Reactor Models

The dataset was simulated in the high-flux isotope reactor (HFIR) [18], which a multi-purpose isotope production and testing reactor with a current operating power of 85 MW. The HFIR operates on a cycle of 22–24 d and is capable of the world’s highest steady-state neutron thermal flux (2.6×10¹⁵ cm^-2·s^-1), making it a significant global production base for medical isotopes. We inserted a target assembly with a length of 50 cm and diameter of 0.85 cm inside the neutron trap to produce twenty medical isotopes (¹⁴C, ³²P, ⁴⁷Sc, ⁶⁰Co, ⁶⁴Cu, ⁶⁷Cu, ⁸⁹Sr, ⁹⁰Y, ⁹⁹Mo, ¹²⁵I, ¹³¹I, ¹⁵³Sm, ¹⁶¹Tb, ¹⁶⁶Ho, ¹⁷⁷Lu, ¹⁸⁶Re, ¹⁸⁸Re, ⁹²Ir, ²²⁵Ac, ²⁵²Cf). The reactor Monte Carlo (RMC) code was utilized for modeling [19], as illustrated in Fig. 2. Moreover, because HFIR is a thermal reactor, we also performed the same simulations in a high-flux fast reactor (HFFR) [17] to enhance the universality of the database, as shown in Fig. 3.

Fig. 2Full-screen View

(Color online) Modeling diagram of HFIR

Fig. 3Full-screen View

(Color online) Modeling diagram of the fast reactor

Calculation Methods

The yields of medical isotopes can be obtained by performing a Monte Carlo burnup calculation that couples the Monte Carlo criticality and point burnup calculations. The point burnup equation describes the transmutation of nuclides with time and can be written as [20]: $\frac{d{n}_{\text{i}}}{dt}={\displaystyle \sum _{i\ne j}{b}_{\text{j,i}}^{\text{eff}}{\lambda }_{\text{j}}^{\text{eff}}{n}_{\text{j}}-{\lambda }_{\text{i}}^{\text{eff}}{n}_{\text{i}}}$ (1) where n_i is the density of the ith nuclide, λ_i^eff is the effective decay constant of the ith nuclide, and b_i,j^eff is the branching ratio for transmuting the ith nuclide to the jth nuclide. Here, λ_i^eff and b_i,j^eff can be calculated from the following formula: $\{\begin{array}{l}{\lambda }_{\text{i}}^{\text{eff}}={\lambda }_{\text{i}}+\varphi {\displaystyle \sum _{\text{j}}{\sigma }_{\text{i,j}}}\\ b_{\text{i,j}}^{\text{eff}}=\left(b_{\text{i,j}}{\lambda }_{\text{i}}+{\sigma }_{\text{i,j}}\varphi \right)/{\lambda }_{\text{i}}^{\text{eff}}\end{array}$ (2) where λ_i is the decay constant of the ith nuclide, ϕ is the neutron flux, and σ_i,j is the one-group cross-section where the reaction of the ith nuclide generates the jth nuclide.

One-group cross sections are required for the Monte Carlo burnup calculation, which is computed by the Monte Carlo criticality calculation: $(L+C-S)\varphi =\frac{1}{k_{\text{eff}}}F\varphi$ (3) where L is the leakage operator, C is the collision operator, S is the scattering operator, F is the fission operator, and k_eff is the effective multiplication factor.

The neutron flux and nuclear reaction rates are determined by solving Equation (3) and are used to determine the one-group cross sections: ${\sigma }_{\text{i}\text{.j}}=\raisebox{1ex}{$R_{\text{i,j}}$}\!\left/ \!\raisebox{-1ex}{$\varphi $}\right.$ (4) where R_ij is the nuclear reaction rate where the i^th nuclide generates the j^th nuclide.

As shown in Eqs. (1)–(4), the Monte Carlo burnup calculation can only quantify the production efficiency of the entire neutron spectrum, not each energy region, which leads to the unknown energy regions that are favorable and unfavorable for in-reactor medical isotope production. Therefore, two methods have been proposed for the high spectral resolution quantification of neutron energy region values: 1) the subgroup burnup analysis method and 2) the extremum burnup analysis method.

(1) Subgroup burnup analysis method: The neutron flux in each energy region is reduced during the Monte Carlo burnup calculation, and the influence of the flux reduction on the yield is observed, quantifying the neutron energy region values (labeled I_i), which are calculated using the following formula: $I_{\text{i}}=\frac{\text{Δ}{Y}_{\text{i}}/Y}{\text{Δ}{\varphi }_{\text{i}}}=\frac{\left(Y-Y_{\text{i}}\right)/Y}{{\varphi }_{\text{i}}-{{\varphi }^{\prime }}_{\text{i}}}=\frac{\left(Y_{\text{i}}-Y_{\text{i}}\right)/Y_{\text{i}}}{M\cdot {\varphi }_{\text{i}}}$ (5) where the subscript “i” is the index of an energy region; Y and Y_i are the yields of medical isotopes before and after flux reduction, respectively; ϕ and ϕˊ are the neutron flux in the ith energy region before and after flux reduction, respectively; and M is the ratio of flux reduction, which takes values of one-eighth (1/8), one-quarter (1/4), one-half (1/2), and one (1/1). Previous research found [17] that M has a minor effect on these results. Therefore, we adopted M=8. We achieved neutron flux reduction in a specific neutron energy region by modifying the source code of the RMC program.

The subgroup burnup analysis method calculates the sensitivity of the relative change in the yield to the absolute change in the neutron flux in each energy region. To ensure the high spectral resolution of the dataset, the entire energy range was divided into 238 energy regions [21]. Moreover, the subgroup burnup analysis method has no restrictions on the energy division and can achieve a higher spectral resolution. Taking the 90 d irradiation production of californium-252 (²⁵²Cf) in the HFIR as an example [17], we obtained a neutron value curve across the entire energy range by calculating the neutron values for all 238 energy regions, as shown in Figure 4. We also provided the top-10 maximum and minimum values along the neutron value curve, as shown in Table 1.

Fig. 4Full-screen View

Neutron values curve for ²⁵²Cf production in HFIR

(2) Extremum burnup analysis method: The irradiation duration significantly affected the neutron energy region values calculated by the subgroup burnup analysis method. For example, in some energy regions, if the irradiation time exceeds what is required for the yield to reach its peak, then a phenomenon in which the yield increases first and then decreases throughout the irradiation process occurs. This phenomenon is known as “over-irradiation.” In other energy regions, irradiation for this duration may not be sufficient to cause the yield to reach its peak. To address this issue, we employed the extreme burnup analysis method to eliminate the influence of the irradiation duration on the neutron energy region values. The extreme burnup analysis method determines the extreme value in terms of time, which is achieved by calculating the maximum derivative between the neutron energy region value and irradiation time, as shown in Eq. (6): $E_{\text{j}}=\frac{\text{Δ}{I}_{\text{j}}}{\text{Δ}t}=\frac{I_{\text{j}+1}-I_{\text{j}}}{\text{Δ}t}$ (6) where the subscript “j” is the index of a time step, I_j is the neutron energy region value in the jth time step, and Δt represents the time step.

We divided the irradiation duration (90 d) into 20 equal time steps and performed calculations using the subgroup burnup analysis method for each time step, resulting in 20 neutron energy region values for all 238 energy regions. We then calculated the maximum derivative E_j,_max between the neutron energy region value and irradiation time and used it as the final result to quantify the neutron values in each energy region. The extremum burnup analysis method also established an extreme neutron value curve, as shown in Fig. 5. The top 10 maximum and minimum values along the curve are listed in Table 2.

Fig. 5Full-screen View

Extreme neutron value curve for ²⁵²Cf production in HFIR

The maximum values indicate that neutrons in these energy regions are conducive. Hence, increasing the neutron flux in these energy regions (referred to as positive-energy regions) increases the yield of medical isotopes. Conversely, the minimum values suggest that neutrons in these energy regions are detrimental and that increasing the neutron flux in these regions (referred to as negative-energy regions) decreases the yield.

Based on the subgroup and extremum burnup analysis methods, we obtained high-resolution neutron energy region values across the entire energy range and built a dataset to determine the positive and negative energy regions. Increasing the neutron flux in positive-energy regions and reducing it in negative-energy regions can improve the yield of medical isotopes. Therefore, this dataset can help guide the neutron spectrum regulation process for in-reactor medical isotope production.

Technical Validation

We constructed numerous neutron spectra by dispersing nuclides into target materials and verified whether the yields of the medical isotopes under these neutron spectra positively correlate with the total value of the neutron spectra (labeled as “T”), which is calculated by: $T={\displaystyle \sum _{\text{i}=1}^N{\overline{\varphi }}_{\text{i}}\cdot I_{\text{i}}}$ (7) where N represents the number of energy regions, which equals 238; ${\overline{\varphi }}_{\text{i}}$ denotes the normalized neutron flux in the ith energy region; and I_i indicates the neutron energy region value of the ith energy bin.

Taking Rhenium-188 (¹⁸⁸Re) as an example, we selected ten nuclides (⁴He, ⁷Li, ⁵⁰Cr, ⁶⁴Ni, ⁷³Ge, ⁷⁵As, ¹³⁰Ba,¹⁴⁵Nd, ¹⁶⁵Ho, ¹⁷⁶Hf) with three added amounts (10¹⁸, 5×10¹⁸, and 1×10¹⁹ atom/cm³), resulting in 30 different neutron spectra. The correlation coefficient between ΔY and ΔT was calculated as follows: $R(\text{Δ}Y,\text{Δ}T)=\frac{\text{Cov}(\text{Δ}Y,\text{Δ}T)}{\sqrt{\text{Var}[\text{Δ}Y]\text{Var}[\text{Δ}T]}}$ (8) where ΔY represents the change in the yield of ¹⁸⁸Re before and after the addition of nuclides, ΔT is the change in the total value of the neutron spectra before and after the addition of nuclides, Cov(x, y) is the covariance between x and y, Var[x] is the variance of x, and R(x, y) is the correlation coefficient between x and y, shown in Table 4.

As shown in Table 4, ΔY exhibits a positive correlation with ΔT, with a correlation coefficient of 0.9409, meaning that as the total value of the neutron spectra increases, the yield of the medical isotopes also increases. Therefore, a dataset of the neutron energy region values can guide neutron spectrum regulation to enhance in-reactor medical isotope production.

Applications

The dataset of the neutron energy region values identified the positive- and negative-energy regions for producing various medical isotopes. Theoretically, the production efficiency of medical isotopes can be increased by reducing the neutron flux in negative-energy regions. We adopted the neutron spectrum filtering technology [18] to reduce the neutron flux in the negative energy regions, which disperses nuclides with resonance peaks in the negative energy regions to the target material. The effects of adding filter nuclides on the final yields, using the production of ⁴⁷Sc, ⁶⁴Cu, ⁶⁷Cu, ⁹⁰Y, and ¹⁸⁸Re as examples, are shown in Table 5.

Table 5 shows that when 5.00×10¹⁸ atom/cm³ of ¹⁵⁵Gd was dispersed into the target for producing ⁴⁷Sc, the yield of ⁴⁷Sc increased by 154%, which is an impressive result. The yields of medical isotopes increased significantly, proving that the dataset of neutron energy region values can be used for neutron spectrum regulation to enhance the production of medical isotopes. This dataset enables the rapid identification of positive- and negative-energy regions for producing these 20 medical isotopes in fast or thermal reactors. The production efficiency of medical isotopes can be effectively improved by increasing the neutron flux in positive-energy regions and reducing it in negative-energy regions. Additionally, this dataset can be used for the rapid evaluation and comparison of the economics of different irradiation schemes. As shown in Eq. (7), the total value of the neutron spectrum of an irradiation scheme can be calculated, and schemes with higher total neutron spectrum values are more economically viable.

Data Availability

The process files generated during the calculation of this dataset can be accessed at https://doi.org/10.57760/sciencedb.j00186.00468. The calculation results of the subgroup burnup analysis method for each medical isotope can be viewed in the “Subgroup burnup” folder, which contains the raw data records and yields of this medical isotope after flux reduction in each energy region along with the corresponding neutron value curves and top 10 maximum and minimum values within the neutron value curves. The calculation results of the extreme burnup analysis method for each medical isotope can be accessed in the “Extremum burnup” folder, which records the yields of this medical isotope after flux reduction in each energy region across the 20 time steps as well as the corresponding extreme neutron value curves and top-10 maximum and minimum values within the curves.

Both the folders named “Subgroup burnup” and “Extremum burnup” contain two subfolders each, named “HFFR” and “HFIR,” which represent the computational data from these two reactors. Furthermore, both the “HFFR” and “HFIR” folders were further divided into 20 subfolders, with each named after the 20 medical isotopes covered in this dataset. These 20 subfolders were not further subdivided and contained the lowest-level data information from the computational process.

The raw data for use in technical validation and applications are stored in the “Spectrum Regulation” folder, which contains two folders named “Technical Validation” and “Application”. The “Technical Validation” folder includes 10 subfolders and an Excel sheet, which provide detailed data for technical validation. The “Application” Folder contains 5 subfolders and an Excel sheet that offer detailed data on the application process.