# Skin Absorbed Dose Coefficients for Human Legs from Beta Radiation as a Function of Height
## 2 Materials and Methods  
The Geant4 toolkit is widely used for health, medical physics, and biomedical applications [37; 38; 39]. In this manuscript, the Geant4 code is utilized to simulate the absorbed dose coefficient to the skin of a human leg from beta-emitter radionuclides located on a flat surface. The main objective of these simulations was to compute the deposited energy from beta decay to the basal layer of skin for the different range of energy sources. The computed energies were converted to absorbed doses, and the result at one meter above the ground was compared with the results from Federal Guidance Report 12. Furthermore, we calculated the energy deposited into the basal layer from secondary electrons and photons only and estimated the portion of this deposited energy from the whole absorbed energy. To set up the environment, a full sphere was created. This sphere formed the boundary of the simulation universe, and it was divided into two separate regions. The bottom half of the sphere was filled with soil to create the ground area of the contaminated environment, and the top half was filled with air. The configuration of the universe is shown in Figure 1. The data from Federal Guidance Report 12 was used to determine the composition of the materials, including air and soil; the mass percentage of each element of the materials used in the simulations, as well as their density- are presented in Table 1. Furthermore, the air and soil distribution in the simulation environment was assumed to be homogeneous. Figure 1: Simulation Universe Configuration. The blue hemisphere shows the soil and the above hemisphere is the air where the leg phantom is located. The beta emission source was considered a cylindrical volume source set to correspond with the surface of the soil, with a depth of 1 \(\upmu\)m, and the radius of the cylindrical source was set equal to the continuous slowing down approximation (CSDA) range of the electron particles in the air. The CSDA range is the average distance charged particles can travel through a medium [40]. The interaction between charged particles and atoms in the surrounding universe causes energy loss from the particles. The loss rate of the charged particle's kinetic energy equals the total stopping power. Thus, the CSDA range is calculated by the integral of the total stopping power for the particles in each medium with respect to the kinetic energy of the charged particle [41]. \[R(T)=\int_{0}^{T}\biggl{(}-\frac{dE}{dx}\biggr{)}^{-1}dE \tag{3}\]  
R(T) is the range of a particle with kinetic energy (T) [42]. The unit for the range, R(T), is g\(\cdot\)cm\({}^{-2}\). The range formula for beta particles with a kinetic energy of 0.01 \(\leq\) T \(\leq\) 2.5 MeV is reduced to:  
\[R(T)=0.412\times T^{(1.27-0.0954\times\ln{(T)})} \tag{4}\]  
And for the kinetic energy of T \(>\) 2.5 MeV:  
\[R(T)=0.530\times T-0.106 \tag{5}\]  
After calculating the range, R(T), for the desired electron energy, the CSDA was calculated by dividing it by the medium density:  
\[CSDA=\frac{R(T)}{\rho} \tag{6}\]  
where \(\rho\) is considered to be the density of air in this manuscript. The beta decay source is assumed to be monoenergetic, emitting uniform and isotropic beta particles with a negative charge (electrons) in all directions (4\(\pi\)). The energy range for electrons varied from 0.1 MeV to 4 MeV, and different simulated values for the source are presented in Table 2. The source particles (electrons) were distributed uniformly in the cylindrical source. During each simulation, one energy range was assigned to the source; for example, the energy of the source was set to 0.1 MeV for all electrons.