Gamma-ray buildup factors study for deep penetration in some silicates

Annals of Nuclear Energy 51 (2013) 81–93

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Gamma-ray buildup factors study for deep penetration in some silicates Kulwinder Singh Mann a,⇑, Turgay Korkut b a b

Department of Physics, DAV College, Dyanad Nagar, Bathinda (Pb.) 151 001, India Department of Physics, Faculty of Science and Art, Ag˘rı Ibrahim Çeçen University, 04100 Ag˘rı, Turkey

a r t i c l e

i n f o

Article history: Received 22 April 2012 Received in revised form 2 August 2012 Accepted 5 August 2012 Available online 9 October 2012 Keywords: Gamma-ray buildup factor Equivalent atomic number Silicates Deep penetration GEANT4

a b s t r a c t The gamma ray buildup factors for six silicate samples have been calculated, in the energy range of 0.015–15 MeV for penetration depths up to 100 mfp (mean free path), using five parameters based geometric progression (G-P) fitting formula with modified expression for dose multiplication factor [K(E, x)]. The computations were done using ANSI/ANS 6.4.3-1991 (American National Standard). The extrapolation to the buildup factors of the selected samples beyond 40 mfp and up to 100 mfp in this energy range are new to the available literature. Calculated buildup factors of water have been shown good agreement with the available standard data. The obtained results for all samples have been compared and verified by using WinXCom software and GEANT4 Monte Carlo simulations. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The energetic gamma rays are hazardous for living cells and tissues. Therefore a detailed study is required for the safe and acceptable use of gamma radiations, radioactive materials and nuclear energy. Additional area of interest is gamma radiation shielding development has been grown considerably due to leakage of radioactive materials, dumping of nuclear waste, increasing demand of radiotherapy and nuclear weapons. The basic objective of the radiation shielding is to protect the living beings from the hazardous effects of the gamma radiations. In order to develop effective shields for radioactive waste management the understanding of deep penetration of gamma rays is required. The present work will be helpful for reducing radioactive radiation pollution to the environment. The buildup factors are used to obtain the corrected response to the uncollided photons by including the contribution of the scattered photons. It can be defined as the ratio of the total detector response to that of uncollided photons. The buildup factor measures the degree of violation of the Lambert–Beer law (I = Io  elx) due to multiple scattering of photons. The modified equation becomes I = B  Io  elx (Singh et al., 2008), where B is the buildup factor for one energy at the shield thickness x, Io is the initial dose rate, I is the shielded dose rate, l is linear attenuation coefficient in cm1 and ‘x’ is the shield thickness in cm. The average distance that photons of a given energy travel before an interaction in a given medium is equal to the reciprocal of the attenuation coefficient. ⇑ Corresponding author. Tel.: +91 9417325696; fax: +91 1642214666. E-mail address: [email protected] (K.S. Mann). 0306-4549/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2012.08.024

The distance x in ordinary units can be converted into the dimensionless quantity lx, termed as mean-free-path (mfp). This parameter ‘B’ is always equal to or greater than unity (B = 1; in case narrow beam geometry, interacting material is thin and the photon is assumed to be mono-energetic, otherwise it is greater than unity). Buildup factor has been classified into two categories named as energy absorption buildup factor (EABF) and exposure buildup factor (EBF). The EABF is the buildup factor in which the quantity of interest is the absorbed or deposited energy in the interacting material and the detector response function is that of absorption in the interacting material. Whereas for the EBF the quantity of interest is the exposure and the detector response function is that of the absorption in air; that is, exposure is assumed to be equivalent to the absorbed dose in air as measured by the nonperturbing detector. Different methods such as G-P fitting method (Harima et al., 1986), invariant embedding method (Shimizu, 2002; Shimizu et al., 2004), iterative method (Suteau and Chiron, 2005) and Monte Carlo method (Sardari et al., 2009) are available for computing buildup factors. It was shown by Shimizu et al. (2004) that by three different approaches (invariant embedding, G-P fitting and Monte Carlo methods) agree well for 18 low-Z materials within small discrepancies. When compared with other available approximations such as Berger, Taylor and three exponential, the geometric progression (G-P) fitting seems to reproduce the buildup factors with better accuracy. American National Standards ANSI/ANS 6.4.3 (American National Standard, 1991) has provided five G-P fitting parameters and buildup factor data for 23 elements (Be, B, C, N, O, Na, Mg, Al, Si, P, S, Ar, K, Ca, Fe, Cu, Mo, Sn, La, Gd, W, Pb and U), one compound (H2O) and two mixtures (air and concrete) at

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

Table 1 Elemental composition of the silicate-samples (Mann and Sidhu, 2012).

Table 3 Equivalent atomic numbers of silicate samples for incident photon energy range 0.015–15 MeV.

Sample

Symbols

Chemical formula and weight fraction in percentage

Kyanite Sodium silicate Datolite

S1 S2

Al2SiO5; O(49.37), Al(33.30), Si(17.33) Na2SiO4;O(39.32), Na(37.67), Si(23.01)

S3

Diopside

S4

Slag

S5

Anorthite

S6

Water

Water

CaBSiO4(OH); H(0.63), B(6.76), O(50.00), Si(17.56), Ca(25.05) CaMgSi2O6; O(44.33), Mg(11.22), Si(25.94), Ca(18.51) Mg3Si4O12H2; H(0.53), O(50.62), Mg(19.22), Si(29.62) CaAl2Si2O8; O(46.01), Al(19.40), Si(20.19), Ca(14.41) H2O; H(11.20), O(88.80)

Energy (MeV)

25 standard energies in the energy range 0.015–15 MeV up to the penetration depth of 40 mean free path (mfp). The data are intended to be standard reference data for use in radiation analyses employing point-kernel methods. Harima et al. (1986) have showed that the absolute value of the maximum deviation of exposure buildup factors for water in the G-P fitting is within 0.5–3%, in the three-exponential approach is within 0.4–9.3%, in the Berger approach is within 0.9–42.7% and in the Taylor approximation is within 0.4–53.2%. ANSI/ANS-6.4.3 (1991) standard has been administratively withdrawn in 2001, but the work is in progress for updating this much used standard by a working group chartered in 2007 by the American Nuclear Society (ANS). The reasons for the revision were (a) lack of self-consistency within the standard in which the mass attenuation coefficients tabulated in the standard were not necessarily used in the calculation of the buildup factors and the correction factors for coherent scattering and tissue dose, and (b) newer, presumably, more accurate attenuation coefficient data have become available (Chadwick et al., 2006) since the issuance of the 1991 standard (Ryman et al., 2008). Recently, a study has been made for the purpose of updating gamma-ray buildup factors for high-Z engineering materials that are presented in the current ANS standard (Ruggieri and Sanders, 2008). The working groups could proceed along a track of reaffirmation of the standards using PINS (Project Initiation Notification System) charter form. The G-P fitting method has been used by different researchers Singh et al. (2008, 2009), Kurudirek and Topcuoglu (2011) and Mann and Sidhu (2012) for calculating buildup factors up to 40 mfp. Brar et al. (1994), using G-P fitting formula has calculated the EABF for water, air and concrete up to 100 mfp. In the present work the data for buildup factors up to 100 mfp for some low-Z silicates have been generated using G-P fitting formula with modified expression for the dose multiplication factor K(E, x) (Sakamoto and Trubey, 1991). This work has been done with following objectives kept in mind. Firstly to check whether the G-P fitting formula is able to generate correctly the buildup factor data for deep penetration of 100 mfp? Secondly study the variations of

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

Equivalent atomic number (Zeq) S1

S2

S3

S4

S5

S6

11.31 11.34 11.39 11.40 11.43 11.43 11.39 11.58 11.98 11.50 11.50 11.50 11.50 11.50 11.50 11.50 11.50 9.79 10.37 10.69 10.62 10.92 10.42 10.63 10.61

10.98 11.04 11.04 11.10 11.16 11.13 11.17 11.15 10.94 12.95 12.50 12.50 12.50 12.50 12.50 12.50 12.50 8.80 10.40 10.40 10.35 10.69 10.48 10.56 10.59

13.71 13.89 14.08 14.16 14.26 14.34 14.42 14.38 14.41 14.94 14.50 14.50 14.50 14.50 14.50 14.50 14.50 12.81 12.48 11.32 11.87 11.83 11.85 11.81 11.77

13.59 13.70 13.83 13.88 13.98 14.08 14.05 14.12 14.43 14.48 14.50 14.50 14.50 14.50 14.50 14.50 14.50 12.83 12.51 12.23 12.44 12.06 12.39 12.13 12.27

11.15 11.19 11.24 11.28 11.34 11.34 11.35 11.55 10.90 10.50 10.50 10.50 10.50 10.50 10.50 10.50 10.50 9.70 10.28 10.57 10.14 10.40 10.49 10.50 10.51

13.17 13.27 13.38 13.46 13.52 13.57 13.52 13.69 13.44 14.49 14.50 14.50 14.50 14.50 14.50 14.50 14.50 9.76 12.58 11.94 11.99 11.92 12.16 11.91 12.01

buildup factors with penetration depths. An attempt has been made to compute EBF and EABF values by using the G-P fitting method for the selected samples in the energy range 0.015– 15 MeV up to 100 mfp. The present study will be of prime importance for radiation shield designing (Suteau and Chiron, 2005) and production of a new materials made from the selected silicate samples for effective gamma radiation shielding. Mortazavi and Mosleh-Shirazi (2010) have been experimented to use datolite (S3) in heavy concrete for shielding nuclear reactors and megavoltage radiotherapy rooms. All the selected materials can be used in concrete and building materials (glass, tile, fibre–glass) for improving gamma-ray shielding properties. 2. Material and methods The elemental compositions by weight percentage of the selected samples (kyanite, sodium silicate, datolite, diopside, slag and anorthite) have been taken from the literature (Mann and Sidhu, 2012) and listed in Table 1. The physical properties of the selected sample have been listed in Table 2. The selected silicates can be used in hardening/waterproofing of concrete, acid-proof cements, thermal insulation, plywood laminating, soil solidification, cement slurry thinners and roofing granules. In extruded brick and clay products, soluble silicate can reduce the force (power) required for extrusion, as well as decrease shrinkage on firing. The

Table 2 Physical properties of the samples (Mann and Sidhu, 2012). Properties

S1

S2

S3

S4

S5

S6

Chemical formula Density (in g/cm3) Molecular-weight (in g) Hardness (in mohs) Si content (% by wt.) Colour

Al2SiO5 3.56–3.67 162.05

Na2SiO4 2.30–2.50 138.06

CaBSiO4(OH) 2.96–3.00 159.98

CaMgSi2O6 3.25–3.55 216.55

Mg3Si4O12H2 3.20–3.60 379.27

CaAl2Si2O8 2.74–2.76 278.21

4.0–7.0 17.33 Blue, White, Gray, Green, Black.

6.0–6.5 23.01 Colourless, white

5.0–5.5 17.56 Colourless, white

6.0–6.5 25.94 Blue, Brown, Colourless, Green, Gray

6.0–7.0 29.62 Gray, Brown

6.0–6.5 20.19 Colourless, reddish grey, white

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93 Table 4 G-P fitting function coefficients for EABF and EBF of kyanite (S1). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.047 1.109 1.358 1.761 2.393 2.992 3.985 4.389 4.072 3.632 3.041 2.737 2.545 2.410 2.233 2.114 1.936 1.845 1.706 1.614 1.542 1.479 1.388 1.324 1.225

0.406 0.416 0.461 0.598 0.686 0.857 1.144 1.367 1.660 1.733 1.735 1.674 1.615 1.556 1.458 1.382 1.253 1.154 1.050 0.983 0.946 0.929 0.903 0.894 0.892

0.201 0.196 0.184 0.126 0.107 0.053 0.021 0.064 0.113 0.123 0.125 0.117 0.110 0.102 0.088 0.076 0.054 0.034 0.009 0.009 0.019 0.024 0.033 0.036 0.038

12.038 14.532 14.517 15.802 14.366 13.821 12.929 13.002 13.404 13.734 14.028 14.341 14.351 14.571 14.838 14.996 14.280 14.750 10.630 13.160 12.670 15.970 12.280 13.930 14.720

0.105 0.105 0.097 0.066 0.066 0.040 0.004 0.017 0.043 0.046 0.045 0.041 0.039 0.035 0.031 0.027 0.021 0.013 0.001 0.012 0.015 0.027 0.023 0.029 0.033

1.047 1.109 1.348 1.710 2.263 2.668 3.135 3.250 3.111 2.911 2.627 2.458 2.331 2.236 2.108 2.013 1.879 1.795 1.684 1.606 1.536 1.494 1.406 1.347 1.263

0.386 0.413 0.468 0.617 0.692 0.857 1.115 1.301 1.514 1.581 1.597 1.563 1.525 1.485 1.410 1.353 1.234 1.154 1.056 0.992 0.956 0.914 0.902 0.876 0.823

0.219 0.200 0.181 0.118 0.104 0.053 0.012 0.049 0.085 0.095 0.100 0.097 0.093 0.088 0.078 0.069 0.049 0.034 0.011 0.006 0.016 0.031 0.033 0.043 0.064

12.209 13.826 14.511 15.879 14.239 14.017 13.631 13.353 14.896 15.326 15.088 15.281 15.468 15.345 15.429 15.695 15.210 14.430 10.470 12.860 15.200 11.380 13.530 13.190 14.350

0.120 0.106 0.096 0.060 0.060 0.044 0.011 0.004 0.020 0.024 0.025 0.026 0.027 0.025 0.024 0.023 0.017 0.012 0.001 0.009 0.021 0.025 0.026 0.034 0.056

Table 5 G-P fitting function coefficients for EABF and EBF of sodium-silicate (S2). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.051 1.120 1.393 1.829 2.499 3.122 4.114 4.471 4.054 3.620 3.030 2.727 2.541 2.405 2.229 2.113 1.936 1.845 1.706 1.614 1.542 1.479 1.388 1.324 1.225

0.409 0.410 0.470 0.622 0.724 0.904 1.197 1.420 1.707 1.767 1.757 1.692 1.628 1.564 1.465 1.385 1.253 1.154 1.050 0.983 0.946 0.929 0.903 0.894 0.892

0.197 0.202 0.181 0.117 0.093 0.038 0.033 0.074 0.120 0.128 0.128 0.120 0.113 0.103 0.089 0.077 0.054 0.034 0.009 0.009 0.019 0.024 0.033 0.036 0.038

11.440 14.549 14.660 16.087 14.823 14.234 12.039 13.248 13.488 13.764 13.926 14.219 14.287 14.564 14.852 14.904 14.280 14.750 10.630 13.160 12.670 15.970 12.280 13.930 14.720

0.091 0.110 0.095 0.061 0.063 0.033 0.003 0.023 0.047 0.049 0.047 0.043 0.040 0.036 0.032 0.028 0.021 0.013 0.001 0.012 0.015 0.027 0.023 0.029 0.033

1.052 1.121 1.381 1.763 2.360 2.790 3.251 3.345 3.164 2.948 2.647 2.473 2.343 2.247 2.114 2.019 1.879 1.795 1.684 1.606 1.536 1.494 1.406 1.347 1.263

0.379 0.399 0.478 0.652 0.728 0.896 1.163 1.347 1.555 1.611 1.620 1.578 1.540 1.493 1.417 1.355 1.234 1.154 1.056 0.992 0.956 0.914 0.902 0.876 0.823

0.223 0.211 0.177 0.103 0.091 0.042 0.023 0.058 0.092 0.100 0.104 0.099 0.096 0.089 0.079 0.070 0.049 0.034 0.011 0.006 0.016 0.031 0.033 0.043 0.064

11.690 13.562 14.585 16.290 14.292 13.474 13.588 14.039 14.772 14.868 14.465 15.159 15.081 15.392 15.271 15.698 15.210 14.430 10.470 12.860 15.200 11.380 13.530 13.190 14.350

0.115 0.113 0.093 0.051 0.056 0.037 0.005 0.009 0.024 0.027 0.027 0.028 0.029 0.026 0.025 0.023 0.017 0.012 0.001 0.009 0.021 0.025 0.026 0.034 0.056

addition of kyanite can improve capacity of concrete shield to bear high-temperature and pressure conditions in nuclear reactors (Winter and Ghose, 1979; Comodi et al., 1997; Rao et al., 1999; Friedrich et al., 2004). The mass attenuation coefficients of the samples in the energy region 0.015–15 MeV have been calculated using WinXCom (computer-software) (Gerward et al., 2004) and GEANT4 (Monte Carlo Simulations). The G-P fitting function coefficients of a sample can be obtained by the method of interpolation which then used to find the equivalent atomic number (Zeq) of the sample. Finally the equivalent atomic number (Zeq) can be used to

interpolate the values of EBF and EABF at specified energy and penetration depth. It is to be noted that the G-P method provides good results for narrow beam condition (narrow beam geometry). 3. Calculations 3.1. Calculation of equivalent atomic number (Zeq) The equivalent atomic number (Zeq) of a particular material has been interpolated by matching the ratio [(l/q)Compton/(l/q)Total] of

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

Table 6 G-P fitting function coefficients for EABF and EBF of datolite (S3). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.024 1.053 1.178 1.387 1.650 2.117 2.945 3.597 3.994 3.697 3.124 2.797 2.583 2.440 2.244 2.118 1.940 1.834 1.697 1.608 1.548 1.475 1.382 1.313 1.229

0.401 0.417 0.395 0.452 0.584 0.582 0.783 0.982 1.300 1.452 1.549 1.543 1.516 1.482 1.414 1.358 1.247 1.165 1.059 0.990 0.927 0.929 0.908 0.912 0.856

0.204 0.186 0.215 0.190 0.128 0.148 0.076 0.021 0.051 0.077 0.094 0.095 0.092 0.088 0.079 0.071 0.052 0.036 0.012 0.007 0.027 0.025 0.032 0.031 0.053

11.965 17.257 14.370 14.619 16.438 13.604 13.807 13.735 17.087 15.385 14.271 14.997 15.155 14.963 15.168 15.000 14.635 14.425 14.174 14.200 12.946 15.545 12.036 14.412 14.304

0.103 0.110 0.117 0.105 0.066 0.081 0.055 0.033 0.005 0.018 0.025 0.027 0.027 0.027 0.025 0.024 0.020 0.013 0.000 0.010 0.025 0.028 0.023 0.026 0.047

1.025 1.053 1.177 1.372 1.613 1.849 2.324 2.542 2.670 2.627 2.451 2.336 2.233 2.157 2.041 1.966 1.860 1.787 1.675 1.600 1.538 1.486 1.405 1.343 1.256

0.361 0.421 0.395 0.456 0.564 0.680 0.806 0.984 1.234 1.343 1.447 1.452 1.440 1.419 1.378 1.323 1.231 1.155 1.063 0.996 0.944 0.929 0.902 0.884 0.841

0.247 0.184 0.216 0.189 0.142 0.100 0.068 0.020 0.034 0.052 0.074 0.077 0.077 0.075 0.072 0.063 0.048 0.033 0.009 0.005 0.022 0.026 0.034 0.041 0.059

12.940 17.207 14.014 14.348 15.010 15.130 13.989 13.835 11.060 8.462 18.298 16.693 16.307 17.172 15.524 16.632 15.405 16.102 15.598 12.988 11.239 11.867 13.774 13.115 14.268

0.168 0.109 0.116 0.107 0.076 0.055 0.051 0.034 0.010 0.007 0.013 0.015 0.017 0.019 0.021 0.020 0.017 0.011 0.002 0.009 0.021 0.020 0.028 0.033 0.052

Table 7 G-P fitting function coefficients for EABF and EBF of diopside (S4). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.025 1.057 1.188 1.410 1.684 2.174 3.035 3.680 4.026 3.685 3.116 2.788 2.576 2.436 2.243 2.121 1.940 1.834 1.695 1.607 1.545 1.472 1.379 1.309 1.230

0.394 0.404 0.398 0.458 0.603 0.603 0.812 1.014 1.331 1.485 1.568 1.559 1.530 1.490 1.420 1.358 1.246 1.163 1.060 0.992 0.927 0.931 0.910 0.915 0.842

0.210 0.197 0.214 0.188 0.120 0.140 0.067 0.012 0.057 0.083 0.098 0.098 0.095 0.090 0.080 0.071 0.052 0.035 0.012 0.006 0.027 0.025 0.032 0.030 0.059

12.367 16.667 14.344 14.629 16.793 13.563 13.952 13.760 15.052 14.759 14.375 14.756 14.997 14.989 15.120 14.989 14.678 14.606 13.647 14.045 13.293 15.337 12.433 14.513 14.168

0.115 0.114 0.116 0.104 0.061 0.078 0.050 0.028 0.008 0.022 0.028 0.029 0.029 0.028 0.026 0.024 0.020 0.012 0.000 0.009 0.026 0.029 0.024 0.026 0.053

1.025 1.056 1.186 1.393 1.643 1.891 2.384 2.603 2.711 2.657 2.470 2.347 2.244 2.168 2.047 1.974 1.857 1.785 1.674 1.600 1.536 1.483 1.403 1.340 1.251

0.367 0.414 0.399 0.464 0.581 0.701 0.831 1.008 1.259 1.363 1.460 1.464 1.449 1.422 1.382 1.323 1.231 1.154 1.062 0.991 0.945 0.934 0.905 0.889 0.851

0.239 0.191 0.213 0.186 0.134 0.093 0.060 0.014 0.039 0.056 0.076 0.079 0.079 0.076 0.072 0.063 0.048 0.033 0.007 0.007 0.023 0.024 0.033 0.040 0.056

13.419 16.540 14.173 14.368 15.152 15.279 14.520 13.701 10.627 8.008 17.481 16.201 16.303 17.998 15.390 16.182 15.714 15.797 14.881 12.827 10.343 12.082 13.856 13.077 14.376

0.163 0.111 0.114 0.105 0.071 0.052 0.051 0.031 0.008 0.006 0.013 0.015 0.017 0.020 0.021 0.019 0.017 0.010 0.002 0.011 0.020 0.020 0.028 0.033 0.050

the material at a specific energy with the corresponding ratios for the elements at the same energy. For the interpolation of Zeq for which the ratio [(l/q)Compton/(l/q)Total] lies between two successive ratios of elements with atomic numbers Z1 and Z2, the following formula (Sidhu et al., 1999a,b) has been used:

Z eq ¼

Z 1 ðlog R2  log RÞ þ Z 2 ðlog R  log R1 Þ log R2  log R1

ð1Þ

where Z1 and Z2 are the atomic numbers of elements corresponding to the ratios R1 and R2, respectively, R is the ratio for the selected sample at a specific energy. Let datolite be an example for calculation of Zeq at the energy of 0.015 MeV. The ratio [(l/q)Compton/(l/ q)Total] of datolite is 0.01384 which lies between the ratios 0.01656 and 0.01292 corresponding to the elements Z1 = 13 and Z2 = 14, respectively. Then, using these values in Eq. (1),

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93 Table 8 G-P fitting function coefficients for EABF and EBF of slag (S5). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.049 1.114 1.373 1.789 2.436 3.044 4.035 4.420 4.065 3.628 3.037 2.733 2.544 2.408 2.232 2.113 1.936 1.845 1.706 1.614 1.542 1.479 1.388 1.324 1.225

0.408 0.413 0.465 0.608 0.702 0.876 1.164 1.387 1.678 1.745 1.743 1.680 1.619 1.559 1.461 1.383 1.253 1.154 1.050 0.983 0.946 0.929 0.903 0.894 0.892

0.199 0.198 0.183 0.122 0.101 0.047 0.025 0.068 0.116 0.125 0.126 0.118 0.111 0.102 0.088 0.076 0.054 0.034 0.009 0.009 0.019 0.024 0.033 0.036 0.038

11.740 14.540 14.578 15.920 14.551 13.986 12.582 13.097 13.436 13.745 13.991 14.297 14.328 14.568 14.843 14.962 14.280 14.750 10.630 13.160 12.670 15.970 12.280 13.930 14.720

0.098 0.107 0.096 0.064 0.065 0.037 0.001 0.020 0.044 0.047 0.046 0.041 0.039 0.036 0.032 0.028 0.021 0.013 0.001 0.012 0.015 0.027 0.023 0.029 0.033

1.050 1.114 1.362 1.732 2.302 2.717 3.180 3.286 3.131 2.925 2.634 2.463 2.336 2.240 2.110 2.015 1.879 1.795 1.684 1.606 1.536 1.494 1.406 1.347 1.263

0.382 0.406 0.472 0.631 0.707 0.872 1.134 1.319 1.530 1.592 1.605 1.568 1.530 1.488 1.413 1.354 1.234 1.154 1.056 0.992 0.956 0.914 0.902 0.876 0.823

0.221 0.205 0.179 0.112 0.099 0.049 0.017 0.052 0.087 0.097 0.101 0.098 0.094 0.088 0.078 0.070 0.049 0.034 0.011 0.006 0.016 0.031 0.033 0.043 0.064

11.950 13.705 14.543 16.049 14.260 13.800 13.615 13.618 14.849 15.155 14.859 15.237 15.327 15.362 15.371 15.696 15.210 14.430 10.470 12.860 15.200 11.380 13.530 13.190 14.350

0.118 0.109 0.095 0.056 0.058 0.041 0.009 0.006 0.021 0.025 0.026 0.027 0.028 0.026 0.024 0.023 0.017 0.012 0.001 0.009 0.021 0.025 0.026 0.034 0.056

Table 9 G-P fitting function coefficients for EABF and EBF of anorthite (S6). E (MeV)

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

G-P energy absorption buildup coefficients

G-P exposure buildup coefficients

b

c

a

Xk

d

b

c

a

Xk

d

1.028 1.066 1.207 1.454 1.797 2.288 3.186 3.806 4.070 3.681 3.108 2.783 2.576 2.432 2.241 2.123 1.940 1.834 1.697 1.608 1.548 1.475 1.381 1.311 1.230

0.373 0.371 0.407 0.474 0.588 0.638 0.860 1.068 1.374 1.525 1.593 1.575 1.539 1.501 1.426 1.360 1.247 1.165 1.060 0.991 0.924 0.929 0.908 0.915 0.850

0.232 0.223 0.210 0.180 0.132 0.126 0.052 0.001 0.064 0.090 0.102 0.101 0.096 0.092 0.081 0.071 0.053 0.036 0.012 0.006 0.028 0.025 0.032 0.030 0.056

13.620 15.219 14.198 14.724 15.822 14.063 14.035 13.689 14.023 14.312 14.412 14.604 14.970 14.975 15.019 14.994 14.597 14.442 14.577 14.347 12.988 15.476 11.995 14.497 14.227

0.155 0.122 0.115 0.100 0.074 0.078 0.043 0.020 0.012 0.026 0.030 0.030 0.029 0.029 0.027 0.025 0.020 0.013 0.000 0.010 0.026 0.028 0.023 0.025 0.049

1.028 1.062 1.205 1.434 1.698 2.020 2.490 2.701 2.773 2.697 2.496 2.364 2.258 2.180 2.057 1.982 1.862 1.787 1.675 1.599 1.538 1.485 1.405 1.343 1.254

0.386 0.396 0.411 0.482 0.619 0.700 0.870 1.049 1.296 1.394 1.477 1.477 1.458 1.428 1.385 1.325 1.232 1.155 1.064 0.997 0.942 0.932 0.902 0.885 0.844

0.215 0.208 0.205 0.177 0.118 0.097 0.049 0.005 0.046 0.062 0.079 0.082 0.080 0.077 0.073 0.063 0.049 0.033 0.012 0.005 0.023 0.025 0.034 0.041 0.058

14.914 14.905 14.649 14.563 15.688 14.271 14.035 13.424 13.540 9.963 16.755 15.933 16.318 18.407 15.427 15.816 15.136 16.073 16.181 13.006 10.636 11.946 13.815 13.101 14.253

0.146 0.117 0.109 0.098 0.061 0.054 0.045 0.026 0.003 0.001 0.014 0.017 0.018 0.022 0.021 0.019 0.017 0.011 0.002 0.009 0.021 0.020 0.028 0.033 0.052

Zeq = 13.72 is obtained. The computed values of Zeq for the samples have been listed in Table 3.

3.2. Interpolation of G-P fitting function coefficients The computed values of Zeq for the selected samples were used to interpolate G-P fitting function coefficients (b, c, a, Xk and d) for the EBF and EABF, at the selected energy range and penetration depth. The formula (Sidhu et al., 1999a,b) used for the purpose of interpolation of the G-P fitting function coefficients is as follows:

P¼

P1 ðlog Z 2  log Z eq Þ þ P2 ðlog Z eq  log Z 1 Þ log Z 2  log Z 1

ð2Þ

HereP1 and P2 are the values of G-P fitting function coefficients corresponding to the elemental atomic numbers Z1 and Z2, respectively at a specific energy, whereas Zeq is the equivalent atomic number of the selected sample at the given energy. Using this formula, G-P fitting function coefficients for EBF and EABF of all the samples were computed for selected energies and penetration depths up to 40 mfp. The exposure and energy absorption G-P

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

Thus the buildup factor data for each sample has been obtained by using the Eqs. (3)–(5). Proposed by Harima et al. (1991) and supported by Sakamoto and Trubey (1991), the extrapolation of the dose multiplication factor K (E, x) for deep penetration above 40 mfp can be obtained as follows:

30

At 10 MeV

Water 25

IE (invariant embedding) EGS4 ANSI/ANS-6.4.3 G-P Fitting Method (WinXCom) G-P Fitting Method (GEANT4)

Exposure Buildup factor

20

 nðxÞ K 40  1 KðE; xÞ ¼ 1 þ ðK 35  1Þ K 35  1  KðE; xÞ ¼ K 35

15

nðxÞ ¼

K 40 K 35

nðxÞfm

ðx=35Þ0:1  1 ð40=35Þ0:1  1

for

;

for 0 6

K 40  1 61 K 35  1

K 40  1 K 40  1 < 0 or >1 K 35  1 K 35  1

f m ¼ 0:8;

ð6Þ

ð7Þ

K 35 ¼ KðE; 35Þ;

 K 40 ¼ KðE; 40Þ 10

Here the ‘fm’ is called adjustment parameter and ‘f(x)’ is called distribution function. 3.4. GEANT4 simulations

5

Geant4 is a toolkit for the simulation of the passage of particles through matter. Its areas of application include high energy, nuclear and accelerator physics, as well as studies in medical and space science (Geant4 Collaboration; Agostinelli et al., 2003; Allison et al., 2006). It has several electromagnetic interaction model such as Livermore, Penelope and Standard Models. For the low energetic gamma interactions Penelope Model has widely used. In the present study 4.9.4.p01 released version was used to simulate interactions between gammas and silicate samples.

0

1

10

100

Penetration Depth (mfp) Fig. 1. Standardisation and verification of the G-P fitting method by comparison of calculated and standard values of EBF of water at 10 MeV.

4. Results and discussion 4.1. Standardisation of the method used

fitting function coefficients of the samples have been listed in Tables 4–9. 3.3. Computation of EBF and EABF The computed G-P fitting function coefficients (b, c, a, Xk and d) were then employed to compute the values of EBF and EABF for selected energies and penetration depths, with the help of G-P fitting formula (Harima et al., 1986), as given by the following equations:

BðE; xÞ ¼ 1 þ

ðb  1ÞðK x  1Þ K 1

for K–1

ð3Þ

BðE; xÞ ¼ 1 þ ðb  1Þx for K ¼ 1 KðE; xÞ ¼ cxa þ d  40 mfp

tan hðx=X k  2Þ  tan hð2Þ 1  tan hð2Þ

ð4Þ for x ð5Þ

where x is source to detector distance in the medium in mfp, b is the value of buildup factor at 1 mfp; K(E, x) is the dose multiplication factor which represents the change in the shape of the dose weighted spectrum with increasing penetration depth and is represented by hyperbolic tangent function of penetration depth in mfp. Here, the mfp represents the average distance between two successive interactions of photons in which the intensity of the incident photon beam is reduced by a factor of 1/e. Where a, c, d, and Xk are G-P fitting function coefficients that depend on the chemical composition of the material, source energy and penetration depth.

In order to standardise the interpolation method discussed above, firstly the values EBF were computed for water up to 100 mfp in the chosen energy range of 0.015–15 MeV with the help of present (G-P) method. Secondly, the results obtained were compared with standard values of EBF for water for 10 MeV provided by (Shimizu and Hirayama, 2003) ANSI/ANS 6.4.3-1991, EGS4 and IE (invariant embedding). The compared values of EBF have been plotted and shown in Fig. 1. It has been warranted from the figure due to good agreement of the results that the G-P method for calculating the values of buildup factors for deep penetration up to 100 mfp is accurate. Thus it can safely be assumed that the present method is appropriate and suitable for calculation of EBF and EABF for the samples up to 100 mfp. 4.2. Dependence of values of EBF and EABF on incident photon energy Fig. 2 shows the variation in of values of EBF and EABF with incident photon energies at fixed penetration depths. All the samples show almost similar variations in the values of EBF and EABF in the selected energy region due to domination of different photon interaction processes in different energy regions. The low values of buildup factors have been observed in lower and upper region of energies of the selected energy range. This may be due to the dominance of photoelectric absorption and pair production, which result in the complete removal of photons. The maximum values of buildup factors have been observed at intermediate energies (0.04–0.70 MeV) where Compton scattering dominates. In this process, the photons are not completely removed but only their energies are degraded. Hence, the Compton scattering results in more multiple scattered photons, which leads to increase in the value

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

(a) 4k

(b)

40mfp

4k

S1

40mfp S1

S2

S2

S3

S3

S4

S4

S5

3k

−−− EABF −−−−>

−−− EBF −−−−>

S6

2k

S6

2k

1k

1k

0

0

0.1 0.01 1 10 −−−−− Incident photon energy (MeV) −−−−>

(c) 50k

S5

3k

0.1 0.01 1 10 −−−−− Incident photon energy (MeV) −−−−>

(d)50k 100mfp

100mfp S1

40k

S1

S2

S2

40k

S3

S3 S4

S4

S5

S5

30k

S6

−−− EABF −−−−>

−−− EBF −−−−>

30k

20k

20k

10k

10k

0

0

-10k

0.01 1 10 0.1 −−−−− Incident photon energy (MeV) −−−−>

S6

-10k

0.01

0.1

1

10

−−−−− Incident photon energy (MeV) −−−−>

Fig. 2. The EBF and EABF values for all samples in the energy region 0.015–15 MeV: (a and b) at 40 mfp, (c and d) at 100 mfp.

of buildup factors in the medium. From Fig. 2a–d it is clear the trend in variation of buildup factors is identical in both cases and the only difference is in the magnitudes of EBF and EABF corresponding to the photon energies. The dominance of partial interaction process with incident photon energies for all samples has been shown in Fig. 3a–f. The variation of mass attenuation coefficient due to photoelectric absorption (l/q)PE, Compton scattering (l/ q)CS, pair production (l/q)PP and total (l/q)Total explains the dominance of the respective interaction process with variation of incident photon energy. At selected energy and penetration depth the magnitude of EABF is approximately double than that of EBF. Table 10 indicates that for energy range 0.06–0.60 MeV the values of EBF

and EABF at 100 mfp are approximately ten times than that of the corresponding values of EBF and EABF at 40 mfp. This may be due the multiplication of photons, due to Compton scattering by interaction with large number of atoms for deep penetration. For energy range 0.08–0.3 MeV there exist broad peaks for all penetration depths showing the maximum value of the buildup factors due to exclusive dominance of Compton scattering. Because of the multiple scattering of photons, they exist for a longer time in the sample which leads to a higher value of buildup factors. This implies that the contribution of secondary gamma ray photons to energy spectra would be maximum in this energy range for all the samples under consideration.

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

(a) 10

2

(b) 10

10 1

S1

10 0

10 0

10 -1

10 -1

10 -2

10 -2

−−− [μ/ρ] (cm2/g) −−−>

−−− [μ/ρ] (cm2/g) −−−>

10 1

2

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -6 10 -7 10 -8 10 -9

0.01

0.1

S2

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -6 10 -7 10 -8

1

10 -9

10

0.01

--- Energy (MeV) ---> 10 2

S3

10 1

(d) 10

10 -1

10 -1

10 -2

10 -2

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -8 10 -9

0.01

0.1

S4

10 1 10 0

10 -7

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -6 10 -7 10 -8

1

10 -9

10

0.01

--- Energy (MeV) --->

(e) 10

2

10

0

2

10

0

10 -1

10 -2

10 -2

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -7 10 -8 10 -9

0.01

0.1

10 -3 10 -4 10 -5

(μ/ρ)PE (μ/ρ)CS (μ/ρ)PP (μ/ρ)Total

10 -6

10 -8

--- Energy (MeV) --->

10

S6

10 -7

1

1

10 1

−−− [μ/ρ] (cm2/g) −−−>

−−− [μ/ρ] (cm2/g) −−−>

(f) 10

10 -1

10 -6

0.1

--- Energy (MeV) --->

S5

10 1

10

2

10 0

10 -6

1

--- Energy (MeV) --->

−−− [μ/ρ] (cm2/g) −−−>

−−− [μ/ρ] (cm2/g) −−−>

(c)

0.1

10

10 -9

0.01

0.1

1

10

--- Energy (MeV) --->

Fig. 3. The mass attenuation coefficient for partial interaction processes in the energy region 0.015–15 MeV: (a) for S1, (b) for S2, (c) for S3, (d) for S4, (e) for S5 and (f) for S6.

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93 Table 10 Ratio of EBF and EABF for chosen samples at 100 mfp to 40 mfp in the selected energy range.

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

(a) 1.35

Ratio of EBF at 100 mfp to 40 mfp S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

1.09 1.12 1.34 1.60 2.89 5.80 7.82 9.98 14.43 15.05 10.58 8.23 6.64 5.70 4.54 4.07 3.23 2.80 2.70 2.94 1.98 6.00 2.55 2.53 2.52

1.11 1.15 1.38 1.63 2.83 7.89 7.70 10.47 15.38 14.73 9.68 8.22 6.47 5.78 4.46 4.05 3.23 2.80 2.70 2.94 1.98 6.00 2.55 2.53 2.52

1.04 1.03 1.19 1.33 1.56 1.93 2.74 5.28 8.87 10.94 9.90 7.99 6.49 5.97 4.58 4.19 3.31 3.05 2.78 2.89 5.08 4.87 2.51 4.87 4.05

1.03 1.03 1.19 1.35 1.58 1.95 2.62 5.76 8.91 10.66 10.03 7.88 6.55 6.20 4.52 4.08 3.36 3.06 3.04 3.00 6.71 4.63 2.45 5.20 4.11

1.10 1.14 1.36 1.61 2.88 6.62 7.84 10.21 14.84 14.98 10.27 8.23 6.59 5.73 4.51 4.06 3.23 2.80 2.70 2.94 1.98 6.00 2.55 2.53 2.52

1.01 1.06 1.19 1.38 1.63 2.48 4.99 6.79 9.05 10.20 10.16 7.89 6.63 6.27 4.52 3.99 3.26 3.05 3.03 2.88 6.19 4.73 2.50 4.94 4.16

1.09 1.10 1.35 1.57 2.72 8.25 7.44 8.75 10.56 11.42 9.21 7.25 5.76 5.24 4.35 3.84 3.04 2.85 2.78 2.91 3.59 1.85 2.48 2.51 2.35

1.11 1.11 1.37 1.61 2.46 6.55 7.01 9.07 11.27 11.67 9.00 7.16 5.62 5.30 4.34 3.81 3.04 2.85 2.78 2.91 3.59 1.85 2.48 2.51 2.35

1.05 1.03 1.17 1.32 1.46 2.78 2.85 6.29 7.26 10.59 8.38 7.57 6.25 5.33 4.43 3.82 3.17 2.91 2.69 2.72 3.71 1.93 5.55 3.16 3.95

1.04 1.03 1.18 1.33 1.47 2.80 2.67 6.08 7.65 10.59 8.63 7.39 6.20 5.37 4.44 3.83 3.18 2.95 2.68 2.75 3.43 1.98 5.03 3.24 4.47

1.10 1.11 1.36 1.59 2.63 7.61 7.40 8.92 10.86 11.53 9.14 7.22 5.71 5.26 4.35 3.83 3.04 2.85 2.78 2.91 3.59 1.85 2.48 2.51 2.35

1.03 1.06 1.20 1.37 1.57 2.61 6.39 6.66 8.23 10.63 8.88 7.28 6.21 5.37 4.44 3.84 3.16 2.92 2.68 2.71 3.72 1.94 5.66 3.13 4.15

(b) 1.35

0.015 MeV S1

1.30

--- EBF ---->

Ratio of EABF at 100 mfp to 40 mfp

1.25 1.20 1.15 1.10

S1 S2 S3

1.25

S4 S5 S6

1.20 1.15 1.10 1.05

1.05 1.00 -10

0.015 MeV

1.30

S2 S3 S4 S5 S6

--- EABF ---->

E (MeV)

0

10

20

30

40

50

60

70

80

90

1.00 -10

100 110

0

10

−−− Penetration depth (mfp) −−−>

(c)

S2 S3

--- EABF ---->

--- EBF ---->

(d)

S1

S4 S5 S6

10 3

10 2

10 1 -10

30

40

50

60

70

80

90

100 110

90

100 110

−−− Penetration depth (mfp) −−−>

0.15 MeV

10 4

20

0.15 MeV

10 4

S1 S2 S3

10 3

S4 S5 S6

10 2 10 1

0

10

20

30

40

50

60

70

80

90

100 110

−−− Penetration depth (mfp) −−−>

10 0 -10

0

10

20

30

40

50

60

70

80

−−− Penetration depth (mfp) −−−>

Fig. 4. The EBF and EABF values for all samples up to penetration depth of 100 mfp; (a and b) at 0.015 MeV, (c and d) at 0.15 MeV.

4.3. Dependence of values of buildup factors on equivalent atomic numbers The variations of values of EBF and EABF with equivalent atomic numbers have been studied also from the Fig. 2a–d. It has been observed for lower energies from 0.015 to 0.70 MeV; all the samples have different values of EBF and EABF. The values of buildup factors are small for those samples with high value of Zeq and buildup fac-

tors values are large for low Zeq samples. Further in energy range 0.70–3 MeV, buildup factors have almost same values for all samples. The values of buildup factors are completely independent of the Zeq of samples in this energy range due to the dominance of Compton scattering process. Fig. 4a–d shows the variation of EBF and EABF with the penetration depths for all the samples at fixed incident photon energies 0.015 MeV and 0.15 MeV. For all samples at lower energies there

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

--- EBF ---->

300

(b)

1.5 MeV

300

S1 S2 S3 S4 S5 S6

200

--- EABF ---->

(a)

100

S1 S2 S3 S4 S5 S6

200

100

0

0 -10

1.5 MeV

0

10

20

30

40

50

60

70

80

90

100 110

-10

0

10

−−− Penetration depth (mfp) −−−>

50

(d)

50

15 MeV S1 S2 S3 S4 S5 S6

40 30

--- EABF ---->

--- EBF ---->

(c)

20

0 20

30

40

50

60

70

50

60

70

80

90

100 110

80

90

100 110

20

0 10

40

80

90

100 110

S1 S2 S3 S4 S5 S6

30

10

0

30

15 MeV

40

10

-10

20

−−− Penetration depth (mfp) −−−>

-10

0

10

−−− Penetration depth (mfp) −−−>

20

30

40

50

60

70

−−− Penetration depth (mfp) −−−>

Fig. 5. The EBF and EABF values for all samples up to penetration depth of 100 mfp; (a and b) at 1.5 MeV, (c and d) at 15 MeV.

is a significant variation of values of EBF and EABF with penetration depths. It has been observed that for the samples with low equivalent atomic numbers (S1, S2 and S5) the values of buildup factors are large than the samples with higher equivalent atomic numbers (S3, S4 and S6). So the values of buildup factors vary inversely with values of Zeq. This dependence has been reduced at the energy 1.5 MeV as shown in Fig. 5a and b. It is clear from the Fig. 5c and d that the values of buildup factors are completely independent of Zeq of the samples for energy 15 MeV up to penetration depth of 30 mfp and again start depending on the Zeq but the trend is reversed. This may be due to the dependence of cross section for absorption process of pair production on Zeq is not as much significant as for photo-electric absorption process. For energies more than 1.022 MeV, the pair production starts dominating in energy absorption process. As a result the buildup factors values have been reduced for all the samples. Finally we have reached to a conclusion that the values of buildup factors vary inversely with Zeq for photon energies below 1.5 MeV. 4.4. Comparison of exposure and energy absorption buildup factors Fig. 6a–d shows that the values of EBF and EABF are equivalent in variation with respect to the incident photon energy and penetration depth. The obtained differences may help in estimation of where the maximum radiation intensity occurs, whether at the surface of the sample or inside it. This means that in some cases the EBF is more than EABF or vice versa for selected sample. The maximum differences exist in the intermediate energy region where Compton scattering is the main photon interaction process thus leading to large buildup factor values of EBF and EABF. For the materials of low Zeq, i.e. sodium silicate (S2) the values of EBF is comparatively larger than the values of EABF whereas for the materials of high Zeq, i.e. datolite (S3), the energy absorption build-

up factor (EABF) is significantly higher than the exposure buildup factor (EBF). The absorption of photons in air is comparable with materials of low Zeq, hence the EBF values are higher than the EABF values. Thus, absorption in air contributes much more to the EBF rather than the EABF. On the other hand, for materials of high Zeq when compared to air, the absorption inside the medium is much more than the absorption in air. Therefore, the EABF values are higher than the EBF values in those energy regions where EABF are higher. From Fig. 7a and b it can be seen that the relative percentage difference between EBF and EABF is high in the intermediate energy region due to the buildup of scattered photons and the difference is not significant at the lower and higher energies due to the absorption processes leading to lower values of buildup factor. 4.5. Comparison of EBF and EABF calculated by G-P formula using mass attenuation coefficients from WinXCom software and GEANT4 Monte Carlo simulations It is clear from Table 11 that the values of total mass attenuation coefficients (Coherent and In Coherent) for all samples calculated by WinXCom software and GEANT4 Monte Carlo simulations show good agreement with small percentage deviations. Fig. 8a and b shows the percentage deviation of the computed buildup factors using mass attenuation coefficients by WinXCom software with Monte Carlo simulation (GEANT4). It has been concluded that the results obtained by both methods agreed with small discrepancies. 5. Conclusions The gamma ray exposure and energy absorption buildup factors for the silicates samples in the energy region 15–15 MeV up to penetration depths of 100 mfp have been calculated for the first

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K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

(a)

S2

5 mfp 15 mfp 25 mfp 40 mfp

10 4

(b)

5 mfp 15 mfp 25 mfp 40 mfp

10 4

70 mfp 100mfp

10 3

70 mfp 100mfp

10 3 −−−EABF −−−>

−−−EBF−−−>

S2

10 2

10 2

10 1

10 1

10 0

10 0 -2

-1

10

10

0

1

10

-2

10

-1

10

10

−−− Photon energy (MeV)−−−>

(c)

(d)

S3

5 mfp 15 mfp

10 4

0

S3

10

5 mfp 15 mfp

10 4

25 mfp 40 mfp

25 mfp 40 mfp

70 mfp

70 mfp 100mfp

100mfp 10 3

10 3 −−−EABF−−−>

−−−EBF−−−>

1

10

−−− Photon energy (MeV)−−−>

10 2

10 2

10 1

10 1

10 0

10 0 -2

10

-1

10

0

10

1

10

−−− Photon energy (MeV) −−−>

-2

10

-1

10

0

10

1

10

−−− Photon energy (MeV) −−−>

Fig. 6. The variation of EBF and EABF with incident photon energies at some selected penetration depths; (a and b) for S2 and (c and d) for S3.

time by using G-P fitting formula with modified expression of the dose multiplication factor K(E, x). From the present study, it has been concluded that for penetrations depth up to 40 mfp the results obtained for buildup factors are in good agreement with those

calculated by I.E. (invariant embedding) method, EGS4 (Monte Carlo code) and ANSI-1991 standard values. For deep penetration up to 100 mfp the agreement of exposure buildup factor for water calculated by the present method with I.E. method certified and

92

K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

Fig. 7. The variation of percentage difference between EBF and EABF for energy range 0.015–15 MeV up to 100 mfp; (a) for S2 and (b) for S3.

0.015 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00

S2

S3

S4

S5

S6

A

B

A

B

A

B

A

B

A

B

A

B

3.58 2.18 1.89 1.66 1.37 1.02 0.33 0.20 0.81 0.82 0.37 0.11 0.41 0.59 0.71 0.64 0.32 0.05 0.51 0.77 0.93 1.02 1.06 1.04 0.92

1.62 1.26 0.39 0.34 0.59 0.52 0.07 0.40 0.86 0.81 0.30 0.15 0.41 0.60 0.71 0.64 0.31 0.10 0.45 0.56 0.55 0.52 0.44 0.38 0.17

3.68 2.27 1.93 1.66 1.35 1.01 0.31 0.21 0.82 0.83 0.37 0.10 0.40 0.59 0.70 0.64 0.31 0.05 0.54 0.78 0.93 1.02 1.07 1.05 0.93

2.25 1.90 0.85 0.05 0.40 0.38 0.03 0.45 0.88 0.85 0.37 0.10 0.40 0.60 0.71 0.64 0.30 0.10 0.47 0.57 0.55 0.53 0.44 0.38 0.17

2.60 1.54 1.50 1.48 1.32 1.03 0.36 0.17 0.80 0.81 0.36 0.11 0.41 0.60 0.72 0.60 0.33 0.03 0.50 0.78 0.91 1.02 1.05 1.03 0.92

1.45 1.13 0.77 0.40 0.16 0.12 0.32 0.15 0.67 0.72 0.32 0.13 0.42 0.61 0.72 0.65 0.32 0.08 0.43 0.55 0.53 0.52 0.45 0.41 0.23

3.65 2.26 1.95 1.75 1.47 1.11 0.37 0.18 0.82 0.83 0.37 0.10 0.41 0.60 0.72 0.66 0.33 0.03 0.50 0.77 0.91 1.01 1.05 1.03 0.91

1.62 1.24 0.75 0.29 0.05 0.03 0.30 0.27 0.76 0.78 0.35 0.12 0.42 0.61 0.72 0.65 0.32 0.09 0.44 0.55 0.53 0.52 0.44 0.41 0.22

3.24 1.97 1.78 1.59 1.32 1.00 0.31 0.21 0.81 0.82 0.37 0.10 0.40 0.59 0.70 0.64 0.30 0.06 0.52 0.79 0.93 1.02 1.07 1.05 0.93

2.09 1.68 0.68 0.15 0.46 0.43 0.01 0.44 0.88 0.85 0.37 0.10 0.40 0.60 0.70 0.63 0.29 0.12 0.46 0.58 0.56 0.54 0.45 0.39 0.18

3.63 2.23 1.93 1.73 1.45 1.09 0.36 0.19 0.82 0.83 0.37 0.11 0.41 0.60 0.72 0.66 0.33 0.03 0.50 0.77 0.91 1.01 1.05 1.03 0.91

1.52 1.16 0.65 0.16 0.09 0.09 0.22 0.29 0.78 0.79 0.35 0.12 0.42 0.61 0.72 0.65 0.32 0.08 0.44 0.55 0.53 0.52 0.44 0.40 0.21

A: In Coherent. B: NonCoherent.

6

Percentage Deviation

E (MeV) S1

(a)

4

EBF (S1) EABF (S1) EBF (S2) EABF (S2) EBF (S3) EABF (S3) EBF (S4) EABF (S4) EBF (S5) EABF (S5) EBF (S6) EABF (S6)

At 40 mfp

2

0

0.01

0.1

1

10

Incident photon energy (MeV)

(b) 12 10

Percentage Deviation

Table 11 Percentage deviation of total mass attenuation coefficients for samples calculated by WinXCom software and GEANT4 Monte Carlo simulations.

8

EBF (S1) EABF (S1) EBF (S2) EABF (S2) EBF (S3) EABF (S3) EBF (S4) EABF (S4) EBF (S5) EABF (S5) EBF (S6) EABF (S6)

At 100 mfp

6 4 2 0

authenticated the present method for calculation of buildup factors. The present method is easy to use and quick as compared to other methods. Effectively of radioactive waste dumped deep inside soil or water can be assessed easily from the present study. Finally it can be concluded that the G-P fitting formula successfully generates the buildup factors for penetration depth up to 100 mfp in energy range 0.015–15 MeV. The study of the buildup factor data so generated indicates that silicate samples for energies between 4 and 6 MeV behaves differently when compared with other energies. This may be due to the modified expression of the dose multiplication factor K(E, x). Much work can be done to use G-P method for calculation of buildup factors for further deep

0.01

0.1

1

10

Incident photon energy (MeV) Fig. 8. Absolute percentage deviations of buildup factors for all samples calculated by G-P formula and GEANT4 Monte Carlo simulations: (a) at 40 mfp and (b) at 100 mfp.

penetrations and for different materials, so that the data collected will be helpful in the early reaffirmation of ANSI/ANS 6.4.3-1991 standards.

K.S. Mann, T. Korkut / Annals of Nuclear Energy 51 (2013) 81–93

Acknowledgements The author is grateful to respected Dr. M.J. Berger and Professor L. Gerward of the Department of Physics, Technical University of Denmark for providing the WinXCom program. The author paid tribute to respected Late Dr. J.H. Hubbell for his wonderful contribution to the field of gamma radiations interaction with matter.

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