Rapid Calculation of Neutron Spectrum

A genetic algorithm [17] was employed for neutron spectrum optimization. A large number of irradiation schemes are screened, necessitating a rapid algorithm for calculating the neutron spectrum, where a previously proposed method [18] was adopted.

Assuming that a neutron with energy E_i undergoes a nuclear reaction with a non-absorbing nuclide, the energy of the neutron then changes to $E_j={\displaystyle \sum _i{P}_{i,j}\left(E_i\right)\cdot }{E}_i,$ (1) where E_j and E_i represent the neutron energies before and after the reaction, respectively, and P_i,j denotes the probability of the energy transfer from E_i to E_j. If the continuous energy is divided into n energy bins, P_i,j can be written as an n × n matrix, $R_k=\left(\begin{array}{cccc}{r}_{0,0}& \cdots & \cdots & r_{0,n}\\ ⋮& \ddots & r_{i,j}& ⋮\\ ⋮& ⋮& \ddots & ⋮\\ r_{n,0}& \cdots & \cdots & r_{n,n}\end{array}\right),$ (2) and Eq. (1) can also be rewritten in matrix form, $V_1=R_k{V}_{\text{0}},$ (3) where V₀ and V₁ represent the neutron spectra before and after the reaction, R_k is the spectrum response matrix for nuclide k; and the matrix element r_i,j represents the probability of energy transfer from energy bin i to energy bin j.

To consider absorption reactions, each row of the matrix is normalized, $a_i+{\displaystyle \sum _{j=1}^n{r}_{i,j}=1},$ (4) where a_i is the neutron absorption probability in energy bin i.

Equation (4) makes the sum of each row of R_k less than 1.0, causing the norm of V₁ also to be less than 1.0, where Eq. (3) fails to yield a stable neutron spectrum. In this case, V₀ is used to supplement the reduction in V₁ after each iteration, $V_1=R_k{V}_{\text{0}}\text{+}\frac{\left|V_{\text{0}}\right|\text{-}\left|R_k{V}_{\text{0}}\right|}{\left|V_{\text{0}}\right|}{V}_{\text{0}}.$ (5)

To consider the presence of multiple nuclides in a material, the spectrum response matrices of each nuclide are used to form a spectrum response matrix of this material, $R_{\text{mat}}={\displaystyle \sum _{k=0}^n{F}_k{R}_k},$ (6) where F_k is a diagonal matrix composed of the weights of nuclide k, $F_k=\left(\begin{array}{cccc}{f}_{k,1}& & & \\ & f_{k,2}& & \\ & & \ddots & \\ & & & f_{k,n}\end{array}\right),$ (7) where the matrix element f_k,i represents the weight of nuclide k in energy bin i. It reflects the importance of nuclide k to the spectrum transfer process in this material, $f_{k,i}=\frac{{\sum }_{\text{total}}^{k,i}}{{\sum }_{\text{total}}^i}$ (8) where ${\text{Σ}}_{\text{total}}^{k,i}$ is the macroscopic total cross section of nuclide k in energy bin i, and ${\text{Σ}}_{\text{total}}^i$ is the macroscopic total cross section of the material in energy bin i.

To consider different geometries, the leakage of neutrons is calculated because some neutrons fly out of the system without reaction. The probability p_i of a neutron not undergoing a collision can be derived based on the exponential decay law, $p_i=e^{-{\sum }_{\text{total}}^i\cdot b},$ (9) where b is the distance to the boundary, which is related to the geometric parameters of the system. Then, the spectrum response matrix is revised to $R_{\text{real}}=R_{\text{k}}\times (I-P),$ (10) where R_real is the true spectrum response matrix for the geometry, I is the identity diagonal matrix, and P is a diagonal matrix composed of the noncollision probabilities, $P=\left(\begin{array}{cccc}{p}_1& & & \\ & p_2& & \\ & & \ddots & \\ & & & p_n\end{array}\right).$ (11) With Eqs. (3), (5), (6), and (10), the output neutron spectrum for any source neutron spectrum under any geometric and material conditions can be calculated rapidly.

High-Resolution Spectrum Regulation

Equation (8) indicates that the response relationship of neutron spectra is correlated with the isotopic composition of the materials. Therefore, the neutron spectrum can be modulated by altering the isotopic composition within a spatial region. The spectrum response matrices of 423 isotopes were built, and the full energy range was divided into 238 energy bins [19] to enhance the spectral resolution and precision of neutron spectrum regulation.

The source neutron spectrum (n energy bins) is denoted as ψ_in = {x₁, x₂, · ·· ·, x_n}, and the output neutron spectrum is denoted as ψ_out = {y₁, y₂, · ·· ·, y_n}. A stable output neutron spectrum can be obtained from the iterative convergence of Eq. (3), ${\psi }_{out}={\displaystyle \sum _{m=1}^{\infty }{\varphi }_m},$ (12) where φ_m represents the neutron spectrum after m collisions, which is calculated by $\begin{array}{l}{\varphi }_m=R_{\text{real}}\cdot {\varphi }_{m-1}=R_{\text{real}}^2\cdot {\varphi }_{m-2}=\cdot \cdot \cdot =\\ R_{\text{real}}^{m-1}\cdot {\varphi }_1=R_{\text{real}}^m\cdot {\text{ψ}}_{\text{in}}.\end{array}$ (13)

Combining Eqs. (12) and (13), one obtains ${\psi }_{\text{out}}={\displaystyle \sum _{m=1}^{\infty }{\varphi }_m}={\psi }_{\text{in}}+R_{\text{real}}\cdot {\psi }_{\text{in}}+R_{\text{real}}^2\cdot {\psi }_{\text{in}}+R_{\text{real}}^3\cdot {\psi }_{\text{in}}+\cdot \cdot \cdot =\frac{{\psi }_{\text{in}}}{I-R_{\text{real}}},$ (14) and simplifying Eq. (14) yields ${\psi }_{\text{out}}=\frac{I}{I-R_{\text{real}}}\cdot {\psi }_{\text{in}}=\text{A}\cdot {\psi }_{\text{in}},$ (15) where A is a spectral matrix that transforms the source neutron spectrum into the output neutron spectrum. Many matrices satisfy Eq. (15), for example, $\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{cccc}\frac{x_1\cdot y_1}{{\displaystyle {\sum }_1^n{x}_i^2}}& \frac{x_2\cdot y_1}{{\displaystyle {\sum }_1^n{x}_i^2}}& \cdots & \frac{x_n\cdot y_1}{{\displaystyle {\sum }_1^n{x}_i^2}}\\ \frac{x_1\cdot y_2}{{\displaystyle {\sum }_1^n{x}_i^2}}& \frac{x_2\cdot y_2}{{\displaystyle {\sum }_1^n{x}_i^2}}& \cdots & \frac{x_n\cdot y_2}{{\displaystyle {\sum }_1^n{x}_i^2}}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{x_1\cdot y_n}{{\displaystyle {\sum }_1^n{x}_i^2}}& \frac{x_2\cdot y_n}{{\displaystyle {\sum }_1^n{x}_i^2}}& \cdots & \frac{x_n\cdot y_n}{{\displaystyle {\sum }_1^n{x}_i^2}}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right].$ (16) If matrix A can be constructed with a certain isotopic composition of these 423 nuclides, then the high-resolution neutron spectrum is achieved without any deviation. However, because the full energy range has been divided into 238 energy bins for high resolution, A is a 238 × 238 matrix, making the construction of A quite challenging. A cannot be obtained solely based on these 423 matrices. Instead, one can use only these 423 matrices to construct a matrix-approximating matrix A.

Yield Maximum with Optimal Spectrum Regulation

Previous studies have demonstrated that transuranic isotope production can be improved greatly by neutron spectrum optimization and have provided the importance curves that quantify the values of neutrons in various energy bins for the production of transuranic isotopes [12], as shown in Fig. 2. To determine the optimal neutron spectrum further, a genetic algorithm was employed to build an optimal spectrum database that provides the optimal neutron spectra under different neutron fluxes and irradiation durations [13]. The theoretical maximum yields of ²³⁸Pu under different irradiation conditions are shown in Fig. 3. Therefore, once this optimal neutron spectrum is constructed within the target, the yields of transuranic isotopes can be maximized.

Fig. 2Full-screen View

Value of neutrons in each energy region for ²⁵²Cf production [12]

Fig. 3Full-screen View

Maximum yields under different flux levels and irradiation durations [13]

To construct the optimal neutron spectrum inside the target based on an initial irradiation scheme, it is necessary to build a spectrum response matrix that converts the initial neutron spectrum into the optimal one. However, it is impossible to construct the optimal neutron spectrum using only the spectrum response matrices of these 423 nuclides. Therefore, a genetic algorithm is employed to search for the optimal proportions of these 423 nuclides, such that the neutron spectrum inside the target is closest to the optimal one, thereby maximizing the yields of transuranic isotopes. As shown in Fig. 4, the specific calculation process is as follows.

Fig. 4Full-screen View

Flowchart for searching the optimal neutron spectrum

(1) Constructing the initial population for the genetic algorithm: Various nuclides are randomly added to the target, constructing different neutron spectrum regulation schemes. Assuming that there are originally i types of nuclide in the target, the spectrum response matrix inside the target at this time is $R_{\text{real}}={\displaystyle \sum _{k=0}^i{F}_k{R}_k}\times (I-P),$ (17) and the spectrum response matrix inside the target is changed after j new types of nuclides are added to the target, ${R^{\prime }}_{\text{real}}={\displaystyle \sum _{k=0}^{i+j}{F}_k{R}_k}\times (I-P).$ (18)

(2) Calculation of the neutron spectrum inside the target: In the case of a known fixed source term, the neutron spectrum can be directly calculated using Eq. (14). However, because the incident neutron spectrum of the irradiation channel also changes after the target material is modified, the incident neutron spectrum must be updated according to the nuclide composition of the target. Linear interpolation is used to provide the incident neutron spectra for various nuclide compositions inside the target, as shown in Fig.5.

Fig. 5Full-screen View

Schematic diagram for determining the incident neutron spectra

To accomplish this, one first must determine two initial source terms: the incident neutron spectra ψ_in,0 and ψ_in,∞ when the cross sections of the material in the irradiation channel are zero and infinity, respectively. Then, the non-collision probability P in Eq. (9) is used as the interpolation factor to calculate the incident neutron spectrum for different nuclide compositions inside the target, ${\psi }_{\text{in,real}}=(I-P)\times {\psi }_{\text{in},0}+P\times {\psi }_{\text{in},\infty }.$ (19)

After the spectrum response matrix of the target and incident initial neutron spectrum are determined, the neutron spectrum inside the target can be quickly calculated using the following equation: ${\psi }_{\text{out}}=\frac{{\psi }_{\text{in},\text{real}}}{I-{R^{\prime }}_{\text{real}}}$ (20)

(3) Calculating the yields of transuranic isotopes: To meet the high computational demand of the genetic algorithm, the yields of transuranic isotopes are calculated by solving the burnup equation [20, 21], $\frac{\text{d}{n}_i}{\text{d}t}={\displaystyle \sum _{i\ne j}{b}_{j,i}^{\text{eff}}{\lambda }_j^{\text{eff}}{n}_j-{\lambda }_i^{\text{eff}}{n}_i},$ (21) where n_i is the density of the i^th nuclide, λ_i^eff is the effective decay constant of the i^th nuclide, and b_i,j^eff is the branching ratio for the transmutation of the i^th nuclide to the j^th nuclide. In addition, λ_i^eff and b_i,j^eff can be calculated from the following formula, $\{\begin{array}{l}{\lambda }_i^{\text{eff}}={\lambda }_i+\varphi {\displaystyle \sum _j{\sigma }_{i,j}}\\ b_{i,j}^{\text{eff}}=\left(b_{i,j}{\lambda }_i+{\sigma }_{i,j}\varphi \right)/{\lambda }_i^{\text{eff}}\end{array},$ (22) where λ_i is the decay constant of the i^th nuclide, ϕ is the neutron flux, and σ_i,j is the one-group cross section where the reaction of the i^th nuclide generates the j^th nuclide.

The one-group macroscopic cross sections ∑_r of a neutron spectrum needed by the point burnup calculation are calculated by ${\text{Σ}}_{\text{r}}={\displaystyle \sum _{i=1}^{238}{\sigma }_{\text{r},i}}\cdot {\overline{\psi }}_{\text{out},i}$ (23) where the subscript i represents an energy bin; subscript r represents a reaction type, ${\overline{\psi }}_{\text{out},i}$ represents the normalized output neutron spectrum, and σ represents the grouped microscopic cross section.

(4) Performing evolutionary operations on the population: With the yield of transuranic isotopes as the optimization objective, the population is screened, crossed, and mutated to obtain a new population. Step (2) is then repeated until the evolution of the population is completed.

(5) Outputting the optimal neutron spectrum regulation scheme: The optimal neutron spectrum regulation scheme is described in terms of the types and proportions of nuclides added to the target, along with the yields of transuranic isotopes under this optimal scheme and its enhancement effect compared with the original production scheme.

The nuclear data, such as decay data, cross-section data, and fission yield data required for the neutron spectrum and burnup calculation in this study, all originate from ENDF/B-VII.1 [9]. The spatial self-shielding effect of the target can be considered during neutron energy spectrum calculation.

State-of-the-Art Irradiation Scheme

The High-Flux Isotope Reactor (HFIR) [22] is currently the primary facility for transuranic isotope production, accounting for 70% of the global ²⁵²Cf supply. HFIR has a steady-state neutron thermal flux of 2.5 × 10¹⁵ cm⁻²·s⁻¹ and a refueling cycle of 25 days. HFIR and the target are modeled using the RMC code [23–25], as shown in Fig. 6.

Fig. 6Full-screen View

(Color online) Modeling diagram of HFIR and the target

The volumetric neutron spectrum inside the target is changed by adding various nuclides to it. The state-of-the-art irradiation scheme (called “SOTA”) employs a target with a diameter of 5 mm, whereas the diameter of the reactor is 110 cm, resulting in an extremely narrow range for neutron spectrum regulation. For comparison, an alternative irradiation scheme (called “ALTER”) was envisioned in which targets are placed throughout the entire neutron flux trap. In this hypothetical scenario, the diameter of the target is 12.6 cm, which has a larger range of neutron spectrum regulation than SOTA. Monte Carlo simulations were employed to obtain the incident neutron spectra when the cross sections of the target are zero and infinity. Furthermore, based on Eqs. (19) and (20), the adjustable range of the neutron spectrum for ²⁵²Cf production using the two irradiation schemes was calculated, as shown in Fig.7. The average relative deviation (Ave.Re) of the neutron spectrum is used to measure its adjustable range, $\text{Ave}.\mathrm{Re}={\displaystyle \sum _{i=1}^{238}\left|\frac{{\varphi }_{0,i}-{\varphi }_{\infty ,i}}{{\varphi }_{\infty ,i}}\right|}/238,$ (24) where ф_0,i and ф_∞,i are the volumetric neutron spectra inside the target when the cross sections of the target are zero and infinity, respectively.

Fig. 7Full-screen View

Adjustable range of the neutron spectrum for ²⁵²Cf production

Table 1 presents the Ave.Re of the incident neutron spectra, the volumetric neutron spectrum inside the target for ²⁵²Cf and ²³⁸Pu (two types of target) production when the cross sections of the target are zero and infinity.

As shown in Fig. 7 and Table 1, the range for neutron spectrum regulation is small, which makes neutron spectrum optimization extremely difficult. The HFIR irradiation scheme of transuranic isotope production, which is currently the optimal irradiation scheme available, is chosen as the initial approach in this study. If the research presented in this paper can further enhance these state-of-the-art irradiation schemes through neutron spectrum regulation, the significance of our work can be demonstrated.

In the subsequent analysis, the neutron spectrum within the flux trap and target is adopted as the starting point for neutron spectrum regulation. The neutron flux for point burnup calculations is 2.5 × 10¹⁵ cm⁻²·s⁻¹, with a burnup time of 25 days, which is divided into 10 subburnup steps. During the calculation process, the genetic algorithm is first used to search for and identify the 20 most important nuclides for enhancing the yield of transuranic isotopes. The concentrations of these 20 nuclides are then optimized. Hence, in the next section, the addition of these 20 most important nuclides for yield improvement is discussed.

Volumetric Neutron Spectrum Regulation for ²⁵²Cf Production

The target for ²⁵²Cf production is a mixture of plutonium, americium, and curium [26, 27], whose nuclide components are listed in Table 2. The calculation procedure outlined in Fig. 4 was used to optimize the neutron spectrum for the SOTA and ALTER irradiation schemes.

For the SOTA irradiation scheme, the yield of ²⁵²Cf after 25-day irradiation of this target in HFIR is 1.33 × 10¹⁸ atom/cm³. After the nuclides listed in Table 3 are added to the target, a maximum yield of 1.49 × 10¹⁸ atom/cm³ can be achieved, a 12.16% increase compared with the original SOTA irradiation scheme.

For the ALTER irradiation scheme, the yield of ²⁵²Cf after 25-day irradiation of this target in HFIR is 1.15 × 10¹⁷ atom/cm³. After the nuclides listed in Table 4 are added to the target, a maximum yield of 1.77 × 10¹⁷ atom/cm³ can be achieved, which is a 54.54% increase compared with the original ALTER irradiation scheme.

The nuclides listed in Tables 3 and 4 are added to the target by dispersion, with the density of the added nuclides not exceeding 1% of the initial nuclide density of the target. After the addition of the nuclides, the density of the original nuclides in the target decreases accordingly. Meanwhile, in this study, a high-resolution neutron spectrum regulation method, achieved by adjusting the proportions of these 423 nuclides within a certain spatial region, was developed. Theoretically, the more the nuclides that can be used for neutron spectrum regulation, the higher the regulation capability. However, many nuclides are unstable and extremely expensive, which makes them difficult to use in practical engineering applications. Therefore, it is recommended that readily available nuclides or natural elements be adopted to develop neutron spectrum regulation schemes in practical engineering.

Volumetric Neutron Spectrum Regulation for ²³⁸Pu Production

²³⁸Pu is produced by the in-reactor irradiation of americium-241 (²⁴¹Am) [28, 29] or neptunium-237 (²³⁷Np) [30, 31], with the nuclide components of the targets listed in Table 5. Experience has shown that the in-reactor irradiation of ²³⁷Np facilitates a higher yield of ²³⁸Pu, whereas the in-reactor irradiation of ²⁴¹Am facilitates a higher abundance of ²³⁸Pu. Therefore, these two irradiation methods are currently the practical solutions adopted, and analyses of both methods were conducted.

Comparison and Analyses

The yields of these irradiation schemes are compared in Table 10, showing that the volumetric neutron spectrum regulation method proposed can enhance the production of transuranic isotopes. Even for the state-of-the-art irradiation schemes that are currently optimal, one can still improve their efficiencies. Furthermore, the spectrum optimization schemes proposed are simple and feasible, requiring only the dispersion of various nuclides inside the target without the need to modify reactor design or irradiation channel parameters, making them highly practical for engineering applications.

A comparison between the SOTA irradiation scheme and the ALTER irradiation scheme reveals the following conclusions. (1) The SOTA irradiation scheme consistently outperforms the ALTER irradiation scheme, demonstrating the significance of neutron spectrum regulation for transuranic isotope production. (2) The larger the range for neutron spectrum regulation, the more pronounced the promotion in enhancing transuranic isotope production. Therefore, the volumetric neutron spectrum regulation method proposed can identify and construct an optimal neutron spectrum within a limited range. The calculated results provided only cover a narrow range and fail to reflect the overall physical process. In this study, only the effectiveness of this regulation scheme and its technical value are emphasized, without delving into its physical significance in detail. If further increasing the production of transuranic isotopes is required, merely altering the nuclide composition of the target is insufficient. At this point, modifications to the reactor design are necessary to broaden the regulation range of the neutron spectrum within the irradiation channels. This will be explored in planned future research.