Specifying the flux and dose-equivalent buildup factors for infinite slabs irradiated by radionuclide neutron sources

Applied Radiation and Isotopes 157 (2020) 109040

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Specifying the flux and dose-equivalent buildup factors for infinite slabs irradiated by radionuclide neutron sources Rahim Khabaz Department of Physics, Faculty of Sciences, Golestan University, Gorgan, 49138-15739, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Radionuclide neutron sources Flux buildup factor Dose-equivalent buildup factor Infinite slabs

In this work, flux and neutron dose-equivalent buildup parameters are calculated for six radionuclide point-like neutron sources having broad energy spectrum which irradiate infinite slab-like common shielding materials of beryllium, concrete, iron, graphite, water, and lead, employing the MCNPX code. The description of the buildup factor is made in a straightforward way which is analogous to that of gamma. The parameters are obtained for thicknesses of shield from 0.5 to 10 mean free paths (mfp). The achieved dose-equivalent buildup parameters are parameterized by polynomial expressions. Using this parameterization, one can determine these factors for the desired thickness of shield material and each neutron source.

1. Introduction Neutrons are treated somewhat differently than other radiations in the shielding and dose calculation. For this reason, estimates are usually made of the source strength, flux energy spectra and dose equivalent conversion factors used (Khabaz, 2018b; Khabaz, 2012; Vega-Carrillo et al., 2010; Vega-Carrillo and Martinez-Ovalle, 2016). The dominant modes of interaction by neutrons are often elastic and inelastic scat­ tering with the former generally being the most important (JANIS, 2010). Estimation of the neutron dose attenuation in shields is of impor­ tance for radiation sources shielding. Recently detailed particle trans­ port simulation codes (Cloth et al., 1988; Waters et al., 1995) have become available. However, these simulation codes need too long a computing time to be applied frequently to practical shielding designs. Usually, therefore, the estimation of neutron dose at in workplaces within nuclear facilities is made by the simplified empirical formula. The point kernel calculation method is widely used for gamma ray attenuation calculations in bulk materials (Shani, 2000; Trubey, 1988). For neutron shielding can also have similar formulation. Buildup factors are key data for this method (Harima, 1993; Alamatsaz and Shirani, 2002). The buildup factor represents the intensity ratio of the overall radiation approaching a point, to the primary radiation’s intensity approaching the same point. If the incident beam, as well as the area one is trying to shield, may be highly narrow, or, if the shielding material may be very thin, merely the un-scattered primary particles may induce effective dose, and

buildup factor would be equal to unity. However, since this is rarely the case, buildup factor will, in general, be larger than 1, as it accounts also for the contribution coming from those particles scattered and from those that have been formed by the primary particles. Similar to what is followed in the gamma ray calculations (Khabaz, 2018a; Vega-Carrillo et al., 2018), it can be defined the flux buildup P factor, BF( t d, E), for neutron at any point d in the shield based on the formula as follows: ! X ΦðdÞ P BF d; E ¼ (1) d t Φ∘ e t where Ф(d) is the total neutron flux at special point d, Ф0 is the initial P flux and t is the total macroscopic cross section of shield material at energy E. Therefore, regarding a point isotropic multi-energy source, emitting one neutron per second, on one side of an infinite slab-like shield of certain thickness and a point indicator on the other side of the shield with the sight line between detector and source common to the surfaces of slab; the flux buildup factor is obtained as: P �X � BF d; E ¼ 4πd2 ΦðdÞ e t d (2) t The dose-equivalent buildup parameter, BH( from:

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.apradiso.2020.109040 Received 13 December 2017; Received in revised form 25 December 2019; Accepted 7 January 2020 Available online 9 January 2020 0969-8043/© 2020 Elsevier Ltd. All rights reserved.

P

t d,

E), is also given

R. Khabaz

Applied Radiation and Isotopes 157 (2020) 109040

Fig. 1. The Monte Carlo simulated spectra of

P hðEi Þ Φi P � � i d 2 t � P d; E ¼ 4 π d e t h Ej Pj

�X BH

�

241

Am–Li, Po–Be,

252

Cf,

241

Am–B,

241

Am–Be and

239

Pu–Be neutron sources (Griffith et al., 1990).

(3)

j

P where, the t is the total macroscopic cross section obtained by aver­ aging over the spectral neutron distribution of the source (ISO 10647, 1996), the Фi is the flux of all neutrons with energy between Ei-1 and Ei, the h(Ei) is the flux-to-dose-equivalent conversion factor and Pj is number of neutrons with energy Ej emitted per nuclear transition. Owing to the relative simplicity and low cost with no requirements for complicated controlling mechanisms, radionuclide neutron sources, today, find increasing applications; therefore, in order to use them safely, determination of the buildup factors is especially important for their shielding. According to our knowledge, neutron buildup factors for radionu­ clide neutron sources with broad energy spectrum (multi-energy sour­ ces) have not been calculated before. However, the buildup factors have been studied for monoenergetic neutrons (Shirani and Shahriari, 2007; Shin et al., 1997; Wyman and Harms, 1985). In this work, flux and dose-equivalent buildup parameters were achieved using MCNPX Monte Carlo simulation code for six common radionuclide neutron sources which irradiate infinite slab-like shields of six shielding materials. The shielding materials were beryllium, con­ crete, iron, graphite, lead and water which exposed to 241Am–Li, Po–Be, 252 Cf, 241Am–B, 241Am–Be and 239Pu–Be point-like neutron sources. 2. Materials and methods

Fig. 2. Schematic representation of setup in Monte Carlo calculation of neutron buildup factors.

Monte Carlo calculations of the buildup factors of six radionuclide neutron sources, i.e., 241Am–Li, Po–Be, 252Cf, 241Am–B, 241Am–Be and 239 Pu–Be were performed for six shields, i.e., beryllium, concrete, iron, graphite, lead and water. Calculations were carried out with the MCNPX2.6 code (Pelowitz, 2008) and cross sections were obtained from the latest cross-section library ENDF/B-VII.0 (Chadwick et al., 2006).

The mean energy values of these sources were 0.56, 2.04, 2.54, 3.27, 4.46 and 5.40 MeV, respectively. The spectra of energy of these radio­ nuclide neutron sources (Fig. 1) have been used in MCNPX simulation (Khabaz, 2015; Griffith et al., 1990). P At the first step, the average total macroscopic cross section ( t ) was 2

R. Khabaz

Applied Radiation and Isotopes 157 (2020) 109040

Table 1

Obtained average macroscopic cross section (

t)

by Monte Carlo simulation.

241

Po–Be

252

241

241

P

Error

P

Error

P

Error

P

Error

P

0.0005 0.0002 0.0002 0.0002 0.0002 0.0005

0.3541 0.2096 0.2561 0.2016 0.2216 0.3229

0.0003 0.0002 0.0002 0.0002 0.0002 0.0003

0.3097 0.1837 0.2538 0.1726 0.2068 0.2732

0.0003 0.0002 0.0002 0.0002 0.0002 0.0002

0.2733 0.1505 0.2797 0.1445 0.2187 0.2019

0.0002 0.0001 0.0002 0.0001 0.0002 0.0002

0.2899 0.1450 0.2793 0.1508 0.2336 0.2152

Am–Li

t ðcm

Be Concrete Fe C Pb Water

P

1

Þ

0.4848 0.2782 0.2413 0.2798 0.2294 0.5474

t ðcm

Cf

1

Þ

t ðcm

Am–B

1

Þ

t ðcm

1

Þ

239

Am–Be

t ðcm

1

Þ

Pu–B

Error

P

0.0003 0.0001 0.0002 0.0002 0.0002 0.0002

0.2580 0.1389 0.2814 0.1337 0.2147 0.1795

t ðcm

1

Þ

Error 0.0002 0.0001 0.0002 0.0001 0.0002 0.0002

Table 2 Flux buildup factors data calculated by MCNPX for six shields by 241Am–Li neutron source with relative error. P mfp ( t d)

Be

0.5 1 2 3 4 5 6 7 8 9 10

Concrete

Fe

C

t

cm

1

�

Water

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.782 2.984 7.959 20.415 52.152 130.252 327.263 831.926 2032.961 5077.649 13150.974

0.001 0.001 0.001 0.002 0.003 0.004 0.006 0.010 0.010 0.010 0.019

1.763 2.953 7.889 20.587 52.570 130.428 322.025 796.130 2054.787 4645.077 10741.033

0.001 0.001 0.002 0.004 0.006 0.007 0.007 0.008 0.033 0.014 0.016

1.710 2.766 7.058 17.744 44.557 111.910 287.032 728.951 1823.828 4601.410 11967.928

0.001 0.001 0.002 0.003 0.004 0.004 0.006 0.007 0.008 0.008 0.010

1.772 2.969 7.892 20.222 51.254 128.855 329.392 828.322 2083.180 5245.357 13492.866

0.001 0.001 0.002 0.003 0.004 0.005 0.007 0.008 0.009 0.012 0.016

1.751 2.903 7.621 19.350 48.710 123.892 307.253 780.295 2024.157 5096.174 12837.217

0.001 0.001 0.002 0.003 0.004 0.006 0.006 0.007 0.010 0.010 0.011

1.805 3.175 9.159 23.638 55.357 118.160 231.683 422.886 797.518 1336.626 2346.953

0.001 0.002 0.003 0.005 0.007 0.010 0.029 0.016 0.022 0.030 0.023

calculated for each configuration of source-shield. For obtaining this parameter, each shield’s slab with 5 cm thickness was irradiated by a plane-parallel beam of each neutron source. The parallel neutrons are tallied on the exit surface of the slab with a current tally (F1) associated with a cosine card. In this status, the buildup parameter in Eq. (1) is unit, and also outputs of MCNPX are normalized to one neutron of the source, P then t ðcm 1 Þ for 5 cm thickness of each shield is given by: X

Pb

BF

lnðΦÞ 5

¼

equivalent buildup parameters were obtained, respectively. For each case, a number of histories were large enough (�5 � 108) to reach an uncertainty less than 4%. For checking the data obtained using MCNPX, the same configura­ tion of Fig. 2 is simulated by GEANT4.10.5 code version to achieve the flux buildup factors of six shields for 252Cf and 241Am–Be neutron sources. Event generation in GEANT4 is handled by the user-defined Primary Generator Action class. This class contains instructions as to the particle types being generated, and how to distribute and transport the particles.

(4)

In the following, for each material, the slab-like shields from 0.5 to 10 mfp thicknesses waere simulated in the MCNPX modeling. The con­ crete density was 2.25 g/cm3 whose elemental composition was 0.453% H, 51.260% O, 1.527% Na, 3.555% Al, 36.036% Si, 5.791% Ca and 1.378% Fe (Harmon et al., 1994). Regarding the geometry of the current study, a point isotropic source of neutron was placed on one side of the infinite slab shield and a point detector located on the other side of the shield along with a sight line between detector and source normal to the surfaces of slab (Fig. 2). The source and detector were on the surface of the shield. For obtaining the buildup parameters, the point detector tally (F5) in the MCNPX code was used, then by Eqs. (2) and (3) flux and dose-

3. Results and discussion P The average total macroscopic cross section ( t ) calculated for each configuration of source-shield in this work with Monte Carlo simulation are presented in Table 1. Using results of Table 1, the flux and dose-equivalent buildup factors for some radionuclide neutron sources, i.e., 241Am–Li, Po–Be, 252Cf, 241 Am–B, 241Am–Be and 239Pu–Be placed one side of six infinite slab-like shields, i.e., beryllium, concrete, iron, graphite, lead and water with different thicknesses 0.5 to 10 mfp. The results of flux buildup param­ eters are shown in Tables 2–7 for 241Am–Li, Po–Be, 252Cf, 241Am–B,

Table 3 Flux buildup factors data calculated by MCNPX for six shields by Po–Be neutron source with relative error. P mfp ( t d) 0.5 1 2 3 4 5 6 7 8 9 10

Be

Concrete

Fe

C

Pb

Water

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.805 3.075 8.492 22.730 59.362 153.204 388.831 1001.833 2454.715 6363.533 15284.137

0.001 0.001 0.002 0.003 0.006 0.006 0.007 0.012 0.014 0.021 0.015

1.746 2.915 7.799 20.128 51.546 128.339 323.142 758.551 1824.717 4176.130 9322.752

0.001 0.001 0.003 0.004 0.008 0.008 0.011 0.015 0.019 0.017 0.020

1.720 2.832 7.395 18.918 47.671 119.519 305.453 761.475 1934.250 4973.197 12500.190

0.001 0.001 0.002 0.004 0.004 0.005 0.006 0.006 0.007 0.009 0.009

1.752 2.917 7.741 19.931 50.834 130.839 328.982 845.283 2104.617 5399.515 13515.069

0.001 0.001 0.002 0.003 0.005 0.006 0.008 0.012 0.011 0.016 0.027

1.745 2.918 7.784 20.113 51.545 131.582 333.203 854.446 2197.987 5671.175 14255.900

0.001 0.001 0.003 0.004 0.005 0.006 0.006 0.007 0.009 0.011 0.010

1.821 3.178 8.485 18.624 36.758 63.603 107.700 200.122 361.660 668.947 1047.739

0.001 0.002 0.005 0.008 0.013 0.017 0.025 0.034 0.035 0.038 0.039

3

Applied Radiation and Isotopes 157 (2020) 109040

R. Khabaz

Table 4 Flux buildup factors data calculated by MCNPX for six shields by 252Cf neutron source with relative error. P mfp ( t d) 0.5 1 2 3 4 5 6 7 8 9 10

Be

Concrete

Fe

C

Pb

Water

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.808 3.117 8.796 23.760 62.581 159.969 412.512 1003.741 2561.889 6461.969 15612.823

0.001 0.001 0.002 0.003 0.005 0.006 0.011 0.011 0.020 0.029 0.027

1.740 2.914 7.859 20.521 52.350 131.023 314.072 724.972 1688.720 3857.285 8057.035

0.001 0.002 0.003 0.005 0.007 0.011 0.012 0.018 0.016 0.029 0.022

1.726 2.866 7.532 19.399 49.326 125.369 316.522 807.302 2016.853 5179.519 13142.283

0.001 0.001 0.002 0.003 0.004 0.006 0.006 0.008 0.007 0.011 0.009

1.743 2.904 7.764 20.237 51.896 132.205 342.364 860.374 2091.256 5366.342 13136.403

0.001 0.001 0.003 0.004 0.006 0.008 0.017 0.013 0.014 0.021 0.020

1.744 2.928 7.913 20.430 52.839 136.320 346.314 879.836 2326.989 5830.463 19566.473

0.001 0.002 0.003 0.003 0.004 0.006 0.007 0.007 0.014 0.012 0.012

1.798 3.175 8.428 18.033 32.360 53.071 90.604 138.597 241.341 402.104 575.649

0.001 0.003 0.005 0.010 0.015 0.022 0.026 0.029 0.030 0.034 0.035

Table 5 Flux buildup factors data calculated by MCNPX for six shields by 241Am–B neutron source with relative error. P mfp ( t d) 0.5 1 2 3 4 5 6 7 8 9 10

Be

Concrete

Fe

C

Pb

Water

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.812 3.147 8.926 24.154 63.019 159.337 401.013 986.572 2389.025 5722.658 13639.164

0.001 0.001 0.002 0.004 0.006 0.008 0.010 0.017 0.020 0.020 0.027

1.741 2.954 8.162 21.699 55.455 136.184 315.087 712.230 1509.185 3203.759 6149.562

0.001 0.002 0.004 0.005 0.010 0.011 0.016 0.018 0.020 0.031 0.021

1.734 2.900 7.776 20.334 51.792 131.194 330.317 836.668 2153.421 5377.423 13816.682

0.001 0.001 0.002 0.003 0.004 0.005 0.006 0.008 0.009 0.008 0.009

1.703 2.811 7.457 19.712 50.644 131.817 325.543 795.413 2042.890 5280.437 12742.848

0.001 0.001 0.003 0.004 0.007 0.014 0.011 0.015 0.019 0.024 0.029

1.743 2.924 7.937 20.642 54.189 138.353 350.078 903.583 2320.731 5873.353 15116.807

0.001 0.002 0.003 0.004 0.007 0.008 0.008 0.009 0.011 0.009 0.010

1.809 3.205 8.201 14.743 21.018 30.056 40.131 52.540 63.923 77.554 107.351

0.001 0.003 0.007 0.015 0.024 0.043 0.019 0.025 0.028 0.032 0.037

Table 6 Flux buildup factors data calculated by MCNPX for six shields by 241Am–Be neutron source with relative error. P mfp ( t d) 0.5 1 2 3 4 5 6 7 8 9 10

Be

Concrete

Fe

C

Pb

Water

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.884 3.400 10.323 29.604 81.123 214.400 551.498 1369.956 3392.699 8382.700 20716.316

0.001 0.001 0.002 0.003 0.005 0.007 0.009 0.010 0.013 0.019 0.018

1.719 2.856 7.589 19.641 48.546 118.554 273.968 636.846 1248.929 2711.129 5068.623

0.001 0.002 0.004 0.006 0.009 0.017 0.020 0.036 0.018 0.032 0.030

1.729 2.879 7.738 20.163 51.508 130.825 331.012 842.825 2165.586 5462.103 13852.415

0.001 0.001 0.003 0.004 0.004 0.005 0.005 0.007 0.008 0.010 0.008

1.726 2.867 7.708 20.431 53.562 138.979 358.018 892.839 2260.377 5450.282 14612.542

0.001 0.001 0.003 0.004 0.007 0.009 0.012 0.013 0.019 0.015 0.029

1.741 2.938 8.083 21.483 56.147 146.990 378.227 990.477 2532.754 6578.913 16856.239

0.001 0.002 0.003 0.004 0.005 0.006 0.006 0.010 0.009 0.013 0.012

1.789 3.131 7.939 15.840 28.082 48.807 86.460 132.798 221.337 325.412 477.627

0.001 0.003 0.006 0.011 0.017 0.022 0.026 0.029 0.031 0.033 0.037

Table 7 Flux buildup factors data calculated by MCNPX for six shields by 239Pu–Be neutron source with relative error. P mfp ( t d) 0.5 1 2 3 4 5 6 7 8 9 10

Be

Concrete

Fe

C

Pb

Water

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

BF

Rel. Er.

1.925 3.543 11.007 31.986 86.188 228.895 588.313 1490.343 3579.586 8655.333 20403.825

0.001 0.001 0.002 0.004 0.005 0.008 0.012 0.014 0.022 0.019 0.021

1.705 2.824 7.581 19.428 49.889 117.921 256.570 587.136 1224.465 2369.108 4851.178

0.001 0.002 0.004 0.006 0.013 0.016 0.011 0.026 0.026 0.026 0.031

1.727 2.889 7.767 20.337 52.246 131.729 331.405 842.022 2176.018 5525.250 13962.154

0.001 0.002 0.003 0.004 0.004 0.005 0.005 0.005 0.008 0.008 0.008

1.716 2.849 7.677 20.378 53.120 137.774 348.438 892.556 2175.436 5587.432 12998.187

0.001 0.002 0.003 0.005 0.007 0.012 0.015 0.018 0.021 0.027 0.025

1.762 3.005 8.297 22.412 58.906 152.317 393.314 1017.046 2622.873 6654.289 17230.404

0.001 0.002 0.003 0.006 0.005 0.005 0.008 0.009 0.009 0.008 0.017

1.787 3.130 7.414 13.669 22.610 35.844 53.933 90.181 113.076 222.434 225.259

0.001 0.002 0.005 0.010 0.018 0.024 0.027 0.030 0.032 0.035 0.039

4

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Applied Radiation and Isotopes 157 (2020) 109040

Table 8 Flux buildup factors achieved by GEANT4 for six shields by 252Cf neutron source and relative difference with the results of MCNPX. mfp P ( t d)

Be BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

0.5 1 2 3 4 5 6 7 8 9 10

1.825 3.157 8.920 24.079 63.530 162.858 420.782 1026.829 2626.936 6630.980 16035.369

0.9 1.2 1.3 1.3 1.5 1.8 2.0 2.3 2.5 2.6 2.7

1.755 2.948 7.960 20.787 53.134 133.380 320.351 741.644 1730.936 3957.571 8274.570

0.8 1.0 1.1 1.3 1.4 1.6 1.6 1.9 2.3 2.5 2.6

1.742 2.894 7.620 19.652 50.064 127.251 322.218 823.446 2063.240 5324.544 13536.550

1.0 1.0 1.2 1.3 1.5 1.5 1.8 2.0 2.3 2.8 3.0

1.756 2.928 7.840 20.462 52.625 134.054 347.499 875.010 2128.879 5479.046 13425.412

0.7 0.8 1.0 1.1 1.4 1.4 1.5 1.7 1.8 2.1 2.2

1.764 2.961 8.018 20.714 53.635 138.511 352.558 897.333 2375.876 5987.286 20133.951

1.1 1.1 1.2 1.4 1.5 1.6 1.8 2.0 2.1 2.7 2.9

1.811 3.202 8.505 18.223 32.749 53.865 92.154 140.963 245.938 410.555 590.042

0.8 0.9 0.9 1.0 1.2 1.5 1.6 1.7 1.9 2.1 2.5

Concrete

Fe

C

Pb

Water

Table 9 Flux buildup factors achieved by GEANT4 for six shields by 241Am–Be neutron source and relative difference with the results of MCNPX. mfp P ( t d)

Be BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

BF

Rel. Diff. (%)

0.5 1 2 3 4 5 6 7 8 9 10

1.894 3.425 10.398 29.831 81.924 216.768 557.566 1389.165 3443.569 8525.216 21192.591

0.5 0.7 0.7 0.8 1.0 1.1 1.1 1.4 1.5 1.7 2.3

1.727 2.872 7.651 19.817 48.984 119.868 276.983 646.395 1268.922 2762.642 5175.164

0.6 0.6 0.8 0.9 0.9 1.1 1.1 1.5 1.6 1.9 2.1

1.742 2.903 7.807 20.366 52.021 132.265 334.654 852.949 2193.737 5549.597 14143.326

0.8 0.8 0.9 1.0 1.0 1.1 1.1 1.2 1.3 1.6 2.1

1.742 2.894 7.792 20.675 54.215 140.776 363.385 908.014 2301.034 5559.286 14963.223

0.8 1.0 1.1 1.2 1.2 1.3 1.5 1.7 1.8 2.0 2.4

1.754 2.964 8.153 21.697 56.766 148.911 383.145 1004.334 2575.821 6703.932 17260.759

0.8 0.9 0.9 1.0 1.1 1.3 1.3 1.4 1.7 1.9 2.5

1.802 3.149 8.019 16.024 28.392 49.345 87.585 134.791 224.879 331.369 486.712

0.6 0.9 1.0 1.1 1.1 1.1 1.3 1.5 1.6 1.8 1.9

Concrete

Fe

C

Fig. 3. Neutron dose-equivalent buildup factors for

5

241

Pb

Water

Am–Li source irradiating different shields.

R. Khabaz

Applied Radiation and Isotopes 157 (2020) 109040

Fig. 4. Neutron dose-equivalent buildup factors for Po–Be source irradiating different shields.

Fig. 5. Neutron dose-equivalent buildup factors for 241

Am–Be and 239Pu–Be sources, respectively. The flux buildup factors of various shields for 252Cf and 241Am–Be sources obtained by GEANT4, and the relative difference with the results of MCNPX are listed in Tables 8 and 9. As it can be seen, the results of GEANT4 are close to the values of flux buildup factors obtained by MCNPX. For 252Cf source the relative errors

252

Cf source irradiating different shields.

are less than 3.0% and for 241Am–Be are less than 2.5%. This agreement shows that the results of simulation by MCNPX are reliable and acceptable. Furthermore, the induced flux was then altered through an ambient dose equivalent response function from the International Commission’s compilation on Radiological Protection (ICRP) Publication 74 (ICRP, 6

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Applied Radiation and Isotopes 157 (2020) 109040

Fig. 6. Neutron dose-equivalent buildup factors for

Fig. 7. Neutron dose-equivalent buildup factors for

1997) to obtain the dose-equivalent buildup factor based on Eq. (3). Figs. 3–8 show the available dose-equivalent buildup factor data ob­ tained by Monte Carlo method for six shielding materials by 241Am–Li, Po–Be, 252Cf, 241Am–B, 241Am–Be and 239Pu–Be sources, for different penetration depths, respectively. It can be seen that the neutron dose-equivalent buildup factor for

241

Am–B source irradiating different shields.

241

Am–Be source irradiating different shields.

each source increases with the thickness of shield. Indeed, as the thickness of shield increases, the proportion of scattered radiation in the total neutron flux increases (and that of un-scattered radiation decreases). The values of this factor for concrete and graphite are very close to each other exposed to sources with lower mean energy i.e., 241Am–Li, 7

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Applied Radiation and Isotopes 157 (2020) 109040

Fig. 8. Neutron dose-equivalent buildup factors for Table 10 Comparison between dose-equivalent buildup factors calculated in this work for

239

Pu–Be source irradiating different shields.

252

Cf source and reported by Shirani and Shahriari (2007) for energy of 2.5 MeV.

mfp P ( t d)

Fe

Pb

By Shirani and Shahriari

This work

Rel. Diff. (%)

By Shirani and Shahriari

This work

Rel. Diff. (%)

Water By Shirani and Shahriari

This work

Rel. Diff. (%)

0.5 1 2 3 4 5 6 7 8 9 10

1.759 2.906 7.566 18.67 45.3 106.2 246.3 578.9 1365 3314 7973

2.34 3.81 9.59 23.47 56.39 135.78 322.60 772.82 1815.90 4447.78 10406.74

24.7 23.7 21.1 20.4 19.7 21.8 23.7 25.1 24.8 25.5 23.4

1.743 2.842 7.627 19.91 49.8 127.1 318.2 749.1 2027 5123 12420

2.38 3.96 10.50 26.49 66.70 166.90 410.70 997.79 2547.84 6048.19 20297.12

26.7 28.2 27.4 24.8 25.3 23.8 22.5 24.9 20.4 15.3 38.8

1.613 2.289 3.947 5.59 6.953 8.527 10.09 12.02 14.89 16.68 18.2

2.08 2.97 5.51 9.68 16.63 28.14 48.95 81.57 138.70 251.18 397.33

22.4 22.9 28.3 42.2 58.2 69.7 79.4 85.3 89.3 93.4 95.4

Po–Be and 252Cf, and also for beryllium and graphite by sources with higher mean energy i.e., 241Am–B, 241Am–Be and 239Pu–Be. Further­ more, for all sources, in the same thickness of shields, the lead and water Table 11 Polynomial fitting coefficients of BH for 241

have highest and lowest value in buildup factor, respectively. With comparison of the results, it is obvious that the neutron doseequivalent buildup factor for each source increases with increasing the

241

Am–Li, Po–Be and 252Cf sources.

Am–Li

252

Po–Be

Cf

α

β

γ

δ

α

β

γ

δ

α

β

γ

δ

Be

6.150E-03

Concrete

1.783E-02

Fe

5.870E-03

C

5.640E-03

Pb

3.950E-03

Water

1.341E02

3.905E01 3.657E01 4.005E01 4.059E01 4.323E01 2.796E01

2.088E02 1.067E02 3.270E03 1.910E02 5.860E03 4.810E03

1.290E03 5.402E04 1.742E04 1.030E03 2.615E04 2.967E04

2.937E02 3.003E02 2.739E02 2.303E02 2.439E02 3.212E02

4.926E01 4.642E01 4.815E01 4.926E01 5.024E01 3.661E01

3.182E02 2.178E02 1.812E02 2.809E02 1.803E02 1.678E02

1.610E03 1.020E03 9.708E04 1.370E03 9.158E04 8.610E04

1.683E01 1.674E01 1.663E01 1.576E01 1.373E01 1.862E01

4.386E01 4.123E01 4.168E01 4.440E01 4.761E01 2.883E01

2.557E02 1.523E02 6.160E03 2.356E02 1.816E02 8.920E03

1.260E03 6.149E04 3.036E04 1.110E03 1.200E03 4.354E04

8

R. Khabaz

Applied Radiation and Isotopes 157 (2020) 109040

Table 12 Polynomial fitting coefficients of BH for 241

241

Am–B, 241Am–Be and 239Pu–Be sources. 241

Am–B

239

Am–Be

Pu–B

α

В

γ

δ

α

β

γ

δ

α

β

γ

δ

Be

2.652E-02 2.720E-02

Fe

2.408E-02

C Pb

1.012E02 2.364E-02

Water

6.698E-02

3.161E02 2.090E02 8.720E03 3.758E02 6.770E03 3.001E02

1.500E03 7.204E04 4.248E04 1.750E03 2.955E04 1.430E03

1.787E-02

Concrete

4.586E01 4.426E01 4.345E01 4.965E01 4.400E01 3.127E01

4.837E01 4.038E01 4.301E01 4.715E01 4.449E01 3.135E01

3.014E02 1.292E02 7.580E03 2.548E02 6.260E03 9.660E03

1.460E03 3.021E04 3.642E04 1.190E03 2.621E04 2.575E04

2.824E02 1.009E02 1.427E02 6.350E03 1.277E02 7.400E02

4.832E01 4.336E01 4.340E01 4.564E01 4.528E01 2.835E01

3.236E02 2.232E02 8.620E03 2.392E02 8.230E03 9.110E03

1.600E03 9.254E04 4.359E04 1.010E03 3.646E04 2.444E04

2.999E-02 1.947E-02 4.850E03 1.541E-02 5.251E-02

effective mass number of shield material; however, for 241Am–Be and 239 Pu–Be sources, this factor for beryllium is higher than concrete. This may be due to (n, 2n) interaction in the beryllium. In Table 10, the calculated dose-equivalent buildup factors of iron, lead and water in this study for 252Cf source (with the mean energy of about 2.5 MeV) are compared to those calculated and reported by Shirani and shahriari (2007) for energy point of 2.5 MeV. As can be seen, the relative differences for lead and iron were about 20–25% and 15–39%, respectively. However, for thick shields of water (>4mfp), this difference reaches to 58–95%. Thus, to obtain the buildup factors for a neutron source having a broad energy spectrum (multienergy), all energies of the source must be considered in the simulation. In other words, the buildup factor of mean energy is not a proper sub­ stitute for all energies of the source. Although some analysts prefer to use data tables directly in computer calculations, others prefer parameterized forms. The results of doseequivalent buildup factor for any source-material configuration were fitted based on a polynomial equation as: �X � �X �2 �X �3 �X � BH d ¼αþβ d þγ d þδ d (5) t t t t

arranged values are useful, especially if one has these shield thicknesses and wants to determine the dose rate, and they are not proper when one wishes to determine dose rate for other thicknesses. Using an analytical form of the buildup factor can be appropriated for any desired thickness. The calculated dose-equivalent buildup factors, in the current study, were fitted by a polynomial formula. Author contribution statements R. Khabaz conceived of the presented idea, and also developed the theory and performed the computations. He verified the analytical methods, then discussed the results and contributed to the final manuscript. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References

where α, β, γ, and δ are constant coefficients for a radionuclide neutron source and shield material. The obtained values of coefficients α, β, γ, and δ for neutron sources are listed in Tables 11 and 12. Using Eq. (5) and presented coefficients in Tables 11 and 12, the neutron dose-equivalent buildup factor can be obtained for each neutron source and the desired thickness of shield material.

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4. Conclusions A neutron buildup factor was defined in a way analogous to the gamma ray buildup factor. Flux and dose-equivalent buildup parameters were obtained using MCNPX code for uniform infinite slabs of beryllium, concrete, iron, graphite, water and lead irradiated by six radionuclide neutron point isotropic sources having different mean energy. A point isotropic source is assumed in this study. Practical sources are broad beam sources. However, the attenuation of the neutron dose equivalent is not much variation between these two except for the region close to the source, when a correction for geometrical attenuation is made to the present data. The buildup factor values presented here have a maximum relative error of 4% which occurs for thick shields of 10 mfp. For thinner shields, the relative error is less than 1%. Since the Monte Carlo simulation as­ sumes all types of interactions of neutrons with matter, the buildup parameters achieved are adequately precise to be employed in neutron shielding and dosimetry calculations measurements for these radionu­ clide neutron sources. The results were tabulated according to shield material, source type, and shield thickness (a number of neutron mean free paths). Such 9

R. Khabaz

Applied Radiation and Isotopes 157 (2020) 109040

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