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Inelastic-collision cross sections for the interactions of totally stripped H, He and C ions with liquid water
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 262 (2007) 1–6 www.elsevier.com/locate/nimb
Inelastic-collision cross sections for the interactions of totally stripped H, He and C ions with liquid water M.A. Bernal *, J.A. Liendo Department of Physics, Simon Bolivar University, P.O. Box 89000, Caracas 1080-A, Venezuela Received 4 April 2007; received in revised form 26 April 2007 Available online 10 May 2007
Abstract The HKS and CDW-EIS models are implemented to determine single ionization cross sections (SICS) corresponding to the impact of H+, He2+ and C6+ ions on liquid water, for incident energies from 0.3 to 10 MeV/u. Corrected expressions for the HKS method have been used. The same kind of initial electron wave functions and binding energies have been used with both methods, in order to compare the formalisms themselves. Double and single differential as well as total SICS of liquid water have been calculated by use of both formalisms and comparisons have been made between their theoretical predictions. Also, these results have been compared with experimental values reported previously for ionization of water vapor due to protons and alpha particles. Despite its sophistication, the CDW-EIS method does not show better results than the more simple HKS model when comparing liquid water single and double differential cross sections with experimental water vapor values. However, once the excitation cross sections are included to determine electronic stopping cross sections in liquid water, the results based on the CDW-EIS method provide the best agreement when compared with corresponding data published in ICRU reports, obtaining discrepancies of about 9%, 16% and 19% for incident protons, alpha particles and carbon ions respectively. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 34.50.Gb; 34.80.Kw Keywords: Ion–molecule collisions; Water ionization cross sections; HKS model; CDW-EIS approximation
1. Introduction The study of the interaction of light ions (Z 6 10) with liquid water is of great interest when biological media are irradiated with these particles. Many specific-purpose Monte Carlo codes have been developed for ion track structure simulation [2–5], mainly for nanodosimetry studies. Most of these codes are based on ionization cross sections (ICS) for water vapor although important phase effects on ICS have been described [6,7].
*
Corresponding author. Present address: School of Physics, Central University of Venezuela, P.O. Box 47586, Caracas 1041, Venezuela. Tel.: +58 414 922 1913; fax: +58 2129063601. E-mail addresses: mbernal@fisica.ciens.ucv.ve (M.A. Bernal), jliendo@ usb.ve (J.A. Liendo). 0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.05.001
In biomedical applications of any kind of ionizing radiation, the secondary electrons play a primordial role, so that the ICS become a very important matter to take into account. Specifically in radiobiology, electron production and transport down to energies of about 10 eV should be accounted for, in order to resolve DNA structures and avoid underestimation of the DNA damage yield. For the same reasons, an event-by-event ion transport simulation is strongly recommended and, consequently, the stopping power data are not enough to carry out this kind of simulation. The irradiation of cancer-patients with light ion beams is carried out in such a way that the Bragg peak and tumor depths match one each other. Near this peak, the ions have energies of the order of a very few tens of MeV/u. Then, if we are interested in studies concerning the biological effectiveness of radiation on target tissues, a non-relativistic
2
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
treatment of cross section calculations is justified, even if high energy ion beams are used. At these energies, most of the phase effects come from changes in electron binding and excitation energies and the Fermi density effect can be safely discarded. Unfortunately, experimental ICS in water due to the impact of ions are very scarce. In fact, the very limited data available are only for water vapor [8–11]. In previous works, calculated water ICS values have been compared with water vapor data [12]. Nevertheless, liquid water stopping powers have been measured for many ions [13,14] and these results have led to the development of semi-empirical methods for building a standard data base for ion stopping powers [15,16]. A great number of methods have been developed to determine SICS corresponding to the impact of light ions on condense media. In this case, phase effects on SICS have been accounted for through the changes in electron binding energies, when comparing liquid and gaseous water [1]. For the heavy particles we are dealing with in this work, many of these formalisms are based on the impact parameter approximation in which the projectile is assumed to be moving along a straight trajectory. This is the case of the HKS [1,17] and CDW-EIS [18] methods. HKS is a relatively simple method whereas CDW-EIS is a sophisticated one. In this work, single and double differential and total ICS have been calculated for H+, He2+ and C6+ projectiles bombarding liquid water with energies ranging from 0.3 to 10 MeV/u, by use of the HKS and CDW-EIS models. It is important to remark that these formalisms are completely theoretical and only the electron binding energies have been taken from experimental works. The cross sections obtained from both methods have been compared to each other and against corresponding experimental values for water vapor. Furthermore, liquid water excitation cross sections for the ions mentioned above have been calculated and combined with the SICS to obtain corresponding electronic stopping cross sections (SCS). The impact of both models on the SCS of liquid water for these projectiles is investigated. The electronic capture process has not been considered in this work because of its negligible impact on SCS at these energies. 2. Methods 2.1. Ionization cross sections The HKS method is based on the impact parameter first Born approximation [19]. It describes the initial and final electron states through a hydrogenic and plane wave functions, respectively. Initially, the HKS model did not take into account the electron momentum in its bound state (peaking approximation) [17] but this parameter was included into this treatment later [20]. Since HKS is based on the first Born approximation, it treats the projectile– electron interaction as a first order perturbation. For these reasons, HKS can be regarded as an impact parameter
Table 1 Oscillator strengths and ionization energies for liquid water sub-shells used in this work Shells
N
(eV)
1b1 3a1 1b2 2a1 1a1
2 2 2 2 2
10.9a 13.5a 17.0a 32.30b 539.0b
a b
Experimental values from Faubel and Steiner [7]. Semi-empirical values from Dingfelder et al. [12].
plane wave Born approximation. However, the CDWEIS approximation accounts for the projectile–electron interaction explicitly, so that it can reproduce the two-center effects [21]. Indeed, CDW-EIS describes the distortion of the electron state due to the interaction with the projectile at both the entrance and exit channels. More details on this approximation can be found in the original work of Crothers and McCann [18]. Although any kind of electron initial wave function can be incorporated into the CDWEIS formalism, a hydrogenic wave function has been used in this work in order to compare the capacities of both methods under similar conditions. This procedure makes possible the study of the potential of each perturbative treatment. Single (SDCS) and double (DDCS) differential as well as total (TCS) single ionization cross sections were calculated using both methods. In the HKS approach, SDCS and DDCS were determined directly by corresponding formulae [1] and TCS were obtained by numerical integration of the appropriate SDCS. To carry out corresponding calculations for the CDW-EIS approximation, a code provided by Crothers et al. has been employed [22]. This code was adapted to fulfill our needs and, in addition, the way in which the exactitude of the iterative integration process is evaluated was changed from an absolute to a relative fashion. Since the CDW-EIS approximation requires the specification of the target nucleus charge as an input parameter, a hydrogenic effective charge obtained from the relation Zeff = (2)1/2 was used, where is the ionization energy of the corresponding liquid water orbital. SDCS were calculated for all liquid water orbitals, weighted by their corresponding oscillator strengths and summed over all the orbitals to obtain the molecular SDCS. Analogue calculations were carried out for DDCS and TCS. Table 1 shows ionization energies and oscillator strengths used in the current investigation for all liquid water orbitals. Values for the three outermost orbitals were taken from experiments [7], while semi-empirical data were used for the others [12]. 2.2. Excitation cross sections A semi-empirical procedure proposed by Green and Stolarski [23] for the determination of the liquid water excitation cross section was used. This method is based
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
and the results have been compared with corresponding values reported by ICRU [15,16].
Table 2 Excitation energies Wn and fitting parameters A, x, c and m for Eq. (1) State
Wn (eV)
A
x
c
m
e 1 B1 A e 1 A1 B Rydberg (A + B) Rydberg (C + D) Diffuse bands Plasmon excitation
8.40 10.10 11.26 11.93 14.10 21.4
0.0302 0.0617 0.0142 0.0590 0.0860 3.7250
0.6537 0.6447 0.6670 0.5796 0.4535 0.7360
1.561 1.537 1.415 1.813 3.000 0.679
3 3 3 3 3 1
on the assumption that the first Born approximation is valid. The cross section for the excitation to the nth state is given by x c m A Wn Wn 1 ; ð1Þ rexc;n ¼ 4pZ 2P 2 T Wn T where Wn is the excitation energy, ZP and T are the projectile charge and reduced kinetic energy respectively, and A, x, c and m are fitting parameters (see Table 2) determined by Kutcher and Green [24] from the results of an experiment where liquid water was irradiated with ultraviolet rays [6]. In this case, collective excitations have been accounted for. 2.3. Stopping cross sections The electronic SCS were calculated from the ionization and excitation cross sections as follows: SCStot ¼ SCSion þ SCSexc X Z Emax X ¼ ErðEÞion;i dE þ W n rexc;n ; i
0
3
n
where E and Emax = 4T are the projectile energy loss and its maximum, respectively. rion,i and rexc,n are the cross sections for the ionization of the ith water orbital and the excitation of the nth state, respectively. SCS have been determined by using both HKS and CDW-EIS formalisms
3. Results and discussion 3.1. Double differential ionization cross sections The predictions of the HKS model were compared in a previous publication with the experimental cross sections used in this work [1]. In this section, the discussion will be focused on the comparison between the CDW-EIS and HKS results and between the CDW-EIS predictions and experimental data. Fig. 1 depicts DDCS for 0.3 and 1.5 MeV protons impacting on liquid water calculated in this work by use of the HKS and CDW-EIS methods. Each curve is for a fixed secondary electron energy, W. Corresponding experimental values for water vapor [8] are shown. Uncertainties of about 20% are included in the W = 12 eV curve for reference purposes. CDW-EIS and HKS use hydrogenic wave functions to describe the electron initial state, which causes an underestimation of DDCS at backward angles [20]. Furthermore, CDW-EIS accounts for the projectile–electron interaction explicitly so that the positive ion attractive effect on the atomic electron tends to reduce the ejection probability of the electron at large angles. This is one of the well known two-center effects [21] and the reason why CDW-EIS underestimates the DDCS at backward angles more than HKS does. The same attraction also causes CDW-EIS to overestimate, for all projectile energies, the DDCS at forward angles with respect to the HKS values when the electron energy is close to WECC, the energy corresponding to the electronic capture into continuum (ECC) peak. This effect is evident for all W values in Fig. 1(a), where the proton velocity is the lowest. However, in Fig. 1(b), this overestimation is only observed for electron energies near WECC since the greater the projectile energy the weaker the influence of the ECC peak over the whole
Fig. 1. DDCS determined by use of the HKS (dashed lines) and CDW-EIS (solid lines) methods for protons bombarding liquid water at (a) 0.3 MeV and (b) 1.5 MeV. Experimental values for water vapor extracted from [8] are included.
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M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
angular distribution. The shift of the CDW-EIS curves binary encounter peaks towards smaller angles is another evidence of the presence of two-center effects. Although CDW-EIS accounts for two-center effects, it is unable to reproduce experimental DDCS near 0°. This behavior could be caused by the use of hydrogenic instead of realistic bound electron wave functions [25]. Neither CDW-EIS nor HKS provide agreements with experimental water vapor DDCS. This lack of accuracy could affect the use of these DDCS in track structure codes designed for nanodosimetry, where the ion transport is a strongly non-homogeneous process. Nevertheless, the authors believe it could have a very little impact on macroscopic dose distribution calculations. DDCS for 0.3 and 0.5 MeV/u alpha particles impinging on liquid water calculated by using the CDW-EIS and HKS models are shown in Fig. 2, together with experimental values reported previously for water vapor [26]. The two-center effects are slightly stronger than in the proton
case. For instance, the shift of the binary encounter peak, the underestimation and overestimation of backward and forward electron emission cross sections respectively, exhibited by CDW-EIS with respect to HKS are more noticeable when the DDCS curves corresponding to W 50 eV and E0 = 0.3 MeV/u (Figs. 1(a) and 2(a)) are compared. This occurs because the alpha particle has a charge twice greater than that of the proton. 3.2. Single differential ionization cross sections Fig. 3 shows SDCS for protons and alpha particles on liquid water, computed by the CDW-EIS and HKS models. Corresponding experimental values previously published for water vapor are included for reference purposes [9,26]. As a general behavior, HKS provides better agreement with the experimental results than CDW-EIS, for protons as well as for alpha particles. The CDW-EIS cross sections are lower than the HKS predictions for all
Fig. 2. DDCS determined by use of the HKS (dashed lines) and CDW-EIS (solid lines) methods for alpha particles bombarding liquid water at (a) 0.3 MeV/u and (b) 0.5 MeV/u. Experimental values for water vapor extracted from [26] are included.
Fig. 3. SDCS for (a) 0.5, 1.5, 3.0 and 4.2 MeV protons and (b) 0.3 and 0.5 MeV/u alpha particles impacting on liquid water calculated by the CDW-EIS (solid lines) and HKS (dashed lines) formalisms. Experimental results corresponding to water vapor are shown (symbols) [9,26].
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
projectile energies probably due to the significant underestimation of the DDCS at large ejection angles, located, in Figs. 1 and 2, to the right of the binary encounter peak. At a projectile energy of 0.5 MeV/u, the CDW-EIS and HKS SDCS are consistent with the experimental data for values of W above 10 eV for a proton beam and for a slightly higher W value for incident alpha particles. This may be due to the fact that both CDW-EIS and HKS methods are first order perturbative approaches, e.g. the greater the projectile charge the worse the expected results. The discrepancies below about 10 eV could have been caused by experimental difficulties to detect such low electron energies, as commented in [9].
5
calculations. It is unfortunate that experimental TCS for alpha particles are limited to an energy as high as 0.4 MeV/u, but a good tendency is shown in this case. 3.4. Excitation cross sections Cross sections for liquid water molecule excitation by protons calculated in this work are shown in Fig. 5 together with similar results reported in the literature from several authors [29,5,30]. Our results are close to those of Paretzke [30]. Proton cross sections have been scaled following the Z 2P rule to obtain corresponding values for He2+ and C6+ projectiles. These cross sections have been combined properly with ICS to find electronic stopping cross sections for the ions treated here.
3.3. Total ionization cross sections 3.5. Electronic stopping cross sections TCS calculated by use of CDW-EIS and HKS approximations for liquid water ionization due to the impact of H+, He2+ and C6+ ions are exhibited in Fig. 4. Experimental results for protons and alpha particles on water vapor are also shown for comparison [10,11,27]. Unfortunately, the authors could not find experimental values for C6+ projectiles. It is important to mention that the experiment of Schutten et al. [27] was carried out with electrons as incident particles and the ion energy was scaled to match the electron velocity [9]. This procedure has been justified by Hooper et al. [28] for protons with energies above 0.5 MeV. However, we believe that the works of Rudd and Goffe [10] and Rudd et al. [11] are more reliable. The CDW-EIS model provides the best agreement with experiments of protons at low impact energies, as expected since this model takes into account the distortion of electron states in the entrance and exit channels. For energies close to 10 MeV/u, the results obtained from the CDW-EIS and HKS models and experiments tend to converge all together within the combined uncertainty of experiments and
Fig. 4. TCS for liquid water ionization caused by the irradiation with H+, He2+ and C6+ ions, calculated by using the CDW-EIS (solid lines) and HKS (dash lines) formalisms. Experimental TCS values obtained in previous investigations for protons and alpha particles are also shown (symbols) [10,11,27].
The impressive agreement between stopping cross sections calculated according to our methodology, based on the CDW-EIS model, for all projectiles, and those extracted from ICRU reports [15,16] can be observed in Fig. 6, for the complete projectile energy interval. CDWEIS-based SCS are consistent with ICRU values within about 9%, 16% and 19% for incident protons, alpha particles and carbon ions respectively. For carbon ions, the results for incident energies below about 1 MeV are not so good as for lighter projectiles, probably due to the failure of the first Born approximation to determine excitation cross sections for such a high projectile charge. The SCS results based on the HKS approximation show a systematic overestimation with respect to the values reported by ICRU. As expected, the greater the ion charge, the worse the agreement between theoretical calculations and reference values.
Fig. 5. Excitation cross section for liquid water molecule due to proton impact determined in this wok (symbols). Calculations from Uehara et al. [29] (solid line), Emfietzoglou et al. [5] (dashed line) and Paretzke [30] (dotted line) are included for comparison.
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M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
Fig. 6. Electronic stopping cross sections for H+, He2+ and C6+ ions impacting on liquid water calculated by the CDW-EIS (solid lines) and HKS (dashed lines) formalisms. Corresponding reference values from ICRU reports are also shown (symbols) [15,16].
4. Conclusions Both the CDW-EIS and HKS methods, using hydrogenic initial wave functions, generate secondary electron angular distributions for the ionization of liquid water which are not consistent with the available experimental data corresponding to water vapor. The HKS approximation produces more consistent single differential ionization cross sections values than the CDW-EIS method when compared with experimental results reported previously for water vapor. Nevertheless, electronic stopping cross sections obtained from the CDW-EIS and the Kutcher and Green formalisms are consistent with the values published by ICRU reports for liquid water [15,16]. Therefore, the authors recommend the use of the latter combination to obtain basic data for Monte Carlo track structure codes in non-relativistic ion transport. References [1] M.A. Bernal, J.A. Liendo, The HKS model for electron production in liquid water by light ions, Nucl. Instr. and Meth. B 251 (2006) 171. [2] M. Zaider, D.J. Brenner, W.E. Wilson, The application of track calculations to radiobiology. Monte Carlo simulation of proton tracks, Rad. Res. 95 (1983) 231. [3] P.A. Aarnio, J. Lindgren, J. Ranft, A. Fasso, G.R. Stevenson, Enhancements to the Fluka 86 program (Fluka 87), report TIS-RP/ 190, European Organization for Nuclear Research, 1987. [4] W.E. Wilson, J.H. Miller, H. Nikjoo, Computational Approaches in Molecular Radiation Biology, Plenum, New York, 1994, Ch. PITS: A code set for positive ion track structure. [5] D. Emfietzoglou, G. Papamichael, K. Kostarelos, M.D. Moscovitch, A Monte Carlo track structure code for electrons (10 eV–10 keV) and protons (0.3–10 MeV) in water: partitioning of energy and collision events, Phys. Med. Biol. 45 (2000) 3171. [6] J.M. Heller, R.N. Hamm, R.D. Birchoff, L.R. Painter, Collective oscillation in liquid water, J. Chem. Phys. 60 (1976) 3483.
[7] M. Faubel, B. Steiner, in: Linking the gaseous and condensed phases of matter, Nato ASI Series, Vol. B326, Plenum Press, New York, 1994. [8] L.H. Toburen, W.E. Wilson, Energy and angular distributions of electrons ejected from water vapour by 0.3-1.5 MeV protons, J. Chem. Phys. 66 (1977) 5202. [9] W.E. Wilson, J.H. Miller, L.H. Toburen, S.T. Manson, Differential cross sections for ionization of methane, ammonia and water vapour by high velocity ions, J. Chem. Phys. 80 (1984) 5601. [10] M.E. Rudd, T.V. Goffe, Cross sections for ionization of water vapor by 7–4000 keV protons, Phys. Rev. A 31 (1) (1985) 492. [11] M.E. Rudd, A. Itoh, T.V. Goffe, Cross sections for ionization, capture, and loss for 5–450 keV He+ on water vapor, Phys. Rev. A 32 (4) (1985) 2499. [12] M. Dingfelder, M. Inokuti, H. Paretzke, Inelastic-collision cross sections of liquid water for interactions of energetic protons, Radiat. Phys. Chem. 59 (2000) 255. [13] M. McInally, The differential stopping power of liquid water for alpha particles, Proc. Roy. Soc. Lon. A 237 (1208) (1956) 28. [14] D.I. Twaites, Stopping cross sections of liquid water and water vapour for alpha particles within the energy region 0.3 to 5.5 MeV, Phys. Med. Biol. 26 (1981) 71. [15] ICRU49, Stopping powers and ranges for protons and alpha particles, Tech. Rep., International Commission on Radiation Units and Measurements, Bethesda, MD, 1993. [16] Stopping of ions heavier than helium., Tech. Rep. 73, International Commission on Radiation Units and Measurements, Bethesda, MD, 2005. [17] J.P. Hansen, L. Kocbach, Ejection angles of fast delta electrons from K-shell ionisation induced by energetic ions, J. Phys. B 22 (1989) L71. [18] D. Crothers, J. McCann, Ionisations of atoms by ion impact, J. Phys. B 16 (1983) 3229. [19] M. McDowell, J. Coleman, Introduction to the Theory of Ion-Atoms Collisions, North-Holland, Amsterdam, 1970. [20] N. Stolterfoht, R. DuBois, R.D. Rivarola, Electron Emission in Heavy Ion-atom Collisions, Springer, Berlin, 1997. [21] P.D. Fainstein, V.H. Ponce, R.D. Rivarola, Two-centre effects in ionization by ion impact, J. Phys. B 24 (1991) 3091. [22] S.F.C. O’Rourke, D.M. McSherry, D.S.F. Crothers, Ion-atom/neon – calculation of ionization cross sections by fast ion impact for neutral target atoms ranging from lithium to neon, Comput. Phys. Commun. 131 (1–2) (2000) 129. [23] A.E.S. Green, R.S. Stolarski, Analytic models of electron impact excitation cross sections, J. Atmos. Terr. Phys. 34 (1972) 1703. [24] G.J. Kutcher, A.E.S. Green, A model for energy deposition in liquid water, Rad. Res. 67 (1976) 408. [25] D.H. Madison, Electron capture in slow collisions, Proc. Roy. Soc. A 245 (1973) 175. [26] L.H. Toburen, W.E. Wilson, R.J. Popovich, Secondary electron emission from ionization of water vapor by 0.3 to 2.0 MeV He+ and He2+ ions, Rad. Res. 82 (1980) 27. [27] J. Schutten, F.J. DeHeer, H.R. Moustafa, A.J.H. Boerboom, J. Kistemaker, Gross and partial ionization cross sections for electrons on water vapor in the energy range 0.1–20 keV, J. Chem. Phys. 44 (1966) 3924. [28] J.W. Hooper, D.S. Harmer, D.W. Martin, E.W. McDaniel, Comparison of electron and proton ionization data with the Born approximation predictions, Phys. Rev. 125 (6) (1962) 2000. [29] S. Uehara, H. Nikjoo, D.T. Goodhead, Cross sections for ionization of water vapor for the Monte Carlo electron track structure code from 10 eV to the MeV region, Phys. Med. Biol. 37 (1992) 1841. [30] H.G. Paretzke, Radiation Track Structure Theory, Kinetic on Nonhomogeneous Processes, Wiley, New York, 1987.
Nuclear Instruments and Methods in Physics Research B 262 (2007) 1–6 www.elsevier.com/locate/nimb
Inelastic-collision cross sections for the interactions of totally stripped H, He and C ions with liquid water M.A. Bernal *, J.A. Liendo Department of Physics, Simon Bolivar University, P.O. Box 89000, Caracas 1080-A, Venezuela Received 4 April 2007; received in revised form 26 April 2007 Available online 10 May 2007
Abstract The HKS and CDW-EIS models are implemented to determine single ionization cross sections (SICS) corresponding to the impact of H+, He2+ and C6+ ions on liquid water, for incident energies from 0.3 to 10 MeV/u. Corrected expressions for the HKS method have been used. The same kind of initial electron wave functions and binding energies have been used with both methods, in order to compare the formalisms themselves. Double and single differential as well as total SICS of liquid water have been calculated by use of both formalisms and comparisons have been made between their theoretical predictions. Also, these results have been compared with experimental values reported previously for ionization of water vapor due to protons and alpha particles. Despite its sophistication, the CDW-EIS method does not show better results than the more simple HKS model when comparing liquid water single and double differential cross sections with experimental water vapor values. However, once the excitation cross sections are included to determine electronic stopping cross sections in liquid water, the results based on the CDW-EIS method provide the best agreement when compared with corresponding data published in ICRU reports, obtaining discrepancies of about 9%, 16% and 19% for incident protons, alpha particles and carbon ions respectively. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 34.50.Gb; 34.80.Kw Keywords: Ion–molecule collisions; Water ionization cross sections; HKS model; CDW-EIS approximation
1. Introduction The study of the interaction of light ions (Z 6 10) with liquid water is of great interest when biological media are irradiated with these particles. Many specific-purpose Monte Carlo codes have been developed for ion track structure simulation [2–5], mainly for nanodosimetry studies. Most of these codes are based on ionization cross sections (ICS) for water vapor although important phase effects on ICS have been described [6,7].
*
Corresponding author. Present address: School of Physics, Central University of Venezuela, P.O. Box 47586, Caracas 1041, Venezuela. Tel.: +58 414 922 1913; fax: +58 2129063601. E-mail addresses: mbernal@fisica.ciens.ucv.ve (M.A. Bernal), jliendo@ usb.ve (J.A. Liendo). 0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.05.001
In biomedical applications of any kind of ionizing radiation, the secondary electrons play a primordial role, so that the ICS become a very important matter to take into account. Specifically in radiobiology, electron production and transport down to energies of about 10 eV should be accounted for, in order to resolve DNA structures and avoid underestimation of the DNA damage yield. For the same reasons, an event-by-event ion transport simulation is strongly recommended and, consequently, the stopping power data are not enough to carry out this kind of simulation. The irradiation of cancer-patients with light ion beams is carried out in such a way that the Bragg peak and tumor depths match one each other. Near this peak, the ions have energies of the order of a very few tens of MeV/u. Then, if we are interested in studies concerning the biological effectiveness of radiation on target tissues, a non-relativistic
2
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
treatment of cross section calculations is justified, even if high energy ion beams are used. At these energies, most of the phase effects come from changes in electron binding and excitation energies and the Fermi density effect can be safely discarded. Unfortunately, experimental ICS in water due to the impact of ions are very scarce. In fact, the very limited data available are only for water vapor [8–11]. In previous works, calculated water ICS values have been compared with water vapor data [12]. Nevertheless, liquid water stopping powers have been measured for many ions [13,14] and these results have led to the development of semi-empirical methods for building a standard data base for ion stopping powers [15,16]. A great number of methods have been developed to determine SICS corresponding to the impact of light ions on condense media. In this case, phase effects on SICS have been accounted for through the changes in electron binding energies, when comparing liquid and gaseous water [1]. For the heavy particles we are dealing with in this work, many of these formalisms are based on the impact parameter approximation in which the projectile is assumed to be moving along a straight trajectory. This is the case of the HKS [1,17] and CDW-EIS [18] methods. HKS is a relatively simple method whereas CDW-EIS is a sophisticated one. In this work, single and double differential and total ICS have been calculated for H+, He2+ and C6+ projectiles bombarding liquid water with energies ranging from 0.3 to 10 MeV/u, by use of the HKS and CDW-EIS models. It is important to remark that these formalisms are completely theoretical and only the electron binding energies have been taken from experimental works. The cross sections obtained from both methods have been compared to each other and against corresponding experimental values for water vapor. Furthermore, liquid water excitation cross sections for the ions mentioned above have been calculated and combined with the SICS to obtain corresponding electronic stopping cross sections (SCS). The impact of both models on the SCS of liquid water for these projectiles is investigated. The electronic capture process has not been considered in this work because of its negligible impact on SCS at these energies. 2. Methods 2.1. Ionization cross sections The HKS method is based on the impact parameter first Born approximation [19]. It describes the initial and final electron states through a hydrogenic and plane wave functions, respectively. Initially, the HKS model did not take into account the electron momentum in its bound state (peaking approximation) [17] but this parameter was included into this treatment later [20]. Since HKS is based on the first Born approximation, it treats the projectile– electron interaction as a first order perturbation. For these reasons, HKS can be regarded as an impact parameter
Table 1 Oscillator strengths and ionization energies for liquid water sub-shells used in this work Shells
N
(eV)
1b1 3a1 1b2 2a1 1a1
2 2 2 2 2
10.9a 13.5a 17.0a 32.30b 539.0b
a b
Experimental values from Faubel and Steiner [7]. Semi-empirical values from Dingfelder et al. [12].
plane wave Born approximation. However, the CDWEIS approximation accounts for the projectile–electron interaction explicitly, so that it can reproduce the two-center effects [21]. Indeed, CDW-EIS describes the distortion of the electron state due to the interaction with the projectile at both the entrance and exit channels. More details on this approximation can be found in the original work of Crothers and McCann [18]. Although any kind of electron initial wave function can be incorporated into the CDWEIS formalism, a hydrogenic wave function has been used in this work in order to compare the capacities of both methods under similar conditions. This procedure makes possible the study of the potential of each perturbative treatment. Single (SDCS) and double (DDCS) differential as well as total (TCS) single ionization cross sections were calculated using both methods. In the HKS approach, SDCS and DDCS were determined directly by corresponding formulae [1] and TCS were obtained by numerical integration of the appropriate SDCS. To carry out corresponding calculations for the CDW-EIS approximation, a code provided by Crothers et al. has been employed [22]. This code was adapted to fulfill our needs and, in addition, the way in which the exactitude of the iterative integration process is evaluated was changed from an absolute to a relative fashion. Since the CDW-EIS approximation requires the specification of the target nucleus charge as an input parameter, a hydrogenic effective charge obtained from the relation Zeff = (2)1/2 was used, where is the ionization energy of the corresponding liquid water orbital. SDCS were calculated for all liquid water orbitals, weighted by their corresponding oscillator strengths and summed over all the orbitals to obtain the molecular SDCS. Analogue calculations were carried out for DDCS and TCS. Table 1 shows ionization energies and oscillator strengths used in the current investigation for all liquid water orbitals. Values for the three outermost orbitals were taken from experiments [7], while semi-empirical data were used for the others [12]. 2.2. Excitation cross sections A semi-empirical procedure proposed by Green and Stolarski [23] for the determination of the liquid water excitation cross section was used. This method is based
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
and the results have been compared with corresponding values reported by ICRU [15,16].
Table 2 Excitation energies Wn and fitting parameters A, x, c and m for Eq. (1) State
Wn (eV)
A
x
c
m
e 1 B1 A e 1 A1 B Rydberg (A + B) Rydberg (C + D) Diffuse bands Plasmon excitation
8.40 10.10 11.26 11.93 14.10 21.4
0.0302 0.0617 0.0142 0.0590 0.0860 3.7250
0.6537 0.6447 0.6670 0.5796 0.4535 0.7360
1.561 1.537 1.415 1.813 3.000 0.679
3 3 3 3 3 1
on the assumption that the first Born approximation is valid. The cross section for the excitation to the nth state is given by x c m A Wn Wn 1 ; ð1Þ rexc;n ¼ 4pZ 2P 2 T Wn T where Wn is the excitation energy, ZP and T are the projectile charge and reduced kinetic energy respectively, and A, x, c and m are fitting parameters (see Table 2) determined by Kutcher and Green [24] from the results of an experiment where liquid water was irradiated with ultraviolet rays [6]. In this case, collective excitations have been accounted for. 2.3. Stopping cross sections The electronic SCS were calculated from the ionization and excitation cross sections as follows: SCStot ¼ SCSion þ SCSexc X Z Emax X ¼ ErðEÞion;i dE þ W n rexc;n ; i
0
3
n
where E and Emax = 4T are the projectile energy loss and its maximum, respectively. rion,i and rexc,n are the cross sections for the ionization of the ith water orbital and the excitation of the nth state, respectively. SCS have been determined by using both HKS and CDW-EIS formalisms
3. Results and discussion 3.1. Double differential ionization cross sections The predictions of the HKS model were compared in a previous publication with the experimental cross sections used in this work [1]. In this section, the discussion will be focused on the comparison between the CDW-EIS and HKS results and between the CDW-EIS predictions and experimental data. Fig. 1 depicts DDCS for 0.3 and 1.5 MeV protons impacting on liquid water calculated in this work by use of the HKS and CDW-EIS methods. Each curve is for a fixed secondary electron energy, W. Corresponding experimental values for water vapor [8] are shown. Uncertainties of about 20% are included in the W = 12 eV curve for reference purposes. CDW-EIS and HKS use hydrogenic wave functions to describe the electron initial state, which causes an underestimation of DDCS at backward angles [20]. Furthermore, CDW-EIS accounts for the projectile–electron interaction explicitly so that the positive ion attractive effect on the atomic electron tends to reduce the ejection probability of the electron at large angles. This is one of the well known two-center effects [21] and the reason why CDW-EIS underestimates the DDCS at backward angles more than HKS does. The same attraction also causes CDW-EIS to overestimate, for all projectile energies, the DDCS at forward angles with respect to the HKS values when the electron energy is close to WECC, the energy corresponding to the electronic capture into continuum (ECC) peak. This effect is evident for all W values in Fig. 1(a), where the proton velocity is the lowest. However, in Fig. 1(b), this overestimation is only observed for electron energies near WECC since the greater the projectile energy the weaker the influence of the ECC peak over the whole
Fig. 1. DDCS determined by use of the HKS (dashed lines) and CDW-EIS (solid lines) methods for protons bombarding liquid water at (a) 0.3 MeV and (b) 1.5 MeV. Experimental values for water vapor extracted from [8] are included.
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angular distribution. The shift of the CDW-EIS curves binary encounter peaks towards smaller angles is another evidence of the presence of two-center effects. Although CDW-EIS accounts for two-center effects, it is unable to reproduce experimental DDCS near 0°. This behavior could be caused by the use of hydrogenic instead of realistic bound electron wave functions [25]. Neither CDW-EIS nor HKS provide agreements with experimental water vapor DDCS. This lack of accuracy could affect the use of these DDCS in track structure codes designed for nanodosimetry, where the ion transport is a strongly non-homogeneous process. Nevertheless, the authors believe it could have a very little impact on macroscopic dose distribution calculations. DDCS for 0.3 and 0.5 MeV/u alpha particles impinging on liquid water calculated by using the CDW-EIS and HKS models are shown in Fig. 2, together with experimental values reported previously for water vapor [26]. The two-center effects are slightly stronger than in the proton
case. For instance, the shift of the binary encounter peak, the underestimation and overestimation of backward and forward electron emission cross sections respectively, exhibited by CDW-EIS with respect to HKS are more noticeable when the DDCS curves corresponding to W 50 eV and E0 = 0.3 MeV/u (Figs. 1(a) and 2(a)) are compared. This occurs because the alpha particle has a charge twice greater than that of the proton. 3.2. Single differential ionization cross sections Fig. 3 shows SDCS for protons and alpha particles on liquid water, computed by the CDW-EIS and HKS models. Corresponding experimental values previously published for water vapor are included for reference purposes [9,26]. As a general behavior, HKS provides better agreement with the experimental results than CDW-EIS, for protons as well as for alpha particles. The CDW-EIS cross sections are lower than the HKS predictions for all
Fig. 2. DDCS determined by use of the HKS (dashed lines) and CDW-EIS (solid lines) methods for alpha particles bombarding liquid water at (a) 0.3 MeV/u and (b) 0.5 MeV/u. Experimental values for water vapor extracted from [26] are included.
Fig. 3. SDCS for (a) 0.5, 1.5, 3.0 and 4.2 MeV protons and (b) 0.3 and 0.5 MeV/u alpha particles impacting on liquid water calculated by the CDW-EIS (solid lines) and HKS (dashed lines) formalisms. Experimental results corresponding to water vapor are shown (symbols) [9,26].
M.A. Bernal, J.A. Liendo / Nucl. Instr. and Meth. in Phys. Res. B 262 (2007) 1–6
projectile energies probably due to the significant underestimation of the DDCS at large ejection angles, located, in Figs. 1 and 2, to the right of the binary encounter peak. At a projectile energy of 0.5 MeV/u, the CDW-EIS and HKS SDCS are consistent with the experimental data for values of W above 10 eV for a proton beam and for a slightly higher W value for incident alpha particles. This may be due to the fact that both CDW-EIS and HKS methods are first order perturbative approaches, e.g. the greater the projectile charge the worse the expected results. The discrepancies below about 10 eV could have been caused by experimental difficulties to detect such low electron energies, as commented in [9].
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calculations. It is unfortunate that experimental TCS for alpha particles are limited to an energy as high as 0.4 MeV/u, but a good tendency is shown in this case. 3.4. Excitation cross sections Cross sections for liquid water molecule excitation by protons calculated in this work are shown in Fig. 5 together with similar results reported in the literature from several authors [29,5,30]. Our results are close to those of Paretzke [30]. Proton cross sections have been scaled following the Z 2P rule to obtain corresponding values for He2+ and C6+ projectiles. These cross sections have been combined properly with ICS to find electronic stopping cross sections for the ions treated here.
3.3. Total ionization cross sections 3.5. Electronic stopping cross sections TCS calculated by use of CDW-EIS and HKS approximations for liquid water ionization due to the impact of H+, He2+ and C6+ ions are exhibited in Fig. 4. Experimental results for protons and alpha particles on water vapor are also shown for comparison [10,11,27]. Unfortunately, the authors could not find experimental values for C6+ projectiles. It is important to mention that the experiment of Schutten et al. [27] was carried out with electrons as incident particles and the ion energy was scaled to match the electron velocity [9]. This procedure has been justified by Hooper et al. [28] for protons with energies above 0.5 MeV. However, we believe that the works of Rudd and Goffe [10] and Rudd et al. [11] are more reliable. The CDW-EIS model provides the best agreement with experiments of protons at low impact energies, as expected since this model takes into account the distortion of electron states in the entrance and exit channels. For energies close to 10 MeV/u, the results obtained from the CDW-EIS and HKS models and experiments tend to converge all together within the combined uncertainty of experiments and
Fig. 4. TCS for liquid water ionization caused by the irradiation with H+, He2+ and C6+ ions, calculated by using the CDW-EIS (solid lines) and HKS (dash lines) formalisms. Experimental TCS values obtained in previous investigations for protons and alpha particles are also shown (symbols) [10,11,27].
The impressive agreement between stopping cross sections calculated according to our methodology, based on the CDW-EIS model, for all projectiles, and those extracted from ICRU reports [15,16] can be observed in Fig. 6, for the complete projectile energy interval. CDWEIS-based SCS are consistent with ICRU values within about 9%, 16% and 19% for incident protons, alpha particles and carbon ions respectively. For carbon ions, the results for incident energies below about 1 MeV are not so good as for lighter projectiles, probably due to the failure of the first Born approximation to determine excitation cross sections for such a high projectile charge. The SCS results based on the HKS approximation show a systematic overestimation with respect to the values reported by ICRU. As expected, the greater the ion charge, the worse the agreement between theoretical calculations and reference values.
Fig. 5. Excitation cross section for liquid water molecule due to proton impact determined in this wok (symbols). Calculations from Uehara et al. [29] (solid line), Emfietzoglou et al. [5] (dashed line) and Paretzke [30] (dotted line) are included for comparison.
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Fig. 6. Electronic stopping cross sections for H+, He2+ and C6+ ions impacting on liquid water calculated by the CDW-EIS (solid lines) and HKS (dashed lines) formalisms. Corresponding reference values from ICRU reports are also shown (symbols) [15,16].
4. Conclusions Both the CDW-EIS and HKS methods, using hydrogenic initial wave functions, generate secondary electron angular distributions for the ionization of liquid water which are not consistent with the available experimental data corresponding to water vapor. The HKS approximation produces more consistent single differential ionization cross sections values than the CDW-EIS method when compared with experimental results reported previously for water vapor. Nevertheless, electronic stopping cross sections obtained from the CDW-EIS and the Kutcher and Green formalisms are consistent with the values published by ICRU reports for liquid water [15,16]. Therefore, the authors recommend the use of the latter combination to obtain basic data for Monte Carlo track structure codes in non-relativistic ion transport. References [1] M.A. Bernal, J.A. Liendo, The HKS model for electron production in liquid water by light ions, Nucl. Instr. and Meth. B 251 (2006) 171. [2] M. Zaider, D.J. Brenner, W.E. Wilson, The application of track calculations to radiobiology. Monte Carlo simulation of proton tracks, Rad. Res. 95 (1983) 231. [3] P.A. Aarnio, J. Lindgren, J. Ranft, A. Fasso, G.R. Stevenson, Enhancements to the Fluka 86 program (Fluka 87), report TIS-RP/ 190, European Organization for Nuclear Research, 1987. [4] W.E. Wilson, J.H. Miller, H. Nikjoo, Computational Approaches in Molecular Radiation Biology, Plenum, New York, 1994, Ch. PITS: A code set for positive ion track structure. [5] D. Emfietzoglou, G. Papamichael, K. Kostarelos, M.D. Moscovitch, A Monte Carlo track structure code for electrons (10 eV–10 keV) and protons (0.3–10 MeV) in water: partitioning of energy and collision events, Phys. Med. Biol. 45 (2000) 3171. [6] J.M. Heller, R.N. Hamm, R.D. Birchoff, L.R. Painter, Collective oscillation in liquid water, J. Chem. Phys. 60 (1976) 3483.
[7] M. Faubel, B. Steiner, in: Linking the gaseous and condensed phases of matter, Nato ASI Series, Vol. B326, Plenum Press, New York, 1994. [8] L.H. Toburen, W.E. Wilson, Energy and angular distributions of electrons ejected from water vapour by 0.3-1.5 MeV protons, J. Chem. Phys. 66 (1977) 5202. [9] W.E. Wilson, J.H. Miller, L.H. Toburen, S.T. Manson, Differential cross sections for ionization of methane, ammonia and water vapour by high velocity ions, J. Chem. Phys. 80 (1984) 5601. [10] M.E. Rudd, T.V. Goffe, Cross sections for ionization of water vapor by 7–4000 keV protons, Phys. Rev. A 31 (1) (1985) 492. [11] M.E. Rudd, A. Itoh, T.V. Goffe, Cross sections for ionization, capture, and loss for 5–450 keV He+ on water vapor, Phys. Rev. A 32 (4) (1985) 2499. [12] M. Dingfelder, M. Inokuti, H. Paretzke, Inelastic-collision cross sections of liquid water for interactions of energetic protons, Radiat. Phys. Chem. 59 (2000) 255. [13] M. McInally, The differential stopping power of liquid water for alpha particles, Proc. Roy. Soc. Lon. A 237 (1208) (1956) 28. [14] D.I. Twaites, Stopping cross sections of liquid water and water vapour for alpha particles within the energy region 0.3 to 5.5 MeV, Phys. Med. Biol. 26 (1981) 71. [15] ICRU49, Stopping powers and ranges for protons and alpha particles, Tech. Rep., International Commission on Radiation Units and Measurements, Bethesda, MD, 1993. [16] Stopping of ions heavier than helium., Tech. Rep. 73, International Commission on Radiation Units and Measurements, Bethesda, MD, 2005. [17] J.P. Hansen, L. Kocbach, Ejection angles of fast delta electrons from K-shell ionisation induced by energetic ions, J. Phys. B 22 (1989) L71. [18] D. Crothers, J. McCann, Ionisations of atoms by ion impact, J. Phys. B 16 (1983) 3229. [19] M. McDowell, J. Coleman, Introduction to the Theory of Ion-Atoms Collisions, North-Holland, Amsterdam, 1970. [20] N. Stolterfoht, R. DuBois, R.D. Rivarola, Electron Emission in Heavy Ion-atom Collisions, Springer, Berlin, 1997. [21] P.D. Fainstein, V.H. Ponce, R.D. Rivarola, Two-centre effects in ionization by ion impact, J. Phys. B 24 (1991) 3091. [22] S.F.C. O’Rourke, D.M. McSherry, D.S.F. Crothers, Ion-atom/neon – calculation of ionization cross sections by fast ion impact for neutral target atoms ranging from lithium to neon, Comput. Phys. Commun. 131 (1–2) (2000) 129. [23] A.E.S. Green, R.S. Stolarski, Analytic models of electron impact excitation cross sections, J. Atmos. Terr. Phys. 34 (1972) 1703. [24] G.J. Kutcher, A.E.S. Green, A model for energy deposition in liquid water, Rad. Res. 67 (1976) 408. [25] D.H. Madison, Electron capture in slow collisions, Proc. Roy. Soc. A 245 (1973) 175. [26] L.H. Toburen, W.E. Wilson, R.J. Popovich, Secondary electron emission from ionization of water vapor by 0.3 to 2.0 MeV He+ and He2+ ions, Rad. Res. 82 (1980) 27. [27] J. Schutten, F.J. DeHeer, H.R. Moustafa, A.J.H. Boerboom, J. Kistemaker, Gross and partial ionization cross sections for electrons on water vapor in the energy range 0.1–20 keV, J. Chem. Phys. 44 (1966) 3924. [28] J.W. Hooper, D.S. Harmer, D.W. Martin, E.W. McDaniel, Comparison of electron and proton ionization data with the Born approximation predictions, Phys. Rev. 125 (6) (1962) 2000. [29] S. Uehara, H. Nikjoo, D.T. Goodhead, Cross sections for ionization of water vapor for the Monte Carlo electron track structure code from 10 eV to the MeV region, Phys. Med. Biol. 37 (1992) 1841. [30] H.G. Paretzke, Radiation Track Structure Theory, Kinetic on Nonhomogeneous Processes, Wiley, New York, 1987.