Klein–Nishina formula and Monte Carlo method for evaluating the gamma attenuation properties of Zn, Ba, Te and Bi elements

Photons (X- and gamma-rays) are extensively used in medical intervention procedures for the sterilization of medical equipment, as well as therapeutic and diagnostic purposes, all over the world [1]. In fact, the health facilities and intervention procedures based on these radiations have increased in recent times. Many other industrial processes utilizing radiations have increased as well, such as medical applications [2], wastewater treatment [3], environmental protection [4], neutron irradiation [5], and applications of thin films [6]. Today, major diagnostic and therapeutic procedures involving the use of photons include diagnostic X-ray radiography, mammography, fluoroscopy, computed tomography (CT), brachytherapy, radiotherapy, and so on, all of which have increased environmental and personnel radiation doses considerably [7]. In view of the health hazards associated with exposing healthy tissues to radiation, the protection of patients, caregivers, and the general public is a fundamental aspect of a good quality assurance program (QAP), needed in all radiation facilities [8, 9]. At the heart of the QAP program are the following: the accurate detection of radiation and its measurement; and shielding systems. While the dosimetry system ensures prescribed dose compliance, a shield provides protection to healthy tissues, personnel, and the general public by confining radiation within a given space. Beyond the shield, doses are kept below the threshold of non-stochastic effects in living tissues.

Proper understanding of photon interaction mechanism and associated parameters is important if good choices of materials for accurate dose measurement and for the provision of an effective shield are to be made. The interaction parameters and mechanism illuminate our knowledge of how much photon penetrates or interacts with a medium and allows the quantification of energy deposited in the medium. Hence, photon interaction parameters are fundamental for determining the radiation dose and shielding competences of materials before their practical use.

Photon interactions leading to energy/photon absorption in the interacting medium come in three major forms, namely, photoelectric effect (PE), Compton scattering (CS), and pair production (PP). While the PE and PP lead to total absorption of photons, CS – on the other hand – offers more complex interaction procedures. Many radiation facilities produce and utilize photon beams whose energy ranges between a few keV and 5 MeV and where CS is a significant interaction process. Consequently, knowledge of the CS interaction processes and parameters of materials utilized for various functions (dosimetry, shielding, and so on) in these facilities is necessary. The CS of photons involves the inelastic collision of photons with atomic electrons. After the encounter, a photon transfers part of its energy to an electron and gets deflected from its initial path. If the energy transfer to the electron is greater than the electron's binding energy, the electron can be assumed to be free and at rest. The free electron approximation of the CS cross section is well described by the famous Klein–Nishina (K–N) formula [10]. The K–N theory is accurate for the evaluation of photon absorption and scattering cross sections of different materials for energies where CS dominates the interaction processes. These cross sections and energy-transfer coefficients are crucial for characterizing materials for dosimetry and shielding purposes [11]. Consequently, the present work describes the electronic, energy-transfer, and Compton cross sections for Zn, Ba, Te, and Bi using the K–N formula and Monte Carlo simulations. The evaluated parameters are useful for understanding and quantifying the photon absorption process of the chosen atoms with the view to assess their potential for shielding applications. The currently selected atoms (Zn, Ba, Te, and Bi) are the main ingredients of shielding materials that have shown superior abilities to many currently used shields [12,13,14,15,16,17,18,19,20,21,22,23].

In 2018, the electronic and CS cross sections of wax were evaluated for shielding purposes using the K–N equation. The study suggested that wax may not be ideal for shielding at high photon energies [24]. Moreover, Alexander et al. [25] adopted the K–N formula for the electronic cross section, CS attenuation, and energy-transfer coefficient of Pb, Cu, Co, Ca, and Al for radiation shielding and dosimetric applications. The study revealed that the cross sections and attenuation coefficients strongly depend on the photon energy and the ratio of the atomic number Z to the atomic weight A (Z/A ratio) of the interacting medium. It was concluded that controlling the Z/A ratio of a material can make it useful for radiation-shielding purposes. More recently, Thiele et al. [26] reported, using a modified K–N formula, the radiation shielding and dosimetry characteristics of Bi, W, Cd, Zn, Ni, and Fe. The report agreed with the study of Alexander et al. [25] that the electronic cross section depends on the Z of the interacting atoms, while CS attenuation strongly depends on the Z/A ratio of the atom. Clearly, these previous studies have shown that the K–N equation is a viable method for obtaining and analyzing photon interaction cross sections and attenuation coefficients.

Recently, there has been a growing continuous demand to develop new shielding materials for radiation applications in nuclear and medical facilities [26, 27]. The main ingredients of these materials are elements such as Zn, Te, Ba, and Bi, which have several superior physical and chemical properties in relation to other elements in the periodic table. For example, Al-Buriahi and Mann [28] studied the radiation shielding effect of some glass systems containing Te, Nb, and W. Alzahrani et al. [29] investigated the radiation protection features of a TeO₂–Na₂O–TiO glass system by using the Particle and Heavy ion Transport Code System (PHITS) Monte Carlo code. The findings of the studies show the importance of Te in the ability of the glass sample to shield the radiation. Al-Buriahi et al. [30] also evaluated the nuclear protection ability of bismuth barium telluroborate glasses as dense and environment-friendly candidates. Their findings show the important role of Bi element in developing the glass system to shield from ionizing nuclear radiation. Therefore, this encouraged us to introduce a new method to study the radiation-shielding properties of the individual elements.

In the present article, the K–N approach has been used to evaluate the electronic, atomic, and energy-transfer cross sections of four elements, namely, zinc (Zn), tellurium (Te), barium (Ba), and bismuth (Bi), for different photon energies (0.662 MeV, 0.835 MeV, 1.170 MeV, 1.330 MeV, and 1.600 MeV). The results obtained by the K–N approach were compared with those determined by Geant4 Monte Carlo simulations.

Table 1 shows the studied elements, namely, zinc (Zn), tellurium (Te), barium (Ba), and bismuth (Bi), along with their atomic numbers, atomic mass, and Z/A ratios. Such elements are very adaptable to the preparation and design of new materials for various applications related to radiation shielding. To use the K–N scattering equation for calculating the electronic cross sections of these elements, one has to evaluate the coupling strength (α) for each photon energy under study. The α values were calculated for six photon energies in the range of 0.662–1.600 MeV. The results of α are shown in Figure 2. Clearly, there is a direct linear relationship between the coupling strength and the photon energy. The maximum values of α were observed at 1.600 MeV with the value of 3.1311, while the minimum values of α were observed at 0.662 MeV, with the value of 1.2955.

Fig. 2

The α values were then used to estimate the electronic, the atomic, and the Compton cross sections (otherwise called the Compton mass attenuation coefficient) for the studied elements according to Eqs. (1), (1a), and (1b). Table 2 shows the photon energy versus coupling constant, the K–N electronic cross section, and the electronic mass energy-transfer cross section. Figure 3 shows the variation of the K–N mass attenuation coefficients (σ/ρ) with the incident photon beam for the involved elements (Zn, Te, Ba, and Bi). It is noted that the σ/ρ values decrease with increasing beam energy. This decrease is due to the partial photon processes that have a strong relation with the energy. This behavior has been observed for several materials, such as glassy materials containing TeO₂/Na₂O/TiO [29], environment-friendly telluroborate glasses [30], and PbO/B₂O₃/Bi₂O₃/Fe₂O₃ glasses [31]. In this context, it is known that the PE (one of the partial photon processes) occurs at an energy of E<0.4 MeV, while CS dominates the partial photon processes in the energy range between 0.4 MeV and 5 MeV. Both PE (~E⁻³) and CS (~E⁻¹) have an inverse proportion with energy. Therefore, the σ/ρ values were high at low energies and were low at high energies. This observation supports the results of the photon attenuation coefficient studies obtained by different methods and simulation codes [33,34,35,36].

Fig. 3

Figure 4 demonstrates the behavior of the Compton mass attenuation coefficients with the Z/A ratio. Clearly, these coefficients increase linearly as the Z/A ratio increases. In Tables 3 and 4, the results from the use of the K–N scattering formula were compared with those extracted from the Geant4 simulations for all of the studied elements at photon energies of 0.662 MeV, 0.835 MeV, 1.170 MeV, 1.330 MeV, and 1.600 MeV. This comparison is presented in terms of the Compton scattering mass coefficients for Zn and Ba elements in Table 3 and for Te and Bi elements in Table 4. Generally, there is good agreement between the results of the K–N scattering formula and the Geant4 simulations. For example, the highest percentage deviation (Dev.%) was < 2% for the photon energy of 0.662 MeV in the case of Bi (see Table 4). From Table 3, in the case of Zn, the Dev.% values were 0.851, 0.944, 0.824, 0.697, and 0.504 for photon energies of 0.662 MeV, 0.835 MeV, 1.170 MeV, 1.330 MeV, and 1.600 MeV, respectively. The maximum Dev.% values were 0.944 at 0.835 MeV for Zn, 1.234 at 0.662 MeV for Ba, 1.204 at 0.835 MeV for Te, and 1.714 at 0.835 MeV for Bi.

Fig. 4

Figure 5 shows the relation between the K–N energy-transfer cross section and the photon energy. It is seen that, for a given element, the energy-transfer cross section decreases with increase in the photon energy. As the photon energy increased from 0.662 MeV to 1.600 MeV, the values of the energy-transfer cross sections decreased from 81.135 cm² to 69.184 cm² in the case of Bi, from 50.832 cm² to 43.344 cm² for Te, from 54.742 cm² to 46.678 cm² for Ba, and from 29.326 cm² to 25.006 cm² for Zn.

Fig. 5

Figure 6 represents the relation between the K–N mass energy-transfer coefficients and the photon energy. This figure provides detailed information about the energy-transfer cross section of the studied elements at different photon energies. Clearly, there is a linear relation with a small slope between the energy-transfer cross section and the photon energy for all of the studied elements. Such behavior can be attributed to the trend of the Z/A ratio, which has to be 0.39 < Z/A < 0.5 for stable elements [25]. Furthermore, there is a remarkable decrease in this cross section with increase in the photon energy from 0.662 MeV to 1.6 MeV. This indicates that the highest attenuation of the photon beams occurs at the low-energy region.

Fig. 6

Moreover, there is a weak dependence on the Z/A ratio for a given element. For example, as the photon energy varies from 0.662 MeV to 1.600 MeV, the energy-transfer coefficients varied from 0.0270 cm²/g to 0.0230 cm²/g for Zn, from 0.0240 cm²/g to 0.0205 cm²/g for Ba, from 0.0240 cm²/g to 0.0204 cm²/g for Te, and from 0.0234 cm²/g to 0.0199 cm²/g for Bi.

In the present study, we have reported the electronic, energy-transfer, and Compton cross sections for Zn, Te, Ba, and Bi obtained by using K–N scattering formula and Geant4 simulations at photon energies of 0.662 MeV, 0.835 MeV, 1.170 MeV, 1.330 MeV, and 1.600 MeV. The highest deviation between the results of the K–N approach and Geant4 simulations was <2% for the photon energy of 0.662 MeV in the case of Bi. The maximum value of α (3.1311) was observed at 1.600 MeV, while the minimum value of α (1.2955) was observed at 0.662 MeV. The energy-transfer coefficients varied from 0.0270 cm²/g to 0.0230 cm²/g for Zn, from 0.0240 cm²/g to 0.0205 cm²/g for Ba, from 0.0240 cm²/g to 0.0204 cm²/g for Te, and from 0.0234 cm²/g to 0.0199 cm²/g for Bi. Knowing and controlling the shielding properties of the studied elements would be very helpful in preparing and developing new advanced materials for gamma-ray-shielding applications.