PREPRINT 2011:23
Analytical Solutions for the Pencil-beam Equation with Energy Loss and Straggling
MOHAMMAD ASADZADEH TOBIAS GEBÄCK Department of Mathematical Sciences Division of Mathematics
CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Gothenburg Sweden 2011
Preprint 2011:23
Analytical Solutions for the Pencil-beam Equation with Energy Loss and Straggling
Mohammad Asadzadeh and Tobias Gebäck
Department of Mathematical Sciences Division of Mathematics
Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Gothenburg, October 2011
ANALYTICAL SOLUTIONS FOR THE PENCIL-BEAM EQUATION 9
z (cm)
E (
MeV
)
FEM
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
0.1
0.2
0.3
0.4
0.5
0.6
z (cm)
E (
MeV
)
Explicit (NESA)
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2. Level curves for the numerical solution to equation (2.30) (top),and the analytical solution (2.36) under the narrow energy spectrum approx-imation (NESA) (bottom). The initial energy was E0 = 50 MeV. The cross-sections for electrons in water were used. The dashed thick lines are the curvesE = Ea(z), and the solid thick lines are the average energies for the respectivesolutions.
References
[1] Asadzadeh M., Brahme A., Kempe J. Ion transport in inhomogeneous media based onthe bipartition model for primary ions Computers & Mathematics With Applications
10 M. ASADZADEH AND T. GEBACK
60(8):2445–2459 (2010).[2] Borgers C., Larsen E.W. Asymptotic derivation of the fermi pencil-beam approximation
Nucl. Sci. Eng. 123:343–357 (1996).[3] Brahme A. Simple relations for the penetration of high energy electrons in matter report
SSI 1975-011 National Institute of Radiation Protection, Stockholm (1975).[4] Carlsson A.K., Andreo P., Brahme A. Monte carlo and analytical calculation of proton
pencil beams for computerized treatment plan optimization Physics In Medicine andBiology 42(6):1033–1053 (1997).
[5] Eriksson K., Estep D., Hansbo P., Johnson C. Computational Differential EquationsStudentlitteratur, Lund (1996).
[6] Eyges L. Multiple scattering with energy loss Phys. Rev. 74:1534–1535 (1948).[7] Hogstrom K.R., Mills M.D., Almond P.R. Electron-beam dose calculations Physics In
Medicine and Biology 26(3):445–459 (1981).[8] ICRU Radiation dosimetry: Electron beams with energies between 1 and 50 mev ICRU
Report 35 Bethesda MD (1984).[9] Kempe J., Brahme A. Solution of the Boltzmann equation for primary light ions and
the transport of their fragments Physical Review Special Topics-accelerators and Beams13(10):104702 (2010).
[10] Rossi B., Greisen K. Reviews of Modern Physics vol. 13 chap. Cosmic-ray theory, pp.241–315 (1941).
[11] Zhengming L., Brahme A. High-energy electron transport Phys. Rev. B 46(24):15739–15751 (1992).
[12] Zhengming L., Brahme A. An overview of the transport theory of charged particlesRadiat. Phys. Chem. 41(4):673–703 (1993).
E-mail address: [email protected]
E-mail address: [email protected]
† Department of Mathematics, Chalmers University of Technology and Goteborg University,
SE–412 96, Goteborg, Sweden
‡ Department of Oncology-Pathology, Karolinska Institute, SE-171 77 Stockholm, Sweden