2.1.1 Nano SiO2

The chemical analysis of Portland concrete and Nano SiO₂ is given by Elsharkawy and Sadawy [1]. Nano concrete is formed by grinding the commercially available grade Portland concrete at a high energy [7]. Tables 1 and 2 show the concrete sample without Nano SiO₂ and the concrete mixed with a weight percentage of 0.5% CaO in Nano SiO₂, respectively. Table 3 shows the densities of Nano SiO₂ at different weight percentages of SiO₂ nanoparticles [1].

To study the nuclear attenuation properties, the compressive strength of hardened concrete cubic samples (7×7×7 cm³) is considered. A beam from both point and disk isotropic gamma-ray sources of ¹³⁷Cs, with one line of energy of 0.662 MeV, ⁶⁰Co, with two lines of energy of 1.17 and 1.33 MeV, and ²²Na, with two lines of energy of 0.511 and 1.274 MeV, was used in the MCNPX code. Figure 1 shows the disk gamma-ray source and cubic concrete sample simulated by the MCNPX code. Figures 2a and b show the tracks of photons obtained from this code for the point and disk gamma-ray sources, respectively.

Fig. 1.Full-screen View

(Colro online)The disk gamma-ray source and the cubic concrete sample simulated using the MCNPX code

Fig. 2.Full-screen View

(Color online) The tracks of photons simulated using the MCNPX code; a: point source, b: disk source

2.1.2 Nano MnFe2O4

We have added MnFe₂O₄ nanoparticles to concrete with a weight percentage of 2.0% CaO at different temperatures, as shown in Tables 4 and 5. MnFe₂O₄ is a magnetic oxide, in which oxygen has a close packing structure with the Mn²⁺ and Fe³⁺ ions. These spinel ferrites are very promising candidates for understanding and controlling the magnetic properties of nanoparticles at the atomic level [11-12].

2.2.1 MCNPX code

The shielding material parameters are the most important for characterizing the penetration and diffusion of gamma-rays [15]. Theoretical modeling of the photon attenuation coefficients of the materials permits more flexibility and simplicity compared to experimental studies. The linear attenuation coefficients, μ, in materials are determined by the transmission method according to Beer- Lambert’s law [1, 15-17]:

where I₀ and I are the incident and attenuation photon intensities, respectively, μ (cm⁻¹) is the linear attenuation coefficient, and X is the thickness of the slab. The mass attenuation coefficient is defined as μ_m (cm² g^-1) = μ (cm⁻¹)/ρ (g/cm³).

In this work, the μ and μ_m coefficients are obtained by the MCNPX code at various photon energies: 511, 662, 1170, 1274, and 1332.5 keV. The concrete material content ratios have been defined relative to a sample material in this code. Calculations were performed for one million histories per run using Intel Core i7 CPU 3.40 GHz computers. The obtained statistical errors were less than 1% for a long run time. The flux integrated over a surface tally (F2) has been used in the μ and μ_m calculations [3, 17-19].

One of two methods can be used for the calculation of the photon dose rate in concrete samples that are useful for converting the flounce quantities to units of dose. The choice is between using a heating number method or one or more flounces-to-dose conversion functions. Both approaches are valid for photon dose rate calculations, but the use of conversion functions is recommended for dose equivalent and effective dose calculations [14, 20].

In the heating number method, the code calculates the absorbed dose on the basis of the KERMA approximation, which is locally deposited [14]. The KERMA approximation dose can be represented using the following equations:

where $N_{\text{a}}$ is Avogadro’s number (6.022×10²³ mol^-1), $\eta$ is the number of atoms per molecule, $M$ is the molar mass in grams, $\varnothing$ is the fluence score in particles/cm², ${\sigma }_{\text{T}}$ is the total atomic cross section at energy of scoring track in barns, $H$ is the heating number in MeV per collision, $T$ is the number of scoring source particle tracks and $N$ is the number of source particles.

The second method is the flounce-to-dose conversion function. In this method, the absorbed dose rate is calculated by using DF_n (Dose Function) and tally F4 operations. Suppose one wanted to compute the dose rate of some type of flux tally, either total or by energy group. This feature allows you to enter a point wise response function (such as flux-to-dose conversion factors) as a function of energy to modify a regular tally. Therefore, the absorbed dose rate is calculated by combining the DF_n and F4 cards. Finally, the DF_n card is defined as follows:

where IU, FAC, IC and INT are the controls units, normalization factor for dose, standard dose function, and energy interpolation, respectively [14, 20].

3.1.1 Calculation of attenuation coefficients in Nano SiO2

The μ value was calculated by the MCNPX code, and is shown in Fig. 3. This figure presents the effects of varying gamma-ray energy with μ at weight percentages of 0.0%, 0.5%, 1.5% and 2.0% SiO₂ in Nano SiO₂, and compares these to the experimental results of ref. 1. Calculations have been carried out using a beam of point isotropic gamma-ray sources of ¹³⁷Cs, with one line of energy of 0.662 MeV, ⁶⁰Co, with two lines of energy of 1.17 and 1.33 MeV, and ²²Na, with two lines of energy of 0.511 and 1.274 MeV. This figure illustrates that the values of μ decrease with increasing gamma-ray energies and that the maximum and minimum values of are found at the energies of 0.511 and 1.33 MeV, respectively. In addition, these parameters increase with increasing weight percentage of SiO₂ nanoparticles until 1.5%, after which they decrease due to decreasing density. In general, the relative error in all these calculations is less than 1.0%.

Fig. 3.Full-screen View

(Color online) Effect of gamma energy on the μ at different SiO₂ nanoparticle percentages

3.1.2 Calculation of attenuation coefficients of Nano MnFe2O4

Figure 4 shows the effect of gamma-ray energy on μ for Nano MnFe₂O₄ with 2.0% CaO at 300, 573, 673, and 773 K. Figure 5 presents the same effect for Nano MnFe₂O₄ at different temperatures, as calculated by the MCNPX code. As shown in Figs. 4 and 5, the values of μ decrease with increasing gamma-ray energies and the maximum and minimum values of μ are found at 0.511 and 1.33 MeV, respectively. In addition, this parameter can be increased by increasing the density up to 5.194 g/cm³, as seen for the Nano MnFe₂O₄ temperature of 673 K.

Fig. 4.Full-screen View

(Color online) Effect of gamma energy on μ at different temperatures for Nano MnFe₂O₄

Fig. 5Full-screen View

(Color online) Effect of temperature of Nano MnFe₂O₄ on μ for different gamma energies

3.1.3 Comparison of SiO2 and MnFe2O4 nanoparticles

Figure 6 shows the μ_m of the SiO₂ and MnFe₂O₄ nanoparticles constituting 2.0 wt.% in concrete for different gamma-ray energies. According to Fig. 6 a fairly good agreement with the experimental results was observed, notably in the simulation of Nano SiO₂ with a density of 2.26 g/cm³, whereas for the theoretical results, a good agreement was obtained at the density of 2.47 g/cm³. The average relative differences between the simulation results and the experimental and theoretical results are 6.4%, 5.5% and 2.1% for the densities of 2.26 g/cm³ and 2.47 g/cm³, respectively. Moreover, as seen in this figure, the μ_m of the MnFe₂O₄ nanoparticles is much higher compared to that of the SiO₂ nanoparticles, which is attributed to a higher density.

Fig. 6.Full-screen View

(Color online) Comparison of SiO₂ and MnFe₂O₄ nanoparticles in concrete on the μ_m at different gamma energies

As listed below, there are sources of uncertainty in the results of the MCNPX code in comparison to the experimental and theoretical results:

1. Inherent error in the probabilistic method of the MCNPX code,

2. Inherent error in this code due to the libraries used, especially the inability to define the cross-section of the nanoparticles,

3. Physical models used in the MCNPX code.

3.2.1 Calculation of dose rate in Nano SiO2 and MnFe2O4 nanoparticles

Table 6 and figure 7 show the effect of adding 2.0 wt.% SiO₂ and MnFe₂O₄ nanoparticles to concrete on the photon dose rate at different distances from the point source for gamma-ray energies of 0.511 and 1.274 MeV. According to Table 6, the photon dose rate decreases at the average values of about 9.2% and 3.7% for the gamma-ray energies of 0.511 and 1.274 MeV, respectively. This difference is due to the larger μ in Nano MnFe₂O₄ in comparison to Nano SiO₂ (as seen in Fig. 6).

Fig. 7.Full-screen View

(Color online) Comparison of SiO₂ and MnFe₂O₄ nanoparticles: the photon dose rate at different distances from the point source