- Email: [email protected]
Influence of ionization cross-section data on the Monte Carlo calculation of nanodosimetric quantities
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84 www.elsevier.com/locate/nima
Influence of ionization cross-section data on the Monte Carlo calculation of nanodosimetric quantities E. Gargioni, B. Grosswendt Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Available online 13 May 2007
Abstract In this work, we discuss the effect of cross-section data in the determination of nanodosimetric quantities, such as ionization clustersize distributions and mean cluster sizes. In particular, two different theoretical approaches to determine secondary electron distributions after proton interactions in nitrogen and propane were tested by means of a Monte Carlo code. The results show that differences of 10–15% in the cross-section data lead to differences of up to 10–20% in the calculated mean cluster sizes, especially at low energies, suggesting the need for new, reliable and consistent experimental cross-section data. r 2007 Elsevier B.V. All rights reserved. PACS: 02.50.N; 87.50.G; 34.50.F; 87.53.Rd Keywords: Monte Carlo; Nanodosimetry; Protons; Ionization
1. Introduction Nanodosimetric quantities, such as size distributions of clustered ionization, are important in understanding the radiation-induced damage in biological targets of nanometric size (such as DNA segments). These quantities, however, are not directly measurable in biological targets and our present knowledge is mostly based on theoretical models. A common practice to overcome this problem is to measure cluster-size distributions using a nanodosimeter, which basically consists of a gas-filled counter operating at low pressure. In the last years, several types of nanodosimeters were developed aided by the important contribution of Monte Carlo simulations [1]. These simulations require an accurate, complete and consistent set of scattering crosssections in the gas of interest. Since experimental results are often fragmentary and contradictory, the use of theoretical models to obtain interaction data becomes crucial. In this work, we use two different theoretical approaches to determine differential ionization cross-sections for proton interactions in nitrogen and propane, which are the gases most commonly used in nanodosimetry, and we Corresponding author. Tel.: +49 531 5926640; fax: +49 531 5926015.
E-mail address: [email protected] (E. Gargioni). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.05.042
show how the choice of a particular model affects the Monte Carlo calculated cluster-size distribution. 2. Determination of nanodosimetric quantities The working principle of a nanodosimeter is to evaluate the clustering of radiation-induced ionization by counting the number of charges deposited in a small volume of lowdensity gas. The cluster-size distribution is defined as the probability Pn that in a well-defined volume exactly n ionizations are produced by a single primary particle (including its secondary electrons). If this probability is known, the mean cluster size is given by the first moment M1 of the distribution M1 ¼
1 X
nPn
n¼0
with
1 X
Pn ¼ 1.
(1)
n¼0
Due to the stochastic nature of the ionization process, it is possible to demonstrate [1,2] that M 1 / 1=ðlrÞion
(2)
where r is the gas density and lion the mean free path length of the primary particle between two ionization events, which is, by definition, inversely proportional to the
ARTICLE IN PRESS E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
82
meter, R ¼ 13.606 eV the Rydberg energy, ai ¼ (Bi/R)1/2 the initial momentum parameter and
ionization cross section sion ðlrÞion ¼ ðM=N A Þs1 ion where M is the gas molar mass and NA the Avogadro constant. Eq. (2) therefore points out the importance of the mean cluster size to characterize the radiation quality. 3. Ionization cross-section data for protons
K~ m;i ¼ ða2i þ k2 Þ=ð2vac;i Þ h pffiffiffiffiffiffiffiffiffii1=2 k~t;i ¼ k2 þ 0:2a2i v=ai =ac;i 2 1=2 kc;i ¼ k þ 2a2i = lnð2v2 =a2i Þ ac;i ¼ ai 1 þ 0:7v2 =ðv2 þ k2 Þ k ¼ ð=RÞ1=2 .
Consistent and reliable interaction data are important for Monte Carlo simulations of particle track structures. In particular, charged particles passing through matter lose most of their energy in ionizing collisions with bound electrons, resulting in energetic free electrons, whose transport has to be followed completely. Therefore, the most important data are ionization cross-sections of primary particles, the spectral and angular distribution of secondary electrons and the data concerning electron degradation via elastic, excitation and ionization processes. The singly differential cross-section (SDCS) of the primary particles gives information about the secondary electron spectral distribution and can be integrated over the electron energy e to obtain the total ionization crosssection: X Z TBi dsi sion ¼ d (3) d 0 i where the sum is taken over all target shells. T is the primary particle energy and Bi the ionization threshold of the corresponding orbital. While some experimental cross-sections are available for proton collisions in nitrogen, those for propane are, up to now, still fragmentary [3,4]. Theoretical models are therefore helpful to obtain the missing information. Here, we present two different approaches for determining secondary electron distributions after proton collisions: the Hansen–Kocbach–Stolterfoht (HKS) model and the Rudd model.
The advantage of using Eq. (4) is that it depends only on parameters related to the kinematic of the collision (like momentum and kinetic energy of protons) and on the target’s binding energies. Nevertheless, a thorough investigation of its applicability to any target is still lacking [3] and would be desirable. 3.2. Rudd model Rudd et al. [4] based their semi-empirical theory on the classical binary-encounter approximation at high proton energy and on the molecular promotion model at low energy [6]. The resulting SDCS is given by dsi ðSi =Bi ÞðF 1 þ F 2 wi Þð1 þ wi Þ3 ¼ . d 1 þ exp ½wðwi wc;i Þ=v
(5)
Here, w is an adjustable parameter and S i ¼ 4pa20 N i ðR=Bi Þ2 wi ¼ =Bi wc;i ¼ 4v2 2v R=ð4Bi Þ F 1 ¼ L1 þ H 1 F 2 ¼ L2 H 2 =ðL2 þ H 2 Þ with Ni being the electron occupation number in a shell and L1 ¼ C 1 vD1 =½1 þ E 1 vðD1 þ4Þ H 1 ¼ A1 lnð1 þ v2 Þ=ðv2 þ B1 =v2 Þ L2 ¼ C 2 vD2
3.1. Hansen–Kocbach–Stolterfoht (HKS) model
H 2 ¼ A2 =v2 þ B2 =v4 .
This model is based on a semi-classical approximation and gives a simple expression for the SDCS of each target shell [3,5] 8 2 2 dsi 4Z 1 a0 < 5ðK~ m;i þ k~t;i Þ þ 3ðK~ m;i þ k~t;i Þ3 ¼ d Rv2 ai k3 k~t;i : 3½1 þ ðK~ m;i þ k~t;i Þ2 2
Although the Rudd model needs 11 parameters (A1, A2, B1, B2, C1, C2, D1, D2, E1, E2, w) to be adjusted on the basis of experimental results, it allows to cover a wide energy range and to extrapolate missing data.
c;i
5ðK~ m;i k~t;i Þ þ 3ðK~ m;i k~t;i Þ3 3½1 þ ðK~ m;i k~t;i Þ2 2 0 19 = ~t;i 2 k A þ arctan@ 2 2 1 þ K~ m;i k~t;i ;
ð4Þ
where a0E5.29 1011 m is the Bohr radius, Z1 the primary particle’s charge, v ¼ (T/R)1/2 its velocity para-
4. Monte Carlo calculation Cluster-size distributions in propane and nitrogen were calculated with an ad hoc code developed at PTB [7,8], which simulates the geometry of an ion-counting nanodosimeter developed in collaboration with the Weizmann Institute (Rehovot, Israel) [7]. The binding energies and occupation numbers for nitrogen and propane were taken from Hwang et al. [9] and the fitting parameters for the Rudd model from Rudd et al. [4]. The proton energy was
ARTICLE IN PRESS E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
0.12 HKS Rudd 0.08 T = 0.1 MeV Pν
varied between 0.1 and 10 MeV, the gas pressure was 1.2 mbar and its temperature 25 1C. Both the HKS and the Rudd model were used to determine the secondary electron distributions. Since there are no experimental SDCS data for propane, the fitting parameters for methane were used in Eq. (5) after scaling by the ratio of the number of weakly bound electrons of both molecules [10].
0.04
5. Results and discussion
0.00 0
10
20 cluster size ν
30
40
Fig. 2. Example of a cluster-size distribution Pn calculated for proton impact at T ¼ 0.1 MeV in propane.
12 HKS Rudd 8 M1
In Fig. 1, the integrated ionization cross-section for nitrogen and propane, calculated numerically using Eq. (3) is illustrated as a function of the proton energy. While the agreement between the two models is within 3–5% in the intermediate energy range, the HKS model gives results that are approximately 15% higher than the Rudd model at low energies and up to 25% lower at high energies. The difference at low proton energies is enhanced in the resulting cluster-size distributions, as can be seen in Figs. 2 and 3. In particular, the mean cluster sizes calculated in propane using the HKS model are, at low energies, about 20% higher than those calculated with the Rudd model. Since both approaches still have several limitations, like the scaling used in Eq. (5) for propane and the lack of systematic verifications of the applicability of
83
propane
4
0 4 HKS Rudd
M1
3
2 nitrogen 1
0 0.1
1 T / MeV
10
Fig. 3. Mean cluster size M1, as a function of proton energy T, calculated in propane and nitrogen using Eq. (1).
Fig. 1. Integrated ionization cross-section sion as a function of the proton impact energy T, calculated for the two theoretical models.
Eq. (4), there is an urgent need for reliable experimental cross-section data. However, some comparisons of measured and calculated cluster-size distributions will also contribute to an evaluation of the two formulae. The results of these experiments will be published elsewhere.
ARTICLE IN PRESS 84
E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
References [1] B. Grosswendt, Radiat. Prot. Dosim. 110 (2004) 789. [2] B. Grosswendt, Radiat. Environ. Biophys. 41 (2002) 103. [3] International Commission on Radiation Units and Measurements, Secondary Electron Spectra from Charged Particles Interactions, ICRU Report 55, Bethesda (MD), ICRU, 1996. [4] M.E. Rudd, Y.-K. Kim, D.H. Madison, T.J. Gay, Rev. Mod. Phys. 64 (1992) 441.
[5] N. Stolterfoht, R.D. DuBois, R.D. Rivarola, Electron Emission in Heavy Ion-Atom Collisions, Springer, Berlin Heidelberg, 1997. [6] M.E. Rudd, Phys. Rev. A 38 (1988) 6129. [7] G. Garty, S. Shchemelinin, A. Breskin, R. Chechik, G. Assaf, I. Orion, V. Bashkirov, R. Schulte, B. Grosswendt B, Nucl. Instr. and Meth. A 492 (2002) 212. [8] B. Grosswendt, S. Pszona, Radiat. Environ. Biophys. 41 (2002) 91. [9] W. Hwang, Y.-K. Kim, M.E. Rudd, J. Chem. Phys. 104 (1996) 2956. [10] W.E. Wilson, L.H. Toburen, Phys. Rev. A 11 (1975) 1303.
Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84 www.elsevier.com/locate/nima
Influence of ionization cross-section data on the Monte Carlo calculation of nanodosimetric quantities E. Gargioni, B. Grosswendt Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Available online 13 May 2007
Abstract In this work, we discuss the effect of cross-section data in the determination of nanodosimetric quantities, such as ionization clustersize distributions and mean cluster sizes. In particular, two different theoretical approaches to determine secondary electron distributions after proton interactions in nitrogen and propane were tested by means of a Monte Carlo code. The results show that differences of 10–15% in the cross-section data lead to differences of up to 10–20% in the calculated mean cluster sizes, especially at low energies, suggesting the need for new, reliable and consistent experimental cross-section data. r 2007 Elsevier B.V. All rights reserved. PACS: 02.50.N; 87.50.G; 34.50.F; 87.53.Rd Keywords: Monte Carlo; Nanodosimetry; Protons; Ionization
1. Introduction Nanodosimetric quantities, such as size distributions of clustered ionization, are important in understanding the radiation-induced damage in biological targets of nanometric size (such as DNA segments). These quantities, however, are not directly measurable in biological targets and our present knowledge is mostly based on theoretical models. A common practice to overcome this problem is to measure cluster-size distributions using a nanodosimeter, which basically consists of a gas-filled counter operating at low pressure. In the last years, several types of nanodosimeters were developed aided by the important contribution of Monte Carlo simulations [1]. These simulations require an accurate, complete and consistent set of scattering crosssections in the gas of interest. Since experimental results are often fragmentary and contradictory, the use of theoretical models to obtain interaction data becomes crucial. In this work, we use two different theoretical approaches to determine differential ionization cross-sections for proton interactions in nitrogen and propane, which are the gases most commonly used in nanodosimetry, and we Corresponding author. Tel.: +49 531 5926640; fax: +49 531 5926015.
E-mail address: [email protected] (E. Gargioni). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.05.042
show how the choice of a particular model affects the Monte Carlo calculated cluster-size distribution. 2. Determination of nanodosimetric quantities The working principle of a nanodosimeter is to evaluate the clustering of radiation-induced ionization by counting the number of charges deposited in a small volume of lowdensity gas. The cluster-size distribution is defined as the probability Pn that in a well-defined volume exactly n ionizations are produced by a single primary particle (including its secondary electrons). If this probability is known, the mean cluster size is given by the first moment M1 of the distribution M1 ¼
1 X
nPn
n¼0
with
1 X
Pn ¼ 1.
(1)
n¼0
Due to the stochastic nature of the ionization process, it is possible to demonstrate [1,2] that M 1 / 1=ðlrÞion
(2)
where r is the gas density and lion the mean free path length of the primary particle between two ionization events, which is, by definition, inversely proportional to the
ARTICLE IN PRESS E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
82
meter, R ¼ 13.606 eV the Rydberg energy, ai ¼ (Bi/R)1/2 the initial momentum parameter and
ionization cross section sion ðlrÞion ¼ ðM=N A Þs1 ion where M is the gas molar mass and NA the Avogadro constant. Eq. (2) therefore points out the importance of the mean cluster size to characterize the radiation quality. 3. Ionization cross-section data for protons
K~ m;i ¼ ða2i þ k2 Þ=ð2vac;i Þ h pffiffiffiffiffiffiffiffiffii1=2 k~t;i ¼ k2 þ 0:2a2i v=ai =ac;i 2 1=2 kc;i ¼ k þ 2a2i = lnð2v2 =a2i Þ ac;i ¼ ai 1 þ 0:7v2 =ðv2 þ k2 Þ k ¼ ð=RÞ1=2 .
Consistent and reliable interaction data are important for Monte Carlo simulations of particle track structures. In particular, charged particles passing through matter lose most of their energy in ionizing collisions with bound electrons, resulting in energetic free electrons, whose transport has to be followed completely. Therefore, the most important data are ionization cross-sections of primary particles, the spectral and angular distribution of secondary electrons and the data concerning electron degradation via elastic, excitation and ionization processes. The singly differential cross-section (SDCS) of the primary particles gives information about the secondary electron spectral distribution and can be integrated over the electron energy e to obtain the total ionization crosssection: X Z TBi dsi sion ¼ d (3) d 0 i where the sum is taken over all target shells. T is the primary particle energy and Bi the ionization threshold of the corresponding orbital. While some experimental cross-sections are available for proton collisions in nitrogen, those for propane are, up to now, still fragmentary [3,4]. Theoretical models are therefore helpful to obtain the missing information. Here, we present two different approaches for determining secondary electron distributions after proton collisions: the Hansen–Kocbach–Stolterfoht (HKS) model and the Rudd model.
The advantage of using Eq. (4) is that it depends only on parameters related to the kinematic of the collision (like momentum and kinetic energy of protons) and on the target’s binding energies. Nevertheless, a thorough investigation of its applicability to any target is still lacking [3] and would be desirable. 3.2. Rudd model Rudd et al. [4] based their semi-empirical theory on the classical binary-encounter approximation at high proton energy and on the molecular promotion model at low energy [6]. The resulting SDCS is given by dsi ðSi =Bi ÞðF 1 þ F 2 wi Þð1 þ wi Þ3 ¼ . d 1 þ exp ½wðwi wc;i Þ=v
(5)
Here, w is an adjustable parameter and S i ¼ 4pa20 N i ðR=Bi Þ2 wi ¼ =Bi wc;i ¼ 4v2 2v R=ð4Bi Þ F 1 ¼ L1 þ H 1 F 2 ¼ L2 H 2 =ðL2 þ H 2 Þ with Ni being the electron occupation number in a shell and L1 ¼ C 1 vD1 =½1 þ E 1 vðD1 þ4Þ H 1 ¼ A1 lnð1 þ v2 Þ=ðv2 þ B1 =v2 Þ L2 ¼ C 2 vD2
3.1. Hansen–Kocbach–Stolterfoht (HKS) model
H 2 ¼ A2 =v2 þ B2 =v4 .
This model is based on a semi-classical approximation and gives a simple expression for the SDCS of each target shell [3,5] 8 2 2 dsi 4Z 1 a0 < 5ðK~ m;i þ k~t;i Þ þ 3ðK~ m;i þ k~t;i Þ3 ¼ d Rv2 ai k3 k~t;i : 3½1 þ ðK~ m;i þ k~t;i Þ2 2
Although the Rudd model needs 11 parameters (A1, A2, B1, B2, C1, C2, D1, D2, E1, E2, w) to be adjusted on the basis of experimental results, it allows to cover a wide energy range and to extrapolate missing data.
c;i
5ðK~ m;i k~t;i Þ þ 3ðK~ m;i k~t;i Þ3 3½1 þ ðK~ m;i k~t;i Þ2 2 0 19 = ~t;i 2 k A þ arctan@ 2 2 1 þ K~ m;i k~t;i ;
ð4Þ
where a0E5.29 1011 m is the Bohr radius, Z1 the primary particle’s charge, v ¼ (T/R)1/2 its velocity para-
4. Monte Carlo calculation Cluster-size distributions in propane and nitrogen were calculated with an ad hoc code developed at PTB [7,8], which simulates the geometry of an ion-counting nanodosimeter developed in collaboration with the Weizmann Institute (Rehovot, Israel) [7]. The binding energies and occupation numbers for nitrogen and propane were taken from Hwang et al. [9] and the fitting parameters for the Rudd model from Rudd et al. [4]. The proton energy was
ARTICLE IN PRESS E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
0.12 HKS Rudd 0.08 T = 0.1 MeV Pν
varied between 0.1 and 10 MeV, the gas pressure was 1.2 mbar and its temperature 25 1C. Both the HKS and the Rudd model were used to determine the secondary electron distributions. Since there are no experimental SDCS data for propane, the fitting parameters for methane were used in Eq. (5) after scaling by the ratio of the number of weakly bound electrons of both molecules [10].
0.04
5. Results and discussion
0.00 0
10
20 cluster size ν
30
40
Fig. 2. Example of a cluster-size distribution Pn calculated for proton impact at T ¼ 0.1 MeV in propane.
12 HKS Rudd 8 M1
In Fig. 1, the integrated ionization cross-section for nitrogen and propane, calculated numerically using Eq. (3) is illustrated as a function of the proton energy. While the agreement between the two models is within 3–5% in the intermediate energy range, the HKS model gives results that are approximately 15% higher than the Rudd model at low energies and up to 25% lower at high energies. The difference at low proton energies is enhanced in the resulting cluster-size distributions, as can be seen in Figs. 2 and 3. In particular, the mean cluster sizes calculated in propane using the HKS model are, at low energies, about 20% higher than those calculated with the Rudd model. Since both approaches still have several limitations, like the scaling used in Eq. (5) for propane and the lack of systematic verifications of the applicability of
83
propane
4
0 4 HKS Rudd
M1
3
2 nitrogen 1
0 0.1
1 T / MeV
10
Fig. 3. Mean cluster size M1, as a function of proton energy T, calculated in propane and nitrogen using Eq. (1).
Fig. 1. Integrated ionization cross-section sion as a function of the proton impact energy T, calculated for the two theoretical models.
Eq. (4), there is an urgent need for reliable experimental cross-section data. However, some comparisons of measured and calculated cluster-size distributions will also contribute to an evaluation of the two formulae. The results of these experiments will be published elsewhere.
ARTICLE IN PRESS 84
E. Gargioni, B. Grosswendt / Nuclear Instruments and Methods in Physics Research A 580 (2007) 81–84
References [1] B. Grosswendt, Radiat. Prot. Dosim. 110 (2004) 789. [2] B. Grosswendt, Radiat. Environ. Biophys. 41 (2002) 103. [3] International Commission on Radiation Units and Measurements, Secondary Electron Spectra from Charged Particles Interactions, ICRU Report 55, Bethesda (MD), ICRU, 1996. [4] M.E. Rudd, Y.-K. Kim, D.H. Madison, T.J. Gay, Rev. Mod. Phys. 64 (1992) 441.
[5] N. Stolterfoht, R.D. DuBois, R.D. Rivarola, Electron Emission in Heavy Ion-Atom Collisions, Springer, Berlin Heidelberg, 1997. [6] M.E. Rudd, Phys. Rev. A 38 (1988) 6129. [7] G. Garty, S. Shchemelinin, A. Breskin, R. Chechik, G. Assaf, I. Orion, V. Bashkirov, R. Schulte, B. Grosswendt B, Nucl. Instr. and Meth. A 492 (2002) 212. [8] B. Grosswendt, S. Pszona, Radiat. Environ. Biophys. 41 (2002) 91. [9] W. Hwang, Y.-K. Kim, M.E. Rudd, J. Chem. Phys. 104 (1996) 2956. [10] W.E. Wilson, L.H. Toburen, Phys. Rev. A 11 (1975) 1303.