1 Introduction

Nowadays nuclear technology is used in the industry, agriculture and medicine. Therefore, protection against harmful effects of external radiations is regarded significant. Due to high penetration of gamma-ray and its widespread application in radiotherapy, shielding of gamma radiation is important. Radiotherapy is used to treat almost every type of solid tumor ^[1-5]. The degree of tissue damage depends not only on quantity of radiation but also individual susceptibility of each organ to radiation damage. Parts of the body are shielded against radiation. For example, in radiotherapy of breast tumor, the nearby healthy tissue is shielded with a cap ^[2]; and in maxilla-facial tumor radiotherapy, poly cyclohexane is used to avoid gamma-rays irradiation of healthy tissues surrounding the tumors ^[6]. In fact, shields are always present for protection of healthy tissue.

However, shields can cause multiple scattering of photons, and some of the gamma rays are inclined towards healthy tissues. The flux received by healthy tissues must be accurately calculated. Using shields to prevent radiation from reaching to healthy tissues is conventional procedure in radiography. For instance, shielding is used to protect the genitalia during pelvis and lower parts radiographic examinations ^[7]. Besides, shields of high atomic number materials are used in CT scans, such as the lead shields in head and neck CT scan to protect the surface tissue ^[8].

The presence of shielding material along with radiations is described by the buildup factor. It is associated with multiple scattering of X- and gamma -rays within the shields. Buildup factor leads to accumulation of additional flux in specific areas (e.g. the healthy tissue). According to its importance in medical ^[9,10], industrial ^[11-13], agriculture ^[14] and nuclear areas ^[15-18], buildup factors have been measured, but precise determination of the coefficients in all materials through experiment is not easy. So, the coefficients are often calculated by using photon absorption and scattering cross sections. The codes to calculate buildup factors include BIGGI-4T, EGS4, PALLAS-PL and Monte Carlo N-particle (MCNP).

In early 1900s, the calculation of buildup factor for different elements was the goal of most studies. In 1993, Y. Harima studied the relationship between flux buildup factor with energy and material compositions ^[19]. Other groups studied parameters associated with buildup factor in recent decades ^[10-17] by using MCNP and the other codes. In this study, we investigated the effect of source geometry on flux buildup factor by Monte Carlo method.

2 Method of calculation

At first, using the MCNP code, the flux buildup factor was calculated for ⁶⁰Co gamma-rays of 1.17 and 1.33 MeV, and their average energy of 1.25 MeV, in different geometries, with a lead shielded detector. Then, three materials of different atomic numbers were used for comparison, at 1.17 and 1.33 MeV. The atomic numbers, aluminum (Z=13), iron (Z=26) and lead (Z=82), were selected so that the changes of buildup factor with atomic number could be studied.

Geometries of photon sources used in the simulations were: 1) isotropic point sources; 2) isotropic linear sources in X, Y and Z axis; and 3) plate and disc isotropic superficial sources of cylindrical, spherical and cubic isotropic volumetric sources.

Fig.1 shows schematically the source-shield-detector geometries. The shield thickness was 1.55 cm and the sources were centered at (0,1.27,0).

Fig. 1Full-screen View

(Color online) Schematics of the source-shield-detector system (in mm).

Simulations were performed using two isotropic and parallel beam models, with the tally of F4, which outputs the photon numbers per unit volume of the detector. The results are shown in Fig.2. In the following, changes of buildup factor at 1.17 and 1.33 MeV for lead, iron and aluminum will be exanimated.

Fig. 2Full-screen View

(Color online) Simulated γ-ray flux buildup factors for the 12 source geometries at 1.17 and 1.33 MeV with Al, Fe and Pb shields of 1.55 cm thickness.

Flux buildup factor changing with energy and shielding materials were simulated, and sources with the lowest buildup factor were determined. The buildup factors at 0.66, 1.25 and 6.20 MeV for the 12 source geometries are shown in Fig.3. In Fig.4, the buildup factors of the three shielding materials at the same energy are compared.

Fig. 3Full-screen View

(Color online) Simulated γ-ray flux buildup factors for the 12 source geometries at 0.66, 1.25 and 6.20 MeV with Al, Fe and Pb shields of 1.55 cm thickness.

Fig. 4Full-screen View

(Color online) Comparison of the flux buildup factors regarding to the source types. Repetition.

3 Results

From Figs. 2, 3 and 4, the flux buildup factor depends on the source geometric shapes, γ-ray energy and atomic number of the shield. Among the photon interactions with shielding materials (photoelectric, Compton and pair production), Compton scattering is dominant effect. Occurrence probabilities of the total photon interactions and the Compton interaction are shown in Fig.5, from which photon cross-sections for Al, Fe and Pb can be obtained, using the Xcom Gamma Cross Section database ^[20].

Fig. 5Full-screen View

(Color online) Mass attenuation coefficient changes according to energy for Compton and total interactions in energy range 1 keV to 100 MeV

In Fig. 5(b), in the energy range of interest (0.66-6.20 MeV), the Compton scattering probability for Pb differs greatly from those for Al and Fe. The Compton scattering probabilities for Al and Fe decrease with increasing energy, while the Compton scattering probability for Pb increases with energy. So, the Pb buildup factor changes differently from those of Al and Fe (Fig.3). Escape probability of scattered energetic photons through the Pb shield is greater than that of low-energy photons. For the Al and Fe shields, Compton interactions of ¹³⁷Cs γ-rays produce more secondary photons than ⁶⁰Co and ¹⁶N, hence the greater buildup factor of ¹³⁷Cs in Fig. 3.

For ⁶⁰Co source, the flux buildup factor of Fe shields is greater than those of Pb and Al. The reason lies in the total interaction probability of photons for these elements. The photon absorption probability increased with scattering interaction in the lead shield, so less scattered photon flux is recorded in the detector than ferrous shield. In parallel beams, decreased buildup factor is related to small surface of the shield that photon scattering interaction is involved.

For Pb, coaxial cylindrical sources with shield and detector have the least buildup factor of all the sources (Fig. 3). After interaction inside the volume source, less energy photons are produced in the shield, which has high absorption probability. The coaxial cylindrical source with shield has the less solid angle than other directions, which affects the buildup factor. For shields of intermediate atomic number, coaxial disc source with shield has the lowest buildup factor, the total flux is the lowest of all the sources, but the non-contact flux is greater than the other sources. In this situation, coaxial disc and plate sources have the best performance in terms of the buildup factor.

At the energies over 6 MeV, the pair production interactions in high atomic number materials can occur and the annihilated photons may reach the detector, and flux buildup factor of Pb is greater than those of Al and Fe.

The error of the results in the Monte Carlo method is a relative quantity in the simulation outputs and is of utmost importance; therefore, we will dedicate this section of our study to this quantity. According to the definition of the buildup factor which is “the ratio of the value of a specific radiation quantity at each point to the un-collided component of that quantity”, the relative error of the flux buildup factor, σ_B/B, is calculated by σ_B/B=[(σ_{φ_uncollided}/φ_uncollided)²+(σ_T/φ_T)²]^1/2, where σ_{φ_uncollided}/φ_uncollided is the relative error of the un-collided flux, and σ_T/φ_T is the relative error of the total flux.

For the 12 source geometries, from Table 1, the relative error for aluminum shielding to ¹³⁷Cs, 60Co and ¹⁶N sources is of the order of one thousandth; the maximum relative error for iron shielding to ⁶⁰Co source is of the order of one hundredth, and of the order of one thousandth to ¹³⁷Cs and ¹⁶N sources; and the maximum relative error is of the order of one hundredth for lead shielding to all the sources. According to the standards of the Monte Carlo method in simulations, all the errors have reasonable values and the calculations have adequate validity.

4 Conclusions

Flux buildup factors of ¹³⁷Cs, ⁶⁰Co and ¹⁶N γ-rays in shielding materials of Al, Fe and Pb are calculated by Monte Carlo method for linear, surface and volume sources types in different axial directions. Secondary photons are the effective factor on buildup factor. The sources cover energy range of 0.66-6.20 MeV, while the 1.17 and 1.33 MeV γ-rays of ⁶⁰Co, and the average energy (1.25 MeV), are used.

According to simulation results, among the point, linear, surface and volume sources along X, Y and Z axis, the coaxial volumetric cylindrical source-shield-detector system has the least flux buildup factor. For low atomic number materials, coaxial disc surface source-shield-detector systems have small flux buildup factor. Then simulation errors are in the order of thousandths.