2.1 Material

The raw materials used in this work were Ludwigite (M1, 3.09 g/cm³), Szaibelyite (M2, 2.58 g/cm³), boron bearing iron concentrate ore (M3, 3.95 g/cm³), boron concentrate (M4, 2.69 g/cm³), boron rich slag (M5, 2.97 g/cm³), and boron mud (M6, 2.51 g/cm³). Furthermore, these boron-containing ores were purchased from Fengcheng Iron and Steel Group Co. Ltd., China. Table 1 lists the composition and densities of six boron-containing ores.

2.2 Methods

2.2.1 Exposure buildup factor (EBF) of boron-containing ores

In order to calculate the EBF, the G-P fitting parameters were obtained from the equivalent atomic number (Z_eq) using a logarithmic interpolation method. The calculations are illustrated step by step as follows:

1. Calculation of equivalent atomic number (Zeq);

2. Calculation of parameters for the G-P fitting method;

3. Calculation of EBFs.

The atomic number of a single element is denoted by Z, whereas ore samples have an equivalent atomic number (Z_eq), because the gamma-ray partial interaction processes with the material depend on the energy. Thus, Z_eq is an energy-dependent parameter. Using the WinXCOM^[22,23] program, the total mass attenuation coefficient of the selected boron-containing ores in the Liaoning province of China and Compton partial mass attenuation coefficient for the elements from Z = 4 to Z= 50 were obtained in the energy range of 0.015–15 MeV. The equivalent atomic number is calculated by matching the ratio of Compton partial mass attenuation coefficient to the total mass attenuation coefficient of the selected ores with an identical ratio for a single element at the same energy. The following formula is used to interpolate Z_eq^[24,25]:

$Z_{\text{eq}}\text{=}\frac{Z_1（\log R_2-\log R)+Z_2(\log R-\log R_1)}{(\log R_2-\log R_1)},$ (1)

where Z₁ and Z₂ are the atomic numbers of elements corresponding to the ratios R₁ and R₂, respectively, and R is the aforementioned ratio of the boron-containing ores at a specific energy. For example, considering that the ratio ${\left(\mu /\rho \right)}_{\text{Compton}}/{\left(\mu /\rho \right)}_{\text{total}}$ of boron bearing iron concentrate ore (M3) at the energy of 0.015 MeV is 0.0052, which lies between R₁= ${\left(\mu /\rho \right)}_{\text{Compton}}/{\left(\mu /\rho \right)}_{\text{total}}$ =0.0057 for Z₁=18 and R₂=${\left(\mu /\rho \right)}_{\text{Compton}}/{\left(\mu /\rho \right)}_{\text{total}}$ = 0.0048 for Z₂=19, using equation (1), Z_eq = 18.53 is calculated. The G-P fitting parameters are calculated similarly using a logarithmic interpolation method for Z_eq. The change in Z_eq for the selected boron-containing ores with the incident photon energy is presented in Fig. 3.

Fig.3.Full-screen View

(Color online) Variation of Z_eq of boron-containing ores in the energy range of 0.015–15 MeV

As the second step, to evaluate the G-P fitting parameters (a, b, c, X_k, and d), a similar interpolation procedure was adopted as in the case of Z_eq. The G-P fitting parameters for the elements were obtained from the standard reference database released by the American National Standards ANSI/ANS-6.4.3^[17]. American Nuclear Society (ANSI/ANS-6.4.3, 1991) used the G-P fitting method and provided EBF data for 23 elements, one compound, and two mixtures e.g., water, air, and concrete at 25 standard energies in the energy range of 0.015 to 15.0 MeV with a suitable interval up to the penetration depth of 40 mfp.

The G-P fitting parameters of the selected boron-containing ores were interpolated according to the following equation^[24,25]:

$P\text{=}\frac{P_1（\log Z_2-\log Z_{\text{eq}})+P_2(\log Z_{\text{eq}}-\log Z_1)}{(\log Z_2-\log Z_1)},$ (2)

where P₁ and P₂ are the values of the G-P fitting parameters at a particular energy, corresponding to the atomic numbers Z₁ and Z₂, respectively.

Finally, the calculated G-P fitting parameters were used to calculate the EBF of the selected boron-containing ores as follows ^[24,25]:

$\begin{array}{l}B(E,x)\text{=}1\text{+}\frac{b-1}{K-1}(K^x-1),\text{ for }K\ne 1\\ B(E,x)\text{=}1\text{+(}b\text{-1)}x,\text{ for }K\text{=1}\end{array}$ (3) $K(E,x)\text{=}c{x}^a\text{+}d\frac{\tanh (x/X_k-2)-\tanh (-2)}{1-\tanh (-2)},\text{ for }x\le \text{40 mfp}$ (4)

where E is the incident photon energy, (a, b, c, X_k, and d) are the G-P fitting parameters, x is the penetration depth in the mean free path, and K is the photon dose multiplication. The mean free path (mfp) is the average distance between two successive interactions of photons wherein the intensity of the incident photon beam is reduced by a factor of 1/e.

3.1 Shielding effectiveness of boron-containing ores against gamma rays

In this work, we evaluated the radiation attenuation properties of boron-containing ores by using MCNP-4B Monte Carlo code to calculate the mass attenuation coefficient for M1-M6 samples. The validation of MCNP-4B code was performed by comparing the results with standard WinXCOM data.

The mass attenuation coefficients (µ/ρ) of the boron-containing ores calculated using WinXCOM in the photon energy range of 0.015–15 MeV are shown in Fig. 1. Furthermore, the calculated values of μ/ρ are shown in Fig. 2 together with MCNP-4B results. The variation of Z_eq of boron-containing ores for the photon energy range of 0.015–15 MeV is shown in Fig. 3. Furthermore, the variation in the EBFs of boron-containing ores is shown in Fig. 4 (M1–M6) at constant penetration depth (0.5, 5, 10, 20, 3, and 40 mfp). Furthermore, Fig. 5 shows the EBFs as a function of penetration depth at constant photon energy (0.015, 0.15, 1.5, and 15 MeV).

Fig.1.Full-screen View

(Color online)Total mass attenuation coefficient of boron-containing ores in the energy range of 0.015–15 MeV

Fig.2.Full-screen View

(Color online) Comparison between the calculated values of mass attenuation coefficients of MCNP-4B and WinXCOM versus photon energy for the boron-containing ores

Fig.4.Full-screen View

(Color online) Variation of EBFs of boron-containing ores (M1–M6)

Fig.5.Full-screen View

(Color online) Variation of EBF of boron-containing ores with the penetration depth (0.015 MeV, 0.15 MeV, 1.5 MeV, and 15 MeV)

From Fig. 1, it can be observed that the mass attenuation coefficients of the selected boron-containing ores decrease rapidly in the energy range of 0.015–0.1 MeV, slowly decrease in the energy range of 0.1–10 MeV, and again increase in the energy range of 10–15 MeV. The variation in the values of mass attenuation coefficients with energy can be explained by partial photon interaction processes (photoelectric absorption, Compton scattering, and pair production) as the interaction cross-section depends on photon energy and atomic number. The interaction cross-section is directly proportional to the atomic number (Z⁴) for photoelectric absorption in the energy region of 0.015–0.1 MeV; therefore, the values of µ/ρ for the boron-containing ores decrease rapidly. In the Compton scattering region (0.1–10 MeV), the interaction cross-section is dependent on Z whereas the cross-section is directly proportional to Z² for pair production (10–15 MeV) ^[26]. According to Fig. 1, it can be observed that the value of µ/ρ of boron bearing iron concentrate ore (M3) is the highest among the selected ores. It can be concluded that M3 is the best boron-containing ore for gamma ray shielding in the energy region of 0.015–0.1 MeV. From Fig. 2, it can be observed that the values of μ/ρ simulated using MCNP-4B for all the samples (M1–M6) are consistent with the theoretical estimation using WinXCOM.

From Fig. 3, it can be observed that the values of Z_eq of the boron-containing ores increase slightly, and subsequently decrease with further increase in the incident photon energy up to 4 MeV. However, the values of Z_eq of all the boron-containing ores are approximately the same in the energy range of 4–15 MeV.

The variation of EBF of the boron-containing ores with the photon energy (0.015–15 MeV) is shown in Fig. 4. From Fig. 4, it can be observed that the EBF increases up to a maximum value (at 0.2 MeV) and decreases thereafter. The variation of EBF with the photon energy can be explained by the photon interaction process (similar to the mass attenuation coefficient). The EBF values of the boron-containing ores are minimal at low energy owing to the dominance of photoelectric effect. With the increase in the incident photon energy, EBF increases owing to multiple scattering as Compton scattering dominates. In the high-energy region, pair production takes over the Compton scattering process, resulting in the reduction of EBF to a minimum value. Notably, the boron-containing ore with the highest Z_eq (i.e., M3) achieves the minimum values of EBF. From Table 1, the content of Fe in M3 is 39.567%. This high content of Fe leads to the highest equivalent atomic number (Z_eq) of M3 as shown in Fig. 3, such that M3 has the lowest EBFs. Hence, it is the best boron-containing ore for gamma ray shielding.

The variation of EBFs of the boron-containing ores with the penetration depth is shown in Fig. 5. It can be observed that the EBF values of boron-containing ores increase with the increase in the penetration depth and they are the lowest for the penetration depth of 1 mfp and highest for the penetration depth of 40 mfp owing to multiple scattering events for large penetration depths. It can be observed that, at 0.015 MeV, the values of EBF of all the boron-containing ores are in the range of 1 to 1.2. Furthermore, at the photon energy of 1.5 MeV, the EBFs of all the ores are approximately the same. At the photon energy of 15 MeV, M3 shows the maximum value of EBF above the penetration depth of 10 mfp owing to pair production in the ores, which generates electron pairs. This trend reversal can be explained on the basis that, at this incident photon energy (i.e., 15 MeV), pair production is the dominant photon interaction process^[30].

3.2 Macro cross-section for thermal neutrons

Fig. 6 shows the data for the neutron micro cross-section of elements contained in the six boron-containing ores at the energy up to 20 MeV. The energy of thermal neutrons is chosen as 0.0253 eV^27-29] in this study as shown in Fig. 6. Moreover, the contents of B-10 isotope and B-11 isotope are calculated according to the natural boron content^[8]. Table 2 presents the specific data for thermal neutrons of the elements contained in six boron-containing ores. From Fig. 6 and Table 2, the micro cross-section of B-10 is approximately 3847.52b and significantly more than that of the other elements; the micro cross-section of Fe is significantly less than that of B-10, but slightly more than that of the other elements. Moreover, the micro cross-sections for thermal neutrons of the other elements are approximately the same.

Fig.6.Full-screen View

Micro cross-section of elements contained in boron-containing ores for thermal neutrons

Fig. 7 shows the macro cross-section of all the boron-containing ores for thermal neutrons. M2 is observed to have the highest macro cross-section and the lowest is observed for M6. Compared with the B element content of ores, it can be concluded that, the more the B element content of ores, the better the shielding properties of boron-containing ores. However, M1 and M3 are the opposite, because the B content of these two ores is approximately the same, but the Fe content of these two ores is quite different, i.e., approximately 22% in M1 and 40% in M3. However, the micro cross-section of the Fe element is slightly higher than that of the other elements and it leads to the difference between removal cross-sections of the two ores. Nevertheless, M2 is the best ore for thermal neutron shielding.

Fig.7.Full-screen View

Macro cross-section of boron-containing ores for thermal neutrons