# Nuclear Analysis of High-Power LIEBE Molten Target at CERN for the Production of Radioisotopes
## 4 Simulation of the LIEBE Target  
Monte Carlo methods are used to solve any system with stochastic properties with realistic objects. Since there is uncertainty in the entire nuclear particle transport problem, it can be said that the movement of particles in a certain space has stochastic properties. This method is able to simulate physical experiments, with only a few restrictions on the geometrical conditions, and its results are more accurate than those of the deterministic method. Deterministic methods can quickly obtain an approximate solution to the transport equation by discretizing direction, energy, space, and time. Systems of nuclear objects are characterized by complex geometry, complex energy spectrum and anisotropic scattering. The Monte Carlo method is often used specifically for nuclear facilities. When the Monte Carlo method is applied to the simulation of particle transport, first, a random movement of a single particle in a known environment is performed, and then random experimental values are obtained by observing the number of particle histories. Finally, an estimate of the numerical characteristics of the random variable is obtained. The Monte Carlo method involves three processes in solving the neutron transport equations: (1) the source is determined using a probability distribution; (2) tracking the particle's local energy and direction; (3) data recording and analysis of results. Particle transport calculations were performed using the 3D Monte Carlo code MCNP, which is widely used in nuclear applications for neutron particle transport calculations, together with various nuclear reaction cross-section libraries and geometry models. The modified model was converted to the MCNP input, as outlined in Section 3. A 1 mm radii, mono-energetic 70 MeV--100 uA or 1.987742 \(\times\) 10\({}^{16}\)\(\#\)/cm\({}^{2}\)/s flux proton beam falling on the liquid lead target LIEBE was modeled. FENDL-3.1 cross-section data library was used, and dose rate maps of 10\({}^{9}\) particle histories were calculated, from which proton-, neutron- and gamma-induced dose rate maps and fluxes were calculated. Nuclear reactions such as \({}^{208}\)Pb(p,n) or \({}^{208}\)Pb(p,\(\gamma\)) have cross sections exceeding 10 barns (see Figure 10 for more information), so the reaction produces a broad spectrum of neutrons and gamma rays. Therefore, to evaluate the radiation processes in the target and the surrounding premises, it is necessary to determine the possible proton-initiated neutron and gamma-ray fluxes. Simulation results of proton scattering allow us to predict the possible distribution of flux during direct interaction of the beam with the lead target (see Figure 11). Figure 12,  
Figure 10: Lead isotope \({}^{208}\)Pb(p,n) and \({}^{208}\)Pb(p,gamma) reaction cross sections [14]. presenting the proton beam penetration depth inside the target, shows that the target effectively absorbs the proton beam, with the maximum flux reaching \(1.885490\times 10^{16}\)\(\#/\)cm\({}^{2}/\)s which is close to the flux of the proton beam: \(1.987742\times 10^{16}\)\(\#/\)cm\({}^{2}/\)s. Additionally, the proton beam penetrates roughly 7 mm inside the target. Consequently, it can be seen that very little scattering of the proton beam occurs before the beam hits the LBE inside the irradiation chamber. It should also be noted that the difference between the maximum flux values of Figures 11 and 12 is due to the fact that the tally mesh in Figure 11 is more coercive than that of Figure 12, so the mean value at a single cell is lower in Figure 11 than in Figure 12.  
The highest proton dose is contained to the beam location (Figure 13) until the target is reached. Most of the proton dose is absorbed inside, with practically no dose present outside the LIEBE target. Figure 11: Proton-induced flux [\(\#/\)cm\({}^{2}/\)s] map cross sections from the side (left) and top (right) of the target along the proton beam.