Investigations of mass attenuation coefficients and exposure buildup factors of some low-Z building materials

Annals of Nuclear Energy 43 (2012) 157–166

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Investigations of mass attenuation coefficients and exposure buildup factors of some low-Z building materials Kulwinder Singh Mann a,⇑, Jyoti Singla b, Vipan Kumar b, Gurdeep Singh Sidhu c a

Department of Physics, Dravidian University, Kuppam 517 425, AP, India Department of Physics, Singhania University, Rajasthan, India c Department of Physics, G.S.S. School, Jodhpur-Romana, Bathinda 151 001, India b

a r t i c l e

i n f o

Article history: Received 30 June 2011 Received in revised form 4 January 2012 Accepted 7 January 2012 Available online 14 February 2012 Keywords: Exposure buildup factor Mass attenuation coefficient Equivalent atomic number Shielding Building materials

a b s t r a c t To check the gamma ray shielding properties of selected low-Z building materials such as Soil-I, Soil-II, Dolomite, Gypsum, Igneous Rock and Lime Stone, some parameters of dosimetric interest have been investigated in the energy range 0.015–15 MeV. The photon interactions with the samples have been discussed mainly in terms of mass attenuation coefficient, equivalent atomic number and exposure buildup factor. From the present investigations, it has been concluded that the values of exposure buildup factors are very large in the medium energy region and Soil-I acts as best gamma ray shielding material among the selected samples. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Exposure to gamma rays can occur in a range of industries, medical diagnostic centers, nuclear research establishments, nuclear reactors and nuclear weapons. Since the energetic gamma rays are hazardous for living cells and tissues, the needed precautions must be taken by shielding the radiations. But in the study of design of the gamma radiations shielding or estimating the exposure dose, there is an undesired situation faced by radiation physicists, oncologists and engineers due to secondary radiations that can occur due to buildup of photons from the collided part of the incident beam. For this reason it is of importance to determine the buildup factors to make corrections for effective energy deposition in different shielding materials. So a detailed study is required for the safe and acceptable use of gamma radiations, radioactive materials and nuclear energy. Due to numerous nuclear accidents (Fukushima, Chernobyl, Three Mile Island etc.) and the possibility of facing such issues in future, due to lost radiation sources, transportation and Abbreviations: G.P., Geometric Progression; mfp, mean free path; Z, atomic number; Zeq, equivalent atomic number; Zeff, effective atomic number; EBF, exposure buildup factor; l, mass attenuation coefficient; Epeak, incident photon energy at which EBF is maximum called peak value. ⇑ Corresponding author. Address: Department of Physics, D.A.V. College, Dyanad Nagar, Bathinda 151 001, PB, India. Tel.: +91 9417325696; fax: +91 1642241666. E-mail addresses: [email protected] (K.S. Mann), [email protected] (J. Singla), [email protected] (V. Kumar), [email protected] (G.S. Sidhu). 0306-4549/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2012.01.004

storage of nuclear waste, radiological terrorism and the possibility of nuclear weapons being used in a war, everybody should be encouraged to use such materials for construction of buildings (shields), which could effectively protect them from the hazardous external radiations. A new area of interest is gamma radiation shielding development has grown considerably due to all aspects related to homeland security. The gamma ray buildup factor is a multiplicative factor used to obtain the corrected response to the uncollided photons by including the contribution of 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), which is often used in the computation of attenuation coefficients. It arises due to multiple scattering of gamma-rays or may be due to the large thickness of the interacting material or due to the divergence of the radiation beam. The modified intensity equation becomes (I = B Io elx), where B is known as buildup factor (Singh et al., 2008). This parameter B is always equal to or greater than unity (B = 1, in case of narrow beam geometry or interacting material is thin and the photon is assumed to be mono-energetic, else B > 1). Buildup factor has been classified into two categories viz. energy absorption buildup factor and exposure buildup factor. The energy absorption buildup factor (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 the exposure buildup factor (EBF) is defined as that

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buildup factor in which the quantity of interest is the exposure and the detector response function is that of absorption in air. There are different methods such as G.P. (Geometric Progression) 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. Harima et al. (1986) has also computed buildup factors using G.P. fitting method and compared the results with PALLAS code (Takeuchi and Tanaka, 1984), a good agreement was observed (discrepancy was within 7%) for penetration depth up to 40 mfp. Similarly, (Sakamoto et al., 1988) interpolated buildup factors for compounds/mixtures and reported good agreement with PALLAS code for low-Z materials (discrepancies were within 10%), whereas for high-Z materials discrepancies were found to be as large as 30%.Shimizu et al. (2004) compared the buildup factor values obtained by three different approaches (G.P. fitting, invariant embedding and Monte Carlo method) and only small discrepancies were observed for low-Z elements up to 10 mfp. Yoshida, 2006 developed a fitting methods using geometric progression formulae of gamma-ray buildup factors. He showed that even with fitting up to 300 mfp, the average standard deviation of 26 materials was 2.9% and acceptable G.P. parameters were extracted. Singh et al., 2008 studied buildup factors for some commonly used solvents. Hence one can use any of these methods/codes for computing buildup factors for low-Z materials. Engineers require these buildup factors while performing calculations for radiation shielding design. The selected low-Z building materials viz. Soil-I (S1), Soil-II (S2), Dolomite (S3), Gypsum (S4), Igneous Rock (S5) and Lime Stone (S6) are low cost and abundantly available in nature. The NBS (National Bureau of Standards) Concrete is considered as good shielding material for the gamma radiations in the selected energy range. The attenuation coefficients of the samples and NBS Concrete has been calculated and compared for verification of shielding properties of the present samples. These samples are called low-Z as the equivalent atomic numbers remain less than 18 in the selected energy range. Since the buildup factors of selected samples are not found in any compilation or tabulation, so in the present study by using G.P. fitting formula an attempt has been made to compute exposure buildup factor values for these building materials in the energy range 0.015–15 MeV and up to penetration depth of 40 mfp. Such data will be of prime importance for those who are working with scattering of photons and related phenomena like radiation shield designing (Suteau and Chiron, 2005) and production of a new materials for gamma radiation shielding (Mortazavi and Mosleh-

Shirazi, 2010). Also an attempt has been made to perform a comparative study on the basis of different properties of selected samples so as to visualize all the study and findings. 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). Recently, the mass attenuation coefficients and exposure buildup factors (EBFs) have been studied by different researchers; Singh et al. (2008), Kurudirek and Topcuoglu (2011), Kurudirek and Ozdemir (2011), Manohara et al. (2011), Mann et al., (2011, 2012).

2. Materials and methods 2.1. Calculation of elemental composition Selected materials samples are used in the cement, sand, bricks, glass and other raw materials used in building materials. The elemental composition of samples are indicated in the Table 1 were analyzed using a wavelength dispersive X-ray fluorescence

Table 2 Mass attenuation coefficients (cm2/g) of the selected samples. E (MeV)

S1

S2

S3

S4

S5

S6

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

11.60 5.07 1.62 0.77 0.46 0.33 0.22 0.18 0.14 0.12 0.11 0.09 0.09 0.08 0.07 0.06 0.06 0.06 0.05 0.04 0.04 0.04 0.03 0.03 0.03

11.30 4.98 1.59 0.76 0.46 0.33 0.22 0.18 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.06 0.06 0.05 0.04 0.04 0.04 0.03 0.03 0.03

8.12 3.54 1.15 0.57 0.36 0.27 0.20 0.17 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.03

10.60 4.58 1.46 0.70 0.43 0.31 0.21 0.18 0.14 0.13 0.11 0.10 0.09 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.04 0.03 0.03 0.03

9.06 3.95 1.27 0.62 0.39 0.28 0.20 0.17 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.06 0.06 0.05 0.04 0.04 0.04 0.03 0.03 0.03

10.70 4.67 1.49 0.71 0.43 0.31 0.21 0.18 0.14 0.13 0.11 0.10 0.09 0.08 0.07 0.06 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.03

Table 1 Elemental composition of the chosen building material samples (% weight). Elements

H C O Na Mg Al Si P S K Ca Ti Mn Fe

Selected building materials S1

S2

S3

S4

S5

S6

NBS Concrete

– – 0.4748 – – 0.0935 0.2844 – – 0.0344 – – – 0.1129

– – 0.4963 – – 0.0660 0.2772 – – 0.0232 0.0395 – – 0.0979

– 0.1303 0.5206 – 0.1318 – – – – – 0.2173 – – –

0.0234 – 0.5576 – – – – – 0.1862 – 0.2328 – – –

– – 0.4731 0.0284 0.0213 0.0822 0.2812 – – 0.0264 0.0366 – – 0.0508

0.0009 0.1134 0.4962 0.0004 0.0476 0.0043 0.0243 0.0002 0.0011 0.0027 0.3043 0.0004 0.0004 0.0040

0.0056 – 0.4983 0.0171 0.0024 0.0456 0.3158 – 0.0012 0.0192 0.0826 – – 0.0122

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K.S. Mann et al. / Annals of Nuclear Energy 43 (2012) 157–166 Table 3 Equivalent atomic numbers (Zeq) of the selected samples. Energy (MeV)

S1

S2

S3

S4

S5

S6

NBS Concrete

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

14.52 14.72 14.93 15.06 15.18 15.27 15.40 15.54 15.49 14.96 14.50 14.50 14.50 14.50 14.50 14.50 14.50 9.79 12.62 12.35 12.53 12.62 12.44 12.42 12.38

14.37 14.61 14.81 14.97 15.04 15.18 15.29 15.39 15.49 14.96 14.50 14.50 14.50 14.50 14.50 14.50 14.50 9.77 13.18 11.94 12.53 12.21 12.44 12.42 12.38

12.89 13.05 13.21 13.33 13.44 13.48 13.57 13.64 13.42 14.48 14.50 14.50 14.50 14.50 14.50 14.50 14.50 9.73 10.69 10.60 10.79 10.74 10.94 10.64 10.87

13.88 14.02 14.25 14.33 14.38 14.46 14.56 14.59 14.33 14.45 14.50 14.50 14.50 14.50 14.50 14.50 14.50 12.64 12.20 11.97 11.88 11.81 11.99 11.89 11.80

13.43 13.58 13.76 13.86 13.97 14.03 13.97 14.14 14.46 14.49 14.50 14.50 14.50 14.50 14.50 14.50 14.50 12.88 11.72 11.46 11.99 11.72 12.02 11.73 11.89

14.03 14.23 14.43 14.54 14.62 14.75 14.81 14.99 14.41 14.94 14.50 14.50 14.50 14.50 14.50 14.50 14.50 9.71 11.31 11.84 11.87 11.83 12.06 11.81 11.90

12.88 13.00 13.14 13.23 13.29 13.33 13.39 13.43 13.50 13.54 13.59 13.61 13.63 13.64 13.64 13.64 12.04 11.65 11.55 11.51 11.50 11.49 11.48 11.48 11.47

Table 4 Values of effective atomic number (Zeff), symbols for samples and Epeak for selected samples.

a

Sample symbol

Building material

Zeff

a

S1 S2 S3 S4 S5 S6

Soil-I Soil-II Dolomite Gypsum Igneous Rock Lime Stone

13.989 13.939 12.840 13.637 13.464 13.583

0.297 0.306 0.296 0.305 0.300 0.296

Epeak (MeV)

Epeak: incident photon energy at which EBF is maximum called peak value.

Table 5 Exposure G.P. fitting parameters for sample S1. Energy (MeV)

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

spectrometer (WDXRFS) and also obtained from literature (http:// webmineral.com/data/Dolomite.shtml). 2.2. Computational work The computations of energy exposure buildup factors have been divided into three parts. Step by step computations are illustrated as follows. 2.2.1. Computation of equivalent atomic number Firstly the values of Compton partial attenuation coefficient (lcomp) and total attenuation coefficients (l)tot in cm2/g were Table 6 Exposure G.P. fitting parameters for sample S2.

G.P. fitting parameters

Energy (MeV)

b

c

a

Xk

d

1.0204 1.0437 1.1436 1.3105 1.5158 1.7189 2.1065 2.3474 2.5517 2.5933 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.8104 1.6734 1.6012 1.5331 1.4821 1.4010 1.3386 1.2495

0.3749 0.4304 0.3941 0.4377 0.5139 0.6172 0.7594 0.9005 1.1656 1.3212 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1577 1.0596 0.9887 0.9496 0.9334 0.9087 0.8929 0.8556

0.2356 0.1761 0.2132 0.1956 0.1646 0.1234 0.0799 0.0420 0.0205 0.0484 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0347 0.0015 0.0079 0.0208 0.0253 0.0322 0.0397 0.0543

11.9959 14.8706 14.3815 14.4405 14.6752 14.7385 13.7433 13.4782 12.1382 8.9793 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 14.7859 12.9026 12.7327 10.5642 12.2114 13.8635 13.0642 14.5010

0.1520 0.1582 0.1151 0.1087 0.0917 0.0675 0.0487 0.0423 0.0166 0.0087 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0128 0.0024 0.0117 0.0184 0.0222 0.0275 0.0332 0.0497

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

G.P. fitting parameters b

c

a

Xk

d

1.0211 1.0448 1.1480 1.3161 1.5292 1.7296 2.1331 2.3712 2.5517 2.5933 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.8106 1.6723 1.5985 1.5331 1.4833 1.4010 1.3386 1.2495

0.3675 0.4297 0.3942 0.4396 0.5186 0.6225 0.7588 0.9114 1.1656 1.3212 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1577 1.0572 0.9967 0.9496 0.9338 0.9087 0.8929 0.8556

0.2431 0.1765 0.2136 0.1948 0.1626 0.1213 0.0810 0.0390 0.0205 0.0484 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0347 0.0100 0.0051 0.0208 0.0244 0.0322 0.0397 0.0543

11.9913 15.2848 14.3324 14.4271 14.6867 14.7523 13.5245 13.6405 12.1382 8.9793 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 14.7906 10.7232 13.0108 10.5642 12.0794 13.8635 13.0642 14.5010

0.1597 0.1506 0.1152 0.1083 0.0905 0.0662 0.0495 0.0418 0.0166 0.0087 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0128 0.0025 0.0091 0.0184 0.0202 0.0275 0.0332 0.0497

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obtained for elements from Z = 1 to 25 and chosen samples in the energy of 0.015–15.0 MeV, using the state-of-the-art and convenient computer program XCOM and WinXCom (Berger and Hubbell, 1987; http://physics.nist.gov/xcom; Gerward et al., 2004). Further, by using a simple computer program, the ratio R(lcomp/ ltot) was obtained for selected samples. Then the value of equivalent atomic number (Zeq) for these samples was calculated by matching the ratio R(lcomp/ltot) of particular sample at a given energy with corresponding ratios of elements at the same energy. For the case the ratio lies in between the two ratios of known elements. The value of Zeq was interpolated by using the following for-

Table 7 Exposure G.P. fitting parameters for sample S3. Energy (MeV)

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

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 elemental atomic numbers. R1 and R2 respectively corresponding to the ratios (lcomp/ltot) and R is the ratio for selected sample at a specified energy. The computed values of mass attenuation coefficients (l) and equivalent atomic numbers

Table 9 Exposure G.P. fitting parameters for sample S5.

G.P. fitting parameters

Energy (MeV)

b

c

a

Xk

d

1.0298 1.0662 1.2135 1.4479 1.7089 2.0447 2.5010 2.7043 2.8152 2.6340 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.8112 1.6864 1.6081 1.5370 1.4948 1.4062 1.3481 1.2631

0.3947 0.3862 0.4162 0.4885 0.6269 0.6994 0.8742 1.0503 1.3210 1.3476 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1579 1.0556 0.9918 0.9555 0.9142 0.9018 0.8756 0.8237

0.2065 0.2179 0.2019 0.1734 0.1152 0.0981 0.0483 0.0046 0.0511 0.0531 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0348 0.0110 0.0058 0.0160 0.0305 0.0330 0.0426 0.0635

15.2786 14.0084 14.8600 14.6278 15.7920 14.0693 13.9570 13.4145 16.8645 8.3601 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 14.8049 10.6863 13.7475 15.1528 11.7000 13.5089 13.2736 14.3754

0.1382 0.1197 0.1068 0.0951 0.0593 0.0549 0.0440 0.0259 0.0014 0.0070 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0128 0.0010 0.0087 0.0199 0.0246 0.0259 0.0326 0.0555

Table 8 Exposure G.P. fitting parameters for sample S4. Energy (MeV)

mula of interpolation (Harima, 1993) given in the following equation.

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

G.P. fitting parameters b

c

a

Xk

d

1.0264 1.0576 1.1893 1.3979 1.6436 1.8896 2.4069 2.5992 2.6758 2.6331 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.7817 1.6767 1.6022 1.5380 1.4867 1.4048 1.3433 1.2541

0.3742 0.4087 0.4013 0.4658 0.5815 0.6999 0.8405 1.0068 1.2379 1.3471 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1532 1.0618 0.9944 0.9411 0.9286 0.9023 0.8834 0.8445

0.2306 0.1959 0.2118 0.1850 0.1340 0.0928 0.0573 0.0147 0.0350 0.0530 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0321 0.0065 0.0055 0.0239 0.0259 0.0339 0.0415 0.0577

13.9599 16.0658 14.2490 14.3910 15.1553 15.2742 14.6117 13.7106 10.9945 8.3732 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 15.4115 14.8565 12.9353 10.2182 11.8391 13.8506 13.1162 14.2516

0.1565 0.1130 0.1135 0.1042 0.0709 0.0520 0.0501 0.0313 0.0097 0.0070 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0093 0.0014 0.0090 0.0216 0.0207 0.0285 0.0333 0.0516

Table 10 Exposure G.P. fitting parameters for sample S6.

G.P. fitting parameters

Energy (MeV)

b

c

a

Xk

d

1.0237 1.0508 1.1693 1.3610 1.5985 1.8270 2.2886 2.5115 2.6918 2.6366 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.7831 1.6738 1.5982 1.5378 1.4858 1.4050 1.3425 1.2549

0.3542 0.4261 0.3947 0.4529 0.5561 0.6697 0.7920 0.9713 1.2477 1.3493 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1537 1.0626 0.9969 0.9427 0.9303 0.9020 0.8849 0.8424

0.2555 0.1789 0.2152 0.1904 0.1455 0.1037 0.0728 0.0234 0.0369 0.0534 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0324 0.0083 0.0050 0.0231 0.0253 0.0340 0.0412 0.0584

12.3900 17.5612 14.0977 14.3638 14.9531 15.0510 13.6822 13.9013 10.8256 8.3207 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 15.5972 15.3129 13.0154 10.7510 11.8944 13.8469 13.1006 14.2612

0.1744 0.1092 0.1160 0.1071 0.0787 0.0572 0.0511 0.0360 0.0087 0.0069 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0097 0.0023 0.0091 0.0215 0.0202 0.0285 0.0332 0.0521

1.50E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 8.00E02 1.00E01 1.50E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 8.00E01 1.00E+00 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 8.00E+00 1.00E+01 1.50E+01

G.P. fitting parameters b

c

a

Xk

d

1.0228 1.0486 1.1623 1.3461 1.5728 1.7858 2.2402 2.4359 2.6820 2.5950 2.4566 2.3397 2.2378 2.1617 2.0438 1.9693 1.8379 1.8114 1.6808 1.5992 1.5377 1.4856 1.4044 1.3429 1.2540

0.3505 0.4274 0.3946 0.4485 0.5419 0.6498 0.7720 0.9408 1.2416 1.3223 1.4507 1.4562 1.4438 1.4199 1.3799 1.3230 1.2280 1.1579 1.0586 0.9962 0.9429 0.9307 0.9029 0.8842 0.8447

0.2604 0.1781 0.2147 0.1918 0.1520 0.1109 0.0792 0.0308 0.0357 0.0486 0.0749 0.0779 0.0780 0.0755 0.0720 0.0630 0.0475 0.0348 0.0033 0.0052 0.0230 0.0251 0.0338 0.0414 0.0577

11.9809 16.7401 14.1742 14.3849 14.8499 14.9042 13.2578 14.0659 10.9297 8.9539 18.0497 16.5066 16.3051 17.5358 15.4631 16.4254 15.2421 14.8096 12.3928 12.9953 10.7997 11.9067 13.8519 13.1083 14.2506

0.1775 0.1242 0.1157 0.1075 0.0832 0.0606 0.0514 0.0401 0.0093 0.0086 0.0130 0.0150 0.0169 0.0197 0.0208 0.0194 0.0158 0.0128 0.0001 0.0091 0.0215 0.0201 0.0284 0.0333 0.0516

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(Zeq) of the chosen samples are listed in Table 2 and Table 3 respectively. 2.2.2. Calculation of effective atomic number (Zeff) To assign a particular atomic number to each sample irrespective of the incident photon energy, the Zeq is averaged over the

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25 incident photon energies and the atomic number so obtained is treated as the effective atomic number Zeff of that sample, that is

Z eff ¼

15:0 X

Zeq=25

B¼0:015

Fig. 1. Difference (%) between calculated values of EBF and ANSI standard values of EBF for water at some energies in the selected energy range up to 40 mfp.

Fig. 2. Comparison of all selected samples and NBS Concrete using variation with incident photon energies of; (a) mass attenuation coefficients and (b) ratio mass attenuation coefficient of samples with NBS Concrete.

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This is done to study the average behavior of different samples at fixed penetration depth and fixed incident photon energy. The calculated values of effective atomic numbers and symbols for all six samples are listed in Table 4.

sample at the same energy. Z1 and Z2 are the elemental atomic numbers between which the equivalent atomic number Z of the chosen samples lies. The computed G.P. parameters of all samples are listed in Tables 5–10.

2.2.3. Computation of G.P. fitting parameters American National Standard has provided the energy exposure G.P. fitting parameters of 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 (water) and two mixtures (air and concrete) in the energy range of 0.015–15.0 MeV and up to a penetration depth of 40 mfp (ANSI/ ANS-6.4.3-1991). Using the interpolation formula, five G.P. fitting parameters (b, c, a, Xk and d) for selected samples were computed at the different incident photon energies using equivalent atomic number (Zeq), in the chosen energy range (0.015–15.0 MeV) up to penetration depth of 40 mfp. The formula used for the purpose of interpolation (Sidhu et al.,1999a,b) is as follows:

2.2.4. Computation of exposure buildup factors The computed G.P. fitting parameters were then used to compute the exposure buildup factors for the selected samples at some standard incident photon energies up to a penetration depth of 40 mean free paths, with the help of G.P. fitting formula, as given by following equations (Harima et al., 1986).

P¼

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

ð2Þ

Here P1 and P2 are the values of G.P. fitting parameters corresponding to the atomic numbers Z1 and Z2 respectively at a fixed energy, whereas Z is the equivalent atomic number of the chosen

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

tanhðx=X k  2Þ  tanhð2Þ 1  tanhð2Þ

ð4Þ for x 6 40 mfp

ð5Þ

where a, b, c, d and Xk are the G-P fitting parameters and x is source to detector distance in the medium (mfp). The parameter K (E, x) represents photon dose multiplication. In order to standardize the interpolation method discussed above, firstly the exposure buildup factors were computed for

Fig. 3. Deviation of exposure buildup factors with incident photon energies for chosen samples at penetration depth of; (a) 1 mfp, (b) 5 mfp, (c) 15 mfp and (d) 40 mfp.

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water (H2O) in the chosen energy range of 0.015–15.0 MeV up to 40 mfp with the help of above method. The results so obtained were compared with standard exposure buildup factor data of the American National Standards (ANSI/ANS-6.4.3-1991) for some randomly selected energy between 0.015 and 15.0 MeV (Harima, 1993). A good agreement was observed indicated by Fig. 1, with discrepancies less than 10%. Thus it can safely be assumed that the present method is appropriate and suitable for calculation of exposure buildup factors of the selected samples. 3. Results and discussion The results of the present research work are discussed under two headings:3.1. Dependence of mass attenuation coefficient and EBF on incident photon energy The mass attenuation coefficient is a measure of the relative dominance of the partial interaction processes (photoelectric effect, Compton scattering and pair production) of gamma rays with the samples, whereas the exposure buildup factor and is the outcome of the gamma rays interactions with the samples.

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Fig. 2a clearly explains the variation of the mass attenuation coefficients of all samples and concrete with incident photon energy. We conclude that for incident photon energy up to 0.1 MeV the values of mass attenuation coefficients is maximum for sample S1 having maximum effective atomic number (Zeff  13.9) among the selected samples. Also for incident photon energies more than 0.1 MeV the values of mass attenuation coefficients are maximum for sample S4 with effective atomic number (Zeff = 13.637). It is evident that selected samples show high values of mass attenuation coefficients for energies more than 1 MeV. Also the Fig. 2b shows the variation of ration of mass attenuation coefficients of the six selected samples with NBS Concrete. It is evident that except small region of energy the selected samples show high value of attenuation coefficient indicating good shielding effectiveness of the samples than NBS Concrete. Fig. 3 shows the variation in exposure buildup factors (EBFs) with incident photon energy in the energy region 0.015–15 MeV at different penetration depths up to 40 mfp. It is of worth noting that all the samples show almost similar variations of EBF in the continuous energy region based on domination of different photon interaction processes in different energy regions. From all these figures it is observed that the value of exposure buildup factors for all samples in the selected energy region up to penetration depth of 40 mfp is always greater than

Fig. 4. Deviation of exposure buildup factors of chosen samples with penetration depths for incident photon energies of; (a) 0.015 MeV, (b) 0.15 MeV, (c) 1.5 MeV and (d) 15 MeV.

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one. This is because of buildup of photons is due to larger penetration depth or due to the beam divergence. At lower energies of the range the dominant photon interaction process is photoelectric absorption, for which the atomic cross section is proportional to fZ aeq  E7=2 g; where the values of a is in between 4 and 5 depending on the photon energy (E). However as the incident photon energy increases the Compton scattering process starts dominating. The maximum values of EBF were 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, this process results with more multiple scattered photons, which leads to increase in buildup factors in the medium. It is indicated that the value EBF is minimum for sample S1 in this region of energy. Further, for energies between 0.08 MeV and 0.3 MeV there exist broad peaks for all samples showing the maximum value of EBF due to exclusive dominance of Compton scattering. This is because of the multiple scattering of photons, as a result they exist for a longer time in a material that 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. These results in a broad peak around a particular value of incident photon energy called peak value of energy (Epeak). For energy region 0.70–1.25 MeV, EBF have almost same values for all samples. It shows that the exposure buildup factor (EBF) is completely independent of the nature of the samples in this region of energy. This is due to the reason that pair-production starts balancing the Compton scattering process in this energy region.

For all samples the EBF values are almost same beyond 0.8 MeV. Hence no significant variation of EBF is observed in the energy region (0.80–3.0 MeV). At higher energies the pair production starts dominating, but there is no significant variation in the EBF with equivalent atomic number (Zeq). Since the cross section of electron–positron pair production (with threshold of 1.022 MeV) is approximately proportional to {Zeq(Zeq + 1)}. The cross section increases slowly with incident photon energy between the thresholds of 1.02 MeV to about 5 MeV. For further higher energies it is proportional to the logarithm of photon energy. Exposure buildup factors have low values at lower and higher energy regions of the selected range. This is due to the dominance of photoelectric absorption and pair production over Compton scattering also at fixed penetration depth, for the selected energy range the EBF values can reach in the order of 100 to the values in the order of 103. Fig. 4 shows the variation of exposure buildup factors with the penetration depths for all the samples at fixed incident photon energy 0.015 MeV. At such a low energy there is a significant variation of EBF for all samples. It is observed that for the samples with low effective atomic numbers (Zeff) (S3,S5 and S4) the values of EBF are large, on the other side the samples with higher effective atomic numbers (S1,S2 and S6) the values of EBF are comparatively small. So EBF depends on nature of the samples at incident photon energy 0.015 MeV. This dependence of EBF on the nature of materials reduces at the incident photon energy 0.15 MeV. Further at incident photon energies 1.50 and 15.00 MeV EBF is completely independent of nature of the samples. This is due to the dominance of the pair-production process over other energy degradation

Fig. 5. Variation of exposure buildup factors of chosen samples with effective atomic number (Zeff) for some incident photon energies between 0.015 and 15 MeV at penetration depth of 15 mfp.

K.S. Mann et al. / Annals of Nuclear Energy 43 (2012) 157–166

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Fig. 6. Variation of exposure buildup factors of chosen samples with effective atomic numbers (Zeff) for some incident photon energies between 0.015 and 15 MeV at penetration depth of 40 mfp.

processes in this energy region. For energy region 3–15 MeV, the pair production exclusively starts dominating in energy absorption process. As a result the EBF values reduce at the same time a significant variation to a small extent for all the samples. It may be due to the dependence of atomic cross section for absorption process (pair production) on equivalent atomic number (Zeq) is not as much significant as for photo-electric absorption process. Finally we find that EBF varies inversely with effective atomic numbers (Zeff) of samples at very low incident photon energies (below 0.15 MeV). It is evident that the trend of dependence of EBF on Zeff is reversed for incident photon energy 15 MeV and for penetration depths more than 10 mfp.

buildup factor values shows a similar trend, however the values of the EBF is comparatively higher at all the incident photon energies. It is observed that for deep penetration of 40 mfp and for incident photon energy range from 2.0 to 5.0 MeV the values of exposure buildup factors (EBFs) remain independent of effective atomic number (Zeff). But for incident photon energy range from 6.0 to 15.0 MeV the values of EBF start increasing with increase in effective atomic number (Zeff). This is because with the increase in penetration depth, the probability of multiple scattering increases creating more photons in the material thereby increasing the exposure buildup factor values.

3.2. Dependence of EBF on Zeff

4. Conclusions

In order to investigate of the dependence of exposure buildup factor (EBF) on the effective atomic number (Zeff), two penetration depths 15 mfp and 40 mfp had been selected for some randomly selected values of incident photon energies. From Fig. 5 it is evident that for lower incident photon energies (0.015–0.6 MeV) at penetration depth of 15 mfp the values of EBF show a markedly decreasing trend with increase in Zeff. This trend is most pronounced at lower energies and for lower Zeff, but for higher Zeff it is also lower. This trend results mainly due to the presence of elements with atomic number more than 18 in these samples. Fig. 6 shows at the higher penetration depth of 40 mfp the exposure

From the present investigations, we have found that among the selected samples, S1 (Soil-I) acts as best gamma ray shielding material, due to its higher values for mass attenuation coefficient and least values for exposure buildup factor in the selected energy range.  Lambert–Beer law’s violation is less in the selected energy region. – Where photon absorption process is dominating over the scattering process. – At shallow penetration depths.

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– For high equivalent atomic numbers of the interacting samples.  The computed data G.P. fitting parameters and exposure buildup factors for selected six low-Z building materials (25 energies and 40 penetration depths) may be useful in the future study of variety of shielding configurations. In the lower energy region the value of exposure buildup factor (EBF) depends strongly on the chemical composition (Zeq) of the samples, in the intermediate energy region it shows a very little dependence and in the higher energy region it becomes almost independent of the chemical composition of the selected samples. The inverse relationship between exposure buildup factor and effective atomic number is justified as the sample S3 (Dolomite) with least effective atomic number (Zeff) possesses the maximum value of exposure buildup factor. On other side the sample S1 (Soil-I) with maximum effective atomic number (Zeff) possesses the minimum exposure buildup factor in majority of the selected energy range. Also the selected samples show high values of attenuation coefficients than concrete for incident photon energy more than 1 MeV. At the same time the dependence of exposure buildup factor on the penetration depth, mass attenuation coefficient and energy of incident gamma ray photon is useful for further study the gamma ray shielding properties of the selected samples. From above points of view, the present study is expected to be helpful in radiation dosimetry, diagnostics and radiotherapy for determining the chemical composition, gamma-ray exposure buildup factors in shielding materials composed of selected samples. Acknowledgements We are grateful to respected Drs. M. J. Berger, J. H. Hubbell and L. Gerward for providing the state-of-the-art and user friendly computer programs XCOM/WinXCom. References ANSI, 1991. American National Standard Gamma-Ray Attenuation Coefficient and Buildup Factors for Engineering Materials. ANSI/ANS-6.4.3. Berger, M.J., Hubbell, J.H., 1987. NBSIR87-3597, XCOM: Photon Cross Sections on a Personal Computer, NIST, Gaithersburg, MD (New version 1995). Gerward, L., Guilbert, N., Jensen, K.B., Levring, H., 2004. WinXCom – a program for calculating X-ray attenuation coefficients. Radiat. Phys. Chem. 71, 653–654.

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