Method for calculating abundance of fission product nuclear decay processes

Abstract

The application provides a method for calculating the abundance of a fission product nuclear decay process. According to the characteristics of the decay differential equation, only the decay process is considered for n fission product nuclei, the differential equation set for the abundance change is represented by a matrix, a program is written by using Python language to solve it, and the result of the abundance change of the fission product nuclei with time is obtained. The method has simple program design, less input data types, and fast speed, and can quickly obtain the abundance data of the fission product nuclear decay process. In the case of a processor i5-1155G7@2.50GHz, the time consumption is less than 1 second, which is much faster than similar programs.

Description

TECHNICAL FIELD

[0001] The present application relates to a calculation method of fissile product nuclei, and particularly relates to a calculation method of abundance of fissile product nuclei in a decay process. BACKGROUND

[0002] The relationship between the abundance of fissile product nuclei and the decay time can be used for the calculation of physical quantities accompanied by the post-decay process of fissile product nuclei, such as decay heat, delayed particles (delayed neutrons, photons, charged particles), and can be applied to the research of nuclear engineering. At the same time, by comparing with experimental results (such as decay heat), the fissile yield data can be verified.

[0003] The abundance of the present application refers to the number density of product nuclei at a certain moment after the nuclear fission of actinide. The fragments generated by a single nuclear fission, after emitting prompt neutrons but before starting to decay, are taken as the initial moment, and the abundance at this moment is the independent yield of product nuclei. Thereafter, the unstable product nuclei start to decay, forming a decay chain, as shown in Figure 2 , which decays to stable nuclei (or approximately stable nuclei) until the decay network composed of the decay chain from A=66-172 is formed. By solving the network equation, the abundance change of nuclides similar to Figure 3 can be obtained.

[0004] After the spontaneous or induced fission of actinide, about 1000 product nuclei are generated. How to simply and quickly calculate the decay process of product nuclei and obtain the abundance change with time has great significance for basic research and scientific engineering research.

[0005] The closest to the present application is the burnup calculation program of a reactor. There are mainly two categories of reactor burnup calculation methods: numerical calculation method based on matrix solution of burnup equation set and analytical method based on single burnup chain. The Origen2 program developed by Los Alamos Laboratory in the 1980s is based on Taylor expansion method. The burnup calculation module in the Serpent system developed by VTT Center in Finland, the burnup program MENDEL developed by France, and the nuclide inventory calculation program FISPACT in the activation calculation system developed by Europe, and the CINDER90 program based on analytical method. For introduction in this regard, please refer to the doctoral thesis of Wu Mingyu, Development and Research of Transport-Burnup Coupling Calculation System Based on STEP1.0 and MCMG-II.

[0006] The numerical calculation method often takes a long time to calculate. The analytical method needs to analyze and split the decay chain into single chains, and it is difficult to split into single chains without bifurcation and circulation. At the same time, these programs are relatively large, and the types of input data are relatively complex. SUMMARY

[0007] The present application aims at solving the problems of the prior art, such as complicated program design, difficult chain breaking, multiple input data types, long calculation time, etc., and providing a method for quickly and accurately calculating the abundance of the decay process of a fission product nucleus.

[0008] The technical scheme of the present application is as follows: a method for calculating the abundance of the decay process of a fission product nucleus, comprising the following steps:

[0009] (1) determining initialization parameters, including a target nucleus TAR, a neutron energy En, a decay time DecT, and a decay data file name FnD and a yield file name FnY;

[0010] (2) reading decay data from the decay data file and reading fission yield data from the yield file. The decay data and the yield data are obtained from an existing database such as ENDF / B-VIII, JEFF-3.2.

[0011] (3) for a product nucleus i after fission, only the decay process is considered, and the differential equation for the change in the abundance of the product nucleus i is as follows:

[0012]

[0013] wherein Y i (t) is a function of the abundance of the product nucleus i with respect to time, -λ i Y i is the amount of disappearance caused by the decay itself, ∑ j λ j r i,j Y j is the sum of the increase amount of the product nucleus i caused by the decay of the product nucleus j, λ is a decay constant, T is a half-life, and r i,j is a branching ratio of the product nucleus j to the product nucleus i;

[0014] For n product nuclei, the differential equation for the change in the abundance is expressed in a matrix as follows, and the differential equation set is solved:

[0015] dY=A Y

[0016] wherein, Y i is the abundance of the product nucleus i,

[0017] Y=(Y1,Y2,…,Y n ) T , the initial abundance Y(0) at zero time is an independent yield,

[0018] A is a coefficient matrix composed of decay constants;

[0019] (4) outputting the result of the change in the abundance of the product nucleus with respect to time.

[0020] Further, in the abundance calculation method of the fission product nucleus decay process as described above, the coefficient matrix A composed of the decay constants in step (3) is as follows:

[0021] A=(λ j r i,j )| i,j＝1,n

[0022] When i=j, r i,j =-1;

[0023] The diagonal elements in the matrix A are the decay constants caused by the disappearance of the self-decay, and the non-diagonal elements are the decay constants caused by the increase of the i nucleus due to the decay of the j nucleus;

[0024] Where λ=ln(2) / T is the decay constant, and T is the half-life,

[0025] r i,j is the branching ratio of the decay of the j nucleus to the i nucleus,

[0026] n is the number of fission product nuclei.

[0027] Further, in the abundance calculation method of the fission product nucleus decay process as described above, in step (2), the decay data includes: identification numbers of parent and daughter nuclei, branching ratios of the decay of the parent nucleus to the daughter nucleus, decay modes, and lifetimes of product nuclei; wherein the identification numbers of the parent and daughter nuclei are ID=1000A+10Z+Isomer, A is the mass number of the product nucleus, Z is the charge number of the product nucleus, and Isomer is the isomer of the product nucleus.

[0028] Further, in the abundance calculation method of the fission product nucleus decay process as described above, in step (2), the fission yield data includes: target nuclei, charge numbers, element symbols, mass numbers, isomers of fission product nuclei, independent yields or cumulative yields, yields and errors, and neutron incident energies.

[0029] The beneficial effects of the present application are as follows:

[0030] The abundance calculation method of the fission product nucleus decay process of the present application has the advantages of simple program design, few types of input data, fast speed, and can quickly obtain the abundance data of the decay process of the product nucleus after fission.

[0031] The present application is designed according to the characteristics of the decay differential equation, and has the characteristics of clear physical process, few input parameters, simple program, and fast speed. Taking the thermal neutron induced 235 U fission as an example, the calculation time is less than 1 second (Windows environment, CPU is i5-1155G7@2.50GHz).

[0032] The calculation result can be further used for calculation of decay heat of fission product nuclei, important data of delayed particles (photons, neutrons, charged particles, neutrinos, etc.), and the like. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 A schematic diagram of the abundance calculation method of the fission product nucleus decay process of the present application;

[0034] Figure 2 A decay chart of the fission product nucleus (taking A=95 as an example);

[0035] Figure 3 A decay chart of the fission product nucleus 95 Sr, 95 Y, 95 Zr, 95 Nb and 95m Nb (from top to bottom in the figure) abundance change with time; the dots are the calculation results of the present application, and the lines are the calculation results of CINDER90, which are consistent. DETAILED DESCRIPTION

[0036] In order to make the objects, technical solutions and advantages of the present application clearer, the present application is further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application, and are not used to limit the present application.

[0037] The present application considers the decay process of the fission product nucleus, without the neutron reaction process. For the fission product nucleus, if only the decay process is considered, the differential equation set of the abundance change is:

[0038]

[0039] Wherein, Y i (t) is the function of the abundance of the product nucleus i with respect to time, the first term on the right side is the disappearance amount caused by the decay itself, the second term is the sum of the increase amount of the decay of other nuclei to the nucleus, λ = ln(2) / T is the decay constant (T is the half-life), r i,j is the branching ratio of the decay of the j nucleus to the i nucleus.

[0040] Suppose there are n fission product nuclei, and the equation set (1) can be expressed by a matrix:

[0041] dY = A Y,(2)

[0042] Wherein,

[0043] Y = (Y1, Y2, …, Y n ) T , the initial abundance Y(0) at zero time is the independent yield.

[0044] A is a coefficient matrix composed of decay constants:

[0045] A = (a i,j ) = (λ j r i,j )| i,j＝1,n When i = j, r i,j = -1(3)

[0046] The diagonal elements of matrix A are the decay constants that disappear due to self-decay, and the non-diagonal elements are the decay constants that increase the i-nucleus due to the j-nucleus decaying to the i-nucleus.

[0047] The coefficient matrix A is related to the decay constant and is independent of time, so it can be processed first when designing the program, and it does not need to be repeatedly calculated in the solving process. This is the main difference from other nuclear reaction processes. Other nuclear reaction processes include neutron or charged particle nuclear reactions, and the matrix A is related to time and is not a constant matrix, which needs to be updated iteratively in the solving process.

[0048] Based on formulas (2) and (3), the DFPY.py (Decaying Fission Product Yield) program is written in Python language in this embodiment, and the flow chart is as Figure 1 shown. The main steps are as follows:

[0049] A. Input initialization parameters:

[0050] a) TAR, En, DecT: target nucleus, neutron energy, decay time

[0051] b) FnD, FnY: decay data file name, yield file name

[0052] B. Read input data:

[0053] a) Read in decay data, the data format is shown in Table 1;

[0054] b) Read in fission independent yield, the data format is shown in Table 2;

[0055] The decay data and yield data are obtained from existing databases such as ENDF / B-VIII, JEFF-3.2, and the website of the Nuclear Data Section of the International Atomic Energy Agency https: / / www-nds.iaea.org / . The website provides ENDF format and list format, and the sorted decay data and yield data are shown in Table 1 and Table 2, respectively;

[0056] C. Calculate the coefficient matrix A in formula (3);

[0057] D. To solve the ODE system of equation (2), the present application uses the function odeint in the Scipy package of Python (Table 3, lines 45-48) with the following input parameters:

[0058] DY: function, name of the differential equation function (Table 3, lines 37-43);

[0059] Y0: array, initial values of the abundances, read from a file (format as shown in Table 2);

[0060] t: array, decay times, user defined;

[0061] E. Output results, shown as Figure 2

[0062] The corresponding main calculation code is shown in Table 3.

[0063] Table 1. Decay data (input)

[0064]

[0065] Table 2. Fission yield data (input)

[0066]

[0067] Table 3. Main program code of DFPY.py

[0068]

[0069]

[0070] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. It is intended that the specification and examples be considered as illustrative only of the preferred embodiments of the application and not as limiting its scope. The scope of the application should be determined by reference to the appended claims and equivalents thereof.

[0071] The above embodiments are only illustrative of the present application, and the present application can be implemented in other specific ways or other specific forms without departing from the spirit or essential characteristics of the present application. Therefore, the described embodiments should be considered as illustrative rather than limiting in any respect. The scope of the present application should be explained by the appended claims, and any changes equivalent to the intent and scope of the claims should be included in the scope of the present application.​

Claims

1. A method for calculating the abundance of fission product nuclei in the process of decay, comprising the steps of: (1) determining initialization parameters, including target nuclei TAR, neutron energy En, decay time DecT, and decay data file name FnD, yield file name FnY; (2) reading decay data from the decay data file and reading fission yield data from the yield file; (3) for the product nuclei i after fission, only considering the decay process, the differential equation for the change in abundance is as follows: wherein Y i (t) is the abundance of the fission product nucleus i as a function of time, -λ i Y i is the amount of disappearance due to decay of the i nucleus itself, is the sum of the increase of the i nucleus due to decay of the j nucleus, λ i , λ j are the decay constants of the i and j nuclei, respectively, λ i = ln(2) / T i , λ j = ln(2) / T j , T i is the half-life of the i nucleus, T j is the half-life of the j nucleus, r i,j is the branching ratio of the decay of the j nucleus to the i nucleus; i, j are the nuclide numbers, taking values [1, n], n is the number of fission product nuclei. For n fission product nuclei, the differential equation for the change in abundance is represented by a matrix as follows, and the differential equation set is solved: dY = A Y wherein, Y i is the abundance of the fission product nucleus i, Y = (Y1, Y2,..., Y n ) T , the initial abundance Y(0) at time zero is the independent yield, A is a coefficient matrix composed of decay constants; (4) outputting the results of the change in abundance of the fission product nuclei over time.

2. The method of claim 1, wherein the method is characterized by: The coefficient matrix A composed of decay constants in step (3) is as follows: A = (λ j r i,j )| i＝[1,n],j＝[1,n] r = -1 when i = j i,j = -1; The diagonal elements in matrix A are the decay constants of the disappearance caused by self-decay, and the non-diagonal elements are the decay constants of the increase of i nuclei caused by the decay of j nuclei to i nuclei; where λ j = ln(2) / T j is the decay constant of the j nucleus, T j is the half-life of the j nucleus, r i,j the branching ratio for decay of a j nucleus to an i nucleus, n is the number of fission product nuclei.

3. The method of claim 1, wherein the method is characterized by: In step (2), the decay data includes the identification numbers of parent and daughter nuclei, the branching ratio of the parent nucleus decaying to the daughter nucleus, the decay mode, and the product nucleus lifetime; wherein the identification numbers of the parent and daughter nuclei are ID = 1000A + 10Z + Isomer, A is the mass number of the product nucleus, Z is the charge number of the product nucleus, and Isomer is the isomeric state of the product nucleus.

4. The method of claim 1, wherein the method is characterized by: In step (2), the fission yield data includes the target nuclei, the charge number, element symbol, mass number, isomeric state of the fission product nuclei, independent yield or cumulative yield, yield and error, and neutron incident energy.

5. The method of calculating the abundance of a decay process of a fission product nucleus according to claim 3 or 4, characterized in that, The decay data and fission yield data are obtained from the database ENDF / B-VIII or JEFF-3.2.

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