## 2 Materials and Methods  
When the shielding-material particles of the medical radiation shield are smaller, the distance between particles (DBP) decreases, and the incidental radiation energy is attenuated because of the collision between the photons and particles of the shielding material [24]. This is effective when small particles have a dense structure inside the shield. The size and arrangement of particles can actively react with radiation; therefore, better technology is required to increase shield durability. As shown in Figure 1, the interaction probability of incidental radiation on particles increases according to the arrangement and size of the particles [25]. In this study, we assumed that uniformly isotropic shielding material particles exist within a certain space of the shielding sheet. At this time, the probability that the radiation particles reach distance \(x\) without interaction is \(P(x)\), and the probability of reaching the distance \(x+dx\) is \(P(x+dx)\). If the probability of interaction between the radiation particles  
Figure 1: Radiation shielding according to particle size and arrangement of shielding material. and the particles of the shielding material at a distance \(dx\) is \(\mu dx\), then the probability of not interacting is \(1-\mu dx\), and Equation (1) is established. \[P(x+dx)=P(1-\mu dx), \tag{1}\]  
where \(dx\) is the thickness of the shielding sheet. Therefore, if \(P(0)=1\), Equation (2) is established as follows:  
\[P(x)=e^{-\mu x}. \tag{2}\]  
Assuming that the radiation flux is a single energy, flux \(f_{0}\) of unscattered particles passing through a certain thickness can be represented by \(x=0\), and the first flux by \(f_{0}\times c^{-\mu x}\). \[f_{0}(x)=f_{0}(0)e^{-\mu x}, \tag{3}\]  
where \(\mu x\) represents the thickness of the shield; the greater the probability of collision is with shielding material particles per unit area when radiation passes inside the shield, the more shielding is achieved. Therefore, the distance when interacting with the shielding-material particles can be described as the mean free path of the medium; this mean free path is related to the thickness of the shield [26]. If the particles have an uneven arrangement, the interaction becomes a function of \(x\) and is subject to Equation (4). \[P(x)=exp\bigg{[}-\int_{0}^{x}\mu(x)dx\bigg{]}. \tag{4}\]  
Lastly, if the interaction while passing through thickness \(dx\) is regarded as the influence of the medium regardless of uniformity, it is expressed as Equation (5). This is a stochastic interaction depending on the thickness of the shield because the particles of the shielding materials are nonuniformly arranged per unit area of the shield. Since the nonuniformity of the medium corresponds to the number of particles in the unit area, the nonuniformity in the medium can be explained as shown in \(\mu_{i}x_{i}\). \[f_{0}(x)=f_{0}(0)exp\bigg{[}-\sum_{i}\mu_{i}x_{i}\bigg{]}. \tag{5}\]  
Therefore, it is theoretically desirable to increase the number of particles, such as in Equation (5), to generate a large amount of interaction through the flux and attenuation media. In this study, experiments were conducted to evaluate the multidispersed arrangement of particles according to the particle size of the shielding material, and the shielding performance according to the clustering effect. Tungsten was selected as the shielding material; tungsten has an atomic number of 74 and density of 19.25 g/cm\({}^{3}\), which makes it an effective alternative to lead [27; 28]. First, as shown in Table 1, the purchased tungsten particles were processed into micro- and nanosizes of 100-400 \(\upmu\)m and 400-900 nm, respectively, by ball-milling and laser-scattering methods (Microtrac Co., model S-3500) [29; 30]. To evaluate the shielding performance of a shielding sheet and quantitatively identify the particle distribution in the shielding sheet, we produced shielding sheets of three particle types (nanoparticles, microparticles, and a mix of nanoparticles and microparticles).