, Particle-Induced X-Ray Emission Spectrometry (PIXE). New York: Wiley-Interscience, 1995. [3] S. Agostinelli et al., “Geant4—A simulation toolkit,” Nucl. Instrum. Methods Phys. Res. A, vol. A506, no. 3, pp. 250–303, 2003. [4] J. Allison et al., “Geant4 development and applications,” IEEE Trans. Nucl. Sci., vol. 53, no. 1, pp. 270–278, Feb. 2006. [5] P. Predehl et al., “eROSITA,” in Proc. SPIE, 2007, vol. 6686, pp. 668617–668617-9. [6] P. Murdin, “Spectrum-X gamma,” in Encyclopedia of Astronomy and Astrophysics . Bristol, U.K.: Inst. of Physics, 2001. [7] C. G. Ryan, D. R. Cousens, S. H. Sie, and W. L. Griffin, “Quantitative analysis of PIXE spectra in geoscience applications,” Nucl. Instrum. Methods Phys. Res. B, vol. B49, no. 1–4, pp. 271–276, 1990. [8] J. A. Maxwell, J. L. Campbell, and W. J. Teesdale, “The Guelph PIXE software package,” Nucl. Instrum. Methods Phys. Res. B, vol. B43, no. 2, pp. 218–230, 1989. [9] J. A. Maxwell, W. J. Teesdale, and J. L. Campbell, “The Guelph PIXE software package II,” Nucl. Instrum. Methods Phys. Res. B, vol. B95, no. 3, pp. 407–421, 1995. [10] J. L. Campbell, T. L. Hopman, J. A. Maxwell, and Z. Nejedly, “The Guelph PIXE software package III: Alternative proton database,” Nucl. Instrum. Methods Phys. Res. B, vol. B170, pp. 193–204, 2000. [11] E. Clayton, PIXAN, The Lucas Heights PIXE Analysis Computer Package, Australian Nuclear Science and Technology Organisation, AAEC/M113, 1986. [12] G. Szabo and I. Borbely-Kiss, “PIXYKLM computer package for PIXE analyses,” Nucl. Instrum. Methods Phys. Res. B, vol. B75, no. 1–4, pp. 123–126, 1993. [13] K. Sera and S. Futatsugawa, “Personal computer aided data handling and analysis for PIXE,” Nucl. Instrum. Methods Phys. Res. B, vol. B109–110, pp. 99–104, 1996. [14] B. Vekemans, K. Janssens, L. Vincze, F. Adams, and P. Van Espen, “Analysis of X-ray Spectra by Iterative Least Squares (AXIL): New developments,” X-ray Spectrom., vol. 23, no. 6, pp. 278–285, 1994. [15] A. D. Lipworth, H. J. Annegarn, and M. A. Kneen, “Advanced, enhanced HEX program for PIXE,” Nucl. Instrum. Methods Phys. Res. B, vol. B75, no. 1–4, pp. 127–130, 1993. [16] K. K. Loh, C. H. Sow, I. Orlic, and S. M. Tang, “Computer simulation of PIXE and -PIXE spectra for inhomogeneous thick target analysis,” Nucl. Instrum. Methods Phys. Res. B, vol. B77, no. 1–4, pp. 132–136, 1993. [17] I. Orlic, S. Zhou, J. L. Sanchez, F. Watt, and S. M. Tang, “Virtual PIXE and RBS laboratory,” Nucl. Instrum. Methods Phys. Res. B, vol. B150, no. 1–2, pp. 83–89, 1999. [18] C. Pascual-Izarra, N. P. Barradas, and M. A. Reis, “LibCPIXE: A PIXE simulation open-source library for multilayered samples,” Nucl. Instrum. Methods Phys. Res. B, vol. B249, pp. 820–822, 2006. [19] W. Brandt and G. Lapicki, “Energy-loss effect in inner-shell Coulomb ionization by heavy charged particles,” Phys. Rev. A, vol. 23, p. 1717, 1981. [20] M. Blaauw, J. L. Campbell, S. Fazinic, M. Jaksic, I. Orlic, and P. Van Espen, “The 2000 IAEA intercomparison of PIXE spectrum analysis software,” Nucl. Instrum. Methods Phys. Res. B, vol. B189, pp. 113–122, 2002. [21] Y. Namito and H. Hirayama, “Implementation of electron-impact ionization into the EGS4 code,” Nucl. Instrum. Methods Phys. Res. A, vol. A423, no. 2–3, pp. 238–246, 1999. [22] I. Kawrakow and D. W. O. Rogers, The ENSnrc Code System: Monte Carlo Simulation of Electron and Photon Transport, NRCC Rep. PIRS701, Sep. 2006. [23] H. Hirayama et al., The EGS5 Code System Stanford Linear Accelerator Center, Stanford, CA, Rep. SLAC-R-730, 2006. [24] J. Baro, J. Sempau, J. M. Fernandez-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. B, vol. B100, no. 1, pp. 31–46, 2005. [25] M. J. Berger, “Monte Carlo calculation of the penetration and diffusion of fast charged particles,” in Methods in Computational Physics, B. Alder, S. Fernbach, and M. Rotenberg, Eds. New York: Academic, 1963, vol. 1, pp. 135–215. [26] S. T. Perkins, D. E. Cullen, and S. M. Seltzer, Tables and Graphs of Electron-Interaction Cross Sections from 10 eV to 100 GeV Derived from the LLNL Evaluated Electron Data Library (EEDL), UCRL50400, vol. 31, 1997. [27] F. Salvat, J. M. Fernandez-Varea, and J. Sempau, “PENELOPE—A code system for Monte Carlo simulation of electron and photon transport,” in Proc. Workshop, Issy-les-Moulineaux, France, 2003, 92-6402145, Data Bank. [28] S. Chauvie et al., “Geant4 low energy electromagnetic physics,” in Proc. Computing in High Energy and Nuclear Physics, Beijing, China, 2001, pp. 337–340. [29] S. Chauvie et al., “Geant4 low energy electromagnetic physics,” in Proc. IEEE Nuclear Science Symp. Conf. Rec., 2004, pp. 1881–1885. PIA et al.: PIXE SIMULATION WITH GEANT4 [30] S. Guatelli, A. Mantero, B. Mascialino, P. Nieminen, M. G. Pia, and S. Saliceti, “Geant4 atomic relaxation,” in Proc. IEEE Nuclear Science Symp. Conf. Rec., 2004, pp. 2178–2181. [31] E. Gamma, R. Helm, R. Johnson, and J. Vlissides, Design Patterns. New York: Addison-Wesley, 1995. [32] S. Guatelli et al., “Geant4 atomic relaxation,” IEEE Trans. Nucl. Sci., vol. 54, no. 3, pp. 585–593, Jun. 2007. [33] M. Gryziński, “Two-particle collisions. I. General relations for collisions in the laboratory system,” Phys. Rev., vol. 138, no. 2A, pp. 305–322, 1965. [34] M. Gryziński, “Two-particle collisions. II. Coulomb collisions in the laboratory system of coordinates,” Phys. Rev., vol. 138, no. 2A, pp. 322–335, 1965. [35] H. Paul and J. Sacher, “Fitted empirical reference cross sections for K-shell ionization by protons,” Atom. Data Nucl. Data Tab., vol. 42, pp. 105–156, 1989. [36] S. Saliceti, “Studio della Composizione del Pianeta Mercurio: Modelli per l’Emissione di Fluorescenza e la loro Validazione Sperimentale,” M.Sc. thesis, Univ. Genova, Genova, Italy, 2004. [37] A. Mantero, “Development and Validation of X-Ray Fluorescence Simulation Models for Planetary Astrophysics Investigations,” Ph.D. dissertation, Univ. of Genova, Genova, Italy, 2008. [38] A. Mantero, B. Mascialino, P. Nieminen, M. G. Pia, and S. Saliceti, “Geant4 atomic relaxation models,” in The Monte Carlo Method: Versatility Unbounded in a Dynamic Computing World. LaGrange Park, IL: American Nuclear Society, 2005. [39] H. Paul and O. Bolik, “Fitted empirical reference cross sections for K-shell ionization by alpha particles,” Atom. Data Nucl. Data Tab., vol. 54, pp. 75–131, 1993. [40] “Geant4 Physics Reference Manual” ver. 9.2 [Online]. Available: http://www.cern.ch/geant4/UserDocumentation/UsersGuides/ Physics-ReferenceManual/fo/PhysicsReferenceManual.pdf [41] H. Abdelhwahed, S. Incerti, and A. Mantero, “New Geant4 cross section models for PIXE simulation,” Nucl. Instrum. Methods Phys. Res. B, vol. B267, no. 1, pp. 37–44, 2009. [42] I. Orlic, C. H. Sow, and S. M. Tang, “Semiempirical formulas for calculation of L subshell ionization cross sections,” Int. J. PIXE, vol. 4, no. 4, pp. 217–230, 1994. [43] Z. Liu and S. J. Cipolla, “ISICS: A program for calculating K-, L-, and M-shell cross sections from ECPSSR theory using a personal computer,” Comput. Phys. Commun., vol. 97, pp. 315–330, 1996. [44] S. J. Cipolla, “An improved version of ISICS: A program for calculating K-, L-, and M-shell cross sections from PWBA and ECPSSR theory using a personal computer,” Comput. Phys. Commun., vol. 176, pp. 157–159, 2007. [45] G. Weidenspointner, M. G. Pia, and A. Zoglauer, “Application of the Geant4 PIXE implementation for space missions—New models for PIXE simulation with Geant4,” in Proc. IEEE Nuclear Science Symp. Conf. Rec., 2008, pp. 2877–2884. [46] I. Jakobson, G. Booch, and J. Rumbaugh, The Unified Development Process. Boston, MA: Addison-Wesley, 1999. [47] M. G. Pia et al., “R&D for co-working condensed and discrete transport methods in Geant4 kernel,” in Proc. Int. Conf. Mathematics, Computational Methods and Reactor Physics, La Grange Park, IL, 2009. [48] IEEE Standard for Software Verification and Validation, IEEE Standard 1012-2004, IEEE Computer Society, Jun. 2005. [49] K. Amako et al., “Comparison of Geant4 electromagnetic physics models against the NIST reference data,” IEEE Trans. Nucl. Sci., vol. 52, no. 4, pp. 910–918, Aug. 2005. [50] A. Lechner, M. G. Pia, and M. Sudhakar, “Validation of Geant4 low energy electromagnetic processes against precision measurements of electron energy deposit,” IEEE Trans. Nucl. Sci., vol. 56, no. 2, pp. 398–416, Apr. 2009. [51] G. Booch, J. Rumbaugh, and I. Jakobson, The Unified Modelling Language User Guide. Boston, MA: Addison-Wesley, 1999. [52] S. Giani, V. N. Ivanchenko, G. Mancinelli, P. Nieminen, M. G. Pia, and L. Urban, Geant4 Simulation of Energy Losses of Slow Hadrons, INFN/AE-99/20, Frascati, 1999. [53] M. H. Chen and B. Crasemann, “Relativistic calculation of atomic N-shell ionization by protons,” Phys. Rev. A, vol. 34, pp. 87–92, 1986. [54] S. J. Cipolla, “The united atom approximation option in the ISICS program to calculate K-, L-, and M-shell cross sections from PWBA and ECPSSR theory,” Nucl. Instrum. Methods Phys. Res. B, vol. B261, pp. 142–144, 2007. 3649 [55] G. Lapicki, “The status of theoretical K-shell ionization cross sections by protons,” X-Ray Spectrom., vol. 34, pp. 269–278, 2005. [56] G. Lapicki, “Scaling of analytical cross sections for K-shell ionization by nonrelativistic protons to cross sections by protons at relativistic velocities,” J. Phys. B, vol. 41, p. 115201, 2008, (13pp). [57] S. Cipolla, ISICS, 2008 Version, private communication. [58] A. Kahoul, M. Nekkab, and B. Deghfel, “Empirical K-shell ionization cross-sections of elements from 4Be to 92U by proton impact,” Nucl. Instrum. Methods Phys. Res. B, vol. B266, pp. 4969–4975, 2008. [59] S. Guatelli, A. Mantero, B. Mascialino, M. G. Pia, and V. Zampichelli, “Validation of Geant4 atomic relaxation against the NIST physical reference data,” IEEE Trans. Nucl. Sci., vol. 54, no. 3, pp. 594–603, Jun. 2007. [60] R. S. Sokhi and D. Crumpton, “Experimental L-shell X-ray production and ionization cross sections for proton impact,” Atom. Data Nucl. Data Tab., vol. 30, pp. 49–124, 1984. [61] I. Orlic, J. Sow, and S. M. Tang, “Experimental L-shell X-ray production and ionization cross sections for proton impact,” Atom. Data Nucl. Data Tab., vol. 56, pp. 159–210, 1994. [62] Y. Miyagawa, S. Nakamura, and S. Miyagawa, “Analytical formulas for ionization cross sections and Coster-Kronig corrected fluorescence yields of the Ll, L2, and L3 subshells,” Nucl. Instrum. Methods Phys. Res. B, vol. B30, pp. 115–122, 1988. [63] C. H. Sow, I. Orlic, K. K. Lob, and S. M. Tang, “New parameters for the calculation of L subshell ionization cross sections,” Nucl. Instrum. Methods Phys. Res. B, vol. B75, pp. 58–62, 1993. [64] C. M. Romo-Kroeger, “An empirical curve describing the ionization cross-sections in atomic collisions involving heavy ions,” Nucl. Instrum. Methods Phys. Res. B, vol. B164–165, pp. 349–352, 2000. [65] S. T. Perkins, D. E. Cullen, M. H. Chen, J. H. Hubbell, J. Rathkopf, and J. Scofield, Tables and Graphs of Atomic Subshell and Relaxation Data Derived From the LLNL Evaluated Atomic Data Library (EADL), –100, Lawrence Livermore National Laboratory, Rep. UCRL-50400, 1991, vol. 30. [66] M. G. Pia, P. Saracco, and M. Sudhakar, “Validation of K and L shell radiative transition probability calculations,” IEEE Trans. Nucl. Sci., vol. 56, no. 6, Dec. 2009. [67] A. Lechner, A. Mantero, M. G. Pia, and M. Sudhakar, “Validation of Geant4 X-ray fluorescence transitions—Validation of Geant4 electromagnetic models against calorimetry measurements in the energy range up to 1 MeV,” in Proc. IEEE Nuclear Science Symp. Conf. Rec., 2008. [68] R. K. Bock and W. Krischer, The Data Analysis BriefBook. Berlin, Germany: Springer, 1998. [69] H. Paul and J. Muhr, “Review of experimental cross sections for K-shell ionization by light ions,” Phys. Rep., vol. 135, no. 2, pp. 47–97, 1986. from contingency tables, and [70] R. A. Fisher, “On the interpretation of the calculation of P,” J. Royal Stat. Soc., vol. 85, no. 1, pp. 87–94, 1922. test,” [71] F. Yates, “Contingency table involving small numbers and the J. Royal Stat. Soc. Suppl., vol. 1, pp. 217–235, 1934. test of Goodness of Fit,” Biometrika, vol. 14, [72] K. Pearson, “On the no. 1–2, pp. 186–191, 1922. [73] M. O. Krause, “Atomic radiative and radiationless yields for K and L shells,” J. Phys. Chem. Ref. Data, vol. 8, pp. 307–327, 1979. [74] A. Denker et al., “High-energy PIXE using very energetic protons: Quantitative analysis and cross-sections,” X-Ray Spectrom., vol. 34, no. 4, pp. 376–380, 2005. [75] M. Peisach and C. A. Pineda, “Prompt analysis of heavy elements by high-energy-induced (p, X) reactions,” Nucl. Instrum. Methods Phys. Res. A, vol. A299, no. 1–3, pp. 618–623, 1990. [76] K. Sera et al., “K-shell ionization of Al and Cu for 0.5-40 MeV-proton bombardment,” Phys. Rev. A, vol. 21, pp. 1412–1418, 1980. [77] E. Pfeffermann et al., “Shielding of cosmic ray induced background in CCD detectors for X-ray astronomy,” in Proc. SPIE, 2004, vol. 5501, pp. 304–311. [78] N. Meidinger et al., “eROSITA camera design and first performance measurements with CCDs,” in “Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray,” Proc. SPIE, 2008, vol. 7011, p. 70110J, 10.1117/12.787793. [79] R. Andritschke, G. Hartner, R. Hartmann, N. Meidinger, and L. Strueder, “Data analysis for characterizing pnCCDs,” in Proc. IEEE Nuclear Science Symp., 2008, pp. 2166–2172.