High-efficiency Doppler broadening method under ultrahigh temperature based on Kernel Broadening coupling Gaussian integral method

Abstract

The invention relates to the field of nuclear reaction, and discloses a Kernel Broadening coupled Gaussian integral-based high-efficiency Doppler broadening method at ultrahigh temperature, electronic equipment and a computer readable storage medium. The method comprises the following steps: S1, calculating parameters required by Doppler broadening of a nuclear reaction cross section; s2, Doppler broadening calculation is carried out on the section under the initial energy point; s3, determining a next energy point conforming to a Doppler broadening condition, and performing broadening calculation; s4, performing linearization processing on the Doppler broadening cross section by adopting an inverted stack algorithm; and S5, based on the determined Doppler broadening energy range, repeating the steps S3 to S4 until Doppler broadening calculation of the nuclear reaction cross section under all energy points is completed. Through the adaptive strategy, intelligent optimization matching of Doppler broadening calculation of all energy sections and different nuclide types is realized, and the efficiency bottleneck in ultra-high temperature calculation is overcome.

Description

Technical Field

[0001] This invention relates to the field of nuclear reaction technology, and more particularly to an efficient Doppler broadening method, electronic device, and computer-readable storage medium based on Kernel Broadening coupled Gaussian integral at ultra-high temperatures. Background Technology

[0002] The neutron nuclear reaction cross section is affected by the relative velocity between the incident neutron and the target nucleus. Different temperatures result in different thermal velocity distributions of the target nucleus, leading to variations in the nuclear reaction cross section at the same incident neutron energy—a phenomenon known as Doppler broadening. Accurately performing Doppler broadening calculations to obtain the neutron nuclear reaction cross section at different temperatures is fundamental and a prerequisite for conducting high-precision numerical simulations of nuclear reactors. Currently available Doppler broadening programs include: NJOY from Los Alamos National Laboratory, AMPX from Oak Ridge National Laboratory, and MC from Argonne National Laboratory. 2 The programs used include the GALILEE program from the French Atomic Energy and Alternative Energies Commission, the FRENDY program from the Japan Atomic Energy Agency, the PREPRO program from the International Atomic Energy Agency, and in my country, the NECP-Atlas program from Xi'an Jiaotong University, the Ruler program from the China Nuclear Data Center, the RXSP program from Tsinghua University, and the AXSP program from North China Electric Power University. Existing Doppler broadening calculation methods can be divided into direct Doppler broadening and online Doppler broadening methods. Direct Doppler broadening methods mainly include the Kernel Broadening precise Doppler broadening method and the Phi-Chi method, while online Doppler broadening methods mainly include cross-sectional temperature interpolation and fitting methods.

[0003] The Kernel Broadening (KB) method, based on first-principles calculations and the Maxwell-Boltzmann distribution function of the target nucleus's thermal velocity, precisely calculates the Doppler broadening cross-section through convolution operations. To ensure the accuracy and efficiency of the numerical convolution operation, the nuclear reaction cross-section at 0 K is piecewise linearized, and incomplete probability integrals are introduced to calculate the Doppler broadening cross-section at arbitrary temperatures. This method boasts high accuracy, versatility, and wide applicability, making it the most widely used Doppler broadening method currently available.

[0004] The Phi-Chi method, starting from the single-level Breit-Wigner formula, neglects the low-energy approximation in the Kernel Broadening method. The relative energy distribution between the incident neutron and the target nucleus plays a major role in the Doppler broadening cross-section. It is assumed that this distribution exists within a small range on both sides of the incident neutron energy, allowing for a Taylor expansion of the relative energy with respect to the incident neutron energy. While simple to implement, numerous assumptions result in low accuracy in calculating the low-energy cross-section and poor performance with strongly absorbing nuclides.

[0005] The cross-sectional temperature interpolation method selects typical temperature points within different temperature ranges. Based on the precise Doppler broadening method, the cross-sections at these typical temperature points are processed offline to create a temperature-cross-section offline interpolation table. When the Monte Carlo program requires a cross-section at a specific temperature point, interpolation is performed based on this table to obtain the cross-sectional value at the desired temperature point. This method is computationally fast, but it consumes a lot of memory, and the accuracy of the cross-section is affected by the fineness of the interpolated temperature points.

[0006] The fitting method calculates the Doppler broadening cross-section at multiple temperature points based on the precise Doppler broadening method, derives the broadening formula for the equivalent cross-section, calculates fitting parameters, and then uses these parameters to fit the cross-section during neutron transport calculations. This method is computationally fast but consumes a significant amount of memory.

[0007] The computation time for the Kernel Broadening (KB) method, which focuses on precise Doppler broadening, increases significantly with increasing temperature. The Phi-Chi method exhibits low accuracy in calculating cross-sections in the low-energy region and performs poorly with strong absorber nuclides. Cross-section temperature interpolation requires offline creation of temperature-cross-section interpolation databases at multiple typical temperatures, consuming significant storage space, and the cross-section accuracy is affected by the fineness of the interpolation temperature points. Fitting methods also require pre-calculation of cross-section data at multiple temperature points, resulting in high memory consumption.

[0008] The above problems arise because, as the computational temperature increases, the thermal motion of the target nucleus becomes more intense, and the Maxwell-Boltzmann velocity distribution function of the target nucleus broadens with increasing temperature. For a neutron at a certain incident energy, the Kernel Broadening exact Doppler broadening method becomes more difficult to converge when calculating incomplete probability integrals, requiring a significant increase in the number of energy points to consider, leading to a substantial increase in computational time. The Phi-Chi method ignores the low-energy approximation in the Kernel Broadening method, assuming that the relative energy distribution between the incident neutron and the target nucleus is within a small range on both sides of the incident neutron energy, thus performing a Taylor expansion of the relative energy with respect to the incident neutron energy, resulting in low accuracy of the cross-section calculation in the low-energy region. The computational accuracy of the cross-section temperature interpolation method depends on the selection of a small temperature range, resulting in a huge storage space required for offline typical temperature-cross-section databases, and the cross-section accuracy is affected by the fineness of the interpolated temperature points. The fitting method requires pre-calculation of a cross-sectional database of more than 3,000 temperature points for each nuclide, and fitting 13 parameters for each incident energy to generate a parameter database. Moreover, when the evaluation nuclide database changes, the parameter database needs to be updated synchronously. Furthermore, the implementation of this method requires direct modification of the Monte Carlo program source code, making the method difficult to apply.

[0009] Traditionally, researchers have focused on room-temperature operating conditions, resulting in insufficient understanding of Doppler broadening under ultra-high temperature conditions. Compared to the Doppler effect at room temperature, the study of Doppler broadening under ultra-high temperature conditions has unique research value and demand due to its distinctive engineering applications. Furthermore, the computational physics and mathematical models for Doppler broadening under ultra-high temperature conditions are extremely complex, leading to high computational costs. Traditional computational methods often struggle to achieve a balance between computational accuracy and efficiency when addressing such problems, posing a significant challenge. Summary of the Invention

[0010] This invention constructs a novel hybrid integration framework based on the characteristics of the Kernel Broadening precise Doppler broadening method and the Gaussian numerical integration method. By combining the variation characteristics of nuclear reaction cross-sections of different nuclides in different energy ranges, the optimal numerical integration method is matched for the Doppler broadening of nuclear reaction cross-sections in each energy range, significantly improving computational efficiency while strictly ensuring the calculation accuracy of Doppler broadening of nuclear reaction cross-sections across the entire energy range.

[0011] This invention addresses the critical technical challenge of simultaneously optimizing accuracy and efficiency in calculating Doppler broadening of nuclear reaction cross-sections under ultra-high temperature conditions. Based on the Kernel Broadening precise Doppler broadening coupled with a Gaussian numerical integration method, and considering the cross-sectional variation characteristics of different nuclides in different energy ranges, it matches the optimal numerical integration method, significantly reducing the number of incomplete probability integrations. For non-resonant nuclides, whose nuclear reaction cross-section changes gradually with energy, this invention introduces the Gauss-Hermite numerical integration method, although its theoretical integration domain is... However, its kernel function exhibits exponential decay. When the incident neutron energy is greater than a certain value, a finite-domain approximation can be performed in the numerical implementation, satisfying the integration accuracy requirements and greatly improving computational efficiency. When the incident neutron energy is lower than this value, the traditional Kernel Broadening precise Doppler broadening method is still used. For resonance nuclides, the nuclear reaction cross-section changes drastically with energy within the resonance energy range. When the energy range is relatively dense, the Gauss-Legendre numerical integration method is used, leveraging its accuracy on higher-order polynomials to accurately capture the resonance peak shape with fewer nodes, thereby significantly improving computational efficiency while ensuring computational accuracy. When the energy range is relatively wide, the traditional Kernel Broadening precise Doppler broadening method is still used.

[0012] This invention provides an efficient Doppler broadening method for ultra-high temperature environments based on a Kernel Broadening coupled Gaussian numerical integration method, comprising the following steps:

[0013] Step 1: Calculate the parameters required for Doppler broadening of the nuclear reaction cross section;

[0014] Step 2: Calculate the Doppler broadening of the cross section at the initial energy point: Based on the nuclide resonance parameter identifier obtained in Step 1, determine whether the nuclide is a resonance nuclide, and for resonance nuclides and non-resonance nuclides, match the optimal numerical integration method according to the characteristics of their nuclear reaction cross section change with energy to realize the Doppler broadening calculation of the cross section at the initial energy point.

[0015] Step 3: Determine the next energy point that meets the Doppler broadening condition and perform broadening calculations on it;

[0016] Step 4: Linearize the Doppler broadening cross section using a stack-off algorithm;

[0017] Step 5: Based on the determined Doppler broadening energy range, repeat steps 3 to 4 until the Doppler broadening calculation of the nuclear reaction cross section is completed at all energy points.

[0018] Specifically, step 1 is as follows:

[0019] (1) Obtain the energy-section interpolation table that can be linearly interpolated at the initial temperature;

[0020] (2) Determine the maximum broadening energy;

[0021] (3) Obtain the nuclide resonance parameter identifier;

[0022] (4) Calculate the temperature correlation coefficient and parameters related to incident neutron energy ;

[0023] (5) Calculate the trend of cross-sectional change corresponding to the initial energy point of Doppler broadening.

[0024] More specifically, in step 1:

[0025] (1) Based on the cross-sectional change characteristics of different reaction types, determine the reaction channels that need to be Doppler broadened and their number, and obtain the linearly interpolable energy-section interpolation table of each reaction channel at the initial temperature;

[0026] (2) For nuclides with a distinguishable resonance region, the upper limit of the distinguishable resonance region energy given in the evaluation nuclear database is the maximum broadening energy; for nuclides without a distinguishable resonance region but with an indistinguishable resonance region, the lower limit of the indistinguishable resonance region is the maximum broadening energy; for non-resonant nuclides, the maximum broadening energy is the smaller of 6.5 MeV and the lowest threshold energy among all threshold energy reactions.

[0027] (3) Obtaining the nuclide resonance parameter identifier: For the subnuclear reaction section data in the ENDF-6 format evaluation nuclide database, the LRP identifier given in evaluation nuclide database file 1 (which is a file in the publicly available ENDF / B database) can be used to determine whether the nuclide is a resonance nuclide: when the LRP identifier is 0, it indicates that the nuclide is a non-resonance nuclide; when the LRP identifier is 1, it indicates that the nuclide is a resonance nuclide; (4) Calculating the temperature correlation coefficient and parameters related to incident neutron energy Temperature correlation coefficient and target mass and temperature Correlation between incident neutron energy and temperature-related parameters Incident neutron rate and the relative motion rate between the incident neutron and the target nucleus The relevant calculation formula is as follows;

[0028] (5) Calculate the trend of cross-sectional change corresponding to the initial energy point of Doppler broadening. The trend of nuclear reaction cross-sectional change is judged based on the positive and negative values ​​of the difference between the initial energy point and the adjacent energy point in the evaluation nuclear database. When the absolute value of the difference between the nuclear reaction cross-sectional values ​​is less than a certain set value, it is considered as no change. When the difference is positive and greater than the set value, it is recorded as 1. When the difference is negative and its absolute value is greater than the set value, it is recorded as -1.

[0029] Specifically, step 2 is as follows:

[0030] (1) Establish the Doppler broadening equation: based on the conservation of nuclear reaction rate and the Maxwell-Boltzmann distribution considering the thermal motion rate of the target nucleus;

[0031] (2) For non-resonant nuclides, when the incident neutron energy-related parameter Y is greater than a certain value, the Gauss-Hermite numerical integration method is used to solve it.

[0032] (3) For resonance nuclides, when the secondary neutron energy-related parameters satisfy When the value is less than a certain specific value, the Gauss-Legendre numerical integration method is used to solve the problem;

[0033] (4) For other energy points that do not meet the conditions in (2) and (3) and require Doppler broadening, the Kernel Broadening precise Doppler broadening method is still used, and the exponential integral is calculated by calculating the incomplete probability integral.

[0034] (5) Based on (2) to (4) above, complete the Doppler broadening calculation of the cross section at the initial energy point.

[0035] More specifically, step 2 is as follows:

[0036] (1) Establishing the Doppler broadening equation: Based on the conservation of nuclear reaction rate and the Maxwell-Boltzmann distribution considering the thermal motion rate of the target nucleus, the Doppler broadening equation can be expressed as:

[0037] in, for The cross section of a nuclear reaction after Doppler broadening at the corresponding energy;

[0038] Preferably, to simplify the cross-section calculation, it can be divided into two parts:

[0039] calculate This solves the calculation of formula (4), where:

[0040] For a linearized nuclear reaction cross section that can be given in the form of an energy-section interpolation table, Equation (6) is expressed as:

[0041] (2) For non-resonant nuclides, when the incident neutron energy-related parameter Y is greater than a certain value, the Gauss-Hermite numerical integration method is used to solve formula (7):

[0042] First, we use variable substitution to let Formula (6) is expressed as:

[0043] Then, the integral of formula (8) is transformed into a sum of finite polynomials using the Gauss-Hermite numerical integration method, expressed as:

[0044] in:

[0045] (3) For resonance nuclides, when the secondary neutron energy-related parameters satisfy When the value is less than a certain specific value, the Gauss-Legendre numerical integration method is used to solve formula (7):

[0046] For each integration interval Replacing the analytical integral with the N-point numerical integral, formula (7) is expressed as:

[0047] in,

[0048] To balance computational accuracy and efficiency, the two-point Gauss-Legendre numerical integration method is adopted, with the integration points selected as follows:

[0049] The Gauss-Legendre numerical integration formula is expressed as:

[0050] (4) For other energy points that do not meet the conditions in (2) and (3) and require Doppler broadening, the Kernel Broadening precise Doppler broadening method is still used, and the exponential integral in formula (7) is calculated by calculating the incomplete probability integral.

[0051] (5) Based on (2) to (4) above, complete the Doppler broadening calculation of the cross section at the initial energy point.

[0052] Specifically, step 3 is as follows:

[0053] (1) Determine the next energy point that needs to be Doppler broadened. The next energy point that needs to be broadened must meet one of the following conditions: a) It is the last energy point in the energy grid; b) The ratio of the next energy point to the current energy point is greater than a certain value; c) The cross-sectional change trend is different from that of the previous energy point.

[0054] (2) For the next energy point that needs to be Doppler broadened, the Doppler broadening cross section is calculated using the numerical integration method of Doppler broadening given in step 2.

[0055] In addition, specifically, step 4 is as follows:

[0056] (1) Based on the Doppler broadening cross section of adjacent energy points calculated in steps 2 and 3, determine the intermediate energy point, and obtain the nuclear reaction cross section corresponding to the intermediate energy point based on linear interpolation. Compare it with the Doppler broadening cross section calculated based on numerical integration method to determine whether it meets the linearization convergence condition.

[0057] (2) If the linearization convergence condition is not met, the cross-sectional value of the intermediate energy point is calculated by numerical integration, and linear interpolation is continued based on the lower energy point and the intermediate energy point. The steps shown in (1) are repeated until all interpolated energy points within the adjacent energy points calculated in steps 2 and 3 meet the linearization convergence condition.

[0058] Step 5 also includes outputting the nuclear reaction cross sections at all energy points obtained after Doppler broadening in a specific format.

[0059] The present invention further provides a method for high-precision numerical simulation of nuclear reactors. Based on the temperature of the materials required for the numerical simulation of nuclear reactor neutron physics, the method is used to calculate the nuclear reaction cross section at different temperatures, including ultra-high temperature, to obtain the high-precision basic input parameters required for the numerical simulation of nuclear reactor neutron physics.

[0060] This invention also provides an efficient Doppler broadening model for ultra-high temperature environments based on the Kernel Broadening coupled Gaussian numerical integration method, including a data input module, a data processing module, and a result output module, wherein the data processing module is used to execute the method described.

[0061] The present invention also provides a computer-readable storage medium, wherein the storage medium stores at least one instruction or at least one program, the at least one instruction or the at least one program being loaded and executed by a processor to implement the method or the model.

[0062] During the research process of this invention, a single Gauss-Hermite numerical integration method was used for Doppler broadening numerical integration calculations. While this algorithm significantly reduces computation time, its accuracy is insufficient for the cross-sectional accuracy requirements of nuclear reactor numerical calculations in the resonance region where cross-sectional changes are drastic. The method ultimately determined in this invention, through the aforementioned adaptive strategy, achieves intelligent optimization and matching for Doppler broadening calculations across the entire energy range and for different nuclide types, overcoming the efficiency bottleneck in ultra-high temperature calculations. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating the technical process of the present invention.

[0064] Figure 2 10 8 K 235 Comparison of different nuclear reaction cross sections and their relative deviations. (a) through (d) represent 10... 8 K 235 A comparison of the cross sections and their relative deviations of elastic scattering, fission reaction, inelastic scattering of the first discrete energy level, and radiative trapping reaction of U.

[0065] Figure 3 10 8 K 238 Comparison of different reaction cross sections and their deviations. (a) to (c) represent 10... 8 K 238 Comparison of cross sections and relative deviations of U in elastic scattering, fission reaction and radiation capture reaction. Detailed Implementation

[0066] Example 1

[0067] See Figure 1 The technical solution flow of the present invention is shown below.

[0068] First, obtain the linearly interpolable energy-section interpolation table at the initial temperature, the nuclide resonance parameter identifiers, and the parameters required to calculate the Doppler broadening of the nuclear reaction cross section, including the maximum broadening energy and temperature correlation coefficient. and the correlation coefficient of incident neutron energy The process begins by analyzing the trend of cross-sectional changes corresponding to the initial energy point of Doppler broadening. Then, depending on whether the nuclide is a resonance nuclide, different Doppler broadening methods are used to calculate the nuclear reaction cross-section at the initial energy point. Next, the next energy point requiring broadening is identified, and its corresponding nuclear reaction cross-section is calculated using Doppler broadening. Then, it is determined whether the broadened nuclear reaction cross-section meets the convergence condition. If not, a stack-reversing algorithm is used to linearize the Doppler broadened cross-section until it converges. Finally, the process continues to identify the next energy point requiring broadening and calculate the broadening of its corresponding nuclear reaction cross-section until the last energy point is reached.

[0069] Therefore, this invention provides an efficient Doppler broadening method for ultra-high temperature environments based on a Kernel Broadening coupled Gaussian numerical integration method, which includes the following steps:

[0070] Step 1: Calculate the parameters required for Doppler broadening of the nuclear reaction cross section.

[0071] (1) Obtain the energy-section interpolation table that can be linearly interpolated at the initial temperature. Based on the cross-sectional change characteristics of different reaction types, determine the reaction channels that need to be Doppler broadened and their number, and obtain the energy-section interpolation table that can be linearly interpolated at the initial temperature of each reaction channel;

[0072] (2) Determine the maximum broadening energy. For nuclides with a distinguishable resonance region, the upper limit of the distinguishable resonance region energy given in the evaluation nuclide database is the maximum broadening energy; for nuclides without a distinguishable resonance region but with an indistinguishable resonance region, the lower limit of the indistinguishable resonance region is the maximum broadening energy; for non-resonant nuclides, the maximum broadening energy is the smaller of 6.5 MeV and the lowest threshold energy among all threshold energy reactions.

[0073] (3) Obtain the nuclide resonance parameter identifier. For subnuclear reaction cross-section data in the ENDF-6 format evaluation nuclear database, the LRP identifier can be used to determine whether the nuclide is a resonance nuclide;

[0074] (4) Calculate the temperature correlation coefficient and parameters related to incident neutron energy Temperature correlation coefficient and target mass and temperature Related parameters, including incident neutron energy correlation coefficients, and temperature correlation coefficients. incident neutron rate and the relative motion rate between the incident neutron and the target nucleus Related;

[0075] (5) Calculate the trend of cross-sectional change corresponding to the initial energy point of Doppler broadening. The trend of nuclear reaction cross-sectional change is judged based on the positive and negative values ​​of the difference between the initial energy point and the adjacent energy point in the evaluation nuclear database. When the absolute value of the difference between the nuclear reaction cross-sectional values ​​is less than a certain set value, it is considered as no change. When the difference is positive and greater than the set value, it is recorded as 1. When the difference is negative and its absolute value is greater than the set value, it is recorded as -1.

[0076] Step 2: Perform Doppler broadening calculations on the cross section at the initial energy point.

[0077] Based on the nuclide resonance parameter identifier obtained in step 1, determine whether the nuclide is a resonance nuclide. For resonance nuclides and non-resonance nuclides, according to the characteristics of their nuclear reaction cross section change with energy, match the optimal numerical integration method to realize the Doppler broadening calculation of the cross section at the initial energy point.

[0078] (1) Establish the Doppler broadening equation. Based on the conservation of nuclear reaction rate and the Maxwell-Boltzmann distribution considering the thermal motion rate of the target nucleus, the Doppler broadening equation can be expressed as:

[0079] in, for The cross section of the nuclear reaction after Doppler broadening at the corresponding energy.

[0080] To simplify the cross-section calculation, it can be divided into two parts:

[0081] calculate This solves the calculation of formula (4), where:

[0082] For a linearized nuclear reaction cross section that can be given in the form of an energy-section interpolation table, equation (6) can be expressed as:

[0083]

[0084] (2) For non-resonant nuclides, when the incident neutron energy-related parameter Y is greater than a certain value, the Gauss-Hermite numerical integration method is used to solve formula (7).

[0085] First, we use variable substitution to let Formula (6) can be expressed as:

[0086]

[0087] Then, the integral of formula (8) is transformed into a sum of finite polynomials using the Gauss-Hermite numerical integration method, which can be expressed as:

[0088] in:

[0089] (3) For resonance nuclides, when the secondary neutron energy-related parameters satisfy When the value is less than a certain specific value, the Gauss-Legendre numerical integration method is used to solve formula (7).

[0090] For each integration interval Replacing the analytical integral with the N-point numerical integral, formula (7) can be expressed as:

[0091] in,

[0092] To balance computational accuracy and efficiency, the two-point Gauss-Legendre numerical integration method is employed. The integration points are selected as follows:

[0093] The Gauss-Legendre numerical integration formula can be expressed as:

[0094] (4) For other energy points that do not meet the conditions in (2) and (3) and require Doppler broadening, the Kernel Broadening precise Doppler broadening method is still used, and the exponential integral in formula (7) is calculated by calculating the incomplete probability integral.

[0095] (5) Based on (2) to (4) above, complete the Doppler broadening calculation of the cross section at the initial energy point.

[0096] Step 3: Determine the next energy point that meets the Doppler broadening condition and perform broadening calculations on it.

[0097] (1) Determine the next energy point that needs to be Doppler broadened. The next energy point that needs to be broadened must meet one of the following conditions: a) It is the last energy point in the energy grid; b) The ratio of the next energy point to the current energy point is greater than a certain value; c) The trend of cross-sectional change is different from that of the previous energy point.

[0098] (2) For the next energy point that needs to be Doppler broadened, the Doppler broadening cross section is calculated using the numerical integration method of Doppler broadening given in step 2.

[0099] Step 4: Linearize the Doppler broadening cross section using the stack-off algorithm.

[0100] (1) Based on the Doppler broadening cross section of adjacent energy points calculated in steps 2 and 3, determine the intermediate energy point, and obtain the nuclear reaction cross section corresponding to the intermediate energy point based on linear interpolation. Compare it with the Doppler broadening cross section calculated based on numerical integration method to determine whether it meets the linearization convergence condition.

[0101] (2) If the linearization convergence condition is not met, the cross-sectional value of the intermediate energy point is calculated by numerical integration, and linear interpolation is continued based on the lower energy point and the intermediate energy point. The steps shown in (1) are repeated until all interpolated energy points within the adjacent energy points calculated in steps 2 and 3 meet the linearization convergence condition.

[0102] Step 5: Based on the determined Doppler broadening energy range, repeat steps 3-4 until the Doppler broadening calculation of the nuclear reaction cross-sections at all energy points is completed. Output the nuclear reaction cross-sections at all energy points obtained after Doppler broadening in a specific format.

[0103] Verification Example

[0104] Compared with the NJOY2016 core data processing program, which is widely recognized in the industry, the program developed using the solution in this patent application was used to conduct preliminary implementation tests on the solution.

[0105] Test case description: For 235 U and 238 U in 10 8 The Doppler broadening cross section at K temperature was calculated, and the accuracy and efficiency of the results were compared with those calculated by the NJOY2016 program.

[0106] The calculation process is as follows: For the above-mentioned nuclides, the method proposed in this invention is used for testing and verification. The specific implementation process is as follows. First, based on step 1 (3) in Example 1, the following is determined: 235 U and 238 U are all resonance nuclides; then, based on step 2 (3) in Example 1, when the secondary neutron energy-related parameters satisfy When the Gauss-Legendre numerical integration method is used, and the Kernel Broadening precise Doppler broadening method is used to perform Doppler broadening calculations when the above conditions are not met, the calculations are performed. Finally, steps 3-4 in Example 1 are repeated to complete the Doppler broadening calculations of the nuclear reaction cross-sections at all energy points. The 10... 8 At K temperature 235 The elastic scattering cross section, fission reaction cross section, inelastic scattering cross section of the first discrete energy level, and radiation trapping reaction cross section are compared with the results and relative deviations calculated by the NJOY2016 program as follows: Figure 2 of (a) Figure 2 of (b) Figure 2 (c) and Figure 2 As shown in (d). Figure 2 of (a) Figure 2 of (b) Figure 2 (c) and Figure 2 The results (d) show that the relative deviations of the elastic scattering cross section, fission reaction cross section, first discrete energy level inelastic scattering cross section, and radiation trapping reaction cross section calculated by the present invention and NJOY2016 program are all less than 0.4%. 238 Calculation results and relative deviations of the U-elastic scattering cross section, fission reaction cross section, and radiation trapping reaction cross section are as follows: Figure 3 of (a) Figure 3 (b) and Figure 3 As shown in (c), Figure 3 of (a) Figure 3 (b) and Figure 3 (c) The results show that the maximum relative deviation between the elastic scattering cross section, fission reaction cross section, and radiation trapping reaction cross section calculated by the present invention and the NJOY2016 program is less than 0.8%. 235 U and 238 The calculation results of the U-Doppler broadened cross section demonstrate the correctness of the method of the present invention.

[0107] The calculation time using the method of this invention and the calculation time using the NJOY2016 program are shown in Table 1.

[0108] Table 1. Comparison of Program Execution Time

[0109]

[0110] The calculation results show that, compared with the NJOY2016 program, the method of this invention exhibits a relative deviation of approximately 0.8% at one energy point, while the maximum relative deviation of the Doppler broadening cross-section at other energy points does not exceed 0.4%. Regarding computational efficiency, [the method is more efficient]. 235 U and 238The computation time for U is significantly reduced. This patent application's solution can significantly reduce Doppler stretching computation time and improve computational efficiency while maintaining computational accuracy.

[0111] In performing numerical integration for Doppler broadening calculations, this invention employs a Kernel Broadening coupled Gaussian numerical integration method. Based on the energy-dependent characteristics of the nuclear reaction cross-sections of non-resonant and resonant nuclides, it utilizes the Gauss-Hermite and Gauss-Legendre numerical integration methods, respectively. This significantly improves the computational efficiency of Doppler broadening calculations of nuclear reaction cross-sections at ultra-high temperatures while ensuring computational accuracy, providing crucial technical support for the accurate and efficient acquisition of nuclear reaction cross-sections at ultra-high temperatures.

Claims

1. A high-efficiency Doppler broadening method under ultra-high temperature based on Kernel Broadening coupled Gaussian-type numerical integration method, comprising the following steps: Step 1: Calculate the parameters required for Doppler broadening of nuclear reaction cross section; Step 2: Calculate the Doppler broadening of the cross section at the initial energy point: according to the nuclear resonance parameter identifier obtained in step 1, determine whether the nuclide is a resonance nuclide, and according to the characteristics of the change of the nuclear reaction cross section with energy for resonance nuclides and non-resonance nuclides, match the optimal numerical integration method to realize the Doppler broadening calculation of the cross section at the initial energy point; Step 3: Determine the next energy point that meets the Doppler broadening condition and perform broadening calculation; Step 4: Linearize the Doppler broadened cross section using the reverse stack algorithm; Step 5: Based on the determined Doppler broadening energy range, repeat steps 3-4 until the Doppler broadening calculation of the nuclear reaction cross section at all energy points is completed.

2. The method of claim 1, wherein step 1 is as follows: (1) Obtain an energy-cross section interpolation table that can be linearly interpolated at the initial temperature; (2) Determine the maximum broadening energy; (3) Obtain the nuclide resonance parameter identifier; (5) Calculate the trend of the change of the cross section corresponding to the initial energy point of the Doppler broadening. (4) calculating a temperature correlation coefficient and an incident neutron energy related parameter ; 3. The method of claim 1, wherein in step 1: (1) According to the characteristics of the change of the cross section for different reaction types, determine the reaction channels that need to be Doppler broadened and the number thereof, and obtain the energy-cross section interpolation table that can be linearly interpolated at the initial temperature for each reaction channel; (2) For nuclides with resolvable resonance regions, the upper limit of the resolvable resonance region given in the nuclear database is the maximum broadening energy; for nuclides without resolvable resonance regions but with non-resolvable resonance regions, the lower limit of the non-resolvable resonance region is the maximum broadening energy; for non-resonance nuclides, the maximum broadening energy is the smaller value of 6.5 MeV and the lowest threshold energy in all threshold energy reactions; (3) Obtain the nuclide resonance parameter identifier: for the ENDF-6 format evaluated nuclear database, the LRP identifier given in the file 1 (which is a file in the public ENDF / B database) can be used to determine whether the nuclide is a resonance nuclide: when the LRP identifier is 0, it indicates that the nuclide is a non-resonance nuclide; when the LRP identifier is 1, it indicates that the nuclide is a resonance nuclide; (5) Calculate the trend of the change of the cross section corresponding to the initial energy point of the Doppler broadening: according to the difference of the nuclear reaction cross section at the initial energy point and the adjacent energy point in the evaluated nuclear database, judge the trend of the change of the nuclear reaction cross section, when the absolute value of the difference is less than a certain set value, it is considered as no change, when the difference is positive and greater than the set value, it is recorded as 1, when the difference is negative and its absolute value is greater than the set value, it is recorded as -1.

4. The method of claim 1, wherein step 2 is as follows: (4) calculating the temperature-dependent coefficient and the incident neutron energy-dependent parameter : the temperature-dependent coefficient and the mass of the target nucleus and the temperature , the incident neutron energy-dependent parameter and the temperature-dependent coefficient , the incident neutron rate , and the relative velocity of the incident neutron and the target nucleus , and the calculation formula is as follows; ； (1) Establish the Doppler broadening equation: based on the conservation of nuclear reaction rate and the Maxwell-Boltzmann distribution considering the thermal motion rate of the target nucleus; ​ ​ ​ (2) For non-resonant nuclides, when the incident neutron energy related parameter Y is greater than a certain value, Gauss-Hermite numerical integral method is used to solve; (3) For the resonant nuclei, when the secondary neutron energy related parameters satisfy less than a certain value, the Gauss-Legendre numerical integration method is used to solve; (4) For other energy points that do not meet the conditions in (2) and (3) and need to be Doppler broadened, KernelBroadening accurate Doppler broadening method is still used to calculate the exponential integral in formula (7) by calculating the incomplete probability integral; (5) Based on the above (2)-(4), the Doppler broadening calculation of the cross section at the initial energy point is completed.

5. The method of claim 4, wherein, Step 2 is as follows: (1) Establish the Doppler broadening equation: Based on the conservation of nuclear reaction rate and considering the Maxwell-Boltzmann distribution of target nuclear thermal motion rate, the Doppler broadening equation can be expressed as: ； wherein, is corresponding to the energy under the Doppler broadening process of the nuclear reaction cross section; Preferably, to simplify the cross section calculation, it can be calculated in two parts: ； Calculations That is, the calculation of equation (4) is solved, where: ； For linearized nuclear reaction cross sections that can be given in the form of energy-cross section interpolation table, formula (6) is expressed as: ； (2) For non-resonant nuclides, when the incident neutron energy related parameter Y is greater than a certain value, Gauss-Hermite numerical integral method is used to solve formula (7): First, variable substitution is used, let Equation (6) is expressed as: ； Then, the integral of formula (8) is converted into a finite polynomial sum by Gauss-Hermite numerical integral method, expressed as: ； wherein: ; (3) For the resonant nuclei, when the secondary neutron energy dependent parameter satisfies less than a certain value, the Gauss-Legendre numerical integration method is used to solve formula (7): for each integration interval The formula (7) is expressed as: ； wherein ; In order to balance the calculation accuracy and efficiency, two-point Gauss-Legendre numerical integral method is used, and the integral points are selected as: ； The Gauss-Legendre numerical integral formula is expressed as: ； (4) For other energy points that do not meet the conditions in (2) and (3) and need to be Doppler broadened, KernelBroadening accurate Doppler broadening method is still used to calculate the exponential integral in formula (7) by calculating the incomplete probability integral; (5) Based on the above (2)-(4), the Doppler broadening calculation of the cross section at the initial energy point is completed.

6. The method of claim 1, wherein, Step 3 is as follows: (1) Determine the next energy point that needs to be Doppler broadened, wherein the next energy point that needs to be broadened needs to meet one of the following conditions: a) the last energy point in the energy grid; b) the ratio of the next energy point to the current energy point is greater than a certain value; c) the cross section change trend is different from that of the previous energy point; (2) For the determined next energy point that needs to be Doppler broadened, the Doppler broadening cross section calculation is carried out by using the numerical integral method for Doppler broadening given in step 2.

7. The method of claim 1, wherein, Step 4 is as follows: (1) Based on the Doppler broadened cross sections at adjacent energy points calculated in steps 2 and 3, determine the intermediate energy point, and obtain the nuclear reaction cross section corresponding to the intermediate energy point based on linear interpolation, and compare it with the Doppler broadened cross section calculated based on the numerical integral method to judge whether it meets the linearization convergence condition; (2) If the linearization convergence condition is not satisfied, the cross section value of the intermediate energy point is calculated by using the numerical integration method, and the linear interpolation is continued based on the lower energy point and the intermediate energy point, and the steps shown in (1) are repeated until all the interpolation energy points in the adjacent energy points calculated by steps 2 and 3 satisfy the linearization convergence condition. Step 5 further comprises outputting the nuclear reaction cross sections at all energy points after Doppler broadening in a specific format.

8. A method for high-precision numerical simulation of a nuclear reactor, comprising calculating nuclear reaction cross sections at different temperatures, including ultra-high temperatures, using the method according to any one of claims 1 to 8 based on the temperature of the material required for the numerical simulation of the neutron physics of the nuclear reactor, to obtain high-precision basic input parameters required for the numerical simulation of the neutron physics of the nuclear reactor.

9. A high-temperature efficient Doppler broadening model based on Kernel Broadening coupled Gaussian-type numerical integration method, characterized in that, The model comprises a data input module, a data processing module and a result output module, wherein the data processing module is used to execute the method according to any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The storage medium stores at least one instruction or at least one program, and the at least one instruction or the at least one program is loaded and executed by the processor to implement the method according to any one of claims 1 to 8, or the model according to claim 9.

Images