A Closed-Form Formulation for the Build-Up Factor and Absorbed Energy for Photons and Electrons in the Compton Energy Range in Cartesian Geometry

Author(s)
Volnei Borges,
Julio Cesar Fernandes,
Bardo Ernest Bodmann,
Marco Túllio Vilhena,
Bárbara Rodriguez

ABSTRACT

In this work, we report on a closed-form formulation for the build-up factor and absorbed energy, in one and two di- mensional Cartesian geometry for photons and electrons, in the Compton energy range. For the one-dimensional case we use the LTSN method, assuming the Klein-Nishina scattering kernel for the determination of the angular radiation intensity for photons. We apply the two-dimensional LTSN nodal solution for the averaged angular radiation evaluation for the two-dimensional case, using the Klein-Nishina kernel for photons and the Compton kernel for electrons. From the angular radiation intensity we construct a closed-form solution for the build-up factor and evaluate the absorbed energy. We present numerical simulations and comparisons against results from the literature.

In this work, we report on a closed-form formulation for the build-up factor and absorbed energy, in one and two di- mensional Cartesian geometry for photons and electrons, in the Compton energy range. For the one-dimensional case we use the LTSN method, assuming the Klein-Nishina scattering kernel for the determination of the angular radiation intensity for photons. We apply the two-dimensional LTSN nodal solution for the averaged angular radiation evaluation for the two-dimensional case, using the Klein-Nishina kernel for photons and the Compton kernel for electrons. From the angular radiation intensity we construct a closed-form solution for the build-up factor and evaluate the absorbed energy. We present numerical simulations and comparisons against results from the literature.

Cite this paper

V. Borges, J. Fernandes, B. Bodmann, M. Vilhena and B. Rodriguez, "A Closed-Form Formulation for the Build-Up Factor and Absorbed Energy for Photons and Electrons in the Compton Energy Range in Cartesian Geometry,"*World Journal of Nuclear Science and Technology*, Vol. 2 No. 1, 2012, pp. 23-28. doi: 10.4236/wjnst.2012.21004.

V. Borges, J. Fernandes, B. Bodmann, M. Vilhena and B. Rodriguez, "A Closed-Form Formulation for the Build-Up Factor and Absorbed Energy for Photons and Electrons in the Compton Energy Range in Cartesian Geometry,"

References

[1] C. F. Segatto, M. T. Vilhena and R. P. Pazos, “On the Con- vergence of the Spherical Harmonics Approximations,” Nuclear Science and Engineering, Vol. 134, No. 1, 2000, pp. 114-119.

[2] M. T. Vilhena, C. F. Segatto and L. B. Barichello, “A Par- ticular Solution for the SN Radiative Transfer Problems,” Journal of Quantitative Spectroscopy and Radiative Tran- sfer, Vol. 53, No. 4, 1995, pp. 467-469. doi:10.1016/0022-4073(95)90020-9

[3] C. Borges and E. W. Larsen, “The Transversely Inte- grated Scalar Flux of a Narrowly Focused Particle Beam,” SIAM Journal on Applied Mathematics, Vol. 55, No. 1, 1995, pp. 1-22.

[4] C. Borges and E. W. Larsen, “On the Accuracy of the Fokker-Planck and Fermi Pencil Beam Equation for Charged Particle Transport,” Medical Physics, Vol. 23, No. 10, 1996, pp. 1749-1759. doi:10.1118/1.597832

[5] B. D. A. Rodriguez, M. T. Vilhena, V. Borges and G. Hoff, “A Closed Form Solution for the Two-Dimensional Fokker-Planck Equation for Electron Transport in the Range of Compton Effect,” Annals of Nuclear Energy, Vol. 35, No. 5, 2008, pp. 958-962. doi:10.1016/j.anucene.2007.09.002

[6] B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “A So- lution for the Two-Dimensional Transport Equation for Photons and Electrons in a Rectangular Domain by the Laplace Transform Technique,” International Journal of Nuclear Energy Science and Technology, Vol. 5, No. 1, 2010, pp. 25-40. doi:10.1504/IJNEST.2010.030304

[7] H. Hirayama and K. Shin, “Application of the EGS4 Monte Carlo Code to a Study of Multilayer Gamma-Ray Expo- sure Buildup Factors,” Journal of Nuclear Science and Technology, Vol. 35, No. 11, p. 816. doi:10.3327/jnst.35.816

[8] D. H. Wright, “Physics Reference Manual,” 2001. http://cern.ch/geant4

[9] G. C. Pomraning, “Flux-Limited Diffusion and Fokker- Planck Equations,” Nuclear Science and Engineering, Vol. 85, No. 2, 1983, p. 116.

[10] J. Wood, “Computational Methods in Reactor Shielding,” Pergamon Press, Oxford, 1982.

[11] S. Agostinelli, et al., “Geant4: A Simulation Toolkit,” Nu- clear Instruments and Methods in Physics Research A, Vol. 506, No. 3, 2003, pp. 250-303.

[12] DOORS 3.1, “One, Two and Three Dimensional Discrete Ordinates Neutron Photon Transport Code System,” Ra- diation Safety Information Computational Center (RSICC), Code Package CCC-650, Oak Ridge, Tennessee, 1996.

[13] B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “The Determination of the Exposure Buildup Factor Formula- tion in a Slab Using the LTSN Method,” Kerntechnik, Vol. 71, No. 4, 2006, pp. 182-184.

[1] C. F. Segatto, M. T. Vilhena and R. P. Pazos, “On the Con- vergence of the Spherical Harmonics Approximations,” Nuclear Science and Engineering, Vol. 134, No. 1, 2000, pp. 114-119.

[2] M. T. Vilhena, C. F. Segatto and L. B. Barichello, “A Par- ticular Solution for the SN Radiative Transfer Problems,” Journal of Quantitative Spectroscopy and Radiative Tran- sfer, Vol. 53, No. 4, 1995, pp. 467-469. doi:10.1016/0022-4073(95)90020-9

[3] C. Borges and E. W. Larsen, “The Transversely Inte- grated Scalar Flux of a Narrowly Focused Particle Beam,” SIAM Journal on Applied Mathematics, Vol. 55, No. 1, 1995, pp. 1-22.

[4] C. Borges and E. W. Larsen, “On the Accuracy of the Fokker-Planck and Fermi Pencil Beam Equation for Charged Particle Transport,” Medical Physics, Vol. 23, No. 10, 1996, pp. 1749-1759. doi:10.1118/1.597832

[5] B. D. A. Rodriguez, M. T. Vilhena, V. Borges and G. Hoff, “A Closed Form Solution for the Two-Dimensional Fokker-Planck Equation for Electron Transport in the Range of Compton Effect,” Annals of Nuclear Energy, Vol. 35, No. 5, 2008, pp. 958-962. doi:10.1016/j.anucene.2007.09.002

[6] B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “A So- lution for the Two-Dimensional Transport Equation for Photons and Electrons in a Rectangular Domain by the Laplace Transform Technique,” International Journal of Nuclear Energy Science and Technology, Vol. 5, No. 1, 2010, pp. 25-40. doi:10.1504/IJNEST.2010.030304

[7] H. Hirayama and K. Shin, “Application of the EGS4 Monte Carlo Code to a Study of Multilayer Gamma-Ray Expo- sure Buildup Factors,” Journal of Nuclear Science and Technology, Vol. 35, No. 11, p. 816. doi:10.3327/jnst.35.816

[8] D. H. Wright, “Physics Reference Manual,” 2001. http://cern.ch/geant4

[9] G. C. Pomraning, “Flux-Limited Diffusion and Fokker- Planck Equations,” Nuclear Science and Engineering, Vol. 85, No. 2, 1983, p. 116.

[10] J. Wood, “Computational Methods in Reactor Shielding,” Pergamon Press, Oxford, 1982.

[11] S. Agostinelli, et al., “Geant4: A Simulation Toolkit,” Nu- clear Instruments and Methods in Physics Research A, Vol. 506, No. 3, 2003, pp. 250-303.

[12] DOORS 3.1, “One, Two and Three Dimensional Discrete Ordinates Neutron Photon Transport Code System,” Ra- diation Safety Information Computational Center (RSICC), Code Package CCC-650, Oak Ridge, Tennessee, 1996.

[13] B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “The Determination of the Exposure Buildup Factor Formula- tion in a Slab Using the LTSN Method,” Kerntechnik, Vol. 71, No. 4, 2006, pp. 182-184.