Reactor and Target

All the analyses were performed using a high-flux isotope reactor (HFIR) [19]. Rated at 100 MW and currently operating at 85 MW, the HFIR has a high steady-state neutron heat flux of 2.6 × 10¹⁵ cm^-2·s^-1. We used a pure ²³⁷Np target, a 50 cm high and 0.85 cm diameter rod placed in an irradiation channel at the neutron flux trap. The HFIR and target were modeled using the RMC code, a self-developed Monte Carlo code [20], as shown in Fig. 3.

Fig. 3Full-screen View

(Color online) Modeling diagram of the high-flux isotope reactor (HFIR) and the target

Computational Method

The yield of ²³⁸Pu can be obtained by performing a Monte Carlo burnup calculation, which is a coupling of the Monte Carlo criticality and point-burnup calculations. The point-burnup equation describes the transmutation of nuclides over time when a target is irradiated in a high-flux reactor. For each nuclide in the burnup chain, the time-dependent point-burnup equation can be written as $\frac{\text{d}{n}_i}{\text{d}t}={\displaystyle \sum _{i\ne j}{b}_{j,i}^{\text{eff}}}{\lambda }_j^{\text{eff}}{n}_j-{\lambda }_i^{\text{eff}}{n}_i,$ (3) 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 transmuting the i^th nuclide to the j^th nuclide. λ_i^eff and b_i,j^eff can be calculated from the following formula: $\{\begin{array}{l}{\lambda }_i^{\text{eff}}={\lambda }_i+\varphi {\displaystyle \sum _j{\sigma }_{i,j}}\\ b_{i,j}^{\text{eff}}=\left(b_{i,j}{\lambda }_i+{\sigma }_{i,j}\varphi \right)/{\lambda }_i^{\text{eff}},\end{array}$ (4) 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-sections where the i^th nuclide’s reaction generates the j^th nuclide.

As shown in Eq. (2), one-group cross-sections of the target are required; therefore, the Monte Carlo criticality calculation should be performed. $\left(L+C-S\right)\varphi =\frac{1}{k_{\text{eff}}}F\varphi ,$ (5) 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 physical parameters around the target, such as the neutron flux, fission reaction rate, and absorption reaction rate, can be obtained by solving Eq. (3), which is used to determine the one-group cross-sections.

The Monte Carlo burnup calculation requires one-group cross sections integrated according to the neutron spectrum over the entire energy range rather than the cross sections in each single energy region, which can only quantify the influence of the entire neutron spectrum on the production efficiency and not a closer analysis of each single energy region. Therefore, we cannot know which energy regions are favorable and which are unfavorable for ²³⁸Pu production, and thus, cannot provide a more refined theory for optimization.

We propose a filter burnup method and a single-energy burnup method for refined spectrum analysis, quantifying the transmutation rate of the nuclide chain in each energy region, and building relationship curves between the production efficiency and neutron spectrum. We also propose an extreme burnup analysis method to investigate the influence of the irradiation time on the relationship curves. Therefore, three relationship curves (importance, yield, and extreme) were obtained using the three analytical methods.

2.2.1

Filter Burnup Method

The filter burnup method seems to perform subtraction on the neutron spectrum, i.e., reducing the neutron flux in a certain energy region to investigate the relationship between the change in grouped flux and the change in yield, quantifying the importance of the flux in a certain energy region on the transmutation rate of the nuclei. Because this is a sister paper to Reference [18], the detailed theory is not repeated and only the definition of importance is provided. $I_i=\frac{\text{Δ}{Y}_i/Y}{\text{Δ}{\varphi }_i}=\frac{\left(Y-Y_i\right)/Y}{{\varphi }_i-\varphi \phi }=\frac{\left(Y-Y_i\right)/Y}{M\cdot {{\varphi }^{\prime }}_i},$ (6) where the subscript “i” is the energy region index, Y and Y_i are the yields of ²³⁸Pu before and after flux reduction in the ith energy region, respectively, ϕ and ϕˊ are the grouped neutron flux before and after flux reduction, respectively, and M is the ratio of flux reduction. Previous analyses [18] have shown that the value of M (1/1, 1/2, 1/4, and 1/8) has little influence on the obtained importance curve; therefore, we take M = 1/8 directly.

The entire energy range was divided into 238 regions [21]. The energy division for the filter burnup method has no limitations, thereby achieving a high spectrum resolution. The importance of each energy region throughout the 90-day irradiation period was calculated, building up the importance curve of ²³⁸Pu production, as shown in Figure 4. The ten maxima and minima in the importance curve are listed in Table 1.

Fig. 4Full-screen View

Importance curve of ²³⁸Pu production for the filter burnup method

Table 1Full-screen View

Maxima and minima in the importance curve

Maximums		Minimums
Energy regions (MeV)	Values	Energy regions (MeV)	Values
[1.45×10-6, 1.50×10-6]	1.91×10-15	[1.85×10-5, 1.90×10-5]	-2.34×10-16
[4.50×10-7, 5.00×10-7]	1.28×10-15	[2.77×10-6, 2.87×10-6]	-1.33×10-16
[5.00×10-7, 5.50×10-7]	6.18×10-16	[2.87×10-6, 2.97×10-6]	-7.46×10-17
[1.30×10-6, 1.35×10-6]	5.94×10-16	[5.00×10-2, 5.20×10-2]	-4.64×10-17
[1.40×10-6, 1.45×10-6]	5.18×10-16	[1.15×10-4, 1.19×10-4]	-3.07×10-17
[1.50×10-6, 1.59×10-6]	3.18×10-16	[1.00×101, 1.28×101]	-2.59×10-17
[1.35×10-6, 1.40×10-6]	2.89×10-16	[8.19×100, 1.00×101]	-2.55×10-17
[4.00×10-7, 4.50×10-7]	2.50×10-16	[6.43×100, 8.19×100]	-2.38×10-17
[1.20×10-9, 1.50×10-9]	2.21×10-16	[1.70×10-5, 1.85×10-5]	-2.29×10-17
[1.25×10-6, 1.30×10-6]	2.14×10-16	[1.85×100, 2.35×100]	-2.10×10-17

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The maximum values indicate that the neutrons in these energy regions are productive, that is, increasing the neutron flux in these energy regions will increase the yield of ²³⁸Pu. Meanwhile, the minima indicate that the neutrons in these energy regions are harmful, that is, increasing the neutron flux in these energy regions will decrease the yield of ²³⁸Pu. Therefore, we should increase the neutron flux in the energy regions with maximum values (referred to as the “positive energy region”) and reduce the neutron flux in the energy regions with minimum values (referred to as the “negative energy region”) to promote the production of ²³⁸Pu.

2.2.2

Single-energy Burnup Method

The single-energy burnup method does addition on the neutron spectrum, assuming that the target was irradiated with a single-energy neutron source. We simulated the ²³⁸Pu production efficiency with single-energy neutron sources, that is, the total neutron flux was the same, but all neutrons were in a certain energy region. The efficiency of a certain energy region was quantified by comparing the yields of ²³⁸Pu with those of single-energy neutron sources. The full energy range was divided into 238 regions, and the yields of ²³⁸Pu in each energy region throughout the 90-day irradiation were calculated, thereby constructing the yield curve for the single-energy burnup method, as shown in Fig. 5. The ten maximum and minimum values in the yield curve are given in Table 2.

Fig. 5Full-screen View

Yield curve of ²³⁸Pu production for the single-energy burnup method

Table 2Full-screen View

Maxima and minima in the yield curve

Maximums		Minimums
Energy regions (MeV)	Values	Energy regions (MeV)	Values
[1.45×10-6, 1.50×10-6]	1.17×101	[1.57×101, 1.73×101]	4.60×10-4
[4.50×10-7, 5.00×10-7]	1.09×101	[1.73×101, 2.00×101]	4.60×10-4
[1.30×10-6, 1.35×10-6]	1.01×101	[1.46×101, 1.57×101]	5.24×10-4
[1.40×10-6, 1.45×10-6]	9.94×100	[1.38×101, 1.46×101]	6.04×10-4
[5.00×10-7, 5.50×10-7]	9.84×100	[1.28×101, 1.38×101]	6.69×10-4
[1.50×10-6, 1.59×10-6]	9.10×100	[1.00×101, 1.28×101]	8.55×10-4
[1.35×10-6, 1.40×10-6]	8.89×100	[8.19×100, 1.00×101]	1.06×10-3
[1.25×10-6, 1.30×10-6]	8.29×100	[6.43×100, 8.19×100]	1.51×10-3
[3.73×10-6, 4.00×10-6]	8.14×100	[4.80×100, 6.43×100]	2.97×10-3
[4.00×10-7, 4.50×10-7]	7.80×100	[4.30×100, 4.80×100]	3.99×10-3

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2.2.3

Burnup Extremum Analysis Method

The importance curve obtained using the filter burnup method and the yield curve obtained using the single-energy burnup method exhibited similar variation trends in the resonance energy range, proving the physical nature of Figs. 4 and 5. However, the specific values shown in Figs. 4 and 5 are inconsistent. Eight of the ten maxima in Tables 1 and 2 correspond to the same energy regions, but only three of the ten minima, which is due to the different physical assumptions of the two analysis methods. The filter burnup method perturbs a particular energy spectrum to perform a perturbation analysis that describes the importance of a particular neutron spectrum, whereas the single-energy burnup method is not limited to a particular spectrum. Taking the high-energy regions as an example, the neutrons in these regions are almost useless for ²³⁸Pu production. Therefore, the yield calculated by the single-energy burnup method is zero, and the importance calculated by the filter burnup method is also zero. However, the filter burnup method can also find the energy regions that are negative for ²³⁸Pu production under a particular spectral environment. Therefore, the importance can be negative numbers, which explains why only three of the ten minimum values in Tables 1 and 2 correspond to the same energy regions.

The ²³⁷Np target [22] was irradiated in the neutron flux trap for 90 days using both the filter burnup and single-energy burnup methods. Significant differences in ²³⁸Pu production efficiency exist among different energy regions, resulting in different irradiation times required. For some energy regions, a 90-day irradiation exceeded the time required for the yield to peak, resulting in an increase first and then a decrease in the yield during the entire irradiation period (which is referred to as “excessive irradiation”), whereas in other energy regions, a 90-day irradiation cannot lead to peak production. We used the single-energy burnup method to calculate the yields of ²³⁸Pu with 5-day and 10-day irradiation to investigate the influence of the irradiation time on the yield, as shown in Fig. 6.

Fig. 6Full-screen View

(Color online) Comparison of curves with different irradiation times

As shown in Fig. 6, different irradiation times affected the curves. We proposed a burnup extreme analysis method to eliminate the influence of irradiation time on the importance curve (Fig. 4) and yield curve (Fig. 5), that is, the maximum derivative between the importance (or yield) and irradiation time was calculated and used to quantify the production efficiency in the 238 energy regions, building two extreme curves, as shown in Fig. 7.

Fig. 7Full-screen View

(Color online) Two extreme curves

It can be observed that the two curves have a similar distribution trend, regardless of the analysis method. Therefore, the importance curve in Fig. 4, the yield curve in Fig. 5, and the extreme curves in Fig. 7 have physical significance and can jointly guide the spectrum optimization for ²³⁸Pu production.

Universality Testing

All curves in Sect. 2.2 are calculated based on the HFIR, a thermal reactor. To test the universality of these curves, we performed the same calculations in another reactor with a fast spectrum [23-32], a high-flux lead-bismuth reactor (HFLBR). Detailed parameters of this reactor can be found in a previous study [13]. As previously discussed, the filter burnup method is based on a particular neutron spectrum, whereas the single-energy burnup method is not limited to a particular neutron spectrum. Therefore, the yield curve is independent of the reactor model, whereas the importance curve is related to it. The importance curve of the HFLBR for ²³⁸Pu production was calculated and compared to that of the HFIR, as shown in Fig. 8. The ten maxima and minima in the importance curves are compared in Table 3.

Fig. 8Full-screen View

Importance curves of the HFLBR and HFIR

The two reactors exhibited different neutron spectra and flux levels. As shown in Fig. 8, the two importance curves coincide well in the resonance-energy region and exhibit large deviations in the high- and low-energy regions, proving that the neutron spectrum influences the importance curve. However, as listed in Table 2, nine of the ten maxima and eight of the ten minima correspond to the same energy regions, proving that the importance curves are not completely dependent on the neutron spectrum and are of physical significance, which can be used to determine the positive and negative energy regions. Therefore, the importance curve is universal and can be used for neutron spectrum optimization.

Neutronics Model Verification

We obtained the importance curve (Fig. 4, marked as “I_i”) by the filter burnup method, the yield curve (Figure 5, marked as “Y_i”) by the single-energy burnup method, and the two extremes curves by the burnup extremum analysis method (Fig. 7, marked as “E_i (I_i)” and “E_i (Y_i)”). These curves were used to build a high-resolution neutronics model for ²³⁸Pu production in high-flux reactors, which could be used to guide neutron spectrum optimization to improve production efficiency.

To demonstrate that these curves can be used to guide the neutron spectrum optimization for ²³⁸Pu production in high-flux reactors, we constructed a variety of irradiation schemes by dispersing nuclides into the target to obtain many different neutron spectra. We determined whether the yields of these irradiation schemes were positively correlated with the corresponding total spectral efficiency. The total spectrum efficiency (marked as “T”) of an irradiation scheme was calculated as follows: $T={\displaystyle \sum _{i=1}^N{\varphi }_i\cdot X_i},$ (7) where N is the total number of divided energy regions, ϕ_i is the neutron flux in the i^th energy region, X_i represents the efficiency of the i^th energy region, preferably I_i, Y_i, E_i (I_i), or E_i (Y_i) where T corresponds to T₁, T₂, T₃, and T₄.

The nuclides we selected included ¹⁰⁷Ag, ¹⁵²Eu, ¹⁵⁷Gd, ¹⁷⁰Tm, ¹⁶¹Dy, ¹⁵³Eu, ¹⁵¹Sm, ¹⁰⁸Pd, ¹⁴⁰Ba, ⁴⁰Ar, ⁷Li, ⁶⁴Ni, ⁴⁹Ti, ¹⁵¹Eu, ⁴He, ¹³⁸La, ¹⁴⁷Sm, and ¹⁸⁶W. The number of nuclides added could be 10^-2, 10^-3, 10^-4, and 10^-5 (in (10²⁴#/cm³)). A total of 56 irradiation schemes were constructed. The correlation coefficients between ΔY and ΔT were calculated as follows: $\begin{array}{cc}r\left(\text{Δ}Y,\text{Δ}{T}_n\right)=\frac{\text{Cov}\left(\text{Δ}Y,\text{Δ}{T}_n\right)}{\sqrt{\text{Var}\left[\text{Δ}Y\right]\text{Var}\left[\text{Δ}{T}_n\right]}}& n=1,2,3,4\end{array}$ (8) where ΔY is the variation of ²³⁸Pu yield, ΔT is the variation of the total spectrum efficiency, Cov(x,y) is the covariance of x and y, and Var[x] is the variance of x. The variation in the total spectrum efficiency T of these radiation schemes and the corresponding variation in ²³⁸Pu yield are listed in Table 4.

As can be observed from Table 4, as the filter burnup method is based on a particular neutron spectrum, ΔY is positively correlated with ΔT₁ and ΔT₃, with correlation coefficients larger than 0.9. The single burnup method is not based on a particular neutron spectrum, and hence ΔY is not positively correlated with ΔT₂ and ΔT₄. Therefore, the importance curve based on the filter burnup method and the extreme curve based on the importance curve can be used to guide the optimization of a particular irradiation scheme, that is, to further increase the yield of ²³⁸Pu of a particular irradiation scheme. The yield curve based on the single-energy burnup method and the extreme curve based on the yield curve cannot be used to guide the optimization of a particular irradiation scheme. However, the single-energy burnup method can better determine the positive- and negative-energy regions, particularly when the neutron flux in the energy region accounts for a small amount of the total neutron flux. Therefore, the single-energy burnup method can amplify the difference in importance of the filter burnup method, which is not sufficiently explicit in some energy regions. Consequently, the two methods complement each other.

Neutronics Model Application

As shown by the importance, yield, and extreme curves, neutrons in low-energy regions are conducive to the production of ²³⁸Pu. Therefore, we slowed the neutrons in the target by dispersing ¹H in the target and constructing thermalized neutron energy spectra. The amount of hydrogen added started from 0.0 (10²⁴#/cm³), increased by 0.2 (10²⁴#/cm³) each time, and ended at 3.0 (10²⁴#/cm³), resulting in a total of 16 irradiation schemes (including the original scheme). The results of ΔY(%), T₁ and T₃ are presented in Fig. 9, drawn as a dot plot.

Fig. 9Full-screen View

(Color online) Results of ΔY(%), T₁, and T₃

As shown in Fig. 9, the yield of ²³⁸Pu can be increased by regulating the neutron spectrum around the target. In the above 16 irradiation schemes, the rate of increase in the yield of ²³⁸Pu was the highest when the amount of ¹H added was 2.6 (10²⁴#/cm³), reaching 18.81%. Meanwhile, the rate of increase was positively correlated with the total spectrum efficiency, implying that the yield of ²³⁸Pu can be further improved by a more refined neutron spectrum optimization. Therefore, a high-resolution neutronics model based on the importance curve, yield curve, and two extreme curves can guide the neutron spectrum optimization to improve the production efficiency of ²³⁸Pu.