Reactor neutron spectrum prior information determination method based on non-extensive statistical theory

Abstract

The invention discloses a method for determining prior information of a reactor neutron spectrum based on a non-extensive statistical theory, belongs to a neutron spectrum measurement technology, and mainly solves the technical problem of low accuracy of prior information determined in the prior art. The method comprises the following steps: 1, acquiring basic parameters of a reactor neutron field, and carrying out multi-group processing on the basic parameters; 2, defining a thermal neutron field as a thermal neutron field system, and establishing an entropy of the thermal neutron field system based on a non-extensive statistical theory; 3, defining the relationship among the average neutron energy, the average particle number, the neutron flux density and the neutron energy of the thermal neutron field system; 4, obtaining the relationship between the neutron flux density and the neutron energy of the thermal neutron field system in combination with the maximum entropy principle; 5, respectively acquiring the relationship between the neutron flux density and the neutron energy of the intermediate energy neutron field system and the fast neutron field system according to the same method; and 6, obtaining the relationship between the neutron flux density and the energy of the reactor neutron field system so as to determine the prior information of the reactor neutron energy spectrum.

Description

Technical Field

[0001] The present invention relates to a neutron energy spectrum measurement method, and in particular to a method for determining reactor neutron energy spectrum priori information based on non-extensive statistical theory. Background Art

[0002] Research reactors are usually equipped with a variety of neutron irradiation experimental chambers inside or outside the core. The neutron energy spectrum of the chamber, that is, the neutron energy distribution, is an important parameter of the reactor. It has a wide range of applications in technical fields such as nuclear physics experiments, neutron dosimetry, radiation effects and damage. Accurate neutron energy spectrum will effectively improve the accuracy of nuclear reaction cross-section measurement and macroscopic inspection of nuclear parameters. Therefore, it is of great significance to carry out energy spectrum measurement of the neutron field.

[0003] In the channel of the reactor internal experiment, the neutron energy range of the neutron field is usually 1×10 -4 eV~2×10 7 eV, also containing thermal, intermediate-energy, and fast neutrons, and the neutron field is also mixed with high-intensity gamma rays. Reactor neutron spectrum measurements are typically achieved using detection techniques, prior information, and spectrum decomposition methods (except for time-of-flight methods). Currently, multi-sphere spectrometers and multi-foil activation techniques are widely used for neutron spectrum measurements in the neutron field of reactor irradiation platforms. However, due to the limitations of these detection techniques and the specific energy group structure requirements, the problem of solving the neutron spectrum based on detector measurement data and response functions becomes underdetermined, and the neutron spectrum cannot be uniquely determined. Based on prior information and measurement data, spectrum decomposition methods formulate an optimization problem with single- or multi-objective functions based on constraints such as measurement data error limits and energy spectrum smoothness constraints. The neutron spectrum is then determined through an iterative optimization algorithm.

[0004] The prior information required for neutron spectrum measurement is closely related to the spectrum interpretation method. Since the 1960s, various effective spectrum interpretation methods have been established. These methods are based on the least squares principle, maximum entropy principle, Monte Carlo algorithm, neural network algorithm, or genetic algorithm. Three main types of prior information are used: 1) An initial spectrum that is close to the actual spectrum and completely known is used as prior information. This type of prior information uses the Monte Carlo method to establish a theoretical model of the neutron source and medium. By simulating the neutron transport process in the medium, an initial spectrum close to the actual spectrum is obtained. However, obtaining this initial spectrum in practice comes at a cost. For example, for reactor test channels, complex models and time-consuming calculations are required. However, due to the difficulty of accurate modeling and low computational efficiency, the initial spectrum is difficult to obtain and the uncertainty of the calculation is large. 2) The energy distribution of the neutron flux based on empirical knowledge and with unknown parameters is used as prior information. Specifically, from an empirical perspective, the neutron energy spectrum is expressed using an empirical formula with unknown parameters based on factors such as the neutron energy range and the radiation environment, thereby obtaining a parameterized empirical distribution of the neutron spectrum. However, the acquisition of this type of prior information relies on the accuracy of empirical knowledge and lacks a systematic and complete physical basis. 3) Prior information based on a large number of actual spectra. By learning a large number of neutron spectra, a suitable neural network structure is established. However, the accuracy of the neutron spectrum solution depends on the rationality of the network structure. Therefore, the accuracy of this type of prior information fundamentally depends on the breadth of the learned spectra.

[0005] In summary, in the process of solving the neutron energy spectrum based on prior information and measurement data, the current numerical simulation or empirical methods for obtaining prior information of the neutron energy spectrum cannot accurately and comprehensively reflect the distribution state of neutrons with energy in the actual neutron field system, resulting in low accuracy of the prior information used in the existing spectrum solution methods. Summary of the Invention

[0006] The purpose of the present invention is to solve the technical problem of low accuracy of prior information determined in the prior art and to provide a method for determining prior information of reactor neutron energy spectrum based on non-extensive statistical theory.

[0007] To achieve the above objectives, the technical solutions provided by the present invention are:

[0008] A method for determining a priori information of a reactor neutron spectrum based on non-extensive statistical theory is characterized in that it comprises the following steps:

[0009] Step 1: Obtain basic parameters of the reactor neutron field and perform multi-group processing on them; the reactor neutron field includes thermal neutron field, intermediate energy neutron field and fast neutron field; the basic parameters include neutron energy, neutron flux density and neutron density;

[0010] Step 2: define the thermal neutron field in step 1 as a thermal neutron field system, and establish the entropy of the thermal neutron field system based on non-extensive statistical theory;

[0011] Step 3: Based on the probability normalization of the non-extensive statistical system and the definition of the average neutron energy and average particle number of the thermal neutron field system, a constraint relationship between the neutron flux density and the neutron energy of the thermal neutron field system is established;

[0012] Step 4: Using the entropy of the thermal neutron field system established in step 2 as the objective function and the constraint relationship defined in step 3 as the constraint condition, based on the maximum entropy principle and combined with the influence factor of the absorption reaction between neutrons and the nuclei of the surrounding medium on the neutron flux density, an explicit relationship between the neutron flux density and the neutron energy of the thermal neutron field system is obtained;

[0013] Step 5: According to the methods of steps 2 to 4, respectively obtain the explicit relationship between the neutron flux density and the neutron energy of the intermediate energy neutron field system and the fast neutron field system corresponding to the intermediate energy neutron field and the fast neutron field in step 1;

[0014] Step 6: Combining the explicit relationship between the neutron flux density and the neutron energy obtained in steps 4 and 5, the relationship between the neutron flux density and the neutron energy of the reactor neutron field system is obtained, thereby determining the prior information of the reactor neutron energy spectrum.

[0015] Furthermore, in step 1, the multi-grouping process refers to dividing the neutrons in the unit volume of the corresponding neutron field into m groups according to their neutron energy E, and defining the neutron flux density of the jth group as The neutron density is n j , where j = 1, 2,…, m.

[0016] Furthermore, in step 2, the entropy of the thermal neutron field system is established based on the Tsallis entropy in the non-extensive statistical theory. for:

[0017]

[0018] Where k is the Boltzmann constant, q1 is the non-extensive parameter of the thermal neutron field system, i represents the microscopic state of the thermal neutron field system, and W represents the total number of microscopic states of the thermal neutron field system. represents the probability that the thermal neutron field system is in microscopic state i.

[0019] Furthermore, in step 3, the constraint relationship between the neutron flux density and neutron energy of the thermal neutron field system is:

[0020]

[0021] in, is the average neutron energy of the thermal neutron field system, is the average neutron number of the thermal neutron field system, is the neutron flux density of the jth group of the thermal neutron field system, E j is the neutron energy of the jth group in the thermal neutron field system.

[0022] Furthermore, in step 4, the explicit relationship between the neutron flux density and the neutron energy of the thermal neutron field system is:

[0023]

[0024] in, is the neutron flux density of the thermal neutron field system, C th is the fraction of thermal neutrons in the reactor neutron field, β1 and g1 are parameters related to the average energy and particle number in the thermal neutron field system, It is the factor affecting the neutron flux density caused by the absorption reaction between neutrons in the thermal neutron field system and the atomic nuclei of the surrounding medium. a1 is the parameter related to the absorption reaction in the thermal neutron field system.

[0025] Furthermore, in step 6, the relationship between the neutron flux density and the neutron energy of the reactor neutron field system is:

[0026]

[0027] Among them, C ep 、C f are the fractions of medium-energy neutrons and fast neutrons in the reactor neutron field, respectively; β2 and g2 are parameters related to the average energy and particle number in the medium-energy neutron field system, respectively; q2 is the non-extensive parameter of the medium-energy neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons and the nuclei of the surrounding medium in the medium-energy neutron field system; a2 is the parameter related to the absorption reaction in the medium-energy neutron field system; β3 and g3 are parameters related to the average energy and particle number in the fast neutron field system, respectively; q3 is the non-extensive parameter of the fast neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons in the fast neutron field system and the nuclei of the surrounding medium; a3 is the parameter related to the absorption reaction in the fast neutron field system; E0, E W are the lower and upper limits of neutron energy in the reactor neutron field, respectively; E1 is the energy node of thermal neutrons and intermediate energy neutrons in the reactor neutron field; E2 is the energy node of intermediate energy neutrons and fast neutrons in the reactor neutron field.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] 1. Based on the basic parameters of the reactor neutron field, the present invention establishes the entropy of the thermal, intermediate, and fast neutron field systems through non-extensive statistical theory. Furthermore, based on the probability normalization of the non-extensive statistical system and the definitions of the average neutron energy and average particle number of the thermal, intermediate, and fast neutron field systems, a constraint relationship between the neutron flux density and neutron energy is established for each of these systems. This leads to an explicit relationship between the neutron flux density and neutron energy, ultimately yielding prior information on the reactor neutron energy spectrum. The prior information obtained using this method is highly accurate, providing strong theoretical and data support for subsequent neutron spectrum calculations and measurements.

[0030] 2. The idea of determining the parameterized neutron spectrum of the present invention provides a new theoretical approach for determining the prior information required for reactor neutron spectrum measurement technology.

[0031] 3. The parameterized neutron energy spectrum provided in the present invention is based on the theory of non-extensive statistics and has a more systematic and complete physical basis compared with the existing parameterized neutron energy spectrum prior information based on empirical knowledge. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 Flowchart of an embodiment of the present invention. DETAILED DESCRIPTION

[0033] In order to make the objects, advantages and features of the present invention more clear, the present invention is further described in detail below with reference to the accompanying drawings and specific examples. It should be understood by those skilled in the art that these embodiments are only used to explain the technical principles of the present invention and are not intended to limit the scope of protection of the present invention.

[0034] The design concept of this invention is as follows: Currently, the Boltzmann-Gibbs distribution (BG distribution) in classical statistical mechanics is suitable for describing systems composed of a large number of particles in equilibrium. The mutual collisions between these particles are the microscopic dynamic mechanism that allows the system to reach equilibrium. While the energy conversion process of the particles does not involve nuclear reactions, the interaction between neutrons and the nuclei of the medium is the primary microscopic mechanism that allows the neutron number to reach a steady state along with the energy distribution, resulting in both absorption and scattering reactions. Current maximum entropy spectral decomposition (MAXED) methods, which are related to statistical properties, treat the neutron energy spectrum as a probability distribution and use the cross-entropy based on the BG distribution as a measure of the degree to which the solution spectrum approximates the initial spectrum. Using the cross-entropy between the initial spectrum and the solution spectrum as the objective function, the neutron energy spectrum is obtained by combining measured data. Therefore, the initial spectrum, as prior information, plays a fundamental role in determining the accuracy of the energy spectrum. In a sense, this method can be viewed as a spectral decomposition method similar to the least squares method. The required prior information, the initial spectrum, must be obtained through methods such as the Monte Carlo algorithm. Therefore, its ability to determine the neutron energy spectrum does not prove that the BG distribution is suitable for describing neutron field systems. Traditional extensive statistics, based on the BG distribution, has a key assumption: the ergodic hypothesis, which states that a system can traverse all possible microscopic states within a given timeframe. However, this assumption is not met in many complex systems discovered to date, making it infeasible for traditional extensive statistics to study them. In fact, the "hardening" of the thermal neutron spectrum in reactor neutron fields demonstrates that the actual neutron spectrum does not follow the BG distribution. This is because thermal neutrons are all slowed down by high-energy neutrons, and because the medium absorbs neutrons to varying degrees, some neutrons are absorbed before reaching equilibrium with the surrounding nuclei. This means that the neutron field system does not meet the ergodic hypothesis, and therefore traditional extensive statistics cannot describe the neutron field energy distribution. In fact, the traditional BG distribution has indeed encountered difficulties in describing complex systems such as plasmas, metastable systems, and dissipative systems.

[0035] Currently, the theory of non-extensive statistics has developed into several branches, such as Tsallis statistics, non-holonomic statistics, and q-deformation statistics. Among them, Tsallis statistics is considered to be a non-extensive theory that can be used to directly analyze real phenomena. Tsallis statistics introduces a non-extensive parameter q on the basis of classical BG entropy to construct non-extensive entropy. Its definition is as follows:

[0036]

[0037] Where k is the Boltzmann constant, q is called the non-extensive parameter, i represents the microscopic state of the system, W is the total number of microscopic states of the system, and p irepresents the probability that the system is in microscopic state i. In the limit of q→1, this entropy returns to the currently well-known Boltzmann entropy. Therefore, this new entropy form is a generalization of the traditional Boltzmann entropy.

[0038] In summary, in view of the current status of non-extensive statistical theory, this embodiment is based on Tsallis entropy, combined with the probability normalization of non-extensive statistical system, and the definition of neutron average energy and average particle number of neutron field system, to establish the constraint relationship between neutron flux density and neutron energy of neutron field system, and provide a method for determining prior information of reactor neutron energy spectrum based on non-extensive statistical theory, such as Figure 1 As shown, the method includes the following steps:

[0039] Step 1: Obtain the basic parameters of the reactor neutron field and perform multi-group processing on them.

[0040] The reactor neutron field of this embodiment includes a thermal neutron field, an intermediate energy neutron field, and a fast neutron field, and the basic parameters refer to the neutron energy, neutron flux density, and neutron density of each neutron field.

[0041] Multi-grouping treatment means dividing the neutrons in the unit volume of the corresponding neutron field into m groups according to the neutron energy E, and defining the neutron flux density of the jth group as Abbreviated as The neutron density is n(E j ), referred to as n j , where j = 1, 2,…, m.

[0042] Step 2: Define the thermal neutron field in step 1 as a thermal neutron field system, and establish the entropy of the thermal neutron field system based on the Tsallis entropy in non-extensive statistical theory. Right now:

[0043]

[0044] Where k is the Boltzmann constant, q1 is the non-extensive parameter of the thermal neutron field system, i represents the microscopic state of the thermal neutron field system, and W represents the total number of microscopic states of the thermal neutron field system. represents the probability that the thermal neutron field system is in microscopic state i.

[0045] Step 3: Based on the probability normalization of the non-extensive statistical system and the definition of the average neutron energy and average particle number of the thermal neutron field system, a constraint relationship between the neutron flux density and the neutron energy of the thermal neutron field system is established. The constraint relationship is as follows:

[0046]

[0047] in, is the average neutron energy of the thermal neutron field system, is the average neutron number of the thermal neutron field system, is the neutron flux density of the jth group of the thermal neutron field system, E j is the neutron energy of the jth group in the thermal neutron field system.

[0048] Step 4: Take the entropy of the thermal neutron field system established in step 2 as the objective function, and the constraint relationship established in step 3 as the constraint condition, and then based on the maximum entropy principle (establish the Lagrangian function L corresponding to the thermal neutron field system, and let ) to obtain the neutron flux density of the jth group of the thermal neutron field system With energy E j The explicit relationship is as follows:

[0049]

[0050] Where C is a constant, β and g are parameters related to the average energy and particle number of the thermal neutron field system respectively.

[0051] Based on the above formula, we can further obtain the explicit relationship between the neutron flux density and neutron energy of the thermal neutron field system, namely:

[0052]

[0053] Taking into account the difference between the properties of the neutron field system and the statistical system, on the basis of the relationship between the neutron flux density and the neutron energy in the above formula, the influence factor Ea of the absorption reaction between neutrons and the atomic nuclei of the surrounding medium on the neutron flux density is introduced. 1 , thus establishing a new explicit relationship between the neutron flux density and neutron energy of the thermal neutron field system as follows:

[0054]

[0055] in, is the neutron flux density of the thermal neutron field system, C th is the proportion of thermal neutrons in the reactor neutron field, β1 and g1 are parameters related to the average energy and particle number in the thermal neutron field system, respectively, and a1 is a parameter related to the absorption reaction in the thermal neutron field system.

[0056] Step 5: According to the methods of steps 2 to 4, the explicit relationship between the neutron flux density and the neutron energy of the intermediate energy neutron field system and the fast neutron field system corresponding to the intermediate energy neutron field and the fast neutron field in step 1 is obtained respectively.

[0057] Step 6: Combine steps 4 and 5 to obtain the explicit relationship between the neutron flux density and neutron energy of each neutron field system, and obtain the neutron flux density of the reactor neutron field system. The relationship with the neutron energy E is:

[0058]

[0059] Among them, C ep 、C f are the fractions of medium-energy neutrons and fast neutrons in the reactor neutron field, respectively; β2 and g2 are parameters related to the average energy and particle number in the medium-energy neutron field system, respectively; q2 is the non-extensive parameter of the medium-energy neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons and the nuclei of the surrounding medium in the medium-energy neutron field system; a2 is the parameter related to the absorption reaction in the medium-energy neutron field system; β3 and g3 are parameters related to the average energy and particle number in the fast neutron field system, respectively; q3 is the non-extensive parameter of the fast neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons in the fast neutron field system and the nuclei of the surrounding medium; a3 is the parameter related to the absorption reaction in the fast neutron field system; E0, E W are the lower and upper limits of neutron energy in the reactor neutron field, respectively; E1 is the energy node of thermal neutrons and intermediate energy neutrons in the reactor neutron field; E2 is the energy node of intermediate energy neutrons and fast neutrons in the reactor neutron field.

[0060] The relationship between the neutron flux density and neutron energy of the above-mentioned reactor neutron field system is the parameterized neutron energy spectrum, thereby determining the prior information of the reactor neutron energy spectrum.

[0061] In order to determine the prior information required for reactor neutron spectrum measurement, the present invention respectively targets thermal neutrons, intermediate energy neutrons and fast neutrons, establishes the entropy of the neutron field system and the corresponding probability normalization, energy and particle number constraints based on the theory of non-extensive statistics, and finally determines the parameterized neutron spectrum based on the maximum entropy principle.

[0062] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the present invention.

Claims

1. A method for determining prior information of reactor neutron energy spectrum based on non-extensive statistical theory, characterized in that: The following steps are involved: Step 1: Obtain basic parameters of the reactor neutron field and perform multi-group processing on them; the reactor neutron field includes thermal neutron field, intermediate energy neutron field and fast neutron field; the basic parameters include neutron energy, neutron flux density and neutron density; Step 2: define the thermal neutron field in step 1 as a thermal neutron field system, and establish the entropy of the thermal neutron field system based on non-extensive statistical theory; Step 3: Based on the probability normalization of the non-extensive statistical system and the definition of the average neutron energy and average particle number of the thermal neutron field system, a constraint relationship between the neutron flux density and the neutron energy of the thermal neutron field system is established; Step 4: Using the entropy of the thermal neutron field system established in step 2 as the objective function and the constraint relationship defined in step 3 as the constraint condition, based on the maximum entropy principle and combined with the influence factor of the absorption reaction between neutrons and the nuclei of the surrounding medium on the neutron flux density, an explicit relationship between the neutron flux density and the neutron energy of the thermal neutron field system is obtained; Step 5: According to the methods of steps 2 to 4, respectively obtain the explicit relationship between the neutron flux density and the neutron energy of the intermediate energy neutron field system and the fast neutron field system corresponding to the intermediate energy neutron field and the fast neutron field in step 1; Step 6: Combining the explicit relationship between the neutron flux density and the neutron energy obtained in steps 4 and 5, the relationship between the neutron flux density and the neutron energy of the reactor neutron field system is obtained, thereby determining the prior information of the reactor neutron energy spectrum.

2. The method for determining a priori information of a reactor neutron spectrum based on non-extensive statistical theory according to claim 1, characterized in that: In step 1, the multi-grouping process is to divide the neutrons in the unit volume of the corresponding neutron field into m groups according to the neutron energy E, and define the neutron flux density of the jth group as The neutron density is n j , where j = 1, 2,…, m.

3. The method for determining a priori information of a reactor neutron spectrum based on non-extensive statistical theory according to claim 2, characterized in that: In step 2, the entropy of the thermal neutron field system is established based on the Tsallis entropy in the non-extensive statistical theory. for: Where k is the Boltzmann constant, q1 is the non-extensive parameter of the thermal neutron field system, i represents the microscopic state of the thermal neutron field system, and W represents the total number of microscopic states of the thermal neutron field system. represents the probability that the thermal neutron field system is in microscopic state i.

4. The method for determining a priori information of a reactor neutron spectrum based on non-extensive statistical theory according to claim 3, characterized in that: In step 3, the constraint relationship between the neutron flux density and neutron energy of the thermal neutron field system is: in, is the average neutron energy of the thermal neutron field system, is the average neutron number of the thermal neutron field system, is the neutron flux density of the jth group of the thermal neutron field system, E j is the neutron energy of the jth group in the thermal neutron field system.

5. The method for determining a priori information of reactor neutron energy spectrum based on non-extensive statistical theory according to claim 4, characterized in that: In step 4, the explicit relationship between the neutron flux density and the neutron energy of the thermal neutron field system is: in, is the neutron flux density of the thermal neutron field system, C th is the fraction of thermal neutrons in the reactor neutron field, β1 and g1 are parameters related to the average energy and particle number in the thermal neutron field system, It is the factor affecting the neutron flux density caused by the absorption reaction between neutrons in the thermal neutron field system and the atomic nuclei of the surrounding medium. a1 is the parameter related to the absorption reaction in the thermal neutron field system.

6. The method for determining a priori information of reactor neutron energy spectrum based on non-extensive statistical theory according to claim 5, characterized in that: In step 6, the neutron flux density of the reactor neutron field system is The relationship with the neutron energy E is: Among them, C ep 、C f are the fractions of medium-energy neutrons and fast neutrons in the reactor neutron field, respectively; β2 and g2 are parameters related to the average energy and particle number in the medium-energy neutron field system, respectively; q2 is the non-extensive parameter of the medium-energy neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons and the nuclei of the surrounding medium in the medium-energy neutron field system; a2 is the parameter related to the absorption reaction in the medium-energy neutron field system; β3 and g3 are parameters related to the average energy and particle number in the fast neutron field system, respectively; q3 is the non-extensive parameter of the fast neutron field system; is the factor affecting the neutron flux density caused by the absorption reaction between neutrons in the fast neutron field system and the nuclei of the surrounding medium; a3 is the parameter related to the absorption reaction in the fast neutron field system; E0, E W are the lower and upper limits of neutron energy in the reactor neutron field, respectively; E1 is the energy node of thermal neutrons and intermediate energy neutrons in the reactor neutron field; E2 is the energy node of intermediate energy neutrons and fast neutrons in the reactor neutron field.

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