- Email: [email protected]
Monte Carlo transport of electrons and positrons through thin foils
Radiation Physics and Chemistry 61 (2001) 549–551
Monte Carlo transport of electrons and positrons through thin foils F. Legarda*, R. Idoeta ! ! Universidad del Pa!ıs Vasco/Euskal Dpto. Ingenier!ıa Nuclear y Mecanica de Fluidos, E.T.S. Ingenieros Industriales y de Telecomunicacion, Herriko Unibertsitatea, Alda Urquijo, s/n, 48013 Bilbao, Spain
Abstract In measurements on electrons traversing matter it is important to know the transmission through that medium, their path-lengths and their angular distribution through matter. This allows one to seek improvement in techniques which employ electrons, including medical applications and materials irradiation. This work presents a simulation of the transport of beams of electrons and positrons through thin foils using an analog Monte Carlo code that simulates in a detailed way every electron movement or interaction in matter. As those particles penetrate thin absorbers, it has been assumed that they interact with matter only through elastic scattering, with negligible energy loss. This type of interaction has been described quite precisely because its angular form influences very much the angular distribution of electrons and positrons in matter. With this code it has been calculated that the number of particles, with energies between 100 and 3000 keV, which are transmitted through different media of various thicknesses as well as their angular distributions, show good agreement with the experimental data. The discrepancies are less than 5% for thicknesses lower than about 30% of the corresponding range in the tested material. As elastic scattering is very anisotropic, its angular distribution resembles a collimated incident beam for very thin foils, becoming slowly more isotropic when absorber thickness is increased. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Transmission of electrons; Positrons; Monte-Carlo
2. Theoretical method
Since this work is devoted to thin foils, it has been assumed that particles interact mainly through elastic scatterings with atoms, producing negligible energy losses. This implies neglect of processes involving loss of energy and photon generation, which is possible in the intermediate energy region of interest here (Legarda and Idoeta, 1995). The elastic scattering differential cross-section has been treated by means of its factorised form. This factorisation arises from the use of the Rutherford cross section corrected for effects of spin, relativity and nuclear charge screening by atomic electrons
The interactions that electrons and positrons undergo can be classified into elastic and inelastic processes.
ds 1@b2 1 ¼ Z2 r2o Fs ðy; EÞFsr ðy; EÞ; 4 dO b ð1@cos yÞ2
*Corresponding author. Tel.: +34-94-601-4278; fax: +3494-601-4159. E-mail address: [email protected] (F. Legarda).
where y is the scattering angle, E is the kinetic energy, b the speed of incident electrons/positrons relative to c, and ro is the classical electron radius. The factor Fs is the
1. Introduction A correct description of the penetration of electrons and positrons through matter is fundamental to their multiple applications. This work presents a simulation of the transport of beams of electrons and positrons from 100 to 3000 keV through thin foils using an analog Monte Carlo code that simulates each particle movement and interaction.
0969-806X/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 3 2 9 - 2
550
F. Legarda, R. Idoeta / Radiation Physics and Chemistry 61 (2001) 549–551
spin-relativistic correction, being the ratio of the unscreened Mott cross section to the Rutherford cross section. This is calculated using the approach of previous work (Idoeta and Legarda, 1992) although for electrons, analytic fitting (Lijian et al., 1995) to those numerical values has been employed in order to accelerate the code. The screening factor, Fsr , is computed through the use of the screening angle, Z, of Goudsmit and Saunderson (1940) Fsr ¼
ð1@cos yÞ2 ; ð1@cos y þ 2ZÞ2
1 1@b2 Z ¼ a2 Z2=3 ; 4 b2
where a is the fine-structure constant. With this factorisation, an accurate and fast calculation of the differential cross section is obtained. The analytical integration of that cross-section allows the calculation of the macroscopic cross-section, S, Z ds S¼N dO 4p dO with N being the number of atoms per unit volume.
3. Simulation procedure The trajectory followed by a particle is made up of straight paths between collisions whose length, x, between subsequent collisions is sampled by applying the inverse transform given by: x ¼ @l ln ðPÞ, P being a random number in the interval (0,1) and l the mean free path, given by S@1 . l is calculated only once during code execution since no energy change is produced. Due to the collision, the direction of flight of particles is changed, the linear momentum being rotated through an angle, y, the scattering angle, which shows the following probability density function (pdf):
pdf ¼
4. Results and discussion Using this code, beams of monoenergetic electrons impinging perpendicularly on thin foils have been simulated. The transmitted fraction of a beam of electrons through different foils as well as its angular distribution has been calculated. With respect to transmitted fraction calculations, 40,000 electron histories have been followed for each sheet thickness and each energy. In the case of angular distributions, for statistical reasons, we have employed 106 particles in each situation. The transmitted fraction of a beam of electrons with energies ranging between 100 and 3000 keV has been calculated through different foils of aluminium, copper, molybdenum, silver, tin, gold and lead in order to compare these with the experimental measurements of Seliger (1955) and Soum et al. (1987). Fig. 1 shows that the agreement is fairly close (with discrepancies of less than 5%) for all thicknesses lower than about 30% of the corresponding electron range. The transmitted fractions of collimated positron beams through different foils have also been calculated. Agreement with the experimental data of Seliger (1955) for positrons of different energies transmitted through foils of aluminium, silver and lead, is fairly good in the same foil thickness range which has been found valid for electron beams (Fig. 1). Fig. 2 shows different comparisons between the code calculations of transmitted angular distributions and experimental values of Soum et al. (1984, 1987). The
1 ds ¼ Af ðyÞgðyÞ; sdO
with s being the microscopic elastic scattering crosssection, A an energy-dependent constant and the functions f and g are defined as f ðyÞ ¼
2ZðZ þ 1Þ ; ð1@cos y þ 2ZÞ2
gðyÞ ¼ Fs =Fsm ; where Fsm is the maximum spin-relativistic factor value for a given energy and element, searched for once during the code execution. The scattering angle, y, is sampled using an inverse transform from f ðyÞ and accepted as a result of the rejection method applied to gðyÞ.
Fig. 1. Transmitted fractions of beams of electrons and positrons of different energies through different foils of lead. The lines are the simulation results and the symbols represent the experimental data by Seliger.
F. Legarda, R. Idoeta / Radiation Physics and Chemistry 61 (2001) 549–551
551
and second, that the analog Monte Carlo code presented here offers a reliable and quick description of transmission studies of beams of electrons and positrons through thin foils.
Acknowledgements Support from the University of Basque Country (UPV/EHU), project no. UPV 149.345FEA 120/98, is acknowledged.
References
Fig. 2. Angular distributions of 3 MeV electrons transmitted through aluminium sheets of the indicated thickness calculated in this work (lines) and experimental measurements (symbols) of Soum et al.
thicknesses for which the smallest disagreements are found are practically the same as those encountered in transmitted fractions studies. The observed angular flux resembles a collimated incident beam for very thin foils, becoming slowly more isotropic when absorber thickness is increased, under the influence of very anisotropic elastic scattering. The good results which have been obtained indicate that first, elastic scattering explains the different behaviours of electrons and positrons travelling through matter of various atomic numbers for thin absorbers,
Goudsmit, G.A., Saunderson, J.L., 1940. Multiple scattering of electrons. Phys. Rev. 57, 24–29. Idoeta, R., Legarda, F., 1992. Review and calculation of Mott scattering cross section by unscreened point nuclei. Nucl. Instr. Meth. B 71, 116–125. Legarda, F., Idoeta, R., 1995. Calculation of beta-ray attenuation coefficients through thin foils. Nucl. Instr. Meth. B 103, 429–434. Lijian, T., Qing, H., Zhengming, L., 1995. Analytical fitting to the Mott cross section of electrons. Radiat. Phys. Chem. 45, 235–245. Seliger, H.H., 1955. Transmission of positrons and electrons. Phys. Rev. 100, 1029–1037. Soum, G., Arnal, F., Jouffrey, B., Verdier, P., 1984. Diffusion multiple delectrons dans les objets amorphes ou polycristallins: distribution angulaire des electrons transmis. J. Microsc. Spectrosc. Electron. 9, 419–430. Soum, G., Mousselli, A., Arnal, F., Verdier, P., 1987. Etude de la transmission et de la retrodiffusion delectrons d’energie 0,05 a 3 MeV dans le domaine de la diffusion multiple. Rev. Phys. Appl. 22, 1189–1209.
Monte Carlo transport of electrons and positrons through thin foils F. Legarda*, R. Idoeta ! ! Universidad del Pa!ıs Vasco/Euskal Dpto. Ingenier!ıa Nuclear y Mecanica de Fluidos, E.T.S. Ingenieros Industriales y de Telecomunicacion, Herriko Unibertsitatea, Alda Urquijo, s/n, 48013 Bilbao, Spain
Abstract In measurements on electrons traversing matter it is important to know the transmission through that medium, their path-lengths and their angular distribution through matter. This allows one to seek improvement in techniques which employ electrons, including medical applications and materials irradiation. This work presents a simulation of the transport of beams of electrons and positrons through thin foils using an analog Monte Carlo code that simulates in a detailed way every electron movement or interaction in matter. As those particles penetrate thin absorbers, it has been assumed that they interact with matter only through elastic scattering, with negligible energy loss. This type of interaction has been described quite precisely because its angular form influences very much the angular distribution of electrons and positrons in matter. With this code it has been calculated that the number of particles, with energies between 100 and 3000 keV, which are transmitted through different media of various thicknesses as well as their angular distributions, show good agreement with the experimental data. The discrepancies are less than 5% for thicknesses lower than about 30% of the corresponding range in the tested material. As elastic scattering is very anisotropic, its angular distribution resembles a collimated incident beam for very thin foils, becoming slowly more isotropic when absorber thickness is increased. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Transmission of electrons; Positrons; Monte-Carlo
2. Theoretical method
Since this work is devoted to thin foils, it has been assumed that particles interact mainly through elastic scatterings with atoms, producing negligible energy losses. This implies neglect of processes involving loss of energy and photon generation, which is possible in the intermediate energy region of interest here (Legarda and Idoeta, 1995). The elastic scattering differential cross-section has been treated by means of its factorised form. This factorisation arises from the use of the Rutherford cross section corrected for effects of spin, relativity and nuclear charge screening by atomic electrons
The interactions that electrons and positrons undergo can be classified into elastic and inelastic processes.
ds 1@b2 1 ¼ Z2 r2o Fs ðy; EÞFsr ðy; EÞ; 4 dO b ð1@cos yÞ2
*Corresponding author. Tel.: +34-94-601-4278; fax: +3494-601-4159. E-mail address: [email protected] (F. Legarda).
where y is the scattering angle, E is the kinetic energy, b the speed of incident electrons/positrons relative to c, and ro is the classical electron radius. The factor Fs is the
1. Introduction A correct description of the penetration of electrons and positrons through matter is fundamental to their multiple applications. This work presents a simulation of the transport of beams of electrons and positrons from 100 to 3000 keV through thin foils using an analog Monte Carlo code that simulates each particle movement and interaction.
0969-806X/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 3 2 9 - 2
550
F. Legarda, R. Idoeta / Radiation Physics and Chemistry 61 (2001) 549–551
spin-relativistic correction, being the ratio of the unscreened Mott cross section to the Rutherford cross section. This is calculated using the approach of previous work (Idoeta and Legarda, 1992) although for electrons, analytic fitting (Lijian et al., 1995) to those numerical values has been employed in order to accelerate the code. The screening factor, Fsr , is computed through the use of the screening angle, Z, of Goudsmit and Saunderson (1940) Fsr ¼
ð1@cos yÞ2 ; ð1@cos y þ 2ZÞ2
1 1@b2 Z ¼ a2 Z2=3 ; 4 b2
where a is the fine-structure constant. With this factorisation, an accurate and fast calculation of the differential cross section is obtained. The analytical integration of that cross-section allows the calculation of the macroscopic cross-section, S, Z ds S¼N dO 4p dO with N being the number of atoms per unit volume.
3. Simulation procedure The trajectory followed by a particle is made up of straight paths between collisions whose length, x, between subsequent collisions is sampled by applying the inverse transform given by: x ¼ @l ln ðPÞ, P being a random number in the interval (0,1) and l the mean free path, given by S@1 . l is calculated only once during code execution since no energy change is produced. Due to the collision, the direction of flight of particles is changed, the linear momentum being rotated through an angle, y, the scattering angle, which shows the following probability density function (pdf):
pdf ¼
4. Results and discussion Using this code, beams of monoenergetic electrons impinging perpendicularly on thin foils have been simulated. The transmitted fraction of a beam of electrons through different foils as well as its angular distribution has been calculated. With respect to transmitted fraction calculations, 40,000 electron histories have been followed for each sheet thickness and each energy. In the case of angular distributions, for statistical reasons, we have employed 106 particles in each situation. The transmitted fraction of a beam of electrons with energies ranging between 100 and 3000 keV has been calculated through different foils of aluminium, copper, molybdenum, silver, tin, gold and lead in order to compare these with the experimental measurements of Seliger (1955) and Soum et al. (1987). Fig. 1 shows that the agreement is fairly close (with discrepancies of less than 5%) for all thicknesses lower than about 30% of the corresponding electron range. The transmitted fractions of collimated positron beams through different foils have also been calculated. Agreement with the experimental data of Seliger (1955) for positrons of different energies transmitted through foils of aluminium, silver and lead, is fairly good in the same foil thickness range which has been found valid for electron beams (Fig. 1). Fig. 2 shows different comparisons between the code calculations of transmitted angular distributions and experimental values of Soum et al. (1984, 1987). The
1 ds ¼ Af ðyÞgðyÞ; sdO
with s being the microscopic elastic scattering crosssection, A an energy-dependent constant and the functions f and g are defined as f ðyÞ ¼
2ZðZ þ 1Þ ; ð1@cos y þ 2ZÞ2
gðyÞ ¼ Fs =Fsm ; where Fsm is the maximum spin-relativistic factor value for a given energy and element, searched for once during the code execution. The scattering angle, y, is sampled using an inverse transform from f ðyÞ and accepted as a result of the rejection method applied to gðyÞ.
Fig. 1. Transmitted fractions of beams of electrons and positrons of different energies through different foils of lead. The lines are the simulation results and the symbols represent the experimental data by Seliger.
F. Legarda, R. Idoeta / Radiation Physics and Chemistry 61 (2001) 549–551
551
and second, that the analog Monte Carlo code presented here offers a reliable and quick description of transmission studies of beams of electrons and positrons through thin foils.
Acknowledgements Support from the University of Basque Country (UPV/EHU), project no. UPV 149.345FEA 120/98, is acknowledged.
References
Fig. 2. Angular distributions of 3 MeV electrons transmitted through aluminium sheets of the indicated thickness calculated in this work (lines) and experimental measurements (symbols) of Soum et al.
thicknesses for which the smallest disagreements are found are practically the same as those encountered in transmitted fractions studies. The observed angular flux resembles a collimated incident beam for very thin foils, becoming slowly more isotropic when absorber thickness is increased, under the influence of very anisotropic elastic scattering. The good results which have been obtained indicate that first, elastic scattering explains the different behaviours of electrons and positrons travelling through matter of various atomic numbers for thin absorbers,
Goudsmit, G.A., Saunderson, J.L., 1940. Multiple scattering of electrons. Phys. Rev. 57, 24–29. Idoeta, R., Legarda, F., 1992. Review and calculation of Mott scattering cross section by unscreened point nuclei. Nucl. Instr. Meth. B 71, 116–125. Legarda, F., Idoeta, R., 1995. Calculation of beta-ray attenuation coefficients through thin foils. Nucl. Instr. Meth. B 103, 429–434. Lijian, T., Qing, H., Zhengming, L., 1995. Analytical fitting to the Mott cross section of electrons. Radiat. Phys. Chem. 45, 235–245. Seliger, H.H., 1955. Transmission of positrons and electrons. Phys. Rev. 100, 1029–1037. Soum, G., Arnal, F., Jouffrey, B., Verdier, P., 1984. Diffusion multiple delectrons dans les objets amorphes ou polycristallins: distribution angulaire des electrons transmis. J. Microsc. Spectrosc. Electron. 9, 419–430. Soum, G., Mousselli, A., Arnal, F., Verdier, P., 1987. Etude de la transmission et de la retrodiffusion delectrons d’energie 0,05 a 3 MeV dans le domaine de la diffusion multiple. Rev. Phys. Appl. 22, 1189–1209.