2.1 Geometry of concretes

Cylindrical geometries were employed for modeling of concrete samples. Eight sections of sub-cylinders of Φ30 cm×5 cm were considered for every type of concretes and set on Z axis in tandem. Figure 1 shows the simulated geometry.

Fig. 1Full-screen View

Geometry of modeled configuration (sizes are not in scale).

2.2 Source specification

Sources were considered as planar, collimated and monoenergetic with uniform distribution of radioactive material that emits gamma-ray beams perpendicular to front face of the samples (in direction of Z axis). A disc source of Φ2 cm, parallel to the X-Y plane and origin on Z axis was defined in data card of the MCNP code with ERG, PAR, POS, and DIR commands for energy and type of particle, position and direction of source respectively.

2.3 Materials specification of concretes

Elemental composition of concrete depends mainly on the mix proportions and chemical composition of the materials. The percentages by weight of the elements in the concretes are presented in Table 2 ^{[6, 10]}.

2.4 Detector geometry and tally definition

A small cylinder of Φ2 cm ×2 cm was considered as detector volume and set inside the collimator at 58 cm from the source. A Φ22 cm ×10 cm lead cylinder with a Φ2 cm ×10 cm hole was used as the collimator (Fig. 1).

Tally F4 was used to obtain MCNP-4C simulated data. This tally calculates average flux in a cell (detector volume) for only one incident gamma photon.

The simulations were performed with 1 to 10 million histories depending on type and thickness of concrete specimens. All simulated results were reported with less than 0.1% error.

3.1 Transmission factor

The transmission factors of a concrete type, can be defined by Eq.(1)

where, E is the gamma-ray energy (keV), d is thickness (cm) of shielding concrete, Φ(E, d) is the flux in the detector attained by the Tally F4, and Φ(E,0) is the flux in absence of any shielding material. The transmission factors for selected gamma-rays as a function of concrete thickness are shown in Fig. 2.

Fig. 2Full-screen View

Transmission factors of concretes to gamma-rays of different energies.

Of all the concrete types, the ordinary concrete with the least density has the least amount of attenuation (the most amount of transmission); while the steel-magnetite concrete with the greatest density has the greatest amount of attenuation (the least amount of transmission), even though the barite and ilmenite-limonite concretes contain high atomic number elements (Ba, Ti). Also, the transmission factors of concretes span from 1 to less than 10⁻⁹.

3.2 Linear and mass attenuation coefficients, HVL and TVL values of concretes

Linear and mass attenuation coefficients of the concretes (µ and µ_m) were derived from transmission factor curves through fitting Lambert law (I = I₀e^−μt) using MATLAB version 7.10.0.499. The results are given in Table 3.

The mass attenuation coefficients of concretes were also calculated using WinXCom program data using Eq.(2)

where, w_i and µ_m,i (obtained directly from WinXCom program) are percentage by weight and mass attenuation coefficient of i^th element in the concrete, respectively. The linear attenuation coefficients were calculated by multiplying mass attenuation coefficient of each type of concrete by its density. The linear and mass attenuation coefficients of concretes obtained by WinXCom program for different photon energies are presented in Table 4.

It is completely clear from both tables that the MC-simulated data and WinXCom-calculated data are very close to each other. The percentage differences between the MCNP and WinXCom results of mass attenuation coefficients for all gamma rays and concrete types range from −1.89% to 1.72%, averaged at 0.53%.

In Tables 3 and 4, at 316.51 keV, the linear attenuation coefficient of barite concrete (density=3.35 g·cm⁻³) is greater than steel-scrap value (density = 4 g·cm⁻³), unlike the general tendency that the linear attenuation coefficient increases with the density. This discrepancy could be due to the photoelectric effect. This effect is favored by low energy photons and high atomic number absorbers, so that cross section for this reaction varies approximately as Zⁿ/E^3.5, where the exponent n varies between 4 and 5 over the gamma-ray energy region of interest ^[21]. According to this fraction, at low energetic gamma rays, barite concrete with components of high atomic number elements (especially barium) has greater linear attenuation coefficient than those of the other concretes. This makes barite aggregate a good material for shielding against gamma-rays.

Figure 3 shows the half value layer (HVL= ln2/μ) and tenth value layer (TVL= ln10/μ) of the concretes, i.e. the HVL and TVL quantities are defined as the concrete thickness which reduces photon intensity to half and tenth of its initial value, respectively.

Fig. 3Full-screen View

Half value layer (a) and tenth value layer (b) of studied concretes for selected photon energies.

Of cause, the HVL and TVL values decrease with increasing density and increase with the incident photon energy, but at 316.51 keV the values of barite concrete are smaller than those of steel-scrap, due to the photoelectric effect of barite concrete, as mentioned above

The calculated data were compared with available experimental results of some concretes ^{[2, 6, 20]} (Tables 5 and 6). Generally, the calculation results are in accordance with the experimental data. The percentage differences between calculated and experimental data of HVL and TVL range from −2.94% to 15.45%, being <5% for ordinary concrete at 662 keV and < 3% for barite concrete at 1332 keV. The percentage differences between calculated and experimental data of linear attenuation coefficients at 1.5 MeV range from −26.83% to 15.45% (Table 6). Due to differences in the densities and elemental compositions among the concretes in this work and Refs.[2, 6, 20], some inequalities between experimental and calculated data were observed. Differences in geometry of simulation relative to experiment, lead to little discrepancy in calculated and measured values. Reported error of experimental work in Table 6 is <±10% ^[6].

3.3 Effective atomic number and electron density of concretes

The total atomic cross sections for concretes are calculated from the simulated values of µ_m using Eq.(3) ^[22]:

where N is the atomic mass of concrete and N_A is the Avogadro’s number. Also the σ_a and the total electronic cross sections, σ_e, for WinXCom program can be calculated from Eqs. (4) and (5) ^[23]:

where f_i denotes the fractional abundance of the i^th element with respect to the number of atoms such that f₁+f₂+f₃+…+f_i = 1, and Z_i and N_i are the atomic number and atomic mass of the i^th element, respectively. Finally, effective atomic number (Z_eff) and effective electron density (N_eff) of the concretes are calculated by Eqs. (6) and (7) ^[3] (Tables 7 and 8):

Both tables show good agreement between MCNP-4C code and WinXCom program results. In Table 7, the Z_eff value approximately increases with the concrete density, and decreases with increasing photon energy. This is due to high atomic number elements ratio in material (especially barium in barite concrete), and concretes of high Z_eff values will effectively absorb incoming photons. Of all the concretes, the effective atomic number of steel-magnetite is the greatest, while that of serpentine concretes is the lowest. Iron (Z=26) and oxygen (Z=8) as heavy and light weight elements constitute the greatest elemental portion in these concretes (about 76 and 56% respectively). In Table 8, the effective electron density of studied concretes varies from 2.83×10²³ to 3.2×10²³ electron/g. It is decreases slowly with increasing photon energy, and serpentine and barite concretes have the greatest and the lowest values of effective electron densities, respectively.