Journal of Nuclear Energy Science & Power Generation TechnologyISSN: 2325-9809

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Research Article, J Nucl Ene Sci Power Generat Technol Vol: 3 Issue: 2

Estimation of Exposure Buildup Factor in Iron Using Different Methods: A Comparative Study

Danial Salehi1,2*, Dariush Sardari1 and M Salehi Jozani3
1Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2Young Researchers and Elites club, Science and Research Branch, Islamic Azad University, Tehran, Iran
3Department of Electrical Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran
Corresponding author : Danial Salehi
Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
Tel: +98-91-9803-3577
E-mail: [email protected]; [email protected]
Received: April 15, 2014 Accepted: June 20, 2014 Published: June 27, 2014
Citation: Salehi D, Sardari D, Jozani MS (2014) Estimation of Exposure Buildup Factor in Iron Using Different Methods: A Comparative Study. J Nucl Ene Sci Power Generat Technol 3:2. doi:10.4172/2325-9809.1000122

Abstract

The gamma exposure buildup factor in iron is computed by using different methods of GP fitting method, invariant embedding, a simulation program written by Monte Carlo method to calculate this factor and MCNP4C code. Coherent scattering effect on this factor it has been studied as well. The results are obtained in the energy range 0.1-10 MeV and penetration depths up to 25 mfp and the result are compared with GP fitting method.

Keywords: Exposure buildup factor; Coherent scattering; Iron; Monte carlo method; GP fitting; Invariant embedding method

Keywords

Exposure buildup factor; Coherent scattering; Iron; Monte carlo method; GP fitting; Invariant embedding method

Introduction

Estimation of photon fields under poor geometry conditions and the effect of scattered photons in addition to unscattered primary photons, is best dealt with by a factor that is called “Buildup factor” (B), which is greater than1.0 to account for photons which are scattered towards the receptor from regions outside the primary beam. When the buildup is included, the radiation intensity is (I = B I0 e-μx).
The buildup factor B and the exponential attenuation factor e-μx, are functions of μx (or number of mean free paths (mfp)) which can complicate calculations of the shield thickness x to reduce a photon intensity from I0 to I(x).
This factor is dependent on the absorbing medium, the photon energy (E), the attenuation coefficient for specific energy photons in the medium (μ), and the absorber thickness (x).
Shimizu [1,2] applied Invariant Embedding (IE) method for calculations of exposure buildup factors for point isotropic sources in infinite homogeneous media up to depths of 100 mean free paths (mfp) without bremsstrahlung in water, iron and lead for typical source energies of 10 MeV, 1.0 MeV and 0.1 MeV.
Durani [3], carried out a study to update gamma-ray buildup factors for iron and water as a function of source energy that are present in ANS Standard by using ENDF/B-VI.8 photo-atomic crosssection library data in Monte Carlo N Particle 5 (MCNP5) version 1.40.In another work Abdi Saray et al. [4] calculated EBF for the pointed and isotropic gamma sources in the uranium, aluminum and iron by using MCNP4C code, in the range of 0.5-25 mean free path (mfp). Shirani and Alamatsaz [5,6] investigated the effects of including incoherent and coherent scattering in exposure buildup factor calculations for point isotropic gamma ray sources, in water and lead and penetrating a two-layer water-lead shield in the gamma ray energy (Eγ) range of 40 keV to 3 MeV.
The buildup factors presented in this work are for exposure inthe air after penetration through the absorber or shielding material that called the “Exposure Buildup factor” (EBF). Other types of buildup factors also exist, in particular energy absorption buildup factors (EABF) for energy deposition in an absorbing medium and dose buildup factors in absorbing media. Since a primary assessment in radiation protection is the exposure field before and after use of a radiation shield, exposure buildup factors (as provided in this study) are of more general use with appropriate adjustments of the air exposure to obtain absorbed dose [7].

Materials and Methods

GP Fitting method
The geometric progression (G-P) fitting formula has been developed by Harima [8].This is a theoretical method and is presented to determine the exposure buildup factors in most of the elements. The fitting parameters obtained by the GP formula and Taylor’s formula are compiled in ANSI/ANS 6.4.3 [9]. GP fitting parameters for the exposure build-up factor (b, c, a, X k and d) of iron were taken from this reference and are shown in Table 1. To calculate the buildup factors, the G-P fitting parameters were obtained by the method of interpolation from the equivalent atomic number (Zeq) [10]. Computations are divided into the following steps:
Table 1: Exposure G-P fitting parameters for iron.
Calculation of equivalent atomic number (Zeq): In the first step, the equivalent atomic number Zeq, of a particular material was calculated by matching the ratio,(μ/ρ)Compton /(μ/ρ)total, of that material at a specific energy with the corresponding ratio of an element at the same energy.
Thus, firstly the Compton partial mass attenuation coefficient (μ/ρ)Compton, and the total mass attenuation coefficients (μ/ρ)total, were obtained for iron in the energy region 0.1-10MeV, using the WinXCom computer program [11,12] initially developed as XCOM [13].
For the interpolation of Zeq for which the ratio (μ/ρ)Compton/ (μ/ρ)total lies between two successive ratios of elements, the value of Zeq was calculated by using the following formula:
(1)
where Z1 and Z2 are the elemental atomic numbers corresponding to the ratios R1 and R2 respectively, and R is the ratio for the iron at the specific energy.
Calculation of the geometric progression (GP) fitting parameters: In the second step, to calculate the G-P fitting parameters, a similar interpolation procedure was adopted as in the case of equivalent atomic number. The G-P fitting parameters for elements were taken from the ANSI/ANS-6.4.3 [9] standard reference database, which provides the G-P fitting parameters for elements from beryllium to iron in the energy region 0.015-15 MeV up to a depth of 40 mfp. G-P fitting buildup factor coefficients of the used materials were interpolated according to the given formula as follows:
(2)
where P is the G-P fitting function coefficient corresponding to Zeq, P1 and P2 are the values of G-P fitting function coefficients corresponding to the element atomic numbers Z1 and Z2, respectively, at a given energy, whereas Zeq is the equivalent atomic number of the chosen material at the given energy.
Calculation of the exposure buildup factor: In the final step, the computed G-P fitting parameters (b, c, a, Xk and d) are used to compute the exposure build-up factors of iron in the energies 0.1-10 MeV and penetration depths 0.5-25 mean free path (Table 1) with the help of the G-P fitting formula, as given by the equations [14]:
(3)
(4)
where
(5)
and E is the incident photon energy, x is the penetration depth in mfp, a, b, c, d and Xk are the G-P fitting parameters and b is the value of the buildupfactorat1mfp.Theparameter K represents photon dose multiplication and a change in the shape of the spectrum.
Invariant Embedding method
The buildup factors are calculated by the IE method. The detailed description of the IE method is found in [1,2,15]. The method of direct numerical integration is used. The solution for the modified transmission at the initial space mesh is obtained by solving its equation by the Runge-Kutta method. The solution for the modified transmission at the initial space mesh is obtained by solving its equation by the Runge-Kutta method. The solutions for extended thickness are obtained by using the functional relation for the modified transmission function. The initial space mesh Δ is chosen as 1/512 mfp for most low-Z materials and as 1/32,768 mfp for high-Z elements with bremsstrahlung. It was confirmed that the error in the buildup factor due to the space mesh is negligible (less than 0.02%) up to depths of 100 mfp. Even starting from a very fine initial mesh, calculations can efficiently be extended to large depths, since the solution for a double thickness can successively be obtained by using the functional relation. This is an excellent feature of the IE method [2]. The complete description of this method exists in mentioned references.
Monte Carlo methods
Monte Carlo simulation program: In the following we give a brief description of the Monte-Carlo simulation program which was written to calculate the exposure buildup factors [5]. In the program a gamma ray point source of definite energy Eγ is assumed to be located at the center of a spherical shield consisting of a thickness of Iron. The initial direction of a source photon is selected by means of internally generated random numbers to simulate an isotropic source. Photons generated in this way travel in the shield and interact with the shielding media according to various types of interactions and their corresponding cross sections. By this method the history of each photon (from the time it is produced by the source until it is absorbed or leaves the system under consideration) is determined. If Ei is the energy of the ith photon that has crossed the surface of a sphere with radius r, and θi is the angle between the direction of this photon and the normal to the surface at the crossing point, the exposure buildup factor for photons of energy Eγ at a distance r from the source is calculated as follows:
(6)
where r is the thickness of the medium (iron), N0 is the total number of the source photons generated with energy Eγ, N’ is the number of photons that has passed the surface of the sphere with radius r, μ1 is total linear attenuation coefficients of the Iron, (μen / ρ) air and (μen / ρ) iair are mass energy absorption coefficients of air for photons of energy Eγ and Ei respectively. In the above relation, θ = π/2 can only occur when a photon passes tangent to a given sphere of radius r. This case is not considered as a hit to that sphere and therefore no singularity can ever happen. The type of interactions considered in the program are photoelectric, either Compton scattering (Klein-Nishina cross section applied to free electrons) or incoherent scattering (which is Compton scattering considering binding effects of electrons), coherent (Rayleigh) scattering and pair production.
MCNP: Utilizing the macroscopic cross sections for iron in the Monte Carlo code MCNP4C, and considering slab geometry, buildup factor for iron was computed for various energies of incident photons. The results of two methods are presented in Table 2. In this case the required data was obtained from the library of MCNP computer code.
Table 2: Exposure buildup factors data obtained by different methods for iron at different energies.

Results and Discussion

Variation of EBF with Energy and Penetration Depths
The variation of EBF with incident photon energy is shown in Figure 1. It can be seen EBF has low values at lower and higher energies. This is due to the dominance of photoelectric absorption and pair production, which result in the complete removal of photons. The maximum values of EBF were observed at intermediate energies, where Compton scattering dominates. In this process, the photons are not completely removed but only their energies are degraded. Besides, their directions are changed. Hence, this process results in more multiple scattered photons, which leads to increase in the buildup of photons in the medium.
Figure 1: Variation of exposure buildup factor (EBF) with incident photon energy (MeV) for Iron.
EBF increases with increasing penetration depth; these various complexities are illustrated in Figure 2. It has been shown that with increasing penetration depths, EBF also increase due to increase in number of scattered photons.The maximum values of the EBF, which are in the order of 50, have been obtained at the largest penetration depth (40mfp) and in 15 MeV [3].
Figure 2: Variation of exposure buildup factor with penetration depth of Iron at energy ranges 0.1-8 MeV.
EBF for Iron obtained using various methods
Table 2 lists the available exposure buildup factor data obtained by different methods and previously published data [16] for iron at different energies for different penetration depths.
Comparisons with results of the literatures show that the absolute deviation of the GP fitting method is less than 3%. So it could be said that the GP fitting method is most suitable method for the calculation of EBF in Iron. These results indicate that the GP fitting method is able to estimate photon exposure build up factors up to depths of 40 mfp [9]. Figures 3-5 are shown comparison of exposure buildup factor data obtained bydifferent methods at energies 0.1, 1 and 10 MeV for iron. The relative differences in exposure buildup factors between the MCNP code the IE method and GP fitting method are shown in Figures 6 and 7 for iron. These results indicate that the deviation of the Monte-Carlo simulation program and MCNP 4C code used in this study with GP fitting method is approximately 6% and the IE method has about 9% difference. It has been observed that the maximum values of the EBF in this range of energy occur in 0.5MeV.
Figure 3: Comparison among exposure buildup factor data obtained in the GP fitting, IE, Monte Carlo method and MCNP for Iron at 0.1 MeV.
Figure 4: Comparison among exposure buildup factor data obtained in the GP fitting, IE, Monte Carlo method and MCNP for Iron at10 MeV.
Figure 5: Comparison among exposure buildup factor data obtained in the GP fitting, IE, Monte Carlo method and MCNP for Iron at 10 MeV.
Figure 6: Relative difference (%) between the buildup factor values for iron obtained through IE and GP fitting approximation.
Figure 7: Relative difference (%) between the buildup factor values for iron obtained through MCNP and GP fitting approximation.
Effect of Coherent Scattering
Because in GP fitting method, total attenuation coefficient with incoherent scattering is used to calculate equivalent atomic number, so coherent scattering effect have been studied by MCNP code in this section. Since no energy change and only a small change in direction is associated with the coherent scattering of photons, the matter that has been ignored in the buildup factor data was coherent scattering. This is true only for photons above a few hundred keV. The probability that the photon is scattered by a large angle increases below a few hundred keV, especially for a high Z material [17,18].
The consequence of coherent scattering is to deflect the photons from their direction, which as a result increases the path length between the source and any point of observation in that geometry. Subbaiah et al. [19] found that the inclusion of coherent scattering in the gamma-ray calculations leads to an enormous reduction in flux and dose in the deep penetration problems and is essential to all materials in low energy zones. Figures 8-13 show that coherent scattering decrease the value of the exposure buildup factor at all energies and this effect is more significant in the energy range 0.02 ≤ Eγ ≤ 0.2 MeV at large energy valuesand large values of mfp.
Figure 8: Exposure buildup factors with coherent scattering effect forEγ=0.02 MeV (without coherent scattering, with coherent scattering).
Figure 9: Exposure buildup factors with coherent scattering effect forEγ=0.04 MeV (without coherent scattering, with coherent scattering).
Figure 10: Exposure buildup factors with coherent scattering effect forEγ=0.06 MeV (without coherent scattering, with coherent scattering).
Figure 11: Exposure buildup factors with coherent scattering effect forEγ=0.08 MeV (without coherent scattering, with coherent scattering).
Figure 12: Exposure buildup factors with coherent scattering effect forEγ=0.1 MeV (without coherent scattering, with coherent scattering).
Figure 13: Exposure buildup factors with coherent scattering effect forEγ=0.2 MeV (without coherent scattering, with coherent scattering).

Conclusion

Exposure buildup factors were calculated for iron using GP fitting and obtained with three other different methods. The coherent scattering effect on these factors was investigated. The results showed that the effects of coherent scattering in iron are considerable for gamma rays of energies up to about 0.2 MeV. The effects of coherent scattering are more significant in R ≤ 8 mfp and decreases the exposure buildup factor values in higher energy and penetration depths.

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