Fast neutron effective removal cross-section

Neutrons interact with matter via three primary mechanisms: elastic scattering, inelastic scattering, and neutron capture reactions. The likelihood of a fast or fission-energy neutron undergoing its first collision is quantified by the removal cross-section [38]. This parameter represents the probability of removing neutrons from an uncollided beam, thereby reducing the flux of penetrating neutrons. The effective removal cross-section remains relatively constant for neutron energies ranging from 2–12 MeV, making it a reliable metric for evaluating shielding materials within this energy range. For compounds or mixtures, the effective removal cross-section can be calculated using the following equation:(1)In this context, ρi represents the partial density of the i^th constituent element in the material. Here, ρi is defined as the density of the specific element, ∑_R/ρi denotes the mass removal cross-section of the i^th element. These parameters are essential for calculating the overall removal cross-section of the composite material, as they quantify the contribution of each element to the neutron shielding performance.

The partial density ρi can be calculated as the product of the weight fraction ωi of the i^th element and the overall density of the sample, as expressed by the following equation:(2)In recent years, several user-friendly computational tools have been developed to facilitate the analysis and design of radiation-shielding materials. Notable examples include Phy-X [39], MERCSF-N [40], MRCsC [41], NXcom [42] and ParShied [43]. These programs leverage extensive databases of elemental removal cross-sections and incorporate empirical models to predict the shielding performance of various materials. By providing accurate and efficient calculations, these tools have become indispensable for researchers and engineers in fields such as nuclear energy, medical physics, and space exploration. The integration of such software into the design process not only enhances the precision of the shielding material selection but also accelerates the development of innovative solutions tailored to specific radiation environments.

thermal neutron transmission ratios and macroscopic cross-section

The thermal neutron shielding properties of selected boron-based composite materials can be evaluated by analyzing their neutron transmission ratios and macroscopic cross-sections (∑). Although the ∑ is expressed in units of cm^-1, which may cause confusion as it resembles a linear attenuation coefficient rather than the traditional cross-section units of cm², it is widely used in the literature to describe neutron shielding performance. The value of ∑ can be derived from neutron transmission measurements, providing a quantitative measure of a material’s ability to attenuate thermal neutrons.

However, the experimental determination of these properties can be challenging, especially for boron-containing compounds such as TiB₂, MgB₂, and SiB₆, which typically exist as solid powders at room temperature. Handling powdered boron compounds under neutron irradiation necessitates specialized facilities and stringent safety protocols, which were inaccessible for this computational study. Several high-performance boron compounds (e.g., ZrB₁₂ and GdB₆) are challenging to synthesize in the bulk quantities required for experimental neutron irradiation tests. To address these limitations, the Monte Carlo method has emerged as a powerful computational tool for simulating neutron interactions and predicting the shielding performance. For instance, Qian et al. [44] highlighted the effectiveness of Monte Carlo simulations in overcoming the experimental constraints.

In this study, the Monte Carlo N-Particle Transport Code, version 5 (MCNP5) [45] was employed to simulate the thermal neutron shielding behavior of boron composites. The simulation geometry is illustrated in Fig. 1, was carefully designed to replicate real-world conditions, ensuring accurate and reliable results. The use of MCNP5 circumvents the practical difficulties associated with handling powdered materials and provides a cost-effective and efficient alternative to experimental testing. By leveraging this computational approach, researchers can gain deeper insights into the shielding mechanisms of boron composites, optimize material compositions, and accelerate the development of advanced shielding solutions for applications in nuclear science and technology.

Fig. 1Full-screen View

(Color online) MCNP5 simulation geometry

In the simulation setup, the thermal neutron source was positioned at the center of the shielding structure. The boron composite samples under investigation were modeled as spherical shells, with a neutron detector placed on the outer surface to measure the transmitted neutron flux. This detector utilizes the F2 tally function in MCNP5 to record neutron and secondary γ-ray flux data. To ensure the reliability of the simulation results, the MCNP5 code utilized the ENDF/B-VI neutron cross-section database and set the total number of incident neutrons (nps) to 10⁸ to reduce the statistical errors to less than 1%. During the simulation, the convergence of the geometric model was verified by adjusting the spherical shell thickness (1–10 μm), and the results showed that the relative deviation of the ∑ value due to thickness variation was less than 2%. More details of the MCNP5 input are presented in Table 1.

Based on the Beer-Lambert law [46], the macroscopic cross-section (∑) of a material can be determined using the following formula:(3)where I is the transmitted neutron flux, I₀ is the incident neutron flux, and x is the thickness of the shielding material. This relationship provides a direct method for calculating the macroscopic cross-section, which quantifies the ability of a material to attenuate thermal neutrons. By combining this theoretical framework with MCNP5 simulation results, researchers can accurately assess the shielding performance of boron composites, even in scenarios where experimental measurements are impractical.

In addition, the macroscopic cross-section (∑) can be calculated manually using the following equation [47]:(4)Here, Wi represents the partial density of the i^th constituent element, while is the neutron mass attenuation coefficient of the i^th constituent for a specific interaction type (j), as defined in Eq. 5.(5)Additionally, N_A stands for Avogadro’s constant, A is the atomic weight of the i^th element, and σ_t is the microscopic cross-section of the i^th element. These microscopic cross-section values can be retrieved from the comprehensive database available at the International Atomic Energy Agency (IAEA) website [48]. This resource provides essential nuclear data for accurate calculations and analyses in neutron-shielding studies.

Fast neutron shielding performance

The recently developed Phy-X software [39], which incorporates an updated database and a refined empirical model, was employed to calculate the macroscopic removal cross-sections (∑_R) for fast neutrons in various boron compounds. These results were compared with those obtained using the NXcom program [42], a well-established tool for radiation shielding analysis. The calculated ∑_R values for the investigated boron compounds are presented in Table 3. Notably, the results generated by the Phy-X software showed excellent agreement with those derived from the NXcom program, demonstrating the reliability and accuracy of both computational tools.

The influence of mass density on the fast neutron shielding properties of various boron compounds is shown in Fig. 2. Among the studied materials, tungsten diboride (WB₂) exhibits the highest values, attributed to its significantly higher mass density of 14.15 g/cm³. In contrast, potassium borohydride (KBH₄), with the lowest mass density of 1.11 g/cm³, demonstrates the lowest ∑_R values. This trend highlights a strong positive correlation between mass density and neutron shielding efficiency, as materials with greater densities tend to provide more effective neutron attenuation owing to increased atomic interactions. As shown in the figure, most of the data points align closely with the fitting curve, indicating a consistent relationship between the mass density and shielding performance. However, a few compounds, such as boron nitride (BN), iron boride (FeB), and potassium borohydride (KBH₄), deviated slightly from the curve. These deviations may arise from differences in the material composition, crystalline structure, or other factors influencing the neutron interaction probabilities. Despite these minor discrepancies, the overall trend confirms that boron compounds with higher mass densities generally exhibit superior fast-neutron shielding capabilities.

Fig. 2Full-screen View

(Color online) Effective removal cross-section with mass density for different the boron compounds

The relationship between the ∑_R values of boron compounds and their boron concentration, as well as boron number density, is illustrated in Figs. 3 and 4. As depicted in Fig. 3, tungsten diboride (WB₂) exhibits the highest ∑_R values despite having a relatively moderate boron concentration of 10.52%. This observation challenges the conventional assumption that materials with higher boron concentrations inherently possess superior neutron-shielding properties. For instance, boron carbide (B₄C), which has the highest boron concentration at 78.26%, demonstrates only average fast neutron shielding behavior compared to other boron compounds under study. This suggests that boron concentration alone is not the sole determinant of the shielding performance. Figure 3 further reveals that the FNSB of boron compounds does not exhibit a direct correlation with boron concentration. Instead, other factors, such as material density, atomic structure, and the presence of additional elements, may play significant roles in influencing the neutron attenuation efficiency. For example, the high density and unique atomic arrangement of WB₂ likely contribute to its exceptional shielding performance, even with a lower boron concentration. These findings underscore the complexity of neutron shielding mechanisms and highlight the need to consider multiple material properties when designing effective shielding solutions for neutron radiation.

Fig. 3Full-screen View

(Color online) Effective removal cross-section with boron concentration for different the boron compounds

Fig. 4Full-screen View

(Color online) Effective removal cross-section with boron number density for different the boron compounds

The boron number density, which varies across boron compounds, is another critical factor that may influence the FNSB of these materials. Figure 4 illustrates the relationship between the values and the boron number density for various boron compounds. As shown in the figure, only a subset of compounds, including potassium borohydride (KBH₄), sodium borohydride (NaBH₄), boron trioxide (B₂O), magnesium diboride (MgB₂), calcium hexaboride (CaB), silicon hexaboride (SiB₆), chromium diboride (CrB₂), gadolinium hexaboride (GdB₆), boron carbide (B₄C), and zirconium dodecaboride (ZrB₁₂), follow the expected trend where higher boron number density corresponds to larger ∑_R values. However, this trend did not hold for all compounds. For instance, tungsten diboride (WB₂) and calcium hexaboride (CaB₆) exhibit nearly identical boron number densities; however, their FNSB properties differ significantly. This discrepancy suggests that the boron number density alone cannot fully account for the variations in the neutron shielding performance. The non-proportional distribution of values with the number density of boron compounds may indicate that the number density is not the only key factor in the FNSB for boron compounds.

To determine whether the density and boron number density are the dominant factors influencing the FNSB of boron compounds, the variations in ∑_R values were analyzed as a function of the product of the density and boron number density, as illustrated in Fig. 5. An exponential fitting curve was also included to model this relationship. In contrast to the scattered distribution observed in Figs. 3 and 4, the data points in Fig. 5 exhibit a much closer alignment with the fitting curve, indicating a strong correlation. Only a few compounds, such as iron boride (Fe₂B), boron nitride (BN), and calcium hexaboride (CaB₆), deviated slightly from this trend. The exponential increase in ∑_R values with the product of the density and boron number density suggests that these two parameters collectively play a significant role in determining the FNSB of boron compounds. This relationship highlights the synergistic effect of density and boron number density on the neutron shielding performance, as higher values of both parameters enhance the probability of neutron interactions and energy absorption. The close fit of most data points to the exponential curve further supports the idea that these factors are the primary drivers of the shielding efficiency.

Fig. 5Full-screen View

(Color online) Effective removal cross-section with the product of density and boron number density for different boron compounds

Furthermore, the ∑_R values of commonly used neutron moderators demonstrate significant variation: water (H₂O) exhibits a ∑_R of 0.103 cm^-1, polyethylene ((C₂H₄)_n) shows 0.122 cm^-1, and graphite possesses 0.113 cm^-1. These materials have been extensively employed as fast neutron moderators in radiation-shielding applications owing to their effective moderation properties. Notably, among boron-containing compounds, although potassium borohydride (KBH₄, ∑_R=0.082 cm^-1) and sodium borohydride (NaBH₄, ∑_R=0.108 cm^-1) exhibit slightly lower ∑_R values than water and graphite, the majority of boron compounds demonstrate superior ∑_R values. This enhanced performance highlights their exceptional fast neutron shielding capabilities and suggests their promising potential for advanced radiation-shielding applications. Although this study shows that tungsten diboride (WB₂) exhibits the best performance in fast neutron shielding (∑_R=0.225 cm^-1), its high cost and complex manufacturing process limit its industrial application. In contrast, boron carbide (B₄C), with its lightweight properties (density of 2.50 g/cm³) and high boron content (78.26%), has become the material of choice for reducing the weight and space of neutron shielding [3]. However, the low density of B₄C may lead to multiple scattering of fast neutrons, requiring the enhancement of the overall shielding efficiency through composite material design (e.g., mixing with high-density tungsten particles). Additionally, the brittleness of boron carbide may affect its stability under dynamic loading conditions, which could be improved in the future through nano-reinforcement or fiber composite technology.

Thermal neutron shielding performance

The thermal neutron transmission ratios (I/I₀) of various boron compounds are presented in Fig. 6. The results indicated that the transmission ratios of the selected boron composites ranged from 0.017 to 0.98. The lowest transmission ratio, observed for gadolinium hexaboride (GdB₆), signifies its exceptional TNSB.

Fig. 6Full-screen View

Thermal neutron transmission ratios of boron compounds

The macroscopic cross-section (∑) values for the boron compounds studied at a neutron energy of 0.025 eV are presented in Table 4. Notably, the ∑ values derived from the MCNP5 simulations show excellent agreement with those calculated manually using Eq. (4). To quantify the consistency between the simulated and calculated results, the percentage relative deviation (RD) was determined using Eq. 6 [50]. The RD values fell within a narrow range of 0.11% to 6.77%, demonstrating a high level of accuracy in the simulation outcomes. This close alignment between the simulated and manually calculated values underscores the reliability of the MCNP5 code in predicting neutron shielding properties and validates the robustness of the computational approach employed in this study. Furthermore, the secondary γ-ray fluences (cm^-2) are also presented in Table 4. As we known, ¹⁰B can absorb the thermal neutrons and release secondary γ-rays, the reaction formula is given as Eq. 7.(6)(7)To gain deeper insights into the factors influencing the TNSB of the selected boron compounds, the ∑ for thermal neutrons was analyzed in relation to the mass density, boron concentration, and boron number density, as illustrated in Figs. 7 and 8. These figures provide a comprehensive visualization of how these parameters collectively affect the shielding performance of boron-containing compounds. The distribution shown in Fig. 7(a) and (b) are similar, indicating that the ∑ of thermal neutrons is not proportional to density and boron concentration.

Fig. 7Full-screen View

(Color online) Macroscopic cross-section of thermal neutrons with mass density and boron concentration for different the boron compounds

Fig. 8Full-screen View

(Color online) Macroscopic cross-section of thermal neutrons with boron number density for different the boron compounds

As shown in Fig. 8, most data points closely align with the fitting curve, except for gadolinium hexaboride (GdB₆). This trend indicates that boron compounds with higher boron number densities generally exhibit larger ∑ values, reflecting their enhanced thermal neutron-shielding capabilities. Notably, GdB₆ exhibits a significantly higher ∑ value than other boron compounds, attributable to the extraordinarily high thermal neutron cross-section of ¹⁵⁷Gd (253909.786 cm^-1) at 0.025 eV, which far exceeds that of ¹⁰B (3844.16 cm^-1), far exceeding the ∑ value of ¹⁰B. Owing to its relatively low boron number density, the TNSB of commercially available polyethylene-boron composites (BPE-5) demonstrates inferior performance compared to other boron compounds. In addition to GdB₆, B₄C (∑=447.37 cm^-1) and ZrB₁₂ (∑=474.67 cm^-1) also demonstrate outstanding TNSB.

Among the various factors examined, the boron number density emerged as the most significant determinant of TNSB in boron compounds. However, further research is essential to fully understand the generation of secondary particles resulting from neutron capture in these materials. As is widely recognized, neutron capture reactions, such as the (n, α) reaction in ¹⁰B and similar reactions in elements such as ¹⁴⁹Sm and ¹⁵⁷Gd, produce secondary gamma rays [51]. The energy of the secondary gamma rays produced by ¹⁰B, ¹⁴⁹Sm, and ¹⁵⁷Gd was 0.48 MeV, 7.89 MeV, and 7.88 MeV, respectively. Furthermore, as TNSB increased, a corresponding increase in secondary gamma-ray production was observed, as shown in Table 4. These secondary emissions must be carefully considered in the design of radiation-shielding systems, as they can compromise the overall shielding effectiveness. To address this, the design of multilayer shielding structures combining an outer layer of high-density materials (e.g., tungsten or lead) with an inner layer of high boron number density materials (e.g., B₄C or GdB₆) could optimize the overall performance by simultaneously attenuating neutrons and minimizing secondary radiation [16, 52-56]. This approach not only improves neutron attenuation but also reduces the secondary radiation burden, making boron-based materials more versatile and effective for radiation shielding applications. Continued exploration of these compositional adjustments is vital for advancing the development of next-generation shielding materials that meet the stringent demands of modern radiation protection.

Future Research Directions

This study primarily focused on the influence of the boron number density, mass density, and boron content on the neutron shielding performance. However, it is important to acknowledge that the structural morphology, such as porosity, grain boundaries, and nanoscale material design, can also play a significant role in neutron attenuation. These microstructural characteristics may affect the real-world performance of shielding materials by altering the neutron interaction probabilities, scattering mechanisms, and overall material integrity. To address these gaps, further investigations are recommended in the following areas: (1) microstructural optimization: systematic studies on how porosity, grain size, and phase distribution influence neutron shielding in boron composites [57, 58]. (2) Advanced fabrication techniques: methods such as spark plasma sintering or additive manufacturing to control the morphology and minimize defects [59, 60].

WB₂ is a refractory ceramic characterized by exceptional hardness (~ 20 GPa) and high thermal stability (melting point ~ 2800 °C) [24], making it ideal for use in extreme environments. Its dense atomic structure and tungsten content enhance radiation shielding performance; however, its high brittleness and poor processability necessitate composite designs (e.g., metal matrix bonding) for practical applications [16]. GdB₆ offers outstanding neutron absorption owing to gadolinium’s high cross-section, coupled with moderate mechanical strength (~ 15 GPa) and thermal stability (melting point ~ 2500 °C) [51, 52]; however, its mechanical properties often require reinforcement (e.g., polyethylene matrix integration). ZrB₁₂ stands out for its exceptional thermal resilience (melting point > 3000 °C) and high boron number density (118.03 atoms/nm³), providing an optimal balance of neutron attenuation and structural integrity for high-temperature nuclear components [23, 61]. Future efforts should focus on optimizing the material composition and structural design to balance the shielding efficiency, cost, and processability, thereby promoting the practical application of boron-based composite materials in the nuclear energy, aerospace, and medical fields.

Fabrication techniques play a pivotal role in optimizing the neutron-shielding performance of boron-containing composites and alloys. Advanced methods such as spark plasma sintering and powder metallurgy have been employed to fabricate dense, homogeneous composites such as B₄C/Al and WB/Al, ensuring uniform boron dispersion and enhanced mechanical integrity [62]. For polymer-based shields (e.g., HDPE/B₄C), melt blending and hot pressing are widely used to achieve optimal particle-matrix adhesion, whereas cold spraying has emerged for creating layered structures with minimal porosity [63]. In alloy systems (e.g., borated steels), arc melting and mechanical alloying are critical for achieving high boron solubility and structural stability under irradiation [24]. Future efforts should focus on scalable techniques, such as additive manufacturing, to address challenges in complex geometries and industrial adoption.

Future research could also explore the development of multi-component boron-based composite materials, such as combining GdB₆ with polyethylene, to leverage the synergistic effects of hydrogen moderation and boron capture for enhanced shielding efficiency. The synergistic use of ZrB₁₂ and GdB₆ in nuclear reactor control rods is worth exploring. The high-temperature stability of ZrB₁₂ and its high boron number density render it suitable for fast neutron absorption, whereas GdB₆ can efficiently capture thermal neutrons. Such combinations can optimize the neutron absorption dynamics of control rods while enhancing their durability in extreme environments.