Rapid diagnostic method for transplutonium isotope production in high flux reactors Enhanced Publication

Rapid diagnostic method for transplutonium isotope production in high flux reactors 1 corresp first-author panqingquan@sjtu.edu.cn, panqingquan@sjtu.edu.cn, panqingquan@sjtu.edu.cn https://orcid.org/0000-0002-9384-0070 1 2 2 2 https://orcid.org/0000-0001-7762-3466 1 corresp xiaojingliu@sjtu.edu.cn xiaojingliu@sjtu.edu.cn xiaojingliu@sjtu.edu.cn School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu 610200, China Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu 610200, China Transplutonium isotopes are scarce and need to be produced by irradiation in high flux reactors. However, their production is inefficient, and optimization studies are necessary. This study analyzes the physical nature of transplutonium isotope production using 252 Cf, 244 Cm, 242 Cm, and 238 Pu as examples. Traditional methods based on the Monte Carlo burnup calculation have the limitations of many calculations and cannot analyze the individual energy intervals in detail; thus, they cannot support the refined evaluation, screening, and optimization of the irradiation schemes. After understanding the physical nature and simplifying the complexity of the production process, we propose a rapid diagnostic method for evaluating radiation schemes based on the concepts "single energy interval value (SEIV)" and "Energy Spectrum Total Value (ESTV)". The rapid diagnostic method not only avoids tedious burnup calculations but also provides a direction for optimization. The optimal irradiation schemes for producing 252 Cf, 244 Cm, 242 Cm, and 238 Pu are determined based on a rapid diagnostic method. Optimal irradiation schemes can significantly improve production efficiency. Compared with the initial scheme, the optimal scheme improved the production efficiency of 238 Pu by 7.41 times; 242 Cm, 11.98 times; 244 Cm, 65.20 times; and 252 Cf, 15.08 times. Thus, a refined analysis of transplutonium isotope production is conducted and provides a theoretical basis for improving production efficiency. Transplutonium isotope Rapid diagnostic method Production optimization Single energy interval value Energy spectrum total value INTRODUCTION 1 Transplutonium isotopes [ 1 1 ] refer to nuclides with an atomic number greater than or equal to 94 (plutonium elements). Most are radioactive isotopes with unstable nuclei that can spontaneously release rays ( α , β , γ ) or neutrons by spontaneous fission. Transplutonium isotopes mainly include the 94 th element plutonium (Pu), 95 th element americium (Am), 96 th element 96 curium (Cm), 97 th element berkelium (Bk), and 98 th element californium (Cf). All transplutonium isotopes are metals ( Figure 1 Figure 1 ). Transplutonium isotopes have attracted extensive scientific attention [ 2 2 - 4 4 2-4 ]. Transplutonium isotopes are scarce and more expensive than medical isotopes [ 5 5 - 8 8 5-8 ]. For example, 252 Cf is the most expensive element, at USD 27m a gram, a price 650,000 times more than that of gold. 252 Cf is a strong neutron source, emitting 2.35×10 6 neutrons per microgram per second with a half-life of 2.65 years. 252 Cf releases neutrons in a spectrum similar to that of a fission reactor; thus, it is an ideal source of neutrons for reactor start-up [ 9 9 ]. 252 Cf is also effective for the treatment of cervical cancer [ 10 10 ], oil exploration [ 11 11 ], and prompt gamma neutron activation analysis (PGNAA) [ 12 12 ]. 238 Pu is widely used in radioisotope thermoelectric generators because of its moderate half-life, high thermal power density, and easy radiation protection from α decay [ 13 13 ]. 242 Cm and 244 Cm are portable α -ray sources that can power satellites and spacecraft and are used as target materials to produce 252 Cf [ 14 14 ]. Most transplutonium isotopes do not exist naturally and must be produced [ 15 15 - 18 18 15-18 ]. The three methods of production are 1) in accelerators [ 19 19 ], 2) during a thermonuclear explosion [ 20 20 ], and 3) during irradiation in a high flux reactor [ 21 21 ]. Among these methods, irradiation in a high flux reactor is the most stable and efficient production method and the only way to produce 252 Cf in commercial operation. Only two laboratories worldwide can stably produce 252 Cf using a high flux irradiated curium target [ 22 22 ]: the Oak Ridge National Laboratory (ORNL) in the United States and the Research Institute of Atomic Reactors (RIAR) in Russia. However, the worldwide production of transplutonium isotopes remains low. In 78 production runs over the past 60 years [ 23 23 ], only 1.2 g 249 Bk, 10.2 g 252 Cf, 39 mg 253 Es-, and 15 pg 257 Fm were produced using the high flux isotope reactor (HFIR) [ 24 24 ] at the Radiochemical Engineering Development Center (REDC) [ 25 25 ] in the United States. Neutron capture and decay reactions occur during production by reactor irradiation [ 26 26 - 28 28 26-28 ]. For each neutron absorbed by a nuclide, the atomic number of the nuclide increases by one, becoming a new nuclide. Therefore, the target material changes from nuclides with small mass numbers to nuclides with large mass numbers through a series of nuclear reactions [ 29 29 - 30 30 29-30 ] (radiation capture, β -decay). Figure 2 Figure 2 shows the nuclide conversion process for producing transplutonium isotopes in a high flux reactor [ 31 31 ]. In addition to the absorption reaction, fission occurs [ 32 32 - 33 33 32-33 ]. Once the fission reaction occurs, the production of transplutonium isotopes stops, causing fission losses and leading to a low conversion rate. Therefore, the fission reaction should be reduced, and the absorption reaction should be increased to improve production efficiency [ 34 34 - 35 35 34-35 ]. Because the cross-section is related to the neutron spectrum [ 36 36 - 37 37 36-37 ], the neutron spectrum around the target should be studied. The neutron spectrum around the target greatly influences production efficiency [ 38 38 ]. Based on our literature review, answering two questions would improve the production efficiency of transplutonium isotopes: 1) how to find the optimal neutron spectrum and 2) how to achieve the optimal neutron spectrum. The background of the optimization of transplutonium isotope production is shown in Fig. 3 Fig. 3 . In answering the first question, determining the optimal neutron spectrum is difficult because of the high complexity of the production process, and no corresponding neutronics model can analyze each energy interval individually. Traditional methods search for the optimal neutron spectrum by a large number of Monte Carlo burnup calculations. The burnup equation is independent of energy, making multi-group burnup calculation impossible. Therefore, traditional methods require many computations and can only macroscopically analyze the whole energy spectrum, not the particular energy intervals in detail; thus, they cannot support the refined evaluation, screening, and optimization of the irradiation scheme. Therefore, a rapid diagnostic method for transplutonium production is necessary. Understanding the essence of the production process and simplifying the complexity are necessary to refine neutronics analysis to support the optimization, improve production efficiency, and reduce production cost. Notable in answering the second question is that the neutron spectrum has a complex relationship with the design parameters; thus, performing reactor physics calculations [ 39 39 - 42 42 39-42 ] and sensitivity analysis is necessary. After the sensitivity analysis of some design parameters for production efficiency, the optimal neutron spectrum can be achieved by optimizing these design parameters, such as the 1) selection of the moderator, 2) layout of the moderator, 3) position of the target, 4) size of the target, and 5) shape of the target. We attempted to adjust these parameters to obtain a production scheme with high efficiency by adjusting these parameters to obtain an optimal neutron spectrum for producing transplutonium isotopes. This study analyzes the optimization method of transplutonium isotope production by using 252 Cf, 244 Cm, 242 Cm, and 238 Pu as examples. By proposing the concepts of “Single Energy Interval Value (SEIV)” and “Energy Spectrum Total Value (ESTV)”, we establish a rapid diagnostic method and realize the detailed analysis of the energy spectrum around the target. The rapid diagnostic method not only avoids tedious burnup calculations but also provides a direction for optimizing the irradiation scheme. Subsequently, an optimal irradiation scheme is proposed after sensitivity analysis based on the rapid diagnostic method. The remainder of this paper is organized as follows: Sect. 2 introduces the rapid diagnostic method, Sect. 3 provides numerical and experimental verification, Sect. 4 introduces the optimal process based on sensitivity analysis, and Sect. 5 provides a conclusion. RAPID DIAGNOSTIC METHOD 1 High Flux Reactor 2 All analyses are performed in a high flux lead-bismuth reactor under design. It has a 90-day refueling cycle with the highest flux of 6.7×10 15 . The inlet temperature is 170℃, the outlet temperature is 536.5℃, and coolant velocity is 4.0 m/s. 208 Pb-Bi is used as the coolant, and 208 Pb is used as the reflector layer. Figure 4 Figure 4 shows the X-Y and X-Z sections of the reactor. The fuel rod is a simplified model consisting of fuel pellets, an air gap, and cladding, with a total length of 50 cm. The fuel assembly has a hexagonal design and contains 91 fuel rods. The geometry of the fuel rod and fuel assembly are shown in Figure 5 Figure 5 . Detailed parameters of the high flux reactor are listed in Table 1 Table 1 . 1 1 Parameters 1 1 Value 1 1 Thermal power (MW) 1 1 150 1 1 Loading capacity of Fuel/U-235 (kg) 1 1 779 /175.3 1 1 Equivalent diameter of active zone (cm) 1 1 58.14 1 1 Height of active zone (cm) 1 1 50 1 1 Average line power density (W/cm) 1 1 302.45 1 1 Maximum line power density (W/cm) 1 1 404.20 1 1 Average volume power density (W/cm 3 ) 1 1 1130 1 1 Fuel material 1 1 U-10% Zr alloy 1 1 Enrichment of 235 U 1 1 25 % 1 1 Inner/outer diameter of fuel rod (mm) 1 1 4/4.6 1 1 Filled gas in the gaps 1 1 Helium 1 1 Fuel rod clearance width (mm) 1 1 0.1 1 1 Cladding material 1 1 T91 1 1 Thickness of the cladding (mm) 1 1 0.2 1 1 Ratio of pitch to diameter (P/D) 1 1 1.1304 1 1 Number of assemblies 1 1 109 1 1 Number of fuel rods per assembly 1 1 91 1 1 Axial/radial thickness of reflector layers (cm) 1 1 80/120 The target for producing 252 Cf is a mixture of plutonium, americium, and curium [ 9 9 ]. The nuclide composition of the target is shown in Table 2 Table 2 . The target is irradiated for five cycles of 25 d of irradiation each. This target irradiation is representative of typical operations during a californium production campaign. 1 1 Nuclide 1 1 Weight (g) 1 1 Nuclide 1 1 Weight (g) 1 1 238 Pu 1 1 0.002 1 1 244 Cm 1 1 8.734 1 1 239 Pu 1 1 0.001 1 1 245 Cm 1 1 0.271 1 1 240 Pu 1 1 0.565 1 1 246 Cm 1 1 22.349 1 1 242 Pu 1 1 0.016 1 1 247 Cm 1 1 0.647 1 1 243 Am 1 1 3.903 1 1 248 Cm 1 1 4.743 Production Evaluation by Monte Carlo Burnup Calculation 2 The yields of transplutonium isotopes can be obtained by performing the Monte Carlo burnup calculation, a coupling of the Monte Carlo criticality calculation and the burnup calculation. Figure 6 Figure 6 shows the flowchart for evaluating transplutonium isotope production using the Monte Carlo burnup calculation. The point burnup equation describes the transmutation of nuclides over time when the 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 <math display="block" id="M1"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>n</mi><mi>i</mi></msub></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow></munder><mrow><msubsup><mi>b</mi><mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow><mrow><mtext>eff</mtext></mrow></msubsup></mrow></mstyle><msubsup><mi>λ</mi><mi>j</mi><mrow><mtext>eff</mtext></mrow></msubsup><msub><mi>n</mi><mi>j</mi></msub><mo>−</mo><msubsup><mi>λ</mi><mi>i</mi><mrow><mtext>eff</mtext></mrow></msubsup><msub><mi>n</mi><mi>i</mi></msub><mo>,</mo></mrow></math> 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 of the transmutation of the i th nuclide to the j th nuclide. λ i eff and b i , j eff can be calculated using the following formula: <math display="block" id="M2"><mrow><mrow><mo>{</mo><mtable columnalign="left"><mtr><mtd><msubsup><mi>λ</mi><mi>i</mi><mrow><mtext>eff</mtext></mrow></msubsup><mo>=</mo><msub><mi>λ</mi><mi>i</mi></msub><mo>+</mo><mi>ϕ</mi><mstyle displaystyle="true"><munder><mo>∑</mo><mi>j</mi></munder><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><msubsup><mi>b</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mtext>eff</mtext></mrow></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>b</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>λ</mi><mi>i</mi></msub><mo>+</mo><msub><mi>σ</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mi>ϕ</mi></mrow><mo>)</mo></mrow><mo>/</mo><msubsup><mi>λ</mi><mi>i</mi><mrow><mtext>eff</mtext></mrow></msubsup></mtd></mtr></mtable></mrow></mrow></math> 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 i th nuclide’s reaction generates the j th nuclide. During the burnup calculation, the one-group cross-section and neutron flux within each burnup step are assumed to be constant so that the point burnup equation can be treated as a first-order linear ordinary differential equation. We wrote this into a matrix form and obtained an expression more concise than the prior expression: <math display="block" id="M3"><mrow><mfrac><mrow><mtext>d</mtext><mover accent="true"><mi>n</mi><mo>→</mo></mover></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mi>A</mi></mstyle><mover accent="true"><mi>n</mi><mo>→</mo></mover><mo>,</mo></mrow></math> where A represents the coefficient matrix of the N order point burnup equation, which is regarded as a constant matrix in a single burnup step. Because of initial boundary conditions, the solution to Eq. (3) can be written in the form of an exponential matrix: <math display="block" id="M4"><mrow><mover accent="true"><mi>n</mi><mo>→</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>A</mi></mstyle><mi>t</mi></mrow></msup><msub><mover accent="true"><mi>n</mi><mo>→</mo></mover><mn>0</mn></msub><mo>,</mo></mrow></math> where matrix A t is referred to as the burnup matrix for the convenience of quotation. Because of the extremely low yields of transplutonium isotopes and the complex nuclide composition of the target, the point burnup model usually adopts a complex nuclide system, which needs to cover more than 1,000 isotopes, including many short half-life nuclides. For example, the ORIGEN-S burnup database [ 43 43 ] contains approximately 1,500 nuclides. Some of these nuclides, such as 212 Po, have half-life of 10 -7 s. Therefore, the point burnup equations for transplutonium isotope production are not only large in number but also rigid, which makes the solution difficult. As shown in Eq. (2), the cross-section of the target is required for the Monte Carlo burnup equation; therefore, a criticality calculation should be performed. The k -eigenvalue neutron transport equation can be written as [ 44 44 ]: <math display="block" id="M5"><mrow><mrow><mo>(</mo><mrow><mi>L</mi><mo>+</mo><mi>C</mi><mo>−</mo><mi>S</mi></mrow><mo>)</mo></mrow><mi>ϕ</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mtext>eff</mtext></mrow></mfrac><mi>M</mi><mi>ϕ</mi><mo>,</mo></mrow></math> where L is the leakage operator, C is the collision operator, S is the scattering operator, M is the fission operator, k eff is the effective multiplication factor, and ϕ is neutron flux. The physical parameters around the target, such as the neutron flux, fission reaction rate, and absorption reaction rate, can be obtained by solving Eq. (5). The Monte Carlo method uses the following formula for calculating the reaction rates: <math display="block" id="M6"><mrow><msub><mi>Σ</mi><mi>g</mi></msub><msub><mi>ϕ</mi><mi>g</mi></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>E</mi><mi>g</mi></msub></mrow><mrow><msub><mi>E</mi><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mtext>d</mtext><mi>E</mi></mrow></mrow></mstyle><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>V</mi></msub><mrow><mtext>d</mtext><mi>V</mi><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><mi>W</mi><mo>⋅</mo><mi>T</mi><msubsup><mi>L</mi><mi>V</mi><mi>i</mi></msubsup></mrow></mstyle><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mrow><mrow><mi>V</mi><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><msubsup><mi>W</mi><mn>0</mn><mi>i</mi></msubsup></mrow></mstyle></mrow></mfrac><mo>,</mo></mrow></math> where ∑ g represents the macroscopic cross-section in the g th energy interval, and ∑ represents the cross-section of nuclides related to position r and energy E . Therefore, we have the following formula for calculating the cross-section required by the Monte Carlo burnup calculation: <math display="block" id="M7"><mrow><msub><mi>Σ</mi><mi>g</mi></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>E</mi><mi>g</mi></msub></mrow><mrow><msub><mi>E</mi><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mtext>d</mtext><mi>E</mi></mrow></mrow></mstyle><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>V</mi></msub><mrow><mtext>d</mtext><mi>V</mi><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><mi>W</mi><mo>⋅</mo><mi>T</mi><msubsup><mi>L</mi><mi>V</mi><mi>i</mi></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>E</mi><mi>g</mi></msub></mrow><mrow><msub><mi>E</mi><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mtext>d</mtext><mi>E</mi></mrow></mrow></mstyle><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>V</mi></msub><mrow><mtext>d</mtext><mi>V</mi><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><mi>W</mi><mo>⋅</mo><mi>T</mi><msubsup><mi>L</mi><mi>V</mi><mi>i</mi></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></mrow></mstyle></mrow></mfrac><mo>.</mo></mrow></math> The aforementioned formulas show that the Monte Carlo burnup calculation is complex and costly [ 45 45 , 46 46 ]. Testing in the high flux reactor ( Fig. 4 Fig. 4 ), with the calculation parameters of 1,000,000 neutrons per cycle, 100 inactive cycles, 300 active cycles, and 10 burnup steps, demonstrated that the duration to perform one Monte Carlo burnup calculation with 64 threads on the 2nd Gen AMD EPYC™ 7742 was 150 min. Even when only considering the production of 252 Cf, 244 Cm, 242 Cm, and 238 Pu, and only analyzing the five design parameters (shown in Figure 9 Figure 9 ), a total of 9720 calculations are necessary, with a computing time of 1458,000 min (1012.5 d). This amount of computation time is unacceptable. Moreover, because the Monte Carlo burnup calculation only requires a one-group cross-section, the calculations only provide the influence of the entire neutron spectrum on the production efficiency, not a closer analysis of each single energy interval. Therefore, when optimizing the neutron spectrum, only the macroscopic design parameters can be obtained, leaving unknown which energy intervals are favorable or unfavorable to production; thus, more detailed guidance for the optimization cannot be provided. Therefore, understanding the physical nature of the production process and finding a new rapid diagnostic method that can simplify the calculation and analyze the production process in more detail are necessary. Moreover, a rapid diagnostic method is helpful for evaluating, screening, and optimizing irradiation schemes. Rapid Diagnostic Method Based on SEIV and ESTV 2 The root cause of the low production of transplutonium isotopes is fission loss [ 14 14 ]. When the target is irradiated in a high flux reactor, the target material changes from nuclides with a small mass number to nuclides with a large mass number through absorption reactions. In addition to the absorption reaction, fission reactions occur. Once the fission reaction occurs, the production of transplutonium isotopes stops, causing fission losses and leading to a low conversion rate. Therefore, the ratio of the absorption rate to the fission rate ( A / F ) is an important physical quantity affecting production efficiency. Production efficiency can be assessed by analyzing the A / F s of nuclides in the nuclide chain. Moreover, A / F can be calculated in individual energy intervals, achieving a refined neutronics analysis. For example, regarding the production of 252 Cf from the target described in Table 2 Table 2 , Figure 7 Figure 7 shows the A / F s of intermediate nuclides 244 Cm, 245 Cm, 246 Cm, 247 Cm, 248 Cm, 249 Bk, 250 Cf, and 251 Cf. Nuclear data were obtained from the JANIS database [ 47 47 ]. As shown in Fig. 7 Fig. 7 , the A/Fs of different nuclides vary greatly, as do the A / F s of different energy intervals; thus, different nuclides and different energy intervals have significantly different effects on production efficiency. Analyzing only the entire energy spectrum will not yield an accurate result; thus, a detailed analysis of each energy interval is required. In this study, a rapid diagnostic method based on the neutron spectrum around the target is proposed to realize a rapid, refined evaluation of different production schemes. The model only needs to obtain the neutron spectrum around the target by using the Monte Carlo criticality calculation [ 48 48 - 49 49 48-49 ] to evaluate the efficiency of different production schemes, avoid the burnup calculation, and greatly reduce the required computing resources. Therefore, the rapid diagnostic method can quickly screen out the optimal scheme for transplutonium isotope production and help provide a direction for optimizing the irradiation scheme. The model is divided into four steps: 1) obtain the neutron spectrum around the target by using the Monte Carlo criticality calculation, which can be described by the 46-group neutron flux, the 46-group fission rate, and the 46-group absorption rate; 2) define the physical quantity “Single Energy Interval Value (SEIV)” and calculate the SEIV of these 46 energy intervals; 3) define the physical quantity “Energy Spectrum Total Value (ESTV)” and use the 46-group neutron flux and the 46-group SEIV to calculate the ESTV of the neutron spectrum; and 4) evaluate the efficiency of different schemes by comparing the ESTV. Those with a higher ESTV are considered to have higher production efficiency and yield. A schematic of this process is shown in Figure 8 Figure 8 . The Single Energy Interval Value (SEIV) is calculated by <math display="block" id="M8"><mrow><msub><mrow><mtext>SEIV</mtext></mrow><mi>i</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mrow><mi>a</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><mrow><msub><mi>R</mi><mrow><mi>f</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><msub><mi>R</mi><mrow><mi>a</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow><mrow><msub><mi>ϕ</mi><mi>i</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> where the subscript “ i ” represents the serial number of the energy intervals, R a,i represents the absorption rate in the i th energy interval, R f,i represents the fission rate in the i th energy interval, and ϕ i represents the neutron flux in the i th energy interval. The SEIV is the product of the two expressions. The first expression is A/F, indicating whether it tends to undergo an absorption or fission reaction within this energy interval. The other expression is the ratio of the absorption rate to the neutron flux, which indicates the probability of an absorption reaction occurring in this energy interval. Therefore, a higher SEIV means that within this energy interval, absorption reactions tend to occur rather than fission reactions, and more absorption reactions occur. An energy interval with a high SEIV is considered important in the production process, and the neutron flux in this energy interval should be increased as much as possible. The energy spectrum total value (ESTV) is calculated by <math display="block" id="M9"><mrow><mtext>ESTV</mtext><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>46</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><msub><mrow><mtext>SEIV</mtext></mrow><mi>i</mi></msub><mo>×</mo><msub><mi>ϕ</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mstyle><mo>.</mo></mrow></math> The ESTV is the integral value of the SEIV along with the neutron spectrum, indicating the target nuclide’s overall trend of absorbing neutrons without fission. We posit that a production scheme with a higher ESTV produces transplutonium isotopes more efficiently. The ESTV is only related to the neutron spectrum around the target, and the burnup calculation is no longer required; therefore, the ESTV can be used for the rapid evaluation of the production scheme. NUMERICAL AND EXPERIMENTAL VERIFICATION 1 Numerical Verification 2 The rapid diagnostic method is verified by comparing the burnup calculation with ESTV. The production efficiency is optimized by adjusting five design parameters: 1) moderator selection, 2) moderator layout, 3) target position, 4) target size, and 5) target shape. The optional values for these five design parameters are shown in Figure 9 Figure 9 . A large hexagonal prism target with Position #1 as the initial scheme and zirconium hydride as the moderator is used. Different irradiation schemes are obtained by modifying the moderator layout. Criticality calculations are used to obtain the neutron spectrum around the target. The SEIV and ESTV are determined and used to evaluate the efficiency of the irradiation schemes. 238 Pu, 242 Cm, 244 Cm, and 252 Cf are examples. The target for producing 238 Pu is 237 Np. The target for producing 242 Cm is 241 Am. The target for producing 244 Cm is 239 Pu. The nuclide composition of the target used for producing 252 Cf is listed in Table 2 Table 2 . The SEIV and ESTV for the target without the moderator (bare target) and the six layouts of the moderator are calculated; thus, there are seven production schemes for each transplutonium isotope to be produced. The SEIV for the production of these four transplutonium isotopes is presented in Table 3 Table 3 . The ESTV for the production of these four transplutonium isotopes is presented in Table 4 Table 4 . 1 1 Energy (MeV) 1 1 238 Pu 1 1 242 Cm 1 1 244 Cm 1 1 252 Cf 1 1 [1.00×10 -11 , 1.00×10 -7 ] 1 1 4.55×10 4 1 1 3.0×10 3 1 1 4.96 1 1 2.42 1 1 [1.00×10 -7 , 4.14×10 -7 ] 1 1 2.78×10 4 1 1 2.10×10 3 1 1 1.10×10 1 1 1 2.53 1 1 [4.14×10 -7 , 8.76×10 -7 ] 1 1 6.99×10 4 1 1 1.88×10 3 1 1 9.02×10 -1 1 1 1.75×10 1 1 1 [8.76×10 -7 , 1.86×10 -6 ] 1 1 1.93×10 4 1 1 4.87×10 2 1 1 7.34×10 -2 1 1 1.16×10 2 1 1 [1.86×10 -6 , 5.04×10 -6 ] 1 1 3.89×10 3 1 1 3.17×10 2 1 1 1.89×10 -2 1 1 1.48 1 1 [5.04×10 -6 , 1.07×10 -5 ] 1 1 6.12×10 3 1 1 1.13×10 2 1 1 1.17×10 -1 1 1 9.41 1 1 [1.07×10 -5 , 3.73×10 -5 ] 1 1 1.50×10 3 1 1 1.24×10 2 1 1 2.15×10 -1 1 1 4.46 1 1 [3.73×10 -5 , 1.01×10 -4 ] 1 1 2.48×10 2 1 1 1.21×10 2 1 1 1.52×10 -1 1 1 1.91 1 1 [1.01×10 -4 , 2.14×10 -4 ] 1 1 1.48×10 2 1 1 1.78×10 2 1 1 1.70×10 -1 1 1 1.90 1 1 [2.14×10 -4 , 4.54×10 -4 ] 1 1 1.61×10 2 1 1 1.34×10 2 1 1 2.62×10 -1 1 1 1.90 1 1 [4.54×10 -4 , 1.58×10 -3 ] 1 1 1.55×10 2 1 1 7.46×10 1 1 1 1.16×10 -1 1 1 1.73 1 1 [1.58×10 -3 , 3.35×10 -3 ] 1 1 7.21×10 1 1 1 4.83×10 1 1 1 1.19×10 -1 1 1 7.63×10 -1 1 1 [3.35×10 -3 , 7.10×10 -3 ] 1 1 4.26×10 1 1 1 2.64×10 1 1 1 8.62×10 -2 1 1 5.65×10 -1 1 1 [7.10×10 -3 , 1.50×10 -2 ] 1 1 2.62×10 1 1 1 2.02×10 1 1 1 3.31×10 -2 1 1 4.08×10 -1 1 1 [1.50×10 -2 , 2.19×10 -2 ] 1 1 1.89×10 1 1 1 1.61×10 1 1 1 1.62×10 -2 1 1 3.14×10 -1 1 1 [2.19×10 -2 , 2.42×10 -2 1 1 1.65×10 1 1 1 1.41×10 1 1 1 1.36×10 -2 1 1 2.83×10 -1 1 1 [2.42×10 -2 , 2.61×10 -2 ] 1 1 1.56×10 1 1 1 1.35×10 1 1 1 1.15×10 -2 1 1 2.67×10 -1 1 1 [2.61×10 -2 , 3.18×10 -2 ] 1 1 1.50×10 1 1 1 1.32×10 1 1 1 9.06×10 -3 1 1 2.46×10 -1 1 1 [3.18×10 -2 , 4.09×10 -2 ] 1 1 1.22×10 1 1 1 1.33×10 1 1 1 6.17×10 -3 1 1 2.12×10 -1 1 1 [4.09×10 -2 , 6.74×10 -2 ] 1 1 8.42 1 1 1.05×10 1 1 1 3.88×10 -3 1 1 1.49×10 -1 1 1 [6.74×10 -2 , 1.11×10 -1 ] 1 1 4.25 1 1 6.26 1 1 2.01×10 -3 1 1 7.85×10 -2 1 1 [1.11×10 -1 , 1.83×10 -1 ] 1 1 1.71 1 1 3.22 1 1 1.48×10 -3 1 1 4.19×10 -2 1 1 [1.83×10 -1 , 2.97×10 -1 ] 1 1 5.54×10 -1 1 1 1.31 1 1 1.19×10 -3 1 1 2.19×10 -2 1 1 [2.97×10 -1 , 3.69×10 -1 ] 1 1 1.89×10 -1 1 1 4.81×10 -1 1 1 8.20×10 -4 1 1 1.34×10 -2 1 1 [3.69×10 -1 , 4.98×10 -1 ] 1 1 4.20×10 -2 1 1 2.08×10 -1 1 1 4.86×10 -4 1 1 7.76×10 -3 1 1 [4.98×10 -1 , 6.08×10 -1 ] 1 1 8.78×10 -3 1 1 7.30×10 -2 1 1 2.75×10 -4 1 1 3.71×10 -3 1 1 [6.08×10 -1 , 7.43×10 -1 ] 1 1 3.46×10 -3 1 1 2.23×10 -2 1 1 1.48×10 -4 1 1 1.85×10 -3 1 1 [7.43×10 -1 , 8.21×10 -1 ] 1 1 1.85×10 -3 1 1 7.91×10 -3 1 1 6.30×10 -5 1 1 1.12×10 -3 1 1 [8.21×10 -1 , 1.00] 1 1 1.23×10 -3 1 1 4.01×10 -3 1 1 2.94×10 -5 1 1 7.19×10 -4 1 1 [1.00, 1.35] 1 1 5.54×10 -4 1 1 1.45×10 -3 1 1 1.12×10 -5 1 1 4.20×10 -4 1 1 [1.35, 1.65] 1 1 2.45×10 -4 1 1 6.23×10 -4 1 1 4.44×10 -6 1 1 3.30×10 -4 1 1 [1.65, 1.92] 1 1 1.39×10 -4 1 1 3.14×10 -4 1 1 2.50×10 -6 1 1 2.82×10 -4 1 1 [1.92, 2.23] 1 1 9.06×10 -5 1 1 1.39×10 -4 1 1 1.43×10 -6 1 1 1.93×10 -4 1 1 [2.23, 2.35] 1 1 6.45×10 -5 1 1 7.85×10 -5 1 1 9.42×10 -7 1 1 1.35×10 -4 1 1 [2.35, 2.37] 1 1 5.95×10 -5 1 1 6.54×10 -5 1 1 8.18×10 -7 1 1 1.20×10 -4 1 1 [2.37, 2.47] 1 1 5.53×10 -5 1 1 5.63×10 -5 1 1 7.20×10 -7 1 1 1.10×10 -4 1 1 [2.47, 2.73] 1 1 4.52×10 -5 1 1 3.94×10 -5 1 1 5.13×10 -7 1 1 8.03×10 -5 1 1 [2.73, 3.01] 1 1 3.26×10 -5 1 1 2.00×10 -5 1 1 3.04×10 -7 1 1 4.89×10 -5 1 1 [3.01, 3.68] 1 1 2.31×10 -5 1 1 7.95×10 -6 1 1 1.85×10 -7 1 1 2.09×10 -5 1 1 [3.68, 4.97] 1 1 1.18×10 -5 1 1 1.17×10 -6 1 1 1.10×10 -7 1 1 3.64×10 -6 1 1 [4.97, 6.07] 1 1 5.27×10 -6 1 1 1.05×10 -7 1 1 3.67×10 -8 1 1 4.23×10 -7 1 1 [6.07, 7.41] 1 1 2.35×10 -6 1 1 1.77×10 -8 1 1 4.22×10 -9 1 1 1.37×10 -7 1 1 [7.41, 8.61] 1 1 1.17×10 -6 1 1 9.91×10 -9 1 1 4.80×10 -9 1 1 8.47×10 -8 1 1 [8.61, 1.00×10 1 ] 1 1 7.86×10 -7 1 1 1.13×10 -8 1 1 2.56×10 -8 1 1 8.52×10 -8 1 1 [1.00×10 1 , 1.22×10 1 1 1 5.51×10 -7 1 1 0 1 1 1.19×10 -7 1 1 8.78×10 -8 1 1 [1.22×10 1 , 1.42×10 1 ] 1 1 0 1 1 6.32×10 -9 1 1 1.32×10 -7 1 1 0 1 1 Schemes 1 1 238 Pu 1 1 242 Cm 1 1 244 Cm 1 1 252 Cf 1 1 Bare target 1 1 4.50×10 -3 1 1 5.64×10 -3 1 1 3.92×10 -6 1 1 9.22×10 -5 1 1 Layout#1 1 1 1.36×10 -1 1 1 1.92×10 -2 1 1 2.55×10 -5 1 1 4.41×10 -4 1 1 Layout#2 1 1 6.78×10 -1 1 1 4.05×10 -2 1 1 5.00×10 -5 1 1 1.08×10 -3 1 1 Layout#3 1 1 1.60 1 1 7.14×10 -2 1 1 7.59×10 -5 1 1 1.72×10 -3 1 1 Layout#4 1 1 5.46 1 1 1.74×10 -1 1 1 1.77×10 -4 1 1 4.39×10 -3 1 1 Layout#5 1 1 6.26 1 1 1.99×10 -1 1 1 2.01×10 -4 1 1 4.94×10 -3 1 1 Layout#6 1 1 1.11×10 1 1 1 2.94×10 -1 1 1 2.42×10 -4 1 1 4.59×10 -3 As shown in Table 3 Table 3 , the optimal neutron spectra for producing 238 Pu, 242 Cm, 244 Cm, and 252 Cf are not always softer. The energy interval [4.14×10 -7 MeV, 8.76×10 -6 MeV] has the highest SEIV for producing 238 Pu. The energy interval [1×10 -11 MeV, 1×10 -7 MeV] has the highest SEIV for producing 242 Cm. The energy interval [1×10 -7 MeV, 4.14×10 -7 MeV] has the highest SEIV for producing 244 Cm. The energy interval [8.76×10 -7 MeV, 1.86×10 -6 MeV] has the highest SEIV for producing 252 Cf. Moreover, the SEIV varied significantly at different energy intervals. For example, the maximum deviation of the SEIV in different energy intervals can be up to 1.37×10 9 times for producing 252 Cf. Therefore, the efficiency of transplutonium isotopes can be improved by modifying the neutron spectrum around a target [ 50 50 - 51 51 50-51 ]. As shown in Table 4 Table 4 , the ESTV varies significantly for the different irradiation schemes. For example, the maximum deviation of the ESTV can be up to 21.3 times for producing 252 Cf. Moreover, the bare target always has the lowest ESTV regardless of the isotope produced. Therefore, the production of transplutonium isotopes requires the placement of a moderator around the target. Layout #6 of the moderator has the highest ESTV for producing 238 Pu, 242 Cm, and 244 Cm, and Layout #5 has the highest ESTV for producing 252 Cf. Because the SEIV and ESTV can be used to analyze the production efficiency of each energy interval in detail, the SEIV and ESTV are helpful in providing direction for optimizing the irradiation scheme. Comparing the different irradiation schemes using only SEIV and ESTV is not rigorous; therefore, this method needs to be verified. To verify the effectiveness of the rapid diagnostic method, burnup calculations are performed with RMC code [ 52 52 ] to calculate the yields of 238 Pu, 242 Cm, 244 Cm, and 252 Cf. Because the neutron spectrum changes during the burnup process, first, the one-second burnup process was stimulated to ensure that the neutron spectrum used to calculate the ESTV was consistent with the neutron spectrum used to calculate the yields. In the 1 s burnup process, the relationship between the ESTV and the yields with a fixed neutron spectrum is clarified. The yields at 1 s for 238 Pu, 242 Cm, 244 Cm, and 252 Cf are presented in Table 5 Table 5 . 1 1 Schemes 1 1 238 Pu 1 1 242 Cm 1 1 244 Cm 1 1 252 Cf 1 1 Bare target 1 1 5.03×10 -13 1 1 1.53×10 -12 1 1 1.68×10 -45 1 1 1.94×10 -37 1 1 Layout#1 1 1 1.07×10 -12 1 1 2.29×10 -12 1 1 4.96×10 -45 1 1 6.90×10 -37 1 1 Layout#2 1 1 1.86×10 -12 1 1 4.20×10 -12 1 1 1.07×10 -44 1 1 1.32×10 -36 1 1 Layout#3 1 1 2.86×10 -12 1 1 6.75×10 -12 1 1 1.92×10 -44 1 1 1.79×10 -36 1 1 Layout#4 1 1 6.78×10 -12 1 1 1.62×10 -11 1 1 2.44×10 -43 1 1 1.38×10 -35 1 1 Layout#5 1 1 7.62×10 -12 1 1 1.80×10 -11 1 1 2.85×10 -43 1 1 1.46×10 -35 1 1 Layout#6 1 1 1.05×10 -11 1 1 2.64×10 -11 1 1 1.60×10 -43 1 1 5.79×10 -36 As shown in Table 5 Table 5 , regardless of the isotope produced, the yield in 1 s increases as the ESTV increases. Therefore, ESTV can be used to evaluate the yield with the same neutron spectrum. However, the one-second burnup only describes the physical process at the early stage of irradiation and cannot describe the entire irradiation cycle. For the assessment of whether the ESTV can indicate the yield of the whole irradiation cycle, the burnup process of the target irradiation for 90 d is also simulated. The yields at 90 d for 238 Pu, 242 Cm, 244 Cm, and 252 Cf are given in Table 6 Table 6 . 1 1 Schemes 1 1 238 Pu 1 1 242 Cm 1 1 244 Cm 1 1 252 Cf 1 1 Bare target 1 1 1.65 1 1 1.28 1 1 1.96×10 -2 1 1 7.95×10 -3 1 1 Layout#1 1 1 3.03 1 1 1.74 1 1 4.51×10 -2 1 1 2.07×10 -2 1 1 Layout#2 1 1 4.64 1 1 2.68 1 1 7.91×10 -2 1 1 3.42×10 -2 1 1 Layout#3 1 1 6.07 1 1 3.58 1 1 2.98×10 -1 1 1 4.20×10 -2 1 1 Layout#4 1 1 9.44 1 1 6.57 1 1 1.00 1 1 8.23×10 -2 1 1 Layout#5 1 1 1.01×10 1 1 1 7.08 1 1 7.08 1 1 9.80×10 -2 1 1 Layout#6 1 1 1.16×10 1 1 1 1.05×10 1 1 1 1.05×10 1 1 1 8.30×10 -2 As shown in Table 6 Table 6 , regardless of the isotope produced, the yield after 90 d increases as the ESTV increases. Therefore, the ESTV not only indicates the production efficiency at the early stage but also indicates the efficiency of the entire irradiation cycle. Therefore, a rapid diagnostic method based on SEIV and ESTV can be used to evaluate the efficiency of transplutonium isotope production. Experimental Verification 2 Because the high flux reactor shown in Fig. 4 Fig. 4 has not been built, experimental studies cannot be conducted with this reactor; thus, the experimental verification of the rapid diagnostic method was conducted using the HFIR [ 53 53 ]. The HFIR has been producing 252 Cf since 1966 and currently accounts for 70% of the world’s supply; therefore, the HFIR has a large amount of measured data on transplutonium isotope production. Rated at 100 MW and currently operating at 85 MW, HFIR has the world’s highest steady-state neutron heat flux (2.6×10 15 cm -2 ·s -1 ). The HFIR is a light-water-cooled moderated flux trap reactor that uses highly enriched 235 U. The reactor core consists of many concentric annular zones, each approximately 61 cm high (fuel height, 51 cm). At the center is a 12.70 cm cylindrical pore called a “neutron flux trap”, containing 37 vertical experimental holes. The neutron flux trap is surrounded by two concentric fuel assemblies. The reactor core and neutron flux trap are shown in Fig. 10 Fig. 10 [ 54 54 ]. Because the rapid diagnostic method uses the ESTV to evaluate the production efficiency of different irradiation schemes, Monte Carlo criticality calculation should be performed to calculate the ESTV. The RMC code is used to model the HFIR, and the input card of the HFIR is provided in the attachment. Figure 11 Figure 11 shows the core modeling and neutron flux trap modeling results. For the verification of the correctness of the modeling, the radial distributions of the thermal neutron flux and total neutron flux were calculated and compared with the reference solutions ( Fig. 12 Fig. 12 ). This shows that the flux distributions calculated by the RMC are in good agreement with the reference solutions, which proves the correctness of the HRIR model. Next, an experimental verification was conducted. The target design shown in Fig. 13 Fig. 13 was irradiated with HFIR for 5 d. Four irradiation schemes were formed with two target nuclide compositions and two irradiated positions: 245 Cm_A, 245 Cm_C, Cf_A, and Cf_C. The yields of heavy nuclei (all nuclides heavier than the nuclides in the target) for the four irradiation schemes were measured [ 9 9 ], and the corresponding ESTV was calculated. The experimental results are listed in Table 7 Table 7 . 1 1 Irradiation schemes 1 1 Nuclide compositionof the target (ng) 1 1 Irradiation duration (d) 1 1 Yields of heavy nuclei(experiments) 1 1 ESTV(simulations) 1 1 245 Cm_A 1 1 245 Cm: 45.9±0.3 1 1 5 1 1 7.861 ng 1 1 1.00×10 -12 1 1 245 Cm_C 1 1 249 Cf: 0.099±0.001 1 1   1 1 9.308 ng 1 1 1.09×10 -12 1 1 Cf_A 1 2 249 Cf: 42.4±0.3 250 Cf: 11.3±0.2 251 Cf: 30.2±0.5 1 1 5 1 1 12.6 ng 1 1 3.23×10 -11 1 1 Cf_C 1 1   1 1 12.7 ng 1 1 3.50×10 -11 Because the target nuclide composition should be the same when comparing the ESTV of different irradiation schemes, we compared 245 Cm_A with 245 Cm_C and Cf_A with Cf_C. As shown in Table 7 Table 7 , the measured yield of heavy nuclei of 245 Cm_A was lower than that of 245 Cm_C, and the corresponding ESTV of 245 Cm_A was also lower than that of 245 Cm_C. Additionally, the measured yield of the heavy nuclei of Cf_A was lower than that of Cf_C, and the corresponding ESTV of Cf_A was also lower than that of Cf_C. The experimental results satisfy the law in the rapid diagnostic method that as the ESTV increases, production efficiency increases. Therefore, the experimental verification proved the correctness of the rapid diagnostic method. In the next chapter, we use a rapid diagnostic method to realize rapid screening and determination of the optimal irradiation scheme. SENSITIVITY ANALYSIS AND OPTIMIZATION APPLICATION 1 Logic of Optimization 2 Five design parameters were optimized: 1) selection of the moderator, 2) layout of the moderator, 3) position of the target, 4) size of the target, and 5) shape of the target. As shown in Table 3 Table 3 , regardless of the isotope produced, the energy interval with the highest SEIV is always in the thermal range; therefore, we had to arrange the moderator to soften the neutron spectrum around the target. We aimed to select the material with the strongest moderating power to achieve the best neutron spectrum by deploying the least amount of moderator. The moderating power of various materials is an inherent property of the material and not affected by other design parameters; therefore, the moderator material can be determined first. We used a large hexagonal prism target with Layout #1 and Position #1 as the initial scheme. Nine moderator materials were tested, and the neutron spectra around the target were tallied ( Figure 14 Figure 14 ). As shown in Fig. 14 Fig. 14 , zirconium hydride has the best moderating power; thus, zirconium hydride was selected as the moderator, and the moderator selection was not considered in all subsequent analyses. To determine an optimal production scheme, we had to determine the optimal design parameters for the layout of the moderator and the position, size, and shape of the target. We used a large hexagonal prism bare target with Position #1 as the initial scheme and then optimized the four design parameters. The optimization order of these four parameters is important. We determined the optimization order using a sensitivity analysis. The sensitivity coefficient of design parameter d to production efficiency is defined as <math display="block" id="M10"><mrow><msub><mi>η</mi><mi>d</mi></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>P</mi></mrow><mrow><mo>∂</mo><mi>d</mi></mrow></mfrac><mo>∝</mo><mfrac><mrow><mo>∂</mo><mtext>ESTV</mtext></mrow><mrow><mo>∂</mo><mi>d</mi></mrow></mfrac><mo>∝</mo><mfrac><mrow><mi>max</mi><msub><mrow><mo stretchy="false">(</mo><mtext>ESTV</mtext><mo stretchy="false">)</mo></mrow><mi>d</mi></msub></mrow><mrow><mi>min</mi><msub><mrow><mo stretchy="false">(</mo><mtext>ESTV</mtext><mo stretchy="false">)</mo></mrow><mi>d</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> where P represents the production efficiency, d represents a design parameter. Therefore, the sensitivity coefficient is positively correlated with the maximum increase in ESTV, which can be obtained by adjusting a design parameter. The larger the maximum increase, the greater the importance of this design parameter to the optimization process; therefore, we optimized it first. After optimizing one of these design parameters, we used the optimized scheme as the starting scheme to optimize the remaining design parameters until all design parameters were optimized. A logical diagram of the optimization process is shown in Fig. 15 Fig. 15 . This paper uses 252 Cf as an example to describe the optimization process of the irradiation scheme in detail. The optimal schemes for 238 Pu, 242 Cm, and 244 Cm are presented in Sect. 3.6. First Round of Sensitivity Analysis and Optimization 2 The first round of sensitivity analysis and optimization had to consider four design parameters: the layout of the moderator and the position, size, and shape of the target. The optional values for the four design parameters are shown in Fig. 9 Fig. 9 . New irradiation schemes can be obtained by adjusting one of the four design parameters based on the initial scheme. The ESTVs for all new schemes are listed in Table 8 Table 8 , and the symbol “ η ” represents the sensitivity coefficient. The bold “η” represents the sensitivity coefficient, its definition is given by Eq. (10) 1 1 Layout 1 1 ESTV 1 1 Position 1 1 ESTV 1 1 Bare target 1 1 9.22×10 -5 1 1 Position#1 1 1 9.22×10 -5 1 1   1 1   1 1 Position#2 1 1 1.07×10 -4 1 1 Layout#1 1 1 4.41×10 -4 1 1 Position#3 1 1 5.43×10 -4 1 1 Layout#2 1 1 1.08×10 -3 1 1 Position#4 1 1 6.97×10 -4 1 1 Layout#3 1 1 1.72×10 -3 1 1 Layout#4 1 1 4.39×10 -3 1 1 Layout#5 1 1 4.94×10 -3 1 1 Layout#6 1 1 4.59×10 -3 1 1 η L 1 1 53.6 1 1 η P 1 1 7.56 1 1 Shape 1 1 ESTV 1 1 Size 1 1 ESTV 1 1 Hexagonal prism 1 1 9.22×10 -5 1 1 Large 1 1 9.22×10 -5 1 1 Sphere 1 1 9.78×10 -5 1 1 Medium 1 1 9.35×10 -5 1 1 Cylinder 1 1 9.22×10 -5 1 1 Small 1 1 9.22×10 -5 1 1 η Sh 1 1 1.06 1 1 η si 1 1 1.01 As shown in Table 8 Table 8 , the moderator layout can improve the ESTV the most; therefore, the moderator layout should be optimized first. Moreover, Layout #5 had the highest ESTV; therefore, the layout of the moderator was selected as Layout #5. After the first round of optimization, the scheme became a large hexagonal prism target with Positions #1 and #5. Second Round of Sensitivity Analysis and Optimization 2 The second round of sensitivity analysis and optimization had to consider three design parameters: the position, size, and shape of the target. All new irradiation schemes were obtained based on the scheme after the first round of optimization. The ESTVs for the second round of optimization are listed in Table 9 Table 9 . The bold “η” represents the sensitivity coefficient, its definition is given by Eq. (10) 1 1 Position 1 1 ESTV 1 1 Shape 1 1 ESTV 1 1 Size 1 1 ESTV 1 1 Position#1 1 1 4.94×10 -3 1 1 Hexagonal prism 1 1 4.94×10 -3 1 1 Large 1 1 4.94×10 -3 1 1 Position#2 1 1 3.45×10 -3 1 1 Sphere 1 1 4.03×10 -3 1 1 Medium 1 1 5.42×10 -3 1 1 Position#3 1 1 2.23×10 -3 1 1 Cylinder 1 1 4.75×10 -3 1 1 Small 1 1 6.28×10 -3 1 1 Position#4 1 1 1.04×10 -3 1 1 η P 1 1 4.75 1 1 η Sh 1 1 1.04 1 1 η Si 1 1 1.27 As shown in Table 9 Table 9 , the position of the target can improve the ESTV the most; therefore, the position of the target should be optimized in the second round of optimization. Moreover, Position #1 has the highest ESTV; therefore, the position of the target is selected as Position #1. After the second round of optimization, the scheme became a large hexagonal prism target with Positions #1 and #5. Third Round of Sensitivity Analysis and Optimization 2 The third round of sensitivity analysis and optimization must consider two design parameters: the size and shape of the target. All new irradiation schemes were obtained based on the scheme after the second round of optimization. The ESTVs for the third round of optimization are listed in Table 10 Table 10 . The bold “η” represents the sensitivity coefficient, its definition is given by Eq. (10) 1 1 Shape 1 1 ESTV 1 1 Size 1 1 ESTV 1 1 Hexagonal prism 1 1 4.94×10 -3 1 1 Large 1 1 4.94×10 -3 1 1 Sphere 1 1 4.03×10 -3 1 1 Medium 1 1 5.42×10 -3 1 1 Cylinder 1 1 4.75×10 -3 1 1 Small 1 1 6.28×10 -3 1 1 η Sh 1 1 1.04 1 1 η Si 1 1 1.27 As shown in Table 10 Table 10 , the target’s shape and size had similar effects on improving the ESTV, with the target’s size being slightly larger; thus, target size should be optimized in the third round of optimization. Moreover, the small target had the highest ESTV; therefore, it was selected. After the third round of optimization, the scheme became a small hexagonal prism target with Positions #1 and #5. Fourth Round of Sensitivity Analysis and Optimization 2 The fourth round of sensitivity analysis and optimization had to consider one design parameter, that is, the shape of the target. All new irradiation schemes were obtained based on the scheme after the third round of optimization. The ESTVs for the fourth round of optimization are listed in Table 11 Table 11 . The bold “η” represents the sensitivity coefficient, its definition is given by Eq. (10) 1 1 Shape 1 1 ESTV 1 1 Hexagonal prism 1 1 6.28×10 -3 1 1 Sphere 1 1 5.19×10 -3 1 1 Cylinder 1 1 6.10×10 -3 1 1 η Sh 1 1 1.21 As shown in Table 11 Table 11 , the hexagonal prism had the highest ESTV; therefore, a hexagonal prism target was selected. After the fourth round of optimization, the scheme became a small hexagonal prism target with Positions #1 and #5. Final Irradiation Scheme 2 The optimized scheme for producing 252 Cf was a small hexagonal prism target with Positions #1 and #5. Similarly, we optimized the schemes for producing 238 Pu, 242 Cm, and 244 Cm. The final optimized irradiation schemes for producing 238 Pu, 242 Cm, 244 Cm, and 252 Cf are listed in Table 12 Table 12 . A comparison of the production efficiency before and after optimization is shown in Fig. 16 Fig. 16 . 1 1 Design parameters 1 1 238 Pu 1 1 242 Cm 1 1 244 Cm 1 1 252 Cf 1 1 Selection of moderator 1 1 zirconium hydride 1 1 zirconium hydride 1 1 zirconium hydride 1 1 zirconium hydride 1 1 Layout of moderator 1 1 Layout #6 1 1 Layout #6 1 1 Layout #6 1 1 Layout #5 1 1 Position of target 1 1 Position #1 1 1 Position #1 1 1 Position #1 1 1 Position #1 1 1 Shape of target 1 1 hexagonal 1 1 hexagonal 1 1 cylinder 1 1 hexagonal 1 1 Size of target 1 1 small 1 1 small 1 1 small 1 1 small After four rounds of sensitivity analysis and optimization, we quickly obtained the optimal scheme for transplutonium isotope production. As shown in Figure 16 Figure 16 , the optimal scheme effectively improves production efficiency more than the initial scheme: efficiencies of 238 Pu, 7.41 times; 242 Cm, 11.98 times; 244 Cm, 65.20 times; and 252 Cf, 15.08 times. Therefore, an optimization strategy based on a rapid diagnostic method is helpful for transplutonium isotope production. CONCLUSION 1 The production of transplutonium isotopes with reactor irradiation is inefficient, and further optimization studies are required. Quantitative analysis of the physical process of transplutonium isotope production is complicated. After understanding the physics nature and simplifying the complexity of the production process, we proposed the concept of “Single Energy Interval Value (SEIV)” and “Energy Spectrum Total Value (ESTV)” to realize the quantitative analysis of the production efficiency of transplutonium isotopes. A rapid diagnostic method for evaluating the radiation scheme based on SEIV and ESTV was established. The optimization design for producing 252 Cf, 244 Cm, 242 Cm, and 238 Pu was performed based on the rapid diagnostic method. The optimal scheme improved the efficiency of 238 Pu by 7.41 times; 242 Cm, 11.98 times; 244 Cm, 65.20 times; and 252 Cf, 15.08 times. This paper provides solutions to two key questions in the production of transplutonium isotopes:1) how to find the optimal neutron spectrum for producing transplutonium isotopes and 2) how to achieve the optimal neutron spectrum. For the first question, we used SEIV and ESTV to perform a detailed analysis of the neutron spectrum around the target. For the second problem, we optimized the design parameters around the target to achieve the optimal neutron spectrum, effectively improving computational efficiency. The work presented in this paper realizes a refined analysis of transplutonium isotope production and provides a theoretical basis for improving production efficiency. 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Soc. 104 ( 1 ) ( 2011 ). https://www.ans.org/pubs/transactions/article-12169/ https://www.ans.org/pubs/transactions/article-12169/ Modeling of the high flux isotope reactor cycle 400 with KENO-VI All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Qing-Quan Pan, Qing-Fei Zhao, Bang-Yang Xia, Yun Cai, and Xiao-Jing Liu. The first draft of the manuscript was written by Qing-Quan Pan and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-023-01185-4 This work is sponsored by Natural Science Foundation of Shanghai (NO. 22ZR1431900) and Science and Technology on Reactor System Design Technology Laboratory. 2022-11-21 2023-01-16 2023-01-18 2023-03-27 2023-04-20 © China Science Publishing & Media Ltd. 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