Radiation Protection Dosimetry Advance Access published May 6, 2015 Radiation Protection Dosimetry (2015), pp. 1–5

doi:10.1093/rpd/ncv295

CROSS SECTIONS FOR TRACK STRUCTURE CODES: VOLUME VERSUS SURFACE TRANSPORT M. Dingfelder* and A. Travia Department of Physics, East Carolina University, Mailstop 563, Greenville, NC 27858, USA *Corresponding author: [email protected]

INTRODUCTION Track structure Monte Carlo (MC) simulations are a useful tool to provide detailed information on radiation action on matter under consideration. They successfully connect results obtained in experiments in Radiation Biology with fundamental physical models describing charged particle interactions with biological targets and are of paramount importance for continuing progress in the field(1). Unfortunately, it is extremely difficult to perform experiments with condensed-phase targets of biological relevance such as water and hydrocarbons, which are often used as surrogate for a biological medium(2). Therefore, eventby-event MC simulations of these processes are the most suitable computational method that adequately balances the accuracy, practicability and resolution demanded in Radiation Biology. MC track structure codes depend on reliable interaction cross sections and transport models of the desired radiation quality with the material under consideration. Cross sections for (liquid) water and other cell constituents like DNA bases, proteins and histones are of special interest. They are often calculated within the plane wave first Born approximation (PWBA) or semi-empirical models and are reliable at moderate and high particle velocities or energies. In addition, transport models may provide information about the interaction type (sub-excitation, excitation, ionisation levels, etc.), elastic scattering, secondary electron emission spectra (energies and angles/directions) and other transport or collision-related quantities. The modelling of low-energy electron transport is a challenging subject from both the theoretical and experimental points of view. Theoretically, standard approximations are no longer valid, and corrections need to be applied; often, semi-empirical correction factors are used. Furthermore, the applicability of the classical trajectory simulation method may be questioned at these low energies as they interfere with quantum mechanical principles(3). Similar arguments

can be applied to the use and accuracy of macromolecule cross sections (i.e. DNA components, large target size) in low-energy electron (very small wavelength) scattering. Experimentally, scattering experiments with low-energy projectiles and liquid- or condensed-phase targets are also very challenging(2). However, MC track structure codes can be used to simulate experimental outcomes like secondary electron emission yields from thin foil and frozen gas targets; this procedure allows an evaluation of the transport models and cross-section data used in the simulation. With this in mind, secondary electron emission yields have been measured(2) and simulated for proton impact on solid amorphous ice(4) and thin metal foils(5). These thin metal foils (mainly copper and gold foils of 1 mm thickness) are cooled and used as substrates to freeze thin layers of gases. Electron emission spectra from the clean metal foils are measured before gases are frozen. The continuous change of the emission spectra from the metal foil to the frozen gas target monitors and evaluates the freezing process. Furthermore, metals are conductors, and experimental difficulties with charge built-ups on insulators do not need to be considered. Also, metals as atomic targets are easier to be described theoretically. In general, simulation results for both thin metal foils and frozen gas targets follow experimental data for higher electron energies (typically above 50 eV), while overestimating them by factors up to 10 in the energy range from 1 to 10 eV. Others using different MC track structure codes and cross-section data also obtained similar results(6). There are in principle three possible factors that influence the simulation results: (i) cross-section data, (ii) transport model and (iii) simulation of the experiment. Cross-section data and transport models for most radiation transport codes are based on bulk or volume transport, assuming the transport medium is isotropic and infinitely extended. However, the experimental setup includes a surface; volume transport may not be an adequate description. This communication explores

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Cross-section calculations and transport models for Monte Carlo track structure codes are discussed as well as the simulation of secondary electron emission yields from thin metal foils. Inelastic cross sections for volume (bulk) and surface transport of electrons in copper are presented and implemented into PARTRAC. Simulations for the volume and surface excitation model are presented and analysed.

M. DINGFELDER AND A. TRAVIA

both volume and surface transport models and their influence on the simulated secondary electron emission yields. DIELECTRIC RESPONSE THEORY

ðcqÞ2 ¼ QðQ þ 2me c2 Þ

ð1Þ

where me is the electron mass, and c is the speed of light in vacuum. The DRF is a complex function 1ðE; QÞ ¼ 11 ðE; QÞþi12 ðE; QÞ

ð2Þ

which can be theoretically calculated for simple atomic systems or modelled/measured for more complex or condensed-phase targets. The DRF is related to the optical constants of the material and can be measured in the optical limit, i.e. Q ¼ 0. The DRF is then modelled using a simple energy-momentum relation as an extension algorithm. Currently used models for the DRF of liquid water can be found in Refs. (8 – 11) and a model for metallic copper in Ref. (5).

1(E,Q) is the same DRF as in Equation (2).

CROSS SECTIONS FOR COPPER Interaction cross sections for copper have been calculated within the first Born approximation using the formalism described in Refs. (5, 9). The DRF of copper is modelled by using the optical constants from the literature together with an Ashley-delta oscillator extension algorithm, as described in Ref. (5). Two sets of interaction cross sections have been calculated. One is based on the volume ELF (Equation 3), and the other one is based on the surface ELF (Equation 4). Both data sets represent the extremes of pure volume and pure surface transport. Energy differential inverse mean free paths (DIMFPs) and total inelastic IMFPs for electron impact on copper are shown in Figures 2 and 3. Also shown in Figure 3 is a mixed model (mix): the region close to the surface consists of both and is best described as a linear combination of volume and surface excitations. The

MODELLING FINITE TARGETS Traditional track structure simulation codes only consider radiation transport in bulk material, i.e. in infinitely extended materials. Transport through different materials is performed in the sense that radiation is transported through material one and stopped at the boundary; then transport in material two is continued there as an independent process. Surfaces are completely neglected. For bulk material, the ELF is related to the DRF as follows:   1 ð3Þ h2 ðE; QÞvolume ¼ Im 1ðE; QÞ

Figure 1. Model geometry of the experimental set-up as used for the current MC simulations in PARTRAC.

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The dielectric response theory (DRT) is being used to describe the interactions between an incident charged particle and the many particles in a solid-state or condensed medium. The macroscopic dielectric response method was introduced by Lindhard, Hubbard and Ritchie (see, e.g. Ref. (7)). The theory offers an accurate and practical way to obtain the energy-loss and momentum-transfer profiles of particles involved in this complex perturbation process. The DRT relies on the dielectric response function (DRF) 1 of the medium as a fundamental and complex quantity that embodies the scattering and energy-loss properties of the particles. The energy-loss function (ELF) h2 is derived from the DRF and is proportional to the inverse mean free path (IMFP) of the particles. The IMFP is the key quantity for the MC simulation of the track structure of the particles. The DRF as well as the ELF are functions of the energy transfer E and the recoil energy Q, which is related to the momentum transfer q by

Model geometries describing the experimental set-up more realistic do have also surfaces, as shown in Figure 1. This influences the structure of the material, and with that the dielectric response of the material to a charged particle. Surfaces also give rise to surface excitations and plasmons. A detailed study of surface excitations and the related dielectric formalism can be found in, e.g. Ref. (12). This work follows the formalism of Li, Tung and co-workers (see, e.g. Refs. (13, 14)). In this approach, the ELF for pure surface excitations is defined as follows: ! ð1ðE; QÞ  1Þ2 h2 ; ðE; QÞsurface ¼ Im ð4Þ 1ðE; QÞð1ðE; QÞ þ 1Þ

CROSS SECTIONS FOR TRACK STRUCTURE CODES

Figure 3. Total IMFP of electrons in copper. Three different cross-section data sets are shown.

mixed model parameter is chosen to yield similar DIMFPs for electrons at 500 eV in copper as reported by Li and co-workers(14).

As shown in Figures 2 and 3, DIMFPs and IMFPs for volume and surface excitations for low-energy electrons differ clearly. Surface excitations contribute

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Figure 2. DIMFP of low-energy electrons in copper using the volume (dash) and surface (solid) ELFs.

M. DINGFELDER AND A. TRAVIA

more to the DIMFPs at small energy transfers, while volume excitations become dominant at larger energy transfers. The same trend is seen for the total cross sections: surface IMFPs are larger than volume IMFPs at electron energies below 100 eV and become smaller above 100 eV. The mixed model follows the surface excitations at lower electron energies and the volume excitations at higher ones. SIMULATION RESULTS

Figure 4. Simulated secondary electron emission yields for an emission angle of 408 after 2 MeV proton impact on copper using the track structure code PARTRAC. Experimental data taken from Ref. (2).

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Transport models based on the earlier presented IMFPs and DIMFPs for electron (1 eV to 10 keV) and proton transport (500 keV to 10 MeV) have been implemented into the track structure code PARTRAC. As described in Ref. (5), four subshells are considered. The outermost subshell includes both ionisation and excitation events. Ionisation thresholds were set to 7.7 eV for the volume transport model and to 4.7 eV (the work function of copper) for the surface transport model. For energies above the ionisation thresholds, it is assumed that 30 % of the outermost shell represents excitations, the other 70 % ionisations. Literature(15, 16) indicates that oscillator strengths for excitation levels in atomic copper can account for 60 % or more. Proton transport considers volume excitations only and neglects elastic scattering. Secondary electron emission spectra have been modelled using

angular distributions of the Bethe theory as described in Ref. (17). Electron transport includes exchange contributions but no explicit low-energy corrections. Elastic scattering cross sections have been taken from ICRU report 771(18) for electron energies above 10 eV. Below 10 eV, elastic scattering cross sections are kept constant at the 10 eV values. Scattering angles for primary electrons are determined within the binary collision theory. Secondary electrons are emitted isotropic for small energy transfers, and under 908 relative to the scattered primary electron for larger energy transfers. Secondary electron emission spectra from a thin (1 mm) copper foil after fast proton (2 MeV kinetic energy) impact have been simulated using the geometry shown in Figure 1. Simulations were run with either the volume excitation model or the surface excitation model. Results for a forward emission angle of 408+58 are shown in Figure 4 together with experimental data taken from Ref. (2). As expected from the DIMFPs, secondary electron yields are smaller for the surface excitation model than for the volume excitation model in the energy transfer range from around 5 to 100 eV, but larger in the energy transfer interval from 1 to 5 eV. Above 100 eV, both models coincide. Yields calculated with the mixed model are almost identical to the yields from the volume excitation model (not shown).

CROSS SECTIONS FOR TRACK STRUCTURE CODES

CONCLUSIONS MC track structure codes not only require reliable cross sections for the radiation quality under consideration but also realistic transport models. This holds especially true if interfaces or surfaces are included. Nonetheless, low-energy electron transport is still not well understood and remains a matter of interest, especially in condensed-phase materials. Here, especially elastic scattering, angular distributions and the applicability of the classical transport scheme remain under discussion. REFERENCES

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1. Friedland, W., Dingfelder, M., Kundra´t, P. and Jacob, P. Track structures, DNA targets and radiation effects in the biophysical Monte Carlo simulation code PARTRAC. Mutat. Res. 711, 28– 40 (2011). 2. Toburen, L. H., McLawhorn, S. L., McLawhorn, R. A., Carnes, K. D., Dingfelder, M. and Shinpaugh, J. L. Electron emission from amorphous solid water induced by passage of energetic protons and fluorine ions. Rad. Res. 174, 1– 12 (2010). 3. Liljequist, D. and Nikjoo, H. On the validity of trajectory methods for calculating the transport of very low energy (,1 keV) electrons in liquids and amorphous media. Radiat. Phys. Chem. 99, 45–52 (2014). 4. Dingfelder, M., Travia, A., McLawhorn, R. A., Shinpaugh, J. L. and Toburen, L. H. Electron emission from foils and biological materials after proton impact. Radiat. Phys. Chem. 77, 1213– 1217 (2008). 5. Travia, A. and Dingfelder, M. Simulation of secondary electron yields from thin metal foils after fast proton impact. Rad. Prot. Dosim. 143, 139 (2011). 6. Bug, M. U., Rabus, H. and Rosenfeld, A. Electron emission from amorphous solid water after proton impact: Benchmarking PTa and Geant4 track structure Monte Carlo simulations. Rad. Phys. Chem. 81, 1804–1812 (2012). 7. Hubbard, J. The dielectric theory of electronic interactions in solids. Proc. Phys. Rev. 57, 485 (1955).

8. Dingfelder, M., Ritchie, R. H., Turner, J. E., Friedland, W., Paretzke, H. G. and Hamm, R. N. Comparisons of calculations with PARTRAC and NOREC: transport of electrons in liquid water. Radiat. Res. 169, 584– 594 (2008). 9. Dingfelder, M., Hantke, D., Inokuti, M. and Paretzke, H. G. Electron inelastic scattering cross sections in liquid water. Radiat. Phys. Chem. 53, 1 –18 (1998). 10. Emfietzoglou, D., Cucinotta, F. A. and Nikjoo, H. A complete dielectric response model for liquid water: a solution of the Bethe ridge problem. Radiat. Res. 164, 202–211 (2005). 11. Dingfelder, M. Updated model for dielectric response function of liquid water. Appl. Radiat. Isot. 83, 142– 147 (2014). 12. Salvat-Pujol, F. and Werner, W. S. M. Surface excitations in electron spectroscopy. Part I: dielectric formalism and Monte Carlo algorithm. Surf. Interface Anal. 45, 873–894 (2013). 13. Tung, C. J., Chen, Y. F., Kwei, C. M. and Chou, T. L. Differential cross sections for Plasmon excitations and reflected electron-energy-loss spectra. Phys. Rev. B 49, 684 (1994). 14. Li, Y. C., Tu, Y. H., Kwei, C. M. and Tung, C. J. Influence of the direction of motion on the inelastic interaction between electrons and solid surfaces. Surf. Sci. 589, 67 (2005). 15. Zatsarinny, O., Bartschat, K., Suvorov, V., Teubner, P. J. O. and Brunger, M. J. Electron-impact excitation of the (3d104 s)2S1/2 ! (3d 94s2)2D5/2, 3/2 transitions in copper atoms. Phys. Rev. A. 81, 062705 (2010). 16. Suvorov, V., Teubner, P. J. O., Karaganov, V., Ratnavelu, K., Zhou, Y. and Brunger, M. J. Integral cross sections for electron-impact excitations of the 4 2P state in copper. Phys. Rev. A. 80, 022711 (2009). 17. Dingfelder, M. Cross section calculations in condensed media: charged particles in liquid water. Radiat. Prot. Dosim. 99, 23–28 (2002). 18. International Commission on Radiation Units and Measurements (ICRU). Elastic scattering of electrons and positrons. ICRU Report 77. Oxford University Press (2007).

Cross sections for track structure codes: volume versus surface transport.

Cross-section calculations and transport models for Monte Carlo track structure codes are discussed as well as the simulation of secondary electron em...
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