Optimal shielding structure design plays a guiding role in the implementation of radiation protection engineering. The achievement of the optimal arrangement and thickness ratio for the layers of materials is the key to attaining a light-weight and small-volume shield but with the best shielding effect. In this research, the optimization design method is established by the genetic algorithm combined with the Monte Carlo N-particle code, and a four-layer neutron shield composed of iron (Fe), boron carbon (B_{4}C), lead (Pb), and polyethylene (PE) is designed. When setting the total thickness of the shield to 20 cm, different arrangements and thickness combinations of these four layers are calculated. It is shown that the arrangement Fe-PE–B_{4}C–Pb is the most radiological optimizing arrangement, and the optimal thickness combination is also obtained. Besides, it seems that the thicker the shield, the higher the requirement for the thickness ratio of Fe and Pb. In order to prove this, an optimal 80 cm thick shield is then designed, and the optimal thickness ratio is also obtained. It is found that the thickness ratio of Fe and Pb should also be increased in order to achieve the best shielding effect.

## I. INTRODUCTION

Most nuclear power devices require their shields to be compact in structure, light in weight, and good in shielding performance. It is known that multilayer shields can be effective in realizing such characteristics, and the top priority for a successful shielding design is the achievement of an optimal arrangement and thickness ratio for the layers of the materials. Many researchers have done a lot of studies on this multi-parameter optimization problem. Barnhart first came up with the concept of a multilayer shielding structure in 1955. He investigated the shielding properties of radiation shields composed of concrete, paraffin, and steel.^{1} Considering that the simulation of multilayer shields is quite complicated, Trontl *et al.* proposed a type of mathematic model to calculate the buildup factors for multilayer shields.^{2} SHIELD is one specialized computational simulation code based on a semiempirical approach for radiation shielding calculation. Van Ngoc carried out a simulation by SHIELD for the multilayer shield of the natural uranium graphite reactor. The simulated shield is composed of three layers of materials: a 76 cm thick graphite reflector, a 20.3 cm thick steel heat shield, and an outer layer of concrete.^{3,4} Asbury applied a multiple grid genetic algorithm to the calculation of radiation shielding, by which a good radiation shield can be designed.^{5} Suteau and Chiron introduced an iterative method for calculating buildup factors of gamma rays for a multilayer shield, which is based on the empirical approach for calculating buildup factors of double-layer radiation shields.^{6} Assad *et al.* put forward a new approximate formula to calculate buildup factors of gamma rays for a multilayer shield. The results were compared with those obtained by the SN1D discrete ordinates code, and the two results are consistent.^{7} Koprivnikar and Schachinger conducted a simulation of a radiation shield for a neutron spallation source by Monte Carlo N-Particle (MCNP) and FULKA.^{8} Hirayama and Shin calculated buildup factors of gamma rays for a three-layer shield by EGS4.^{9} Shouop *et al.* carried out a shielding structure design for a Cf-252 neutron source and an ^{241}Am/Be source through the PHITS software.^{10,11} Hu *et al.* studied the design of multilayer shields against fission neutrons and gamma rays.^{12,13} Chen *et al.* studied the design of the radiation shield and collimator for neutrons at EAST.^{14} For 14 MeV neutrons, the energy is high and the shielding is complex, so the shielding effect of a single layer of polymer material or boron carbide material is not satisfactory, and multilayer shielding combination is much more effective. In this research, the genetic algorithm (GA) and MCNP are adopted to establish a method for the optimal shielding structure design for a 14 MeV neutron source.

## II. METHOD AND MODELING

### A. Establishment of GA mathematical model

GA is a random search method derived from imitating the laws of biological evolution, which allows direct manipulation of structural objects and exhibits inherent implicit parallelism and global optimization capabilities. This method aims at the neutron spectrum and considers the secondary gamma rays produced by the interaction of neutrons with shielding materials. The optimal objective is the total dose equivalent of neutrons, with the secondary gamma rays being the lowest. The optimized objective function can be written as

where H(*L*) refers to the total dose equivalent of neutrons and secondary gamma rays penetrating the shielding material, H_{n}(*L*) is the dose equivalent of neutrons, and H_{γ}(*L*) is the dose equivalent of secondary gamma rays. Equation (1) represents the minimum total dose equivalent after neutron penetration of the shield. The constraints of the optimization objective are as follows:

Equation (2) indicates that the sum of all the thickness ratios is one, where *L*_{i} represents the thickness of each layer and *L*_{all} refers to the total thickness of the shield. Equation (3) indicates that the shielding equivalent density is limited within a certain range, where *ρ*_{eff} represents the shielding equivalent density, and *ρ*_{x} and *ρ*_{y} refer to the minimum and maximum density, respectively. Equation (4) as a domain constraint means that each layer of the material is within a special thickness ratio, where *L*_{x} and *L*_{y} denote the minimum and maximum thickness, respectively.

### B. MCNP model for shielding structure design

The MCNP-4C software is used for the shielding calculations. Figure 1 shows the geometry of the model. The shield is 100 cm long, 100 cm wide, and 200 cm thick. The layers in the shield, from left to right, are layer 1, layer 2, layer 3, and layer 4. The surface source is 20 cm from the left side of the shield with a radius of 1 cm. The point flux detector is located 200 cm from the surface source. The centers of the surface source, the shield, and the detector are in a straight line.

### C. Flow chart for structure optimal design

Figure 2 shows a flow chart of the optimal design of the thickness ratio. The equivalent density is first estimated, and a range of density constraints is initially defined. After setting the thickness ratio of each layer in the input file “inp” of MCNP, GA calls the executable program to calculate the “inp” file to produce the output file “outp.” By processing the result, the dose equivalent of neutrons h_{1}, the secondary gamma photon dose equivalent h_{2}, and the total dose equivalent H are obtained, indicating that the calculation of the file “eval.c” has been completed. Then, we can get an optimized objective function value, which becomes one daughter in GA. A new shielding material thickness ratio is then given and used again to calculate the file “eval.c” after GA has properly processed this daughter (e.g., operation crossover and mutation). Through circular calculations, the optimal value of the objective is obtained. When the calculation reaches the pre-set minimum value, the program will give the optimal combination of thickness.

## III. RESULTS AND DISCUSSION

### A. Results of different arrangements

The arrangement can affect the shielding performance of the shield. Thus, some typical arrangements are simulated by MCNP. The results are shown in Table I. The thickness of each layer is the same. The arrangement of iron (Fe)–polyethylene (PE)–boron carbon (B_{4}C)–lead (Pb) outperforms other arrangements, as analyzed from the principle of the interaction of neutrons and gamma rays with the material. The shielding layers are arranged according to the cross section data. When the neutron energy is large, Fe with a high inelastic cross section is chosen as the first layer. As the energy slows down, PE with a high elastic cross section is chosen as the second layer. Element B of high thermal capture cross section can be selected as the third layer. Secondary gamma rays are produced in the front layer. Thus, Pb is chosen as the fourth layer for shielding the secondary gamma rays.

. | . | . | . | . | Shielding performance . | ||
---|---|---|---|---|---|---|---|

Different arrangements . | Neutron (Sv) . | Gamma ray (Sv) . | n-γ all (Sv) . | ||||

1 | Fe | PE | B_{4}C | Pb | 1.0679 × 10^{−15} | 1.2454 × 10^{−17} | 1.0803 × 10^{−15} |

2 | PE | Fe | Pb | B_{4}C | 1.3934 × 10^{−15} | 8.0663 × 10^{−18} | 1.4015 × 10^{−15} |

3 | PE | B_{4}C | Pb | Fe | 1.5036 × 10^{−15} | 7.2572 × 10^{−18} | 1.5109 × 10^{−15} |

4 | B_{4}C | PE | Fe | Pb | 1.3584 × 10^{−15} | 6.1700 × 10^{−18} | 1.3645 × 10^{−15} |

5 | B_{4}C | Pb | Fe | PE | 1.2002 × 10^{−15} | 1.9354 × 10^{−17} | 1.2195 × 10^{−15} |

6 | Pb | B_{4}C | PE | Fe | 1.3494 × 10^{−15} | 9.8543 × 10^{−18} | 1.3592 × 10^{−15} |

. | . | . | . | . | Shielding performance . | ||
---|---|---|---|---|---|---|---|

Different arrangements . | Neutron (Sv) . | Gamma ray (Sv) . | n-γ all (Sv) . | ||||

1 | Fe | PE | B_{4}C | Pb | 1.0679 × 10^{−15} | 1.2454 × 10^{−17} | 1.0803 × 10^{−15} |

2 | PE | Fe | Pb | B_{4}C | 1.3934 × 10^{−15} | 8.0663 × 10^{−18} | 1.4015 × 10^{−15} |

3 | PE | B_{4}C | Pb | Fe | 1.5036 × 10^{−15} | 7.2572 × 10^{−18} | 1.5109 × 10^{−15} |

4 | B_{4}C | PE | Fe | Pb | 1.3584 × 10^{−15} | 6.1700 × 10^{−18} | 1.3645 × 10^{−15} |

5 | B_{4}C | Pb | Fe | PE | 1.2002 × 10^{−15} | 1.9354 × 10^{−17} | 1.2195 × 10^{−15} |

6 | Pb | B_{4}C | PE | Fe | 1.3494 × 10^{−15} | 9.8543 × 10^{−18} | 1.3592 × 10^{−15} |

### B. Results of the thickness ratio of each layer

The Fe–PE–B_{4}C–Pb arrangement is selected to be studied and analyzed, with density constraint from 5.2 to 6.2 g/cm^{3}. Table II shows the calculation results of the shielding performance for 14 MeV neutrons, and the thickness ratio is designed at the shield thickness of 20 cm. The objective of the optimal calculation is to achieve the lowest dose of neutron and secondary gamma rays. Table II compares the performance of the optimal thickness ratio combination with other combinations. It is obvious that the optimal combination outperforms other combinations.

. | Thickness ratio . | Density . | Neutron . | γ ray . | Total . | |||
---|---|---|---|---|---|---|---|---|

Combinations . | Fe . | PE . | B_{4}C
. | Pb . | (g·cm^{−3})
. | (Sv) . | (Sv) . | (Sv) . |

Optimal | 0.5765 | 0.1925 | 0.2305 | 0.0005 | 5.295 | 9.87 × 10^{−16} | 1.73 × 10^{−17} | 1.00 × 10^{−15} |

1 | 0.55 | 0.2 | 0.2 | 0.05 | 5.578 | 1.07 × 10^{−15} | 1.03 × 10^{−17} | 1.08 × 10^{−15} |

2 | 0.5 | 0.2 | 0.25 | 0.05 | 5.311 | 1.07 × 10^{−15} | 1.04 × 10^{−17} | 1.08 × 10^{−15} |

3 | 0.5 | 0.2 | 0.2 | 0.1 | 5.752 | 1.13 × 10^{−15} | 7.15 × 10^{−18} | 1.14 × 10^{−15} |

4 | 0.25 | 0.25 | 0.25 | 0.25 | 5.660 | 1.46 × 10^{−15} | 3.21 × 10^{−18} | 1.46 × 10^{−15} |

5 | 0.6 | 0.3 | 0.0 | 0.1 | 6.126 | 1.14 × 10^{−15} | 9.05 × 10^{−18} | 1.15 × 10^{−15} |

6 | 0.6 | 0.15 | 0.2 | 0.05 | 5.925 | 1.06 × 10^{−15} | 9.37 × 10^{−18} | 1.07 × 10^{−15} |

7 | 0.6 | 0.2 | 0.15 | 0.05 | 5.845 | 1.06 × 10^{−15} | 1.05 × 10^{−17} | 1.07 × 10^{−15} |

. | Thickness ratio . | Density . | Neutron . | γ ray . | Total . | |||
---|---|---|---|---|---|---|---|---|

Combinations . | Fe . | PE . | B_{4}C
. | Pb . | (g·cm^{−3})
. | (Sv) . | (Sv) . | (Sv) . |

Optimal | 0.5765 | 0.1925 | 0.2305 | 0.0005 | 5.295 | 9.87 × 10^{−16} | 1.73 × 10^{−17} | 1.00 × 10^{−15} |

1 | 0.55 | 0.2 | 0.2 | 0.05 | 5.578 | 1.07 × 10^{−15} | 1.03 × 10^{−17} | 1.08 × 10^{−15} |

2 | 0.5 | 0.2 | 0.25 | 0.05 | 5.311 | 1.07 × 10^{−15} | 1.04 × 10^{−17} | 1.08 × 10^{−15} |

3 | 0.5 | 0.2 | 0.2 | 0.1 | 5.752 | 1.13 × 10^{−15} | 7.15 × 10^{−18} | 1.14 × 10^{−15} |

4 | 0.25 | 0.25 | 0.25 | 0.25 | 5.660 | 1.46 × 10^{−15} | 3.21 × 10^{−18} | 1.46 × 10^{−15} |

5 | 0.6 | 0.3 | 0.0 | 0.1 | 6.126 | 1.14 × 10^{−15} | 9.05 × 10^{−18} | 1.15 × 10^{−15} |

6 | 0.6 | 0.15 | 0.2 | 0.05 | 5.925 | 1.06 × 10^{−15} | 9.37 × 10^{−18} | 1.07 × 10^{−15} |

7 | 0.6 | 0.2 | 0.15 | 0.05 | 5.845 | 1.06 × 10^{−15} | 1.05 × 10^{−17} | 1.07 × 10^{−15} |

Figures 3–5 show the results of neutrons, secondary gamma rays, and total, respectively, at the thickness of 10, 30, 40, 50, 60, 70, and 80 cm. The results indicate that the optimal combination is the most radiological optimizing arrangement when the thickness is 20 cm. As the shield thickness increases, the shielding performance of the optimal combination deteriorates. Meanwhile, the shielding performance of combination 5, which has more Fe in the first layer and more Pb in the last layer, is getting better. Generally speaking, there are more high-energy neutrons and gamma rays left as the shield thickness increases. Therefore, more Fe and Pb are needed to shield these fast neutrons and gamma rays. To verify this point, we further carry out the thickness optimization design at the shield thickness of 80 cm. Table III shows the performance comparison between the shields with a thickness of 20 cm and that of 80 cm. At the thickness of 80 cm, the thickness proportion of Fe and Pb increases, while that of PE and B_{4}C decreases. At the thickness of 20 cm, the shield mainly shields fast neutrons and produces relatively few secondary gamma rays, so less Pb is needed. When the thickness increases to 80 cm, there are more high-energy neutrons and gamma rays; therefore, more Fe and Pb are required at the thickness of 80 cm.

Thickness . | Thickness ratio . | Density . | Neutron . | Gamma . | Total . | |||
---|---|---|---|---|---|---|---|---|

(cm) . | Fe . | PE . | B_{4}C
. | Pb . | (g·cm^{−3})
. | (Sv) . | ray (Sv) . | (Sv) . |

20 | 0.577 | 0.193 | 0.229 | 0.001 | 5.295 | 9.87 × 10^{−16} | 1.73 × 10^{−17} | 1.00 × 10^{−15} |

80 | 0.632 | 0.141 | 0.103 | 0.122 | 6.745 | 3.97 × 10^{−19} | 5.60 × 10^{−20} | 4.53 × 10^{−19} |

Thickness . | Thickness ratio . | Density . | Neutron . | Gamma . | Total . | |||
---|---|---|---|---|---|---|---|---|

(cm) . | Fe . | PE . | B_{4}C
. | Pb . | (g·cm^{−3})
. | (Sv) . | ray (Sv) . | (Sv) . |

20 | 0.577 | 0.193 | 0.229 | 0.001 | 5.295 | 9.87 × 10^{−16} | 1.73 × 10^{−17} | 1.00 × 10^{−15} |

80 | 0.632 | 0.141 | 0.103 | 0.122 | 6.745 | 3.97 × 10^{−19} | 5.60 × 10^{−20} | 4.53 × 10^{−19} |

## IV. CONCLUSION

The shielding structure for the typical 14 MeV neutron source is designed, with the simulation and optimization done by GA combined with MCNP. The arrangement (Fe–PE–B_{4}C–Pb) is superior to other arrangements consisting of these four materials. The optimal thickness ratio is gained at the thickness of 20 cm, and the results prove that the optimal combination is better than other combinations at this thickness. Besides, with the increase in the shield thickness, the shielding performance of the optimal combination deteriorates. It seems a higher thickness ratio of Fe and Pb is required as the shield thickness increases. In this research, an optimal 80 cm thick shield is then designed, and the optimal thickness ratio is also obtained. It can be found by comparison at the thickness of 20 cm, the shield mainly shields fast neutrons and produces relatively few secondary gamma rays; therefore, less Pb is required. When the thickness increases to 80 cm, there are more high-energy neutrons and gamma rays, which require more Fe and Pb.

## ACKNOWLEDGMENTS

This research was supported by the National Natural Science Foundation of China (Grant No. 11975182); the Foundation of Key Laboratory of Nuclear Reactor System Design Technology, Chinese Academy of Nuclear Power, General Projects of Shaanxi Provincial Nature Fund (Grant No. 2020JM-030); and the NSAF Joint Fund set up by the National Natural Science Foundation of China and the Chinese Academy of Engineering Physics (Grant No. U1830128).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

^{241}Am/Be source optimum geometry for DSRS management-based Monte Carlo simulations